PI and PID Controller Tuning Rules Handbook, 3rd Edition

HANDBOOK OF
PI AND PID CONTROLLER
TUNING RULES
3rd Edition
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HANDBOOK OF
PI AND PID CONTROLLER
TUNING RULES
3rd Edition
Aidan O’Dwyer
Dublin Institute of Technology, Ireland
ICP
P575.TP.indd 2
Imperial College Press
3/20/09 5:16:18 PM
Published by
Imperial College Press
57 Shelton Street
Covent Garden
London WC2H 9HE
Distributed by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
HANDBOOK OF PI AND PID CONTROLLER TUNING RULES
(3rd Edition)
Copyright © 2009 by Imperial College Press
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or
mechanical, including photocopying, recording or any information storage and retrieval system now known or to
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the publisher.
ISBN-13 978-1-84816-242-6
ISBN-10 1-84816-242-1
Typeset by Stallion Press
Email: [email protected]
Printed in Singapore.
Steven - Hdbook of PI & PID (3rd).pmd
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Dedication
Once more, this book is dedicated with love to Angela, Catherine and Fiona and
to my parents, Sean and Lillian.
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Preface
Proportional integral (PI) and proportional integral derivative (PID) controllers
have been at the heart of control engineering practice for over seven decades.
However, in spite of this, the PID controller has not received much attention from
the academic research community until the past two decades, when work by K.J.
Åström, T. Hägglund and F.G. Shinskey, among others, sparked a revival of
interest in the use of this ‘workhorse’ of controller implementation.
There is strong evidence that PI and PID controllers remain poorly understood
and, in particular, poorly tuned in many applications. It is clear that the many
controller tuning rules proposed in the literature are not having an impact on
industrial practice. One reason is that the tuning rules are not accessible, being
scattered throughout the control literature; in addition, the notation used is not
unified. The purpose of this book, now in its third edition, is to bring together and
summarise, using a unified notation, tuning rules for PI and PID controllers. The
author restricts the work to tuning rules that may be applied to the control of
processes with time delays (dead times); in practice, this is not a significant
restriction, as most process models have a time delay term. In this edition, the
structure of the book has been modified from the previous edition, with controller
tuning rules for non-self-regulating process models being summarised in a
different chapter from those for self-regulating process models.
It is the author’s belief that this book will be useful to control and instrument
engineering practitioners and will be a useful reference for students and
educators in universities and technical colleges.
I wish to thank the School of Electrical Engineering Systems, Dublin Institute
of Technology, for providing the facilities needed to complete the book. Finally,
I am deeply grateful to my mother, Lillian, and my father, Sean, for their
inspiration and support over many years and to my family, Angela, Catherine and
Fiona, for their love and understanding.
Aidan O’Dwyer
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Contents
Preface
........................................................................................................
vii
1. Introduction ........................................................................................................
1.1 Preliminary Remarks ......................................................................................
1.2 Structure of the Book......................................................................................
1
1
2
2. Controller Architecture...........................................................................................
2.1 Introduction ..................................................................................................
2.2 Comments on the PID Controller Structures . ................................................
2.3 Process Modelling ..........................................................................................
2.3.1 Self-regulating process models . ........................................................
2.3.2 Non-self-regulating process models...................................................
2.4 Organisation of the Tuning Rules ...................................................................
4
4
11
12
12
14
16
3. Controller Tuning Rules for Self-Regulating Process Models ...............................
3.1 Delay Model .................................................................................................
3.1.1 Ideal PI controller – Table 2 ..............................................................
3.1.2 Ideal PID controller – Table 3 ..........................................................
3.1.3 Ideal controller in series with a first order lag – Table 4 ................
3.1.4 Classical controller – Table 5 ...........................................................
3.1.5 Generalised classical controller – Table 6 ........................................
3.1.6 Two degree of freedom controller 1 – Table 7 ..................................
3.2 Delay Model with a Zero ...............................................................................
3.2.1 Ideal PI controller – Table 8 . ............................................................
3.3 FOLPD Model ...............................................................................................
3.3.1 Ideal PI controller – Table 9 ..............................................................
3.3.2 Ideal PID controller – Table 10 ........................................................
3.3.3 Ideal controller in series with a first order lag – Table 11 ................
3.3.4 Controller with filtered derivative – Table 12....................................
3.3.5 Classical controller – Table 13 ........................................................
3.3.6 Generalised classical controller – Table 14 . .....................................
3.3.7 Two degree of freedom controller 1 – Table 15 .............................
3.3.8 Two degree of freedom controller 2 – Table 16 .............................
3.3.9 Two degree of freedom controller 3 – Table 17 .............................
18
18
18
23
24
25
26
27
28
28
30
30
78
118
122
134
149
152
168
170
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3.4 FOLPD Model with a Zero ............................................................................
3.4.1 Ideal PI controller – Table 18 ............................................................
3.4.2 Ideal controller in series with a first order lag – Table 19 ................
3.5 SOSPD Model ..............................................................................................
3.5.1 Ideal PI controller – Table 20 ..........................................................
3.5.2 Ideal PID controller – Table 21 ........................................................
3.5.3 Ideal controller in series with a first order lag – Table 22 ................
3.5.4 Controller with filtered derivative – Table 23....................................
3.5.5 Classical controller – Table 24 ........................................................
3.5.6 Generalised classical controller – Table 25 . .....................................
3.5.7 Two degree of freedom controller 1 – Table 26 .............................
3.5.8 Two degree of freedom controller 3 – Table 27 .............................
3.6 SOSPD Model with a Zero ............................................................................
3.6.1 Ideal PI controller – Table 28 .........................................................
3.6.2 Ideal PID controller – Table 29 ........................................................
3.6.3 Ideal controller in series with a first order lag – Table 30 ...............
3.6.4 Controller with filtered derivative – Table 31....................................
3.6.5 Classical controller – Table 32 .........................................................
3.6.6 Generalised classical controller – Table 33 .......................................
3.6.7 Two degree of freedom controller 1 – Table 34 ..............................
3.6.8 Two degree of freedom controller 3 – Table 35 ..............................
3.7 TOSPD Model ..............................................................................................
3.7.1 Ideal PI controller – Table 36 . ..........................................................
3.7.2 Ideal PID controller – Table 37 .........................................................
3.7.3 Ideal controller in series with a first order lag – Table 38 ................
3.7.4 Controller with filtered derivative – Table 39 ....................................
3.7.5 Two degree of freedom controller 1 – Table 40 .................................
3.7.6 Two degree of freedom controller 3 – Table 41 . ...............................
3.8 Fifth Order System Plus Delay Model ...........................................................
3.8.1 Ideal PID controller – Table 42 .........................................................
3.8.2 Controller with filtered derivative – Table 43 ....................................
3.8.3 Two degree of freedom controller 1 – Table 44 . ...............................
3.9 General Model . ..............................................................................................
3.9.1 Ideal PI controller – Table 45 ...........................................................
3.9.2 Ideal PID controller – Table 46 .........................................................
3.9.3 Ideal controller in series with a first order lag – Table 47 ................
3.9.4 Controller with filtered derivative – Table 48 ..................................
3.9.5 Two degree of freedom controller 1 – Table 49 . ..............................
3.10 Non-Model Specific ......................................................................................
3.10.1 Ideal PI controller – Table 50 ............................................................
3.10.2 Ideal PID controller – Table 51 .........................................................
3.10.3 Ideal controller in series with a first order lag– Table 52 . ................
3.10.4 Controller with filtered derivative – Table 53 ...................................
3.10.5 Classical controller – Table 54 .........................................................
3.10.6 Generalised classical controller – Table 55 ......................................
180
180
182
183
183
206
232
236
238
251
253
264
277
277
279
282
284
286
288
289
292
293
293
296
297
298
299
302
303
303
305
308
310
310
312
315
316
317
318
318
324
332
336
341
343
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xi
3.10.7 Two degree of freedom controller 1 – Table 56 . .............................. 346
3.10.8 Two degree of freedom controller 3 – Table 57 . .............................. 349
4. Controller Tuning Rules for Non-Self-Regulating Process Models .......................
4.1 IPD Model .....................................................................................................
4.1.1 Ideal PI controller – Table 58 . ..........................................................
4.1.2 Ideal PID controller – Table 59 ......................................................
4.1.3 Ideal controller in series with a first order lag – Table 60 ................
4.1.4 Controller with filtered derivative – Table 61 ..................................
4.1.5 Classical controller – Table 62 .........................................................
4.1.6 Generalised classical controller – Table 63 .....................................
4.1.7 Two degree of freedom controller 1 – Table 64 . ..............................
4.1.8 Two degree of freedom controller 2 – Table 65 . ..............................
4.1.9 Two degree of freedom controller 3 – Table 66 . ..............................
4.2 IPD Model with a Zero .................................................................................
4.2.1 Ideal PI controller – Table 67 . ..........................................................
4.3 FOLIPD Model ..............................................................................................
4.3.1 Ideal PI controller – Table 68 .........................................................
4.3.2 Ideal PID controller – Table 69 ........................................................
4.3.3 Ideal controller in series with a first order lag – Table 70 ................
4.3.4 Controller with filtered derivative – Table 71 ...................................
4.3.5 Classical controller – Table 72 .........................................................
4.3.6 Generalised classical controller – Table 73 ......................................
4.3.7 Two degree of freedom controller 1 – Table 74 ..............................
4.3.8 Two degree of freedom controller 2 – Table 75 ..............................
4.3.9 Two degree of freedom controller 3 – Table 76 ..............................
4.4 FOLIPD Model with a Zero ........................................................................
4.4.1 Ideal PID controller – Table 77 . .......................................................
4.4.2 Ideal controller in series with a first order lag – Table 78 ...............
4.4.3 Classical controller – Table 79 .........................................................
4.5 I 2 PD Model.....................................................................................................
4.5.1 Ideal PID controller – Table 80 . .......................................................
4.5.2 Classical controller – Table 81 ........................................................
4.5.3 Two degree of freedom controller 1 – Table 82 ..............................
4.5.4 Two degree of freedom controller 2 – Table 83 ................................
4.5.5 Two degree of freedom controller 3 – Table 84 ..............................
4.6 SOSIPD Model .............................................................................................
4.6.1 Ideal PI controller – Table 85 . ..........................................................
4.6.2 Two degree of freedom controller 1 – Table 86 . ..............................
4.7 SOSIPD Model with a Zero............................................................................
4.7.1 Classical controller – Table 87 .........................................................
4.8 TOSIPD Model...............................................................................................
4.8.1 Two degree of freedom controller 1 – Table 88 ..............................
350
350
350
359
364
366
368
371
372
378
381
383
383
385
385
388
392
394
395
397
399
416
418
420
420
422
423
424
424
425
426
427
429
430
430
431
436
436
437
437
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Handbook of PI and PID Controller Tuning Rules
4.9 General Model with Integrator .......................................................................
4.9.1 Ideal PI controller – Table 89 . ..........................................................
4.9.2 Two degree of freedom controller 1 – Table 90 ................................
4.10 Unstable FOLPD Model ................................................................................
4.10.1 Ideal PI controller – Table 91 ............................................................
4.10.2 Ideal PID controller – Table 92 ........................................................
4.10.3 Ideal controller in series with a first order lag – Table 93 ...............
4.10.4 Classical controller – Table 94 .......................................................
4.10.5 Generalised classical controller – Table 95 .......................................
4.10.6 Two degree of freedom controller 1 – Table 96 ................................
4.10.7 Two degree of freedom controller 2 – Table 97 ..............................
4.10.8 Two degree of freedom controller 3 – Table 98 ..............................
4.11 Unstable FOLPD Model with a Zero ...........................................................
4.11.1 Ideal PI controller – Table 99 ............................................................
4.11.2 Ideal controller in series with a first order lag – Table 100 .............
4.11.3 Generalised classical controller – Table 101 ....................................
4.11.4 Two degree of freedom controller 1 – Table 102 ............................
4.12 Unstable SOSPD Model (one unstable pole) ................................................
4.12.1 Ideal PI controller – Table 103 ........................................................
4.12.2 Ideal PID controller – Table 104 . .....................................................
4.12.3 Ideal controller in series with a first order lag – Table 105 ..............
4.12.4 Classical controller – Table 106 ........................................................
4.12.5 Two degree of freedom controller 1 – Table 107 ..............................
4.12.6 Two degree of freedom controller 3 – Table 108 ..............................
4.13 Unstable SOSPD Model (two unstable poles) ...............................................
4.13.1 Ideal PID controller – Table 109 ......................................................
4.13.2 Generalised classical controller – Table 110 . ...................................
4.13.3 Two degree of freedom controller 2 – Table 111 ..............................
4.14 Unstable SOSPD Model with a Zero ..............................................................
4.14.1 Ideal PI controller – Table 112 . ........................................................
4.14.2 Ideal controller in series with a first order lag – Table 113 ..............
4.14.3 Generalised classical controller – Table 114 ...................................
4.14.4 Two degree of freedom controller 1 – Table 115 ..............................
4.14.5 Two degree of freedom controller 3 – Table 116 ..............................
438
438
439
440
440
447
455
458
462
463
473
475
480
480
481
483
484
486
486
488
490
491
497
503
506
506
508
509
511
511
513
516
518
520
5. Performance and Robustness Issues in the Compensation of FOLPD
Processes with PI and PID Controllers ...................................................................
5.1 Introduction ...................................................................................................
5.2 The Analytical Determination of Gain and Phase Margin .............................
5.2.1 PI tuning formulae ...............................................................................
5.2.2 PID tuning formulae ...........................................................................
5.3 The Analytical Determination of Maximum Sensitivity ................................
5.4 Simulation Results ..........................................................................................
521
521
522
522
525
529
529
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Contents
5.5 Design of Tuning Rules to Achieve Constant Gain and Phase Margins,
for All Values of Delay...................................................................................
5.5.1 PI controller design..............................................................................
5.5.1.1 Processes modelled in FOLPD form......................................
5.5.1.2 Processes modelled in IPD form ...........................................
5.5.2 PID controller design ..........................................................................
5.5.2.1 Processes modelled in FOLPD form – classical controller....
5.5.2.2 Processes modelled in SOSPD form – series controller.........
5.5.2.3 Processes modelled in SOSPD form with a negative
zero – classical controller ......................................................
5.5.3 PD controller design . ..........................................................................
5.6 Conclusions . ..................................................................................................
xiii
534
534
534
536
539
539
541
542
542
543
Appendix 1 Glossary of Symbols and Abbreviations................................................. 544
Appendix 2 Some Further Details on Process Modelling........................................... 551
Bibliography
........................................................................................................ 565
Index
........................................................................................................ 599
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Chapter 1
Introduction
1.1 Preliminary Remarks
The ability of proportional integral (PI) and proportional integral derivative (PID)
controllers to compensate most practical industrial processes has led to their wide
acceptance in industrial applications. Koivo and Tanttu (1991), for example,
suggest that there are perhaps 5–10% of control loops that cannot be controlled
by single input, single output (SISO) PI or PID controllers; in particular, these
controllers perform well for processes with benign dynamics and modest
performance requirements (Hwang, 1993; Åström and Hägglund, 1995). It has
been stated that 98% of control loops in the pulp and paper industries are
controlled by SISO PI controllers (Bialkowski, 1996) and that in process control
applications, more than 95% of the controllers are of PID type (Åström and
Hägglund, 1995). PI or PID controller implementation has been recommended
for the control of processes of low to medium order, with small time delays,
when parameter setting must be done using tuning rules and when controller
synthesis is performed either once or more often (Isermann, 1989).
However, despite decades of development work, surveys indicating the state
of the art of control in industrial practice report sobering results. For example,
Ender (1993) states that in his testing of thousands of control loops in hundreds
of plants, it has been found that more than 30% of installed controllers are
operating in manual mode and 65% of loops operating in automatic mode
produce less variance in manual than in automatic (i.e. the automatic controllers
are poorly tuned). The situation does not appear to have improved more recently,
as Van Overschee and De Moor (2000) report that 80% of PID controllers are
badly tuned; 30% of PID controllers operate in manual with another 30% of the
controlled loops increasing the short term variability of the process to be
controlled (typically due to too strong integral action). The authors state that 25%
1
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Handbook of PI and PID Controller Tuning Rules
of all PID controller loops use default factory settings, implying that they have
not been tuned at all.
These and other surveys (well summarised by Yu, 1999, pp. 1–2) show that
the determination of PI and PID controller tuning parameters is a vexing problem
in many applications. The most direct way to set up controller parameters is the
use of tuning rules; obviously, the wealth of information on this topic available in
the literature has been poorly communicated to the industrial community. One
reason is that this information is scattered in a variety of media, including journal
papers, conference papers, websites and books for a period of over seventy years.
The purpose of this book is to bring together, in summary form, the tuning
rules for PI and PID controllers that have been developed to compensate SISO
processes with time delay.
1.2 Structure of the Book
Tuning rules are set out in the book in tabular form. This form allows the rules to
be represented compactly. The tables have four or five columns, according to
whether the controller considered is of PI or PID form, respectively. The first
column in all cases details the author of the rule and other pertinent information.
The final column in all cases is labelled ‘Comment’; this facilitates the inclusion
of information about the tuning rule that may be useful in its application. The
remaining columns detail the formulae for the controller parameters.
Chapter 2 explores the range of PI and PID controller structures proposed in
the literature. It is often forgotten that different manufacturers implement
different versions of the PID controller algorithm (in particular); therefore,
controller tuning rules that work well in one PID architecture may work poorly in
another. This chapter also details the process models used to define the controller
tuning rules.
Chapters 3 and 4 detail, in tabular form, PI and PID controller tuning rules
(and their variations), as applied to a wide variety of self-regulating process
models and non-self-regulating process models, respectively. One-hundred-andsixteen such tables are provided altogether. To allow the reader to access data
readily, the author has arranged that each table starts on its own page; each table
is preceded by the controller used, together with a block diagram showing the
unity feedback closed loop arrangement of the controller and process model.
In Chapter 5, analytical calculations of the gain and phase margins of a large
sample of PI and PID controller tuning rules are determined, when the process
is modelled in first order lag plus time delay (FOLPD) form, at a range of ratios
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Chapter 1: Introduction
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3
of time delay to time constant of the process model. Results are given in
graphical form.
An important feature of the book is the unified notation used for the tuning
rules; a glossary of the symbols used is provided in Appendix 1. Appendix 2
outlines the range of methods used to determine process model parameters; this
information is presented in summary form, as this topic could provide data for a
book in itself. However, sufficient information, together with references, is
provided for the interested reader.
Finally, a comprehensive reference list is provided. In particular, the author
would like to recommend the contributions by McMillan (1994), Åström and
Hägglund (1995), (2006), Shinskey (1994), (1996), Tan et al. (1999a), Yu
(1999), Lelic and Gajic (2000) and Ang et al. (2005) to the interested reader,
which treat comprehensively the wider perspective of PID controller design and
application.
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Chapter 2
Controller Architecture
2.1 Introduction
The ideal continuous time domain PID controller for a SISO process is expressed
in the Laplace domain as follows:
U (s ) = G c (s ) E (s )
(2.1)
with
G c (s) = K c (1 +
1
+ Td s)
Ti s
(2.2)
and with K c = proportional gain, Ti = integral time constant and Td =
derivative time constant. If Ti = ∞ and Td = 0 (i.e. P control), then it is clear
that the closed loop measured value y will always be less than the desired value r
(for processes without an integrator term, as a positive error is necessary to keep
the measured value constant, and less than the desired value). The introduction of
integral action facilitates the achievement of equality between the measured
value and the desired value, as a constant error produces an increasing controller
output. The introduction of derivative action means that changes in the desired
value may be anticipated, and thus an appropriate correction may be added prior
to the actual change. Thus, in simplified terms, the PID controller allows
contributions from present controller inputs, past controller inputs and future
controller inputs.
Many variations of the PID controller structure have been proposed (indeed,
the PI controller structure is itself a subset of the PID controller structure). As
Tan et al. (1999a) suggest, one important reason for the non-standard structures
is due to the transition of the controllers from pneumatic implementation through
electronic implementation to the present microprocessor implementation. A
substantial number of the variations in the controller structures used may be
4
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Chapter 2: Controller Architecture
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5
summarised by controller structure ‘supersets’. In this book, nine such supersets
are specified, which allows a sensible restriction on the number of tables that
need to be detailed. The controller structures specified are detailed below.
1. The ideal PI controller structure:

1 

G c (s) = K c 1 +
 Ti s 
(2.3)
2. The ideal PID controller structure:


1
G c (s) = K c 1 +
+ Td s 
 Ti s

(2.4)
This controller structure has also been labelled the ‘non-interacting’ controller
(McMillan, 1994), the ‘ISA’ algorithm (Gerry and Hansen, 1987) or the
‘parallel non-interacting’ controller (Visioli, 2001). A variation of the
controller is labelled the ‘parallel’ controller structure (McMillan, 1994).
G c (s ) = K c +
1
+ Td s
Ti s
(2.5)
This variation has also been labelled the ‘ideal parallel’, ‘non-interacting’,
‘independent’ or ‘gain independent’ algorithm.
The ideal PID controller structure is used in the following products:
(a) Allen Bradley PLC5 product (McMillan, 1994)
(b) Bailey FC19 PID algorithm (EZYtune, 2003)
(c) Fanuc Series 90–30 and 90–70 Independent Form PID algorithm
(EZYtune, 2003)
(d) Intellution FIX products (McMillan, 1994)
(e) Honeywell TDC3000 Process Manager Type A, non-interactive mode
product (ISMC, 1999)
(f) Leeds and Northrup Electromax 5 product (Åström and Hägglund,
1988)
(g) Yokogawa Field Control Station (FCS) PID algorithm (EZYtune,
2003).
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Handbook of PI and PID Controller Tuning Rules
3. Ideal controller in series with a first order lag:

 1
1
+ Td s 
G c (s) = K c 1 +
 Ti s
 Tf s + 1
(2.6)
4. Controller with filtered derivative:


Td s
1
G c (s ) = K c  1 +
+
 Ti s 1 + s Td

N







(2.7)
This structure is used in the following products:
(a) Bailey Net 90 PID error input product with N = 10 (McMillan, 1994) and
FC156 Independent Form PID algorithm (EZYtune, 2003)
(b) Concept PIDP1 and PID1 PID algorithms (EZYtune, 2003)
(c) Fischer and Porter DCU 3200 CON PID algorithm with N = 8 (EZYtune,
2003)
(d) Foxboro EXACT I/A series PIDA product (in which it is an option
labelled ideal PID) (Foxboro, 1994)
(e) Hartmann and Braun Freelance 2000 PID algorithm (EZYtune, 2003)
(f) Modicon 984 product with 2 ≤ N ≤ 30 (McMillan, 1994; EZYtune,
2003)
(g) Siemens Teleperm/PSC7 ContC/PCS7 CTRL PID products with N = 10
(ISMC, 1999) and the S7 FB41 CONT_C PID product (EZYtune, 2003).
5. Classical controller: This controller is also labelled the ‘cascade’ controller
(Witt and Waggoner, 1990), the ‘interacting’ or ‘series’ controller (Poulin and
Pomerleau, 1996), the ‘interactive’ controller (Tsang and Rad, 1995), the
‘rate-before-reset’ controller (Smith and Corripio, 1997), the ‘analog’
controller (St. Clair, 2000) or the ‘commercial’ controller (Luyben, 2001).

1  1 + sTd

G c (s) = K c 1 +
 Ti s  1 + s Td
N
(2.8)
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Chapter 2: Controller Architecture
7
The structure is used in the following products:
(a) Honeywell TDC Basic/Extended/Multifunction Types A and B products
with N = 8 (McMillan, 1994)
(b) Toshiba TOSDIC 200 product with 3.33 ≤ N ≤ 10 (McMillan, 1994)
(c) Foxboro EXACT Model 761 product with N = 10 (McMillan, 1994)
(d) Honeywell UDC6000 product with N = 8 (Åström and Hägglund, 1995)
(e) Honeywell TDC3000 Process Manager product – Type A, interactive
mode with N = 10 (ISMC, 1999)
(f) Honeywell TDC3000 Universal, Multifunction and Advanced
Multifunction products with N = 8 (ISMC, 1999)
(g) Foxboro EXACT I/A Series PIDA product (in which it is an option
labelled series PID) (Foxboro, 1994).
A subset of the classical PID controller is the so-called ‘series’ controller
structure, also labelled the ‘interacting’ controller or the ‘analog algorithm’
(McMillan, 1994) or the ‘dependent’ controller (EZYtune, 2003).

1 
(1 + sTd )
G c (s) = K c 1 +
 Ti s 
(2.9)
The structure is used in the following products:
(a) Turnbull TCS6000 series product (McMillan, 1994)
(b) Alfa-Laval Automation ECA400 product (Åström and Hägglund, 1995)
(c) Foxboro EXACT 760/761 product (Åström and Hägglund, 1995).
A further related structure is one labelled the ‘interacting’ controller (Fertik,
1975).


Td s
1 
 1 +
G c (s) = K c 1 +
 Ti s  1 + s Td
N







The structure is used in the following products:
(a) Bailey FC156 Classical Form PID product (EZYtune, 2003)
(b) Fischer and Porter DCI 4000 PID algorithm (EZYtune, 2003).
(2.10)
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6. Generalised classical controller:


Td s
1
+
G c (s) = K c 1 +
T

Ti s
1+ d

N



2
 b f 0 + b f 1s + b f 2 s
 1 + a f 1s + a f 2 s 2
s 





(2.11)
7. Two degree of freedom controller 1:





[1 − β]Td s R (s) − K 1 + 1 + Td s
1
+
U (s) = K c  [1 − α] +
c
Td 
T


Ti s
Ti s
1
1+ d
+
s



N 
N




Y(s) (2.12)

s

This controller is also labelled the ‘m-PID’ controller (Huang et al., 2000), the
‘ISA-PID’ controller (Leva and Colombo, 2001) and the ‘P-I-PD (only P is
DOF) incomplete 2DOF algorithm’ (Mizutani and Hiroi, 1991).
This structure is used in the following products:
(a) Bailey Net 90 PID PV and SP product (McMillan, 1994)
(b) Yokogawa SLPC products with α = −1 , β = −1 , N = 10 (McMillan,
1994)
(c) Omron E5CK digital controller with β = 1 and N = 3 (ISMC, 1999).
Notable subsets of this controller structure are:



Td s
1 
1
R (s) − K c 1 +
• U (s) = K c 1 +
+
T

Ti s 
Ti s

1+ d

N



Y(s)

s

(2.13)
which is used in the following products:
(a) Allen Bradley SLC5/02, SLC5/03, SLC5/04, PLC5 and Logix5550
products (EZYtune, 2003)
(b) Modcomp product with N = 10 (McMillan, 1994).
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


1 
1
R (s) − K c 1 +
• U (s) = K c 1 +
+ Td s Y (s)
Ti s 
Ti s



9
(2.14)
Also labelled the ‘PI+D’ controller structure (Chen, 1996) or the
dependent, ideal, non-interacting controller structure (Cooper, 2006a), it
is used in the following products:
(a) ABB 53SL6000 product (ABB, 2001)
(b) Genesis product (McMillan, 1994)
(c) Honeywell TDC3000 Process Manager Type B, non-interactive
mode product (ISMC, 1999)
(d) Square D PIDR PID product (EZYtune, 2003).
 1 


1
R (s) − K c 1 +
• U (s) = K c 
+ Td s Y(s)
Ti s
 Ti s 


(2.15)
which is used in the following products:
(a) Toshiba AdTune TOSDIC 211D8 product (Shigemasa et al., 1987)
(b) Honeywell TDC3000 Process Manager Type C non-interactive
mode product (ISMC, 1999).
8. Two degree of freedom controller 2:




T
s
1
d
 1 + b f 1s (F(s)R (s) − Y(s) ) − K Y(s) ,
+
U(s) = K c 1 +
0

T  1 + a f 1s
Ti s
1+ d s

N 

1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4
F(s) =
(2.16)
1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5 s 4
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9. Two degree of freedom controller 3:
U (s) = F1 (s)R (s) − F2 (s)Y(s) ,




(
)
1
−
β
T
s
1 + b f 1s
[1 − χ] + δ +
d 
,
F1 (s) = K c  [1 − α ] +

Td  1 + a f 1s + a f 2 s 2
Ti s 1 + Ti s
1+
s 

N 





1+ b f 2s
[
1 − φ] [1 − ϕ]Td s 

[
]
+
F2 (s) = K c 1 − ε +


T
Ti s
1 + a f 3s + a f 4 s 2
1+ d s 

N 

K (b + b f 4 s )
− 0 f3
(2.17)
1 + a f 5s
Notable subsets of this controller structure are:




Td s
1 
 R (s) − 1 +
• U (s) = K c 1 +
Y(s) 


Ti s 

 1 + (Td N )s 

(2.18)
Also labelled the ‘reset-feedback’ controller structure (Huang et al., 1996),
it is used in the following products:
(a) Bailey Fisher and Porter 53SL6000 and 53MC5000 products (ISMC,
1999)
(b) Moore Model 352 Single-Loop Controller product (Wade, 1994).


1 + Td s
1 
 R (s) −
• U(s) = K c 1 +
Y(s) 
1 + (Td N )s
Ti s 


(2.19)
Also labelled the ‘industrial controller’ structure (Kaya and Scheib,
1988), it is used in the following products:
(a) Fisher-Rosemount Provox product with N = 8 (ISMC, 1999;
McMillan, 1994)
(b) Foxboro Model 761 product with N = 10 (McMillan, 1994)
(c) Fischer-Porter Micro DCI product (McMillan, 1994)
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Chapter 2: Controller Architecture
11
(d) Moore Products Type 352 controller with 1 ≤ N ≤ 30 (McMillan,
1994)
(e) SATT Instruments EAC400 product with N = 8.33 (McMillan, 1994)
(f) Taylor Mod 30 ESPO product with N = 16.7 (McMillan, 1994)
(g) Honeywell TDC3000 Process Manager Type B, interactive mode
product with N = 10 (ISMC, 1999).


Kc
1  1 + sTd

R (s) − K c 1 +
• U (s) =
Tis
 Tis  1 + sTd
N



Y (s)



(2.20)
which is used in the following products:
(a) Honeywell TDC3000 Process Manager Type C, interactive mode
product with N = 10 (ISMC, 1999)
(b) Honeywell TDC3000 Universal, Multifunction and Advanced
Multifunction products with N = 8 (ISMC, 1999).
2.2 Comments on the PID Controller Structures
In some cases, one controller structure may be transformed into another; clearly,
the ideal and parallel controller structures (Equations 2.4 and 2.5) are very
closely related. It is shown by McMillan (1994), among others, that the
parameters of the ideal PID controller may be worked out from the parameters of
the series PID controller, and vice versa. The ideal PID controller is given in
Equation (2.22) and the series PID controller is given in equation (2.23).


1
G cp (s) = K cp 1 +
+ Tdp s 
 Tip s



(2.22)

1 
(1 + sTds )
G cs (s) = K cs 1 +
 Tis s 
(2.23)
Then, it may be shown that
 T
K cp = 1 + ds
Tis


 Tis
 K cs , Tip = (Tis + Tds ) , Tdp = 

 Tis + Tds

Tds .

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Similarly, it may be shown that, provided Tip > 4Tdp ,


T
 , T = 0.5T 1 + 1 − 4 dp
is
ip


Tip



Tdp 
.
Tds = 0.5Tdp 1 − 1 − 4

Tip 



Tdp
K cs = 0.5K cp 1 + 1 − 4

Tip


,


Åström and Hägglund (1996) point out that the ideal controller admits complex
zeroes and is thus a more flexible controller structure than the series controller,
which has real zeroes; however, in the frequency domain, the series controller
has the interesting interpretation that the zeroes of the closed loop transfer
function are the inverse values of Tis and Tds . O’Dwyer (2001b) developed a
comprehensive set of tuning rules for the series PID controller, based on the ideal
PID controller; as these are not strictly original tuning rules, only representative
examples are included in the relevant tables.
In a similar manner, it is straightforward to show that controller structures
(2.6), (2.7) and (2.8) are subsets of a more general controller structure, and
controller parameters may be transformed readily from one structure to another.
2.3 Process Modelling
Processes with time delay may be modelled in a variety of ways. The modelling
strategy used will influence the value of the model parameters, which will in turn
affect the controller values determined from the tuning rules. The modelling
strategy used in association with each tuning rule, as described in the original
papers, is indicated in the tables (see Chapters 3 and 4). These modelling
strategies are outlined in Appendix 2. Process models may be classified as selfregulating or non-self-regulating, and the models that fall into these categories
are now detailed.
2.3.1 Self-regulating process models
1.
2.
Delay model: G m (s) = K m e −sτm
Delay model with a zero:
G m (s) = K m (1 + Tm 3 s)e − sτ m or G m (s) = K m (1 − Tm 4 s)e − sτ m
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Chapter 2: Controller Architecture
13
3.
First order lag plus time delay (FOLPD) model: G m (s) =
4.
FOLPD model with a zero:
K m e −sτ m
1 + sTm
K m (1 − sTm 4 )e − sτ m
K m (1 + sTm 3 )e −sτm
or G m (s) =
G m (s ) =
1 + sTm1
1 + sTm1
5.
Second order system plus time delay (SOSPD) model:
K m e − sτ m
or G m (s) =
G m (s) =
2
(1 + Tm1s )(1 + Tm 2 s )
Tm1 s 2 + 2ξ m Tm1s + 1
K m e − sτ m
6.
SOSPD model with a zero:
G m (s ) =
G m (s) =
7.
K m (1 + sTm 3 )e − sτm
2
Tm1 s 2 + 2ξ m Tm1s + 1
or G m (s) =
K m (1 − sTm 4 )e −sτ m
K m (1 − sTm 4 )e − sτ m
or G m (s) =
2
(1 + sTm1 )(1 + sTm 2 )
Tm1 s 2 + 2ξ m Tm1s + 1
Third order system plus time delay (TOSPD) model:
G m (s ) =
K m (1 + b1s + b 2 s 2 + b 3s 3 )e − sτ m
or
1 + a 1s + a 2 s 2 + a 3s 3
(
G m (s ) =
)
K m e − sτ m
or
(1 + sTm1 )(1 + sTm 2 )(1 + sTm3 )
G m (s) =
(T
m1
8.
K m (1 + sTm 3 )e −sτm
(1 + sTm1 )(1 + sTm 2 )
K m (Tm 3 s + 1)e −sτ m
)
s + 2ξ m Tm1s + 1 (Tm 2 s + 1)
2 2
Fifth order system plus delay model:
G m (s ) =
K m (1 + b1s + b 2 s 2 + b 3s 3 + b 4 s 4 + b s s 5 )e − sτm
1 + a 1s + a 2 s 2 + a 3s 3 + a 4 s 4 + a 5s 5
(
)
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9. General model.
10. Non-model specific. Corresponding tuning rules may also apply to non-selfregulating processes.
2.3.2 Non-self-regulating process models
1.
Integral plus time delay (IPD) model: G m (s) =
2.
IPD model with a zero:
G m (s) =
3.
K m e −sτ m
s
K m (1 + sTm 3 )e −sτ m
K (1 − sTm 4 )e −sτ m
or G m (s) = m
s
s
First order lag plus integral plus time delay (FOLIPD) model:
K m e −sτ m
G m (s) =
s(1 + sTm )
4.
FOLIPD model with a zero:
G m (s) =
5.
6.
7.
8.
K m (1 + sTm 3 )e −sτ m
K (1 − sTm 4 )e −sτ m
or G m (s) = m
s(1 + sTm1 )
s(1 + sTm1 )
Integral squared plus time delay ( I 2 PD ) model: G m (s) =
K m e − sτ m
s2
Second order system plus integral plus time delay (SOSIPD) model:
K m e −sτ m
K m e −sτ m
G
(
s
)
or
=
G m (s ) =
m
2
2
s(1 + sTm1 )
s(Tm1 s 2 + 2ξ m Tm1s + 1)
K m (1 + sTm 3 )e − sτ m
s(1 + sTm1 )(1 + sTm 2 )
Third order system plus integral plus time delay (TOSIPD) model:
SOSIPD model with a zero: G m (s) =
G m (s) =
K m e − sτ m
K m e − sτ m
or G m (s) =
3
s(1 + sTm1 )(1 + sTm 2 )(1 + sTm 3 )
s(1 + sTm1 )
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Chapter 2: Controller Architecture
9.
15
General model with integrator:
G m (s) =
K m e − sτ m
s(1 + sTm1 )
n
or
K
G m (s ) = m
s
∏ (T
∏ (T
i
i
)∏ (T
s + 1)∏ (T
m1i s + 1
m 3i
i
i
m 2i
m 4i
2 2
)
e
s + 1)
s + 2ξ m ni Tm 2i s + 1
2 2
s + 2ξ mdi Tm 4i
−sτ m
K m e −sτm
Tm s − 1
11. Unstable FOLPD model with a zero:
10. Unstable FOLPD model: G m (s) =
G m (s) =
K m (1 + sTm 3 )e −sτm
K (1 − sTm 4 )e − sτ m
or G m (s) = m
Tm s − 1
Tm1s − 1
12. Unstable SOSPD model (one unstable pole):
G m (s ) =
K m e − sτm
(Tm1s − 1)(1 + sTm 2 )
G m (s ) =
K m e − sτ m
(Tm1s − 1)(Tm 2 s − 1)
13. Unstable SOSPD model (two unstable poles):
14. Unstable SOSPD model with a zero:
G m (s) =
K m (1 + Tm 3 s )e −sτ m
2
Tm1 s 2 − 2ξ m Tm1s + 1
or
K m (1 + Tm 3 s )e − sτ m
2
− Tm1 s 2 + 2ξ m Tm1s + 1
G m (s ) =
or
K m (1 − sTm 4 )e − sτ m
2
Tm1 s 2 − 2ξ m Tm1s + 1
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16
Table 1 shows the number of tuning rules defined for each PI/PID controller
structure and types of process model. The following data is key to the process
model type:
Model 1: Stable FOLPD model (with or without a zero)
Model 2: Stable SOSPD model (with or without a zero)
Model 3: Other stable models
Model 4: Non-model specific
Model 5: Models with an integrator
Model 6: Open loop unstable models
Table 1: PI/PID controller structure and tuning rules – a summary
Process model
Controller
Equation
(2.3)
(2.4)
(2.6)
(2.7)
(2.8)
(2.11)
(2.12)
(2.16)
(2.17)
Total
1
2
3
4
5
6
261
140
20
36
74
7
74
7
28
649
63
82
23
9
53
7
36
3
15
291
58
20
3
5
1
1
14
0
1
103
59
63
9
11
12
4
9
0
2
169
90
66
16
6
26
5
106
16
8
339
32
35
17
0
17
6
37
8
30
182
Total
563
406
88
67
183
30
276
34
84
1731
Table 1 shows that for the tuning rules defined, 60% are based on a selfregulating (or stable) model, 10% are non-model specific, with the remaining
30% based on a non-self-regulating model.
2.4 Organisation of the Tuning Rules
The tuning rules are organised in tabular form in Chapters 3 and 4. Within each
table, the tuning rules are classified further; the main subdivisions made are as
follows:
(i) Tuning rules based on a measured step response (also called process reaction
curve methods).
(ii) Tuning rules based on minimising an appropriate performance criterion,
either for optimum regulator or optimum servo action.
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17
(iii) Tuning rules that give a specified closed loop response (direct synthesis
tuning rules). Such rules may be defined by specifying the desired poles of
the closed loop response, for instance, though more generally, the desired
closed loop transfer function may be specified. The definition may be
expanded to cover techniques that allow the achievement of a specified gain
margin and/or phase margin.
(iv) Robust tuning rules, with an explicit robust stability and robust performance
criterion built into the design process.
(v) Tuning rules based on recording appropriate parameters at the ultimate
frequency (also called ultimate cycling methods).
(vi) Other tuning rules, such as tuning rules that depend on the proportional gain
required to achieve a quarter decay ratio or to achieve magnitude and
frequency information at a particular phase lag.
Some tuning rules could be considered to belong to more than one subdivision,
so the subdivisions cannot be considered to be mutually exclusive; nevertheless,
they provide a convenient way to classify the rules. All symbols used in the
tables are defined in Appendix 1.
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Chapter 3
Controller Tuning Rules for Self-Regulating
Process Models
3.1 Delay Model G m (s) = K m e − sτm

1 

3.1.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
R(s)
+
U(s)

1 

K c 1 +
Ti s 

E(s)
−
K m e −sτ
Y(s)
m
Table 2: PI controller tuning rules – G m (s) = K m e −sτ m
Rule
1
Callender et al.
(1935/6).
Model: Method 1
Kc
Ti
0.568 K m τ m
Process reaction
3.64τ m
2
0.65 K m τ m
2.6τ m
3
0.79 K m τ m
3.95τ m
4
0.95 K m τ m
3.3τ m
1
Decay ratio = 0.015; period of decaying oscillation = 10.5τ m .
2
Decay ratio = 0.1; period of decaying oscillation = 8τ m .
3
Decay ratio = 0.12; period of decaying oscillation = 6.28τ m .
4
Decay ratio = 0.33; period of decaying oscillation = 5.56τ m .
18
Comment
Representative results
deduced from graphs.
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Comment
Two constraints
method – Wolfe
(1951).
Model: Method 1
0.2
K mτm
0.3τ m
Decay ratio is as
small as possible.
Minimum ISE –
Haalman (1965).
Model: Method 1
Minimum error: step
load change – Gerry
(1998).
Minimum IAE –
Gerry and Hansen
(1987).
Minimum IAE –
Shinskey (1988),
p. 123.
Minimum IAE –
Shinskey (1994).
Model: Method 1
Minimum IAE –
Shinskey (1988),
p. 148.
Minimum IAE –
Shinskey (1996),
p. 167.
Minimum IAE –
Huang and Jeng
(2002).
Minimum ISE –
Keviczky and Csáki
(1973).
Model: Method 1
Fertik (1975).
Model: Method 1
Åström and Hägglund
(2000).
Model: Method 1
Minimum error integral (regulator mode).
Minimum performance index: regulator tuning
undefined
1.5τ m
G c (s) = 1 Tis
M s = 1.9 ; A m = 2.36 ; φ m = 50 0 .
0.3
Km
0.42τ m
Model: Method 1
0.345
Km
0.455τ m
Model: Method 1
0.43
Km
0.5τ m
Model: Method 1
0. 4 K m
0.5τ m
p. 67.
undefined
1.6K m τ m
G c (s) = 1 Tis ;
p. 63.
0.4255
Km
0.25Tu
Model: Method 1
0. 4 K u
0.25Tu
Model: Method 1
Minimum performance index: servo tuning
Model: Method 1
0.36 K m
0.47τ m
Minimum IAE = 1.377 τ m . A m = 2 ; φ m = 60 0 .
undefined
1.33τ m
G c (s) = 1 Tis ;
Minimum ISE =
1.53τ m ; K m = 1 .
0.5
0.635τ m
Km = 1
Minimum performance index: other tuning
0.35 K m
0.4 τ m
K c , Ti values
deduced from graph.
K c Ti maximised
0.25
0.35τ m
Km
subject to
M max = 2 .
19
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20
Rule
Kc
Ti
Comment
Skogestad (2001).
Model: Method 1
Skogestad (2003).
Model: Method 1
Skogestad (2001).
Model: Method 1
0.16 K m
0.340τ m
M max = 1.4
0.200 K m
0.318τ m
M max = 1.6
0.26 K m
0.306τ m
M max = 2.0
x1 K m
x 2τm
Åström and Hägglund
(2004).
Model: Method 1
x1
Kc
Ti
undefined
2.70K m τ m
undefined
2.50K m τ m
undefined
x 2τm
1
OS
x2
0.64
105%
x1
1.00
x1
OS
x1
5%
0.72
0.68
Kc =
2
Kc =
(
Stochastic
disturbance.
Critically damped
dominant pole.
Damping factor
(dominant pole) =
0.6.
G c (s) = 1 Tis
Coefficient values:
OS
OS
x2
x2
x2
OS
17%
2
4%
Coefficient values:
OS
OS
x1
x1
OS
50%
1.5
2τ m
0.75
Kc
Ti
15%
0.78
20%
Model: Method 1
0.223607 K m
0.309017 τm
Model: Method 1
Kuwata (1987).
1
2
)
10%
(
)
m
1 − 0.267 τm − 0.25τm .
e − ωd τ m
τ
1 − 0.5e − ωd τ m , Ti = − ωmτ 1 − 0.5e − ωd τ m .
Km
e d m
(
M max
0.400
1.1
0.211
0.342
1.6
0.389
1.2
0.227
0.334
1.7
0.376
1.3
0.241
0.326
1.8
0.363
1.4
0.254
0.320
1.9
0.352
1.5
0.264
0.314
2.0
Direct synthesis: time domain criteria
Step disturbance.
0. 5 K m
0.5τ m
Bryant et al. (1973).
Model: Method 1;
G c (s) = 1 Tis
Nomura et al. (1993).
x2
0.057
0.103
0.139
0.168
0.191
Van der Grinten
(1963).
Model: Method 1
Keviczky and Csáki
(1973).
Model: Method 1;
representative
coefficient values
estimated from
graphs; K m = 1 .
Coefficient values:
M max
x1
x2
0.4τm
K m 1 − 1 − 0.267 τm
) , T = 1.25τ
i
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kamimura et al.
(1994), Manum
(2005).
Model: Method 1;
G c (s) = 1 Tis .
Åström and Hägglund
(1995), pp. 187–189.
Model: Method 1
Kc
Ti
Comment
undefined
2K m τ m
OS = 10%
Also given by Schlegel (1998); M s = 1 .
undefined
2.6667 K m τ m
undefined
2.3529 K m τ m
0.305 K m
0.279τm
‘Quick’ servo
response; ‘negligible’
overshoot.
ξ = 0.3
0.273 K m
0.285τm
ξ = 0.4
OS = 0%
0.218 K m
0.288τm
ξ = 0.6
0.174 K m
0.276τm
ξ = 0.8
0.154 K m
0.265τm
ξ = 0.9
Åström and Hägglund
(1995), pp. 187–189,
Åström and Hägglund
(2006), pp. 185–186.
Model: Method 1
0.388 K m
0.258τm
ξ = 0.1
0.343 K m
0.270τm
ξ = 0.2
0.244 K m
0.288τm
ξ = 0.5
0.195 K m
0.284τm
ξ = 0.707
0.135 K m
0.250τm
ξ = 1.0
Hansen (2000).
0. 2 K m
0.3τ m
Model: Method 1
Buckley (1964).
Model: Method 1;
M max = 2dB .
Schlegel (1998).
Model: Method 1
Hägglund and Åström
(2004).
Model: Method 1
Åström and Hägglund
(2006), p. 243.
Direct synthesis: frequency domain criteria
undefined
1.45K m τ m
G c (s) = 1 Tis
0.075 K m
0.1τ m
0.53 K m
τm
0.55 K m
1.59 τ m
0.318 K m
0.405τm
0.15 K m
0.35τ m
0.15 K u
0.17Tu
0.1677 K m
0.363τm
Representative
results.
Ms = 1
M s = 1.4
Model: Method 1;
M s = M t = 1. 4 .
Robust
Bequette (2003),
p. 300.
τm
K m (2λ + τ m )
0.5τ m
Skogestad (2003).
Model: Method 1
0.294 K m
0.5τ m
Model: Method 1;
λ > τm (Yu, 2006,
p. 19).
IMC based rule.
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22
Rule
Kc
Ti
Comment
1
0.5τm
K m 0.5τm + λ
0.5τm
Model: Method 1
undefined
x1K m τm
G c (s) = 1 Tis ;
Skogestad (2003).
0.25 K m
Other tuning
0.333τ m
Model: Method 1
McMillan (2005),
p. 35.
0.25 K m
0.25τ m
Model: Method 1
Yu (2006), p. 18.
0.30K u
Ultimate cycle
0.25Tu
Model: Method 1
Åström and Hägglund
(2006), p. 189.
Urrea et al. (2006).
Model: Method 1
3
17-Mar-2009
3
1.5 ≤ x1 ≤ 2 .
λ = Tm (aggressive tuning). λ = 3Tm (robust tuning).
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models


1
+ Td s 
3.1.2 Ideal PID controller G c (s) = K c 1 +
 Ti s

R(s)
E(s)
−
+


1
K c 1 +
+ Td s 
T
s
i


Y(s)
U(s)
K me
− sτ m
Table 3: PID controller tuning rules – delay model G m (s) = K m e − sτ m
Rule
Callender et al.
(1935/6).
Model: Method 1
Ream (1954).
Model: Method 1
Nomura et al.
(1993).
Model: Method 1
Kc
Ti
Td
Comment
0.749 K m τm
Process reaction
2.734 τ m
0.303τ m
0.2269 K m
0.3109τm
0.0264τm
‘Butterworth’.
0.2635 K m
0.3610τm
0.1911τm
0.3548 K m
0.4860τm
0.2735τm
‘Minimum
ITAE’.
‘Binomial’.
0.511 K m
0.511τm
0.1247 τm
Model: Method 1
0.40 K m
0.46τm
0.149τm
Regulator.
Representative
results deduced
2
1.31τ m
0.303τ m
1.24 K m τm
from graphs.
Direct synthesis: time domain criteria
Decay ratio
0.61 K m
0.491τm
0.015τm
≈ 33% .
‘Kitamori’.
0.230 K m
0.315τm
0.0086τm
1
Robust
Gong et al.
(1998).
Wang (2003),
pp. 15–17.
Model: Method 1
Servo; ±10%
overshoot.
Deduced from graphs; robust to ±25% uncertainty in process model
parameters.
0.29 K m
0.40τm
0.096τm
1
Decay ratio = 0.015; period of decaying oscillation = 10.5τ m .
2
Decay ratio = 0.12; period of decaying oscillation = 6.28τ m .
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24
3.1.3 Ideal controller in series with a first order lag
 1

1
G c (s) = K c 1 +
+ Td s 
 Tf s + 1
 Ti s
R(s)
+
E(s)
−


1
K c 1 +
+ Td s 
Ti s


1
1 + Tf s
Y(s)
U(s)
K m e −sτ
m
Table 4: PID controller tuning rules – delay model G m (s) = K m e −sτ m
Rule
Kc
Ti
Td
Comment
Robust
Bequette (2003),
p. 300.
Model: Method 1
τm
K m (4λ + τ m )
0.5τ m
0.167 τ m
λ > 1.7 τm (Yu, 2006, p. 19).
Tf =
2
2λ − 0.167τ m
4λ + τ m
2
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models



1  1 + Td s 


3.1.4 Classical controller G c (s) = K c 1 +
 Ti s  1 + Td s 
N 

E(s)
R(s)
−
+





1  1 + Td s  U(s)

K c 1 +
Td 
Ti s 

s
1 +
N 

Y(s)
K m e − sτ
m
Table 5: PID controller tuning rules – delay model G m (s) = K m e −sτ m
Rule
Kc
Ti
Td
Comment
Process reaction
Hartree et al.
(1937).
Model: Method 1
1
2
0.70
K mτm
2.66τ m
τm
0.78
K mτm
2.97 τ m
0.5τ m
1
N = 2. Decay ratio = 0.15; period of decaying oscillation = 4.49 τ m .
2
N = 2. Decay ratio = 0.042; period of decaying oscillation = 5.03τ m .
Representative
results.
25
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26
3.1.5 Generalised classical controller


Td s
1
+
G c (s) = K c 1 +
T

Ti s
1+ d

N

R(s)
E(s)
+
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N

Rule
Yu (2006),
p. 18.
Model: Method 1


2
 b f 0 + b f 1s + b f 2 s
2
 1 + a f 1s + a f 2 s
s

U(s)


2
 b f 0 + b f 1s + b f 2 s
 1 + a f 1s + a f 2 s 2
s 

K m e − sτ
Y(s)
m
Table 6: PID controller tuning rules – delay model K me −sτ m
Comment
Kc
Ti
Td
0.30K u
Ultimate cycle
0
0.23Tu
b f 0 = 0 , b f 1 = 0.12Tu , b f 2 = 0 , a f 1 = 0.012Tu , a f 2 = 0 .




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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
3.1.6 Two degree of freedom controller 1


Td s
1
+
U (s ) = K c  1 +
 Ti s 1 + Td

N






βTd s 
 E (s ) − K c  α +
 R (s )
Td 


s
1+ s 

N 






T
s
β
d
α +
K
Td  c

1+
s

N 

E(s)
R(s)
−
+


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N




 +
s

−
U(s)
Y(s)
K me
− sτ m
Table 7: PID controller tuning rules – delay model G m (s) = K m e −sτ m
Rule
Kc
Kuwata (1987).
Model: Method 1
Nomura et al.
(1993).
Model: Method 1
1
Kc =
(
1
Ti
Kc
0.2 K m
0.278τm
1 Km
1.667 τm − 2.5
0.8τm
K m 1 − 1 − 0.267 τm
).
Comment
Td
Direct synthesis: time domain criteria
0.278τm
0
1.667 τm − 4.167
0.1 K m
27
0
α =1
α = 1 ; also given
by Kuwata
(1987).
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28
3.2 Delay Model with a Zero
G m (s) = K m (1 + Tm 3 s)e −sτ m or G m (s) = K m (1 − Tm 4 s)e −sτ m

1 

3.2.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
E(s)
R(s)
+

1 

K c 1 +

T
is 

−
Y(s)
U(s)
Model
Table 8: PI controller tuning rules – delay model with a zero
G m (s) = K m (1 + Tm 3 s)e −sτ m or G m (s) = K m (1 − Tm 4s)e −sτ m
Rule
Kc
Ti
Comment
Direct synthesis: time domain criteria
1
‘Smaller’ τm Tm 4 .
Ti
undefined
Chidambaram (2002),
p. 157.
Model: Method 1;
G c (s) = 1 Tis .
2
(
1 + 4.712(T
1
Ti = 0.637 τm 1 + 3.142 Tm 4 τm
2
2
Ti = 0.424τm
2
m4
τm
).
).
Ti
‘Larger’ τm Tm 4 .
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Jyothi et al. (2001).
Model: Method 1;
G c (s) = 1 Tis .
Jyothi et al. (2001).
Model: Method 1;
G c (s) = 1 Tis .
Kc
Ti
Comment
Direct synthesis: frequency domain criteria
3
τm Tm3 < 1
Ti
Ti
τm Tm3 > 1
5
Ti
0 < τm Tm 3 < 1
6
Ti
0 < τm Tm 3 < 10
4
undefined
1.5K m (τ m − Tm 3 )
2K m Tm 4
undefined
3
Ti = 0.64K m τm 1 + 9.870(Tm3 τm ) .
4
Ti = 1.27 K m τm 1 + 2.467(Tm3 τm ) .
5
Ti = 2.0309K mTm3e0.419692 τ m
6
Ti =
7
Ti = 1.27 K m τm 1 + 2.467(Tm 4 τm ) .
8
Ti =
9
Ti =
τm Tm 4 < 1
Ti
τm Tm 4 > 1
8
Ti
0 < τm Tm 4 < 1
9
Ti
0 < τm Tm 4 < 10
7
1.5K m (τm + Tm 4 )
2
2
Tm 3
.
K mTm3
.
exp[− 0.706636 ln (τm Tm3 ) − 0.962232]
2
K m Tm 4
0.5 + 0.001945(τm Tm 4 ) − 0.03091(τm Tm 4 )2
K m Tm 4
.
0.6178 − 0.1452(τm Tm 4 ) + 0.0141(τm Tm 4 )2 − 0.0004883(τm Tm 4 )3
.
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3.3 FOLPD Model G m (s) =
K m e −sτm
1 + sTm

1 

3.3.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
R(s)
E(s)
+
−
U(s)

1 

K c 1 +
Ti s 

Y(s)
K m e − sτ
1 + sTm
m
Table 9: PI controller tuning rules – FOLPD model G m (s) =
Rule
Callender et al.
(1935/6).
Model: Method 1
Ziegler and Nichols
(1942).
Model: Method 2
Hazebroek and Van
der Waerden (1950).
Model: Method 2
K m e − sτ m
1 + sTm
Comment
Kc
Ti
1
0.568 K m τ m
Process reaction
3.64τ m
2
0.690 K m τ m
2.45τ m
τm
= 0.3
Tm
0.9Tm
K mτm
3.33τ m
Quarter decay ratio;
τm Tm ≤ 1 .
x1Tm K m τm
x 2τm
τm
Tm
x1
x2
Coefficient values:
x1
x2
τm
Tm
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.68
0.70
0.72
0.74
0.76
0.79
0.81
0.84
0.87
7.14
4.76
3.70
3.03
2.50
2.17
1.92
1.75
1.61
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.90
0.93
0.96
0.99
1.02
1.06
1.09
1.13
1.17
1.49
1.41
1.32
1.25
1.19
1.14
1.10
1.06
1.03
1
Decay ratio = 0.015; period of decaying oscillation = 10.5τ m .
2
Decay ratio = 0.043; period of decaying oscillation = 6.28τ m .
τm
Tm
x1
x2
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
1.20
1.28
1.36
1.45
1.53
1.62
1.71
1.81
1.00
0.95
0.91
0.88
0.85
0.83
0.81
0.80
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Hazebroek and Van
Tm
der Waerden (1950) –
K m τm
continued.
Oppelt (1951).
Model: Method 2
Moros (1999).
Model: Method 1
3
4
Kc ≤
1
Km
τm Tm > 3.5
1.66τ m
τ m >> Tm
Kc
3.32τ m
τ m << Tm
0.8Tm K m τ m
3τ m
0.91Tm K m τ m
3.3τ m
Attributed to
Rosenberg.
0.6Tm K m τm
Regulator
4τ m
0% overshoot.
0.7Tm K m τm
2.33τ m
20% overshoot.
0.35Tm K m τm
Servo
1.17Tm
0% overshoot.
0.6Tm K m τm
Tm
20% overshoot.
0.20 K m
Regulator
0.20τ m
0% overshoot.
0.30 K m
0.22τ m
20% overshoot.
0.15 K m
Servo
0.16τ m
0% overshoot.
0.20 K m
0.14τ m
20% overshoot.
Reswick (1956).
Model: Method 2;
K c , Ti deduced
from graphs;
τm Tm = 6.25 .
Two constraints
method – Wolfe
(1951).
Model: Method 4
2
τm
1.6τm − 1.2Tm
Attributed to Oppelt.
Chien et al. (1952).
Model: Method 2;
0.1 < τm Tm < 1.0 .
Cohen and Coon
(1953).
Model: Method 2


τ
 0.5 m + 1


T
m


3
Comment
Ti
1
Km


T
 0.9 m + 0.083
τm


4
x1Tm K m τm
x 2τm
τm
Tm
Coefficient values:
x1
x2
x1
τm
Tm
0.2
0.5
4.4
1.8
3.23
2.27
Quarter decay ratio;
0 < τm Tm ≤ 1.0 .
Ti
1.0
5.0
0.78
0.30
x2
1.28
0.53


1 
Tm
Tm 
+ 1 ; recommended K c =
− 1 .
1.5708
0.77
τ
K
τm
m
m 



 3.33(τm Tm ) + 0.31(τm Tm ) 2 
.
Ti = Tm 


1 + 2.22(τm Tm )


Decay ratio is as
small as possible;
minimum error
integral (regulator
mode).
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32
Rule
Two constraints
criterion – Murrill
(1967), p. 356.
Model: Method 5
Heck (1969).
Model: Method 2
Kc
0.928  Tm

K m  τ m
Comment
Ti



0.946
Tm  τ m

1.078  Tm



0.583
Quarter decay ratio;
minimum error
integral (servo mode).
0.1 ≤ τm Tm ≤ 1.0 .
0.8Tm K m τm
3τ m
Regulator.
undefined
x 2K mτm
G c = 1 (Tis)
Regulator – representative values (deduced from graph):
τm Tm
x2
τm Tm
x2
τm Tm
x2
τm Tm
x2
0.11
0.13
0.14
0.17
τm
2.94
0.20
2.50
0.40
2.17
0.71
1.96
2.86
0.25
2.38
0.50
2.13
2.78
0.29
2.35
0.56
2.08
2.63
0.33
2.30
0.63
2.00
Servo – representative values (deduced from graph):
Tm
x2
τm Tm
x2
τm Tm
x2
τm Tm
x2
0.11
0.13
0.14
0.17
3.92
3.85
3.64
3.51
0.56 K m
0.20
0.25
0.29
0.33
3.33
0.40
3.08
0.50
2.94
0.56
2.82
0.63
0.65Tm
2.63
0.71
2.27
2.47
2.41
2.35
Model: Method 3
Fertik and Sharpe
(1979).
Parr (1989).
0.91Tm K m τm
3.3τm
Model: Method 2
Sakai et al. (1989).
1.2408Tm K m τm
0.5τ m
Model: Method 2
Borresen and Grindal
(1990).
Model: Method 2
Klein et al. (1992).
Model: Method 1
1.0Tm
K mτm
3τ m
Also described by
ABB (2001).
McMillan (1994),
p. 25.
Model: Method 5
St. Clair (1997),
p. 22.
Model: Method 5
0.28Tm
K m (τ m + 0.1Tm )
0.53Tm
Km
3
τm
Time delay dominant
processes.
0.333Tm
K mτm
Tm
τm
≥ 0.33
Tm
x1 K m
x 2τm
Hay (1998),
Coefficient values – servo tuning:
pp. 197–198.
τ m Tm
x1
x2
τ m Tm
x1
Model: Method 1;
0.1
6.0
9.5
0.1
4.0
coefficients of K c , Ti
4.5
7.5
0.125
3.2
deduced from graphs. 0.125
0.167
3.0
5.4
0.167
2.2
0.25
2.0
3.4
0.25
1.3
0.5
0.8
1.7
0.5
0.5
x2
10.0
8.0
6.0
4.0
2.0
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Hay (1998),
pp. 197–198 –
continued.
Hay (1998), p. 200.
Model: Method 2;
K c , Ti deduced from
graphs.
Shinskey (2000),
(2001).
Lipták (2001).
Kc
Ti
x1 K m
x 2τm
τ m Tm
0.1
0.125
0.167
0.25
0.5
x1 K m
Coefficient values – regulator tuning:
x1
x2
τ m Tm
x1
x2
9.5
7.0
5.0
3.4
1.8
2.8
2.8
2.6
2.3
1.6
3.2
3.2
3.0
2.8
2.0
0.1
0.125
0.167
0.25
0.5
Minimum IAE –
Frank and Lenz
(1969).
Model: Method 1
6.8
5.6
4.3
2.8
1.2
x 2 τm
x1
τm Tm
x2
4
1.6
1.7
1.0
1.2
0.8
1.0
0.7
0.667Tm K m τm
4.9
4
3.0
2.6
2.1
1.9
1.76
1.51
0.2
0.3
0.4
0.5
3.78τ m
0.95Tm K m τm
Regulator.
Servo.
Regulator.
Servo.
Regulator.
Servo.
Regulator.
Servo.
Model: Method 2;
τm Tm = 6 .
Model: Method 2
4τ m
Chidambaram (2002), 1 
Tm 
p. 154.
0.4 + 0.665 
Km 
τm 
Model: Method 1
Faanes and Skogestad
0.71Tm
(2004).
K m τm
Model: Method 1
PMA (2006).
0.39Tm K m τ m
Minimum IAE –
Murrill (1967),
pp. 358–363.
Comment
‘Improved’ Ziegler–
Nichols method.
3.4τm
Equivalent Ziegler–
Nichols ultimate
cycle method.
Model: Method 2
3.3τm
6τ m
Minimum performance index: regulator tuning
0.986
0.707
Model: Method 5;
Tm  τ m 
0.984  Tm 




0.1 ≤ τm Tm ≤ 1.0 .
0.608  Tm 
K m  τm 
1
Km
τm

T 
T 
τm 
 x 1 + x 2 m 
 x 1 + x 2 m 
x
τ
τm 
2 
m 

Representative coefficient values (deduced from graph):
Tm
x1
x2
τ m Tm
x1
x2
0.2
0.4
0.6
0.8
0.37
0.42
0.44
0.45
0.77
0.85
0.91
0.98
1.0
1.2
1.4
0.46
0.46
0.46
1.05
1.11
1.17
33
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
34
Rule
Kc
Ti
Comment
Minimum IAE –
Pemberton (1972a),
Smith and Corripio
(1997), p. 345.
Tm
K mτm
Tm
Model: Method 1;
τ
0.1 ≤ m ≤ 0.5 .
Tm
Minimum IAE –
Murata and Sagara
(1977).
Model: Method 2
x1 K m
τm Tm
0.05
0.075
x 2Tm
x1
x2
τm Tm
x1
x2
τm Tm
4.5
2.3
1.5
0.6
0.9
1.1
0.2
0.4
0.6
1.2
0.9
1.4
1.6
0.8
1.0
5
Minimum IAE –
Kosinsani (1985),
pp. 43–44.
Model: Method 1
5
17-Mar-2009
Ti
Kc
0.05 ≤
τm
T
≤ 1.0 ; 0.05 ≤ L ≤ 7.0 .
Tm
Tm
Load disturbance
1
model =
.
1 + sTL
Values of x1 to x 10 for varying τm Tm provided.
0.1
0.15
0.2
0.3
0.4
0.5
0.75
1.0
x1
19.160 12.886 9.6994 6.5357 4.9749 3.3775 2.5985 2.1285 1.4947
1.1714
x2
4.7284 3.5703 4.2713 10.412 3.9767 2.3329 1.4059 1.1867 0.8493
0.7991
x3
9.4147 4.9974 3.7827 2.5854 1.9599 1.3112 1.0318 0.8244 0.5444
0.3968
x4
5.5495 6.1065 6.4339 0.1239 0.0182 0.1101 -3.0E-5 -5.0E-9 -8.0E-9 -3.0E-7
x5
-11.859 -2.1296 0.4052 200.86 375.61 13.563 2.8052 0.2503 -0.1163 0.8593
x6
1.0356 0.9598 0.8911 0.8734 0.8989 0.8537 0.8128 0.7849 0.7199
0.6711
x7
5.5638 3.9556 3.1631 2.2782 1.7580 1.3671 1.0831 0.9053 0.6215
0.5131
x8
0.2390 0.3740 0.4972 0.7729 1.1390 1.8367 2.4317 3.1877 3.2503
4.1661
x9
0.1162 0.1452 0.1667 0.1883 0.1789 0.2327 0.2775 0.2695 0.5488
0.6450
x 10
-1.6815 -1.5586 -1.4805 -1.4149 -1.5115 -1.2973 -1.4476 -1.5194 -1.4692 -1.5578
Kc =
1 
 x1 − e − x 2 TL
Km 


 T 
 T  
x cosx 4 L  + x 5 sin x 4 L   ,

 3
 Tm 
 Tm  

Tm 
Ti =
Tm
x6 +
1 + x8
.
x7
TL
− x9
Tm
x 10
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Minimum IAE –
Shinskey (1988),
p. 123.
Model: Method 1
6
0.685  TL 


Kc =
K m  Tm 
Ti
6
Kc
Ti
7
Kc
Ti
8
Kc
Ti
9
Kc
Ti
τ m Tm > 0.35
1.00Tm K m τ m
3.0τ m
τ m Tm = 0.2
1.04Tm K m τ m
2.25τ m
τ m Tm = 0.5
1.08Tm K m τ m
1.45τ m
τ m Tm = 1
1.39Tm K m τ m
τm
τ m Tm = 2
0.214 − 0.346
τm
Tm
 τm 


T 
 m
,
1.977
0.874  TL 


Kc =
K m  Tm 
− 0.099 + 0.159
τm
Tm
 τm 


T 
 m
0.871  TL 


Kc =
K m  Tm 
− 0.015 + 0.384
τm
Tm
 τm 


T 
 m
τm
− 0.55
Tm

0.513  TL

Kc =
K m  Tm




0.218
τm
Tm
 τm

T
 m




−1.451
,
TL
τ
≤ 2.641 m + 0.16 .
Tm
Tm
,
− 4.515
τm
Tm
+ 0.067
 τm 


T 
 m
0.876
, 2.641
τm
T
+ 0.16 ≤ L ≤ 1 .
Tm
Tm
−1.055
,
T T 
Ti = m  L 
0.444  Tm 
9
1.123
τ 
 m
T 
 m
−1.041
T T 
Ti = m  L 
0.415  Tm 
8
τm
≤ 0.35
Tm
−1.256
T T 
Ti = m  L 
0.214  Tm 
7
Comment
Kc
Minimum IAE – Yu
(1988).
Model: Method 1.
Load disturbance
K e −sτ L
.
model = L
1 + sTL
35
T T
, Ti = m  L
0.670  Tm




− 0.003
− 0.217
τm
Tm
τm
− 0.213
Tm

− 0.084
τ 
 m
T 
 m
 τm

T
 m




0.56
.
0.867
, 1<
TL
≤ 3.
Tm
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
36
Rule
Kc
Minimum IAE – Hill
and Venable (1989).
Model: Method 1
x1 K m
τm Tm
Ti
Comment
x 2Tm
Load disturbance
1
model =
.
1 + sTL
x1 values (this page) and x 2 values (next page) deduced from
graphs; robust to ±30% process model parameter changes.
0.05 0.07 0.1 0.15 0.2
0.3
0.4
0.5 0.75 1.0
TL Tm = 0.05
14.0
9.8
6.2
4.0
3.0
2.08
1.60
1.32
0.95
0.78
TL Tm = 0.1
16.8
10.3
7.2
4.3
3.2
2.14
1.63
1.33
0.97
0.77
TL Tm = 0.3
17.6
11.8
8.8
5.4
4.0
2.48
1.83
1.46
1.00
0.80
TL Tm = 0.5
17.0
11.4
8.6
5.6
4.2
2.67
1.96
1.58
1.06
0.84
TL Tm = 0.7
16.4
11.0
8.2
5.4
4.1
2.74
2.04
1.64
1.11
0.88
TL Tm = 1.0
15.7
10.5
7.8
5.2
4.0
2.70
2.07
1.68
1.17
0.92
TL Tm = 1.2
15.7
10.5
7.8
5.1
4.0
2.69
2.05
1.68
1.18
0.94
TL Tm = 1.5
15.8
10.5
7.8
5.1
3.9
2.68
2.01
1.68
1.18
0.95
TL Tm = 2.0
15.8
10.5
7.6
5.1
3.9
2.65
2.02
1.65
1.18
0.95
TL Tm = 2.5
15.9
10.5
7.8
5.1
3.9
2.64
1.95
1.63
1.17
0.95
TL Tm = 3.0
15.8
10.5
7.8
5.1
3.9
2.62
2.01
1.62
1.16
0.94
TL Tm = 3.5
15.8
10.4
7.8
5.1
3.9
2.59
1.98
1.61
1.15
0.93
TL Tm = 4.0
15.7
10.5
7.8
5.1
3.9
2.60
2.00
1.63
1.13
0.91
TL Tm = 4.5
15.6
10.4
7.8
5.1
3.8
2.54
1.98
1.60
1.13
0.92
TL Tm = 5.0
15.7
10.3
7.8
5.1
3.8
2.56
1.98
1.61
1.15
0.93
TL Tm = 5.5
15.5
10.3
7.8
5.1
3.9
2.55
1.98
1.62
1.17
0.92
TL Tm = 6.9
15.5
10.3
7.6
5.1
3.9
2.52
1.98
1.60
1.17
0.92
TL Tm = 6.5
15.5
10.3
7.8
5.2
4.0
2.60
2.00
1.62
1.18
0.93
TL Tm = 7.0
15.6
10.3
7.8
5.2
3.9
2.56
2.00
1.62
1.18
0.92
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Minimum IAE – Hill
and Venable (1989).
Model: Method 1
x1 K m
τm Tm
Ti
Comment
x 2Tm
Load disturbance
1
model =
.
1 + sTL
x1 values (previous page) and x 2 values (this page) deduced
from graphs; robust to ±30% process model parameter changes.
0.05 0.07 0.1 0.15 0.2
0.3
0.4
0.5 0.75 1.0
TL Tm = 0.05
1.0
1.0
1.0
1.0
1.0
0.92
0.90
0.88
0.80
0.74
TL Tm = 0.1
1.0
1.0
1.0
1.0
1.0
0.92
0.90
0.88
0.80
0.74
TL Tm = 0.3
3.4
1.8
1.3
1.0
1.0
0.93
0.88
0.86
0.79
0.73
TL Tm = 0.5
4.4
2.8
2.0
1.7
1.2
1.00
0.93
0.87
0.78
0.72
TL Tm = 0.7
4.8
3.3
2.5
1.7
1.4
1.10
1.00
0.90
0.79
0.72
TL Tm = 1.0
5.4
3.8
3.0
2.2
1.7
1.32
1.11
0.99
0.82
0.74
TL Tm = 1.2
5.6
4.0
3.2
2.3
1.8
1.42
1.19
1.06
0.85
0.76
TL Tm = 1.5
6.0
4.2
3.4
2.5
2.0
1.54
1.32
1.13
0.91
0.80
TL Tm = 2.0
6.3
4.6
3.7
2.6
2.2
1.68
1.42
1.26
1.00
0.86
TL Tm = 2.5
6.7
4.7
3.8
2.7
2.3
1.76
1.53
1.32
1.06
0.90
TL Tm = 3.0
6.9
4.9
3.9
2.8
2.3
1.84
1.55
1.38
1.11
0.96
TL Tm = 3.5
7.1
5.1
4.0
3.0
2.4
1.91
1.62
1.43
1.16
0.99
TL Tm = 4.0
7.3
5.2
4.1
3.0
2.5
1.93
1.63
1.46
1.20
1.04
TL Tm = 4.5
7.4
5.3
4.2
3.1
2.5
2.01
1.68
1.50
1.21
1.05
TL Tm = 5.0
7.5
5.4
4.3
3.1
2.5
2.02
1.70
1.51
1.23
1.06
TL Tm = 5.5
7.6
5.4
4.2
3.2
2.6
2.04
1.72
1.52
1.23
1.09
TL Tm = 6.0
7.7
5.4
4.4
3.2
2.6
2.08
1.73
1.56
1.25
1.10
TL Tm = 6.5
7.9
5.5
4.3
3.2
2.6
2.05
1.74
1.56
1.26
1.10
TL Tm = 7.0
7.9
5.5
4.5
3.2
2.7
2.10
1.75
1.57
1.27
1.12
37
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
38
Rule
Kc
Ti
Comment
Minimum IAE –
Marlin (1995),
pp. 301–307.
Model: Method 1;
K c , Ti deduced from
graphs; robust to
±25% process model
parameter changes.
1.4 K m
0.24τ m
τ m Tm = 0.11
1.8 K m
0.52τ m
τ m Tm = 0.25
Minimum IAE –
Huang et al. (1996).
Model: Method 1
Minimum IAE –
Shinskey (1996),
p. 38.
Model: Method 1
Minimum IAE –
Edgar et al. (1997),
pp. 8-14, 8-15.
Model: Method 1
Minimum IAE –
Edgar et al. (1997),
p. 8-15.
10
Kc =
1.4 K m
0.75τ m
τ m Tm = 0.43
1.0 K m
0.68τ m
τ m Tm = 0.67
0. 8 K m
0.71τ m
τ m Tm = 1.0
0.55 K m
0.60τ m
τ m Tm = 1.50
0.45 K m
0.54τ m
τ m Tm = 2.33
0.35 K m
0.49τ m
τ m Tm = 4.0
10
Kc
0. 1 ≤
Ti
τm
≤1
Tm
0.95Tm K m τ m
3.4 τ m
τ m Tm = 0.1
0.95Tm K m τ m
2.9 τ m
τ m Tm = 0.2
0. 4 K m
0.5τ m
Time delay dominant.
0.94Tm K m τ m
4τ m
Time constant
dominant.
0.67Tm K m τ m
3.5τ m
Model: Method 2
−0.9077
−0.063 
τ 
τ 
1 
τ

6.4884 + 4.6198 m + 0.8196 m 
− 5.2132 m 
Km 
Tm

 Tm 
 Tm 


+
τ
1 
− 7.2712 m
Km 
 Tm




0.5961
τm 
− 0.7241e Tm  ,


2
3
4

 τm  
 τm 
 τm 
τm








Ti = Tm 0.0064 + 3.9574
− 10.7619
+ 9.4348
− 6.4789
Tm
Tm 
Tm 
Tm  







τ
+ Tm 7.5146 m

 Tm

5
τ

 − 2.2236 m
 Tm




6
.


b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Minimum IAE –
Arrieta Orozco
(2003).
Model: Method 10
Minimum IAE –
Shinskey (2003).
Minimum IAE –
Harriott (1988).
Model: Method 1;
coefficients of K c , Ti
deduced from graphs.
Kc
11
Kc
Minimum IAE –
Shinskey (1994),
p. 167.
Minimum IAE –
Shinskey (1996),
p. 121.
Minimum ISE –
Hazebroek and Van
der Waerden (1950).
11
12
13
Kc =
1
Km
Comment
Ti
0.1 ≤ τm Tm ≤ 1.2
Also given by Arrieta Orozco and Alfaro Ruiz (2003).
0.74Tm K m τm
4.06τm
x1K u
x 2Tu
Model: Method 2
x1
x2
τm Tm
x1
x2
τm Tm
0.55
0.52
0.50
0.91
0.81
0.64
0.1
0.2
0.5
0.43
0.42
0.45
0.33
1.0
2.0
0.5K u
τm Tm
x2
Minimum IAE –
Shinskey (1988),
p. 148.
Model: Method 1
Ti
0.80
0.1
0.77
0.2
0.58K u
x 2Tu
x2
τm Tm
x2
τm Tm
0.64
0.53
0.5
1.0
0.43
2.0
0.81Tu
τ m Tm = 0.2
0.54 K u
0.66Tu
τ m Tm = 0.5
0.48K u
0.47Tu
τ m Tm = 1
0.46K u
0.37Tu
τ m Tm = 2
Ku
3.05 − 0.35(Tu τm )
0.55K u
13
Kc
12
Ti
Model: Method 1
0.78Tu
Model: Method 1;
τm Tm = 0.2 .
1.43Tm
Model: Method 2;
τ m Tm < 0.2 .
1.1251 


τ


0.4485 + 0.6494 Tm 
 , Ti = Tm − 0.2551 + 1.8205 m


T



 τm 
 m



2

T  
T
Ti = Tu  0.87 − 0.855 u + 0.172  u   .
τm
 τm  

τ 
Tm 
 0.74 + 0.3 m  .
Kc =
K m τ m 
Tm 




0.4749 
.


39
b720_Chapter-03
17-Mar-2009
Handbook of PI and PID Controller Tuning Rules
40
Rule
Hazebroek and Van
der Waerden (1950) –
τm
continued.
Tm
Minimum ISE –
Haalman (1965).
Model: Method 1
Minimum ISE –
Murrill (1967),
pp. 358–363.
Model: Method 5
Minimum ISE –
Frank and Lenz
(1969).
Model: Method 1
Ti
x1Tm K m τm
x 2τm
x1
1
Km



0.959
Coefficient values:
x1
τm
x2
Tm
0.7
1.0
1.5
0.96
1.07
1.26
τm
Tm
2.44
1.85
1.41
2.0
3.0
5.0
M max
Tm
x1
x2
1.46
1.89
2.75
= 1.9;
1.18
0.95
0.81
Am =
2.36; φ m = 50 0 .
Tm  τ m

0.492  Tm



0.739
0. 1 ≤
τm
≤ 1.0
Tm

T 
T 
τm 
 x 1 + x 2 m 
 x 1 + x 2 m 
x
τ
τm 
m 
2 

Representative coefficient values (deduced from graph):
Tm
x1
x2
τ m Tm
x1
x2
0.2
0.4
0.6
0.8
Minimum ISE – Yu
(1988).
Model: Method 1
x2
0.80 7.14
0.83 5.00
0.89 3.23
0.67Tm
Kmτm
1.305  Tm

K m  τ m
τm
Comment
Kc
0.2
0.3
0.5
14
FA
0.53
0.58
0.61
0.62
14
0.82
0.88
0.97
1.05
1.0
1.2
1.4
K e −sτ L
0.921  TL 


Load disturbance model = L
. Kc =
1 + sTL
K m  Tm 




0.954
τm
Tm
1.12
1.17
1.20
τm
≤ 0.35
Tm
Ti
Kc
T T
Ti = m  L
0.430  Tm
0.63
0.63
0.62
0.181− 0.205
− 0.49
 τm

T
 m
τm
Tm




 τm 


T 
 m
0.639
,
−1.214
,
τ
TL
≤ 2.310 m + 0.077 .
Tm
Tm
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Minimum ISE – Yu
(1988) – continued.
Minimum ISE –
Zhuang (1992),
Zhuang and Atherton
(1993).
Minimum ITAE –
Murrill (1967),
pp. 358–363.
Model: Method 5
Ti
16
Kc
Ti
17
Kc



Tm  τ m

0.535  Tm
1.07  TL 


Kc =
K m  Tm 



0.586
0.32
0.35
0.37
0.38
− 0.045 + 0.344
− 0.065 + 0.234
1.289  TL 


Kc =
K m  Tm 
τm
≤ 1. 0
Tm

T 
T 
τm 
 x 1 + x 2 m 
 x 1 + x 2 m 
x
τ
τm 
m 
2 

Representative coefficient values (deduced from graph):
Tm
x1
x2
τ m Tm
x1
x2
τm
Tm
 τm 


T 
 m
0.77
0.80
0.86
0.93
τm
Tm
 τm 


T 
 m
1.0
1.2
1.4
0.04 + 0.067
τm
Tm
 τm 


T 
 m
0.38
0.39
0.39
1.00
1.05
1.10
−1.014
,
− 2.532
τm
Tm
− 0.292
 τm 


T 
 m
0.899
, 2.310
τm
T
+ 0.077 ≤ L ≤ 1 .
Tm
Tm
−1.047
,
T T 
Ti = m  L 
0.347  Tm 
17
0. 1 ≤
0.675
0.438
τ

Tm  τ m 
1.1 ≤ m ≤ 2.0



T
m
0.552  Tm 

Model: Method 1 or Method 5 or Method 6 or Method 15.
0.977
0.680
τ
Tm  τ m 
0.859  Tm 
0.1 ≤ m ≤ 1.0




T
m
K m  τm 
0.674  Tm 
T T 
Ti = m  L 
0.359  Tm 
16
τm Tm > 0.35
Ti
0.945
1.346  Tm

K m  τ m
τm
1.157  TL 


Kc =
K m  Tm 
τm
≤ 0.35
Tm
Kc
0.2
0.4
0.6
0.8
15
Comment
15
1.279  Tm

K m  τ m
1
Km
Minimum ITAE –
Frank and Lenz
(1969).
Model: Method 1
Ti
41
− 0.889
T T
, Ti = m  L
0.596  Tm




−1.112
0.372
τm
Tm
τm
− 0.094
Tm

− 0.44
τ 
 m
T 
 m
 τm

T
 m




0.898
, 1<
0.46
.
TL
≤ 3.
Tm
b720_Chapter-03
17-Mar-2009
Handbook of PI and PID Controller Tuning Rules
42
Rule
Kc
0.598  TL 


Kc =
K m  Tm 
0.272 − 0.254
τm
Tm
 τm 


T 
 m
x 2 (τm + Tm )
τm
Tm
x1
x2
0.67
1.00
1.50
0.94
0.66
0.54
Ti
0.76
0.71
0.65
− 0.011−1.945
0.735  TL 


Kc =
K m  Tm 
τm
Tm
0.787  TL 


Kc =
K m  Tm 
0.084 + 0.154
τm
Tm
0.878  TL 


Kc =
K m  Tm 
0.172 − 0.057
τm
Tm
−1.341
,
 τm 


T 
 m
 τm 


T 
 m
 τm 


T 
 m
x2
τm Tm > 0.35
Ti
0.304
τm
− 0.112
Tm

τ 
 m
T 
 m
0.196
,
TL
τ
≤ 2.385 m + 0.112 .
Tm
Tm
−1.055
,
− 5.809
τm
Tm
+ 0.241
 τm 


T 
 m
0.901
, 2.385
τm
T
+ 0.112 ≤ L ≤ 1 .
Tm
Tm
−1.042
,
T T 
Ti = m  L 
0.431  Tm 
21
x1
2.33 0.47 0.59
4.00 0.42 0.52
9.00 0.375 0.47
Model: Method 1;
τm
≤ 0.35 .
Tm
Ti
T T 
Ti = m  L 
0.425  Tm 
20
τm
Tm
Ti
T T 
Ti = m  L 
0.805  Tm 
19
Comment
Ti
Minimum ITAE with
x1 K m
OS ≤ 8% – Fertik
x1
x2
τm
(1975).
T
m
Model: Method 3;
5.0 0.53
x 1 , x 2 deduced from 0.11
0.25 2.6 0.74
graphs.
0.43 1.45 0.80
18
Minimum ITAE –
Kc
Yu (1988).
19
Kc
Load disturbance
20
K L e −sτ L
Kc
.
model =
1 + sTL
21
Kc
18
FA
− 0.909
T T
, Ti = m  L
0.794  Tm




− 0.148
0.228
τm
Tm
τm
− 0.365
Tm

τ 
 m
T 
 m
− 0.257
 τm

T
 m




0.901
, 1<
0.489
.
TL
≤ 3.
Tm
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Minimum ITAE –
Arrieta Orozco
(2003).
Model: Method 10.
Minimum ITAE –
Arrieta Orozco
(2006), pp. 90–92.
Minimum ITSE –
Frank and Lenz
(1969).
Model: Method 1
22
23
Kc =
0.1 ≤
Ti
τm
≤ 1.2
Tm
Also given by Arrieta Orozco and Alfaro Ruiz (2003).
23
1
Km
τm
2 ≤ Am ≤ 4 ;
Model: Method 10
Ti
Kc

T 
T 
τm 
 x 1 + x 2 m 
 x 1 + x 2 m 
x2 
τm 
τm 

Representative coefficient values (deduced from graph):
Tm
x1
x2
τ m Tm
x1
x2
0.2
0.4
0.6
0.8
22
Comment
Ti
Kc
43
0.46
0.50
0.53
0.55
0.84
0.89
0.97
1.04
1.0
1.2
1.4
0.55
0.54
0.54
1.1055 

τ
T 
1 
 , Ti = Tm − 1.5926 + 2.9191 m
0.2607 + 0.6470 m 
T

Km 
τ

 m
 m



1.09
1.15
1.20




0.1789 
.


−1.5345 + 0.2865 A m − 0.0389 A m 2 

 τm 
1 
2
,

Kc =
1.2585 − 0.6179A m + 0.0764A m + x1 

Km 
Tm 



2
x1 = 0.8653 − 0.1783A m + 0.0127 A m , 0.1 ≤
τm
≤ 2.0 ;
Tm
7.7448 − 4.4925 A m + 0.6454 A m 2 

τ 
2
,
Ti = Tm 24.5244 − 17.4081A m + 2.7267 A m + x 2  m 


Tm 



2
x 2 = −21.8710 + 16.5760A m − 2.6258A m , 0.1 ≤
τm
< 1.0 ;
Tm
1.3703 − 0.2639 A m − 0.0110 A m 2 

 τm 
2

,

Ti = Tm 1.1097 − 0.2739A m + 0.0272A m + x 2 


Tm 



2
x 2 = 1.1821 − 0.3504A m + 0.0417 A m , 1.0 ≤
τm
≤ 2.0 .
Tm
b720_Chapter-03
Rule
Minimum ISTSE –
Zhuang (1992),
Zhuang and Atherton
(1993).
Minimum ISTSE –
Zhuang (1992),
Zhuang and Atherton
(1993).
Minimum ISTES –
Zhuang (1992),
Zhuang and Atherton
(1993).
Nearly minimum
IAE, ISE, ITAE –
Hwang (1995).
Model: Method 44
Nearly minimum IAE
and ITAE – Hwang
and Fang (1995).
Model: Method 30
25
26
27
FA
Handbook of PI and PID Controller Tuning Rules
44
24
17-Mar-2009
Kc
Comment
Ti
1.015  Tm

K m  τ m



0.957
Tm  τ m

0.667  Tm



0.552
0. 1 ≤
τm
≤ 1. 0
Tm
0.673
0.427
τ

Tm  τ m 
1.1 ≤ m ≤ 2.0



Tm
0.687  Tm 

Model: Method 1 or Method 5 or Method 6 or Method 15.
24
Model: Method 1
Ti
K
1.065  Tm

K m  τ m
c
25
1.021  Tm

K m  τ m
Model: Method 37
Ti
Kc



0.953
Tm  τ m

0.629  Tm



0.546
0. 1 ≤
τm
≤ 1. 0
Tm
0.648
0.442
τ

Tm  τ m 
1.1 ≤ m ≤ 2.0



T
m
0.650  Tm 

Model: Method 1 or Method 5 or Method 6 or Method 15.
Decay ratio = 0.15;
26
K
µ
K
ω
τ
(1 − µ1 )K u
c
2 u u
0.1 ≤ m ≤ 2.0 .
Tm
1.076  Tm

K m  τ m
27
Ti
Kc
Decay ratio = 0.12;
τ
0.1 ≤ m ≤ 2.0 .
Tm
 1.892K m K u + 0.244 
 0.706K m K u − 0.227 
τm
 K u , Ti = 

K c = 

 0.7229K K + 1.2736 Tu , 0.1 ≤ T ≤ 2.0 .
+
3
.
249
K
K
2
.
097
m
m u
m u




∧
∧




τm
 4.126K m K u − 2.610  ∧
 5.352K m K u − 2.926  ∧
K
,
=
Kc = 
T
u
i


 T u , 0.1 ≤ T ≤ 2.0 .
∧
∧
 5.848K K u − 1.06 
 5.539K K u + 5.036 
m
m
m




µ1 =
(
)
(
)
1.14 1 − 0.482ωu τm + 0.068ω2u τ2m
0.0694 − 1 + 2.1ωu τm − 0.367ω2u τ2m
, µ2 =
.
K u K m (1 + K u K m )
K u K m (1 + K u K m )
2

τ
τ 
(K c / K u ωu )
K c =  0.515 − 0.0521 m + 0.0254 m 2 K u , Ti =
2 


T

Tm 
m

 0.0877 + 0.0918 τm − 0.0141 τm 
2

Tm
Tm 

b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Comment
Minimum error: step
load change – Gerry
(2003).
0.3 K m
0.42τ m
Model: Method 1;
τm Tm ≥ 5 .
x1 K m
Minimum IAE or ISE
– ECOSSE Team
τ m Tm
x1
(1996a).
0.11
1.1
Model: Method 1;
0.25
1.8
coefficients of K c , Ti
0.43
1.1
deduced from graphs.
0.67
1.0
1.0
0.8
1.50
0.59
2.33
0.42
4.0
0.32
28
Minimum IE –
Devanathan (1991).
Model: Method 1
Minimum IAE –
Rovira et al. (1969).
Model: Method 5
28
Coefficient values:
x2
τ m Tm
0.23
0.23
0.72
0.72
0.70
0.67
0.60
0.53
0.11
0.25
0.43
0.67
1.0
x1
x2
5.8
3.1
2.1
1.7
0.91
0.5
0.6
0.7
0.8
0.9
x 2 Tu
Kc
x 2 coefficient values (deduced from graphs):
τ m Tm
0.2
0.4
0.6
Minimum IAE –
Gallier and Otto
(1968).
Model: Method 58
x 2 (τ m + Tm )
x2
τ m Tm
x2
τ m Tm
x2
0.35
0.8
0.23
1.4
0.29
1.0
0.21
1.6
0.26
1.2
0.19
1.8
Minimum performance index: servo tuning
x1 K m
x 2 (τ m + Tm )
0.18
0.18
0.17
Representative coefficient values (deduced from graphs):
τ m Tm
x1
x2
τ m Tm
x1
x2
0.053
0.11
0.25
0.758  Tm

K m  τ m
12
6.4
2.9



0.861


0.16  

ξ cos  tan −1 1.57 D R −
D R  



Kc =
.
 −1  6.28Tm 

K m cos  tan 


 Tu 
0.95
0.91
0.84
0.67
1.50
4.0
Tm
τ
1.020 − 0.323 m
Tm
1.15
0.65
0.45
0.1 ≤
0.75
0.66
0.55
τm
≤ 1.0
Tm
45
b720_Chapter-03
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FA
Handbook of PI and PID Controller Tuning Rules
46
Rule
Kc
Ti
Comment
Minimum IAE –
x1 K m
x 2Tm
Murata and Sagara
x1
x2
τm Tm
x1
x2
τm Tm
(1977).
2.7
0.9
0.2
0.9
1.3
0.8
Model: Method 2;
1.5
1.1
0.4
0.8
1.4
1.0
x1 , x 2 deduced from
1.1
1.2
0.6
graphs.
Minimum IAE – Bain 1 
τm
Tm 
> 0.2
and Martin (1983).

0.3 + 0.3
Tm + 0.4 τ m
T
Km 
τm 
m
Model: Method 1
Minimum IAE –
1.4 K m
0.72τ m
τm Tm = 0.11
Marlin (1995),
K c , Ti deduced from graphs; robust to ±25% process model
pp. 301–307.
parameter changes.
Model: Method 1
Minimum IAE –
τ
0.1 ≤ m ≤ 1
29
Huang et al. (1996).
T
Kc
i
Tm
Model: Method 1
Minimum IAE –
Model: Method 1;
0.6Tm
Smith and Corripio
T
0
.1 ≤ τm Tm ≤ 1.5 .
m
K mτm
(1997), p. 345.
Minimum IAE –
Minimum IAE =
0.59Tm
Huang and Jeng
T
2.1038τ m ;
m
Kmτm
(2002).
τ m Tm < 0.2 .
Model: Method 1
29
Kc =
−1.0169
3.5959 
τ 
τ 
1 
τ

+ 1.1075 m 
− 13.0454 − 9.0916 m + 0.3053 m 
Km 
Tm

 Tm 
 Tm 


+
τ
1 
− 2.2927 m
Km 
 Tm




3.6843
τm 
+ 4.8259e Tm  ,


2
3
4

τ  
τ 
τ 
τ
Ti = Tm 0.9771 − 0.2492 m + 3.4651 m  − 7.4538 m  + 8.2567 m  

Tm
 Tm  
 Tm 
 Tm 


τ
+ Tm  − 4.7536 m

 Tm

5
τ

 + 1.1496 m
 Tm




6
.


b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Minimum IAE –
Arrieta Orozco
(2003).
Model: Method 10
Minimum IAE –
Tavakoli and Fleming
(2003).
Model: Method 1
Minimum IAE –
Tavakoli et al.
(2006).
Minimum IAE –
Harriott (1988).
Ti coefficients
deduced from graph.
30
1
Kc =
Km
Kc
30
Comment
Ti
0.1 ≤
Ti
Kc
τm
≤ 1.2
Tm
Also given by Arrieta Orozco and Alfaro Ruiz (2003).
31
0.1 ≤
Ti
Kc
τm
≤ 10
Tm
Minimum A m = 6dB ; minimum φ m = 60 0 .
32
Kc
0.5K u
Ti
Model: Method 42;
A m ≥ 3 ; φ m ≥ 60 0 .
x 2Tu
Model: Method 1
x2
τm Tm
x2
τm Tm
x2
τm Tm
1.61
0.94
0.2
0.5
0.61
1.0
0.48
2.0
1.2099 
0.8714 






0.2438 + 0.5305 Tm 
 , Ti = Tm 0.9377 + 0.4337 τm 
.
τ 
T 




m 
m 






31
Kc =



1 
Tm
τ
+ 0.3047 , Ti = Tm 0.4262 m + 0.9581 .
0.4849
Km 
τm
Tm



32
Kc =
1
Km
 Tm

+ 0.071 , Ti = min[Tm + 0.143τ m ,9τ m ] .
0.5
τ
m


47
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
48
Rule
Small IAE – Hwang
(1995).
Model: Method 44
Minimum ISE –
Zhuang (1992),
Zhuang and Atherton
(1993).
Model: Method 1 or
Method 5 or Method
6 or Method 15.
Minimum ISE –
Khan and Lehman
(1996).
Model: Method 1
Minimum ITAE –
Rovira et al. (1969).
Model: Method 5
33
17-Mar-2009
µ1 =
(
Ti
Comment
K c µ 2 K u ωu
Decay ratio = 0.1;
τ
0.1 ≤ m ≤ 2.0 .
Tm
Kc
33
(1 − µ1 )K u
0.980  Tm

K m  τ m



0.892
1.072  Tm

K m  τ m



0.560
Tm
τ
0.690 − 0.155 m
Tm
Tm
τ
0.648 − 0.114 m
Tm
0.01 ≤ τm Tm ≤ 0.2
35
Kc
Ti
0.2 ≤ τm Tm ≤ 20
0.586  Tm

K m  τ m



Tm
0.916
τ
1.030 − 0.165 m
Tm
0.1 ≤
τm
≤ 1.0
Tm
)
)
1.11 1 − 0.467ωu τm + 0.0657ω2u τ2m
,
K u K m (1 + K u K m )
(
τm
≤ 2.0
Tm
Ti
µ2 =
µ1 =
1.1 ≤
Kc
1.07 1 − 0.466ωu τm + 0.0667ω2u τ2m
,
K u K m (1 + K u K m )
(
τm
≤ 1.0
Tm
34
µ2 =
µ1 =
0.1 ≤
)
(
)
0.0328 − 1 + 2.21ωu τm − 0.338ω2u τ2m
, r = 0.5;
K u K m (1 + K u K m )
(
)
0.0477 − 1 + 2.07ωu τm − 0.333ω2u τ2m
, r = 0.75;
K u K m (1 + K u K m )
1.14 1 − 0.466ωu τm + 0.0647ω2u τ2m
,
K u K m (1 + K u K m )
(
)
0.0609 − 1 + 1.97ωu τm − 0.323ω2u τ2m
, r = 1.0;
K u K m (1 + K u K m )
r = parameter related to the position of the dominant real pole.
µ2 =
34
35
 0.7388 0.3185  Tm
 0.7388Tm + 0.3185τ m 

K c = 
+
 K , Ti = τ m  − 0.0003082T + 0.5291τ  .
T
τ
m
m 
m
m


 m
 0.808 0.511
 0.808Tm + 0.511τm − 0.255 Tm τm 
0.255  Tm
.
Kc = 
+
−
, Ti = τm 
 τm

Tm
 0.095Tm + 0.846τm − 0.381 Tm τm 
Tm τ m  K m

b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
49
Comment
Minimum ITAE with
x1 K m
x 2 (τm + Tm )
OS ≤ 8% – Fertik
x1
x2
x1
x1
x2
τm
τm
τm
x2
(1975).
T
T
T
m
m
m
Model: Method 3;
x 1 , x 2 deduced from 0.11 3.05 0.95 0.67 0.71 0.76 2.33 0.42 0.59
0.25 1.70 0.88 1.00 0.55 0.71 4.00 0.39 0.52
graphs.
0.43 1.05 0.82 1.50 0.47 0.65 9.00 0.365 0.47
Minimum ITAE –
τ
0.1 ≤ m ≤ 1.2
36
Arrieta Orozco
Ti
Kc
Tm
(2003).
Also given by Arrieta Orozco and Alfaro Ruiz (2003).
Model: Method 10
Minimum ITAE –
2 ≤ Am ≤ 4 ;
37
Arrieta Orozco
Ti
Kc
Model: Method 10
(2006), pp. 90–92.
τ
Minimum ITAE –
− 0.9707 m
Tm
38
Model: Method 1
Barberà (2006).
0
.
9894
e
Kc
36
37
1
Kc =
Km
1.5426 
1.0382 






.
0.1440 + 0.5131 Tm 
 , Ti = Tm 0.9953 + 0.2073 τ m 
τ 
T 




m 
m 






−1.0871+ 0.0048 A m + 0.0047 A m 2 

 τm 
1 
2
,

Kc =
0.9639 − 0.4436A m + 0.0496A m + x1 

Km 
Tm 



2
x1 = 1.3296 − 0.4355A m + 0.0509A m , 0.1 ≤ τm Tm ≤ 2.0 ;
6.1845 − 3.4585 A m + 0.4764 A m 2 

 τm 
2
,


Ti = Tm 2.7836 − 1.2511A m + 0.1692A m + x 2 


Tm 



2
x 2 = 0.8298 − 0.1193A m + 0.0083A m , 0.1 ≤ τm Tm < 1.0 ;
1.6705 − 0.2509 A m − 0.0447 A m 2 

 τm 
2

,

Ti = Tm 2.0174 − 0.5089A m + 0.0311A m + x 2 


Tm 



2
x 2 = 1.0065 − 0.4874A m + 0.0889A m , 1.0 ≤ τm Tm ≤ 2.0 .
38
Kc =
1
.
2


 τm 
τ
 + 2.0214 m + 0.00911
K m 1.5853
Tm 
Tm





b720_Chapter-03
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Handbook of PI and PID Controller Tuning Rules
50
Rule
Kc
Ti
Comment
Smith (2002).
0.586Tm K m τm
Tm
Model: Method 1
Approximately minimum ITAE.
Minimum ISTSE –
Model: Method 1;

1 
Tm
+ 0.2
Fukura and Tamura
0.56
Tm + 0.36τm
0.05 ≤ τm Tm ≤ 1 .
Km 
τm

(1983).
0.921
Minimum ISTSE –
τ
Tm
0.712  Tm 
0.1 ≤ m ≤ 1.0


Zhuang (1992),
T
0
.
968
−
0
.
247
τ
T
m
m
m
K m  τm 
Zhuang and Atherton
(1993).
0.559
τ
Tm
0.786  Tm 
1.1 ≤ m ≤ 2.0
Model: Method 1 or


T
0
.
883
−
0
.
158
τ
T
m
m
m
K m  τm 
Method 5 or Method
6 or Method 15.
39
Minimum ISTSE –
Model: Method 1;
Ti
Kc
Zhuang (1992).
0.1 ≤ τm Tm ≤ 2.0 .
Minimum ISTSE –
Zhuang (1992),
Zhuang and Atherton
(1993).
Minimum ISTSE –
Majhi (2005).
Model: Method 52
39
40
41
0.361K u
41
40
Ti
Ti
Kc
Model: Method 1
Model: Method 37
0.712  Tm

K m  τ m



0.921
0.712Tm
0.694 − 0.182 τm Tm
0.1 ≤
τm
≤ 1.0
Tm
0.780  Tm

K m  τ m



0.543
0.780Tm
0.683 − 0.120 τm Tm
1.1 ≤
τm
≤ 2.0
Tm
0.673  Tm

K m  τ m



0.331
0.673Tm
0.511 − 0.066 τm Tm
2.1 ≤
τm
≤ 3.5
Tm
 4.264 − 0.148K m K u 
 K u , Ti = 0.083(1.935K m K u + 1)Tu .
K c = 

 12.119 − 0.432K m K u 
τ
Ti = 0.083(1.935K m K u + 1)Tu , 0.1 ≤ m ≤ 2.0 .
Tm
∧


∧
τm

∧
 1.506K m K u − 0.177  ∧
Kc = 
 K u , Ti = 0.055 3.616K m K u + 1 T u , 0.1 ≤ T ≤ 2.0 .
∧
 3.341K K u + 0.606 


m
m


b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Minimum ISTES –
Zhuang (1992),
Zhuang and Atherton
(1993).
Model: Method 1 or
Method 5 or Method
6 or Method 15.
Kc
0.569  Tm

K m  τ m



0.951
0.628  Tm

K m  τ m



0.583
Nearly minimum IAE
and ITAE – Hwang
and Fang (1995).
43
44
τ
1.023 − 0.179 m
Tm
Tm
τ
1.007 − 0.167 m
Tm
0.1 ≤
τm
≤ 1.0
Tm
1.1 ≤
τm
≤ 2.0
Tm
Ti
Kc
x1 K m
44
42
Tm
Model: Method 30;
0.1 ≤ τ m Tm ≤ 2.0 ;
decay ratio = 0.03.
Minimum performance index: other tuning
Model: Method 30;
43
Ti
0.1 ≤ τ m Tm ≤ 2.0 ;
Kc
decay ratio = 0.05.
42
Simultaneous
servo/regulator tuning
– Hwang and Fang
(1995).
Minimum
performance index –
Wilton (1999).
Model: Method 1;
coefficients of K c
estimated from
graphs.
Comment
Ti
Tm
Coefficient values:
b
τ m Tm
τ m Tm
x1
x1
b
0.1
0.1
0.1
0.1
750.0
17.487
4.8990
2.1213
0.0001
0.04
0.2
1
0.2
0.2
0.2
0.2
0.2
3.3675
2.2946
1.7146
1.1313
0.8764
0.008
0.04
0.2
1
5
0.4
0.4
0.4
0.4
0.4
1.6837
1.5297
1.2247
0.9192
0.7668
0.008
0.04
0.2
1
5
0.5
0.5
0.5
0.5
150.01
7.1386
2.6944
1.1314
0.0001
0.04
0.2
1
2

τ
τ 
(K c / K u ωu )
K c =  0.438 − 0.110 m + 0.0376 m 2 K u , Ti =
.
2 


T

Tm 
m

 0.0388 + 0.108 τm − 0.0154 τm 
2

Tm
Tm 

2

τ
τ 
(K c / K u ωu )
K c =  0.46 − 0.0835 m + 0.0305 m 2 K u , Ti =
.
2 


T

T
m
m 

 0.0644 + 0.0759 τm − 0.0111 τm 
2

Tm
Tm 

Performance index =
∫ [e (t) + bK
∞
0
2
2
m
(y(t ) − y∞ )2 ]dt .
51
b720_Chapter-03
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FA
Handbook of PI and PID Controller Tuning Rules
52
Rule
Wilton (1999) –
continued.
Minimum ITAE –
ABB (2001).
Model: Method 7
Ream (1954).
Model: Method 1;
decay ratio ≈ 33% .
Bryant et al. (1973).
Model: Method 1
Kc
Ti
x1 K m
Tm
τ m Tm
x1
b
τ m Tm
x1
b
0.6
0.6
0.6
0.6
0.6
1.1225
1.1218
0.9798
0.7778
0.6572
0.008
0.04
0.2
1
5
0.8
0.8
0.8
0.8
0.8
0.8980
0.9178
0.7348
0.7071
0.6025
0.008
0.04
0.2
1
5
1.0
1.0
72.004
1.6657
0.0001
0.2
1.0
1.0
3.6713
0.04
0.8485
1
τm
< 0.5
Tm
0.68

τ 

1.4837Tm  m 

 Tm 
Direct synthesis: time domain criteria
1.56 K m
0.368(τm + Tm )
τm Tm = 1.12
0.85 K m
0.454(τm + Tm )
τm Tm = 2.86
0.37Tm K m τm
Tm
0.40Tm K m τm
Tm
undefined
x 2K mτm
Critically damped
dominant pole.
Damping ratio
(dominant pole) =
0.6.
G c = 1 (Tis )
Coefficient values (deduced from graph):
x2
τ m Tm
x2
0.33
0.40
0.50
0.67
0.33
0.40
0.50
0.67
Mollenkamp et al.
(1973).
Model: Method 1
0.977
0.8591  Tm

K m  τ m
τ m Tm
Chiu et al. (1973).
Model: Method 1
Comment
12.5
11.1
11.1
9.1
7.1
6.3
5.6
4.3
1.0
2.0
10.0
6.7
4.2
3.0
Critically damped
dominant pole.
1.0
2.0
10.0
3.2
2.1
1.6
Damping ratio
(dominant pole) =
0.6.
λ variable;
suggested values: 0.2,
0.4, 0.6, 1.0.
λ = 1.0 , OS < 10%, Model: Method 11 (Pomerleau and Poulin,
2004).
Appropriate for
Tm
T
‘small τ m ’.
m
K (T + τ )
λTm
K m (1 + λτ m )
m
CL
m
Tm
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Comment
0.44Tm
Kmτm
Tm
0.04 ≤ τ m Tm ≤ 1.4 ;
deduced from graph.
0.43Tm
K mτm
Tm
1% overshoot – servo;
τm Tm > 0.2 .
1
Km
Tm
53
undefined
1 (x 2 τm )
G c = 1 Tis
Keviczky and Csáki
x 2 coefficient values (estimated from graph):
(1973);
0%
5%
10%
15%
20%
OS values in first τ m Tm OS
row; τ m Tm values 0.1
30
22
16
12
10
0.2
18
11
9
7.5
6
in first column;
0.5
8
6
5
4
3.5
Km = 1.
1.0
5
4
3.5
3
2.5
Model: Method 1
2.0
4
3
2.5
2.2
2
5.0
3
2.4
2
1.8
1.6
10.0
2.5
2.1
1.8
1.7
1.5
5% overshoot – servo
0.52Tm K m τm
Tm
0.04 ≤ τm Tm ≤ 1.4
– Smith et al. (1975).
Model: Method 1. Also given by Bain and Martin (1983).
1% overshoot – servo
– Smith et al. (1975).
Model: Method 1
Bain and Martin
(1983).
Model: Method 1
Ti
‘Very conservative
tuning for small
delay’.
Model: Method 1
0.5Tm K m τm
Tm
OS = 10%
0.5Tm K m τm
Tm
Model: Method 1
Kuwata (1987).
Suyama (1992).
Model: Method 1
5% overshoot – servo
– Smith and Corripio
(1997), p. 346.
Vítečková (1999),
Vítečková et al.
(2000a)
(Model: Method 9 –
Vítečková et al.,
2000b).
45
Kc =
45
Kc
φ m = 60 0 ; 0.25 ≤ τm Tm ≤ 1 (Voda and Landau, 1995).
x1Tm K m τm
x1
OS (%)
0.368
0.514
0.581
0.641
0
5
10
15
Tm
Coefficient values:
OS (%)
x1
0.696
0.748
0.801
0.853
20
25
30
35
x1
OS (%)
0.906
0.957
1.008
40
45
50
0.4(Tm + τm )
0.5
−
, Ti = 1.25x1 - 0.25(Tm + τm ) , with
K m (Tm + τm − x1 ) K m
x1 =
(Tm + τm )2 − 0.267τm 2 (3Tm + τm ) .
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
54
Rule
Okada et al. (2006).
Model: Method 1
x1
Kc
Ti
x1Tm K m τm
Tm
ξ
Comment
Coefficient values:
ξ
x1
ξ
x1
x1
ξ
x1
ξ
0.368 1.0 0.454 0.8 0.578 0.6 0.765 0.4 1.06 0.2
0.407 0.9 0.510 0.7 0.661 0.5 0.896 0.3 1.29 0.1
Mann et al. (2001).
0.51Tm K m τm
Tm
0 < τm Tm < 2
Model: Method 37 or
46
Ti
Kc
Method 41.
Closed loop response overshoot (step input) < 5%.
47
undefined
G c = 1 (Tis )
Ti
Vítečková and
48
Ti
Kc
Víteček (2002).
Model: Method 1
For 0.05 < τ T < 0.8 , T = T ; overshoot is under 2%
m
m
i
m
(Vítečková and Víteček, 2003).
Pinnella et al. (1986).
Model: Method 24
46
undefined
49
Gc = Ki s
Ki
Critically damped servo response.




0.56 + 1.02 Tm − Tm 1.0404 Tm − x1  + x 2  ,


τm
τm 
τm




 0.56 + 1.02(Tm τm ) − (Tm τm )(1.0404(Tm τm ) − x1 ) + x 2
τ
Ti =  m − 1 
T
  0.04 − 1.02(Tm τm ) + (Tm τm )(1.0404(Tm τm ) − x1 ) + x 2
 m
x 1 = 0.4, x 2 = 3.68, 1 < τ m Tm < 2 ; x 1 = 0.8, x 2 = 3.28, 2 < τ m Tm < 4 ;
Kc =
1
Km

 Tm ;

x 1 = 1.2, x 2 = 2.88, 4 < τ m Tm .
47
Km
Ti = −
2


 1
0.25
1 0.5    Tm 
Tm
e


+
−
−
+
−
+
0
.
25
0
.
5


2
2
τ 


τ
τ
T
τ
T
m
m 
m
m
 m
   m 

1
48
τm Tm x12 + (2Tm + τm )x1 + 1 e τ m x1 ,
Kc = −
Km
[
Ki =
1
4K m τ m 2
.
]
x1 = −
49
2
 τ  0.5 τ m
1+ 0.25 m  −
−1
Tm
 Tm 
2−
τm  τm 

+
Tm  Tm 
2
2
0.5
−
+
τ m Tm
2
τm
2
+
0.25
Tm
2
, Ti = −

τm
τm Tm x12 + (2Tm + τm )x1 + 1
x12 (τm Tm x1 + Tm + τm )
2
 τm  



2
 0.5  2 − Tm +  Tm 

τ 
τ


Tm  2 − m +  m  + 1 e 


Tm  Tm 


.
.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Comment
Schneider (1988),
Bekker et al. (1991),
Khan and Lehman
(1996).
Model: Method 1
Bekker et al. (1991).
Model: Method 1;
ξ = 1 ; Tv = valve
opening time.
0.368Tm K m τm
Tm
ξ =1
0.403Tm K m τm
Tm
ξ = 0.6
1.571Tm K m τm
Tm
ξ = 0.0
0.368Tm
K m τm
Tm
Regulator – Gorecki
et al. (1989).
Model: Method 1
Hang et al. (1991).
Model: Method 1
50
Kc =
τm > 0.5Tm or
T v < 20Tm .
τm < 0.5Tm or
50
Kc
51
Kc
Tv > 20Tm .
Ti
undefined
53
52
Ti
Ti
Kc
Pole is real and has
maximum attainable
multiplicity.
0.16 ≤ τm Tm < 0.96
Servo response: 10% overshoot, 3% undershoot.
0.311Tm

τ
 0.1092 ln  m
 Tm
τm  


K m τm 

 Tm 

T
 − 0.0760 ln  v

T

 m
2


 τ 
2 Tm 
51
Kc =
2 +  m  − 1 e

K m τm 
 2Tm 




 − 0.2685 


 
 τ
2 +  m
 2 Tm

 Tv
 0.0121 ln 
 Tm
Tv  


τ 
 m
.


 + 0.1026 



2

τ
 − 2− m

2 Tm

,
2
52
G c = 1 (T i s ) , Ti =
55
 τ 
1 +  m 
 2Tm 
.
Ti = τm
2
3
2
2
 τm   τ m   τm    τ m  
 τm 
+
 +
 − 2 + 
  2+

3 + 
 



 
 2T 
 2Tm   Tm   2Tm    2Tm  
 m
0.5K m τm
.
2


 1 + (τ 2T )2 − 1 e
m
m


 1+  τ m
 2T

 m


τ
 −1− m 

2 Tm 





  τm
 
 12 + 2  11(τm Tm ) + 13  
 11
+ 13  


(
)
τ
−
37
T
4
4
5
T

m
m

 K , T = 0.2
53
m

 + 1Tu .
Kc = 
 u i
 15  τm

6
 11(τm Tm ) + 13  
− 4  
  37
+ 14
15




 
  Tm
 37(τm Tm ) − 4  

b720_Chapter-03
Rule
Kc
McAnany (1993).
Model: Method 29
Sklaroff (1992),
Åström and Hägglund
(1995).
54
0.5Tm K m τm
55
56
Kamimura et al.
(1994).
Model: Method 1;
servo tuning.
Zhang (1994), Smith
(2002).
Model: Method 1
Kc =
Kc
Ti
Comment
Ti
TCL = 1.67Tm
Model: Method 32 or
3
Method 33.
T
m
K m (1 + 3τ m Tm )
pp. 250–251, 253. Honeywell UDC 6000 controller.
0.5556Tm K m τm
5τ m
τm Tm < 0.33
Fruehauf et al.
(1993).
Model: Method 2
Nomura et al. (1993).
Model: Method 1
Gonzalez (1994) –
pp. 114–115.
Model: Method 1
55
FA
Handbook of PI and PID Controller Tuning Rules
56
54
17-Mar-2009
Kc
Tm
τm Tm ≥ 0.33
Ti
0.01 ≤ τm Tm ≤ 10
0.5τm
δ0 = 0.21, 0.5, 0.7
(pp. 142–143).
2
x 2 Tm τm
2K m
57
Kc
Ti
10% overshoot.
58
Kc
Ti
0% overshoot.
59
Kc
Ti
‘Quick’ response:
‘negligible’
overshoot.
Tm K m τm
Tm
‘Good’ servo response; τ m is ‘small’.
(1.44Tm + 0.72Tm τm − 0.43τm − 2.14) , T
(
2
K m 0.72τm + 0.36τm + 2
)
i
=
(
)
K m 5.56 + 2τm + τm 2
.
4Tm + 1.28τm − 2.4
0.5x1 − 0.1(Tm + τm )
Kc =
, Ti = 1.25x1 − 0.25(Tm + τm ) , with
K m (Tm + τm − x1 )
2
2
x1 = 0.2τm + 0.4τmTm + Tm .
56
x2 = −
2x1
+
τm
4 x12
τm
2
+
2
1
, x1 =
,
Tm τm
1 + (2π loge [b a ])2
b a = desired closed loop response decay ratio.
57
Kc =
58
Kc =
59
Kc =
0.5(Tm + τm )
0.5τm (2Tm + τm ) + 0.10
, Ti = Tm + τm −
.
K m [0.5τm (2Tm + τm ) + 0.10]
Tm + τm
0.375(Tm + τm )
0.5τm (2Tm + τm ) + 0.0781
, Ti = Tm + τm −
.
K m [0.5τm (2Tm + τm ) + 0.0781]
Tm + τm
0.425(Tm + τm )
0.5τm (2Tm + τm ) + 0.083
, Ti = Tm + τm −
.
K m [0.5τm (2Tm + τm ) + 0.083]
Tm + τm
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Davydov et al.
(1995).
Model: Method 6
Kuhn (1995).
Model: Method 17
Khan and Lehman
(1996).
Model: Method 1
Abbas (1997).
Model: Method 1
Ti
Comment
Ti
ξ = 0.9;
0.2 ≤ τ m Tm ≤ 1 .
0.5 K m
0.5(Tm + τ m )
‘Normal tuning’.
1 Km
0.7(Tm + τ m )
‘Rapid tuning’.
Kc
60
Kc
61
Kc
Ti
0.01 ≤ τm Tm ≤ 0.2
62
Kc
Ti
0.2 ≤ τm Tm ≤ 20
63
Kc
Tm + 0.5τ m
0 ≤ V ≤ 0.2 ;
0.1 ≤ τm Tm ≤ 5.0 .
Valentine and
Chidambaram
(1997a).
Bi et al. (1999).
τm 
1 
0.43 − 0.97

Km 
Tm 
0.5064Tm K m τm
Tm
Model: Method 16
Bi et al. (2000).
0.5064Tm K m τm
1.9747 K m Tm
Model: Method 16
Ettaleb and Roche
(2000).
1 Km
Tm + τ m
Model: Method 34;
TCL = Tm + τ m .
Ti
x1 > 0
Prokop et al. (2000).
Model: Method 1
60
Kc =
61
Kc =
65
Kc
0.413 + 0.8
τm
Tm
Model: Method 1;
ξ = 0.707 64.


τ
1
, Ti =  0.153 m + 0.362 Tm .


T
τm
m


K m 1.905
+ 0.826 
Tm


τ 
Tm  0.3852 0.723
+
− 0.404 m2  ,
K m  τm
Tm
Tm 
Ti = τm
− 0.404τm 2 + 0.723Tm τm − 0.3852Tm 2
− 0.525τm 2 + 0.4104Tm τm − 0.00024Tm 2
62
Kc =
0.404Tm + 0.256τm − 0.1275 Tm τm
Tm  0.404 0.256 0.1275 
+
−
, T = τm
.
 i
K m  τm
Tm
τ
T
0.0808Tm + 0.719τm − 0.324 Tm τm
m m 

63
Kc =
0.148 + 0.186(Tm τm )
.
K m (0.497 − 0.464V 0.590 )
64
±1% TS = 2.5Tm , 0.1 ≤ τ m Tm ≤ 0.4 ; ±1% TS = 5Tm , 0.5 ≤ τ m Tm ≤ 0.9 .
65
57
1.045

K m 2 τ m 2  K m 2
x1 +
+ Tm  x1 − 1
2


Tm
T τ T x + 2Tm x1 − 1
 Tm

Kc = m m m 1
, Ti =
.
K m (τm + Tm )(1 + x1τm )
x1 (Tm + τ m )
(
)
.
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
58
Rule
Kc
Chen and Seborg
(2002).
Chidambaram (2002),
p. 155.
Skogestad (2003).
Model: Method 56
69
Kc =
Comment
Kc
Ti
Model: Method 1
67
Kc
0.413Tm + 0.8τm
Model: Method 1
min (Tm ,8τ m )
Ti = 8τm (Faanes and
Skogestad, 2004).
0 ≤ τm Tm ≤ ∞
Kc
Ti
Tm (x1K m τm )
Tm
68
Gorez (2003).
Model: Method 1
Ti
66
0.5Tm K m τm
Klán (2000).
Model: Method 18
66
17-Mar-2009
70
(1 − x 2 )τm + Tm
Tm
x1K m τm
τm ≤
τm
≤1
τ m + Tm
T τ + 2Tm TCL − TCL 2
1 Tm τm + 2Tm TCL − TCL 2
,
, Ti = m m
2
Tm + τm
Km
(TCL + τm )
−τ s
Tise m
 y
0 < TCL < Tm + Tm 2 + Tm τ m ,  
.
=
2
 d desired K c (TCLs + 1)
67
1
Kc =
Km
68
Kc =
69
x1 =
2


 
3.113 − 5.73 τm + 3.24 τm   .
T  
Tm

 m 


1 
2τm (Tm + τm )
1−
K m  1 + 1 + 2τ 2 (τ + T )2
m
m
m

[
]

 , T = τm + Tm
 i
2

2
1
1.5M s − 2  τm 


+
2
M s (M s − 1) 0.32M s (M s − 1)  τm + Tm 
2
2

1 + 1 + 2 τm   − τ .
τ +T   m

m 
 m




5τ m

3 −


(
)
T
M
τ
+
m
m
s 



2.5τ m
0.5
1 −

−

M s (M s − 1) 
(
)
T
M
τ
+
m
m
s


(
2 M s 2 −1
)
; 0≤
2
2

 τm
 
 τm


− 0.4 M s  

− 0.4 M s 

0.1M s 
 τ + Tm
 
 τ m + Tm
 .
70
x1 =
7.5 −  m
  , x 2 = 1 − 0.5

Ms −1 
1 − 0.4 M s
1 − 0.4 M s


 






 

τm
≤ τm .
τ m + Tm
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Comment
Klán and Gorez
(2005a).
Klán and Gorez
(2005b).
71
Kc
Ti
Model: Method 1
72
Kc
Ti
Model: Method 1;
0 ≤ τm Tm ≤ ∞ .
Skoczowski (2004).
73
Kc
Tm
Model: Method 1
Sree et al. (2004),
Chidambaram et al.
(2005).
Foley et al. (2005).
0.9179  Tm 


K m  τm 
Haeri (2005).
Model: Method 1
McMillan (2005),
pp. 60–61.
Model: Method 13
71
72
73
74
0.14Tm
Kc =
e
K m τm
Model: Method 1;
0.1 ≤ τm Tm ≤ 1 .
0.8915
74
Ti
75
Kc
Ti
Model: Method 1
76
Kc
0.96Tm + 0.46τ m
0 ≤ τm Tm ≤ 4
Tm + 0.25τ m
τm Tm > 1
0.25 K m

 τm
τm
1.96
+1.65

τ m + Tm
 τ m + Tm





2




, Ti = 5.38τ m e
59

 τm
τm
 − 3.94
+1.80

τ m + Tm
 τ m + Tm





2




.
2
2
2
 
 
1 + 1 − τm   , Ti = (τm + Tm ) + Tm .
2(τm + Tm )
  τm + Tm  


2T
ln (1 OS)
K c ≤ m 1 + 2x12 − 2 x1 1 + x12  , with x1 ≈
.
2
2
τm 

π + [ln (1 OS)]
0.5
Kc =
Km
2


 τm 
τ
τm

 , 0.1 ≤ m ≤ 0.4 ;


−
+
Ti = Tm 10.59
2
.
3588
0
.
8985

T
T
T


m
m
 m


4
3
2


τ 
τ 
τ 
τ
Ti = Tm 0.7719 m  − 3.6608 m  + 6.5791 m  − 5.1652 m + 2.8059 ,
Tm


 Tm 
 Tm 
 Tm 


0.4 ≤
75
76
Kc =
Kc =
(1 + 2T
m
λK mω90 0
1
Km
) 1+ T
[T + τ (1 + T
2
ω90 0
m
2
2
m
m
m
ω90 0
2
2
ω90 0
2
2
)]
, Ti =


6.282
.
0.407 +
1.195 
1 + 14.956(τm Tm )


2
[
1 + 2Tm ω90 0
(
2
2
ω90 0 Tm + τm 1 + Tm ω90 0
2
)] .
τm
≤ 1.5 .
Tm
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
60
Rule
Trybus (2005).
Model: Method 1
x1
Kc
Ti
Comment
0.6Tm
K m τm
0.8τm + 0.5Tm
x1Tm K m τm
Tm
Servo response: small
overshoot, short
settling time. 77
τm Tm ≤ 2
OS
x1
OS
x1
OS
OS
x1
x1
OS
x1
OS
0.34 0% 0.54 5% 0.64 10% 0.74 15% 0.82 20% 0.90 25%
x1 K m
0.5τm
τm Tm ≥ 2
x1
Xu et al. (2005).
OS
78
Cuesta et al. (2006).
Model: Method 1
77
78
x1
OS
x1
OS
OS
x1
x1
OS
x1
OS
0.17 0% 0.27 5% 0.32 10% 0.37 15% 0.41 20% 0.45 25%
Model: Method 1
(0.3K m τm ) Tm
3.33τm
0.4 K m
79
Ti
Kukal (2006).
Model: Method 1
undefined
80
Ti
Mesa et al. (2006).
Model: Method 1
Tm + 0.5(1 − x1 )τm
K m τm (x1 + 1)
81
40% overshoot; unit
step servo response.
0% overshoot; unit
step servo response.
0.01 ≤ τm Tm ≤ ∞
Ti
Kc
Kc
Ti
0.01 ≤ τm Tm ≤ ∞
Tm + 0.5(1 − x1 )τm
Sample x1 values:
0.3, 0.5, 0.9.
The author quotes the following source for this rule: Findeisen, W. (Ed.), Control Engineering
Handbook (2nd Edition). WNT, Warsaw, 1974 (in Polish).
1 100τm + 6.25Tm
T (100τm + 6.25Tm )(1000τm + 463Tm )
Kc =
, Ti = m
.
(100τm + 46.25Tm )(5.2τm + 684Tm )
K m 100τm + 46.25Tm
0.4Tm (1000τm + 463Tm )
.
(5.2τm + 684Tm )
79
Ti =
80
Gc =
2


T 
T 
1
, Ti = K m 2 m  + 4 m  + 1  e − x1 , x1 =
 τ

Tis
 τm 
  m 

81
Kc =
2


T 
1   Tm 
 + 1 − 2 m   e x 1 , x1 =
8
 τ 
K m   τm 
 m 


4(Tm τm )2 + 1 − 1
2(Tm τm )
8(Tm τm )2 + 1 − 1
2(Tm τm )
3
−1 .
−2;
2
1.5
T 
T 
T 
 T

24 m  + 4 m  + 8 m  + 1 +  8 m + 1
τ
 τm 
 τm 
 τm

Ti = τm  m 
3
T

8 m + 1
τ
 m

.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Skogestad (2006a).
Skogestad (2006b).
Model: Method 1
Visioli (2006),
p. 157.
Vítečková (2006).
Hougen (1979),
p. 335.
Model: Method 1;
coefficients of K c
estimated from
graphs.
Latzel (1988).
Gp = Gm ;
Model: Method 1
Latzel (1988).
Gp =
K pe
− sτ p
(1 + sTp1 ) 2
;
Model: Method 2
Latzel (1988).
Gp =
K pe
− sτ p
(1 + sTp1 )3
;
Model: Method 2
82
Kc ≥
∆d 0
∆y max
82
Kc
1
61
Comment
Ti
min[Tm ,4(TCL + τm )]
Model: Method 1
min[Tm , 4Tm K m ]
Slow control; ‘acceptable’ disturbance rejection performance.
Model: Method 1
0.3Tm K m τm
min[Tm ,8τm ]
≥ 0.368Tm K m τm
τm
Model: Method 9
Tm
Direct synthesis: frequency domain criteria
Maximise crossover
x1 K m
Tm
frequency.
Tm
x1
τ m Tm
x1
τ m Tm
x1
0.1
7.0
0.2
3.5
0.3
2.2
0.4
1.8
0.785Tm K m τm
0.5
0.6
0.7
1.4
1.1
1.0
0.8
0.9
1.0
0.8
0.75
0.7
Tm
φm = 450
0.524Tm K m τm
Tm
φm = 600
0.45Tm
K m (τm + 0.03Tm )
0.57Tm
φm = 450
0.30Tm
K m (τm + 0.03Tm )
0.57Tm
φm = 600
0.42Tm
K m (τm + 0.10Tm )
0.53Tm
φm = 450
0.28Tm
K m (τm + 0.10Tm )
0.53Tm
φm = 600 ; also given
by Klein et al. (1992).
, τm ≤ TCL ≤
Tm ∆y max
− τm ; ∆d 0 = maximum magnitude of a
K m ∆d 0
sinusoidal disturbance over all frequencies, ∆y max = maximum controlled variable change,
corresponding to the sinusoidal disturbance.
b720_Chapter-03
Rule
Kc
Regulator – Gorecki
et al. (1989).
Model: Method 1
Regulator – Gorecki
et al. (1989).
Model: Method 1
Somani et al. (1992).
Model: Method 1;
suggested A m :
1.75 ≤ A m ≤ 3.0 .
83
Kc =
Comment
Ti
2K m (τm + Tm )
Low frequency part of the magnitude Bode diagram is flat.
x 2τm
x 1 K m A m 84
τ m Tm
x1
0.0375
0.1912
0.3693
0.5764
0.8183
1.105
1.449
36.599
7.754
4.361
3.055
2.369
1.950
1.672
Coefficient values:
x2
τ m Tm
3.57
3.33
3.13
2.94
2.78
2.63
2.50
[
]
τm
0.9806
< 1 , Kc ≈
Tm
K mAm
Kc =
1.476
2.38
1.333
2.27
1.226
2.17
1.146
2.08
1.086
2.00
1.042
1.92
1.011
1.85
0.1 ≤ τm Tm ≤ 1
Ti
Kc
x2
1 1 + 3(Tm τm ) + 6(Tm τm )2 + 6(Tm τm )3
,
Km
4 1 + 3(Tm τm ) + 3(Tm τm )2
Kc ≈
85
x1
1.869
2.392
3.067
3.968
5.263
7.246
10.870
Ti = τm
For
Low frequency part
of magnitude Bode
diagram is flat.
G c = 1 (Tis )
Ti
Kc
undefined
85
84
FA
Handbook of PI and PID Controller Tuning Rules
62
83
17-Mar-2009
( 1.886 + (4τ
4
m
Tm ) − 1.373
T
0.9806
1 + 1.886 m
KmAm
 τm
1 + 3(Tm τm ) + 6(Tm τm )2 + 6(Tm τm )3
[
3 1 + 2(Tm τm ) + 2(Tm τ m )2
)
2
]
.
+ 1 or
2
T

τ
0.9806
 ; for m > 1 , K c ≈
1 + 8.669 m
Tm
KmAm
 τm

5Tm
1 
Tm 
.
0.80 + 1.33  , Ti =
K mAm 
τm 
1.04[0.80 + 1.33(Tm τm )]2 − 1
2

 .

b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
86
Lee et al. (1992).
Model: Method 1
x1 = 0.5 + 0.207
Hang et al. (1993a),
(1993b), pp. 61–65.
O’Dwyer (2001a).
Model: Method 1;
τm
> 0.3
Tm
(Yang and Clarke,
1996).
O’Dwyer (2001a).
Model: Method 1
2.42
86
Kc =
88
Comment
Ti
M s = 1.25
τm
, τm Tm < 2.42 ; x1 = 1 , τm Tm ≥ 2.42 .
Tm
ωp Tm
87
Model: Method 37
or Method 41.
Ti
KmAm
x 1Tm
K mτm
A m = π 2x 1 ;
Tm
φ m = π 2 − x 1 .88
Representative results
φm
x1
x1
Am
Am
φm
1.048
1.5
30
0
0.524
3
60 0
0.7854
2
450
0.393
4
x1
τm
Tm
Tm K u Tu
2
2
Tu + 4 π Tm
2
Am
67.50
= π 2x 1 ;
φm = π 2 − x 1 .

τ 
τm
τ
T  τ
− 4 + 1.21 m + 4.4x1 m  m + 2 − 2.2 x1 m 

τm  Tm
Tm
Tm
Tm 

,

τm
τm 

+ 1.21
K m 4 + 4.4 x1

Tm
Tm 


τ
τm
τ
T  τ
− 4 + 1.21 m + 4.4 x1 m  m + 2 − 2.2 x1 m

 Tm
τ
Tm
T
T
m
m
m


Ti =
τ 
T τ
0.96 + 2.42 m  m + 2 − 2.2 x1 m 
τm  Tm
Tm 
τ
1
Ti =
; m ≤ 0.3 (Yang and Clarke, 1996).
2
Tm
4ωp τ m
1
2ωp −
+
π
Tm
2.42
87
Kc
Ti
63



.
Given A m , ISE is minimised when φ m = 68.8884 − 34.3534 A m + 9.1606
τm
Tm
for servo tuning (Ho et al., 1999); 2 ≤ A m ≤ 5 ; 0.1 ≤ τ m Tm ≤ 1 ;
Given A m , ISE is minimised when φ m = 45.9848A m 0.2677 (τ m Tm )0.2755 for
regulator tuning (Ho et al., 1999); 2 ≤ A m ≤ 5 ; 0.1 ≤ τ m Tm ≤ 1 .
b720_Chapter-03
17-Mar-2009
Handbook of PI and PID Controller Tuning Rules
64
Rule
Kc
x1
Am
0.50
3.14
0.61
2.58
0.67
2.34
0.70
2.24
Comment
Ti
x1Tm K m τm
Chen et al. (1999b).
Model: Method 1
Tm
Coefficient values:
Ms
x1
Am
φm
61.4
0.72
2.18
48.7
0
1.30
55.0
0
1.10
0.76
2.07
46.50
1.40
51.6
0
1.20
0.80
1.96
0
1.50
50.0
0
1.26
Symmetrical
optimum principle –
Voda and Landau
(1995).
Model: Method 1
3.5 G p ( jω1350 )
4.6
ω1350
1
4
2.828 G p ( jω1350 )
ω1350
90
Ti
44.1
Model: Method 1
τm
≤ 0.1
Tm
0.1 <
τm
≤ 0.15
Tm
Kc
Ti
0.15 < τm Tm ≤ 1
0.1751
1
ω1350
τm
> 2 ; A m > 3.
Tm
G p ( jω1350 )
1
ωr − 1.273τm ωr + (1 Tm )
Kc =
89
1
90
Ti =
Ms
1.00
r1ωr Tm K m
Friman and Waller
(1997).
Model: Method 1
φm
0
Chen et al. (1999a).
89
FA
2
1
4.6 G p ( jω1350 ) − 0.6K m
, ωr =
, Ti =
π 2 r1 A m − 1
.
4 τ m (r1 A m )2 − 1
1.15 G p ( jω1350 ) + 0.75K m
[
ω135 0 2.3 G p ( jω1350 ) − 0.3K m
].
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Schaedel (1997),
Henry and Schaedel
(2005) – optimal
servo response;
T1 , T2 given in
footnote 91.
91
0.5Ti
K m (T1 − Ti )
2
T1 − 2T2
0.375Ti
K m (T1 − Ti )
2
T1 =
Tm
1 + 3.45
T1 =
τ am
2
Tm − 0.7 τ am
4T2
T1
2
Ti
2 

1 − 2 T2 
2

T1 

2
 3.86T2 2  1.46T2 2 
 Ti 
1 
1 −


0
.
69
−
1
 
T1 

T12 
Km 
T
 2


93
Kc
94
Kc
Tm
+ τ m , T2 2 = 1.25Tm τ am +
1 + 3.45
2
+ τ m , T2 = Tm τ am +
0.7Tm
2
Tm − 0.7 τ am
τ am
93
94
Minimum ITAE.
Minimum φ m = 50 0 .
Tm
τ m + 0.5τ m 2 , 0 <
τma
≤ 0.104 ;
Tm
Tm
+ 0.5τ m 2 ,
2
92
‘Normal’ design –
Butterworth filter.
2
 3.11T2 2  1.43T2 2  ‘Sharp’ design –
1   Ti 
1 −
 Tschebyscheff filter
0.7   − 1
2
T1 
T1 
K m   T2 
(0.1 dB).


Tm
0.7Tm
92
2

1   Ti 

0
.
5
  − 1

Km
T
  2

Leva (2001).
Model: Method 1
91
‘Normal’ design –
Butterworth filter.
Model: Method 25
‘Sharp’ design –
Tschebyscheff filter
(0.5dB).
Model: Method 25
Minimum ITAE
(Henry and Schaedel,
2005).
Model: Method 26
2
T2
T1
T1 −
0.375Ti
K m (T1 − Ti )
Henry and Schaedel
(2005) – optimal
regulator response.
Model: Method 26;
T1 , T2 given in
footnote 91;
τam Tm < 0.2 .
Comment
Ti
T 
Ti = −0.64T1 + 1.64T1 1 − 1.2 2  .
 T1 
 0.6981Tm

20Tm
20
.
K c = min 
,
 , ωc =
τ
+
5Tm
K
τ
K
(
τ
+
5
T
)
m
m m
m m
m 

 0.6981Tm

50Tm
20
.
K c = min 
,
 , ωc =
τ
+
5Tm
m
 K m τm K m (τm + 5Tm ) 
τ am
> 0.104 .
Tm
65
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
66
Rule
Kc
Potočnik et al.
(2001).
Model: Method 48
Cluett and Wang
(1997), Wang and
Cluett (2000), p. 177.
Model: Method 1;
τ
0.01 < m ≤ 10 .
Tm
Ti
Comment
95
Kc
Ti
τm Tm ≤ 0.1
96
Kc
Ti
0.1 < τm Tm < 0.15
97
Kc
Ti
0.1 < τm Tm < 1
98
Kc
Ti
99
Kc
Ti
100
Kc
Ti
7.50 ≤ A m ≤ 8.28 ,
78.50 ≤ φm ≤ 79.70 .
4.35 ≤ A m ≤ 5.02 ,
70.7 0 ≤ φm ≤ 74.00 .
TCL = 1.33τ m ,
3.29 ≤ A m ≤ 3.89 ,
64.40 ≤ φm ≤ 70.60 .
101
95
Kc =
(
G p jω135 0
)
Kc
0.5

G ( jω 0 )
1.14 + 13.24 p 135
Km


97
98
99
(
K m − 2 G p jω1350
[
Kc =
)
2K m K m − 2 G p ( jω135 0 )
(
]
, Ti =
) 

G jω 0
0.75 + 1.15 p 135

Km

0.019952τm + 0.20042Tm
Kc =
, Ti
K m τm
[K
m
57.80 ≤ φm ≤ 68.50 .
,




Ti =
96
2.74 ≤ A m ≤ 3.30 ,
Ti
(
4
ω135 0
(
+ 2 G p jω135 0
(
)
(

1 + 2.48 G p jω1350 + 17.40 G p jω1350
2

Km
Km

ω135 0 G p jω135 0
) ][2 G p (jω135 ) − K m ]
0
) [ G p ( jω135 ) − K m ]
G p (jω135 ) 

.
0

0
0.75 + 1.15
.


Km
ω135 0 



0.099508τm + 0.99956Tm
=
τm , TCL = 4τ m .
0.99747 τm − 8.7425.10−5 Tm
Kc =
2K m
ω135 0
Kc =
0.16440τm + 0.99558Tm
0.055548τm + 0.33639Tm
τm , TCL = 2τ m .
, Ti =
K m τm
0.98607 τm − 1.5032.10− 4 Tm
, Ti =
1
100
Kc =
0.20926τm + 0.98518Tm
0.092654τm + 0.43620Tm
τm .
, Ti =
K m τm
0.96515τm + 4.2550.10− 3 Tm
101
Kc =
0.24145τm + 0.96751Tm
0.12786τm + 0.51235Tm
τm , TCL = τ m .
, Ti =
K m τm
0.93566τm + 2.2988.10− 2 Tm
) 2 


.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Comment
2.39 ≤ A m ≤ 2.93 ,
Cluett and Wang
(1997), Wang and
Cluett (2000) –
continued.
102
Kc
Ti
103
Kc
Ti
38.10 ≤ φm ≤ 66.30 .
Modulus optimum
principle – Cox et al.
(1997).
Model: Method 20
Smith (1998).
Model: Method 1
104
Kc
Ti
τm Tm ≤ 1
0.5Tm K m τm
Tm
τm Tm > 1
0.35
Km
0.42τ m
6 dB gain margin –
dominant delay
process.
Tm
1.3 ≤ M s ≤ 2
Wang et al. (2000a).
Model: Method 37 or
38.
102
103
104
105
Kc
67
49.60 ≤ φm ≤ 67.10 .
2.11 ≤ A m ≤ 2.68 ,
0.26502τ m + 0.94291Tm
0.16051τm + 0.57109Tm
τ m , TCL = 0.8τ m .
, Ti =
K m τm
0.89868τ m + 6.9355.10 − 2 Tm
0.19067 τm + 0.61593Tm
0.28242τm + 0.91231Tm
Kc =
, Ti =
τm , TCL = 0.67τ m .
K m τm
0.85491τm + 0.15937Tm
Kc =
Kc =
0.5  Tm 3 + Tm 2 τ m + 0.5Tm τ m 2 + 0.167 τm 3 
,
2
2
3

K m 
Tm τm + Tm τm + 0.667 τm

 T 3 + Tm 2 τm + 0.5Tm τm 2 + 0.167 τm 3 
.
Ti =  m
2
2


Tm + Tm τm + 0.5τm


105

1.508  Tm
K c = 1.451 −
.

Ms  K m τm

b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
68
Rule
Kc
Wang and Shao
(2000a).
Model: Method 1
Chen and Yang
(2000).
106
0.7Tm K m τm
(1 + ω
1.5 ≤ x1 ≤ 2.5
Model: Method 8
Tm
0.8Tm
0.2 ≤ τm Tm < 1
0.1Tm 0.15
+
K m τm K m
0.3τ m + 0.5Tm
τm
>1
Tm
0.15 K m
0.3τ m
τm Tm → ∞
107
Kc
Ti
108
Kc
Tm
Km
2
90
0
Tm
)
2 1.5
[(T
m
1
2
1 + ω900 Tm
2
[(T
m
M s = 1.4 ;
τm
≥ 0.5 .
Tm

,

+ {1 + ω900 2 Tm 2 }τ m ) sin( − ω900 τ m − tan −1 ω900 Tm )
−
f 2 ( ω900 ) = −
Ti
0.25Tm K m τm

1
1
f 2 (ω90 0 ) −
λf1 (ω90 0 ) 
ω90 0
f1 ( ω900 ) =
Comment
M s = 1.26 ; A m = 2.24 ; φ m = 50 0 .
Hägglund and Åström
(2002).
Model: Method 1
Leva et al. (2003).
Model: Method 1
Kc =
Kc
Ti
A m > x1 ; φm > cos −1 (1 x1 ) (Wang and Shao, 1999b).
Hägglund and Åström
(2002).
Model: Method 5;
M s = 1.4 .
106
17-Mar-2009
Km
(1 + ω
900
2
2
2
Tm 2
)
1.5
[ω
900
]
Tm 2 cos( − ω900 τ m − tan −1 ω900 Tm ) ,
+ {1 + ω900 Tm }τ m ) cot( − ω900 τ m − tan −1 ω900 Tm )
]
−
Ti =
[
2
2
]
2
2
2
2
ω900 Tm 2
1 + ω900 2 Tm 2
;
2
ω90 0 Tm + (1 + ω90 0 Tm )τm cos θ + (1 + 2ω90 0 Tm ) sin θ
3
]
2
− ω90 0 Tm cos θ + ω90 0 [Tm + (1 + ω90 0 Tm )τm ] sin θ
,
θ = − ω900 τ m − tan −1 ω900 Tm .
107
108
Kc =
1
Km

T
 0.14 + 0.28 m

τm


6.8τ m Tm
 , Ti = 0.33τ m +
.

10τ m + Tm

 T (0.5π − φm ) 5Tm 
K c = minimum  m
,
 ; minimum φ m , maximum ωc .
K m τm
K m TCL 

b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Leva et al. (2003) –
continued.
Mataušek and
Kvaščev (2003).
Model: Method 1
Kristiansson and
Lennartson (2002b).
Model: Method 1
110
Comment
Kc
Tm
minimum φ m = 50 0 .
x 1Tm
K mτm
Tm
x 1 = 0.5π − φ m .110
109
Kristiansson (2003).
Model: Method 12
Kristiansson and
Lennartson (2006).
109
Ti
Kc
111
Kc
0.57Tm + 0.29τm
0.2 ≤ τm Tm ≤ 5
112
Kc
Ti
τm Tm < 1
113
Kc
0.60Tm + 0.28τm
τm Tm ≥ 1
114
Kc
1.25 Tm
1.6 ≤ M s ≤ 1.9
115
Kc
1.4 τ m + 0.8Tm
τm Tm < 0.2
69
Model: Method 12; 1.65 ≤ M max ≤ 1.80 .
 0.698Tm

τm + 5Tm
5x1Tm
K c = minimum 
,
, 4 ≤ x1 ≤ 10 .
 , TCL =
x1
 K m τm K m (τm + 5Tm ) 
‘Acceptable’
values
of
0
x1 :
0.5 ≤ x 1 ≤ 3 .
‘Recommended’
values
of
x1 :
0
0.32 ≤ x 1 ≤ 0.54 ( 59 ≤ φ m ≤ 72 ). Choose x 1 = 0.32 when large variations in process
parameters are expected; choose x 1 = 0.54 to give 6% ≤ OS < 13% and 1.4 < M s ≤ 2 .
111
112
113
114
115
Kc =
0.57(0.57 + 0.29[τm Tm ])
, 1.65 ≤ M max ≤ 1.75 .
K m ([τm Tm ] − 0.055)
2

 τm  
0.57
τm

,


Kc =
0.37 + 0.94
− 0.42
K m ([τm Tm ] − 0.055) 
Tm
Tm  



Kc =
0.57(0.60 + 0.28[τm Tm ])
.
K m ([τm Tm ] − 0.055)

ω150 0 
0.075
.
 0.2 +
K m 
K150 0 K m + 0.05 
0.57(0.8Tm + 1.4τm )
Kc =
.
K m Tm ([τm Tm ] − 0.055)
Kc =
[
]
2

τ  
τ
Ti = Tm 0.37 + 0.94 m − 0.42 m   .
Tm

 Tm  

b720_Chapter-03
Rule
Kc
Kristiansson and
Lennartson (2006) –
continued.
Bai and Zhang
(2007).
Model: Method 59
Leva et al. (1994).
Hägglund and Åström
(2004).
Model: Method 1
116
Kc =
117
Kc =
Kc =
Ti
Comment
116
Kc
τ m + Tm
0.2 ≤ τm Tm ≤ 1
117
Kc
0.6τ m + 1.2Tm
τ m Tm ≥ 1
0.526Tm
K m τm
Tan et al. (1996).
Kc =
A m = 2.98 ,
Tm
φm = 59.80 .
118
Kc
Ti
Model: Method 54
119
Kc
Ti
Model: Method 36
120
Kc
Ti
1.920 < φ < 2.356 ;
M s = 1.2 .
121
Kc
Ti
φ = 2.09 radians
(good choice).
0.57(Tm + τm )
.
K m Tm ([τm Tm ] − 0.055)
0.57(1.2Tm + 0.6τ m )
.
K m Tm ([τ m Tm ] − 0.055)
ωcn Ti
Km
1 + ωcn 2Tm 2
2
1 + ωcn Ti
, Ti =
2
[
]
tan φm − 1.571 + τm ωcn + tan −1 (ωcn Tm )
ωcn
with ωcn =
119
FA
Handbook of PI and PID Controller Tuning Rules
70
118
17-Mar-2009
(
x1Ti ωφ 1 + x1Tmωφ
(
A m 1 + x1Ti ωφ
)
2
)2
, Ti =
2.82 − φm − tan −1[(Tm τm )(1.571 − φm )]
.
τm
1
, ωφ < ω u .
x1ωφ tan − tan x1Tm ωφ − x1τm ωφ − φ
[
]
−1
x1 = 0.8, τm Tm < 0.5 ; x1 = 0.5, τm Tm > 0.5 .
120
121
Kc =
Kc =
2.01 − 0.80φ

G jω
1 + (15.45 − 6.30φ) p φ
Km


( )
0.33
( ) 

G jω
1 + 2.26 p φ
Km




( )
G p jωφ

 G p jωφ


6.283(1.72φ − 2.55)
, Ti =
( ) 

G jω
1 + (3.44φ − 5.20 ) p φ
Km


6.61
, Ti =
.
2

G p jωφ 
1 + 2.01
 ωφ
Km 



( )
( )


.
2
ωφ
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Hägglund and Åström
(2004) – continued.
Hägglund and Åström
(2004).
Xu et al. (2004).
Model: Method 37
Ti
Comment
122
Kc
Ti
2.094 < φ < 2.443 ;
M s = 1.4 .
123
Kc
Ti
φ = 2.269 radians
(reasonable choice).
124
Kc
Ti
2.356 < φ < 2.705 ;
M s = 2.0 .
125
Kc
Ti
126
Kc
Ti
φ = 2.53 radians
(good choice).
Model: Method 5;
‘AMIGO’ tuning rule.
1.5 ≤ x1 ≤ 2.5
Tm x1K m τm
Tm
x1 = 2 implies A m > 2, φm > 600 (Xu and Shao, 2004b).
Huang et al. (2005a).
Model: Method 51
0.50Tm + 0.22 τ m
Kmτm
0.9Tm + 0.4τ m
Huang and Jeng
(2005).
0.495Tm + 0.22τm
K m τm
0.9Tm + 0.4 τ m
122
123
124
125
126
Kc =
2.50 − 0.92φ

G jω
1 + (10.75 − 4.01φ) p φ
Km


( )

 G p jωφ


Model: Method 1
6.283(1.72φ − 3.05)
, Ti =
( ) 
2
( ) 
2

G jω
1 + (3.44φ − 6.10 ) p φ
Km


5.36
0.41
, Ti =
.
Kc =
2

G p jωφ 

G p jωφ 
1 + 1.65
 G p jωφ
1 + 1.71
 ωφ
Km 

Km 





6.283(0.86φ − 1.50)
3.43 − 1.15φ
, Ti =
Kc =

G p jωφ 

G jω
1 + (11.30 − 4.01φ)
 G p jωφ
1 + (1.72φ − 2.90) p φ
Km 

Km




4.25
0.52
, Ti =
.
Kc =
2

G p jωφ 

G p jωφ 
1 + 1.15
 G p jωφ
1 + 1.45
 ωφ
Km 

Km 





( )
( )
( )
( )
.
ωφ


( )
( )
( )
Kc =
71
( )
( )
0.15 
τm Tm  Tm
6.7 τm Tm 2
+ 0.35 −
, Ti = 0.35τm + 2
.
2
K m 
(τm + Tm )  K m τm
Tm + 2Tm τm + 10τm 2


.
ωφ
b720_Chapter-03
Rule
Kc
Åström and Hägglund
(2006), pp. 228–229.
Åström and Hägglund
(2006), pp. 258–260.
Model: Method 5;
detuned ‘AMIGO’
tuning rule.
Clarke (2006).
Model: Method 27
K c (1) =
Comment
Ti
(1)
Model: Method 5;
‘AMIGO’ tuning
rule; M s = M t = 1.4 .
Ti
( 2)
τm Tm > 0.11
( 3)
Ti
( 3)
( 4)
Ti
( 4)
τm
< 0.11
Tm
(1)
Ti
( 2)
127
Kc
128
Kc
129
Kc
130
Kc
Tm τm
0.5
T
1+ m
Km
τm
τ m Tm  Tm
13τ m Tm 2
0.15 
(1)
+ 0.35 −
,
T
=
0
.
35
τ
+
.

i
m
K m 
(τ m + Tm )2  K m τ m
Tm 2 + 12Tm τ m + 7 τ m 2
K c (1)
128
129
FA
Handbook of PI and PID Controller Tuning Rules
72
127
17-Mar-2009
Kc
( 2)
= x1K c
K c ( 3) ≥
(1)
, x1 = 0.5,0.1 or 0.01; Ti
( 2)
= Ti
(1)
K c ( 2) Ti (1)
K m + 1 − M s −1
K c (1) K c ( 2) K m + 1 − M s −1
2K c (1) (τm + Tm )
1
+
−1 ,
2
(1)
(1)
−1
M


s
M s  M s + M s − 1 Ti 1 − M s + K c K m


(
)
Ti (3) = Ti (1)
130
K c ( 4) <
.
K c (3) K c (1) K m + 1 − M s
Kc
(1)
−1
K c K m + 1 − M s −1
( 3)
.
2K c (1) (τm + Tm )
1
+
−1,
2
(1)
1
(1)
Ms
M s  M s + M s − 1 Ti 1 − M s − + K c K m


(
)
Ti ( 4) = 2K c (1) K m
τm + Tm
.
2
2

M s  M s + M s + 1  K c ( 4) K m + 1 − M s −1


(
)
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
131
Ti
Comment
Kc
Ti
0.2 ≤ τm Tm ≤ 10 ;
Kc
Ti
Model: Method 1
x1Tm K m τm
Ti
Model: Method 40
Kc
King (2006).
Model: Method 1
131
Lavanya et al.
(2006b).
Padhy and Majhi
(2006b).
132
Kc =
133
73
1.2 ≤ M s ≤ 2.0 .
2
3
4




τ 
 τm 
 τ 
 τ 
1  
  + x 2 log10  m  + x 3 log10  m   + x 4 log10  m  + x 5  ,
x1 log10 
T 

T 
T 
K m  


 m
 Tm  
 m 
 m  


2
2
x1 = −0.2236M s + 1.247 M s − 0.9838 , x 2 = 0.738M s − 3.8471M s + 3.0252 ,
2
2
x 3 = −0.945M s + 4.8577 M s − 3.8272 , x 4 = 0.605M s − 3.2148M s + 2.5509 ,
2
x 5 = −0.2998M s + 1.6155M s − 1.276 ;
2
  τ 3

τ 
τ 
Ti = Tm  x 6  m  + x 7  m  + x 8  m  + x 9  ,
  Tm 

 Tm 
 Tm 


x 6 = −0.001003M s − 0.001185 , x 7 = 0.0155M s + 0.0264 ,
2
x 8 = −0.0772M s + 0.197 M s − 0.0677 , x 9 = 0.1257 M s + 0.8159 .
132
Kc =
33.3
Km
Kc =
Tm
K m τm
∧ 2
∧ 2



1 + ωu Tm 2 1 + ωu TCL 2  , τm > 1 ;






∧ 2
∧ 2



1 + ωu Tm 2 1 + ωu TCL 2  , τm < 1 ;






Ti =
133
x1 =
2φm + π(A m − 1)
(
)
2 Am2 − 1
∧


tan − τm ωu + π − φm  , φm measured in radians.


ωu
1
∧
Tm x1
K m τm
.
; Ti =
A m x1 
A m x1 
1
+
1 −

τm 
π 
Tm
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
74
Rule
Kc
Ti
Comment
Robust
Tm
λK m
Tm
2Tm + τ m
2λ K m
Tm + 0.5τ m
Tm
K m (τ m + λ)
Tm
Tm + 0.5τ m
K m (λ + τ m )
Tm + 0.5τ m
λ ≥ 1.7 τ m ,
λ > 0.1Tm .
Rivera et al. (1986).
λ
≥
1
.
7
τ
,
λ
>
0
.
2
T
(Luyben,
2001);
λ
=
2τ
(minimum
ISE –
Model: Method 1
m
m
m
de Oliveira et al., 1995); λ = max[2.3τm ,0.15Tm ] (Visioli, 2005).
Chien (1988).
Model: Method 1
Brambilla et al.
(1990).
Model: Method 1
134
λ ≥ 1.7 τ m ,
λ > 0.1Tm .
134
No model uncertainty: λ = τ m , 0.1 ≤ τ m Tm ≤ 1 ;
λ = [1 − 0.5 log10 (τ m Tm )]τ m , 1 < τ m Tm ≤ 10 .
λ = Tm (Chien, 1988); (‘aggressive’ tuning) (Åström and Hägglund, 2006).
λ = 3Tm (‘robust’ tuning) (Åström and Hägglund, 2006).
λ = 2τ m (Bialkowski, 1996); (‘aggressive, less robust’ tuning) (Gerry, 1999).
λ > Tm + τ m (Thomasson, 1997).
0.45τm ≤ λ ≤ 0.8τm (Huang et al., 1998).
λ = 2(Tm + τ m ) (‘more robust’ tuning) (Gerry, 1999).
λ = a. max(Tm , τ m ) : a > 3 (‘slow’ design), 2 ≤ a ≤ 3 (‘normal’ design),
1 ≤ a < 2 (‘fast’ design), a < 1 (‘faster’ design) (Andersson (2000), p. 11).
Model: Method 10.
0.1Tm ≤ λ ≤ 0.5Tm , τm Tm ≤ 0.25 ; λ = 1.5(τm + Tm ),0.25 < τm Tm ≤ 0.75 ;
λ = 3(τm + Tm ), τm Tm > 0.75 (Leva, 2001).
Tm ≤ λ ≤ τm or τm ≤ λ ≤ Tm (Smith, 2002).
λ = 1.7Tm (Faanes and Skogestad, 2004).
λ = max[0.25τ m ,0.2T m
] (Leva, 2007).
λ > [0.1Tm ,0.8τm ] (‘aggressive’ tuning), λ > [Tm ,8τm ] (‘moderate’ tuning),
λ > [10Tm ,80τm ] (‘conservative’ tuning) (Arbogast et al., 2006).
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Pulkkinen et al.
(1993).
0.7Tm K m τm
λ (τ m + Tm )
x1 K m
x 2 Tm
Ogawa (1995).
τm
Model: Method 1;
coefficients of K c , Ti Tm
deduced from graphs. 0.5
1.0
0.5
1.0
0.5
1.0
0.5
1.0
Thomasson (1997).
Model: Method 1
2K
Chen et al. (1997).
Model: Method 31
Comment
Ti
x1
x2
0.9
1.3
0.6
1.6
0.7
1.3
0.47
1.7
0.47
1.3
0.36
1.8
0.4
1.3
0.33
1.8
τm
m (τ m + λ)
0.5Tm
K m max( τ m ,0.5)
Model: Method 1
τm
Tm
x1
x2
2.0
10.0
2.0
10.0
2.0
10.0
2.0
10.0
0.45
0.4
0.4
0.35
0.32
0.3
0.3
0.29
2.0
7.0
2.2
7.5
2.4
8.5
2.4
9.0
0.5τ m
λ = TCL .
minimum (Tm ,6τ m )
Isaksson and Graebe
(1999).
Model: Method 1
Chun et al. (1999).
Model: Method 43
τm
2(λ + τ m )
K m (λ + τ m )
Tm +
τ m < 0.5
Desired closed loop
Tm +
τm2
2(λ + τ m )
response =
e − τ ms
.
(λs + 1)
Tm + 0.25τ m
K mλ
Tm + 0.25τ m
Tm > τ m ; λ not
specified.
Tm (τ m + 2λ ) − λ2
K m (τ m + λ )
Tm (τ m + 2λ ) − λ2
τ m + Tm
Representative
λ = 0.4Tm .
Tm
K m (λ + τm )
Tm
x1 K m
x 2 (τm + Tm )
2
Zhong and Li (2002).
Model: Method 1
Wang (2003), p. 14.
Model: Method 1.
Regulator;
x1 , x 2 deduced from
graphs; robust to
±25% uncertainty in
process model
parameters.
τ m > 0.5
3
2
Lee et al. (1998).
Model: Method 1
20% uncertainty in
process parameters.
33% uncertainty in
process parameters.
50% uncertainty in
process parameters.
60% uncertainty in
process parameters.
τ m >> Tm ;
∆τm ≤ λ ≤ 1.4∆τm ;
x1
x2
τ
x1
x2
τ
5.1
4.5
3.6
2.9
2.3
1.8
1.4
0.23
0.34
0.44
0.50
0.57
0.60
0.63
0.05
0.11
0.18
0.25
0.33
0.43
0.54
1.1
0.9
0.8
0.8
0.7
0.6
0.5
0.65
0.64
0.63
0.62
0.61
0.58
0.56
0.67
0.82
1.0
1.22
1.5
1.9
2.3
∆τ m = time delay
uncertainty.
τ = τm Tm
τ
x1
x2
0.4
0.4
0.4
0.4
0.3
0.50
0.50
0.46
0.44
0.42
3.0
4.0
5.7
9.0
19.0
75
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
76
Rule
Kc
Wang (2003), p. 13.
Model: Method 1.
Servo; ±10%
overshoot, step input.
x1 , x 2 deduced from
graphs; robust to
±25% uncertainty in
process model
parameters
x1 K m
Wang (2003),
pp. 50–54.
Model: Method 1.
Servo; ±10%
overshoot, step input.
Wang (2003),
pp. 50–54.
Model: Method 1.
Regulator; x1
deduced from graph;
robust to ±25%
uncertainty in
process model
parameters.
Åström and
Hägglund (2006),
p. 187.
x1
Comment
Ti
x 2 (τm + Tm )
τm
Tm
x2
2.94
2.73
2.20
1.65
1.29
1.04
0.85
0.79 0.05
0.82 0.11
0.81 0.18
0.70 0.25
0.75 0.33
0.72 0.43
0.73 0.54
Tm
K m (τ m + λ)
x1
x2
τm
Tm
x1
x2
τm
Tm
0.81
0.68
0.65
0.59
0.50
0.43
0.38
0.67
0.63
0.63
0.60
0.55
0.51
0.47
Tm
0.67
0.82
1.0
1.22
1.5
1.9
2.3
0.34
0.31
0.29
0.29
0.27
0.43
0.43
0.39
0.39
0.38
3.0
4.0
5.7
9.0
19.0
λ = 0.0132(τm + Tm ) , τm Tm ≤ 0.05 ;
λ = 0.286τm + 0.0034(τm + Tm ) , 0.05 ≤ τm Tm ≤ 19.0 ;
λ = 0.27(τm + Tm ) , τm Tm ≥ 19.0 ;
robust to ±25% uncertainty in process model parameters.
Tm
λ = x1 (τm + Tm )
Tm
K m (τ m + λ)
x1
τm Tm
0.01
0.03
0.04
0.06
0.07
0.05
0.11
0.18
0.25
0.33
Tm
K m (τ m + λ)
λ = Tm
x1
τm Tm
x1
0.08
0.11
0.12
0.13
0.15
0.43
0.54
0.67
0.82
1.00
0.16
0.18
0.19
0.21
0.22
τm Tm
x1
τm Tm
1.22
0.25
4.0
1.5
0.26
5.7
1.9
0.27
9.0
2.3
0.27
19.0
3.0
Model: Method 1;
delay dominated
max [Tm ,3τm ]
processes.
(aggressive tuning); λ = 3Tm (robust tuning).
Ultimate cycle
McMillan (1984).
Model: Method 2 or
Method 55.
135
135
Kc
Ti
Tuning rules
developed from
K u , Tu .




0.65
  T
 

1.881 Tm 
1
 
m

=
τ
+
T
1
.
67
1
Kc =
,



i
m
0.65

 .
K m τm   T
 
  Tm + τm  
m
 
1 + 

  Tm + τm  
b720_Chapter-03
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FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
77
Comment
Ti
x 1K u
x 2 Tu
Kuwata (1987).
x1
x2
x1
x2
x1
x2
τm
τm
τm
Model: Method 1;
T
T
Tu
x 1 , x 2 deduced from
u
u
0.29 0.84 0.28 0.24 0.24 0.35 0.225 0.17 0.43
graphs.
0.27 0.46 0.30 0.23 0.20 0.38 0.225 0.155 0.45
0.25 0.31 0.33 0.225 0.18 0.40 0.225 0.115 0.48
0.654
Boe and Chang
τ 
(1988).
Quarter decay ratio.
1.935Tm  m 
0.54 K u
 Tm 
Model: Method 1
τ
0.876
Claimed equivalent to
Hill and Adams
1.36 − 0.245 m
0.961  Tm 
Tm
Ziegler–Nichols
(1988).


0
.
83
τ
e
m
K m  τm 
ultimate cycle
Model: Method 1
method.
Geng and Geary
0.71Tm
τm
< 0.167
(1993).
3.33τ m
K mτm
Tm
Model: Method 1
Luyben (1993).
0.225K u
0.415Tu
τm Tm = 0.05
Model: Method 1
^
^
Perić et al. (1997).
Model: Method 49
Matsuba et al. (1998).
Model: Method 1
Huang et al. (2005a).
Model: Method 51
136
K u tan 2 φ m
0.1592 Tu
tan φ m
1 + tan 2 φ m
0.99  Tm

K m  τ m
136



0.9
Kc
τ
2.63τ m  m
 Tm
Ti



0.097
0.1 ≤
τm
≤ 1.0
Tm


2


 ^ 


0.9  K u  − 1
0.55 

 
+
Kc =
0
.
4
,
2


K m 
^

 
−1
 π − tan   K u  − 1 

 










2
2

 
^ 
 ^ 


−1  
 
Ti = 0.159 Tu 0.9  K u  − 1 + 0.4π − tan
 K u  − 1   .

 
 



 

^
b720_Chapter-03
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FA
Handbook of PI and PID Controller Tuning Rules
78


1
+ Td s 
3.3.2 Ideal PID controller G c (s) = K c 1 +
 Ti s

E(s)
R(s)
+

 U(s)
1
K c 1 +
+ Td s 
 Ti s

−
Y(s)
K m e − sτ
1 + sTm
m
Table 10: PID controller tuning rules – FOLPD model G m (s) =
Rule
Callender et al.
(1935/6).
Model: Method 1
Ziegler and
Nichols (1942).
Model: Method 2
Chien et al.
(1952) –
regulator.
Model: Method 2
Chien et al.
(1952) – servo.
Model: Method 2
Cohen and Coon
(1953).
Model: Method 2
Kc
1
1.066
K mτm
Ti
K m e − sτ m
1 + sTm
Comment
Td
Process reaction
0.353τ m or
1.418τ m
0.47τ
m
τm
= 0.3
Tm
x1Tm
K m τm
2τ m
0.5τ m
0.95Tm
K mτm
2.38τ m
0.42τ m
1.2 ≤ x1 ≤ 2 ;
quarter decay
ratio.
0% overshoot;
0.1 < τm Tm < 1 .
1.2Tm
K mτm
2τ m
0.42τ m
20% overshoot;
0.1 < τm Tm < 1 .
0.6Tm
K mτm
Tm
0.5τ m
0% overshoot;
0.1 < τm Tm < 1 .
0.95Tm
K mτm
1.36Tm
0.47τ m
20% overshoot;
0.1 < τm Tm < 1 .
Ti
0.37 τm
1 + 0.19(τm Tm )
2
Kc
1
Decay ratio = 0.043; period of decaying oscillation = 6.28τ m .
2
Kc =
 2.5(τm Tm ) + 0.46(τm Tm ) 2 

1 
T
.
1.35 m + 0.25  , Ti = Tm 




Km 
τm
1 + 0.61(τm Tm )



Quarter decay
ratio;
0.1 < τm Tm ≤ 1 .
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Comment
Td
Ti
0.950
0.738
Three constraints
K c K m Td
Tm  τ m 
1.370  Tm 
= 0.5 ;
3




method – Murrill
T
Tm
d
1.351  Tm 
(1967), p. 356. K m  τ m 
0.1 < τm Tm < 1 .
Model: Method 5
Quarter decay ratio; minimum integral error (servo mode).
Åström and
Model:
3 Km
T90%
0.5τ m
Hägglund (1988),
Method 28
pp. 120–126.
T90% = 90% closed loop step response time (Leeds and Northrup
Electromax V controller).
Parr (1989),
Model: Method 2
1.25Tm K m τm
2.5τ m
0.4 τ m
p. 194.
Borresen and
Model: Method 2
Tm K m τm
3τ m
0.5τ m
Grindal (1990).
4
Sain and Özgen
Model:
Ti
Td
Kc
(1992).
Method 29
Connell (1996),
Approximate
1.6Tm
p. 214.
quarter decay
1.6667 τ m
0.4 τ m
K mτm
Model: Method 2
ratio.
x1 K m
x 2τm
x 3τ m
Hay (1998), pp.
Coefficient values – servo tuning:
197,198.
τ
T
x
x2
x3
τm Tm
x1
x2
x3
m
m
1
Model:
0.125
6.5
9.0
0.47
0.125
5.0
9.8
0.34
Method 1;
5.0
6.5
0.45
0.167
3.7
7.0
0.32
coefficients of 0.167
0.25
3.5
4.2
0.40
0.25
2.5
4.8
0.30
K c , Ti , Td
0.5
1.8
2.2
0.35
0.5
1.1
2.5
0.27
deduced from
Coefficient values – regulator tuning:
graphs.
τm Tm
x1
x2
x3
τm Tm
x1
x2
x3
0.125
0.167
0.25
0.5
3
4
τ 
Td = 0.365Tm  m 
 Tm 
Kc =
9.8
7.2
4.5
2.2
2.0
1.8
1.6
1.4
0.52
0.50
0.45
0.35
0.1
0.125
0.167
0.25
0.5
10.0
8.0
6.0
4.0
2.0
2.2
2.2
2
1.8
1.4
0.35
0.35
0.34
0.32
0.28
0.950
.

0.8647Tm + 0.226τm
0.0565Tm
1 
T
 0.6939 m + 0.1814  , Ti =
, Td =
.


T
T
Km 
τm
m

+ 0.8647
0.8647 m + 0.226
τm
τm
79
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
80
Rule
Hay (1998),
p. 200.
Model:
Method 2;
coefficients of
K c , Ti , Td
deduced from
graphs.
Lipták (2001).
Chidambaram
(2002), p. 154.
Alfaro Ruiz
(2005a).
ControlSoft Inc.
(2005).
PMA (2006).
Model: Method 2
Kc =
1
Km
Comment
Kc
Ti
Td
x1 K m
x 2 τm
x 3τm
x1
4.7
2.4
2.8
1.5
1.8
0.5
1.2
0.8
1.0
0.7
1.2Tm K m τm
Moros (1999).
Model: Method 1 1.2Tm K m τm
5
17-Mar-2009
0.85Tm K m τm
5
Kc
x2
x3
0.87
1.87
0.95
1.55
1.00
1.33
1.00
1.15
0.95
1.00
1.32
1.28
1.07
1.0
0.90
0.78
0.78
0.64
0.72
0.54
τm Tm
0.3
0.4
0.5
0.6
0.7
2τ m
0.42τ m
2τ m
0.44τ m
1.6τ m
0.6τ m
2.4τm
0.38τm
Regulator.
Servo.
Regulator.
Servo.
Regulator.
Servo.
Regulator.
Servo.
Regulator.
Servo.
Attributed to
Oppelt.
Attributed to
Rosenberg.
Model: Method 2
Model: Method 1. ‘Improved’ Ziegler–Nichols method.
x1Tm K m τm
2.5τm
0.4τm
2.0 ≤ x1 ≤ 3.33 ;
Model: Method 2
6
Ti
Td
1.0 ≤ x1 ≤ 1.67 ;
Kc
Model: Method 1
7
Model:
2 Km
τ m + Tm
Td
Method 11
2τm
0.59Tm
2τm
K m τm
Alternative.
τm

 Tm
+ 0.45 .
1.8

 τm

τ

τ 
Tm  0.40 + 0.153 m
Tm  2.50 + 0.957 m 
Tm
Tm 

x 
T


6
, Td =
K c = 1 1.56 m + 0.68 , Ti =
(
)
1
+
0
.
787
τ
T
Km 
τm
(
1
0
.
787
T
+
τ
m
m
m
m)

τ T 
τ T 
7
Td = max  m , m  (slow loop); Td = min  m , m  (fast loop).
3
6


 3 6 




.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Td
Comment
Minimum performance index: regulator tuning
0.921
0.749
Minimum IAE –
Tm  τ m 
1.435  Tm 
8




Model: Method 5
Murrill (1967),
Td
K m  τm 
0.878  Tm 
pp. 358–363.
Minimum IAE –
12.52 K m
0.20Tm
0.04Tm
τm Tm = 0.10
Lopez (1968),
2.68 K m
0.69Tm
0.22Tm
τm Tm = 0.50
p. 50.
1.47 K m
1.04Tm
0.42Tm
τm Tm = 1.00
Minimum IAE –
Marlin (1995),
pp. 301–307.
Model: Method 8
Minimum IAE –
Peng and Wu
(2000).
Model:
Method 21
1.3 K m
0.7 τ m
0.01τ m
τ m Tm = 0.43
1.0 K m
0.68τ m
0.05τ m
τ m Tm = 0.67
0.85 K m
0.65τ m
0.08τ m
τ m Tm = 1.0
0.65 K m
0.65τ m
0.10τ m
τ m Tm = 1.5
0.45 K m
0.55τ m
0.14τ m
τ m Tm = 2.33
0.4 K m
0.50τ m
0.21τ m
τ m Tm = 4.0
Coefficients of K c , Ti , Td estimated from graphs; robust to ±25%
process model parameter changes.
6.558K m
0.925Tm
0.061Tm
τ m Tm = 0.1
2.311K m
0.677Tm
0.189Tm
τ m Tm = 0.2
1.479 K m
0.633Tm
0.233Tm
τ m Tm = 0.3
1.117 K m
0.647Tm
0.250Tm
τ m Tm = 0.4
0.947 K m
0.705Tm
0.268Tm
τ m Tm = 0.5
0.825K m
0.752Tm
0.289Tm
τ m Tm = 0.6
0.731K m
0.792Tm
0.312Tm
τ m Tm = 0.7
0.695K m
0.871Tm
0.316Tm
τ m Tm = 0.8
0.630K m
0.895Tm
0.326Tm
τ m Tm = 0.9
0.651K m
1.030Tm
0.306Tm
τ m Tm = 1.0
1.137
8
τ 
Td = 0.482Tm  m 
 Tm 
, 0.1 ≤
τm
≤1.
Tm
81
b720_Chapter-03
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FA
Handbook of PI and PID Controller Tuning Rules
82
Rule
9
Minimum IAE –
Kosinsani
(1985),
pp. 46–48.
Model: Method 1
Kc
Ti
Td
Comment
Kc
Ti
Td
Load disturbance
model =
1
.
1 + sTL
0.05 ≤
τm
T
≤ 1.0 ; 0.05 ≤ L ≤ 7.0 .
Tm
Tm
Values of x1 to x 15 for varying τm Tm provided. The controller
tuning formula is not recommended when τm Tm = 0.4, TL Tm > 6.5 ;
τm Tm = 0.5, TL Tm > 5.5 ; τm Tm = 0.75, TL Tm > 5.0 ;
τm Tm = 1.0, TL Tm > 4.5 .
τm Tm
0.075
0.1
0.15
0.2
0.3
0.4
0.5
0.75
1.0
x1
27.567 18.817 13.984 9.3318 7.0210 4.7110 3.6156 2.9398 2.0164
1.5805
x2
3.2174 1.2714 7.6166 4.9233 3.9864 2.9383 2.2643 1.8975 2.0290
0.6154
x3
-2.3336 -1.8823 6.8896 4.5267 3.2537 2.0947 1.5546 1.2108 0.7548
0.5345
x4
-31.416 -31.416 -1.9E-6 -3.4E-7 -1.0E-5 -5.6E-8 -9.0E-9 -9.5E-9 0.4277
0.3603
x5
-7.2662 -6.2990 -9.9212 -0.0978 1.9228 1.6537 2.3143 1.5861 1.5845 -0.6779
x6
7.8891 3.7533 1.1676 0.9631 1.1989 0.5403 2.5070 2.6381 0.6400
1.1821
x7
8.1491 8.0893 8.5751 6.3996 4.9687 4.0498 7.8855 0.7385 2.5128
2.6522
x8
-0.4706 -0.5880 -0.6459 -0.6104 -0.4955 -0.5196 -0.0233 -0.2809 -0.1774 -0.0445
x9
14.020 11.353 5.2382 1.7098 0.6395 0.1405 1.7941 3.5610 0.0825
x 10
9
0.05
0.6528
-3.6987 -3.4514 -1.7079 -1.3466 -1.3080 -0.2366 -0.5171 -0.5509 -0.0100 -0.4917
x 11
4.6405 3.5079 3.1794 3.5195 7.5219 1.8090 0.5537 0.1952 1.3773
x 12
-0.8986 -0.4818 -0.2947 -0.4508 -2.7213 -4.4931 -1.8157 -2.5837 -4.7205 -2.3611
x 13
0.0270 -1491.0 -4829.2 -1224.6 -13635 -31129 -31539 -40901 -52408
x 14
-0.0090 1491.1 4829.3 1224.7
52408
59325
x 15
-4.1759 6.2E-7 3.8E-7 3.2E-6 5.0E-7 3.9E-7 5.6E-7 5.7E-7 6.8E-7
7.5E-7
Kc =
Ti =
1 
 x1 − e − x 2 TL
Km 

(
x6 + x7 1 − e
13635
31130
31539
40901

 T 
 T  
x 3 cosx 4 L  + x 5 sin x 4 L   ,


 Tm 
 Tm  

Tm
, Td = Tm x13 + x14e x15 TL


TL
x 10 TL Tm
sin  x11
+ x 9e
+ x12 
Tm


0.5675
-59324
Tm 
x 8 Tl Tm
)
(
Tm
).
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Comment
Td
Ti
Load disturbance
Minimum IAE –
x1 K m
x 2Tm
x 3Tm
1
model =
.
Hill and Venable
1 + sTL
(1989).
Model: Method 1 x1 values (this page), x 2 values (p. 84) and x 3 values (p. 85) deduced
from graphs: robust to ±30% process model parameter changes.
0.05 0.07 0.1 0.15 0.2
0.3
0.4
0.5 0.75 1.0
τm Tm
TL Tm = 0.05
19.0
11.8
8.6
TL Tm = 0.1
21.2
13.6
9.8
TL Tm = 0.3
22.0
14.6
10.9
TL Tm = 0.5
22.2
14.7
11.0
TL Tm = 0.7
22.1
14.7
TL Tm = 1.0
22.0
14.8
TL Tm = 1.2
22.2
TL Tm = 1.5
5.5
4.2
2.88
2.20
1.80
1.32
1.08
6.1
4.5
3.00
2.26
1.83
1.33
1.09
7.1
5.3
3.44
2.58
2.06
1.40
1.12
7.2
5.5
3.66
2.76
2.22
1.51
1.19
11.0
7.3
5.6
3.76
2.85
2.33
1.59
1.24
11.1
7.4
5.7
3.81
2.94
2.39
1.66
1.32
14.9
11.2
7.4
5.7
3.84
2.95
2.41
1.70
1.35
22.4
15.0
11.3
7.5
5.8
3.88
2.98
2.43
1.72
1.37
TL Tm = 2.0
22.5
15.0
11.2
7.5
5.8
3.91
3.00
2.45
1.74
1.40
TL Tm = 2.5
22.6
15.2
11.2
7.6
5.9
3.93
3.01
2.46
1.75
1.40
TL Tm = 3.0
22.8
15.0
11.2
7.7
5.9
3.95
3.02
2.48
1.76
1.40
TL Tm = 3.5
22.7
15.1
11.0
7.5
5.9
3.94
3.02
2.48
1.75
1.40
TL Tm = 4.0
22.7
15.0
10.9
7.6
5.9
3.94
3.02
2.48
1.76
1.40
TL Tm = 4.5
22.7
15.1
11.2
7.6
5.9
3.95
3.03
2.48
1.77
1.41
TL Tm = 5.0
22.6
15.1
11.2
7.5
5.9
3.94
3.03
2.50
1.78
1.42
TL Tm = 5.5
22.7
15.1
11.3
7.6
5.9
3.94
3.03
2.51
1.79
1.42
TL Tm = 6.0
22.6
15.0
11.5
7.6
5.9
3.94
3.03
2.51
1.80
1.44
TL Tm = 6.5
22.6
15.0
11.3
7.5
5.8
3.94
3.04
2.52
1.80
1.45
TL Tm = 7.0
22.7
14.9
11.5
7.6
5.9
3.95
3.05
2.54
1.81
1.46
83
b720_Chapter-03
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FA
Handbook of PI and PID Controller Tuning Rules
84
Rule
Kc
Comment
Td
Ti
Load disturbance
Minimum IAE –
1
x1 K m
x 2Tm
x 3Tm
model =
.
Hill and Venable
1 + sTL
(1989).
Model: Method 1 x1 values (p. 83), x 2 values (this page) and x 3 values (p. 85) deduced
from graphs: robust to ±30% process model parameter changes.
0.05 0.07 0.1 0.15 0.2
0.3
0.4
0.5 0.75 1.0
τm Tm
1.0
1.0
1.0
1.0
1.0
0.92
0.90
0.90
0.82
0.78
TL Tm = 0.1
1.0
1.0
1.0
TL Tm = 0.3
10.5
6.6
4.6
1.0
1.0
0.92
0.90
0.90
0.82
0.78
2.3
1.1
1.00
0.96
0.92
0.82
0.79
TL Tm = 0.5
10.7
7.0
5.1
3.2
2.3
1.48
1.18
1.04
0.86
0.80
TL Tm = 0.7
10.4
TL Tm = 1.0
9.7
7.0
5.2
3.4
2.7
1.80
1.40
1.20
0.95
0.82
6.7
5.1
3.5
2.8
2.00
1.62
1.40
1.06
0.92
TL Tm = 1.2
10.3
7.0
5.4
3.8
2.9
2.14
1.75
1.48
1.13
0.96
TL Tm = 1.5
11.0
7.6
5.8
4.0
3.1
2.32
1.86
1.59
1.20
1.00
TL Tm = 2.0
12.3
8.4
6.5
4.6
3.6
2.60
2.10
1.79
1.30
1.06
TL Tm = 2.5
13.1
9.0
7.0
4.9
3.8
2.80
2.26
1.90
1.40
1.15
TL Tm = 3.0
13.7
9.6
7.3
5.2
4.1
3.00
2.39
2.01
1.50
1.22
TL Tm = 3.5
14.1
9.8
7.8
5.4
4.2
3.12
2.50
2.13
1.57
1.26
TL Tm = 4.0
14.6
10.2
8.0
5.6
4.4
3.21
2.59
2.20
1.61
1.30
TL Tm = 4.5
15.0
10.5
8.0
5.7
4.5
3.32
2.66
2.23
1.63
1.33
TL Tm = 5.0
15.4
10.8
8.2
5.9
4.7
3.40
2.72
2.30
1.65
1.35
TL Tm = 5.5
15.6
10.9
8.3
6.0
4.8
3.45
2.76
2.33
1.67
1.37
TL Tm = 6.0
15.9
11.1
8.3
6.1
4.9
3.50
2.79
2.36
1.70
1.39
TL Tm = 6.5
16.1
11.2
8.6
6.2
5.0
3.54
2.80
2.38
1.70
1.40
TL Tm = 7.0
16.4
11.5
8.5
6.2
5.0
3.57
2.83
2.40
1.72
1.40
TL Tm = 0.05
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Comment
Td
Load disturbance
Minimum IAE –
1
x1 K m
x 2Tm
x 3Tm
model =
.
Hill and Venable
1 + sTL
(1989).
Model: Method 1 x1 values (p. 83), x 2 values (p. 84) and x 3 values (this page) deduced
from graphs: robust to ±30% process model parameter changes.
0.05 0.07 0.1 0.15 0.2
0.3
0.4
0.5 0.75 1.0
τm Tm
TL Tm = 0.05
0.022 0.034 0.045 0.067 0.089 0.13
0.17
0.21
0.30
0.39
TL Tm = 0.1
0.020 0.030 0.042 0.064 0.088 0.13
0.17
0.21
0.30
0.40
TL Tm = 0.3
0.027 0.038 0.050 0.068 0.081 0.13
0.17
0.21
0.31
0.41
TL Tm = 0.5
0.027 0.039 0.051 0.074 0.094 0.13
0.17
0.22
0.32
0.41
TL Tm = 0.7
0.026 0.038 0.050 0.075 0.097 0.14
0.18
0.22
0.32
0.41
TL Tm = 1.0
0.024 0.037 0.050 0.073 0.097 0.14
0.19
0.23
0.33
0.42
TL Tm = 1.2
0.026 0.038 0.051 0.076 0.100 0.14
0.19
0.23
0.34
0.43
TL Tm = 1.5
0.027 0.040 0.052 0.079 0.104 0.15
0.20
0.24
0.34
0.44
TL Tm = 2.0
0.028 0.041 0.057 0.083 0.110 0.16
0.21
0.26
0.37
0.46
TL Tm = 2.5
0.028 0.042 0.058 0.085 0.113 0.17
0.22
0.27
0.38
0.48
TL Tm = 3.0
0.029 0.045 0.059 0.088 0.118 0.17
0.23
0.28
0.40
0.50
TL Tm = 3.5
0.029 0.045 0.062 0.090 0.120 0.18
0.23
0.28
0.41
0.52
TL Tm = 4.0
0.029 0.046 0.064 0.090 0.122 0.18
0.24
0.29
0.42
0.52
TL Tm = 4.5
0.030 0.045 0.062 0.092 0.123 0.18
0.28
0.30
0.42
0.52
TL Tm = 5.0
0.030 0.044 0.062 0.095 0.125 0.18
0.24
0.30
0.42
0.52
TL Tm = 5.5
0.030 0.047 0.062 0.094 0.126 0.19
0.24
0.30
0.42
0.52
TL Tm = 6.0
0.031 0.048 0.061 0.094 0.128 0.19
0.25
0.30
0.42
0.52
TL Tm = 6.5
0.031 0.048 0.063 0.095 0.133 0.19
0.25
0.30
0.42
0.52
TL Tm = 7.0
0.031 0.049 0.062 0.095 0.130 0.19
0.25
0.30
0.42
0.52
85
b720_Chapter-03
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FA
Handbook of PI and PID Controller Tuning Rules
86
Rule
Kc
Minimum IAE –
Arrieta Orozco
(2006).
10
Kc
Ti
Td
Ti
Td
Comment
0.1 ≤
τm
≤ 2.0
Tm
pp. 23–24. Model: Method 10
Minimum IAE –
τ
0.1 ≤ m ≤ 1.0
Pessen (1994).
0.7 K u
0.4Tu
0.149Tu
Tm
Model: Method 1
0.749
0.921
Modified
Tm  τ m 
3  Tm 
11


minimum IAE –
Model:


Td
0.878  Tm 
Method 14
Cheng and Hung K m  τ m 
(1985).
1.006 0.1 ≤ τ
0.945
0.771
Minimum ISE –
m Tm ≤ 1 ;
τ 
Tm  τ m 
1.495  Tm 
0.56Tm  m 




Murrill (1967),
Model: Method 5
K m  τm 
1.101  Tm 
 Tm 
pp. 358–363.
Minimum ISE –
13.41 K m
0.14Tm
0.05Tm
τm Tm = 0.10
Lopez (1968),
2.84 K m
0.55Tm
0.28Tm
τm Tm = 0.50
p. 50.
1.52 K m
0.86Tm
0.56Tm
τm Tm = 1.00
0.970
0.753
τ
1.473  Tm 
Tm  τ m 
0.1 ≤ m ≤ 1.0
12


Minimum ISE –


T
T
d
m
1115
.
Zhuang (1992), K m  τ m 
 Tm 
Zhuang and
0.735
0.641
τ
Tm  τ m 
Atherton (1993). 1.524  Tm 
1.1 ≤ m ≤ 2.0
13




T
T
d
m
K m  τm 
1.130  Tm 
Model: Method 1 or Method 5 or Method 6 or Method 15.
10
1
Kc =
Km
1.0158 
0.5022 






 , Ti = Tm − 0.2228 + 1.3009 τm 
,
0.2068 + 1.1597 Tm 
τ 
T 




m 
m 






τ 
Td = 0.3953Tm  m 
 Tm 
11
12
13
τ
Td = 0.482Tm  m
 Tm
1.137




τ 
Td = 0.550Tm  m 
 Tm 
0.948
τ 
Td = 0.552Tm  m 
 Tm 
0.851
.
.
.
0.8469
.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Minimum ISE –
14
Ho et al. (1998).
Ti
Kc
Model: Method 1
0.947
0.738
Minimum ITAE
Tm  τ m 
1.357  Tm 




– Murrill (1967),
K m  τm 
0.842  Tm 
pp. 358–363.
Minimum ITAE
11.80 K m
0.22Tm
– Lopez (1968),
2.59 K m
0.75Tm
p. 50.
1.39 K m
1.15Tm
Minimum ITAE
– Arrieta Orozco
(2006),
pp. 90–92.
14
16
0.116
 τm 
1.0722 φm −


Kc =
0.8432 

K m Am
 Tm 
−0.908
15
1.0082
, Ti =
1.2497Tm φm
A m 0.2099
Td =
2 ≤ Am ≤ 5 ,
Td
300 ≤ φm ≤ 600 .
0.1 ≤ τm Tm ≤ 1 ;
Model: Method 5
Td
0.03Tm
τm Tm = 0.10
0.19Tm
τm Tm = 0.50
0.37Tm
τm Tm = 1.00
Td
2 ≤ Am ≤ 4 ;
Model:
Method 10
Ti
Kc
Comment
Td
Ti
 τm 


T 
 m
0.3678
,
0.4763Tm φm −
Given A m , ISE is minimised when φ m = 46.5489A m
Am
0.2035
0.328
0.0961
(τ m
87
Tm )
1.0317
 τm 


T 
 m
0.3693
, 0. 1 ≤
τm
≤ 1.0 ;
Tm
;
2 ≤ A m ≤ 5 ; 0.1 ≤ τ m Tm ≤ 1.0 (Ho et al., 1999).
15
16
τ 
Td = 0.381Tm  m 
 Tm 
0.995
.
Tuning rule continued onto p. 88.
−0.8757 − 0.0851A m + 0.0120 A m 2 

 τm 
1 
2
,

Kc =
0.6791 − 0.2272A m + 0.0253A m + x1 

Km 
Tm 



2
x1 = 1.9742 − 0.7126A m + 0.0813A m , 0.1 ≤ τm Tm ≤ 2.0 ;
−5.1852 + 4.4910 A m − 0.7891A m 2 

 τm 
2

,

Ti = Tm 7.1461 − 4.8748A m + 0.6843A m + x 2 


Tm 



2
x 2 = −5.6764 + 4.7562A m − 0.6850A m , 0.1 ≤ τm Tm < 1.0 ;
0.6969 + 0.0948 A m − 0.0231A m 2 

 τm 
2

,

Ti = Tm 0.2931 + 0.2270A m − 0.0427 A m + x 2 


Tm 



2
x 2 = 0.9924 − 0.2202A m + 0.0217 A m , 1.0 ≤ τm Tm ≤ 2.0 .
b720_Chapter-03
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FA
Handbook of PI and PID Controller Tuning Rules
88
Rule
Kc
Minimum ISTSE
– Zhuang (1992),
Zhuang and
Atherton (1993).
Ti
Td
17
Kc
Ti
Td
18
Kc
Ti
Td
Comment
Model: Method 1 or Method 5 or Method 6 or Method 15.
Model: Method 1
Ti
0.144Tu
K
19
20
c
0.960
Model:
Method 37
τ
0.1 ≤ m ≤ 1.0
Tm
∧
Ti
Kc
0.15 T u
0.746
Tm  τ m 
1.531  Tm 




Minimum ISTES
K
τ
0
.971  Tm 
m  m 
– Zhuang (1992),
0.705
0.597
Zhuang and
Tm  τ m 
Atherton (1993). 1.592  Tm 


K m  τ m 
0.957  Tm 
23
Td
24
Td
1.1 ≤
τm
≤ 2. 0
Tm
Model: Method 1 or Method 5 or Method 6 or Method 15.
(
Td = Tm 0.4718 − 0.1251A m + 0.0202A m
(
2
)
1.8232 − 0.5092 A m − 0.0721A m 2
)
1.4576 − 0.4119 A m + 0.0813A m 2
 τm 


T 
 m
τ 
Td = Tm 0.5312 − 0.1662A m + 0.0271A m 2  m 
 Tm 
17
Kc =
1.468  Tm 


K m  τm 
0.970
, Ti =
0.730
Tm  τm 


0.942  Tm 
0.725
τ 
, Td = 0.443Tm  m 
 Tm 
, 0.1 ≤
τm
< 1.0 ;
Tm
, 1.0 ≤
τm
≤ 2.0 .
Tm
, 0. 1 ≤
τm
≤ 1.0 .
Tm
0.939
0.847
0.598
τ 
τ
, 1.1 ≤ m ≤ 2.0
, Td = 0.444Tm  m 
Tm
 Tm 
τ
4.434K m K u − 0.966
1.751K m K u − 0.612
19
Kc =
K u , Ti =
Tu , 0.1 ≤ m ≤ 2.0 .
Tm
5.12K m K u + 1.734
3.776K m K u + 1.388
18
Kc =
20
Kc =
1.515  Tm 


K m  τm 
, Ti =
∧
23
24
6.068K m K u − 4.273
∧
5.758K m K u − 1.058
τ 
Td = 0.413Tm  m 
 Tm 
0.933
τ 
Td = 0.414Tm  m 
 Tm 
.
0.850
.
∧
Tm  τm 


0.957  Tm 
K u , Ti =
∧
τ
1.1622K m K u − 0.748 ∧
T u , 0. 1 ≤ m ≤ 2. 0 .
∧
Tm
2.516K m K u − 0.505
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Td
Comment
Minimum error –
step load change
– Gerry (1998).
Nearly minimum
IAE, ISE, ITAE
– Hwang (1995).
Model: Method
44; decay ratio =
0.15.
0.3
Km
0.5τ m
Tm
τm Tm > 5 ;
Model: Method 1
Hwang and Fang
(1995).
25
26
Kc
Ti
26
Kc
Ti
27
Kc
Ti
28
Kc
Ti
29
Kc
25
3 ≤ ε1 < 20
Nearly minimum
IAE, ITAE.
Model: Method 30; decay ratio = 0.12.
Td
[
]
2
2

0.674 1 − 0.447ωH1τm + 0.0607ωH1 τm 
,
K c =  K H1 −


K m (1 + K H1K m )


τ
K c (1 + K H1K m )
Ti =
, 0. 1 ≤ m ≤ 2. 0 .
2
2
Tm
ωH1K m 0.0607 1 + 1.05ωH1τm − 0.233ωH1 τm

K c =  K H1 −


[
(
2
2
0.778 1 − 0.467ωH1τm + 0.0609ωH1 τm
K m (1 + K H1K m )
[
[
)
] ,


(
K c (1 + K H1K m )
ωH1K m 0.0309 1 + 2.84ωH1τm − 0.532ωH12 τm 2
2
ω τ

1.31(0.519 ) H1 m 1 − 1.03 ε1 + 0.514 ε1
K c =  K H1 −

K m (1 + K H1K m )

] ,


K c (1 + K H1K m )
(
ωH1K m 0.0603(1 + 0.929 ln[ωH1τm ]) 1 + 2.01 ε1 − 1.2 ε12
2
2

1.14 1 − 0.482ωH1τm + 0.068ωH1 τm
K c =  K H1 −

K m (1 + K H1K m )

Ti =
29
2.4 ≤ ε1 < 3
ε1 ≥ 20
Ti
Ti =
28
ε1 < 2.4
0.471K u
K m ωu
Ti =
27
89
] ,


(
K c (1 + K H1K m )
ωH1K m 0.0694 − 1 + 2.1ωH1τm − 0.367ωH12 τm 2
2

τ  
τ
K c = 0.802 − 0.154 m + 0.0460 m   K u ,
Tm

 Tm  

Kc
Ti =
,
K u ωu 0.190 + 0.0532(τm Tm ) − 0.00509(τm Tm )2
[
]
2

τ 
τ
Ku
τ
Td = 0.421 + 0.00915 m − 0.00152 m  
, 0.1 ≤ m ≤ 2.0 .
T
Tm
T
ω
K


m
 m  u c

).
).
).
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
90
Rule
Kc
O’Connor and
Denn (1972).
Model:
Method 1;
K m = Tm .
Comment
Td
Ti
∞
∫[
Minimum 0.5 x 2 y( t ) + (du ( t ) dt )
2
2
] dt .
0
30
Syrcos and
Kookos (2005).
Model: Method 1
Minimum IE –
Devanathan
(1991).
Model:
Method 1;
coefficients of
K c , Ti provided
are deduced from
graphs.
2
31
τm Tm < 0.2
x1τm
2
x 2 τ m + 2 x1
( 2 + x1 ) τ m
x 2 τm + 2 x1
2τ m x 2
τm x 2 + 2 x1
Ti
Td
Kc
2
2 ≤ τm Tm ≤ 5
Minimum performance index with constraints.
32
0.32Tu
0.08Tu
x 2 Tu
x 3Tu
Kc
τ m Tm
Coefficient values:
x3
τ m Tm
x2
x2
0.2
0.4
0.6
0.8
1.0
0.31
0.08
1.2
0.26
0.29
0.07
1.4
0.26
0.29
0.07
1.6
0.24
0.29
0.07
1.8
0.24
0.28
0.07
Minimum performance index: servo tuning
x1 K m
x 2 (Tm + τ m )
x 3 (Tm + τ m )
x3
0.07
0.07
0.06
0.06
Minimum IAE –
Representative coefficient values (deduced from graphs):
Gallier and Otto
τ
T
x1
x2
x3
τ m Tm
x1
x2
x3
(1968).
m
m
Model:
0.053
12.3
0.95 0.0215 0.67
1.45
0.75
0.155
Method 1
0.11
6.8
0.93
0.041
1.50
0.85
0.63
0.21
0.25
3.4
0.88
0.083
4.0
0.57
0.53
0.19
0.869
Minimum IAE –
T
m
1.086  Tm 
33


Model: Method 5
Rovira et al.
τ
Td
K m  τm 
0.740 − 0.13 m
(1969).
Tm
30
x1 =
x 2 τm 2 − 2(τm Tm ) + 2 (τm Tm )2 + 2τm 2 x 2
2 + (τm Tm )
0. 6
τm
2
≤ x2 ≤
1
τm 2
.
2.2



 Tm  
Tm 
τm 




T
T
0
.
777
0
.
45
,
0
.
31
0
.
6
,
+
=
+
T
T
0
.
44
0
.
56
=
−

 i
 d
m
m
τ  .
Tm 
τm 



 m 

31
Kc =
1
Km
32
Kc =
ξ cos tan −1 (1.57 D R − (0.16 D R ))
.
K m cos tan −1 (6.28Tm Tu )
33
; recommended
[
[
τ 
Td = 0.348Tm  m 
 Tm 
]
0.914
, 0.1 <
]
τm
≤1.
Tm
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Minimum IAE –
Marlin (1995),
pp. 301–307.
Model:
Method 1
Minimum IAE –
Sadeghi and
Tych (2003).
Minimum IAE –
Arrieta Orozco
(2006).
Wang et al.
(1995a).
Kc
Ti
Td
Comment
1.3 K m
0.93τ m
0.02τ m
τ m Tm = 0.25
1.0 K m
0.80τ m
0.05τ m
τ m Tm = 0.43
0.8 K m
0.71τ m
0.06τ m
τ m Tm = 0.67
0.7 K m
0.73τ m
0.09τ m
τ m Tm = 1.0
0.55 K m
0.64τ m
0.12τ m
τ m Tm = 1.5
0.48 K m
0.56τ m
0.15τ m
τ m Tm = 2.33
0.4 K m
0.52τ m
0.21τ m
τ m Tm = 4.0
91
Coefficients of K c , Ti , Td estimated from graphs; robust to ±25%
process model parameter changes.
Model: Method 1
1.45933τ m
34
Ti
τ
τm
Kc
0.1 ≤ m ≤ 1.0 .
+ 3.27873
T
Tm
m
35
Kc
Ti
Td
pp. 23–24. Model: Method 10
36
Kc
Tm + 0.5τ m
0.5Tm τ m
Tm + 0.5τ m
0.05 < τm Tm < 6;
Model: Method 1
34
Kc =


1 
T 
τ
 0.26266 + 0.82714 m  , Ti = Tm  0.28743 m + 1.35955  .


K m 
τm 
T
m


35
Kc =
0.9971 

τ
T 
1 
 , Ti = Tm 0.9781 + 0.3723 m
0.3295 + 0.7182 m 
T

Km 

 τm 
 m







0.8456 
τ
Td = 0.3416Tm  m
 Tm
,






0.9414
, 0.1 ≤
τm
≤ 2.0 .
Tm

x2 
 x1 +
(Tm + 0.5τm )


τ
m Tm 

36
, x1 = 0.7645 , x 2 = 0.6032 , Minimum IAE;
Kc =
K m (Tm + τm )
x1 = 0.9155 , x 2 = 0.7524 , Minimum ISE.
b720_Chapter-03
Rule
Kc
1.048  Tm

Minimum ISE –
Zhuang (1992), K m  τ m
Zhuang and
Atherton (1993). 1.154  Tm
K m  τ m
38
39
Ti =
1.195 − 0.368
Tm
1.047 − 0.220
Ti =
0.897
37
0. 1 ≤
Td
Ti
τm
≤ 1.0
Tm
m
Tm
Kc =



Comment
Td
Ti
0.567
τ

1.1 ≤ m ≤ 2.0
38

T
Ti
d
Tm

Model: Method 1 or Method 5 or Method 6 or Method 15.
2 ≤ Am ≤ 5 ,
39
T
T
Kc
i
d
300 ≤ φ ≤ 600 .
Minimum ISE –
Ho et al. (1998).
Model: Method 1
Minimum ISE –
Xing et al.
(2001).
Minimum ISE –
Sadeghi and
Tych (2003).
Ti =
FA
Handbook of PI and PID Controller Tuning Rules
92
37
17-Mar-2009
τm
Tm
τm
Tm
40
Kc
Ti
41
Kc
Ti
1.93576τ m
τm
+ 3.83528
Tm
τ 
, Td = 0.489Tm  m 
 Tm 
0.888
τ 
, Td = 0.490Tm  m 
 Tm 
0.708
0.0821
 Tm 
1.8578 φm


0.9087 

K m Am
 τm 
0.9471
, Td =
Model: Method 1
Td
τm
≤ 1.0 ;
Tm
Model: Method 1
0. 1 ≤
.
.
0.4899Tm φm
A m 0.0845
0.0211Tm [1 + 0.3289A m + 6.4572φm + 25.1914(τm
1 + 0.0625A m − 0.8079φm + 0.347(τm Tm )
1.0264
 τm 


,
T 
 m
τ
Tm )]
, 0.1 ≤ m ≤ 1.0 ;
Tm
0.1457
Given A m , with 2 ≤ A m ≤ 5 and 0.1 ≤ τ m Tm ≤ 1.0 , ISE is minimised when
φ m = 29.7985 +
40
62.1189 40.3182 τ m 76.2833τ m
+
−
(Ho et al., 1999).
Am
Tm
A m Tm
2

 τm  
 τm 

,




Kc =
, Ti = Tm 1.126 + 0.168
+ 0.043
2
Tm 
Tm  

 τm 
 τm 




 − 0.293

0.022 + 1.558
T 

 Tm 
 m
Km
[
]
Td = Tm 0.005 + 0.202(τm Tm ) − 0.016(τm Tm ) .
41
Kc =


1 
T 
τ
 0.40455 + 0.96441 m  , Ti = Tm  0.43770 m + 1.39588  .


K m 
τm 
T
m


2
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Minimum ITAE
0.965  Tm

– Rovira et al.
K m  τ m
(1969).
Minimum ITAE
43
– Wang et al.
Kc
(1995a).
Minimum ITAE
44
– Sadeghi and
Kc
Tych (2003).



Comment
Td
Model: Method 5
Tm + 0.5τ m
0.5Tm τ m
Tm + 0.5τ m
Model: Method 1
Ti
1.55237 τ m
τm
+ 4.00000
Tm
0.85
0.855
Modified
1.2  Tm 
minimum ITAE


K m  τm 
– Cheng and
Hung (1985).
0.855
Modified
0.965  Tm 


minimum ITAE
– Smith (2003). K m  τ m 
Minimum ITAE 1.034Tm 0.8733
– Zhang et al.
K m τm 0.8823
(2005).
42
Td
Ti
42
45
Ti
Ti
1.26Tm
46
Ti
τ 
Tm
Ti =
, Td = 0.308Tm  m 
0.796 − 0.1465(τm Tm )
 Tm 
0.1 ≤ τm Tm ≤ 1;
Model: Method 1
ξ = 0.707;
Td
Model:
Method 14
0.308τm
Model: Method 1
Td
Model: Method 1
0.929
, 0. 1 ≤
τm
≤1.
Tm

0.5307 
 0.7303 +
(Tm + 0.5τm )

τm Tm 
τ

43
, 0.05 < m < 6 .
Kc =
Tm
K m (Tm + τm )


1 
T 
τ
 0.20360 + 0.80451 m  , Ti = Tm  0.29349 m + 1.29110  .




Km 
τm 
Tm


44
Kc =
45
Ti =
46
Ti = 1.6013Tm
τ 
Tm
, Td = 0.308Tm  m 
0.796 − 0.147(τm Tm )
 Tm 
0.9011
τm
0.0884
0.929
.
, Td = 0.3088Tm 0.1131τm 0.9047 , 0.02 ≤ τm Tm ≤ 4 .
93
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
94
Rule
Kc
Minimum ITAE
– Arrieta Orozco
(2006),
pp. 90–92.
Minimum ITAE
– Barberà (2006).
47
17-Mar-2009
47
Kc
48
Kc
Ti
Td
Comment
Ti
Td
2 ≤ Am ≤ 4 ;
Model:
Method 10
Td
Model: Method 1
1.3167e
−1.1734
τm
Tm
−1.3612 + 0.2192 A m − 0.0342 A m 2 

 τm 
1 
2
,

Kc =
1.5985 − 0.7918A m + 0.1088A m + x1 

Km 
Tm 



2
x1 = 0.7655 + 0.0142A m − 0.0253A m , 0.1 ≤ τm Tm ≤ 2.0 ;
−4.9021+ 4.2515 A m − 0.7519 A m 2 

 τm 
2
,


Ti = Tm − 0.8444 + 1.4211A m − 0.2677 A m + x 2 


Tm 



2
x 2 = 2.3292 − 1.4155A m + 0.2428A m , 0.1 ≤ τm Tm < 1.0 ;
4.9240 −1.8437 A m + 0.1863A m 2 

 τm 
2

,

Ti = Tm 1.3028 + 0.1528A m − 0.0837 A m + x 2 


Tm 



2
x 2 = 0.2535 − 0.1997 A m + 0.0678A m , 1.0 ≤ τm Tm ≤ 2.0 .
(
)
1.0743 − 0.2489 A m + 0.0595 A m 2
2
(
)
−0.7838 +1.1971A m − 0.1797 A m 2
2
Td = Tm 1.1385 − 0.5597 A m + 0.0839A m
Td = Tm 1.1532 − 0.5945A m + 0.0929A m
48
Kc =
[
1
 τm 


T 
 m
 τm 


T 
 m
, 0.1 ≤
K m 2.2132(τm Tm )2 + 1.0802(τm Tm ) + 0.001045
, 1.0 ≤
τm
< 1.0 ;
Tm
τm
≤ 2.0 .
Tm
],
2



 τm 
τm

 0.2371 − 0.2088 τm  .


Td = 2.2132
1
.
0802
0
.
001045
+
+

T
T
T


m
m
 m


b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Minimum ISTSE
– Zhuang (1992),
Zhuang and
Atherton (1993).
Ti
Td
49
Kc
Ti
Td
50
Kc
Ti
Td
95
Comment
0. 1 ≤
τm
≤ 1.0
Tm
1.1 ≤
τm
≤ 2. 0
Tm
Model: Method 1 or Method 5 or Method 6 or Method 15.
51
Model: Method 1
0.509 K u
0.125Tu
Ti
∧
52
0.604 K u
Minimum ISTSE
– Xing et al.
(2001).
53
0.130 T u
Model:
Method 37
Td
Model: Method 1
∧
Ti
Ti
Kc
0.904
0.968  Tm 


Minimum ISTES
– Zhuang (1992), K m  τ m 
0.583
Zhuang and
Atherton (1993). 1.061  Tm 
K m  τ m 
54
Ti
Td
55
Ti
Td
0. 1 ≤
τm
≤ 1.0
Tm
1.1 ≤
τm
≤ 2. 0
Tm
Model: Method 1 or Method 5 or Method 6 or Method 15.
Kc =
1.042  Tm 


K m  τm 
0.897
49
Kc =
1.142  Tm 


K m  τm 
0.579
50
51
Ti = 0.051(3.302K m K u + 1)Tu , 1 ≤ K m K u ≤ 16 , 0.1 ≤ τm Tm ≤ 2.0 .
52
53
τ 
Tm
, Td = 0.385Tm  m 
0.987 − 0.238(τm Tm )
 Tm 
0.906
, Ti =
τ 
Tm
, Td = 0.384Tm  m 
0.919 − 0.172(τm Tm )
 Tm 
0.839
, Ti =
.
.
∧

∧
Ti = 0.04 4.972K m K u + 1 T u , 0.1 ≤ τm Tm ≤ 2.0 .


2

 τm  
 τm 

,




, Ti = Tm 1.154 + 0.193
Kc =
+ 0.042
2
Tm 
Tm  

 τm 
 τm 




 − 0.272

0.019 + 1.439
T 

 Tm 
 m
Km
[
]
Td = Tm 0.003 + 0.236(τm Tm ) − 0.022(τm Tm ) .
54
Ti =
τ 
Tm
, Td = 0.316Tm  m 
0.977 − 0.253(τm Tm )
 Tm 
55
Ti =
τ 
Tm
, Td = 0.315Tm  m 
0.892 − 0.165(τm Tm )
 Tm 
0.892
.
0.832
.
2
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
96
Rule
Kc
Minimum ISTES
– Xing et al.
(2001).
Xing et al.
(2001).
Model: Method 1
Kc =
Ti
Td
Comment
Model: Method 1
56
Kc
Ti
Td
57
Kc
Ti
Td
Kc
Ti
59
Kc
Ti
60
Kc
Ti
Minimise
∞
6 2
∫ t e (t)dt .
0
Nearly minimum
IAE, ISE, ITAE
– Hwang (1995).
Model:
Method 44
56
17-Mar-2009
58
Km
0.028 + 1.254(τm Tm ) − 0.239(τm Tm )2
[
= T [0.002 + 0.274(τ
ε1 < 2.4
0.471K u
K m ωu
2.4 ≤ ε1 < 3
3 ≤ ε1 < 20
,
]
Ti = Tm 1.191 + 0.243(τm Tm ) + 0.042(τm Tm ) ,
Td
57
58
Kc =
m
2
m
]
Tm ) − 0.032(τm Tm ) .
2
2

 



1.329 + 0.413 τm  + 0.027 τm   ,
T
T
,
=
i
m
2
T  
T 

τ 
τ 
 m 
 m

0.030 + 1.036 m  − 0.210 m 
 Tm 
 Tm 
Km
[

0.822 1 − 0.549ωH1τm +
K c =  K H1 −

K m (1 + K H1K m )

2
2
0.112ωH1 τm
] ,
[
2
2

0.786 1 − 0.441ωH1τm + 0.0569ωH1 τm
K c =  K H1 −

K m (1 + K H1K m )

Ti =
60
[
K c (1 + K H1K m )
(
ωH1K m 0.0142 1 + 6.96ωH1τm − 1.77ωH12 τm 2
] ,


(
K c (1 + K H1K m )
ωH1K m 0.0172 1 + 4.62ωH1τm − 0.823ωH12 τm 2

1.28(0.542 )ωH1τ m 1 − 0.986 ε1 + 0.558 ε12
K c =  K H1 −

K m (1 + K H1K m )

Ti =
2


Ti =
59
[
]
Td = Tm 0.014 + 0.246(τm Tm ) − 0.031(τm Tm ) .
] ,


K c (1 + K H1K m )
(
ωH1K m 0.0476(1 + 0.996 ln[ωH1τm ]) 1 + 2.13 ε1 − 1.13 ε12
).
).
).
b720_Chapter-03
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FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
61
Hwang (1995) –
continued.
Kc
Ti
Td
Comment
Ti
0.471K u
K m ωu
ε1 ≥ 20
97
Decay ratio = 0.1; 0.1 ≤ τm Tm ≤ 2.0 .
Nearly minimum
IAE, ITAE –
Hwang and Fang
(1995).
62
Ti
Kc
Model:
Method 30
Td
0.1 ≤ τm Tm ≤ 2.0 ; decay ratio = 0.03.
63
Minimum performance index: other tuning
Ti
Td
Kc
Hwang and Fang
Nearly minimum IAE, ITAE – simultaneous servo/regulator tuning.
(1995).
Decay ratio = 0.05. Model: Method 30
Servo or
regulator tuning
Model:
Tm
– minimum IAE 0.36 + 0.76 τ
Method 46;
0.47Tm τ m
0.47τ m + Tm
m
– Huang and
τm Tm < 0.33
0.47τ m + Tm
K
m
Jeng (2003).
61
[
2
2

1.14 1 − 0.466ωH1τm + 0.0647ωH1 τm
K c =  K H1 −

K m (1 + K H1K m )

Ti =
62
[
] ,


(
]
K c = 0.537 − 0.0165(τm Tm ) + 0.00173(τm Tm ) K u ,
Ti =
2
[
Kc
K c (1 + K H1K m )
ωH1K m 0.0609 − 1 + 1.97ωH1τm − 0.323ωH12 τm 2
K u ωu 0.0503 + 0.163(τm Tm ) − 0.0389(τm Tm )2
],
2

 τ   Ku
τ
Td = 0.350 − 0.0344 m + 0.00644 m  
.
Tm

 Tm   ωu K c

63
[
]
K c = 0.713 − 0.176(τm Tm ) + 0.0513(τm Tm ) K u ,
Ti =
2
[
Kc
K u ωu 0.149 + 0.0556(τm Tm ) − 0.00566(τm Tm )2
],
2

 τm   K u
τm
τm




Td = 0.371 − 0.0274
+ 0.00557
  ω K , 0.1 ≤ T ≤ 2.0 .
Tm
T

m
 m  u c

).
b720_Chapter-03
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FA
Handbook of PI and PID Controller Tuning Rules
98
Rule
Kc
Huang and Jeng
(2003) –
continued.
64
Kc
Ti
Td
Comment
Ti
Td
τm Tm ≥ 0.33
x 1 τ m + x 2 Tm x 3 τ m + x 4 Tm
x 6 Tm τ m
τm
Åström and
τ m + α 5 Tm
K mτm
τ
m + x 7 Tm
Hägglund (2004).
Model:
x1
x2
x3
x4
x5
x6
x7
Method 1
0.057 0.139 0.400 0.923 0.012 1.59 4.59
Coefficient
values given.
M max = 1.1
0.103 0.261 0.389 0.930 0.040
1.62
4.44
M max = 1.2
0.139 0.367 0.376 0.900 0.074
1.66
4.39
M max = 1.3
0.168 0.460 0.363 0.871 0.111
1.70
4.37
M max = 1.4
0.191 0.543 0.352 0.844 0.146
1.74
4.35
M max = 1.5
0.211 0.616 0.342 0.820 0.179
1.78
4.34
M max = 1.6
0.227 0.681 0.334 0.799 0.209
1.81
4.33
M max = 1.7
0.241 0.740 0.326 0.781 0.238
1.84
4.32
M max = 1.8
0.254 0.793 0.320 0.764 0.264
1.87
4.31
M max = 1.9
0.264 0.841 0.314 0.751 0.288
1.89
4.30
M max = 2.0
Direct synthesis: time domain criteria
Tm τ m
T 
1 
 0.5 + m 
Van der Grinten
Step disturbance.
Tm + 0.5τ m
τ
m + 2Tm
Km 
τm 
(1963).
65
Stochastic
Model: Method 1
Ti
Td
Kc
disturbance.
64
Kc =
0.8194Tm + 0.2773τm
K m τm 0.9738Tm 0.0262
, Ti = 1.0297Tm + 0.3484τm ,
 0.4575Tm + 0.0302τm 
τm .
Td = 

 1.0297Tm + 0.3484τm 
65
Kc =
e − ωd τ m
Km


T
1 − 0.5e − ωd τ m + m e − ωd τ m  ,


τm


Ti =
e
τm
− ωd τ m
(
)


1 − 0.5e− ωd τ m Tm
T
1 − 0.5e− ωd τ m + m e− ωd τ m  , Td =
.


τm
1 − 0.5e − ωd τ m + (Tm τm )e − ωd τ m


b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Td
Ti
99
Comment
66
Regulator –
Ti
Td
τm Tm < 2
Kc
Gorecki et al.
(1989).
Pole is real and has maximum attainable multiplicity.
Model: Method 1
Saito et al.
0.315Tm τ m
τm Tm < 1
(1990).
0.215τ m + Tm
0.315τ m + Tm
0.315τ m + Tm
Model:
1.37 K m τ m
67
1 ≤ τm Tm < 10
Td
Method 22
68
Suyama (1992).
OS=10%
Tm + 0.309 τ m
Td
Kc
Model: Method 1
Fruehauf et al. 0.56Tm K m τm
≤ 0.5τ m
5τ m
τm Tm < 0.33
(1993).
≤ 0.5τ m
0.5Tm K m τm
Tm
τm Tm ≥ 0.33
Model: Method 2
69
Nomura et al.
‘Kitamori’.
Tm + 0.315τm
Td
Kc
(1993).
Model:
70
‘Butterworth’.
Tm + 0.3109τm
Td
Kc
Method 1;
τ
0.01 ≤ m ≤ 10 .
71
Minimum ITAE.
Tm
Tm + 0.3610τm
Td
Kc
66
2Tm
K m τm
Kc =
Ti = τm
τm
2

  x1 − 3− 2 Tm
 

6 + τm x1 − 9 −  τm  −  τm   e
,
 2T   2T  
2Tm

 m  m 

[21 + 3(τ
Td = 0.5τm
m
[6 + (τm 2Tm )]x1 − 9 − (τm 2Tm ) − (τm 2Tm )2
Tm ) + (τm 2Tm )2 ]x1 − 36 − 4.5(τm Tm ) − 1.5(τm Tm )2 − (τm
[6 + (τm
x1 − 1
2Tm )]x1 − 9 − 0.5(τm Tm ) − (τm 2Tm )2
 τ
, x 1 = 3 +  m
 2Tm
2
67
Td =
0.315τm Tm + 0.003τm
.
0.315τm + Tm
68
Kc =

2.236Tm τm
1 
Tm
.
+ 0.2236 , Td =
0.7236
7.236Tm + 2.236τm
Km 
τm

69
Kc =
0.0027 τm + 0.315Tm
0.73 
T 
 0.315 + m  , Td =
τm .
τm 
0.315τm + Tm
K m 
70
Kc =
0.0082τm + 0.3109Tm
0.73 
T 
 0.3109 + m  , Td =
τm .


τm 
0.3109τm + Tm
Km 
71
Kc =
0.0690τm + 0.3610Tm
0.73 
T 
 0.3610 + m  , Td =
τm .
τm 
0.3610τm + Tm
K m 
2Tm )3
2

 .

,
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
100
Rule
Ti
Td
Comment
Tm + 0.4860τm
Td
‘Binomial’.
Kc
Nomura et al.
(1993) –
continued.
Gonzalez (1994),
p. 119.
Model: Method 1
72
Kc
73
Kc
Juang and Wang
(1995).
74
Kc
Ti
Td
Abbas (1997).
75
Kc
Tm + 0.5τ m
Tm τ m
2Tm + τ m
Model: Method 1
4Tm τ m
Tm + τ m
Tm τ m
Tm + τ m
Model: Method 1
Camacho et al.
(1997).
72
Kc =
73
Kc =
Kc =
Tm + τm
K mTm τm
Tm Ti x 2
(
2
Kc =
1 − 4 x1
m
4
]
x1 = 0.2 ;
x 3 = 0.21
(p. 121).
Model: Method 1
TCL = x1Tm .
− x 3Tm + x 32Tm 2 + Tm (τm − Ti ) − x1Ti 2
2
K m 1 + x 2 x1Ti
2
)
, x2 =
Tm (τm − Ti ) − x1Ti 2
1
1 + (2π log e [b a ])
2
,
x1 + (τm Tm ) + 0.5(τm Tm )2
K m (x1 + (τm Tm ))
2
b
= desired closed loop response decay ratio.
a
x1 + (τm Tm ) + 0.5(τm Tm )
,
x1 + (τm Tm )
2
, Ti = Tm
Td =
75
4x1
] τ [1 −
0.1329τm + 0.4860Tm
0.73 
T 
 0.4860 + m  , Td =
τm .


τm 
0.4860τm + Tm
Km 
with x 3 =
74
[
τ m 1 − 1 − 4 x1
0.5Tm τm 2 (x1 + (τm Tm ) − 0.5x1 (τm Tm ))
(x1 + (τm
(
Tm )) x1 + (τm Tm ) + 0.5(τm Tm )2
1.002
τ
0.177 + 0.348(Tm τm )
, 0 ≤ V ≤ 0. 2 , 0. 1 ≤ m ≤ 5. 0 .
0.713
Tm
K m (0.531 − 0.359V
)
), 0< x
1
<1.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Cluett and Wang
(1997), Wang
and Cluett
(2000).
Model:
Method 1;
τ
0.01 ≤ m ≤ 10 .
Tm
76
Kc =
Kc =
Kc =
Kc
Ti
Td
TCL = 4τ m
77
Kc
Ti
Td
TCL = 2τm
78
Kc
Ti
Td
TCL = 1.33τ m
79
Kc
Ti
Td
TCL = τ m
80
Kc
Ti
Td
TCL = 0.8τ m
81
Kc
Ti
Td
TCL = 0.67τ m
−0.0069905τm + 0.029480Tm
τm , 7.97 ≤ A m ≤ 8.56 , 79.67 0 ≤ φm ≤ 79.700 .
0.029773τm + 0.29907Tm
0.16440τm + 0.99558Tm
0.055548τm + 0.33639Tm
τm ,
, Ti =
K m τm
0.98607 τm − 1.5032.10− 4 Tm
Kc =
Kc =
Kc =
−0.030407τ m + 0.27480Tm
τm , 3.30 ≤ A m ≤ 3.56 , 65.860 ≤ φm ≤ 68.500 .
0.25285τm + 1.0132Tm
0.26502τm + 0.94291Tm
0.16051τm + 0.57109Tm
τm ,
, Ti =
K m τm
0.89868τm + 6.9355.10− 2 Tm
Td =
81
−0.024442τm + 0.17669Tm
τm , 3.81 ≤ A m ≤ 4.16 , 69.840 ≤ φm ≤ 70.700 .
0.17150τm + 0.80740Tm
0.24145τm + 0.96751Tm
0.12786τm + 0.51235Tm
τm ,
, Ti =
K m τm
0.93566τm + 2.2988.10− 2 Tm
Td =
80
−0.016651τm + 0.093641Tm
τm , 4.84 ≤ A m ≤ 5.31 , 73.90 ≤ φm ≤ 74.140 .
0.093905τm + 0.56867Tm
0.20926τm + 0.98518Tm
0.092654τm + 0.43620Tm
τm ,
, Ti =
K m τm
0.96515τm + 4.2550.10− 3 Tm
Td =
79
Comment
76
Td =
78
Td
0.099508τm + 0.99956Tm
0.019952τm + 0.20042Tm
τm ,
, Ti =
K m τm
0.99747 τm − 8.7425.10−5 Tm
Td =
77
Ti
101
−0.035204τm + 0.38823Tm
τm , 3.00 ≤ A m ≤ 3.19 , 60.600 ≤ φm ≤ 67.210 .
0.33303τm + 1.1849Tm
0.19067 τm + 0.61593Tm
0.28242τm + 0.91231Tm
τm ,
, Ti =
K m τm
0.85491τm + 0.15937Tm
Td =
−0.039589τm + 0.51941Tm
τm , 2.80 ≤ A m ≤ 2.95 , 52.350 ≤ φm ≤ 66.450 .
0.40950τm + 1.3228Tm
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
102
Rule
Kc
Valentine and
Chidambaram
(1997a).
Morilla et al.
(2000).
Model:
Method 33
Mann et al.
(2001).
Model:
Method 37 or
Method 44.
82
17-Mar-2009
Ti
Td
Comment
Ti
Td
ξ = 0.707 ;
Model: Method 1
82
Kc
83
τm
x1τm
x 1 = 0. 1 ;
Kc
1 + 1 − 4 x1
1 + 1 − 4 x1
0.2 ≤ x 3 ≤ 0.5 .
84
Kc
Ti
Td
OS < 5%;
0 < τ m Tm < 1 .
85
Kc
Ti
Td
OS < 5%;
1 ≤ τ m Tm < 2 .
2


 τ 
τ  
τ
0.746 − 1.242 ln m  , Ti = Tm 0.3106 + 1.372 m − 0.545 m   ,
Tm


 Tm 
 Tm  

Kc =
1
Km
Td =
Tm
.
16.015Tm − 11.702τm
2
±1% TS = 2.5Tm , 0.1 ≤ τm Tm ≤ 0.5 ; ±1% TS = 3.0Tm , τm Tm = 0.6 ;
±1% TS = 3.75Tm , τm Tm = 0.7; ±1% TS = 4.3Tm , τm Tm = 0.8;
±1% TS = 5.0Tm , τm Tm = 0.9 .
83
Kc =
− x 3Tm + x 32Tm 2 + Tm (τm − Ti ) − x1Ti 2
Ti x 2
, x2 =
K m [2 x 3 + x 2 (τm − Ti )]
Tm (τm − Ti ) − x1Ti 2
with x 3 =
84
Kc =
1
1 + (2π log e [b a ])
2
[0.770 + 0.245(τ
Tm )
0.854
m
K m τm
b
= desired closed loop response decay ratio.
a
,
]T
m
0.854 

τ 
 Tm ,
, Ti = 1.262 + 0.147 m 
Tm 





Td =
85
Kc =
[0.603 + 0.275(T
m
K m τm
τm )
2 .4
]T
m

T
, Ti = 1.1618 + 0.165 m

 τm





0.262 + 0.147(τm Tm )0.854
0.770 + 0.245(τm Tm )0.854
τm .
2.4 
 Tm ,


Td =
0.1618 + 0.165(Tm τm )2.4
0.603 + 0.275(Tm τm )2.4
τm .
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Chen and Seborg
86 K c
(2002).
τ
Chidambaram
−1.763 m
Tm
(2002), p. 155. 4.105 e
Km
Vítečková and
Víteček (2002).
Model: Method 1
86
88
Comment
Ti
Td
Model: Method 1
Td
Model: Method 1
Ti
Ti
Td
For 0.05 < τm Tm < 1.6 , Ti = 1.2Tm ; overshoot is under 2%
(Vítečková and Víteček, 2003).
(
)
(2T τ
m m
)
+ 0.5τm (3TCL + 0.5τm ) − 2TCL − 3TCL τm
,
τm (2Tm + τm )
2
3
2
3TCL τm Tm + 0.5Tm τm (3TCL + 0.5τm ) − 2(Tm + τm )TCL
2
Td =
2
(2T τ
m m
88
Td
1 2Tm τm + 0.5τm 2 (3TCL + 0.5τm ) − 2TCL 3 − 3TCL 2 τm
,
Km
2TCL3 + 3TCL 2 τm + 0.5τm 2 (3TCL + 0.5τm )
Kc =
Ti =
87
Ti
87
Kc
103
)
+ 0.5τm (3TCL + 0.5τm ) − 2TCL − 3TCL τm
2
3
2
3
T s(1 + 0.5τms )e −sτ m
 y
= i
;  
.
K c (TCLs + 1)3
 d desired
2

τ  
Tm
τ
Ti = Tm 0.311 + 1.372 m − 0.545 m   , Td =
.
Tm
16.016 − 11.702(τm Tm )

 Tm  

Kc =
[
(
]
)
eτ m x1
3
0.5
2
3
2
2
τm Tm x1 + 3τm Tm + τm x1 + τm x1 − 1 , x 1 = −
−
+
τ m Tm
Km
(
)
 τ 2T x 3 + 3τ T + τm 2 x12 + τm x1 − 1 
Ti = −2  m m 13 2 m m
,
2
x1 τm Tm x1 + 2τm Tm + τm


(
)
(
3
τm2
+
)
)
0.25
Tm 2
;
 τ 2T x 2 + 4τm Tm + τm 2 x1 + 2τm + 2Tm 
Td = −0.5 m 2 m 1 3
.
2
2
 τm Tm x1 + 3τm Tm + τm x1 + τm x1 − 1 
(
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
104
Rule
Kc
Gorez (2003). (1 − x1 )τm + Tm
Model: Method 1
x 2 K m τm
Sree et al.
(2004).
Model: Method 1
Kukal (2006).
89
17-Mar-2009
For 0 ≤
Ti
Comment
Td
(1 − x1 )τm + Tm
(1 − x1 )τmTm
(1 − x1 )τm + Tm
90
Kc
Tm + 0.5τ m
Td
91
Kc
Ti
Td
0.1 ≤ τm Tm ≤ 1
92
Kc
Ti
Td
Model: Method 1
τm
τ T + Tm 2 + 0.5τm 2
,
≤ τ m , x1 = m m
τ m + Tm
(τm + Tm )2
x2 =
For τ m ≤
1
(1.3M s 2 − 1) 1.56τm 2 (0.5τm + 3Tm )
+
;
2M s (M s − 1) 2M s (M s − 1)
(τm + Tm )3
τm
0.65M s
τ T + Tm 2 + 0.5τm 2
, x2 =
≤ 1 , x1 = m m
.
τ m + Tm
Ms − 1
(τm + Tm )2
90
Kc =

0.5τm (Tm + 0.1667 τm )
1  Tm

+ 0.5  , Td =
.
Tm + 0.5τm
K m  τm

91
Kc =
1.377  Tm 


K m  τm 
92
Kc =
2


12(Tm τm )2 + 1 − 1  x1
T 
1 
T
6 12 m  + 1 − 18 m − 1 +
e ,

Km 
2Tm τm
τm
 τm 


x1 =
89
0.8422
τ 
, Ti = 1.085Tm  m 
 Tm 
12(Tm τm )2 + 1 − 1
2(Tm τm )
− 3 ; Ti = τm
4
0.4777


τ
, Td = Tm 0.3899 m + 0.0195 .
Tm


x 2 + x 3 12(Tm τm )2 + 1
3
27(2(Tm τm ) + 1)4
,
2
T 
T 
T 
T
x 2 = 864 m  + 432 m  + 252 m  + 72 m + 5 ,
τm
 τm 
 τm 
 τm 
3
2
T 
T 
T
x 3 = 216 m  + 60 m  + 18 m + 5 ;
τm
τ
τ
 m
 m
Td = τm
3
2
2
  T 2 
T 
T 
T 
T
72 m  + 8 m  + 10 m + 1 + 12 m  + 1 12 m  + 1
τm
  τm 

 τm 
 τm 
 τm 


2
  T 3

 Tm 
Tm
m 





+
+
+
4 108
36
20
5
τ 
τm
  τm 

m 



.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Ti
Td
Comment
1.282Tm
0.321Tm
τm
> 0.368
Tm
Kc
Vítečková
(2006).
Model: Method 9
93
Gorecki et al.
(1989).
94
Kc
Kc
Direct synthesis: frequency domain criteria
Model: Method 1
Ti
Td
Regulator – low frequency part of magnitude Bode diagram is flat.
95
x1Td
Td
Kc
Li et al. (1994).
Model:
Method 48
∧
4hx 2
πA p
93
Kc ≥
1.282Tm
.
K m (2.718τm − Tm )
94
Kc =
1
,
2K m (τm Ti )(1 + (Tm τm )) − 2K m
Ti = τm
Td = τm
95
Kc =
∧
0.318 T u
x 2 given in
footnote 95.
0.080 T u
7 + 42(Tm τm ) + 135(Tm τm )2 + 240(Tm τm )3 + 180(Tm τm )4
,
1 + 7(Tm τ m ) + 27(Tm τm )2 + 60(Tm τm )3 + 60(Tm τm )4
.
[
15[2(Tm τm ) + 1]1 + 3(Tm τm ) + 6(Tm τm )2
]
7 + 42(Tm τm ) + 135(Tm τm )2 + 240(Tm τm )3 + 180(Tm τm )4
4h
πµ
x 2 cos φm
2
Ap − µ
µ
2
+
2
υ + υ2 + (4 x1 )

4 
− 0.5υ + υ2 + 
x

1 

, µ = hysteresis value,
tan φm −  µ A p 2 − µ 2 

4 
2

.
Td = 0.080 T u  υ + υ +  , υ =
2
x1 

2 

1 +  µ Ap − µ 


The values of x 1 and x 2 are given below.
∧
φm
x1
150
2.8
200
3.2
250
3.5
300
3.8
350
4.0
400
4.2
450
4.6
500
4.9
550
5.2
600
5.4
650
5.5
x2
0.5
0.6
0.6
0.6
0.7
0.7
0.8
0.8
0.8
0.9
0.9
105
b720_Chapter-03
Rule
Kc
96
Lee et al. (1992).
Model: Method 1
Kc =
[3.08(T
m
Kc
Ti
Td
Comment
Ti
Td
M s = 1.25
1.1
τ 
x1 = 0.55 + 0.163 m 
 Tm 
97
Somani et al.
(1992).
Model:
Method 1;
suggested A m :
1.75 ≤ A m ≤ 3.0 .
x2
0.0375
0.1912
0.3693
0.5764
0.8183
1.105
1.449
3.57
3.33
3.13
2.94
2.78
2.63
2.50
0.1
x 3τ m
Coefficient values:
x3
τ m Tm
0.71
0.67
0.63
0.59
0.56
0.53
0.50
− 4 + 9.92 x1 (Tm τm )
[
τm
τ
< 2.52 ; x1 = 1 , m ≥ 2.52 .
Tm
Tm
x 2τm
Kc
τ m Tm
τm )
,
0.55
K m 1.54(τm Tm )
0.9
1.869
2.392
3.067
3.968
5.263
7.246
10.870
For
τm
0.7809
< 1 , Kc ≈
Tm
K mAm
+ 4 + 4.96 x1 (τm Tm )
( 0.803 + 4(τ
0.45
4
m
Tm ) + 0.896
2
Kc ≈
x2
x3
2.38
2.27
2.17
2.08
2.00
1.92
1.85
0.48
0.45
0.43
0.42
0.40
0.38
0.37
+ 0.77(4 + 2(τm Tm ))(Tm τm )
0 .1
0.55
 τ
3.08(Tm τ m ) − 4 + 9.92 x1 (Tm τm ) 
Ti = Tm 0.5 m +
,
1 .1
1.54(4 + 2(τm Tm ))(Tm τm )
 Tm

Tm
Td =
.
2Tm
1.54(Tm τm )1.1 (4 + 2(τm Tm ))
+
0.1
0.55
τm
3.08(Tm τm ) − 4 + 9.92 x1 (Tm τm )
97
FA
Handbook of PI and PID Controller Tuning Rules
106
96
17-Mar-2009
)
2
]
0.1
],
+ 1 or
T 
T
0.7809
0.7809
1 + 0.803 m  ; K c ≈
1 + 0.455 m
KmAm
KmAm
 τm 
 τm
2

τ
 , m >1.

T
m

b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Minimum ISTSE
– Zhuang and
Atherton (1993).
Model: Method 1
Chidambaram
(1995a).
98
98
99
mK u cos(φ m )
Ti
Td
Comment
Ti
Ti
x1
A m = 2,
φ m = 60 0 .
Ti
Td
Model: Method 1
mK u cos(φ m )
100
Kc
(
tan (φm ) +
x1 = 0.413(3.302K m K u + 1) , Ti = x1
99
100
)
m = 0.614(1 − 0.233e −0.347 K m K u ) , φ m = 33.8 0 1 − 0.97e −0.45K m K u , 0.1 ≤ τm Tm ≤ 2.0 ,
4
+ tan 2 (φm )
x1
2ωu
, Model: Method 1.
∧
∧
∧


m = 0.613(1 − 0.262e −0.44 K m K u ) , φ m = 33.2 0 1 − 1.38e −0.68 K m K u  , x1 = 1.687 K m K u ,


0.1 ≤ τm Tm ≤ 2.0 , Model: Method 37.
Kc ≈
Ti ≈
Td ≈

1 
K mAm 


1.17 


(

1.17 


(
2

 or 0.85 1 + 4.45 Tm  ;
+
0
.
85
2
τ 
 KmAm
 m
4.45 + 4(τm Tm ) − 2.11

5Tm
5Tm
or
;
2
2

 Tm 
3.42
 − 0.15
4.45
+ 0.85 − 1
2
 τm 

4.45 + 4(τm Tm ) − 2.11

0.8Tm
0.8Tm
or
.
2
2

 Tm 
3.42
 − 0.15
4.45
+ 0.85 − 1
2
 τm 

4.45 + 4(τm Tm ) − 2.11

(
3.42
)
)
)
107
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
108
Rule
Chidambaram
(1995a) –
continued.
Kc
Ti
Td
Comment
x1 K m A m
x2τm
x 3τ m
Alternative.
τm Tm
x1
0.062
0.083
0.150
0.196
0.290
0.339
1.20Tm
29.770
22.100
12.470
9.670
6.686
5.784
K m τm
x2
2.34
0.37
2.33
0.37
2.29
0.37
2.27
0.36
2.23
0.36
2.21
0.35
2.4 τ m
Chidambaram
(1995a).
1.03Tm K m τm
Model: Method 1
0.90Tm K m τm
101
Åström and
Hägglund (1995),
pp. 208–210.
τ
0.14 ≤ m ≤ 5.5.
Tm
102
Coefficient values:
x3
τm Tm
Kc
x2
x3
2.19
2.14
2.00
1.72
1.67
0.35
0.34
0.32
0.28
0.27
A m = 1.5
2.4 τ m
0.38τ m
A m = 1.75
2.4 τ m
0.38τ m
A m = 2.0
Ti
Td
M s = 1.4
Td
Ti
Ti
Kc
x1
5.104
3.781
2.262
1.209
1.105
0.38τ m
103
104
0.389
0.543
1.03
2.92
3.69
105
Td
M s = 2.0
Td
Ti
Model: Method 5 or Method 17.
Ku
cos φ m
Am
Tan et al. (1996).
Model:
Method 36
Kφ
107
Am
101
x1Td
106
Td
Ti
x1 chosen
arbitrarily.
Td
Arbitrary A m ,
φ m at ωφ .


1
Tuning rules converted from formulae developed for G c (s) = K c  [1 − α ] +
+ Tds  .
T
s
i


2
2
2
1.52e −8.22 τ +10.1τ Tm
, Ti = 2.08τme −2.32 τ +1.4 τ , Td = 2.23τm e −0.55τ − 6.9 τ .
K m τm
102
Kc =
103
Ti = 0.184Tm e 2.98τ + 0.7 τ , Td = 0.193Tm e 4.82 τ − 7.6 τ .
104
Kc =
105
Ti = 0.06Tme 4.45τ −1.549 τ , Td = 0.345Tm e −2.75τ −1.151τ .
106
Td = Tu
2
2
2
107
2
2
1.85e −8.95 τ + 9.851τ Tm
, Ti = 0.704τm e −0.85τ − 0.879 τ , Td = 3.91τme −2.55τ − 0.491τ .
K m τm
2
Ti =
tan φm +
(4 x1 ) + tan 2 φm
4π
(
r2 K φ ωu 2 − ωφ 2
ωu ωφ
2
2
2
2
)
K u − r2 K φ
2
, Td =
; recommended A m = 2 , φ m = 450 .
ωu K u 2 − r2 2 K φ 2
(
r2 K φ ωu 2 − ωφ 2
)
, r2 = 0.1 + 0.9
Ku
.
Kφ
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Friman and
Waller (1997).
Model:
Method 1;
Am > 2
Comment
Kc
Ti
Td
0.25K u
1.1879Tu
0.2970Tu
2.6064
ω1500
0.6516
ω1500
0.4830
(
G p jω1500
)
109
τ m Tm < 0.25 ,
φ m > 60 0 .
τm
≤ 2.0,
Tm
0.25 ≤
φ m > 450 .
Branica et al.
(2002).
Model:
Method 1;
M s = 1.8
108
Kc
Ti
Td
109
Kc
Ti
Td
Kc
Ti
Td
110
111
Gorez (2003).
Model: Method 1
Åström and
Hägglund (2006),
pp. 261–262.
108
109
110
111
112
Kc =
x 2 − x1
x 2K m
113
Kc
(1)
Ti
(x 2 − x1 )
(1)
Td
T
1 
T 
 0.021 + 0.788 m  , Ti = 1.534τ m  m
τ
τm 
K m 
 m




0.326
T 
1 
T 
 0.292 + 0.671 m  , Ti = 1.239τm  m 
Kc =
τ 
τm 
K m 
 m
0.554
T 
1 
T 
 0.338 + 0.623 m  , Ti = 1.228τm  m 


τ 
τm 
Km 
 m
0.480
Kc =
τm
≤ 0.5
Tm
0.5 <
τm
≤ 1.1
Tm
1.1 <
τm
≤ 2.0
Tm
Tm
(1)
112
Model:
Method 5;
M s = M t = 1. 4 .
T
, Td = 0.233τ m  m
 τm
T 
, Td = 0.230τm  m 
 τm 
T 
, Td = 0.231τm  m 
 τm 




0.055
.
0.069
.
0.120
.


1
Tuning rules converted from formulae developed for G c (s) = K c  [1 − α ] +
+ Tds  .
T
s
i


For 0 ≤
τm
τ T + Tm 2 + 0.5τm 2
≤ τ m , x1 = m m
τm + Tm
(τm + Tm )2
x2 =
For τ m ≤
113
(1 − x1 )
(x 2 − x1 )τm
0.1 ≤
K c (1) =
1
(1.3M s 2 − 1) 1.56τm 2 (0.5τm + 3Tm )
+
;
2M s (M s − 1) 2M s (M s − 1)
(τm + Tm )3
τm
0.65M s
τ T + Tm 2 + 0.5τm 2
.
≤ 1 , x1 = m m
, x2 =
2
τ m + Tm
Ms − 1
(τm + Tm )
1
Km

0.4τm + 0.8Tm
0.5Tm τm
Tm 
(1)
(1)
τm , Td =
.
0.2 + 0.45  , Ti =
τ
0
.
1
T
0
.3τm + Tm
τ
+
m
m
m 

b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
110
Rule
Kc
Åström and
Hägglund (2006),
pp. 261–262.
114
Kc
Ti
( 2)
Ti
Comment
Td
( 2)
Td
τm Tm > 0.11
( 2)
Model: Method 5; detuned ‘AMIGO’ tuning rule.
(5)
(5)
τm Tm < 0.11
115
Ti
Td
Kc
( 5)
114
Kc
( 2)
= Kc
(3)
( 2)
K c Td
+
( 2)
(1)
K c Td (1)
(K
(1)
− Kc
c
( 4)
), K
( 3)
c
= x1K c
( 4)
, x1 = 0.5,0.1 or 0.01;
K c ( 4) =
τmTm  Tm
0.15 
+ 0.35 −
;

K m 
(τm + Tm )2  K m τm
Kc
Ti
( 2)
=
Kc
Kc
( 3)
( 2)
+
Ti (3)
Ti
115
( 4)
Kc
(5)
K c Td
2
Tm 2 + 12Tm τm + 7 τm 2
K c Td
+
 K c (1) K c ( 4) 

− ( 4) 
 T (1)
Ti 
 i
13τm Tm
(5)
( 6)
( 2)
K c (1) Td (1)
= 0.35τm +
= Kc
( 2)
(5)
K c (1)Td (1)
(K
(1)
c
− Kc
(7)
; Td
, Ti
( 3)
( 2)
=
), K
K c(7) ≥
= Ti
(6)
Ti
Kc
( 4)
K c K m + 1 − Ms
0.1K c Td
(7)
( 4)
( 3)
(1)
or Td
K c ( 2)
= x1K c
K m + 1 − Ms
Kc
( 4)
(1)
c
( 4)
( 3)
( 2)
Ti
(5)
=
Kc
( 6)
Ti ( 6 )
Td
( 5)
(5)
+
K c Td
K c (1) Td (1)
(1)
=
0.1K c Td
Kc
(5)
(1)
 K c (1) K c ( 7 ) 

− ( 4) 
 T (1)
Ti 
 i
or Td (5) =
,
(1)
=
0.01K c Td
(1)
.
K c ( 2)
2K c ( 4 ) (τm + Tm )
1
+
−1 ;
2
( 4)
( 4)
−1
M


s
M s  M s + M s − 1 Ti 1 − M s + K c K m


(5)
(5)
−1
, x1 = 0.5,0.1 or 0.01;
(
K c ( 4)
Kc
−1
, Ti ( 6) = Ti ( 4)
0.01K c (1) Td (1)
K c ( 5)
.
Kc
( 6)
Ti ( 4 )
)
K m + 1 − M s −1
K c ( 4 ) K c ( 6 ) K m + 1 − M s −1
;
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Åström and
Hägglund (2006),
pp. 261–262 –
continued.
116
Kc
( 8)
= Kc
K c (10) <
Ti
(8 )
=
(9)
116
+
Kc
Ti
( 8)
K c Td
(8 )
(8 )
(1)
(1)
K c Td
Ti
(K
(1)
c
− Kc
(10 )
Td
(8 )
Td
), K
(9 )
c
= x1K c
(
, x1 = 0.5,0.1 or 0.01;
Kc
(9)
(9)
K c (8 )
,
( 8) (8 )
(1)
(10 ) 
K T K
K
+ c (1) d (1)  c(1) − c( 4) 
K c Td  Ti
Ti 
(1)
=
τm Tm < 0.11
)
Ti (9) = 2K c (9) K m
(8 )
Comment
2K c ( 4 ) (τm + Tm )
1
+
−1 ;
2
( 4)
( 4)
−1
M


s
M s  M s + M s − 1 Ti 1 − M s + K c K m


Ti
Td
(10 )
(8 )
111
0.1K c Td
Kc
( 8)
(1)
or Td
( 8)
(1)
=
0.01K c Td
Kc
(8 )
(1)
.
τm + Tm
;
2
2

M s  M s + M s + 1  K c (9) K m + 1 − M s −1


(
)
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
112
Rule
Kc
King (2006).
Model: Method 1
117
Kc
Ti
Td
Ti
Td
Comment
0.2 ≤
τm
≤ 10 ;
Tm
1.4 ≤ M s ≤ 1.9 .
Robust
Rivera et al.
Tm + 0.5τ m
(1986).
K m (λ + 0.5τ m )
Model: Method 1
Brambilla et al.
(1990).
Model: Method 1
117
118
Tm + 0.5τ m
λ > 0.1Tm ,
Tm τ m
2Tm + τ m
λ ≥ 0.8τ m .
0.5τ m ≤ λ ≤ 3τ m (Zou et al., 1997).
Tm + 0.5τ m
K m (λ + τ m )
Tm + 0.5τ m
Tm τ m
2Tm + τ m
0.1 ≤
τm
≤ 10
Tm
No model uncertainty: λ ≈ 0.35τ m .
4
3
2



 τm  
 τm 
 τm   
1  
  + x 2 log10 



Kc =
x1 log10 

 T  + x 3 log10  T   
K m  
 Tm  
 m 
 m   



+

 τm 
1 
 + x 5  ,
x 4 log10 

K m 
T

 m
x1 = −0.724M s 2 + 2.767 M s − 1.987 , x 2 = 0.329M s 2 − 2.513M s + 1.707 ,
x 3 = −0.509M s 2 + 3.697 M s − 2.9131 , x 4 = 1.166M s 2 − 5.389M s + 4.264 ,
x 5 = −0.753M s 2 + 3.216M s − 2.407 ;
  τ 2

τ 
Ti = Tm  x 6  m  + x 7  m  + x 8  , x 6 = −0.0486M s 2 + 0.1603M s − 0.1327 ,
  Tm 

 Tm 


x 7 = 0.2061M s 2 − 0.7414M s + 0.9609 , x 8 = −0.1759M s 2 + 0.5994M s + 0.4937 ;
2

  τ 3
 τm 
 τm 
m 





 + x12  ,
+
+
Td = Tm x 9 
x
x
10
11
T 
T 
  Tm 

 m
 m


x 9 = −0.0165M s 2 + 0.0518M s − 0.0387 , x10 = 0.1733M s 2 − 0.5468M s + 0.3938 ,
x11 = −0.2998M s 2 + 0.7705M s − 0.1951 , x 12 = −0.0326M s 2 + 0.1475M s − 0.1426 .
118
0.1Tm ≤ λ ≤ 0.5Tm , τm Tm ≤ 0.25 ;
λ = 1.5(τm + Tm ),0.25 < τm Tm ≤ 0.75 ; λ = 3(τm + Tm ), τm Tm > 0.75 (Leva, 2001).
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Lee et al. (1998).
Model: Method 1
Ti
Ti
119
K m (λ + τ m )
Comment
Td
Td
Ti
TCL = λ ; recommended TCL = 0.5τm (Lee et al., 2006).
120
Kc
Tm
K m (λ + 0.5τ m )
Gerry (1999).
Model:
Method 11
113
Ti
Td
Tm
0.5τ m
λ = 2τ m (aggressive, less robust tuning);
λ = 2(Tm + τ m ) (more robust tuning).
Gong (2000).
Model: Method 1
121
Kc
Wang (2003),
x1 K m
pp. 16–17.
τm Tm
x1
Model:
0.05
13.2
Method 1.
0.11
5.3
Regulator;
0.18
4.3
x1 , x 2 , x 3
0.25
3.3
deduced from
2.6
graphs; robust to 0.33
0.43
2.0
±25%
0.54
1.6
uncertainty in
0.67
1.4
process model
0.82
1.3
parameters.
1.0
1.1
119
Ti = Tm +
120
Kc =
x 2 (τm + Tm )
0.3866τm
τ
1 + 0.3866 m
Tm
x 3 (τm + Tm )
x2
x3
τm Tm
x1
x2
x3
0.18
0.33
0.43
0.49
0.51
0.56
0.59
0.61
0.61
0.61
0
0
0
0
0.044
0.074
0.091
0.106
0.120
0.124
1.2
1.5
1.9
2.3
3.0
4.0
5.7
9.0
19.0
0.9
0.8
0.7
0.6
0.5
0.5
0.4
0.4
0.4
0.60
0.59
0.57
0.56
0.51
0.50
0.47
0.46
0.46
0.136
0.139
0.144
0.146
0.147
0.147
0.148
0.149
0.149
τm 2
τm 2 
τ 
1 − m  .
, Td =

2(λ + τm )
2(λ + τm )  3Ti 
Ti
T 2 + x1τm − 0.5τm 2
, Ti = Tm + x1 − CL
,
K m (2TCL + τm − x1 )
2TCL + τm − x1
Tm x1 −
Td =
Tm + 0.3866τm
0.167 τm 3 − 0.5x1τm 2
T 2 + x1τm − 0.5τm 2
2TCL + τm − x1
− CL
,
Ti
2TCL + τm − x1
2
τm
 T  −
x1 = Tm − Tm 1 − CL  e Tm .
Tm 

121
Kc =
τ 
0.7556Tm 
1 + 0.3866 m  .
K m τm 
Tm 
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Handbook of PI and PID Controller Tuning Rules
114
Rule
Kc
Ti
x 2 (τm + Tm )
Wang (2003),
x1 K m
pp. 15–16.
τm Tm
x1
Model:
0.05
2.89
Method 1.
0.11
2.61
Servo; ±10%
2.28
overshoot, step 0.18
0.25
1.96
input; x1 , x 2 , x 3
0.33
1.58
deduced from
1.40
graphs; robust to 0.43
0.54
1.19
±25%
0.67
1.04
uncertainty in
0.82
0.90
process model
1.0
0.85
parameters.
Shamsuzzoha et
122
al. (2006c).
Kc
Model: Method 1
122
FA
Kc =
x 3 (τm + Tm )
x2
x3
τm Tm
x1
x2
x3
0.81
0.86
0.86
0.87
0.85
0.84
0.78
0.77
0.73
0.74
0
0
0
0.034
0.043
0.066
0.075
0.094
0.100
0.120
1.2
1.5
1.9
2.3
3.0
4.0
5.7
9.0
19.0
0.72
0.68
0.56
0.50
0.43
0.38
0.34
0.32
0.29
0.68
0.65
0.60
0.55
0.52
0.48
0.44
0.44
0.40
0.130
0.142
0.149
0.140
0.137
0.136
0.122
0.116
0.096
Ti
Td
Ti
λ2 + x1τm − 0.5τm 2
, Ti = Tm + x1 −
,
K m (2λξ + τm − x1 )
2λξ + τm − x1
Tm x1 −
Td =
Comment
Td
0.167 τm 3 − 0.5x1τm 2
λ2 + x1τm − 0.5τm 2
2λξ + τm − x1
,
Ti
2λξ + τm − x1
(
)

λ2 − 2λξTm + Tm 2 e − τ m
x1 = Tm 1 −
2
Tm

Tm

.

λ , ξ not
defined.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Comment
Td
λ = x1Tm
123
Ti
Td
Kc
Shamsuzzoha
Coefficients of x1 (deduced from graph):
and Lee (2007a).
Model: Method 1 M s = 1.6 M s = 1.7 M s = 1.8 M s = 1.9 M s = 2.0
123
Kc =
0.11
0.10
0.08
0.07
0.06
0.20
0.17
0.15
0.14
0.12
τm = 0.1Tm
0.35
0.31
0.27
0.25
0.22
τm = 0.2Tm
0.46
0.41
0.37
0.34
0.30
τm = 0.3Tm
0.55
0.49
0.44
0.40
0.36
τm = 0.4Tm
0.62
0.55
0.50
0.46
0.42
τm = 0.5Tm
0.68
0.60
0.55
0.51
0.47
τm = 0.6Tm
0.72
0.65
0.59
0.55
0.50
τm = 0.7Tm
0.77
0.69
0.63
0.59
0.53
τm = 0.8Tm
0.81
0.72
0.66
0.62
0.56
τm = 0.9Tm
0.84
0.76
0.69
0.65
0.59
τm = 1.0Tm
0.91
0.82
0.75
0.70
0.64
τm = 1.25Tm
τm = 0.05Tm
0.97
0.87
0.80
0.75
0.70
τm = 1.5Tm
undefined
0.91
0.84
0.78
0.73
τm = 1.75Tm
undefined
0.96
0.87
0.81
0.76
τm = 2.0Tm
Ti
3λ2 + 2x 2 τm − 0.5τm 2 − x 2 2
, Ti = Tm + 2 x 2 −
,
K m (3λ + τm − 2x 2 )
3λ + τm − 2 x 2
2Tm x 2 + x 2 2 −
Td =

x 2 = Tm 1 −

λ3 + 0.167 τm3 − x 2 τm 2 + x 2 2 τm
3λ2 + 2x 2 τm − 0.5τm 2 − x 2 2
3λ + τm − 2 x 2
,
Ti
3λ + τm − 2 x 2

λ
1 −
 T
m

3
 −τm
 e


Tm

.


115
b720_Chapter-03
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Handbook of PI and PID Controller Tuning Rules
116
Rule
Kc
Td
Comment
Ti
0.25Ti
Tuning rules
developed from
K u , Tu .
x 2 Tu
x 3 Tu
Ti
Ultimate cycle
McMillan
(1984).
Model: Method 2
or Method 55.
124
Kc
x 1K u
Kuwata (1987).
x 1 , x 2 deduced
from graphs.
x1
x2
x3
τm
Tu
x1
x2
x3
τm
Tu
x1
x2
x3
τm
Tu
0.44 0.80 0.08 0.28 0.37 0.26 0.06 0.39 0.28 0.18 0.03 0.50
0.42 0.43 0.08 0.32 0.33 0.22 0.05 0.43
0.40 0.32 0.07 0.35 0.30 0.18 0.04 0.47
Geng and Geary
τm
0.94Tm
< 0.167
(1993).
2τ m
0.5τ m
Tm
Kmτm
Model: Method 1
Wojsznis and
1.3 ≤ x1 ≤ 1.6 ;
Tm τ m
Tm + 0.5τm
Blevins (1995).
Tm + 0.5τ m
2Tm + τ m
Model:
x1K m τm
Method 45
Perić et al.
^
Recommended
(1997).
x1Td
Td
125
K u cos φ m
Model:
x1 = 4 .
Method 49
Matsuba et al.
τ
0.1 ≤ m ≤ 1.0
126
(1998).
Ti
Td
Kc
Tm
Model: Method 1
127
Wojsznis et al.
Model: Method 1
x1Ti
K u cos φ m
Ti
(1999).
A
m
124
125
126
127




0.65
  T
 

1.415 Tm 
1
 
m

=
τ
+
T
1
Kc =
,

i
m
0.65 

 .
K m τm   T
 
  Tm + τm  
m
 
1 + 

  Tm + τm  

Td =  tan φm +


1.31  Tm 


K m  τm 
0.9
^
T
4
+ tan 2 φm  u .
 4π
x1

2.19  Tm 


≤ Kc ≤
K m  τm 
0.9
T 
, Ti = 1.59τm  m 
 τm 
T
Ti =  tan φ m + 4 x1 + tan 2 φ m  u .

 4πx1
0.097
T 
, Td = 0.40τm  m 
 τm 
0.097
.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Td
Comment
Ti
0.15Ti
Default values.
0.38K u
1.2Tu
0.18Tu
0.27 K u
0.87Tu
0.13Tu
Kc
Wojsznis et al.
(1999) –
continued.
Ti
0.5K u cos φ m
128
φ m = 450 ;
τ m Tm < 0.2 .
Am = 3 ,
φ m = 330 ;
τ m Tm ≈ 0.2 5 .
Wojsznis et al.
(1999).
Model:
Method 45
Gu et al. (2003).
128
129
130
129
Ti
Kc
0.125Ti
0.3 ≤ x1 ≤ 0.4 ;
0.25 ≤ x 2 ≤ 0.4 ;
0.2 ≤
130
Ti
Kc
τm
≤ 0.7 .
Tm
Model: Method 1
Td
Ti = 0.53Tu  tan φm + 0.6 + tan 2 φm  .












0.6
0.6



.
K c = K u x 2 + 0.251.0 − x1 −
,
T
T
x
=
+

i
u
1
T
 
T



−  u − 7.0  
−  u − 7.0  


τ
τ

 
 

1+ e  m
1+ e  m




Representative tuning rule: 0.6K u ≤ K c ≤ K u , Ti = 0.5Tu , Td = 0.125Tu ,
with K u ≈
Tu ≈ 2π
τm 3 + 5τm 2Tm + 12τm Tm 2 + 12Tm3
[
(
K m τm τm 2 + 5τm Tm + 8Tm 2
(
)
,
)
τm K m K u τm τm 2 + 6τm Tm + 12Tm 2 + τm 3 + 6τm 2Tm + 18τm Tm 2 + 24Tm 3
(
12(K m K u + 1) τm 2 + 4τm Tm + 6Tm 2
)
].
117
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Handbook of PI and PID Controller Tuning Rules
118
3.3.3 Ideal controller in series with a first order lag
 1

1
G c (s) = K c 1 +
+ Td s 
 Tf s + 1
 Ti s
R(s)
E(s)
−
+


1
K c 1 +
+ Td s 
Ti s


U(s)
1
1 + Tf s
Y(s)
K m e − sτ
1 + sTm
m
Table 11: PID controller tuning rules – FOLPD model G m (s) =
Rule
Kc
Lim et al. (1985).
Schaedel (1997).
Model:
Method 25
1
2
Kc
Kc
Ti
K m e − sτ m
1 + sTm
Comment
Td
Direct synthesis: time domain criteria
0
Model: Method 1
Tm
Direct synthesis: frequency domain criteria
2
2
Tf = Td N ;
T1 − T2
T2 2 T33
− 2
5
≤ N ≤ 20 .
T1 − Td
T1 T2
2
1
Kc =
TCL 2
Tm
.
, Tf =
2ξTCL 2 + τ m
K m (2ξTCL 2 + τm )
2
Kc =
0.375Ti
Tm
, T1 =
+ τm ,
K m (T1 + (Td N ) − Ti )
1 + 3.45 τam Tm
(
2
T2 = 1.25Tm τ am +
T1 =
0.7Tm
2
Tm − 0.7 τ am
T3 3 =
(
τam
Tm
2
τ
+
0
.
5
τ
,
T
=
0
,
0
<
≤ 0.104 ;
3
m
m
Tm
1 + 3.45 τam Tm
(
0.7Tm
2
+ τ m , T2 = Tm τ am +
)
Tm − 0.7 τ am
)+T
0.952Tm 2 τ am − 0.1
Tm −
2
0.7 τ am
)
+ 0.5τ m 2 ,
a
mτmτm
+
0.35Tm 2 τ m 2
Tm −
0.7 τ am
+ 0.167τ m 3 ,
τ am
> 0.104 .
Tm
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kristiansson and
Lennartson
(2002b).
Kc =
Ti
Td
Comment
Kc
Ti
Td
Model: Method 1
3
0.2 ≤ τm Tm ≤ 5 , 1.65 ≤ M max ≤ 1.85 .
Kristiansson and
Lennartson
(2006).
3
Kc
4
Kc
Ti
Td
Model: Method 1
5
Kc
Ti
Td
Model:
Method 12
0.05 ≤ τm Tm ≤ 2 , 1.65 ≤ M max ≤ 1.85 .
[
]
1.8 0.13 + 0.5(τm Tm ) − 0.04(τm Tm )2
,
τ


0.12
K m  m − 0.05 0.7 +
(τm Tm ) + 0.05 
 Tm

2


 τ  
τ
0.12
Ti = 2Tm 0.13 + 0.5 m − 0.04 m   0.7 +
(τm Tm ) + 0.05 
Tm

 Tm   

2
2
2


 τm  
 τm  
τm
τm








0.5Tm 0.13 + 0.5
− 0.04
0.9Tm 0.13 + 0.5
− 0.04
Tm
Tm  
Tm
Tm  





 ,T =

 .
Td =
f



 τm


0.12
Tm
,25
− 0.05 min 5 + 2
0.7 +

(τm Tm ) + 0.05 
τm



 Tm









0
.
54
T
(
)
0
.
12
+
τ
0
.
12
4
m
m
0.7 +
 , Ti = 0.6(τm + Tm )0.7 +
,
Kc =
τm
τm
τ



+ 0.05 
+
0
.
05
K m τm  m − 0.05 


T
T
m
m
T




 m

Td =
5
Kc =
Ti =
0.15(τm + Tm )


0.12
0.7 +

(
)
τ
T
+
0
.
05
m
m


[
, Tf =
0.081(τm + Tm )2
τ
, 0.7 ≤ m ≤ 4 .
 τm

Tm


T
− 0.05 min 5 + 2 m ,25

T
τ
m
 m



2[1.1(Tm τm ) − 0.1] 0.06 + 0.68(τm Tm ) − 0.12(τm Tm )2
[
K m 0.68 + 0.84(τm Tm ) − 0.25(τm Tm )
[
[0.68 + 0.84(τ
2
2Tm 0.06 + 0.68(τm Tm ) − 0.12(τm Tm )
m
Tm ) − 0.25(τm Tm )2
]
2
],
]
],
2
2

 τ  
τ  
τ
τ
Td = 0.5Tm 0.06 + 0.68 m − 0.12 m   0.68 + 0.84 m − 0.25 m   ,

Tm
Tm
 Tm   
 Tm  

[
]
T 0.06 + 0.68(τm Tm ) − 0.12(τm Tm ) [1.1(Tm τm ) − 0.1]
Tf = m
.
min{5 + 2(Tm τm ),25}
2 2
119
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
120
Rule
Kristiansson and
Lennartson
(2006).
Kc
Ti
Td
Comment
Kc
Ti
Td
Model:
Method 12
6
0.7 < τm Tm < 2 ; 1.65 ≤ M max ≤ 1.8 .
Robust
Tm + 0.5τ m
K m (λ + τ m )
Horn et al.
(1996).
Model: Method 1
Tm + 0.5τ m
Tm τ m
2Tm + τ m
λ = max (0.25τ m ,0.2Tm ) (Luyben, 2001);
Tf =
λτ m
,
2(λ + τ m )
λ ∈ [τ m , Tm ]
λ > 0.25τ m (Bequette,
2003); λ > [0.1Tm ,0.8τm ] (‘aggressive’ tuning), λ > [Tm ,8τm ]
(‘moderate’ tuning), λ > [10Tm ,80τm ] (‘conservative’ tuning) (Cooper,
2006b).
7
H ∞ optimal –
Tm + 0.5τ m
τ m Tm
Kc
τ m + 2Tm
Tan et al.
(1998b).
λ = 2 : ‘fast’ response; λ = 1 : ‘robust’ tuning; λ = 1.5 : recommended.
Model: Method 1
Rivera and Jun
Tm
τmλ
Tf =
;
(2000).
0
Tm
K m (2 τ m + λ )
2τ m + λ
Model: Method 1
λ not specified.
 T

0.6667(Tm + τm )1.1 m − 0.1
0.6667(Tm + τm )
 τm

6
Kc =
, Ti =
,
2
2






τ
τ
τ
τ
0.68 + 0.84 m − 0.25 m  
K m τm 0.68 + 0.84 m − 0.25 m  
T  
Tm
Tm  
Tm



 m 





− 0.1
2

 τm
 , T = 0.1667(τ + T )0.68 + 0.84 τm − 0.25 τm   .
d
m
m
T  
Tm


T 
 m 

9Tm  5 + 2 m 
τm 

(Tm + τ m )2 1.1 Tm
Tf =
7
Kc =

τm
0.265λ + 0.307  Tm


 τ + 0.5  , Tf = 5.314λ + 0.951 .
Km
 m

b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Td
Comment
Kc
Ti
Td
Model: Method 1
Tf = Td .
Tm + 0.5τ m
K m (λ + τ m )
Tm + 0.5τ m
Tm
2Tm + τ m
Lee and Edgar
(2002).
Huang et al.
(2002).
Model: Method 1
8
Tf =
λτ m
2(λ + τ m )
λ = 0.65τ m : minimum ISE; λ = 0.5τ m : overshoot to a step input
≤ 5% , A m = 3 . In general, A m = 2(λ τ m + 1) .
Zhang et al.
(2002b).
9
Vilanova (2006).
Model: Method 1
10
Tang et al.
(2007).
Model: Method 1
121
Kc
Tm + 0.5τ m
Tm τ m
2Tm + τ m
Kc
Tm + 1.75τm
1.72τm Tm
Tm + 1.75τm
τm + 2Tm
K m (4λ + τm )
0.5τm + Tm
Tm τm
τm + 2Tm
Model: Method 1
φm ≥ 450
Tf =
2λ2
4λ + τm
Labelled the H ∞ Smith predictor; 0.3 ≤ λ τm ≤ 0.8 .
τm + 2Tm
2K m (λ + τm )
0.5τm + Tm
Tm τm
τm + 2Tm
Tf =
λτ m
2(λ + τm )
Labelled the H 2 Smith predictor or the Dahlin controller;
0.3 ≤ λ τm ≤ 0.8 .
2Tm + τm
3K m τm
8
Kc =
2

τm 2
1
3Tm − τm
Tm + τm
+ τm
+
K m (λ + τm ) 
2(λ + τm )
6(λ + τm ) 4(λ + τm )2

2
Ti = Tm +
9
Kc =
10
Kc =
Tm τm
τm + 2Tm
Tm + 0.5τm
Recommended
λ = 0.5 ;
Tf = 0.167 τm .

,


2
2
τm
3Tm − τm
τm
3Tm − τm
τm
+ τm
+
, Td = τm
+
.
2
2(λ + τm )
6(λ + τm ) 4(λ + τm )
6(λ + τm ) 4(λ + τm )2
0.1Tm τ m
1  Tm + 0.5τm 
λ2

 , Tf =
.
or Tf =
2λ + 0.5τ m
τ m + 2Tm
K m  2λ + 0.5τm 
0.377(Tm + 1.75τm )
, Tf = 1.72τm .
K m τm
b720_Chapter-03
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FA
Handbook of PI and PID Controller Tuning Rules
122


Td s
1
+
3.3.4 Controller with filtered derivative G c (s) = K c 1 +
 Ti s 1 + s Td

N

R(s)
E(s)
−
+


Td s
1
+
K c 1 +
T

Ti s
1+ s d

N







U(s)
Rule
Kc
Ti
Y(s)
K m e − sτ
1 + sTm
m
Table 12: PID controller tuning rules – FOLPD model G m (s) =






K m e − sτ m
1 + sTm
Comment
Td
Minimum performance index: regulator tuning
Ti
Td
1
Kc
Minimum IAE –
Alfaro Ruiz
Model: Method 10; N = 10.
(2005a).
Minimum IAE –
Ti
Td
2
Kc
Arrieta Orozco
Model: Method 10; N = 10. Also given by Arrieta Orozco and Alfaro
(2003).
Ruiz (2003).
1
Kc =
1.0158 
0.5022 


T 

1 
 , Ti = Tm  − 0.2228 + 1.3009 τm 
,
0.2068 + 1.1597 m 
T 
Km 



 τm 
 m




τ 
Td = 0.3953Tm  m 
 Tm 
2
Kc =
1
Km
0.8469
, 0.05 ≤
τm
≤ 2.0 .
Tm
0.9946 
0.4796 






0.1050 + 1.2432 Tm 
 , Ti = Tm − 0.2512 + 1.3581 τm 
,
τ 
T 




m 
m 







τ 
Td = Tm  − 0.0003 + 0.3838 m 

 Tm 

0.9479 
 , 0.1 ≤ τm ≤ 1.2 .
Tm


b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Td
Comment
Ti
Td
3
τ
Kc
0.1 ≤ m ≤ 1.2
Minimum ITAE
Tm
– Arrieta Orozco
Model:
Method
10
;
N
=
10.
Also
given
by
Arrieta
Orozco
and
Alfaro
(2003).
Ruiz (2003).
Minimum performance index: servo tuning
Tavakoli and
Minimum IAE.
Ti
4
Kc
Tavakoli (2003).
0.0111Tm
Minimum ISE.
Ti
5
Model:
Kc
Method 1;




τ




0.1 ≤ m ≤ 2 ;


T
m
1
0
.
8
 τ m  0.3 +

Tm
 τ  0.06  Minimum ITAE.

m

τm 

τm
K m  τm

N = 10
+ 0.1 
+ 0.04 



 Tm
 Tm

3
1
Kc =
Km
1.0191 
0.4440 






0.1230 + 1.1891 Tm 
 , Ti = Tm  − 0.3173 + 1.4489 τm 
,
τ 
T 




m 
m 






0.9286 

τ 
 (Arrieta Orozco, 2003);
Td = Tm 0.0053 + 0.3695 m 
Tm 






τ 
Td = Tm − 0.0053 + 0.3695 m 

 Tm 

4
5
0.9286 
 (Arrieta Orozco and Alfaro Ruiz, 2003).


τ



 



 0.3 m + 1.2  

1 
1
0.06 
Tm
2



Kc =
, Ti = τm
.
 τm
  τm

K m  τm + 0.2 
+
0
.
08
+
0
.
04



 

 Tm

 Tm
  Tm

τm




+ 0.75 
 0.3


1
1 
Tm


.
, Ti = 2.4τm
Kc =
 τm

K m  τm + 0.05 
+
0
.
4




T
T
m


 m

123
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
124
Rule
Kc
Ti
Comment
Td
Minimum IAE –
Ti
Td
6
Kc
Alfaro Ruiz
Model: Method 10; N = 10.
(2005a).
Minimum IAE –
Ti
Td
7
Kc
Arrieta Orozco
Model: Method 10; N = 10. Also given by Arrieta Orozco and Alfaro
(2003).
Ruiz (2003).
Minimum ITAE
T
Td
8
i
Kc
– Arrieta Orozco
Model: Method 10; N = 10. Also given by Arrieta Orozco and Alfaro
(2003).
Ruiz (2003).
6
Kc =
0.9971 
0.8456 

T 


1 
 , Ti = Tm 0.9781 + 0.3723 τm 
,
0.3295 + 0.7182 m 
T 
τm 
Km 



m 






τ 
Td = 0.3416Tm  m 
 Tm 
7
Kc =
1
Km
0.9414
Kc =
τm
≤ 2.0 .
Tm
0.9597 
1.0092 






0.2268 + 0.8051 Tm 
 , Ti = Tm 1.0068 + 0.3658 τm 
,
τ 
T 




m 
m 







τ 
Td = Tm − 0.0146 + 0.3500 m 

 Tm 

8
, 0.05 ≤
0.8100 
 , 0.1 ≤ τm ≤ 1.2 .
Tm


0.9462 
0.6884 

T 


1 
 , Ti = Tm 0.9581 + 0.3987 τm 
,
0.1749 + 0.8355 m 
T 
τm 
Km 



m 






0.7417 

 τm 

 , 0.1 ≤ τm ≤ 1.2 .


Td = Tm − 0.0169 + 0.3126

Tm
T


 m


b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
K c = x1
Comment
9
12
13
1.473  Tm 


K m  τm 
T τ 
Ti = x1 m  m 
1.115  Tm 
Td
Minimum performance index: servo/regulator tuning
Td
τ
Kc
0.1 ≤ m ≤ 1.0
Ti
T
10
11
m
Kc
Td
Minimum ISE –
Arrieta and
Vilanova (2007).
Model:
Method 10;
N = 10.
9
Ti
125
Kc
Kc
0.970
0.753
τ 
Td = 0.550x1Tm  m 
 Tm 
Td
Ti
14
τm
≤ 2.0
Tm
0.897
+ (1 − x1 )
1.048  Tm

K m  τ m
+ (1 − x1 )
Tm
;
1.195 − 0.368(τm Tm )
0.948



Td
1.1 ≤
;
τ 
+ 0.489(1 − x1 )Tm  m 
 Tm 
0.888
;
x1 = 0.5778 − 0.5753(τm Tm ) + 0.5528(τm Tm ) .
2
10
11
12
Kc =
1.473x1 + 1.048(1 − x1 )  Tm 


τ 
Km
 m
K c = x1
1.524  Tm 


K m  τm 
Tm  τm 


1.130  Tm 
0.735
0.641
τ 
Td = 0.552x1Tm  m 
 Tm 
14
.
τ 
Td = [0.550x1 + 0.489(1 − x1 )]Tm  m 
 Tm 
Ti = x1
13
0.970 x 1 + 0.897 (1− x 1 )
Kc =
+ (1 − x1 )
+ (1 − x1 )
0.851
0.948 x 1 + 0.888(1− x 1 )
.
1.154  Tm

K m  τ m



0.567
Tm
1.047 − 0.220
;
τ 
+ 0.490(1 − x1 )Tm  m 
 Tm 
1.524 x1 + 1.154((1 − x1 )  Tm 


τ 
Km
 m
;
τm
Tm
0.708
T 
; x1 = 0.2382 + 0.2313 m 
 τm 
0.735 x 1 + 0.567 (1− x 1 )
τ 
Td = [0.552 x1 + 0.490(1 − x1 )]Tm  m 
 Tm 
.
0.851x 1 + 0.708(1− x 1 )
.
7.1208
.
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
126
Rule
Kc
Minimum ISTSE
– Arrieta and
Vilanova (2007).
Model:
Method 10;
N = 10.
15
17-Mar-2009
K c = x1
1.468  Tm 


K m  τm 
T τ 
Ti = x1 m  m 
0.942  Tm 
15
Kc
16
Kc
18
Kc
19
Kc
0.970
0.725
τ 
Td = 0.443x1Tm  m 
 Tm 
Ti
Td
Ti
Td
17
Ti
τm
≤ 1.0
Tm
1.1 ≤
τm
≤ 2.0
Tm
Td
0.897
+ (1 − x1 )
1.042  Tm

K m  τ m
+ (1 − x1 )
Tm
;
0.987 − 0.238(τm Tm )
0.939



0.1 ≤
Td
Td
20
Comment
;
τ 
+ 0.385(1 − x1 )Tm  m 
 Tm 
0.906
;
x1 = 0.5778 − 0.5753(τm Tm ) + 0.5528(τm Tm ) .
2
16
17
18
Kc =
1.468x1 + 1.042(1 − x1 )  Tm 


τ 
Km
 m
K c = x1
1.515  Tm 


K m  τm 
Tm  τm 


0.957  Tm 
0.730
0.598
τ 
Td = 0.444x1Tm  m 
 Tm 
20
.
τ 
Td = [0.443x1 + 0.385(1 − x1 )]Tm  m 
 Tm 
Ti = x1
19
0.970 x 1 + 0.897 (1− x 1 )
Kc =
+ (1 − x1 )
+ (1 − x1 )
0.847
0.939 x 1 + 0.906 (1− x 1 )
.
1.142  Tm

K m  τ m



0.579
Tm
0.919 − 0.172
;
τm
Tm
τ 
+ 0.384(1 − x1 )Tm  m 
 Tm 
1.515x1 + 1.142(1 − x1 )  Tm

τ
Km
 m




;
0.839
T 
; x1 = 0.2382 + 0.2313 m 
 τm 
0.730 x 1 + 0.579 (1− x 1 )
τ
Td = [0.440x1 + 0.384(1 − x1 )]Tm  m
 Tm
.




0.847 x 1 + 0.839 (1− x 1 )
.
7.1208
.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Minimum ISTES
– Arrieta and
Vilanova (2007).
Model:
Method 10;
N = 10.
21
K c = x1
1.531  Tm 


K m  τm 
T τ 
Ti = x1 m  m 
0.971  Tm 
21
Kc
22
Kc
24
Kc
25
Kc
0.960
0.746
τ 
Td = 0.413x1Tm  m 
 Tm 
Ti
Td
Ti
Td
23
Ti
26
0.1 ≤
τm
≤ 1.0
Tm
1.1 ≤
τm
≤ 2.0
Tm
Td
0.904
+ (1 − x1 )
0.968  Tm

K m  τ m
+ (1 − x1 )
Tm
;
0.977 − 0.253(τm Tm )
0.933



Comment
Td
Td
127
;
τ 
+ 0.316(1 − x1 )Tm  m 
 Tm 
0.892
;
x1 = 0.5778 − 0.5753(τm Tm ) + 0.5528(τm Tm ) .
2
22
23
24
Kc =
1.531x1 + 0.968(1 − x1 )  Tm 


τ 
Km
 m
K c = x1
1.592  Tm 


K m  τm 
Tm  τm 


0.957  Tm 
0.705
0.597
τ 
Td = 0.414x1Tm  m 
 Tm 
26
.
τ 
Td = [0.413x1 + 0.316(1 − x1 )]Tm  m 
 Tm 
Ti = x1
25
0.960 x 1 + 0.904 (1− x 1 )
Kc =
0.850
.
1.061  Tm

K m  τ m
+ (1 − x1 )
+ (1 − x1 )
0.933 x 1 + 0.892 (1− x 1 )



0.583
Tm
0.892 − 0.165
;
τm
Tm
τ 
+ 0.315(1 − x1 )Tm  m 
 Tm 
1.592 x1 + 1.061(1 − x1 )  Tm 


τ 
Km
 m
T
, x1 = 0.2382 + 0.2313 m
 τm
0.832
.
0.705 x 1 + 0.583(1− x 1 )
τ 
Td = [0.414 x1 + 0.315(1 − x1 )]Tm  m 
 Tm 
.
0.850 x 1 + 0.832 (1− x 1 )
.




7.1208
;
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
128
Rule
Kc
Davydov et al.
(1995).
Model: Method 6
Kuhn (1995).
Model:
Method 17;
N = 5.
Schaedel (1997).
27
17-Mar-2009
Kc =
Ti
Comment
Td
27
Kc
28
Kc
Direct synthesis: time domain criteria
Ti
0.25Ti
ξ =0.9;
0
.
2
≤
τm Tm ≤ 1.
T
0.4T
i
i
1 Km
0.66(τ m + Tm )
0.167(τ m + Tm )
‘Normal tuning’.
2 Km
0.8(τ m + Tm )
0.194(τ m + Tm )
‘Rapid tuning’.
T2 2 T33
−
T1 T2 2
Model:
Method 25
29
2
T1
− T2
T1 − Td
Kc
2


1
τ
, Ti =  0.186 m + 0.532 Tm , N = K m .
T
K m (1.552(τm Tm ) + 0.078)
m




1
τ
, Ti =  0.382 m + 0.338 Tm , N = K m .
T
K m (1.209(τm Tm ) + 0.103)
m


Tm
0.375Ti
29
, T1 =
+ τm ,
Kc =
K m (T1 + (Td N ) − Ti )
1 + 3.45 τam Tm
28
Kc =
(
2
T2 = 1.25Tm τ am +
T1 =
0.7Tm
2
Tm − 0.7 τ am
T3 3 =
N=
(
Tm
τa
τm + 0.5τm 2 , T3 = 0 , 0 < m ≤ 0.104 ;
a
Tm
1 + 3.45 τm Tm
(
0.7Tm
2
+ τ m , T2 = Tm τ am +
)
2
Tm − 0.7 τ am
)+T
0.952Tm 2 τ am − 0.1
Tm − 0.7 τ am
(a + τm − Ti )2 + 0.375 TiTd
4
+ 0.5τ m 2 ,
a
m τm τm +
Td
 a + τm − Ti 
−
+
2


)
0.35Tm 2 τ m 2
Tm − 0.7τ am
+ 0.167τ m 3 ,
τ am
> 0.104 .
Tm
, 0.05 ≤ a ≤ 0.2 , K v = prescribed
Kv
overshoot in the manipulated variable for a step response in the command variable.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Goodwin et al.
(2001).
Latzel (1988).
Model: Method 1
Latzel (1988).
Model: Method 2
30
Kc =
Ti =
Ti
Comment
p. 426;
Model: Method 1
Direct synthesis: frequency domain criteria
0.84Tm
0.067Tm
φm = 450
Ti
Td
Kc
0.84Tm
0.067Tm
φm = 600
33
Kc
0.76Tm
0.13Tm
φm = 450
34
Kc
0.76Tm
0.13Tm
φm = 600
30
Kc
31
Kc
32
(4ξTCL 2 + τm )(τm + 2Tm ) − 4TCL 2 2
(
K m 16ξ2TCL 2 2 + 8ξTCL 2 τm + τm 2
),
(4ξTCL 2 + τm )(τm + 2Tm ) − 4TCL 2 2
(
(
2 16ξ 2TCL 2 2 + 8ξTCL 2 τm + τm 2
2
Td = 16ξ 2TCL 2 + 8ξTCL 2 τm + τm
2
)
[(4ξT
)
CL 2
N=
Td
129
4ξTCL 2 + τm
2TCL 2 2
Td with G CL (s) =
(4ξTCL 2 + τm ) ,
+ τ m )2 τ m Tm − 2TCL 2 2 (4ξTCL 2 + τ m )(τ m + 2Tm ) + 8TCL 2 4
(4ξTCL2 + τ m )3 [(4ξTCL2 + τ m )(τ m + 2Tm ) − 4TCL2 2 ]
1
TCL 2 2 s 2 + 2ξTCL 2 s + 1
31
Kc =
0.66Tm
, N = 8.38; G p = G m
K m (τm − 0.02Tm )
32
Kc =
0.44Tm
, N = 8.38; G p = G m
K m (τm − 0.02Tm )
33
Kc =
K pe p
0.60Tm
, N = 6.19; G p =
.
K m (τm − 0.07Tm )
(1 + sTp1 ) 2
34
Kc =
K pe p
0.40Tm
, N = 6.19; G p =
.
K m (τm − 0.07Tm )
(1 + sTp1 ) 2
− sτ
− sτ
.
],
b720_Chapter-03
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FA
Handbook of PI and PID Controller Tuning Rules
130
Rule
Latzel (1988).
Model: Method 2
Kc
Ti
Td
Comment
35
Kc
0.74Tm
0.14Tm
φm = 450
36
Kc
0.74Tm
0.14Tm
φm = 600
Tm τ m
2Tm + τ m
N = 10.
Robust
Chien (1988).
Model: Method 1
Tm + 0.5τ m
K m (λ + 0.5τ m )
Tm + 0.5τ m
λ ∈ [τm , Tm ] , Chien and Fruehauf (1990);
λ > [0.8τm ,0.2Tm ] , Yu (2006), p. 19.
Morari and
Zafiriou (1989).
Model: Method 1
Gong et al.
(1996).
Maffezzoni and
Rocco (1997).
37
Kc
Tm + 0.5τ m
38
Kc
Tm + 0.3866τ m
39
Kc
Ti
λ > 0.25τ m ,
Tm τ m
2Tm + τ m
λ > 0.2Tm .
3 ≤ N ≤ 10 ;
0.3866Tm τ m
Tm + 0.3866τ m Model: Method 1
Model:
Method 14
Td
− sτ p
35
Kc =
K pe
0.58Tm
, N = 5.83; G p =
K m (τm − 0.08Tm )
(1 + sTp1 )3
36
Kc =
K pe p
0.39Tm
, N = 5.83; G p =
K m (τm − 0.08Tm )
(1 + sTp1 )3
37
Kc =
2T (λ + τ m )
1  Tm + 0.5τm 

, N= m
.
λ (2Tm + τ m )
K m  λ + τm 
38
Kc =
39
Kc =
− sτ
N=
(0.1388 + 0.1247 N )Tm + 0.0482 Nτ m τ .
Tm + 0.3866τm
, λ=
m
0.3866( N − 1)Tm + 0.1495Nτ m
K m (λ + 1.0009τm )
2Tm (τm + x1 ) + τm 2
2K m (τm + x1 )2
2Tm (τ m + x 1 ) − τ m x 1
[
τm
τm 2 [2Tm (τm + x1 ) − τm x1 ]
, Td =
,
2(τm + x1 )
2(τm + x1 ) 2Tm (τm + x1 ) + τm 2
2
, Ti = Tm +
]
[
]
τ
. x 1 = 0.25τ m , uncertainty on τ m ; x1 > 0.1Tm , m < 0.4 ,
Tm
x 1 2Tm (τ m + x 1 ) + τ m 2
uncertainty on the minimum phase dynamics within the control band.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
40
Gong et al.
(1998).
Model:
Method 1;
λ = x1τm ; x1
values deduced
from graph.
Kc =
41
Comment
Ti
Td
N = 3, 5 or 10.
x1
τm Tm
x1
τm Tm
x1
τm Tm
1
2
3
4
1
2
3
4
1
2
3
4
0.46
0.44
0.43
0.41
0.41
0.40
0.39
0.38
0.37
0.36
0.36
0.35
5
6
7
8
5
6
7
8
5
6
7
8
0.41
0.40
9
10
N = 3.
0.38
0.37
9
10
N = 5.
0.35
0.35
9
10
N = 10.
Kc
Ti
Td
42
Kc
Ti
Td
43
Kc
Ti
Td
41
(Tm + 0.3866τm )(λ + τm ) − (0.3866λ − 0.1247τm )τm
K m (λ + τm )2
Ti = Tm + 0.3866τm −
Td =
Td
0.63
0.57
0.51
0.48
0.52
0.48
0.45
0.43
0.42
0.40
0.38
0.38
Kasahara et al.
(1999).
Model:
Method 1;
N = 10;
0 < τm Tm < 1 .
40
Kc
Ti
131
Robust to 20%
change in plant
parameters.
Robust to 30%
change in plant
parameters.
Robust to 40%
change in plant
parameters.
,
0.3866λ − 0.1247 τm
τm ,
λ + τm
0.3866Tm τ m
0.3866λ − 0.1247 τ m
−
τm .
Tm + 0.3866τ m
λ + τm
K c = 0.6K u [0.718 − 0.057(τm Tm )] , Ti = 0.5Tu [0.296 + 0.838(τm Tm )] ,
Td = 0.125Tu [0.635 − 1.091(τm Tm )] .
42
K c = 0.6K u [0.690 − 0.076(τm Tm )] , Ti = 0.5Tu [0.274 + 0.788(τm Tm )] ,
Td = 0.125Tu [0.526 − 0.057(τm Tm )] .
43
K c = 0.6K u [0.331 + 0.149(τm Tm )] , Ti = 0.5Tu [0.041 + 0.857(τm Tm )] ,
Td = 0.125Tu [0.495 − 0.064(τm Tm )] .
b720_Chapter-03
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Handbook of PI and PID Controller Tuning Rules
132
Rule
Kc
Kasahara et al.
(1999) –
continued.
Leva and
Colombo (2000).
44
Kc
45
Kc
Leva (2001).
Model: Method 1
46
Kc
Leva et al.
(2003), (2006).
47
Kc
48
Kc
Ti
Td
Ti
Td
Comment
Robust to 50%
change in plant
parameters.
λ not specified;
Model: Method 1
2
Zhang et al.
(2002b).
Model: Method 1
44
17-Mar-2009
Tm +
τm
2(λ + τ m )
Td
Tm +
τm2
2(λ + τ m )
Td
Tm +
τm2
2(λ + τ m )
Td
Model: Method 1
τm Tm Td
−
2Ti
N
λ > 0.0735τm for
stability (Ou et
al., 2005c).
Ti
K c = 0.6K u [0.195 + 0.266(τm Tm )] , Ti = 0.5Tu [− 0.013 + 0.731(τm Tm )] ,
Td = 0.125Tu [0.489 − 0.065(τm Tm )] .
45
46
Kc =
2Tm (λ + τ m )
Tm (λ + τm )
1 2(λ + τm )Tm + τm
,N=
.
, Td =
2
Km
2( λ + τ m ) 2
2(λ + τm )Tm + τm
λ 2(λ + τ m )Tm + τ m 2
Kc =
2Tm (λ + τ m )
1 2(λ + τm )Tm + τm
, N=
−1,
Km
2( λ + τ m ) 2
λ 2(λ + τ m )Tm + τ m 2
2
[
]
2
2
Td = τm 2
2
[
2λTm + (2Tm − λ )τm
[
2(λ + τm ) 2Tm (λ + τm ) + τm 2
[
]
] . 0.1T
m
≤ λ ≤ 0.5Tm , τm Tm ≤ 0.25 ;
λ = 1.5(τ m + Tm ) , 0.25 < τ m Tm ≤ 0.75 ; λ = 3(τ m + Tm ) , τm Tm > 0.75 .
47
Kc =
2
τm Tm (λ + τm )
1 2(λ + τm )Tm + τm
, Td =
,
Km
2( λ + τ m ) 2
2(λ + τm )Tm + τm 2
2Tm (λ + τ m )
[
]
2
N=
[
λ 2(λ + τ m )Tm + τ m 2
] . 0.1T
m
≤ λ ≤ 0.5Tm , τm Tm ≤ 0.33 ;
λ = 1.5(τ m + Tm ) , 0.33 < τm Tm ≤ 3 ; λ = 3(τ m + Tm ) , τm Tm > 3 .
48
Kc =
λ2
0.5τm + Tm − (Td N )
T T
or N = 10.
, Ti = 0.5τm + Tm − d , d =
K m (2λ + 0.5τm )
N N 2λ + 0.5τm
]
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Vilanova and
Balaguer (2006).
Ti
K m (x1 + x 2 )
50
Tang et al.
(2007).
Model:
Method 1;
Td N = Tf .
49
Kc
Ti
49
Ti
Ti
Td
x2
x1 + x 3
N
x1 + x 2
133
Comment
Model: Method 1
Td
Labelled the H ∞ Smith predictor.
51
Kc
Ti
Td
Labelled the H 2 Smith predictor or the Dahlin controller.
2(3Tm + τm )
9K m τm
Tm + 0.333τm
(6Tm − τm )τm
6(3Tm + τm )
Recommended
λ = 0.5 ;
Tf = 0.167 τm .
 x + x3 
T x x + x2
, N= m 1 1
Ti = Tm + τm + x 3 − x1 − x 2  1
−1 .

Ti τm x1 + x 3
 x1 + x 2 
For minimum ISE, servo response: x 3 = τm ;
for minimum IE, regulator response: x 2 = τm ; x1 = real constant.
50
51
(τm + 2Tm )(4λ + τm ) − 4λ2 , T = 0.5τ + T − 2λ2 ,
i
m
m
4λ + τ m
K m (4λ + τm )2
2
Tm τm (4λ + τm )
2λ
2λ2
λ
Td =
−
T
=
,
; 0.3 ≤
≤ 0.8 .
f
2
4λ + τm
τm
(τm + 2Tm )(4λ + τm ) − 4λ 4λ + τm
(τ + 2Tm )(λ + τm ) − λτm , T = 0.5τ + T − λτm ,
Kc = m
i
m
m
2(λ + τm )
2K m (λ + τm )2
λτ m
Tm τ m (λ + τ m )
λτ m
λ
,T =
−
Td =
≤ 0.8 .
; 0.3 ≤
(τ m + 2Tm )(λ + τ m ) − λτ m 2(λ + τm ) f 2(λ + τm )
τm
Kc =
b720_Chapter-03
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Handbook of PI and PID Controller Tuning Rules
134



1  1 + Td s 


3.3.5 Classical controller G c (s) = K c 1 +
 Ti s  1 + Td s 
N 

E(s)
R(s)
−
+



 U(s)

1  1 + Td s 


K c 1 +
Td 
Ti s 

s
1 +
N 

Y(s)
K m e − sτ
1 + sTm
m
Table 13: PID controller tuning rules – FOLPD model G m (s) =
Rule
Kc
Kraus (1986).
Model:
Method 23
0.833Tm K m τm
Witt and
Waggoner
(1990).
Comment
Td
Process reaction
1.5τ m
0.25τm
N=∞
Foxboro EXACT controller pre-tuning. Also given by Hang et al.
(1993b), p. 76.
x1Tm K m τm
τm
τm
10 ≤ N ≤ 20 ;
0.6 ≤ x1 ≤ 1.0 .
Model: Method 2. ‘Equivalent to’ Ziegler and Nichols (1942).
1
Ti
Td
10 ≤ N ≤ 20
Kc
Witt and
Waggoner
(1990).
1
Ti
K m e − sτ m
1 + sTm
Model: Method 2. ‘Preferred’ tuning.
Tuning ‘equivalent to’ Cohen and Coon (1953).
1.350
Kc =
τ 
Tm
T
τ
+ 0.25 + m 0.7425 + 0.0150 m + 0.0625 m 
τm
τm
Tm
 Tm 
2K m
Ti =
2
,
Tm
τ 
τ
T
T
1.350 m + 0.25 − m 0.7425 + 0.0150 m + 0.0625 m 
τm
τm
Tm
 Tm 
2
,
Tm
Td =
T
T
1.350 m + 0.25 + m
τm
τm
τ 
τ
0.7425 + 0.0150 m + 0.0625 m 
Tm
 Tm 
2
.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
2
Witt and
Kc
Waggoner (1990)
– continued.
St. Clair (1997),
Tm K m τm
p. 21.
Model: Method 0.5Tm K m τm
2; N ≥ 1 .
Harrold (1999).
1 Km
Model:
0.5 K m
Method 11;
0.25 K m
N not specified.
Tan et al.
0.6Tm
(1999a), p. 25.
K mτm
Shinskey (2000).
Model: Method 2
Ti
Td
Ti
Td
135
Comment
‘Alternate’ tuning. 10 ≤ N ≤ 20 .
5τ m
0.5τ m
5τ m
0.5τ m
Tm
≤ 0.25Tm
‘Aggressive’
tuning.
‘Conservative’
tuning.
τ m Tm ≤ 0.25
τ m Tm ≈ 0.5
τ m Tm ≥ 1
0.889Tm
K mτm
τm
τm
N =∞;
Model: Method 2
1.75τ m
0.70τ m
τm Tm = 0.167 ;
N not specified.
x1Tm K m τm
x 2τm
x 3τ m
N=∞
O’Dwyer
Representative results: equivalent to Chien et al. (1952) – regulator.
(2001b).
x2
x3
Model: Method 2 x 1
0.7236 1.8353 0.5447
0% overshoot; 0.1 < τ m Tm < 1 .
0.84
1.350
2
Kc =
1.4
0.6
20% overshoot; 0.1 < τ m Tm < 1 .
τ 
Tm
T
τ
+ 0.25 − m 0.7425 + 0.0150 m + 0.0625 m 
τm
τm
Tm
 Tm 
2K m
Ti =
2
,
Tm
τ 
τ
T
T
1.350 m + 0.25 + m 0.7425 + 0.0150 m + 0.0625 m 
τm
τm
Tm
 Tm 
Td =
2
,
Tm
τ 
τ
T
T
1.350 m + 0.25 − m 0.7425 + 0.0150 m + 0.0625 m 
τm
τm
Tm
 Tm 
2
.
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Handbook of PI and PID Controller Tuning Rules
136
Rule
O’Dwyer
(2001b).
Model:
Method 2;
N =∞.
Ti
Td
Comment
3
Kc
Ti
Td
0% overshoot;
τ m Tm < 0.5 .
4
Kc
Ti
Td
20% overshoot;
τ m Tm < 0.7234
Representative results: equivalent to Chien et al. (1952) – servo.
Model:
0.47Tm
Method
1;
τ
τ
m
m
K m τm
N = 10.
Minimum performance index: regulator tuning
8.1 K m
0.29(τm + Tm )
0.13(τm + Tm )
τm Tm = 0.11
Faanes and
Skogestad
(2004).
Minimum ITAE
with OS ≤ 8% –
Fertik (1975).
Model:
Method 3;
N = 11.
Minimum IAE –
Huang and Chao
(1982).
Minimum IAE –
Kaya and Scheib
(1988).
Kc
3.8 K m
0.46(τm + Tm )
0.23(τm + Tm )
0.46(τm + Tm )
0.40(τm + Tm )
0.50(τm + Tm )
1.85 K m
0.93 K m
0.33(τm + Tm )
0.40(τm + Tm )
0.56 K m
0.44(τm + Tm )
τm Tm = 0.25
τm Tm = 0.43
τm Tm = 0.67
τm Tm = 1.00
Coefficients of K c , Ti , Td deduced from graphs and converted to this
equivalent controller architecture.
0.982
N = 8;
0.817  Tm 
5
Model: Method 1


Td
Ti
K m  τm 
Model: Method
6
5; N = 10;
Ti
Td
Kc
0 < τm Tm ≤ 1 .
3
Kc =
0.3Tm 
2τ
1 − 1 − m
K m τm 
Tm
4
Kc =
0.475Tm
K m τm

T 
2τ
 Ti = m 1 + 1 − m
2
Tm

 ,

τ
1 + 1 − 1.3824 m
Tm


T 
2τ 
 Td = m 1 − 1 − m 
2
Tm 

 ,


τ
 Ti = 0.68Tm 1 + 1 − 1.3824 m
Tm
 ,



 ,

τ 
Td = 0.68Tm 1 − 1 − 1.3824 m 
T
m 


5
6
τ 
Ti = 0.903Tm  m 
 Tm 
Kc =
0.780
0.98089  Tm 


K m  τm 
τ
, Td = 0.602Tm  m
 Tm
0.76167




0.954
.
1.05211
, Ti =
Tm  τm 


0.91032  Tm 
τ 
, Td = 0.59974Tm  m 
 Tm 
0.89819
.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Minimum IAE –
Witt and
Waggoner
(1990).
Kc
Ti
Td
7
Kc
Ti
Td
8
Kc
Ti
Td
Comment
0.1 < τm Tm ≤ 0.258 ; 10 ≤ N ≤ 20 ; Model: Method 2.
0.95Tm K m τ m
1.43τ m
0.52τ m
τ m Tm = 0.2
1.17 τ m
0.48τ m
τ m Tm = 0.5
1.03τ m
0.40τ m
τ m Tm = 1
0.77τ m
0.35τ m
τ m Tm = 2
Minimum IAE –
Shinskey (1996), 0.96Tm K m τ m
p. 119.
1.43τ m
0.52τ m
Model:
Method 1;
τ m Tm = 0.2 .
Minimum IAE –
Edgar et al.
0.94Tm K m τ m
(1997), pp. 8-14,
8-15.
1.5τ m
0.55τ m
Minimum IAE –
Shinskey (1988), 0.95Tm K m τ m
p. 143.
1.14Tm K m τ m
Model: Method 1
1.39Tm K m τ m
7
‘Preferred’ tuning. K c =
0.718  Tm 


K m  τm 
0.921 
1 + 1 − 1.693 τm 
T 

 m

−1.886
Model:
Method 1;
Time constant
dominant;
N = 10.

,


1.137
Ti =
8
137
‘Alternate’ tuning. K c =
0.718  Tm 


K m  τm 
τ 
0.964Tm  m 
 Tm 
1.886
τ 
1 − 1 − 1.693 m 
 Tm 
0.921 
Ti =
1 − 1 − 1.693 τm 
T 

 m

τ
0.964Tm  m
 Tm
−1.886
1.137
, Td =
τ
1 − 1 + 1.693 m
 Tm
1.886




1.886
.
τ 
1 + 1 + 1.693 m 
 Tm 

,


1.137




τ 
0.964Tm  m 
 Tm 
1.137
, Td =
τ 
0.964Tm  m 
 Tm 
1.886
τ 
1 + 1 − 1.693 m 
 Tm 
.
b720_Chapter-03
Rule
Kc
Minimum IAE –
Edgar et al.
0.88Tm K m τ m
(1997), p. 8–15.
Minimum IAE –
Smith and
Tm
Corripio (1997),
K mτm
p. 345.
Minimum IAE – 0.89Tm K m τm
Shinskey (2003).
0.56K u
Minimum IAE –
0.49 K u
Shinskey (1988),
0.51K u
p. 148.
Model: Method 1
0.46K
u
Minimum IAE –
K u τm
Shinskey (1994), 3τ − 0.32T
m
u
p. 167.
Minimum ISE – 1.101  T  0.881
 m
Huang and Chao
K m  τ m 
(1982).
Minimum ISE –
Kaya and Scheib
(1988).
11
11
12



Comment
Ti
Td
1.8τ m
0.70τ m
Tm
0.5τ m
1.75τm
0.70τ m
Model: Method 2
0.39Tu
0.14Tu
τ m Tm = 0.2
0.34Tu
0.14Tu
τ m Tm = 0.5
0.33Tu
0.13Tu
τ m Tm = 1
0.28Tu
0.13Tu
τ m Tm = 2
9
Ti
0.14Tu
10
Ti
Td
Ti
Kc
Minimum ITAE 0.745  T
 m
– Huang and
K
m
 τm
Chao (1982).
10
FA
Handbook of PI and PID Controller Tuning Rules
138
9
17-Mar-2009
Td
1.036
12
Td
Ti
Model:
Method 2;
N = 10.
Model:
Method 1;
τ
0.1 ≤ m ≤ 1.5 .
Tm
Model:
Method 1;
N not specified.
Model:
Method 1;
N = 8.
Model:
Method 5;
0 < τm Tm ≤ 1 ;
N = 10.
Model:
Method 1;
N = 8.


T
Ti = Tu  0.15 u − 0.05  .
τ
m


τ 
Ti = 1.134Tm  m 
 Tm 
Kc =
0.883
1.11907  Tm 


K m  τm 
τ 
Ti = 0.771Tm  m 
 Tm 
τ 
, Td = 0.563Tm  m 
 Tm 
0.89711
0.595
, Ti =
0.881
.
Tm  τm 


0.7987  Tm 
0.9548
1.006
τ 
, Td = 0.597Tm  m 
 Tm 
.
τ
, Td = 0.54766Tm  m
 Tm




0.87798
.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Minimum ITAE
– Kaya and
Scheib (1988).
Minimum ITAE
– Witt and
Waggoner
(1990).
Minimum ITAE
– Ou and Chen
(1995).
Minimum ITSE –
Huang and Chao
(1982).
13
Kc
Ti
Td
Comment
Ti
Td
0 < τm Tm ≤ 1
N = 10; Model: Method 5.
14
Kc
Ti
Td
Preferred tuning.
15
Kc
Ti
Td
Alternate tuning.
Model: Method 2; 10 ≤ N ≤ 20 .
1.357  Tm

K m  τ m



0.947
0.994  Tm 
 
Km  τm 
Kc =
0.77902  Tm 


K m  τm 
14
Kc =
0.679  Tm 


K m  τm 
Tm  τ m

0.842  Tm
, Ti =
17
0.947 
1 + 1 − 1.283 τm 
T 

 m

0.947 
1 − 1 − 1.283 τm 
T 

 m

Ti =
16
17
τ 
Td = 0.381Tm  m 
 Tm 
τ 
Ti = 1.032Tm  m 
 Tm 
0.995
0.869
Model:
Method 57
0.738
16
Td
Model:
Method 1;
N = 8.
, N=±
Td
Ti
Tm  τm 


1.14311  Tm 
Ti =
0.679  Tm 


Kc =
K m  τm 



0.907
1.06401
13
15
139
−1.733
0.70949
1.03826
τ 
, Td = 0.57137Tm  m 
 Tm 

 , 0.1 < τ m < 0.379 ;

Tm

τ 
0.762Tm  m 
 Tm 
0.995
1.733
τ 
1 − 1 − 1.283 m 
 Tm 
−1.733
, Td =
τ 
0.762Tm  m 
 Tm 
0.995
1.733
.
τ 
1 + 1 + 1.283 m 
 Tm 

,


τ
0.762Tm  m
 Tm




0.995
τ
1 − 1 + 1.283 m
 Tm
1.733




, Td =
0.75K m K cTd
, N>0.
Ti + K m K c (Td + Ti − τm − Tm )
τ 
, Td = 0.570Tm  m 
 Tm 
.
0.884
.
τ
0.762Tm  m
 Tm




0.995
τ
1 + 1 − 1.283 m
 Tm
1.733




.
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
140
Rule
Kc
Minimum ITAE
with OS ≤ 8% –
Fertik (1975).
Model:
Method 3;
N = 11.
Ti
5.0 K m
Td
Comment
Minimum performance index: servo tuning
0.63(τm + Tm )
0.13(τm + Tm )
τm Tm = 0.11
2.4 K m
1.2 K m
0.69 K m
0.51 K m
0.57(τm + Tm )
0.23(τm + Tm )
0.51(τm + Tm )
0.33(τm + Tm )
0.46(τm + Tm )
0.40(τm + Tm )
0.40(τm + Tm )
0.44(τm + Tm )
τm Tm = 0.25
τm Tm = 0.43
τm Tm = 0.67
τm Tm = 1.00
Coefficients of K c , Ti , Td deduced from graphs and converted to this
equivalent controller architecture.
1.00
Minimum IAE –
Model:


0.673 Tm
18


Huang and Chao
Method
1;
T
Ti
d
K m  τm 
(1982).
N = 8.
Minimum IAE –
0.65  Tm
Kaya and Scheib

K m  τ m
(1988).
Minimum IAE –
Smith and
Corripio (1997),
p. 345.



1.04432
0.83Tm
K mτm
19
Td
Ti
0.5τ m
Tm
1.032
18
Ti =
19
Ti =
τ 
Tm
, Td = 0.502Tm  m 
τ 
 Tm 
0.148 m  + 0.979
T
 m
Tm
0.9895 + 0.09539
τm
Tm
τ
, Td = 0.50814Tm  m
 Tm
.
1.08433




.
Model:
Method 5;
0 < τm Tm ≤ 1 ;
N = 10.
Model:
Method 1;
τ
0.1 ≤ m ≤ 1.5 ;
Tm
N not specified.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Minimum IAE –
Witt and
Waggoner
(1990).
Kc =
1.086  Tm 


K m  τm 
Td
Comment
20
Kc
Ti
Td
Preferred tuning.
21
Kc
Ti
Td
Alternate tuning.
0.1 ≤ τm Tm ≤ 1 ; 10 ≤ N ≤ 20 ; Model: Method 2.
Minimum ISE – 0.787  T
 m
Huang and Chao
K m  τ m
(1982).
20
Ti



22
1 + 1 − 1.392 τm 
T 

 m

Td
Ti
0.914


τm  
0.74 − 0.13   ,
Tm 


τ 
0.696Tm  m 
 Tm 
τ 
1 − 1 − 1.392 m 
 Tm 
0.869
1.086  Tm 


K m  τm 
0.869 
1 − 1 − 1.392 τm 
T 

 m

Ti =
0.914
1.028
T 
Ti = 1.678Tm  m 
 τm 
τ 
0.696Tm  m 
 Tm 
τ 
1 + 1 − 1.392 m 
 Tm 
τ 
, Td = 0.594Tm  m 
 Tm 
τ 
0.696Tm  m 
 Tm 
τ 
1 + 1 − 1.392 m 
 Tm 
0.869
0.914
.

τm 
0.74 − 0.13 
Tm 



τm  
0.74 − 0.13   ,
Tm 


0.869
0.971
.
0.914
,

τm 
0.74 − 0.13 
T
m

Td =
22
0.914

τm 
0.74 − 0.13 
T
m

Td =
Kc =
Model:
Method 1;
N = 8.
0.964
0.869 
Ti =
21
141
τ 
0.696Tm  m 
 Tm 
τ 
1 − 1 − 1.392 m 
 Tm 
0.869
0.914

τm 
0.74 − 0.13 
Tm 

.
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
142
Rule
Kc
Minimum ISE –
23
Kaya and Scheib
Kc
(1988).
Model: Method 5
0.995
Minimum ITAE
0.684  Tm 
– Huang and


K m  τm 
Chao (1982).
Minimum ITAE
25
– Kaya and
Kc
Scheib (1988).
Model: Method 5
0.85
Minimum ITAE
0.965  Tm 


– Ou and Chen
K m  τm 
(1995).
1.03092
23
17-Mar-2009
Kc =
0.71959  Tm 


K m  τm 
, Ti =
Ti
Td
Ti
Td
24
Td
Ti
Ti
26
Td
Td
Ti
Tm
τm
Tm
1.12666 − 0.18145
Comment
τm
≤ 1;
Tm
N = 10.
0<
Model:
Method 1;
N = 8.
τ
0 < m ≤ 1;
Tm
N = 10.
Model:
Method 57
τ 
, Td = 0.54568Tm  m 
 Tm 
0.86411
τ
, Td = 0.42844Tm  m
 Tm
1.0081
.
1.049
24
25
26
τ 
Tm
Ti =
, Td = 0.491Tm  m 
 τm 
 Tm 
 + 0.986
0.114

 Tm 
Kc =
Ti =
1.12762  Tm 


K m  τm 
Tm
0.796 − 0.1465
0.80368
, Ti =
τm
Tm
.
Tm
0.99783 + 0.02860
τ 
, Td = 0.308Tm  m 
 Tm 
τm
Tm




.
0.929
,
N=±
0.75K m K cTd
, N>0.
Ti + K m K c (Td + Ti − τm − Tm )
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Minimum ITAE
– Witt and
Waggoner
(1990).
Kc
0.965  Tm 


Kc =
K m  τm 
Td
Comment
27
Kc
Ti
Td
Preferred tuning.
28
Kc
Ti
Td
Alternate tuning.
0.1 ≤ τm Tm ≤ 1 ; 10 ≤ N ≤ 20 ; Model: Method 2.
Minimum ITSE –
0.718  Tm

Huang and Chao
K m  τ m
(1982).
27
Ti



29
1 + 1 − 1.232 τm 
T 

 m

Td
Ti
0.929


τm  
0.796 − 0.1465   ,
Tm 


τ 
0.616Tm  m 
 Tm 
τ 
1 − 1 − 1.232 m 
 Tm 
0.85
0.85 
1 − 1 − 1.232 τm 
T 

 m

Ti =
0.929
Ti =
τ 
0.616Tm  m 
 Tm 
τ 
1 + 1 − 1.232 m 
 Tm 
τ 
0.616Tm  m 
 Tm 
τ 
1 + 1 − 1.232 m 
 Tm 
τ
Tm
, Td = 0.596Tm  m
τ 
 Tm
0.063 m  + 1.061
T
 m
,
0.85
0.929
.

τm 
0.796 − 0.1465 
T
m



τm  
0.796 − 0.1465   ,
Tm 


0.85
1.028




0.929
,

τm 
0.796 − 0.1465 
T
m

Td =
29
0.929

τm 
0.796 − 0.1465 
T
m

Td =
0.965  Tm 


Kc =
K m  τm 
Model:
Method 1;
N = 8.
1.00
0.85 
Ti =
28
143
.
τ 
0.616Tm  m 
 Tm 
τ 
1 − 1 − 1.232 m 
 Tm 
0.85
0.929

τm 
0.796 − 0.1465 
T
m

.
b720_Chapter-03
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144
Rule
Kc
x1Tm K m τm
Tsang et al.
(1993).
Model:
Method 19
x1
ξ
1.6818
1.3829
1.1610
0.0
0.1
0.2
x1
ξ
Comment
Td
Ti
Direct synthesis: time domain criteria
Tm
0.25τ m
Coefficient values: N = 2.5.
x1
ξ
x1
ξ
0.9916
0.3
0.6693
0.6
0.8594
0.4
0.6000
0.7
0.7542
0.5
0.5429
0.8
Coefficient values: N = ∞ .
x1
ξ
x1
ξ
x1
ξ
0.4957
0.4569
0.9
1.0
x1
1.8194
0.0
1.0894
0.3
0.7482
0.6
0.5709
1.5039
0.1
0.9492
0.4
0.6756
0.7
0.5413
1.2690
0.2
0.8378
0.5
0.6170
0.8
Tsang and Rad 0.809Tm K m τm
Tm
0.5τ m
(1995).
Overshoot = 16%; N = 5; Model: Method 15.
ξ
0.9
1.0
Smith and
0.5τ m
0.5Tm K m τm
Tm
Corripio (1997),
Servo – 5% overshoot; N not specified; Model: Method 1.
p. 346.
2
Schaedel (1997).
T
T1 2 − 2T2 2
T1 − 2
30
Model:
N not defined.
Kc
T1
Method 25
Wang et al.
Tm
1
(2002).
Labelled ‘IMC’.
0.5τ m



τ
τ 
Model:
K m 1 + m  K m 1 + m 
2Tm 
2Tm 


Method 1;
N = 10.
30
Kc =
T1 =

T2
0.375 T1 − 2 

T1 

 T 2 − 2T 2

2
T

2
2 
2
Km  1
+ 2 − T1 − 2T2 
N
T1




Tm
1 + 3.45
T1 =
0.7Tm
τ am
Tm
2
Tm − 0.7 τ am
+ τ m , T2 2 = 1.25Tm τ am +
,
Tm
1 + 3.45
2
+ τ m , T2 = Tm τ am +
0.7Tm 2
Tm − 0.7 τ am
τ am
Tm
τ m + 0.5τ m 2 , 0 <
+ 0.5τ m 2 ,
τ am
Tm
τ am
> 0.104 .
Tm
≤ 0.104 ;
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Wang et al.
(2002) –
continued.
Kc
0.65  Tm

K m  τ m
Ti



Td
Comment
Td
Labelled ‘IAE
setpoint’.
1.04432
31
Ti
145
32
Kc
Ti
Td
33
Kc
Ti
Td
34
Kc
Ti
Td
35
Kc
Ti
Td
36
Kc
Ti
Td
Labelled ‘IAE
load’.
Labelled ‘ISE
setpoint’.
Labelled ‘ISE
load’.
Labelled ‘ITAE
setpoint’.
Labelled ‘ITAE
load’.
Åström and
1
Tm
Hägglund (2006), K 0.5τ + λ
Model: Method 1
Tm
0.5τm
m
m
pp. 188–189.
λ = Tm (aggressive tuning), λ = 3Tm (robust tuning). N = ∞ .
31
32
33
34
35
36
Ti =
1.08433
Tm
0.9895 + 0.09539
τm
Tm
0.98089  Tm

K m  τ m



0.76167
0.71959  Tm

Kc =
K m  τ m



1.03092
1.11907  Tm

K m  τ m



0.89711
1.12762  Tm

Kc =
K m  τ m



0.80368
0.77902  Tm

K m  τ m



Kc =
Kc =
Kc =
τ 
, Td = 0.50814 m 
 Tm 
1.05211
τ 
, Ti = 1.09851Tm  m 
 Tm 
, Ti =
Tm
1.12666 − 0.18145
τ 
, Ti = 1.2520Tm  m 
 Tm 
, Ti =
1.06401
.
τm
Tm
0.95480
τ 
, Td = 0.54568 m 
 Tm 




0.89819
.
0.86411
τ 
, Td = 0.54766Tm  m 
 Tm 
.
0.87798
.
1.0081
Tm
0.9978 + 0.02860
τ 
, Ti = 0.87481Tm  m 
 Tm 
τ
, Td = 0.59974Tm  m
 Tm
τm
Tm
0.70949
τ 
, Td = 0.42844 m 
 Tm 
τ
, Td = 0.57137Tm  m
 Tm
.
1.03826




.
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Handbook of PI and PID Controller Tuning Rules
146
Rule
Hougen (1979) –
p. 335.
O’Dwyer
(2001a).
Model: Method 1
Huang et al.
(2005a).
Model:
Method 51
Kc
37
Kc
Td
Ti
Comment
Direct synthesis: frequency domain criteria
Tm
0.45τm
Model: Method 1; maximise crossover frequency; 10 ≤ N ≤ 30 .
x1Tm K m τm
Tm N
Tm
A m = π 2 x 1 N ; φ m = 0.5π − x 1 N ; representative result below.
0.079Tm
K mτm
0.1Tm
Tm
0.65Tm K m τm
Tm
0.4 τ m
A m = 2.0;
φ m = 450 .
N = 20, A m = 2.7 , φ m = 650 .
The above is a representative tuning rule (also given by Huang et al.
(2003), N not defined, Model: Method 37). Other rules may be deduced
from a graph provided by the authors for other A m , φ m values.
Huang and Jeng
(2005).
0.65Tm K m τm
Harris and
Tyreus (1987).
Model: Method 1
Tm
K m (τ m + λ )
τm
N = 40;
Model: Method 1
0.5τ m
N = (τ m + λ ) λ
Tm
Robust
Chien (1988).
Model: Method 1
Tm
λ ≥ maximum of [0.45τ m ,0.1Tm ]
Tm
K m (λ + 0.5τ m )
Tm
0.5τ m
0.5τ m
K m (λ + 0.5τ m )
0.5τ m
Tm
Tm
0.55τm
0.5τ m
1.1Tm
Tm
Chien (1988). K (λ + 0.5τ )
m
m
Model: Method 1
0.5τ m
K m (λ + 0.5τ m )
λ ∈ [τ m , Tm ]
(Chien and
Fruehauf, 1990);
N = 10.
λ ∈ [τ m , Tm ]
(Chien and
Fruehauf, 1990);
N = 11.
Tuning rules converted to this equivalent controller architecture.
37
K c = 9.0 K m , τm Tm = 0.1 ; K c = 4.2 K m , τm Tm = 0.2 ; K c = 2.8 K m , τm Tm = 0.3 ;
K c = 2.1 K m , τm Tm = 0.4 ; K c = 1.7 K m , τm Tm = 0.5 ; K c = 1.45 K m , τm Tm = 0.6 ;
K c = 1.2 K m , τm Tm = 0.7 ; K c = 1.1 K m , τm Tm = 0.8 ; K c = 0.95 K m , τm Tm = 0.9 ;
K c = 0.85 K m , τm Tm = 1.0 ; values deduced from graph.
b720_Chapter-03
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FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Zhang et al.
38
(1996b).
Kc
Model:
39
Kc
Method 47
40
Zhang et al.
Kc
(2002b).
Shi and Lee
41
(2002).
Kc
Model: Method 1
Lee and Shi
42
(2002).
Kc
Model: Method 1
Faanes and
0.77Tm K m τm
Skogestad
0.5Tm K m τm
(2004). N = 10.
Tm
Tang et al.
K m (2λ + 0.5τm )
(2007).
τm
Model: Method 1
K m (4λ + τm )
Comment
Ti
Td
0.5τ m
Tm
Tm
0.5τ m
≤ 1. 2 τ m .
Tm
0.5τ m
Model: Method 1
Tm
 ωg τ m
1
tan
ωg
 2
Tm
43
Ti
0.2τm ≤ λ



Suggested
ωg = 0.6 τm .
0.5τ m
Tm
0.5τ m
8τm
0.5τ m
Tm
0.5τm
0.5τm
Tm
Suggested
ωg = 0.6 τm .
Model: Method 1
Tf =
2λ2
,
4λ + τm
0.3 ≤
λ
≤ 0.8 .
τm
Tf =
λτ m
,
2(λ + τm )
0.3 ≤
λ
≤ 0.8 .
τm
Labelled the H ∞ Smith predictor.
Tm
K m (λ + τm )
Tm
0.5τm
τm
2K m (λ + τm )
0.5τm
Tm
Labelled the H 2 Smith predictor or the Dahlin controller.
T (2λ + 0.5τ m )
0.5τm
, N= m
.
K m (0.5τm + 2λ )
λ2
38
Kc =
39
Kc =
0.5τ m (2λ + 0.5τ m )
Tm
, N=
.
K m (0.5τm + 2λ )
λ2
40
Kc =
Tm
λ2
.
, N = 10 or N =
0.5τ m (2λ + 0.5τ m )
K m (2λ + 0.5τm )
41
42
43
 ωg τ m
2ω bw
ωbw Tm
tan
, N = 1+
ωg
K m 1 + 2ωbw ωg tan 0.5ωg τm
 2
 ωg τ m
2ω bw
ωbw Tm
tan
Kc =
, N = 1+
ωg
K m 1 + 2ωbw ωg tan 0.5ωg τm
 2
T
1
T
.
Ti = m , 1 < x1 < 1 + m +
x1
τm ωbw τm
Kc =
[ (
) (
)]

.


[ (
) (
)]

.


147
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Handbook of PI and PID Controller Tuning Rules
148
Rule
Kc
Ti
Td
Comment
Tang et al.
(2007) –
continued.
0.667Tm K m τm
Tm
0.5τm
0.333 K m
0.5τm
Tm
Recommended
λ = 0.5 ;
Tf = 0.167 τm .
Pessen (1994).
Model: Method 1
0.35K u
44
Huang et al.
(2005a).
Model:
Method 51
44
Kc
Ultimate cycle
0.25Tu
0.25Tu
Ti
N =∞;
0.1 ≤ τm Tm ≤ 1 .
Td
N = 20, A m = 2.7 , φ m = 650 .
The above is a representative tuning rule; other rules may be deduced
from a graph provided by the authors for other A m , φ m values.


2


 ^ 


 K u  − 1
2
^
0.65 
 ^ 

 
, Ti = 0.159 Tu  K u  − 1 ,
Kc =


2


Km 
 
^ 
−1  

π
−


tan
K

u  −1 

  



 

^ 
Td = 0.064 Tu π − tan −1


2

 ^ 
 K u  − 1  .

 

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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
149
3.3.6 Generalised classical controller


Td s
1
+
G c (s) = K c 1 +
T

Ti s
1+ d

N

R(s) E(s)
+
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N



2
 b f 0 + b f 1s + b f 2 s
2
 1 + a f 1s + a f 2 s
s

U(s)


2
 b f 0 + b f 1s + b f 2 s
 1 + a f 1s + a f 2 s 2
s 

K m e − sτ
1 + sTm
m
Table 14: PID controller tuning rules – FOLPD model G m (s) =
Rule
Tsang et al.
(1993).
Model:
Method 19
Kristiansson
(2003).
Model:
Method 12
1
Kc
x1Tm
Ti
Y(s)
K m e − sτ m
1 + sTm
Comment
Td
Direct synthesis: time domain criteria
0
K m τm
Tm
2
2
b f 0 = 1 , b f 1 = 0.5τm , b f 2 = 0.0833τm , a f 1 = 0.2 τ m , a f 2 = 0.01τ m .
ξ
x1
1.851
1.552
1.329
1
ξ
x1
ξ
x1
x1
0.0
1.160
0.3
0.841
0.6
0.622
0.1
1.028
0.4
0.768
0.7
0.553
0.2
0.925
0.5
0.695
0.8
Direct synthesis: frequency domain criteria
Kc
Ti
Equations continued into the footnote on the next page;
2
 T

τ  
τ
21.1 m − 0.1 0.06 + 0.68 m − 0.12 m  
Tm
 τm
 
 Tm  
Kc =
,
2



τ
τ
K m 0.68 + 0.84 m − 0.25 m  
Tm

 Tm  

Td
0.05 ≤
ξ
0.9
1.0
τm
≤2;
Tm
1.6 ≤ M s ≤ 1.9 .




b720_Chapter-03
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Handbook of PI and PID Controller Tuning Rules
150
Rule
Kristiansson
(2003) –
continued.
Shamsuzzoha
and Lee (2007b).
Kc
Ti
Td
2
Kc
Ti
Td
3
Kc
0.5τm
0.167 τm
Comment
Model: Method 1
2
2


τ  
τ  
τ
τ
2Tm 0.06 + 0.68 m − 0.12 m  
0.5Tm 0.06 + 0.68 m − 0.12 m  
Tm
Tm


 Tm  
 Tm  


Ti =
, Td =
,
2
2


 τm  
 τm  
τ
τ
m
m
0.68 + 0.84
0.68 + 0.84
 
 
− 0.25
− 0.25
Tm
Tm  
Tm
Tm  








N = ∞ , b f 0 = 1 , b f 1 = 0 , b f 2 = 0 , a f 1 = 0.8Tf , a f 2 = Tf 2 with
Tf =
2
[
2 2
0.6667(Tm + τm )[1.1(Tm τm ) − 0.1]
0.6667(Tm + τm )
, Ti =
,
2
2


 τm 
 τm  
τ
τ
m
m
0.68 + 0.84
 
 
K m τm 0.68 + 0.84
− 0.25
− 0.25
Tm
Tm  
Tm
Tm  








0.1667(Tm + τm )
Td =
, N = ∞ , b f 0 = 1 , b f 1 = 0 , b f 2 = 0 , a f 1 = 0.8Tf ,
2

 τm  
τ
m
0.68 + 0.84
 
− 0.25
Tm
Tm  




Kc =
a f 2 = Tf 2 with Tf =
3
]
Tm 0.06 + 0.68(τm Tm ) − 0.12(τm Tm ) [1.1(Tm τm ) − 0.1]
.
min{5 + 2(Tm τm ),25}
Kc =
(Tm + τm )2 [1.1(Tm τm ) − 0.1] ;
9Tm (5 + 2(Tm τm ))
0. 7 <
τm
< 2 ; 1.6 ≤ M s ≤ 1.9 .
Tm
2
 
τm
λ  − τm
 e
, N = ∞ , b f 0 = 1 , b f 1 = Tm 1 − 1 −
2K m (2λ − b f 1 + τm )
  Tm 

af1 =
Tm

 , bf 2 = 0 ,


0.5b f 1τm + λτ m + λ2
− Tm , a f 2 = 0 ; λ = x1Tm , λ values deduced from a graph
2λ − b f 1 + τm
for M s = 1.4, 1.5, 1.6, 1.8 and 1.9; λ values defined for 0.01 ≤ τm Tm ≤ 5 .
b720_Chapter-03
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FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Td
Comment
Model:
Method 1;
λ > τ m , λ < Tm .
151
Robust
Horn et al.
(1996).
Lee and Shi
(2002).
Model: Method 1
Shamsuzzoha et
al. (2004).
Model: Method 1
Shamsuzzoha
and Lee (2006a).
Model: Method 1
4
Kc
Tm + 0.5τ m
τ m Tm
τ m + 2Tm
5
Kc
Tm
x1
0
6
Kc
Tm + 0.5τm
Tm τm
2Tm + τm
0.5τm
K m (λ + τm )
0.5τm
0.167 τm
af1 =
1 < x1 <
2Tm
τm
λ not explicitly
specified.
2
τmλ
τm λ
, af 2 =
, b f 0 = 1 , b f 1 = Tm ,
2(τm + λ )
12(τm + λ )
b f 2 = 0 , N = ∞ . λ = x1τm ; x1 = 1.360 , M s = 1.4 ; x1 = 0.677 ,
M s = 1.6 ; x1 = 0.372 , M s = 1.8 ; x1 = 0.337 , M s = 2.0 .
Yu (2006), p. 18.
Model: Method 1
4
Kc =
0.6K u
Ultimate cycle
0
0.5Tu
0.2 < τm Tm < 2
b f 0 = 0 , b f 1 = 0.125Tu , b f 2 = 0 , a f 1 = 0.0125Tu , a f 2 = 0 .
2Tm + τm
λ2 τm + 2Tm τm (τm − λ ) 2λ (2Tm − λ )
, bf 0 = 1 , bf 1 =
+
,
(τ m + 2λ )
2(2λ + τm − b f 1 )K m
Tm (τm + 2λ )
bf 2 = 0 , a f 1 =
2λτ m + 2λ2 + b f 1 τ m
λ2 τ m
, af 2 =
, N =∞.
2(2λ + τ m − b f 1 )
2(2λ + τ m − b f 1 )
ωbw Tm
; N = ∞ ; b f 0 = 1 , b f 1 = Tm + 0.5τm ;
 2ωbw
 ωg τm  

K m (ωbw x1τm + 1)1 +
tan 

ωg

 2  
τ + 2ωbw Tm τm + 2Tm
Tm τm
0. 6
; af 2 =
; ωg =
b f 2 = 0.5τm Tm , a f 1 = m
.
2(ωbw x1τm + 1)
2(ωbw x1τm + 1)
τm
5
Kc =
6
Kc =
2
2Tm + τm
2T (2ξλ + τm ) + τm − 2λ2
, bf 0 = 1 ; bf 1 = m
, bf 2 = 0 ,
2K m (2ξλ + τm − bf 1 )
2(τm + Tm )
af1 =
2λξτ m + 2λ2 + bf 1τm
λ2 τm
, af 2 =
, N =∞.
2(2ξλ + τm − b f 1 )
2(2ξλ + τm − bf 1 )
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152
3.3.7 Two degree of freedom controller 1


Td s
1
+
U (s) = K c 1 +
T

Ti s
1+ d

N







T
s
β
d
E (s) − K c  α +
R (s)
Td 


s
1+
s

N 






β
T
s
d
α +
K
Td  c

1+ s 

N 

E(s)
+
R(s)
−


Td s
1
+
K c 1 +
T

Ti s
1+ d

N





s

−
U(s)
Y(s)
K m e − sτ
1 + sTm
m
+
Table 15: PID controller tuning rules – FOLPD model G m (s) =
K m e − sτ m
1 + sTm
Rule
Kc
Hiroi and
Terauchi (1986)
– regulator.
τ
0.1 < m < 1 .
Tm
0.95Tm
K mτm
2.38τ m
0.42τ m
0% overshoot.
1.2Tm
K mτm
2τ m
0.42τ m
20% overshoot.
VanDoren
(1998).
1.5Tm K m τm
Ti
Td
Comment
Process reaction
ABB (2001).
OMEGA Books
(2005).
α = 0.6 , β = 1 ; Model: Method 2.
2.5τ m
0.4 τ m
Model: Method 2
α = 0 , β =1, N = ∞ .
1.25Tm K m τm
2τ m
0.5τ m
Model: Method 2
α = 0 , β =1, N = ∞ .
Tm K m τm
Minimum IAE –
Shinskey (1996), 1.32Tm
p. 117.
2.5τ m
0.4 τ m
Model: Method 2
α = 0 , β =1, N = ∞ .
Minimum performance index: regulator tuning
τ m Tm = 0.1 ;
K mτm
1.80τ m
0.44τ m
Model: Method 1
α = 0 , β =1, N = ∞ .
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Td
Comment
Minimum IAE –
Shinskey (1988),
p. 143. Model:
Method 1;
α = 0 , β =1,
N =∞.
Minimum IAE –
Edgar et al.
(1997),
pp. 8–14, 8–15.
1.32Tm K m τ m
1.77 τ m
0.41τ m
τ m Tm = 0.2
1.35Tm K m τ m
1.43τ m
0.41τ m
τ m Tm = 0.5
1.49Tm K m τ m
1.17 τ m
0.37τ m
τ m Tm = 1
1.82Tm K m τ m
0.92τ m
0.32τ m
τ m Tm = 2
1.30Tm K m τ m
1.8τ m
0.45τ m
Model: Method 1
2.1τ m
0.63τ m
Model: Method 2
1.33Tm K m τ m
α = 0 , β =1, N = ∞ .
2
∞
 du 
Kotaki et al. Minimise (ISTSE+w.ISTC), ISTC =   dt , w is chosen so that
0  dt 
(2005a).
Model: Method 1
ISTSE
= 0.6 . p = % allowed perturbation on individual model
w.ISTC
parameters; α = 1 , β = 1 , N = ∞ .
∫
x1 K m
x1
Kotaki et al.
(2005a), (2005b).
Model:
Method 1;
coefficients of
K c , Ti , Td and
p obtained from
plots.
5.9988
3.0722
1.2389
0.6885
0.4305
x1 K m
x 2Tm
x2
x 3Tm
x3
τm Tm
0.3338
0.0018
0.5594
0.0013
0.7901
0.00072
0.9000
0.00007
0.8715
0.00026
x 2Tm
x 3Tm
p
0.1
0.2
0.4
0.6
0.8
29
30
44
61
89
τm Tm
x1
x2
x3
p
6.0
5.4
4.2
3.3
2.3
1.7
1.1
0.6
4.1
3.1
2.3
1.7
1.4
1.2
1.0
0.8
0.5
0.35
0.35
0.4
0.4
0.5
0.5
0.55
0.6
0.55
0.55
0.6
0.6
0.6
0.65
0.65
0.65
0.65
0
0
0
0
0
0
0
0
0.01
0
0
0
0
0
0
0
0
29
33
42
50
63
72
83
100
20
30
42
53
61
67
73
81
100
0.1
0.2
153
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154
Rule
Kotaki et al.
(2005a), (2005b)
– continued.
Kc
Ti
Td
x1 K m
x 2Tm
x 3Tm
x1
2.3
1.7
1.4
0.9
0.8
0.7
0.6
0.4
0.4
2.0
1.7
1.4
1.0
0.8
0.7
0.5
0.4
1.7
1.1
0.8
0.7
0.6
0.4
0.4
1.5
1.0
0.8
0.6
0.6
0.5
0.4
0.4
x2
x3
0.85
0.02
0.9
0
0.8
0
0.75
0
0.8
0
0.75
0
0.75
0
0.75
0
Kotaki et al. (2005a).
0.6
0
Kotaki et al. (2005b).
1.0
0.14
1.0
0
1.0
0
1.0
0
0.9
0
0.9
0
0.85
0
0.7
0
1.1
0.24
1.2
0
1.1
0
1.05
0
1.05
0
0.90
0
0.85
0
1.25
0.32
1.35
0
1.25
0
1.2
0
1.1
0
1.05
0
0.95
0
0.95
0
Comment
p
20
30
41
55
61
74
82
100
τm Tm
0.4
100
20
30
40
50
61
74
82
100
21
30
43
52
66
83
100
22
30
42
55
60
74
83
100
0.6
0.8
1.0
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17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kotaki et al.
(2005b).
Model:
Method 1;
α = 1, β = 1 ,
N =∞.
Kc
2.0
7.5
30
180
40
200
800
1900
800
2000
8000
14000
2000
8000
20000
38000
4000
20000
80000
20000
38000
80000
95000
Kc
1.2787
Kc =
0.5249  Tm 
1 +

K m 
τm 
∫
∞
0
(du dt )2dt . p = % allowed
perturbation on individual model parameters.
p
w
p
τm Tm
w
1
Comment
Td
Minimise (ISTSE+w.ISTC), ISTC =
Madhuranthakam
et al. (2008).
Model: Method 1
1
Ti
33
42
50
63
30
42
53
61
31
40
55
61
30
41
50
61
30
43
52
35
44
55
60
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
1.0
1.0
1.0
1.0
Ti
155
τm Tm
750
3800
42000
72
83
100
0.1
0.1
0.1
8000
20000
60000
73
81
100
0.2
0.2
0.2
40000
56000
160000
74
82
100
0.4
0.4
0.4
80000
105000
200000
73
81
100
0.6
0.6
0.6
95000
165000
250000
190000
400000
450000
66
83
100
74
99
100
0.8
0.8
0.8
1.0
1.0
1.0
Td
0.1 ≤
τm
≤2
Tm

 τm 
 ,
, Ti = τm  2.1356 − 1.9167

 τm + Tm  

2

 τm 
 τm 





Td = Tm 1.1321
0
.
1788
+
 τ + T  , N = 10, α = 0 , β = 1 .
τ m + Tm 

m
m 




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156
Rule
Minimum ISE –
Argelaguet et al.
(1997), (2000).
Kc
Ti
Comment
Td
Minimum performance index: servo tuning
2Tm + τ m
Tm τ m
Model: Method 1
Tm + 0.5τ m
2K m τ m
2Tm + τ m
α = 1, β = 1 , N = ∞ .
Madhuranthakam
et al. (2008).
Model: Method 1
Araki (1985) –
servo/regulator
x1
optimisation.
Model: Method 1 12.5
6.1
4.1
3.1
2.5
2
Ti
Kc
x1 K m
x2
x3
0.22 0.04
0.41 0.08
0.57 0.11
0.71 0.15
0.83 0.18
x1 K m
1.2299
Kc =
0.4967  Tm 

1 +
τm 
K m 
τm
≤2
Tm
Minimum performance index: other tuning
x 2Tm
x 3Tm
N=∞
Taguchi et al.
(1987) –
x1
x 2 x3
servo/regulator
optimisation. 6.30 0.40 0.08
Model: Method 1 3.20 0.69 0.16
2.18 0.92 0.23
1.67 1.10 0.30
1.38 1.25 0.36
1.18 1.38 0.42
1.05 1.50 0.48
Minimum IAE –
0.77 K u
Shinskey (1988),
0.70K u
p. 148. Model:
Method 1;
0.66K u
α = 0 , β =1,
0.60K u
N =∞.
2
0.1 ≤
Td
α
β
0.68
0.63
0.62
0.59
0.58
0.75
0.70
0.70
0.69
0.69
x 2Tm
α
β
0.60
0.56
0.51
0.47
0.43
0.39
0.36
0.63
0.60
0.57
0.54
0.52
0.51
0.50
0.48Tu
τm
Tm
x1
x2
x3
α
β
0.1
0.2
0.3
0.4
0.5
2.11
1.82
1.61
1.44
1.33
0.94
1.05
1.13
1.22
1.26
x 3Tm
0.21
0.24
0.28
0.31
0.34
0.56
0.54
0.51
0.48
0.49
0.69
0.69
0.68
0.69
0.66
N=∞
τm
Tm
x1
x2
x3
α
β
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.95
0.87
0.81
0.71
0.64
0.56
0.51
1.61 0.54
1.72 0.59
1.82 0.64
2.07 0.75
2.32 0.85
2.82 1.00
3.33 1.09
0.11Tu
0.34
0.31
0.30
0.26
0.24
0.20
0.15
τm
τm
Tm
0.6
0.7
0.8
0.9
1.0
τm
Tm
0.50 1.6
0.51 1.8
0.52 2.0
0.57 2.5
0.63 3.0
0.73 4.0
0.78 5.0
Tm = 0.2
0.42Tu
0.12Tu
τ m Tm = 0.5
0.38Tu
0.12Tu
τ m Tm = 1
0.34Tu
0.12Tu
τ m Tm = 2
1.317
 T 
, Ti = 0.6739τm 1 + m 
τm 

,
2

 τm  
 τm 





+
Td = Tm 1.138
0
.
1992
 τ + T  , N = 10, α = 0 , β = 1 .
τm + Tm 

m
m 




b720_Chapter-03
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FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Minimum IAE –
Shinskey (1994),
p. 167.
Taguchi and
Araki (2000).
Model: Method 1
3
Ti
Kc
0.125Tu τm
2
Td
Comment
0.12Tu
Model: Method 1
α = 0 , β =1, N = ∞ .
4
Kc
Ti
0
5
Kc
Ti
Td
τ m Tm ≤ 1.0
Servo/regulator optimisation; overshoot (servo step) ≤ 20% .
6
Model: Method 7
Ti
Td
K
ABB (2001).
τm Tm ≥ 0.5 .
Taguchi and
Araki (2002) –
servo/regulator
optimisation.
Model: Method 1
Kc
157
c
α = 0 , β =1, N = ∞ .
x1 K m
x1
x2
x 2Tm
x3
α
β
N=∞
x 3Tm
x4
x1
x2
x3
α
β
x4
4.38 1.13 0.091 0.03 0.38 0.1 6.68 0.31 0.089 0.65 0.55 1.5
6.36 0.44 0.078 0.62 0.71 0.5 7.18 0.23 0.097 0.68 0.38 2.0
6.32 0.40 0.081 0.61 0.64 1.0
Load disturbance model = K m e −sτ m (1 + sx 4Tm ) ; τm Tm = 0.2 .
3
Kc =
T
T
Ku
Ku
, u < 2.7 ; K c =
, u ≥ 2.7 .
3.73 − 0.69(Tu τm ) τ m
2.62 − 0.35(Tu τm ) τ m
4
Kc =

1 
0.7382
,
 0.1098 +
K m 
(τm Tm ) − 0.002434 
(
)
Ti = Tm 0.06216 + 3.171(τm Tm ) − 3.058(τm Tm ) + 1.205(τm Tm ) ,
2
α = 0.6830 − 0.4242(τm Tm ) + 0.06568(τm Tm ) .
3
2
5
Kc =

1 
1.224
,
 0.1415 +
K m 
(τm Tm ) − 0.001582 
2
3

τ 
τ  
τ
Ti = Tm  0.01353 + 2.200 m − 1.452  m  + 0.4824 m   ,
Tm
 Tm 
 Tm  

2

τ  
τ
Td = Tm  0.0002783 + 0.4119 m − 0.04943 m   , N fixed (but not specified);
Tm
 Tm  

2
α = 0.6656 − 0.2786
6
Kc =
1.3570  Tm 


K m  τm 
2
τ 
τ 
τm
τ
+ 0.03966  m  , β = 0.6816 − 0.2054 m + 0.03936  m  .
Tm
Tm
 Tm 
 Tm 
0.947
τ 
, Ti = 1.1760Tm  m 
 Tm 
0.738
τ 
, Td = 0.3810Tm  m 
 Tm 
0.995
.
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Handbook of PI and PID Controller Tuning Rules
158
Rule
Tavakoli et al.
(2005).
Model: Method 1
Minimum ITAE
– Setiawan et al.
(2000).
Model: Method 1
7
Kc
Ti
Td
Comment
Kc
Ti
0
M max = 2
Servo/regulator optimisation – minimum IAE; 0.1 ≤ τm Tm ≤ 2 .
x1Tm K m τm
Coefficient values:
x1
x2
τm
Tm
τm
Tm
x1
x2
0.5
0.75
1.0
1.5
2.0
0.554
0.570
0.594
0.645
0.720
1.805
1.373
1.117
0.822
0.679
Shen (2002).
Model:
Method 5;
N = ∞ ,β = 0.
8
Kc
Kuwata (1987).
9
Kc
0
x 2τm
3
4
5
6
0.900
1.090
1.290
1.520
∞
Minimise
∫
0.547
0.480
0.445
0.428
α =1
τm
Tm
x1
x2
7
8
9
10
1.740
1.930
2.120
2.410
0.413
0.398
0.385
0.389
∞
r ( t ) − y( t ) dt +
0
∫ d( t) − y(t) dt ; M
s
≤2.
0
Ti
Td
0 < τm Tm < ∞
Direct synthesis: time domain criteria
Model: Method 1
0
Ti
7
Kc =
0.450(τ m Tm ) + 1.083
1 
Tm 
τ
, α = 0.417 − 0.225 m .
0.273 + 0.571  , Ti =
Km 
τm 
(τ m Tm ) + 0.176
Tm
8
Kc =
2

 τm  
 τm 
Tm
 ,
 + 11.15
exp 2.94 − 11.63

τ +T  
K m τm

m 
 m
 τm + Tm 


2

 τm  
 τm 
 ,
 + 0.86
Ti = τm exp 1.88 − 3.63

τ +T  

m 
 m
 τm + Tm 


2

 τm  
 τm 

,




Td = τm exp − 0.25 − 0.06
− 1.99
τm + Tm 
τm + Tm  






 τm
α = 1 − exp − 0.22 − 0.90

 τ m + Tm

9
Kc =

 τm
 + 1.45

 τ m + Tm



2
.


 0.833τm (2Tm + τm )  1.667 τm (2Tm + τm )
0.8(Tm + τm )
1

−
, Ti = 1 −

K m (Tm + τm − x1 ) K m
Tm + τm
(Tm + τm )2 

with x1 =
(Tm + τm )2 − 0.267τm 2 (3Tm + τm )
, α = 1.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Kuwata (1987),
Suyama (1993).
10
Shigemasa et al.
(1987),
Suyama (1993).
α = 1, β = 1 ,
N =∞.
11
Kc
Ti
Td
Ti
Td
159
Comment
Model: Method 1
th
Kc
Closed loop transfer function is 4 order.
Model: Method 1
Ti
Td
Closed loop transfer function is 4th order Butterworth.
12
Ti
Td
Kc
Toshiba AdTune
th
Closed loop transfer function is 4 order ITAE.
TOSDIC 211D8
13
Ti
Td
product.
Kc
Closed loop transfer function is 4th order Bessel.
14
Ti
Td
Kc
Closed loop transfer function is 4th order Binomial.
10
Kc =
1
Km
 0.2τm 3 + 1.2τm 2Tm + 5.4τm Tm 2 + 9.6Tm 3 
 , α = 1, β = 1 , N = ∞ ;

2
τm (τm + 3Tm )


Ti = 1.667 τm
3
τm + 3Tm
2 (τ m + 3Tm )
− 1.389τm
,
τm + 2Tm
(τm + 2Tm )4
Td =
11
Kc =
1.3319(τm + 2Tm )3
K m τm (τm + 3Tm )2
−
Kc =
2.6242(τm + 2Tm )3
K m τm (τm + 3Tm )2
−
Kc =
0.9450(τm + 2Tm )3
K m τm (τm + 3Tm )2
−
Kc =
0.5622(τm + 2Tm )3
K m τm (τm + 3Tm )2
−
1.5095(τm + 2Tm )2 − (τm + Tm )(τm + 3Tm )
1.3319(τm + 2Tm )3 − τm (τm + 3Tm )2
2.3130(τm + 2Tm )2 − (τm + Tm )(τm + 3Tm )
.
2.6242(τm + 2Tm )3 − τm (τm + 3Tm )2
.
3
1
τ + 3Tm
2 (τ m + 3Tm )
, Ti = 3.3333τm m
− 3.527 τm
,
Km
τm + 2Tm
(τm + 2Tm )4
Td = τm (τm + 3Tm )
14
.
3
1
τ + 3Tm
2 (τ m + 3Tm )
, Ti = 1.8898τm m
− 0.720τm
,
Km
τm + 2Tm
(τm + 2Tm )4
Td = τm (τm + 3Tm )
13
1.2(τm + 2Tm )3 − τm (τm + 3Tm )2
3
1
τ + 3Tm
2 (τ m + 3Tm )
, Ti = 2.2532τm m
− 1.692τm
,
Km
τm + 2Tm
(τm + 2Tm )4
Td = τm (τm + 3Tm )
12
Tm 2 τm (τm + 3Tm )
1.3444(τm + 2Tm )2 − (τm + Tm )(τm + 3Tm )
1.9450(τm + 2Tm )3 − τm (τm + 3Tm )2
.
3
1
τ + 3Tm
2 (τ m + 3Tm )
, Ti = 5.3337 τm m
− 9.488τm
,
Km
τm + 2Tm
(τm + 2Tm )4
Td = τm (τm + 3Tm )
1.1244(τm + 2Tm )2 − (τm + Tm )(τm + 3Tm )
0.5622(τm + 2Tm )3 − τm (τm + 3Tm )2
.
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
160
Rule
Kc
Kc =
Comment
Td
Ti
Kc
Ti
17
Kc
Ti
Td
18
Kc
Ti
0
16
Kasahara et al.
(1997).
Model: Method 1
Chien et al.
(1999).
Ti
Kc
15
Nomura et al.
(1993), Kuwata
(1987).
15
17-Mar-2009
Model:
Method 1;
τ
0.01 ≤ m ≤ 10 .
Tm
0
Closed loop
transfer function
is 4th order.
Model: Method 1
 T + τ m − x1 
0.8(Tm + τm )(Tm + 0.5τm )
1
,
−
, Ti = 2.5(1.25x1 − 0.25) m

K m τm (Tm + 0.333τm )
Km
 Tm + τ m 
with x1 = 0.2τm 2 + 0.4τmTm + Tm 2 ; α = 1 .
16
Kc =
(1 + (τm
Tm )) − 1.667 τm (1 + 0.5(τm Tm ))
,
1.667 K m τm (1 + 0.5(τm Tm ))
2
2
 3.333τm (1 + 0.5(τm Tm ))

10(1 + 0.5(τm Tm ))
, α = 1.
Ti = 1 −
2
1 + (τm Tm )
 (3Tm + τm )(1 + (τm Tm )) 
17
Kc =
3
2
2
3
1  0.2τm + 1.2τm Tm + 5.4τm Tm + 9.6Tm 
,

2
K m 
τm (τm + 3Tm )

Ti = 1.667 τm
Td =
3
τm + 3Tm
2 (τ m + 3Tm )
− 1.389τm
,
τm + 2Tm
(τm + 2Tm )4
Tm 2 τm (τm + 3Tm )
1.2(τm + 2Tm )3 − τm (τm + 3Tm )2
 τ + 3Tm
α = 1 − 0.723τ m  m
 τ m + 2Tm
18
Kc =

 τ + 3Tm
 1.667 τ m  m
 
 τ m + 2Tm
− TCL 2 2 + 1.414TCL 2Tm + τm Tm
(
, β = 1 − 0.261τm
K m TCL 2 2 + 1.414TCL 2 τm + τm 2
−1

τ 2 (τ + 3Tm )3 
 − 1.389 m m
 .
(τ m + 2Tm )4 

2
)
, Ti =
2
τ + 3Tm  1

 m
 τ + 2T  T T , N = ∞ ,
m 
i d
 m
2
− TCL 2 + 1.414TCL 2Tm + Tm τm
;
Tm + τm
Underdamped system response: ξ = 0.707 ; τ m > 0.2Tm ; α = 1 .
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Chien et al.
(1999).
Chidambaram
(2000a).
Kasahara et al.
(2001).
Model: Method 1
19
Kc
Ti
Td
Comment
Ti
Td
Model: Method 1
161
α = 1, β = 1 , N = ∞ .
Tm Ti
20
TCL 2 + Ti τm
Ti
0
Model:
Method 1;
choose TCL , ξ .
21
Kc
Ti
Td
OS = 0%
22
Kc
Ti
Td
OS = 10%
α = 1, β = 1 , N = ∞ .
19
Underdamped system response: ξ = 0.707 . τ m > 0.2Tm .
Kc =
1.414TCL 2Tm + τmTm + 0.25τm 2 − TCL 2 2
(
K m TCL 2 2 + 0.707TCL 2 τm + 0.25τm 2
2
Ti =
)
,
2
1.414TCL 2Tm + τmTm + 0.25τm − TCL 2
,
Tm + 0.5τm
0.707TmTCL 2 τm + 0.25Tm τm 2 − 0.5τmTCL 2 2
Td =
Tm τm + 0.25τm 2 + 1.414TCL 2Tm − TCL 2 2
.
2
Tm τm + 2TCL ξ − TCL
T − ξTCL
K K [T + τm ] − Ti
, α= i
or α = c m i
.
Tm + τm
Ti
2K c K m Ti
20
Ti =
21
Kc =
0.281(τm + 2Tm )3
K m τm (τm + 3Tm )
2
−
3
1
τ + 3Tm
2 (τ m + 3Tm )
, Ti = 5.33τm m
− 18.98τm
,
Km
τm + 2Tm
(τm + 2Tm )4
Td = τm (τm + 3Tm )
22
Kc =
0.562(τm + 2Tm )2 − (τm + Tm )(τm + 3Tm )
0.281(τm + 2Tm )3 − τm (τm + 3Tm )2
.
1.066Tm − 0.467 τm
4.695τm (1.066Tm − 0.467 τm )
, Ti =
,
K m τm
τm + 2Tm
 0.331Tm − 0.334τm 
.
Td = τm 

 1.066Tm − 0.467 τm 
b720_Chapter-03
Rule
Ogawa and
Katayama
(2001).
Srinivas and
Chidambaram
(2001).
Srinivas and
Chidambaram
(2001).
Chidambaram
(2002),
pp. 157–158.
Kc
23
Kc
Ti
Td
Comment
Ti
Td
N = 10
Minimum ISE: servo. α = 1 , β = 1 ; Model: Method 1.
24
0.9Tm
K mτm
3.33τ m
0
Kc
Ti
Td
1.2Tm
K m τm
2τ m
0.5τ m
25
26
Desired closed loop response =
e −sτCL
. Kc =
2

τ
 + 0.6974 CL

 Tm
2
0.01 ≤
TCL
≤ 1.00 ;
Tm
Sample K c , Ti taken by the authors correspond to the process reaction curve
τm
1
τ
−
= 0.30 − 1.11 m for
Ti K c K m
Tm
this tuning rule.
Sample K c , Ti , Td taken by the authors correspond to the process reaction tuning
rule of Ziegler and Nichols (1942); α =
β=
26
Model:
Method 1;
N=∞.
Model:
Method 2;
β =1, N = ∞ .

τ
 + 0.007393 , 0.05 ≤ CL ≤ 1.00 .
Tm

tuning rule of Ziegler and Nichols (1942); α =
25
Model: Method 1
1  (τCL Tm ) − 2(TCL Tm ) + 4 

,
K m  (τCL Tm ) + 2(TCL Tm ) 
(1 + sTCL )
 ((τ T ) + 2(TCL Tm ))((τCL Tm ) − 2(TCL Tm ) + 4 ) 
,
Ti = Tm  CL m

2(τCL Tm ) + 4


 (τCL Tm )((τCL Tm ) + 4(TCL Tm ) − 2(τCL Tm )2 ) 
;
Td = Tm 
 ((τCL Tm ) + 2(TCL Tm ))((τCL Tm ) − 2(TCL Tm ) + 4 ) 


τ
TCL
= −0.1902 CL
Tm
 Tm
24
FA
Handbook of PI and PID Controller Tuning Rules
162
23
17-Mar-2009
1
Td
τm
1
τ
−
= 0.5 − 0.83 m ,
Ti K c K m
Tm
2

Tm
τ 
− m  = −0.16 for this tuning rule.
τm −
K c K m 2Ti 

α = 0.6762 − 0.4332(τm Tm ) , τm Tm ≤ 0.8 ;
or α = 0.207 + 3.149(τm Tm ) − 8.405(τm Tm )2 + 8.431(τm Tm )3 , τm Tm < 0.5 ;
α = 0.9416 − 0.44(τm Tm ) , 0.5 ≤ τm Tm ≤ 0.9 .
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
27
Kc
Haeri (2002).
27
Kc =
1
Km
Td
Comment
Ti
Td
Model: Method 1
α = 0 , β =1, N = ∞ .
Ozawa et al.
(2003).
28
Kim et al.
(2004).
Model: Method 1
Ti
163
Kc
x1 K m
τ m Tm
Ti
Td
x 2 Tm
x 3Tm
Model:
Method 1;
0% ≤ OS ≤ 10%.
Coefficient values:
x2
x3
x1
α
β
0.1
12.5
0.22
0.04
0.68
0.75
0.2
6.1
0.41
0.08
0.63
0.70
0.3
4.1
0.57
0.11
0.62
0.70
0.4
3.1
0.71
0.15
0.59
0.69
0.5
2.5
0.83
0.18
0.58
0.69
0.6
2.11
0.94
0.21
0.56
0.69
0.7
1.82
1.05
0.24
0.54
0.69
0.8
1.61
1.13
0.28
0.51
0.68
0.9
1.44
1.22
0.31
0.48
0.69
1.0
1.33
1.26
0.34
0.49
0.66
OS < 20% (servo step); settling time ≤ that of corresponding rule of
Chien et al. (1952); a cost function is minimised.


6.84
+ 0.64 ,

0.7
3.5
1 + 2.33(τm Tm ) + 7.82(τm Tm )

[
[2.15 − 0.76(τ
]
T ) ] , 0.29 < τ
Ti = τm 0.95 + 2.58(τm Tm ) + 3.57(τm Tm ) , 0 < τm Tm ≤ 0.29 ,
Ti = τm
2
Tm ) + 0.33(τm
m
2
m
m
Tm < 4 ;
2.8

 τm  
 τm 
 τm 
τ
  , 0 < m ≤ 1.79 ;
 + 3.94
 − 4.65
Td = τm 0.29

T +τ  
T +τ 
T
T
+
τ

m
m 
m 
m 
 m
 m
 m


2
[
Td = τm 0.87(τm Tm ) − 0.49(τm Tm ) + 0.09(τm Tm )
28
2
1.6193(τm + 2Tm )
3
Kc =
K m τm (τm + 3Tm )2
−
2.8
] , 1.79 < τ
m
Tm < 4 .
3
1
τ + 3Tm
2 (τ m + 3Tm )
, Ti = 2.7051τm m
− 1.6706τm
,
Km
τm + 2Tm
(τm + 2Tm )4
Td = τm (τm + 3Tm )
1.7793(τm + 2Tm )2 − (τm + Tm )(τm + 3Tm )
1.6193(τm + 2Tm )3 − τm (τm + 3Tm )2
, α = 1, β = 1 , N = ∞ .
b720_Chapter-03
Rule
Kotaki et al.
(2005a).
Model:
Method 1;
α = 1, β = 1 ,
N =∞.
Kc =
Kc
Ti
x1 K m
x 2Tm
Td
x 3Tm
Closed loop transfer function is 4 order; p = % allowed perturbation
on individual model parameters.
p
x1
x2
x3
τm Tm
6.8016
3.2091
1.4202
0.8298
0.5383
0.3661
29
Kc
30
Kc
31
0.3482
0.0243
0.1
0.6001
0.0453
0.2
0.9000
0.0754
0.4
1.0194
0.0850
0.6
1.0293
0.0656
0.8
0.9685
0.0042
1.0
Direct synthesis: frequency domain criteria
0
Ti
Ms
Ti
0
0.7Tm
0
Kc
0.4Tm
K mτm
= 1.25
0.1 ≤
τm
≤2
Tm
α = 0.5; Model: Method 5 or 17.
(
K m 4 + 4 x1x 2 (τm Tm ) + x 2 2 (τm Tm )2
)
,
2 x 2 2 (τm Tm ) − 4 + x 2 (x 2 (τm Tm ) + 4x1 )((τm Tm ) + 2 − 2 x1x 2 (τm Tm ))
2 x 2 2 (τm Tm ) − x 2 4 (τm Tm )2 + 2 x 2 2 ((τm Tm ) + 2 − 2 x1x 2 (τm Tm ))
x1 = 0.5 +
0.25
x2
2
,
0.486 + 0.675(Tm τm )
τm
τ
;
< 1.91 ; x1 = 1 , m ≥ 1.91 ; x 2 =
1 + 0.0972(τm Tm )
Tm
Tm
,
α = 0.488 − 0.985(τm Tm ) , τm Tm ≤ 0.495 ; α = 0 , τm Tm > 0.495 .
2
2
2
0.29e −2.7 τ + 3.7 τ Tm
, Ti = 8.9τm e−6.6 τ + 3.0 τ or 0.79Tm e −1.4 τ+ 2.4 τ ,
K m τm
2
α = 1 − 0.81e 0.73τ+1.9 τ , M s = 1.4.
2
31
32
36
46
59
77
100
Model: Method 5
or 17;
τ
0.14 ≤ m ≤ 5.5.
Tm
2 x 2 2 (τm Tm ) − 4 + x 2 (x 2 (τm Tm ) + 4 x1 )((τm Tm ) + 2 − 2x1x 2 (τm Tm ))
Ti = Tm
Kc =
Comment
th
Lee et al. (1992).
Model: Method 1
Dominant pole
design – Åström
and Hägglund
(1995), pp. 204–
208.
Modified
Ziegler–Nichols
– Åström and
Hägglund
(1995), p. 208.
30
FA
Handbook of PI and PID Controller Tuning Rules
164
29
17-Mar-2009
2
0.78e −4.1τ + 5.7 τ Tm
, α = 1 − 0.44e 0.78 τ−0.45τ , M s = 2.0.
Kc =
K m τm
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Huang et al.
(2000).
Model: Method 1
Kc
λ (Tm + 0.4τm )
K m τm
α=
Ti
Td
Comment
Tm + 0.4 τ m
0.4Tm τ m
Tm + 0.4τ m
For λ = 0.80 ,
A m = 2.13 .
Tm
− 1 , β = 0 , N = min[20,20Td ] .
Tm + ατ m
Hägglund and
Åström (2002).
Model:
Method 5;
M s = 1.4;
0.35Tm 0.6
−
K mτm
Km
7τ m
0.35Tm 0.6
−
K mτm
Km
0.8Tm
0 < α < 0.5 .
0.25Tm
K mτm
0.8Tm
Hägglund and
Åström (2002).
32
Leva and
Colombo (2004).
33
τm
< 0.11
Tm
0
0
Ti
Kc
0.11 <
τm
< 0.17
Tm
0.17 <
τm
< 0.2
Tm
τm Tm < 0.5
M s = 1.4; 0 < α < 0.5 , Model: Method 1.
Robust
Ti
Kc
Model:
Method 50
α = 0 , β =1,
N =∞.
Td
Tm + 0.5τm
Tm + 0.5τm
Tm τm
Cooper (2006a). K (λ + 0.5τ )
2Tm + τm
m
m
Model: Method 1
λ > [0.1Tm ,0.8τm ] (‘aggressive’ tuning); λ > [Tm ,8τm ] (‘moderate’
tuning); λ > [10Tm ,80τm ] (‘conservative’ tuning).
32
Kc =
6.8τm Tm
1 
T 
 0.14 + 0.28 m  , Ti = 0.33τm +
.
τm 
10τm + Tm
K m 
33
Kc =
2
2


1
τm
τm
,
 , Ti = Tm +
Tm +
K m (τm + λ ) 
2(τm + λ ) 
2(τm + λ )
Td =
τm Tm (τm + λ )
2Tm (τm + λ ) + τm 2
−
2Tm (τ m + λ )2
λτ m
,N=
−1 .
2(τm + λ )
λ 2Tm (τ m + λ ) + τ m 2
[
]
3∆τ m
, ∆τ m = maximum variation in the time delay,
π
with only variation in the time delay considered (Leva and Colombo, 2004); λ
α not specified; β = 1 . λ ≥
^
chosen so that nominal cut-off frequency = ωu (Leva, 2005).
165
b720_Chapter-03
17-Mar-2009
Handbook of PI and PID Controller Tuning Rules
166
Rule
Kc
Ti
Comment
Td
34
Ti
Td
Kc
Shamsuzzoha
and Lee (2007a). M s = 1.6 M s = 1.7 M s = 1.8 M s = 1.9 M s = 2.0
Model:
0.11
0.10
0.08
0.07
0.06
Method 1;
0.20
0.17
0.15
0.14
0.12
N =∞,
0.35
0.31
0.27
0.25
0.22
α = 0.6 , β = 0 ;
0.41
0.37
0.34
0.30
coefficients of x1 0.46
0.55
0.49
0.44
0.40
0.36
(deduced from
graph) indicated 0.62
0.55
0.50
0.46
0.42
in columns.
0.68
0.60
0.55
0.51
0.47
Kc =
λ = x1Tm
τm = 0.05Tm
τm = 0.1Tm
τm = 0.2Tm
τm = 0.3Tm
τm = 0.4Tm
τm = 0.5Tm
τm = 0.6Tm
0.72
0.65
0.59
0.55
0.50
0.77
0.69
0.63
0.59
0.53
τm = 0.8Tm
0.81
0.72
0.66
0.62
0.56
τm = 0.9Tm
0.84
0.76
0.69
0.65
0.59
τm = 1.0Tm
0.91
0.82
0.75
0.70
0.64
τm = 1.25Tm
0.97
0.87
0.80
0.75
0.70
τm = 1.5Tm
τm = 0.7Tm
-
0.91
0.84
0.78
0.73
τm = 1.75Tm
-
0.96
0.87
0.81
0.76
τm = 2.0Tm
Ultimate cycle
0
x 2 Tu
x 1K u
Kuwata (1987);
x 1 , x 2 deduced
from graphs;
α = 1.
34
FA
x1
x2
τm
Tu
x1
x2
τm
Tu
x1
x2
τm
Tu
0.37
0.32
0.29
0.26
0.03
0.07
0.08
0.09
0.25
0.28
0.30
0.33
0.24
0.22
0.21
0.20
0.10
0.11
0.12
0.13
0.35
0.38
0.40
0.43
0.20
0.20
0.14
0.14
0.45
0.48
2
2
3λ2 + 2x1τm − 0.5τm − x1
Ti
,
, Ti = Tm + 2 x1 −
K m (3λ + τm − 2x1 )
3λ + τm − 2x1
2
2Tm x1 + x1 −
Td =
x1 = Tm 1 −

λ3 + 0.167 τm 3 − x1τm 2 + x12 τm
2
2
3λ2 + 2x1τm − 0.5τm − x1
3λ + τm − 2 x1
,
Ti
3λ + τm − 2x1
(1 − (λ Tm ))3 e− τ
m
Tm
.

b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Kuwata (1987);
x 1K u
x 1 , x 2 deduced
x1 x 2 x 3
from graphs;
α = 1, β = 1 ,
0.69 0.63 0.08
N =∞.
0.57 0.48 0.08
0.47 0.37 0.08
35
Hang and
0. 6 K u
Åström (1988a).
36
0. 6 K u
Model:
Method 2;
β = 1 , N = 10 .
0. 6 K u
Hang et al.
(1991) – servo.
37
38
Hang and Cao
(1996).
Model:
Method 37
35
36
37
x1
τm
Tu
39
x 3 Tu
x2
x3
τm
Tu
x1
x2
x3
τm
Tu
0.25 0.38 0.28 0.07 0.36 0.20 0.14 0.01 0.48
0.28 0.31 0.20 0.05 0.40 0.20 0.14 0 0.50
0.32 0.26 0.18 0.04 0.42
0.5Tu
0.125Tu
Ti
0.125Tu
0.335Tu
0.125Tu
0. 6 K u
0.5Tu
0.125Tu
0. 6 K u
'
0.125Tu
0.222 K Tu
0. 6 K u
39
Ti
Td
τm Tm > 1.0 ;
α = 0.2 .
Model:
Method 2;
β = 1 , N = 10 .
0.1 ≤
τm
< 0.5 ;
Tm
N = 10; β = 1 .
τ
τ
1.66τm τ m
,
< 0.3 ; α = 1 − 2OS − m , 0.3 ≤ m < 0.6 ;
Tm
Tm
Tm
Tm
OS = % overshoot.

τm
τm
τm
τm 
Tu . α =
Ti = 0.51.5 − 0.83
− 0.6 , 0.6 ≤
< 0.8 ; α = 0.2 , 0.8 ≤
< 1.0 .
Tm
Tm
Tm
Tm 

α = 1 − 2(OS − 0.1) −
0.16 ≤
τm
15 − K '
< 0.57 ; α = 1 −
, 10% overshoot;
Tm
15 + K '
α = 1−
38
Comment
Td
x 2 Tu
167
0.57 ≤
 11[τ m Tm ] + 13 
36
 .
, 20% overshoot with K ' = 2
'
27 + 5K
 37[τ m Tm ] − 4 
τm
8 4

< 0.96 ; α = 1 −  K ' + 1 , 20% overshoot, 10% undershoot.
17  9
Tm



τ 
τ T
Ti =  0.53 − 0.22 m Tu , Td =  0.53 − 0.22 m  u . α equals 1.0, 0.2 and
Tm  4
Tm 


0.40(τ m Tm ) − 0.05(τ m Tm ) + 0.58 in different circumstances.
2
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Handbook of PI and PID Controller Tuning Rules
168
3.3.8 Two degree of freedom controller 2




T
s
1
d
 1 + b f 1s (F(s) R (s) − Y(s) ) − K Y(s) ,
U(s) = K c 1 +
+
0
 Ti s
T  1 + a f 1s
1+ d s 

N 

1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4
F(s) =
1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5 s 4
R(s)
F(s)
+
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N



 1 + b f 1s
 1 + a f 1s
s

+ U(s)
−
K m e − sτ
1 + sTm
m
K0
Table 16: PID controller tuning rules – FOLPD model G m (s) =
Rule
Kc
Ti
Td
Y(s)
K m e − sτ m
1 + sTm
Comment
Direct synthesis: time domain criteria
0.375(τ m + 2Tm )
N=∞
Tm τ m
Normey-Rico et
Tm + 0.5τ m
K mτm
2Tm + τ m
al. (2000).
a
=
0
.
13
τ
,
b
=
0
;
a
=
0. 5τ m , b f 2 = 0. 2 τ m ;
f1
m
f1
f2
Model: Method 1
a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , bf 3 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 .
Robust
Lee and Edgar
(2002).
Model: Method 1
1
Kc
0
Ti
a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , a f 4 = 0 ,
a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0.25K u .
1
2
Kc =
1 + 0.25K u K mTi
T − 0.25K u K m τm
τm
.
, Ti = m
+
K m (λ + τm )
1 + 0.25K u K m
2(λ + τm )
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Lee and Edgar
(2002).
Model: Method 1
2
Kc
Ti
Td
Comment
Kc
Ti
Td
N=∞
169
a f 1 = Td , b f 1 = 0 ; a f 2 = 0 , b f 2 = 0 ; a f 3 = 0 , a f 4 = 0 ,
a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0.25K u .
Zhang et al.
(2002c).
0.1τm ≤ λ1 ≤ τm ,
0.1τm ≤ λ 2 ≤ τm .
Tm
K m (λ 2 + τm )
0
Tm
Model: Method 1
a f 1 = 0.5τm λ 2 (λ 2 + τm ) , b f 1 = 0.5τm ; a f 2 = λ1 , b f 2 = λ 2 ; a f 3 = 0 ,
a f 4 = 0 , a f 5 = 0 , bf 3 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 .
0.5τm
0.167 τm
0 ≤ x1 ≤ 1
Kc
Shamsuzzoha
and Lee (2007c). a f 2 = b f 1 , b f 2 = x1bf 1 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 ,
Model: Method 1 b f 5 = 0 , K 0 = 0 . λ is given as a multiple of Tm (from a graph, for
3
0.01 ≤ τm Tm ≤ 5 ), for M s = 1.4,1.5,1.6,1.8 and 1.9.
Zhang et al.
(2003).
Model: Method 1
4
Ultimate cycle
0.5Tu
0.125Tu
0.6K u
N=∞
a f 1 = 0 , b f 1 = 0 ; a f 2 = Ti , b f 2 = x 2Ti ; a f 3 = Ti Td , b f 3 = x1Ti Td ,
a f 4 = 0 , a f 5 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 .
2
Kc =
2

1 + 0.25K u K m  Tm − 0.25K u K m τm
τm
+
+ Td  ,

K m (λ + τm )  1 + 0.25K u K m
2(λ + τm )

Ti =
Tm − 0.25K u K m τm
τm
+
+ Td ,
1 + 0.25K u K m
2(λ + τm )
2
2
 0.25K u K m
τm
τm
(T − 0.25K u K m τm )τm 2 
Td = τm 
−
+
+ m

2
2(1 + 0.25K u K m )(λ + τm ) 
 2(1 + 0.25K u K m ) 6(λ + τm ) 4(λ + τm )
3
Kc =
τm 
 
λ  − Tm 
τm
e
, b f 1 = Tm 1 − 1 −
,
  Tm 

2K m (2λ − b f 1 + τm )


af1 =
4
x1 =
0.5
0.5b f 1τm + λτ m + λ2
− Tm , N = ∞ .
2λ − b f 1 + τ m
2
T + K m K c [Ti − τm ]
2Tm Ti + 2K m K cTi [Td − τm ] + K m τm K c
.
, x2 = i
2K m K cTi Td
K m K c Ti
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Handbook of PI and PID Controller Tuning Rules
170
3.3.9 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s)




(
)
1
−
β
T
s
1 + b f 1s
[
]
1
−
χ
δ
d

,
F1 (s) = K c  [1 − α ] +
+
+


T
Ti s
1 + Ti s
1 + a f 1s + a f 2 s 2
d
1
+
s


N 





[
]
K (b + b f 4 s )
1
−
ϕ
T
s
1+ bf 2s
[
]
−
φ
1
d 
F2 (s) = K c  [1 − ε] +
+
− 0 f3
2

 1 + a f 3s + a f 4 s
T
Ti s
1+ a f 5s
1+ d s 

N 

R(s)
+
F1 (s)
U(s)
Y(s)
Model
−
F2 (s)
Table 17: PID controller tuning rules – FOLPD model G m (s) =
Rule
Kc
Hiroi and
Terauchi (1986)
– regulator.
Model:
Method 2;
τ
0.1 < m < 1 .
Tm
0.95Tm K m τm
1.2Tm K m τm
Ti
Td
K m e − sτ m
1 + sTm
Comment
Process reaction
2.38τm
0.42τm
0% overshoot
0.42τm
20% overshoot
2τm
α = 0.6 ; β = 1 or β = −1.48 ; χ = 0 ; δ = 0.15 ; N not specified;
bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 ;
af 4 = 0 , K0 = 0 .
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kaya and Scheib
(1988).
Model:
Method 5;
τ
0 < m ≤ 1.
Tm
Kc
Comment
Td
Ti
Minimum performance index: regulator tuning
Minimum IAE.
x1K c
Kc
0
2
Minimum ISE.
x1K c
K
1
c
3
Minimum ITAE.
x1K c
Kc
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 , K 0 = 1 ; b f 3 = 0 ; b f 4 = x 2 ; b f 5 = 0.1x 2 .
τ
 Tm 
Tm  τ m 
0 < m ≤1




0
T
m
x 3  Tm 
 τm 
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ;
x2
x1
Km
Kaya and Scheib
(1988).
Model: Method 5
x4
φ = 0 ; ϕ = 0 ; b f 2 = x 5Tm (τm Tm )x 6 ; a f 3 = 0.1bf 3 ; a f 4 = 0 ; K 0 = 0.
Coefficient values:
x1
x2
x3
x4
x5
x6
Minimum IAE
Minimum ISE
Minimum ITAE
0.91
1.1147
0.7058
0.7938
0.8992
0.8872
0.8826
1
Kc =
1.31509  Tm 


K m  τm 
Kc =
1.3466  Tm 


K m  τm 
0.9308
2
Kc =
1.3176  Tm 


K m  τm 
0.7937
3
1.01495
0.9324
1.03326
1.3756
, x1 =
Tm  τm 


1.2587  Tm 
1.25738
, x1 =
Tm  τm 


1.6585  Tm 
, x1 =
Tm  τm 


1.12499  Tm 
1.00403
0.8753
0.99138
0.5414
0.56508
0.60006
τ 
, x 2 = 0.5655Tm  m 
 Tm 
0.4576
τ 
, x 2 = 0.79715Tm  m 
 Tm 
1.42603
0.7848
0.91107
0.971
.
0.41941
τ
, x 2 = 0.79715Tm  m
 Tm
.




0.41941
.
171
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Handbook of PI and PID Controller Tuning Rules
172
Rule
Kc
Kaya and Scheib
(1988).
Model:
Method 5;
τ
0 < m ≤1
Tm
4
Kc
5
Kc
6
Kc
Ti
Comment
Td
Minimum performance index: servo tuning
Minimum IAE.
x1K c
0
Minimum ISE.
x1K c
Minimum ITAE.
x1K c
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 , K 0 = 1 ; b f 3 = 0 ; b f 4 = x 2 ; b f 5 = 0.1x 2 .
x1
Km
Kaya and Scheib
(1988).
Model: Method 5
 Tm

 τm



Tm
x2
τ
x3 − x4 m
Tm
0<
0
τm
≤1
Tm
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ;
φ = 0 ; ϕ = 0 ; b f 2 = x 5Tm (τm Tm )x 6 ; a f 3 = 0.1bf 3 ; a f 4 = 0 ; K 0 = 0.
Coefficient values:
x1
x2
x3
x4
x5
x6
Minimum IAE
Minimum ISE
Minimum ITAE
0.81699
1.1427
0.8326
Minimum ISE –
Zhuang (1992),
Zhuang and
Atherton (1993).
7
Kc
1.09112
0.99223
1.00268
Ti
8
Kc
Ti
0.22387
0.35269
0.00854
0
0.44278
0.35308
0.44243
0 .1 ≤
0.97186
0.78088
1.11499
τ m Tm ≤ 1
1.1 ≤ τm Tm ≤ 2
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 , K 0 = 1 ; b f 3 = 0 ; b f 4 = x1 ; b f 5 = 0.1x1 .
Model: Method 1 or Method 5 or Method 6 or Method 15.
0.81314
4
Kc =
1.13031  Tm 


K m  τm 
5
Kc =
1.26239  Tm 


K m  τm 
6
1.004
0.9365
0.7607
τ 
Tm
, x 2 = 0.32175Tm  m 
5.7527 − 5.7241(τm Tm )
 Tm 
, x1 =
0.8388
0.98384  Tm 


Kc =
K m  τm 
, x1 =
τ 
Tm
, x 2 = 0.47617Tm  m 
6.0356 − 6.0191(τm Tm )
 Tm 
0.49851
, x1 =
0.17707
.
0.24572
.
Tm
,
2.71348 − 2.29778(τ m Tm )
x 2 = 0.21443Tm (τm Tm )
0.16768
7
Kc =
1.260  Tm 


K m  τm 
8
Kc =
1.295  Tm 


K m  τm 
0.887
0.886
, Ti =
τ 
Tm
, x1 = 0.375Tm  m 
0.701 − 0.147(τm Tm )
 Tm 
, Ti =
τ 
Tm
, x1 = 0.378Tm  m 
0.661 − 0.110(τm Tm )
 Tm 
0.619
.
0.756
.
.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Kc
Ti
Kc
Ti
9
Minimum ISTSE
– Zhuang (1992),
Zhuang and
Atherton (1993).
10
Comment
Td
0
173
0.1 ≤ τ m Tm ≤ 1
1.1 ≤ τm Tm ≤ 2
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 , K 0 = 1 ; b f 3 = 0 ; b f 4 = x1 ; b f 5 = 0.1x1 .
Model: Method 1 or Method 5 or Method 6 or Method 15.
11
0.1 ≤ τ m Tm ≤ 1
Ti
Kc
0
Minimum ISTES
12
Ti
1.1 ≤ τm Tm ≤ 2
Kc
– Zhuang (1992),
Zhuang and
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
Atherton (1993).
b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 , K 0 = 1 ; b f 3 = 0 ; b f 4 = x1 ; b f 5 = 0.1x1 .
Model: Method 1 or Method 5 or Method 6 or Method 15.
13
Model: Method 1
Ti
Kc
0
Minimum ISTSE
∧
∧
14
Model:
Kc
– Zhuang (1992),
0.271 K u T u K m
Method 37
Zhuang and
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
Atherton (1993).
b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 , K 0 = 1 ; b f 3 = 0 ; b f 4 = x1 ; b f 5 = 0.1x1 .
0.930
9
Kc =
1.053  Tm 


K m  τm 
10
Kc =
1.120  Tm

K m  τ m
11
Kc =
0.942  Tm 


K m  τm 
12
13
1.001  Tm 


Kc =
K m  τm 
Kc =
τ 
Tm
, x1 = 0.349Tm  m 
0.736 − 0.126(τm Tm )
 Tm 
0.907
τ 
Tm
, x1 = 0.350Tm  m 
0.720 − 0.114(τm Tm )
 Tm 
0.811
Ti =
, Ti =
τ 
Tm
, x1 = 0.308Tm  m 
0.770 − 0.130(τm Tm )
 Tm 
, Ti =




0.625
0.933
0.624
τ 
Tm
, Ti =
, x1 = 0.308Tm  m 
0.754 − 0.116(τm Tm )
 Tm 
.
.
0.897
.
0.813
.
4.437 K m K u − 1.587
K u , Ti = 0.037(5.89K m K u + 1)Tu , x1 = 0.112Tu
8.024K m K u − 1.435
0.1 ≤ τm Tm ≤ 2.0 .
∧
14
Kc =
2.354K m K u − 0.696
∧
3.363K m K u + 0.517
∧
∧
K u , x1 = 0.116 T u , 0.1 ≤
τm
≤ 2.0 .
Tm
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174
Rule
Kc
Ti
Td
Comment
Minimum performance index: other tuning
15
Åström and
Ti
Td
Kc
Hägglund (2004).
Model: Method 1 α = 0 , τm Tm ≤ 1 ; α = 1 , τm Tm > 1 ; β = 1 ; χ = 0 ; δ = 0 ; N = ∞ ;
bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 ; K 0 = 0 .
16
Åström and
Ti
Td
Kc
Hägglund (2004).
α = 0 , β = 1 ; χ = 0 ; δ = 0 ; 4 ≤ N ≤ 10 ; b f 1 = 0 , a f 1 = 0 , a f 2 = 0 ;
Model:
Method 56
ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 ; a f 3 = 2Td N ; a f 4 = Td 2 N 2 ; K 0 = 0 .
15
Kc =
0.4τm + 0.8Tm
0.5τm Tm
1 
Tm 
τm ; Td =
.
0.2 + 0.45  , Ti =
Km 
τm 
τm + 0.1Tm
0.3τm + Tm
2
a f 3 = 0.2τm , a f 4 = 0.01τm (Åström and Hägglund, 2004);
a f 3 = 0.17 τm , a f 4 = 0 (Eriksson and Johansson, 2007);
2
a f 3 = 0.333δmax , a f 4 = 0.028δmax , τm < Tm ,
δmax = jitter margin (Eriksson and Johansson, 2007);
2
a f 3 = 4.6δmax − 6τm (with a f 3 ≥ 1 ), a f 4 = (2.3δ max − 3τm ) , τm > Tm
(Eriksson and Johansson, 2007).


[
]
1
T
0
.
5
T
N
+
16
m
d
Kc =
0.2 + 0.45
,
Km 
τm + 0.5[Td N ] 
0.4τm + 0.8Tm + 0.6[Td N ]
Ti =
(τm + 0.5[Td N ]) ,
τm + 0.1Tm + 0.55[Td N ]
Td =
0.5(τm + 0.5[Td N ])(Tm + 0.5[Td N ]) τ m
>1.
;
Tm
0.3τm + Tm + 0.65[Td N ]
b720_Chapter-03
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Majhi and
Atherton (1999),
Atherton and Boz
(1998).
Model: Method 1
Kc
m m m
α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ;
ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ;

(2Tm + τ m )3 x  , b = 1 , b = x 5 , a = 0 .
− 1 −
3
f3
f5
f4
Tm
2Tm 2 τ m


Coefficient values (deduced from graphs). Model: Method 1
x1
x2
x3
x4
x1
x2
x3
x4
1.2
2.6
4.5
6.4
Minimum ISTE – 1.8
servo – Atherton 4.2
and Boz (1998).
6.9
10.0
Minimum ITSE – 1.8
servo – Majhi
4.2
and Atherton
6.9
(1999).
10.0
Minimum ISTSE 2.3
– servo –
5.2
Atherton and Boz 8.4
(1998).
12.4
Minimum ISTSE 2.3
– servo – Majhi
5.2
and Atherton
8.7
(1999).
12.4
17
Comment
Td
Direct synthesis: time domain criteria
τm
(2Tm + τm )3 x1 τmTm
17
≤ 0.8
x2
0
2
T
2
T
+
τ
m
m
m
2K τ T
K0 =
Minimum ISE –
servo – Atherton
and Boz (1998).
Ti
1
Km
0.58
0.91
0.89
0.98
0.17
0.22
0.25
0.26
0.17
0.22
0.25
0.26
0.08
0.11
0.14
0.13
0.08
0.11
0.12
0.13
-0.81
-0.55
-0.30
-0.22
-0.31
-0.16
-0.11
-0.08
-0.29
-0.16
-0.11
-0.08
-0.16
-0.10
-0.07
-0.04
-0.16
-0.10
-0.06
-0.04
2.36
2.49
2.18
2.23
1.08
0.96
0.95
0.92
1.05
0.96
0.95
0.92
0.71
0.65
0.68
0.63
0.69
0.65
0.62
0.64
8.5
10.8
13.3
16.0
13.0
16.2
20.3
24.0
13.0
16.8
20.3
1.02
1.03
1.02
1.00
0.28
0.31
0.29
0.30
0.28
0.27
0.29
-0.16
-0.14
-0.12
-0.09
-0.07
-0.06
-0.04
-0.04
-0.06
-0.05
-0.04
2.18
2.16
2.13
2.10
0.94
0.99
0.93
0.95
0.94
0.90
0.97
16.5
20.4
25.2
31.2
16.5
20.4
25.2
0.14
0.15
0.15
0.14
0.14
0.15
0.15
-0.04
-0.03
-0.02
-0.02
-0.03
-0.03
-0.02
0.64
0.65
0.64
0.61
0.64
0.67
0.65
Equations developed from information provided by Majhi and Atherton (1999).
 (2T + τ )2

m
m

x 4 − 2(Tm + τ m ) 
T 
Tm

x5 = m 
.
3
2
(2Tm + τm ) x 3


−1−
2


2Tm τ m
175
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176
Rule
Kc
Td
Kc
Ti
Td
1.2τ m
K m Tm
0.5τ m
0.15τ m
K m Tm
18
Chidambaram
(2000a).
Model: Method 1
19
Comment
Ti
2
β = 1 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = α ;
φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = K c ; bf 3 = 1 ;
0.8(Tm + 0.4 τ m )
Huang et al.
Kmτm
(2000).
(
0
.
6
T
m + 0.4 τ m )
Model: Method 1
Kmτm
bf 4 = 0 ; a f 5 = 1 .
Regulator tuning.
Tm + 0.4 τ m
0
Servo tuning.
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ;
φ = 0 ; ϕ = 0 ; bf 2 =
af 3 =
18
Kc =
1
Km
0.4Tm τm
+ af 3 ,
Tm + 0.4τm
0.4Tm τm
; af 4 = 0 ; K0 = 0 .


(Tm + 0.4τm )min 20, 0.4Tm τm 
 Tm + 0.4τm 
2

 

4.114 − 6.575 τm + 3.342 τm   ,
T  
Tm

 m 

2

τ  
K cTi
τ
Ti = Tm 0.311 + 1.372 m − 0.545 m   , Td =
;
[
Tm
T
16
.
016
−
11.7(τm Tm )]


m
 


α = 0.207 + 3.149(τm Tm ) − 8.405(τm Tm ) + 8.431(τm Tm ) , τm Tm ≤ 0.5 ,
2
3
α = 0.9416 − 0.44(τm Tm ) , 0.5 ≤ τm Tm ≤ 0.9 ; K 0 = K c .
19
α = 0.6762 − 0.4332
τ
τm τ m
1. 2 τ m
,
.
≤ 0.8 ; b = 0.1493 , m = 0.9 ; K 0 =
Tm
Tm Tm
K m Tm
b720_Chapter-03
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FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
20
Lee et al. (1992).
Model: Method 1
Ti
Kc
177
Comment
Td
Direct synthesis: frequency domain criteria
Ti
Td
M s = 1.25
β = 1 ; χ = 0 ; δ = 0 ; N not specified; b f 1 = 0 , a f 1 = 0 , a f 2 = 0 ;
ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0 .
Åström and
Hägglund (2006),
pp. 252, 265.
Model:
Method 56
21
Kc
Ti
Td
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ;
2
ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = x1 , a f 4 = 0.5x1 ; K 0 = 0 .
Padhy and Majhi
(2006a).
Model:
Method 39
22
Kc
x1
1 + x1x 2 τm
0
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ;
φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 2Tm τm K m ;
b f 3 = 1 ; b f 4 = 0.5τm ; a f 5 = 0 .
20
Kc =
2 x 2 2 (τm Tm ) − 4 + 8x1x 2 + 0.5(τm Tm )x 2 2 (4 + 2 τm Tm )
[
]
K m x 2 2 (τm Tm )2 + 4 + 4 x1x 2 (τm Tm )
,


2 τ

2 x 2 m − 4 + 8x1x 2 


Tm
τ
Tm
,
Ti = Tm  0.5 m +
 , Td =
T


τ
(4 + 2 τ m Tm )x 2 2
Tm
m

 4 + 2 m x 22 
+
2



Tm 
τ m 2 x 2 2 [τ m Tm ] − 4 + 8x1x 2



x1 = 0.5 +
0.25
x2
2
,
0.416 + 0.956 Tm τm
τm
τ
;
≤ 2.51 ; x1 = 1 , m > 2.51 ; x 2 =
1 + 0.0498 τm Tm
Tm
Tm
α = 1 − 0.517(τm Tm )
0.279
, τm Tm ≤ 10.64 ; α = 0 , τm Tm > 10.64 .
0.4τm + 0.8Tm + 0.6 x1
1 
Tm + 0.5x1 
21
Kc =
(τm + 0.5x1 ) ,
0.2 + 0.45
 , Ti =
Km 
τm + 0.1Tm + 0.55x1
τm + 0.5x1 
0.5(τm + 0.5x1 )(Tm + 0.5x1 )
T
T
Td =
, d ≤ x1 ≤ d .
0.3(τm + 0.5x1 ) + (Tm + 0.5x1 ) 30
3
22
Kc =
(A − 1) [T
2φm + π(A m − 1)
2K m τm
2
m
m
]
− 0.7071 Tm τm , x1 =
Tm − 0.7071 Tm τm
1.4142 Tm τm + 1
and
 2φ + π(A m − 1)   2A m
x 2 = 0.7854A m  m
 1 −
2
π
 2 A m − 1  
(
)
 2φm + π(A m − 1) 

 .
 2 A m 2 − 1 
(
)
b720_Chapter-03
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178
Rule
Kc
Ti
Td
Comment
0.1 ≤ τm ≤ 10 ,
23
Eriksson and
Ti
Td
Kc
0.1 ≤ Tm ≤ 10 .
Johansson
α
=
1
,
τ
T
≤
1
,
α
=
0
,
τ
T
>
1
;
β
=
1
;
χ
=
0
;
δ
=0 ; N =∞;
(2007).
m
m
m
m
Model: Method 2
bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 ,
a f 3 = 0.1 ωg , a f 4 = 0.0025 ωg 2 , τm Tm ≤ 0.25 ; a f 3 = 0.2τm ,
2
a f 4 = 0.01τm , τm Tm > 0.25 , ωg = gain crossover frequency;
K0 = 0 .
Robust
Tm
K m (0.5τ m + λ )
Harris and
Tyreus (1987).
Model: Method 1
0
Tm
λ ≥ maximum
[0.8τ m , Tm ]
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ;
φ = 0 ; ϕ = 0 ; b f 2 = 0.5τm ; a f 3 = x1b f 3 , x1 not specified; a f 4 = 0 ;
K 0 = 0.
24
Ho et al. (2001).
Model: Method 1
Kc
0
Tm
1.551Tm
KmAmτm
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ;
φ = 0 ; ϕ = 0 ; b f 2 = 0.5τm ; a f 3 = x1b f 3 , x1 not specified; a f 4 = 0 ;
K 0 = 0.
25
Lee and Edgar
(2002).
Model: Method 1
0
Ti
Kc
α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ;
φ = 0 ; ϕ = 1 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0.25Tu , b f 3 = 1 ;
bf 4 = 0 ; a f 5 = 0 .
23
Kc =
Ti =

0.01Tm (0.4Tm + 11)
1  0.4Tm − 0.04

,
+ 0.16  , Td =
(
(
0
.
4
Tm − 0.04 ) τm ) + 0.16
K m 
τm

(0.35T
m
2
100((0.4Tm − 0.04) τ m ) + 0.16
) (
3
2
+ 4Tm + 50 τm − 0.11Tm − 1.5Tm + 1.5
)
τm 2 .
Tm
24
Kc =
, TCL = −0.5τ m + 0.25τ m 2 + ωg −2 ,
K m (0.5τm + TCL )
ωg = frequency at which the Nyquist curve has a magnitude of unity.
25
Kc =
(1 + 0.25K u K m )Ti
K m (λ + τm )
2
, Ti =
Tm − 0.25K u K m τm
τm
+
.
1 + 0.25K u K m
2(λ + τm )
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
26
Lee and Edgar
(2002).
Model: Method 1
Kc
Ti
Td
Ti
Td
Comment
α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ;
ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0.25Tu ,
bf 3 = 1 , bf 4 = 0 , a f 5 = 0 .
27
Visioli (2002).
Model: Method 1
Tm + 0.5τ m
Kc
Tm τ m
2Tm + τ m
α = 0 , β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 =
0.05 ≤
τm
≤1
Tm
Tm
,
1 + K0
 λτ m

λτ m Tm
; ε = 0;φ = 0; ϕ = 0;
af1 = 
+ Tm  , a f 2 =
2(λ + τm )
 2(λ + τm )

b f 2 = b f 1 , a f 3 = a f 1 , a f 4 = a f 2 ; for minimum IAE (regulator),
K 0 = 0.2171(Tm τm )
1.481
; b f 3 = 1 ; b f 4 = 0 ; a f 5 = Tm .
Tm
Cooper (2006a), K (λ + 0.5τ )
Tm
0.5τm
m
m
(2006b).
α = −0.5τm Tm , χ = 0 , a f 1 = 0 (Cooper, 2006a);
Model: Method 1
Tm λ
(Cooper, 2006b).
α = 0 , χ = −0.5τm Tm , a f 1 =
2(λ + τm )
Tm + 0.5τm
K m (λ + 0.5τm )
Tm + 0.5τm
α = 0 , χ = 0 , af1 =
Tm τm
2Tm + τm
Cooper (2006b)
Tm λ
(Cooper, 2006b).
2(λ + τm )
β = 1 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 2 = 0 ; a f 3 = 0 ; ε = α ; φ = χ ;
ϕ = 0 ; bf 2 = 0 , a f 3 = a f 1 , a f 4 = 0 ; K 0 = 0 .
λ > [0.1Tm ,0.8τm ] (‘aggressive’ tuning); λ > [Tm ,8τm ] (‘moderate’
tuning); λ > [10Tm ,80τm ] (‘conservative’ tuning).
26
2
K c = x1Ti , Ti =
1 + 0.25K u K m
Tm − 0.25K u K m τm
τm
+
, x1 =
1 + 0.25K u K m
2(λ + τm )
K m (λ + τm )
3
4
2
 0.25K u K m τm 2
τm
τm
τ (T − 0.25K u K m τm ) 
Td = x1 
−
+
+ m m
.
2
 2(1 + 0.25K K ) 6(λ + τ ) 4(λ + τ )
2(1 + 0.25K u K m )(λ + τm ) 
u m
m
m

2Tm + τm
1
27
Kc =
2K m (λ + τm ) (1 + K 0 )
179
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180
3.4 FOLPD Model with a Zero
G m (s ) =
K (1 − sTm 4 )e −sτ m
K m (1 + sTm 3 )e − sτm
or G m (s) = m
1 + sTm1
1 + sTm1

1 

3.4.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
E(s)
R(s)
−
+

1 

K c 1 +

T
is 

Y(s)
U(s)
Model
Table 18: PI controller tuning rules – FOLPD model with a zero
K (1 + sTm3 )e −sτ m
K (1 − sTm 4 )e −sτ m
G m (s ) = m
or G m (s) = m
1 + sTm1
1 + sTm1
Rule
Slätteke (2006).
Model: Method 3
Kc
Comment
Ti
Minimum performance index: regulator tuning
Minimum IE, subject
x1 (Tm1 + 0.333τm )
x T + x 4 τm
x 2 τm 3 m1
to M s .
K m Tm 3τm
x 5Tm1 + x 6 τm
x1
x2
x3
x4
x5
x6
Ms
0.09
0.16
0.23
0.28
4
5
1
1
6
23
100
88
1
4
17
15
1
9
11
12
21
105
94
85
1.1
1.2
1.3
1.4
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
1
3
3
Kc =
Ti
Model: Method 1
Tm1
K m (τm + Tm3 )
2
Direct synthesis: frequency domain criteria
Model: Method 1;
Ti
4
Kc
Tm1 > Tm 4 .
Tm1 + 0.5τm
(λ + τm )K m
Marchetti and Scali
(2000).
Model: Method 1
2
x1Tm1
K mTm 4
4Tm1
Desbiens (2007),
p. 90.
Kc =
Comment
Direct synthesis: time domain criteria
‘Smaller’ τm Tm 4 .
Kc
Tm1
2
‘Larger’ τm Tm 4 .
Kc
Chidambaram (2002),
p. 157.
Model: Method 1
Sree and
Chidambaram
(2003a).
Slätteke (2006).
Model: Method 3
1
Ti
Tm1 + 0.5τm
(λ + τm + 2Tm 4 )K m
1.571
(
)
.
(
)
.
K m τm 1 + 3.142 Tm 4 τm 2
2.358
K m τm 1 + 4.712 Tm 4 τm 2
Robust
Tm1 + 0.5τm
Negative zero.
Tm1 + 0.5τm
Positive zero.
x1 = value of the ‘inverse jump’ of the closed loop system; x1 < K m Tm 4 Tm1 .
Ti =
(x1Tm1
Tm 4 )[Tm 4 (1 − x 2 ) − 0.5τm (1 + x 2 )]
;
(x1Tm1 Tm 4 )(1 − x 2 ) − x 2
T
0.98x1Tm1
0.95x1Tm1
≤ x2 ≤
, 0.1 ≤ m 4 ≤ 0.3 ;
x1Tm1 + Tm 4
x1Tm1 + Tm 4
Tm1
0.3x1Tm1
T
0.1x1Tm1
≤ x2 ≤
, m4 > 1 .
x1Tm1 + Tm 4 Tm1
x1Tm1 + Tm 4
4
Kc =
Ti ω
Km
2
Tm1 ω2 + 1
, with ω given by solving (for
(TiTm 4 ) ω4 + (Ti 2 + Tm 42 )ω2 + 1
φm = 58.13 ): 1.015 = 0.5π + tan −1 (Ti ω) − tan −1 (Tm 4ω) − tan −1 (Tm1 ω) − τ m ω ;
2
Ti = Tm1 + 0.175τ m , τm Tm1 < 2 ; Ti = 0.65Tm1 + 0.35τ m , τ m T m1 > 2 .
181
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182
3.4.2 Ideal controller in series with a first order lag

 1
1
+ Td s 
G c (s) = K c 1 +
Ti s

 1 + Tf s
E(s)
R(s)
−
+

 1
1
K c 1 +
+ Td s 
 Ti s
 1 + Tf s
U(s)
Model
Y(s)
Table 19: PID controller tuning rules – FOLPD model with a zero
K (1 + sTm 3 )e −sτm
K (1 − sTm 4 )e −sτ m
G m (s) = m
or G m (s) = m
1 + sTm1
1 + sTm1
Rule
Kc
Ti
Comment
Td
Direct synthesis: time domain criteria
Sree and
Chidambaram
(2003a).
Jyothi et al.
(2001).
Model: Method 1
1
Kc
2
Kc
3
Kc
0
Tm1
Model: Method 1
Direct synthesis: frequency domain criteria
τm Tm3 < 1
0
Tm1
τm Tm3 > 1
Robust
Chang et al.
Tm1
(1997).
K m (TCL + τ m )
Model: Method 2
1
Kc =
2
Kc =
Tm1
T (λ − τ m )
.
, Tf = m 4
K m (2Tm 4 + λ + τm )
2Tm 4 + λ + τm
1.57Tm1
K m τm
3
Tm1
Kc =
K m τm
T 
1 + 9.870 m3 
 τm 
0.79Tm1
T 
1 + 2.467 m3 
 τm 
2
2
.
.
0
Tf = Tm 3
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
K m e − sτ m
3.5 SOSPD model
2
Tm1 s 2 + 2ξ m Tm1s + 1
or
183
K m e −sτ m
(1 + Tm1s )(1 + Tm 2 s )

1 

3.5.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
E(s)
R(s)
+
−

1 

K c 1 +
Ti s 

U(s)
Y(s)
SOSPD
model
Table 20: PI controller tuning rules – SOSPD model
K m e − sτ m
K m e − sτ m
or
(1 + Tm1s)(1 + Tm 2 s)
Tm1 2 s 2 + 2ξ m Tm1s + 1
Rule
Kc
Comment
Ti
Minimum performance index: regulator tuning
Model: Method 1
x1 K m
x 2Tm1
Minimum IAE –
Lopez (1968),
pp. 63, 69–74.
Coefficients of K c and Ti deduced from graphs. These are
representative results.
0.5
0.6
0.8
1.0
1.5
2.0
4.0
ξm
x1
x2
τm Tm1 = 0.1
4.8 4.2 5.7 3.4 7.8 2.8 9.7 2.3 15 1.7 21 1.4 35
x1
x2
x1
x2
x1
x2
x1
x2
x1
x2
x1
x2
-
τm Tm1 = 0.2
2.2 3.3 2.7 3.0 3.9 2.7 5.1 2.4 8.3 1.9 11.5 1.7 27 1.2
τm Tm1 = 0.5
0.76 2.1 1.0 2.3 1.6 2.4 2.1 2.4 3.7 2.4 5.4 2.4 12.5 2.4
τm Tm1 = 1.0
0.33 1.2 0.50 1.6 0.82 2.2 1.2 2.5 2.1 2.8 3.0 3.3 6.6 3.7
τm Tm1 = 2.0
0.23 1.2 0.34 1.6 0.52 2.2 0.70 2.8 1.15 3.8 1.65 4.4 3.4 5.9
τm Tm1 = 5.0
0.32 2.6 0.34 2.9 0.38 3.3 0.46 3.8 0.62 5.0 0.80 6.3 1.5 9.6
τm Tm1 = 10.0
0.34 5.0 0.35 5.3 0.38 5.6 0.39 5.9 0.46 7.1 0.52 8.3 0.90 -
Minimum IAE –
Murata and Sagara
(1977).
Model: Method 4;
K c , Ti deduced from
graphs.
x1 K m
Tm1 = Tm 2
x 2Tm1
x1
x2
τm Tm1
x1
x2
τm Tm1
2.6
1.3
1.0
2.5
2.5
2.6
0.2
0.4
0.6
0.8
0.7
2.7
2.8
0.8
1.0
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
184
Rule
Minimum IAE –
Harriott (1988).
Model: Method 1;
coefficients of Ti
deduced from graphs.
Minimum IAE –
Harriott (1988).
Model: Method 1;
coefficients of Ti
deduced from graphs.
Minimum IAE –
Harriott (1988).
Model: Method 1
Minimum IAE –
Shinskey (1994),
p. 158.
Minimum IAE –
Shinskey (1996),
p. 48.
Model: Method 1;
τm Tm1 = 0.2 .
1
17-Mar-2009
Kc
Ti
Comment
0.5K u
0.4Tu
0.1 ≤ τm Tm1 ≤ 2 .
0.5K u
x 2Tu
Alternative.
Tm1 = Tm 2 ;
x2
τm Tm1
x2
τm Tm1
1.10
0.85
1.45
1.10
1.75
1.25
0.1
0.2
0.1
0.2
0.1
0.2
0.65
0.52
0.75
0.55
0.90
0.5
1.0
0.5
1.0
0.5
0.5K u
τm Tm1
x2
Tm 2 Tm1 = 0.2
Tm 2 Tm1 = 0.5
Tm 2 Tm1 = 1.0
x 2Tu
τm Tm1
x2
1.35
0.1
0.65
0.5
Tm 2 Tm1 = 0.2
0.85
0.2
2.00
0.1
0.80
0.5
Tm 2 Tm1 = 0.5
1.10
0.2
0.60
1.0
2.25
0.1
1.00
0.5
Tm 2 Tm1 = 1.0
1.25
0.2
Disturbance input subsequent to the smaller time constant and
delay terms.
Tm1 = Tm 2 ;
0.45K u
1.1Tu
τ T = 0.2 .
m
m1
Random load disturbance through a filter with time constant =
2Tm1 .
Kc
Ti
Model: Method 1
0.77Tm1 K m τm
2.83(τ m + Tm 2 )
Tm 2 Tm1 = 0.1
1
0.70Tm1 K m τm
0.80Tm1 K m τm
0.80Tm1 K m τm
2.65(τ m + Tm 2 )
2.29(τ m + Tm 2 )
1.67(τ m + Tm 2 )
Tm 2 Tm1 = 0.2
Tm 2 Tm1 = 0.5
Tm 2 Tm1 = 1.0
3Tm1



−

(τ m + Tm 2 )  

Kc =
, Ti = τ m  0.5 + 3.5 1 − e
.
Tm1  

 



−




τ
+
T


K m (τm + Tm 2 ) 50 + 551 − e m m 2  






100Tm1
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Minimum IAE –
Huang et al. (1996).
Model: Method 1
2
2
Kc
Ti
Kc
Ti
Comment
0 < Tm 2 Tm1 ≤ 1 ;
0.1 ≤ τm Tm1 ≤ 1 .
Note: equations continued into the footnote on p. 186.
τm
τ T 
1 
T
− 3.491 m 2 − 25.3143 m m2 2 
Kc =
6.4884 + 4.6198
K m 
Tm1
Tm1
Tm1 
+
−0.9077
−0.063
0.5961 
 τm 
 τm 
τ 
1 

0.8196 m 




5
.
2132
7
.
2712
−
−
T 
T 
T 
Km 

 m1 
 m1 
 m1 


+
T
1 
 − 18.0448 m 2
T
Km 
 m1

+
T
T
1 
0.4937 m 2 + 19.1783 m 2
Km 
Tm1
τm

+
τ
1 
8.4355 m
Km 
Tm1

+
τ m Tm 2 
Tm 2
τm

2
1 
− 0.7241e Tm1 − 2.2525e Tm1 + 5.4959e Tm1  ;

Km 





 Tm 2

 Tm1
0.7204



T
+ 5.3263 m 2
 Tm1
 τm 


 Tm1 
0.557
− 17.6781



1.0049
T
+ 13.9108 m 2
 Tm1
0.8529
+ 12.2494
τm
Tm1
 Tm 2

 Tm1



Tm 2
Tm1



1.005 
 τm 


 Tm1 



0.5613 



1.1818 



2

τ  
T
τ
Ti = Tm1 0.0064 + 3.9574 m + 4.4087 m 2 − 6.4789 m  

Tm1
Tm1
 Tm1  

2
3

T 
τ  
τ T
+ Tm1  − 12.8702 m m2 2 − 1.5083 m 2  + 9.4348 m  

Tm1
 Tm1 
 Tm1  


T
+ Tm1 17.0736 m 2
Tm1


2
 τm 
τ
 + 15.9816 m

T
T
m1
 m1 
 Tm 2

 Tm1
2
T

 − 3.909 m 2
 Tm1




3



4
3
2

 τ  T
τ 
T τ 
+ Tm1 − 10.7619 m  − 10.864 m 2  m  − 22.3194 m   m 2
Tm1  Tm1 

 Tm1   Tm1
 Tm1 

3
4
5

T 
τ  
τ T 
+ Tm1 − 6.6602 m  m 2  + 6.8122 m 2  + 7.5146 m  
Tm1  Tm1 

 Tm1 
 Tm1  


T
+ Tm1 2.8724 m 2
Tm1


4
T
 τm 

 + 11.4666 m 2
T
 Tm1
 m1 



2
3
τ 
 τm 

 + 11.1207 m 
T
 Tm1 
 m1 



2
2



 Tm 2

 Tm1



3



185
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
186
Rule
Minimum IAE –
Huang et al. (1996).
Model: Method 1
3

 τ  T
+ Tm1  − 1.2174 m  m 2

 Tm1  Tm1


T
+ Tm1  − 0.112 m 2
Tm1


Kc
Ti
Comment
Kc
Ti
0.05 ≤ τm Tm1 ≤ 1 .
4
T

 − 4.3675 m 2
 Tm1

5
T
 τm 
 + 1.0308 m 2

T
 Tm1
 m1 
3

 τ  T
+ Tm1 − 3.4994 m   m 2

 Tm1   Tm1

3
17-Mar-2009
3
0.4 ≤ ξ m ≤ 1 ;
5
τ 

 − 2.2236 m 
 Tm1 

6
τ 

 − 1.9136 m 
 Tm1 

τ 

 − 1.5777 m 
 Tm1 

2
 Tm 2

 Tm1
4
6



 Tm 2

 Tm1



2



4

τ
 + 1.1408 m
T
m1

 Tm 2

 Tm1
Note: equations continued into the footnote on p. 187.
τm
τ 
1 
− 5.5175ξ m − 26.5149ξ m m 
Kc =
 − 10.4183 − 20.9497
Km 
Tm1
Tm1 
1
.
4439
0
.
1456
0.3157 
τ 
 τm 
 τm 
1 
42.7745 m 





+
+
+
10
.
5069
15
.
4103
T 
T 
T 
Km 

 m1 
 m1 
 m1 


1
3.7057
4.5359
1.9593
+
34.3236ξ m
− 17.8860ξ m
− 54.0584ξ m
Km
[
1
+
Km
]
−0.0541
4.7426 

 τm 
τ 

22.4263ξ m  m 


+ 2.7497ξ m 
T 
Tm1 


m1 




+
1
Km

T 
τ
1.8288  τ m 

 − 17.1968 m ξ m 2.7227 + 1.0293ξ m m1 
50.2197ξ m
Tm1
τ m 

 Tm1 
+
1
Km
τ
τm

ξm m 
 − 16.7667e Tm1 + 14.5737e ξm − 7.3025e Tm1  ,




2

τ 
τ 
τ
Ti = Tm1 11.447 + 4.5128 m − 75.2486ξm − 110.807 m  − 12.282ξ m m 
Tm1
Tm1 

 Tm1 


3
2

τ 
τ  
+ Tm1 345.3228ξ m 2 + 191.9539 m  + 359.3345ξ m  m  

 Tm1 
 Tm1  

4

 τm  
τm
2
3




+ Tm1 − 158.7611
ξ m − 770.2897ξ m − 153.633
Tm1
Tm1  







5
.


b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Minimum IAE –
Shinskey (1996),
0.48K u
p. 121.
Model: Method 1
Minimum ISE –
x1 K m
McAvoy and Johnson
(1967).
Model: Method 1; ξ m Tm1 x 1
τm
coefficients of K c
1
0.5 0.8
and Ti deduced from
1 4.0 5.7
graphs.
1 10.0 13.6
Comment
Ti
τm Tm1 = 0.2 ,
0.83Tu
Tm 2 Tm1 = 0.2 .
x 2τm
x2
1.82
12.5
25.0
Coefficient values:
ξ m Tm1 x 1 x 2
τm
4
4
0.5 4.3 3.45
4.0 27.1 6.67
ξ m Tm1
τm
7
7
x2
0.5 7.8 3.85
4.0 51.2 5.88
3
2


τ 
τ 
τ
+ Tm1  − 412.5409ξ m  m  − 414.7786ξ m 2  m  + 485.0976 m ξ m 3 
Tm1


 Tm1 
 Tm1 


5
4

τ  
τ 
+ Tm1 864.5195ξ m 4 + 55.4366 m  + 222.2685ξ m  m  

 Tm1  
 Tm1 

3
2


τ 
τ 
τ 
+ Tm1 275.116ξ m 2  m  + 205.2493 m  ξ m 3 − 479.5627 m ξ m 4 


 Tm1 
 Tm1 
 Tm1 


6
5


τ 
τ 
+ Tm1  − 473.1346ξ m 5 − 6.547 m  − 43.2822ξ m  m  + 99.8717ξ m 6 


 Tm1 
 Tm1 


4
3
2


τ 
τ 
τ 
+ Tm1  − 73.5666 m  ξ m 2 − 56.4418 m  ξ m 3 − 37.497 m  ξ m 4 


 Tm1 
 Tm1 
 Tm1 




τ
+ Tm1 160.7714 m ξ m 5  .
T
m1


x1
187
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
188
Rule
Kc
Ti
Comment
Minimum ISE –
Lopez (1968),
pp. 63, 69–74.
x1 K m
x 2Tm1
Model: Method 1
Coefficients of K c and Ti deduced from graphs. These are
representative results.
0.5
0.6
0.8
1.0
1.5
2.0
4.0
ξm
x2
x1
x2
τm Tm1 = 0.1
6.5 4.8 7.7 4.2 10.5 3.4 13 2.9 20.5 2.1 28 1.6
x1
x2
x1
x2
x1
x2
x1
-
-
τm Tm1 = 0.2
3.1 4.2 3.9 3.8 5.4 3.3
7
x2
x1
x2
x1
2.9 11 2.4 15 2.0 33 1.4
τm Tm1 = 0.5
1.25 2.9 1.55 2.9 2.2 2.9 2.9 2.9 4.8 2.9 6.7 2.7 14.5 2.5
τm Tm1 = 1.0
0.63 2.2 0.83 2.5 1.2 2.9 1.65 3.1 2.7 3.7 3.8 3.8
τm Tm1 = 2.0
0.42 1.9 0.54 2.4 0.75 3.0 1.0 3.6 1.6 4.5 2.3 5.9 4.5 6.9
8
4.2
τm Tm1 = 5.0
0.45 3.5 0.48 3.8 0.56 4.4 0.65 5.3 0.85 6.7 1.05 7.7 2.0 11.6
τm Tm1 = 10.0
0.46 6.3 0.48 6.7 0.53 7.4 0.56 8.0 0.63 9.1 0.75 10.6 1.15 16.1
Minimum ITAE –
Lopez (1968),
pp. 63, 69–74.
x1 K m
Model: Method 1
x 2Tm1
Coefficients of K c and Ti deduced from graphs. These are
representative results.
0.5
0.6
0.8
1.0
1.5
2.0
4.0
ξm
x2
x1
x2
τm Tm1 = 0.1
2.9 3.0 3.8 2.7 5.5 2.3 7.4 2.0 11.5 1.5 16.5 1.2
x1
x2
x1
x2
x1
x2
x1
x2
x1
x2
x1
-
0.8
τm Tm1 = 0.2
1.35 2.4 1.9 2.4 2.8 2.2 3.8 2.0 6.5 1.7 9.5 1.5 23 1.1
τm Tm1 = 0.5
0.42 1.3 0.65 1.6 1.15 1.9 1.7 2.0 3.0 2.1 4.5 2.1 10.5 2.0
τm Tm1 = 1.0
0.21 0.8 0.34 1.2 0.64 1.8 0.93 2.1 1.7 2.7 2.5 2.9 5.6 3.4
τm Tm1 = 2.0
0.18 0.9 0.26 1.3 0.42 1.9 0.58 2.4 0.98 3.3 1.4 4.0 3.2 5.4
τm Tm1 = 5.0
0.25 2.3 0.29 2.5 0.34 2.9 0.38 3.4 0.52 4.1 0.66 5.6 1.4 8.7
τm Tm1 = 10.0
0.28 4.3 0.30 4.5 0.32 4.9 0.33 5.6 0.38 6.3 0.46 7.4 0.77 -
Minimum ITAE –
Lopez et al. (1969).
Model: Method 8;
coefficients of K c
x1 K m
ξm
τm
Tm1
x1
x 2Tm1
x2
and Ti deduced from 0.5 0.1 3.0 2.86
0.5 1.0 0.2 0.83
graphs.
0.5 10.0 0.3 4.0
Coefficient values:
ξm τ m x1 x 2
Tm1
1
1
1
0.1 7.0 2.00
1.0 0.95 2.22
10.0 0.35 5.00
ξm
4
4
4
τm
Tm1
x1
x2
0.1 40.0 0.83
1.0 6.0 3.33
10.0 0.75 10.0
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Minimum ITAE –
Chao et al. (1989).
Model: Method 1
Minimum ITAE –
Hassan (1993).
Model: Method 8
4
Kc =
4
Kc
Ti
Kc
Ti
Comment
0.7 ≤ ξ m ≤ 3 ; 0.05 ≤ 0.5τm ξ m Tm1 ≤ 0.5 .
5
0.5 ≤ ξ m ≤ 2 ,
Ti
Kc
0.09578 + 0.6206ξ m − 0.1069ξ m 2  τm 


 2ξ T 
Km
 m m1 
0.1 ≤ τm Tm1 ≤ 4 .
−1.108 + 0.2558ξ m − 0.0579 ξ m 2
,
(
Ti = 2ξ m Tm1 1.31 − 0.0914ξ m + 0.07464ξ m
5
189
2
)
 τm 


 2ξ T 
 m m1 
−0.6441+ 0.7626 ξ m − 0.1193ξ m 2
.
Formulae correspond to graphs of Lopez et al. (1969). K c is obtained as follows:
log[K m K c ] = −0.0099458 + 1.9743700ξm − 3.8984310
τm
− 1.1225270ξ m 2
Tm1
2
τ 

τ
2 τ
+ 2.3493280 m  + 1.9126490ξ m m + 0.0754568ξ m  m 
T
T
T
m1
 m1 
 m1 
2
2
τ 
2 τ
3
− 0.5594610ξ m  m  − 0.7930846ξ m m + 0.2412770ξ m
T
T
m1
 m1 
3
3
τ 
τ 

2 τ
− 0.3567954 m  − 0.0024730ξ m  m  + 0.0478593ξ m  m 
T
T
T
 m1 
 m1 
 m1 
2
2
− 0.1354322ξ m
3
3


τm
3 τ
3 τ
− 1.1756470ξ m  m  − 0.0096974ξ m  m  ;
Tm1
T
T
 m1 
 m1 
Ti is obtained as follows:
 T 
τ
log  i  = 0.9321658 − 0.7200651ξ m − 3.4227010 m + 0.0432084ξ m 2
T
T
m1
 m1 
2
τ 

τ
2 τ
+ 1.4965440 m  + 5.3636520ξ m m − 0.0848431ξ m  m 
T
T
T
m1
 m1 
 m1 
2
2
τ 
2 τ
3
− 1.6011510ξ m  m  − 2.1777800ξ m m + 0.0404586ξ m
T
T
m1
 m1 
3
3
τ 
τ 

2 τ
− 0.1769621 m  + 0.0912008ξ m  m  + 0.1501474ξ m  m 
T
T
T
 m1 
 m1 
 m1 
2
+ 0.3033341ξ m
3
2
3


τm
3 τ
3 τ
+ 0.1753407ξ m  m  − 0.0622614ξ m  m  .
Tm1
T
T
 m1 
 m1 
b720_Chapter-03
Rule
Minimum ITSE –
Chao et al. (1989).
Model: Method 1
Nearly minimum
IAE, ISE, ITAE –
Hwang (1995).
Kc =
6
Kc
Ti
Kc
Ti
Comment
0.7 ≤ ξ m ≤ 3 ; 0.05 ≤ 0.5τm ξ m Tm1 ≤ 0.5 .
7
Decay ratio = 0.2;
Model: Method 14
Ti
Kc
0.3366 + 0.6355ξ m − 0.1066ξ m 2  τm 


 2ξ T 
Km
 m m1 
−1.0167 + 0.1743ξ m − 0.03946 ξ m 2
,
(
Ti = 2ξ m Tm1 1.9255 − 0.3607ξ m + 0.1299ξ m
7
FA
Handbook of PI and PID Controller Tuning Rules
190
6
17-Mar-2009
0.2 ≤ τm Tm1 ≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2 .
[
 0.622 1 − 0.435ωH τ m + 0.052(ωH τm )2
K c = 1 −

K H K m (1 + K H K m )

Ti =
[
]K


[
)
 τm 


 2ξ T 
 m m1 
−0.5848 + 0.7294 ξ m − 0.1136 ξ m 2
.
,
(
K c (1 + K H K m )
0.0697ωH K m 1 + 0.752ωH τm − 0.145ωH 2 τm 2
 0.724 1 − 0.469ωH τm + 0.0609(ωH τm )2
K c = 1 −

K H K m (1 + K H K m )

Ti =
H
2
]K


H
,
K c (1 + K H K m )
(
0.0405ωH K m 1 + 1.93ωH τm − 0.363ωH 2 τm 2
]
) , ε < 2.4 ;
) , 2.4 ≤ ε < 3 ;
 1.26(0.506)ωH τ m 1 − 1.07 ε + 0.616 ε 2 
K H ,
K c = 1 −


K H K m (1 + K H K m )


K c (1 + K H K m )
Ti =
, 3 ≤ ε < 20 ;
0.0661ωH K m (1 + 0.824 ln[ωH τm ]) 1 + 1.71 ε − 1.17 ε 2
[
 1.09 1 − 0.497ωH τm + 0.0724(ωH τm )2
K c = 1 −

K H K m (1 + K H K m )

Ti =
]K


(
H
)
,
(
K c (1 + K H K m )
0.054ωH K m − 1 + 2.54ωH τm − 0.457ωH 2 τm 2
) , ε > 20 .
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Minimum IAE –
Gallier and Otto
(1968).
Model: Method 25.
Minimum IAE –
Murata and Sagara
(1977).
Model: Method 4;
K c , Ti deduced from
graphs.
Minimum IAE –
Huang et al. (1996).
Model: Method 1
8
Kc
Comment
Ti
Minimum performance index: servo tuning
x1 K m
x 2 (Tm1 + Tm 2 + τ m )
Tm1 = Tm 2
Representative coefficient values (deduced from graphs):
τm 2Tm1
x1
x2
τm 2Tm1
x1
x2
0.053
3.6
0.11
2.3
0.25
1.35
x1 K m
1.35
1.17
0.93
0.67
1.50
4.0
0.76
0.72
0.53
0.60
0.43
0.51
Tm1 = Tm 2
x 2Tm1
x1
x2
τm Tm1
x1
x2
τm Tm1
1.4
1.0
0.8
2.3
2.3
2.4
0.2
0.4
0.6
0.6
0.5
2.4
2.5
0.8
1.0
8
0 < Tm 2 Tm1 ≤ 1 ;
Ti
Kc
0.1 ≤ τm Tm1 ≤ 1 .
Note: equations continued into the footnote on p. 192.
1 
T
τm
τ T 
Kc =
+ 2.6647 m 2 + 9.162 m m2 2 
− 13.0454 − 9.0916
K m 
Tm1
Tm1
Tm1 
+
−1.0169
3.5959
3.6843 
 τm 
τ 
 τm 
1 
0.3053 m 





+
−
1
.
1075
2
.
2927
T 
T 
T 
Km 

 m1 
 m1 
 m1 


+
T
1 
− 31.0306 m 2
T
Km 
 m1

+
T
T
1 
 − 0.6418 m 2 + 18.9643 m 2
τm
Km 
Tm1

+
τ
1 
28.155 m
Km 
Tm1

+
τ m Tm 2 
τm
Tm 2

2
1 
4.8259e Tm1 + 2.1137e Tm1 + 8.4511e Tm1  ,

Km 





 Tm 2

 Tm1
0.8476



T
− 13.0155 m 2
 Tm1
 τm 


 Tm1 
0.801
− 2.0067



2.6083
T
+ 9.6899 m 2
 Tm1
−0.2016
τm
Tm1
− 39.7340
 Tm 2

 Tm1



Tm 2
Tm1



2.9049 
 τm 


 Tm1 



1.3293 
3.956 



2

τ 
τ T 
T
τ
Ti = Tm1 0.9771 − 0.2492 m + 0.8753 m 2 + 3.4651 m  − 3.8516 m m2 2 

Tm1
Tm1
Tm1 
 Tm1 





191
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
192
Rule
Minimum IAE –
Huang et al. (1996).
Model: Method 1

T
+ Tm1 7.5106 m 2

 Tm1

9
Kc
Ti
Kc
Ti
2
Comment
0.4 ≤ ξ m ≤ 1 ;
0.05 ≤ τm Tm1 ≤ 1 .
3

τ 
T
 − 7.4538 m  + 11.6768 m 2
Tm1
T
m
1




τ
+ Tm1  − 10.9909 m
Tm1


 Tm 2

 Tm1

T
+ Tm1  − 18.1011 m 2
Tm1


 τm 
τ 

 + 6.2208 m 
T
m
1


 Tm1 
2
 τm 


 Tm1 
2
 Tm 2

 Tm1

T
+ Tm1  − 2.2691 m 2

 Tm1




3

T
+ Tm1 − 4.728 m 2
Tm1


4

T
 − 7.1990 m 2

 Tm1
5
 τm 
T

 + 1.1395 m 2
T
m
1


 Tm1
3

 τ  T
+ Tm1 1.0885 m   m 2

 Tm1   Tm1

3
2
 Tm 2

 Tm1



 τm 


 Tm1 
+
1
Km
6
4
6



 Tm 2

 Tm1



4

τ
 + 4.5398 m
T
m1

2



 Tm 2

 Tm1
3

 

− 10.95 − 1.8845 τm − 3.4123ξ m + 4.5954ξm τm − 1.7002 τm  
T  
Tm1
Tm1

 m1  

2

τ 
τ 
1 
 − 2.1324ξ m  m  − 14.4149ξ m 2  m  − 0.7683ξ m 3 
T 
T 
Km 

 m1 
 m1 


0.421
0.1984
1.8033 
 τm 
τ 
 τm 
1 
7.5142 m 





3
.
7291
5
.
3444
+
+
T 
T 
T 
Km 

 m1 
 m1 
 m1 


1
19.5419
1.0749
1.1006
+
− 0.0819ξ m
− 3.603ξ m
+ 7.1163ξ m
Km
+
[






Note: equations continued into the footnote on p. 193.
Kc =
3



5
 Tm 2

 Tm1



4
τ 

 + 1.1496 m 
 Tm1 

2
 Tm 2

 Tm1
3
τ 

 + 0.6385 m 
 Tm1 


τ 
 + 3.1615 m 

 Tm1 



2
5
τ 
 τm 

 − 8.387 m 
T
m
1
 Tm1 


4

τ
 + 21.9893 m
T
m1


τ 
T
 − 4.7536 m  + 14.5405 m 2
T
Tm1
m
1



2



3

T
+ Tm1 15.8538 m 2

 Tm1

4
2
τ 

 + 8.2567 m 
 Tm1 


T
 − 16.1461 m 2

 Tm1
3

 τ  T
+ Tm1 − 16.651 m  m 2

 Tm1  Tm1

9
17-Mar-2009
]



5
.


b720_Chapter-03
17-Mar-2009
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
+
−0.6753
−0.1642

τ 
 τm 
1 
1.9948  τ m 
3.206ξ m  m 





7
.
8480
11
.
3222
−
ξ
+
ξ
m
m
 T 
T 
T 
Km 
 m1 
 m1 
 m1 

+
τm
ξm τm
T 
1 
2.4239e Tm1 + 3.4137e ξm + 1.0251e Tm1 − 0.5593ξ m m1  ,
Km 
τm 


2

τ  
τ
Ti = Tm1 2.4866 − 23.3234 m + 5.3662ξm + 65.6053 m  
Tm1

 Tm1  

3

τ  
τ
+ Tm1 29.0062ξ m m − 24.1648ξ m 2 − 83.6796 m  
Tm1

 Tm1  

2


 τm 
τ



+ 43.1477 m ξ m 2 + 51.9749ξ m 3 
+ Tm1 − 135.9699ξ m 

Tm1


 Tm1 


4
3
2

 τm 
 τm 
2  τm 








+ Tm1 86.0228
+ 70.4553ξ m 
+ 153.4877ξ m 
Tm1 
Tm1 
Tm1  






5

 τm  
τm
3
4




+ Tm1 − 125.0112
ξ m − 68.5893ξ m − 62.7517
Tm1
Tm1  




4
3
2


 τm 
 τm 
2  τm 







ξm3 
+ Tm1 27.6178ξ m 
− 152.7422ξ m 
+ 20.8705





 Tm1 
 Tm1 
 Tm1 


6

 τm  
 τm  4
5






+ Tm1 54.0012
ξ m + 58.7376ξ m + 13.1193
Tm1 
Tm1  





5
4


 τm 
 τm 
6





ξm 2 
+ Tm1 20.2645ξ m 
− 23.2064ξ m − 61.6742




 Tm1 
 Tm1 


3
2


 τm 
 τm 
τ
3





ξ m 4 + 20.4168 m ξ m 5  .
+ Tm1 136.2439
ξ m − 95.4092


Tm1


 Tm1 
 Tm1 


FA
193
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
194
Rule
Kc
Comment
Ti
x1
x 2τm
Minimum ISE –
Coefficient values:
Keviczky and Csáki
0.1
0.2
0.5
1.0
ξ
(1973).
τ m Tm1
x1 x 2 x1 x 2 x1 x 2 x1 x 2
Model: Method 1;
representative K c , Ti
0.1
0.26 1.5 0.55 2 2.2 3.5 6
6
0.2
0.21 1.3 0.4 1.5 1.5 2.8 3 4.5
coefficients estimated
0.5
0.15 1.2 0.25 1.3 0.8 2
2 3.5
from graph; K m = 1 .
1.0
0.12 1.1 0.2 1.2 0.65 1.8 1.2 3.3
2.0
0.1 1.0 0.16 1.2 0.4 1.9 0.9 3.5
5.0
0.1 1.2 0.16 1.4 0.4 3.0 0.65 5.0
10.0
0.15 1.6 0.2 2.0 0.45 5.5 0.6 7.5
Other coefficient values:
τm
τm
x1
x2
ξ
x1
x2
T
T
m1
m1
0.2
0.5
0.5
1.0
1.0
Minimum ITAE –
Chao et al. (1989).
Model: Method 1
Minimum ITSE –
Chao et al. (1989).
Model: Method 1
10
Kc =
30
13
30
7
14
10
15
14
30
14
30
5.0
5.0
10.0
5.0
10.0
2.0
2.0
5.0
5.0
10.0
10.0
15
30
17
32
19
34
x2
14 9
9
8
5
7
3 6.5
1.8 7
1.0 8
0.7 10
ξ
5.0
10.0
5.0
10.0
5.0
10.0
Ti
Kc
0.7 ≤ ξ m ≤ 3 ; 0.05 ≤ 0.5τm ξ m Tm1 ≤ 0.5 .
11
Ti
Kc
0.7 ≤ ξ m ≤ 3 ; 0.05 ≤ 0.5τm ξ m Tm1 ≤ 0.5 .
0.2763 + 0.3532ξ m − 0.06955ξ m
Km
2
 τm 


 2ξ T 
 m m1 
(
−0.3772 − 0.2414 ξ m + 0.03342 ξ m 2
,
Ti = 2ξ m Tm1 0.9953 + 0.07857ξ m + 0.005317ξ m
11
4
8
1.9
3.4
1.1
2.0
2.0
x1
0.3970 + 0.4376ξ m − 0.08168ξ m
Kc =
Km
(
2
 τm 


 2ξ T 
 m m1 
2
)
 τm 


 2ξ T 
 m m1 
−0.3276 + 0.3226 ξ m − 0.05929 ξ m 2
.
−0.5629 − 0.1187 ξ m + 0.01328ξ m 2
,
)
 τm 

Ti = 2ξ m Tm1 1.3414 − 0.1487ξ m + 0.07685ξ m 2 

 2ξ m Tm1 
−0.4287 + 0.1981ξ m − 0.01978 ξ m
2
.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
12
Nearly minimum
IAE, ISE, ITAE –
Hwang (1995).
Model: Method 14
12
195
Comment
Ti
Ti
Kc
2
For ξ m ≤ 0.776 + 0.0568
τ 
τm
+ 0.18 m  :
Tm1
 Tm1 
Coefficient values:
x3
x4
x5
x1
x2
x6
x7
0.822
x8
-0.549
x9
0.122
x 10
0.0142
x 11
6.96
x 12
-1.77
x 13
0.786
x 14
-0.441
x 15
0.0569
x 16
0.0172
x 17
4.62
x 18
-0.823
x 19
1.28
x 20
0.542
x 21
-0.986
x 22
0.558
x 23
0.0476
x 24
0.996
x 25
2.13
x 26
-1.13
1.14
-0.466
0.0647
0.0609
1.97
-0.323
Decay ratio = 0.1; 0.2 ≤ τm Tm1 ≤ 2.0 ; 0.6 ≤ ξ m ≤ 4.2 .
[
 x 1 + x 2ωH τm + x 3 (ωH τm )2
K c = 1 − 1

K H K m (1 + K H K m )

]K


H
,
Ti =
[
 x 1 + x 8ωH τm + x 9 (ωH τm )2
K c = 1 − 7

K H K m (1 + K H K m )

]K


H
(
x 4ωH K m 1 + x 5ωH τm + x 6ωH 2 τm 2
) , ε < 2.4 ;
,
Ti =
[
K c (1 + K H K m )
]
K c (1 + K H K m )
(
x10ωH K m 1 + x11ωH τm + x12ωH 2 τm 2
) , 2.4 ≤ ε < 3 ;
 x x ωH τ m 1 + x15 s + x16 ε 2 
K H ,
K c = 1 − 13 14


K H K m (1 + K H K m )


Ti =
[
K c (1 + K H K m )
, 3 ≤ ε < 20 ;
x17 ωH K m (1 + x18 ln[ωH τm ]) 1 + x19 s + x 20 s 2
 x 1 + x 22ωH τm + x 23 (ωH τm )2
K c = 1 − 21

K H K m (1 + K H K m )

]K


(
H
)
,
Ti =
(
K c (1 + K H K m )
x 24ωH K m − 1 + x 25ωH τm + x 26ωH 2 τm 2
) , ε > 20 .
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
196
Rule
Kc
Comment
Ti
2
Nearly minimum
IAE, ISE, ITAE –
Hwang (1995) –
continued.
For ξ m > 0.889 + 0.496
τ 
τm
+ 0.26 m  :
Tm1
 Tm1 
Coefficient values:
x3
x4
x5
x1
x2
x6
x7
0.794
x8
-0.541
x9
0.126
x 10
0.0078
x 11
8.38
x 12
-1.97
x 13
0.738
x 14
-0.415
x 15
0.0575
x 16
0.0124
x 17
4.05
x 18
-0.63
x 19
1.15
x 20
0.564
x 21
-0.959
x 22
0.773
x 23
0.0335
x 24
0.947
x 25
1.9
x 26
-1.07
1.07
-0.466
0.0667
0.0328
2.21
-0.338
2
τ 
τ
For ξ m > 0.776 + 0.0568 m + 0.18 m  and
Tm1
 Tm1 
2
τ 
τ
ξ m ≤ 0.889 + 0.496 m + 0.26 m  :
Tm1
 Tm1 
Coefficient values:
x3
x4
x5
x1
x2
x6
x7
0.789
x8
-0.527
x9
0.11
x 10
0.009
x 11
9.7
x 12
-2.4
x 13
0.76
x 14
-0.426
x 15
0.0551
x 16
0.0153
x 17
4.37
x 18
-0.743
x 19
1.22
x 20
0.55
x 21
-0.978
x 22
0.659
x 23
0.0421
x 24
0.969
x 25
2.02
x 26
-1.11
1.11
-0.467
0.0657
0.0477
2.07
-0.333
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Minimum ISTSE –
Fukura and Tamura
(1983).
Model: Method 1
13
τm Tm1 = 2.11
0.462(τm + Tm1 )
undefined
14
τm Tm1 = 3.92
G c = 1 (Tis ) ;
x 2 K m Tm1
Tm1 > Tm 2 .





1
1
Tm 2 
+ 0. 2  
+
,
1.5
 Tm1  Tm1 
  1 + 2 Tm 2


2 + 0.2



τm
 τm 







1
1
Tm 2 
+
Ti = [Tm1 + 0.36τm ]

1.5
 Tm1  Tm1 
 1 + 2 Tm 2


2 + 0.2



τm
 τm 


14
τm Tm1 = 1.27
0.420(τm + Tm1 )
1.14 K m
Bryant et al. (1973).
Model: Method 1
τm Tm1 = 0.60
0.368(τm + Tm1 )
1.69 K m
0.82 K m
1 
Tm1
0.56
τm
Km 
0.05 ≤ τm Tm1 ≤ 1 .
0.268(τm + Tm1 )
3.61 K m
Decay ratio ≈ 33% .
Kc =
Tm 2 ≤ Tm1 ;
Ti
Direct synthesis: time domain criteria
13.9 K m
0.140(τm + Tm1 )
τm Tm1 = 0.22
Ream (1954).
Model: Method 1;
Tm1 = Tm 2 .
13
Comment
Ti
Kc
197



3.6
Tm 2 
1 +

τm
T


m
1
+ 0.05


T
m1


T 
0.3 m1 
 τm 
0.25
.
Representative x 2 values, summarised as follows, are deduced from graphs:
τ m Tm1
x 2 value
ξm
0.5
1.0
2.0
3.0
4.0
5.0
6.0
0.4
0.6
0.8
0.9
1.0
1.1
1.2
1.5
1.9
7.94
8.93
9.80
12.2
15.4
5.13
5.99
5.81
9.01
9.90
10.8
13.3
16.7
5.81
6.94
7.30
7.30
11.5
12.3
13.2
15.4
18.9
7.46
8.40
9.17
10.0
13.9
14.5
15.4
18.2
21.3
9.17
10.0
11.9
14.1
16.4
16.9
17.9
20.0
23.8
10.9
11.8
15.4
19.6
18.9
19.6
20.4
22.7
25.6
12.7
14.1
20.0
28.6
-
b720_Chapter-03
17-Mar-2009
Handbook of PI and PID Controller Tuning Rules
198
Rule
Bryant et al. (1973)
– continued.
Kc
Ti
Comment
x1 K m τ m
Tm1
Tm1 > Tm 2
τ m Tm1
Coefficient values (deduced from graph):
x1
τ m Tm1
x1
0.33
0.40
0.50
0.67
0.33
0.40
0.50
0.67
15
Kuwata (1987).
Model: Method 1
15
FA
0.08
0.09
0.09
0.11
0.14
0.16
0.18
0.23
1.0
2.0
10.0
0.15
0.24
0.33
Critically damped
dominant pole.
1.0
2.0
10.0
0.31
0.47
0.64
Damping ratio
(dominant pole) =
0.6.
x1 K m
x 2 K m Tm1
Tm1 > Tm 2
Tm1 = Tm 2
16
Kc
Ti
17
Kc
Ti
Representative x 1 values, summarised as follows, are deduced from graphs:
τ m Tm1
x 1 value
ξm
1.0
2.0
3.0
4.0
5.0
0.4
0.006
0.076
0.103
0.115
0.6
0.039
0.093
0.114
0.122
0.8
0.055
0.110
0.127
0.114
0.093
0.9
0.139
0.147
0.122
0.093
0.070
Representative x 2 values, summarised as follows, are deduced from graphs:
τ m Tm1
x 2 value
ξm
1.0
0.4
0.6
5.13
0.8
5.99
0.9
5.81
0
.
4
2
T
+
τ
0. 5
(
)
16
m1
m
Kc =
−
, Ti
K m (2Tm1 + τm − x1 ) K m
2.0
3.0
4.0
5.0
6.0
5.81
6.94
7.30
7.30
7.46
8.40
9.17
10.0
9.17
10.0
11.9
14.1
10.9
11.8
15.4
19.6
12.7
14.1
20.0
28.6
= 1.25x1 - 0.25(2Tm1 + τm ) , with
x1 =
17
Kc =
6.0
0.122
0.120
0.075
0.052
(2Tm1 + τm )2 − 0.267τm [6Tm12 + 6Tm1τm + τm 2 ] .
0.4(2ξ m Tm1 + τm )
0. 5
−
, Ti = 1.25x1 - 0.25(2ξm Tm1 + τm ) , with
K m (2ξ m Tm1 + τm − x1 ) K m
x1 =
(2ξmTm1 + τm )2 − 0.267τm [6Tm12 + 6ξmTm1τm + τm 2 ] .
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Rotach (1995).
Model: Method 9
3.35 K m
3.32τm
Pomerleau and Poulin
(2004).
Model: Method 6
Vítečková (2006).
Model: Method 5
Damping factor for oscillations to a disturbance input = 0.75.
Tm1 = Tm 2 ;
Tm1
1.5Tm1
K m (Tm1 + τ m )
OS < 10%.
18
19
Kc ≥
20
Five criteria are
fulfilled.
Ti
Kc
1.571Tm1
.
K m (2.718τm + 1.5Tm1 )
2


x1 
1  0.7Tm 2 (Tm1 + Tm 2 ) + Tm12  
τ 
, Ti = (Tm1 + 0.7Tm 2 )1 + 0.1 m  ;
Kc =
1+




Km
Tm1Tm 2 
Tm1 + 0.7Tm 2
Tm1 

 

x 1 values are obtained as follows:
τ m Tm1
x 1 value
20
Tm1 = Tm 2
1.571Tm1
Kc
Direct synthesis: frequency domain criteria
Maximise crossover
19
frequency.
Ti
Kc
Hougen (1979),
pp. 345–346.
Model: Method 1
Hougen (1988).
Model: Method 1
18
Comment
Tm 2 Tm1
0.1
0.2
0.3
0.4
0.5
0.1
0.3
0.5
1.0
0.31
0.6
0.65
0.7
0.19
0.42
0.5
0.6
0.15
0.35
0.45
0.5
τ m Tm1
0.11
0.28
0.46
0.45
0.09
0.23
0.32
0.38
Tm 2 Tm1
0.6
0.7
0.8
0.9
1.0
0.1
0.3
0.5
1.0
0.085
0.2
0.29
0.34
0.075
0.18
0.26
0.32
0.07
0.16
0.24
0.31
0.065
0.155
0.23
0.29
0.06
0.15
0.22
0.28
Equations for K c deduced from graph;


1
Kc =
10 
Km


1
Kc =
10 
Km
T
0.73 log10  m 2
 τm


 + 0.65 



T
0.48 log10  m 2
 τm


 + 0.05 



Ti = Tm1 + 0.7Tm 2 + 0.12τm .

 Tm 2

τm


 + 0.29 



 Tm 2

τm


 − 0.06 



 0.61 log10 
T
1

, m 2 = 0.1 ; K c =
10 
Tm1
Km

 0.41 log10 
T
1

, m 2 = 0.5 ; K c =
10 
Tm1
Km
,
Tm 2
= 0.2 ;
Tm1
,
Tm 2
=1.
Tm1
199
b720_Chapter-03
17-Mar-2009
Handbook of PI and PID Controller Tuning Rules
200
Rule
Kc
21
Somani et al. (1992).
Model: Method 1;
suggested A m :
1.75 ≤ A m ≤ 3.0
Ti
Kc
τm
Tm1
Alternative.
x 2τm
x1 K m A m
τm
Tm 2
Comment
Ti
x1
Coefficient values:
τm
τm
x2
Tm 2
Tm1
0.1
0.147 11.707 12.5
0.1
0.440 10.091 8.33
0.1
0.954 10.315 6.25
0.1
1.953 11.070 5.00
0.1
4.587 12.207 4.17
0.1
13.89 13.337 3.70
0.125 0.121 11.318 12.5
0.125 0.404 8.613 8.33
0.125 0.897 8.510 6.25
0.125 1.842 8.997 5.00
0.125 4.219 9.840 4.17
0.125 11.628 10.707 3.70
0.25 0.064 14.494 11.11
21
FA
1.00
1.00
1.00
1.00
1.67
1.67
1.67
1.67
1.67
2.5
2.5
2.5
2.5
x1
0.082 12.389
0.437 3.462
1.016 2.370
2.075 2.087
0.169 6.871
0.581 2.773
1.242 1.930
2.445 1.624
4.651 1.957
0.156 7.607
0.338 4.015
0.559 2.797
1.182 1.848
x2
6.25
5.00
4.17
3.57
5.00
4.17
3.57
3.13
3.13
4.55
4.17
3.85
3.33
For 25Tm1Tm 2 Ti << 1 ,
2
T
T  
T
0.9806
Kc ≈
1 +  m1  2.944 − tan −1  m1 + m 2
τm
KmAm
 τm
 τ m  



2
T
1 +  m 2
 τm
2
T  
T
T 
0.9806
Kc ≈
1 +  m1  2.944 − m1 − m 2 
τ
KmAm
τ
τm 
m
 m  
2
For 25Tm1Tm 2
T 
0.9806
Ti >> 1 , K c ≈
1 +  m1  x 1 2
KmAm
 τm 
2



2

T
−1  Tm1
+ m2
2.944 − tan 
τ
τm
 m

T
1 +  m 2
 τm
T
1 +  m 2
 τm
2

 or

2
2
 
T
T 
 2.944 − m1 − m 2  ;
τ
τm 
m
 
2
 2
 x 1 , with


τ
τ
x 1 = 0.981 + 0.5 m + m

T
T
m2
  m1

τ
τ
 + 0.945 + 0.25 m + m
T
T
m2

 m1

τ
τ
 + 0.945 m + m
T
T
m2

 m1

τ
τ
+ 0.5 m + m

Tm1 Tm 2



τ
τ
 + 0.945 − 0.25 m + m

 Tm1 Tm 2

τ
τ
 + 0.945 m + m

 Tm1 Tm 2
2
2












0.333
0.333
.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Somani et al. (1992)
– continued.
Ti
x1 K m A m
x 2τm
τm
Tm1
0.25
0.25
0.25
0.25
0.25
0.5
0.5
0.5
0.5
0.5
Tan et al. (1996).
Model: Method 2
x1
Coefficient values:
τm
τm
x2
Tm 2
Tm1
0.256 6.504
0.667 5.136
1.406 4.961
3.021 5.174
6.452 5.502
0.062 14.918
0.376 4.353
0.909 3.274
1.880 3.025
3.401 3.038
8.33
6.25
5.00
4.17
3.70
8.33
6.25
5.00
4.17
3.70
Ti
Tm1
K m (Tm1 + τ m )
Tm1
−ωTm1 τm
Schaedel (1997).
Model: Method 1
22
Kc =
Ti =
23
Ti =
(
x1Ti ωφ 1 + x1Tm1ωφ
(
A m 1 + x1Ti ωφ
x 1ωφ
tan [− 2 tan
)
)2
2
23
x1
x2
0.075 16.187 4.17
0.238 5.623 3.85
0.433 3.448 3.57
0.667 2.52
3.33
0.951 2.014 3.13
0.073 17.650 3.85
0.235 5.996 3.57
0.424 4.524 3.33
0.649 2.641 3.13
0.917 2.093 2.94
Tm1 = Tm 2
Tm1 > 5Tm 2 ;
minimum φ m = 58 0 .
Minimum φ m = 250
Ti
24
Kc
Ti
‘Normal’ design.
25
Kc
Ti
‘Sharp’ design.
, x1 = 0.8,
τm
τ
< 0.5 ; x1 = 0.5, m > 0.5 ; ωφ < ωu ;
Tm1
Tm1
1
−1
5.0
5.0
5.0
5.0
5.0
10.0
10.0
10.0
10.0
10.0
Kc
22
Poulin et al. (1996).
Model: Method 1
Comment
Kc
τm
Tm 2
201
x1Tm ωφ − x1τ m ωφ − φ m
].
4Tm1 (τm + Tm 2 )
, 0.16Tm1 ≤ Tm 2 + τ m ≤ 0.3Tm1 ; ω = frequency
1.3Tm1 − 4(τm + Tm 2 )
where phase of G m ( jω)G c ( jω) is maximum; G m =
− K m e − sτ m
.
(1 + Tm1s )(1 + Tm 2s )
(2ξmTm1 + τm )2 − 2(Tm12 + 2ξmTm1τm + 0.5τm 2 ) .
24
Kc =
0.5Ti
, Ti =
K m (2ξ m Tm1 + τm − Ti )
25
Kc =
0.375 4ξm 2Tm12 + 2ξ mTm1τm + 0.5τm 2 − Tm12
,
Km
Tm12 + 2ξ m Tm1τm + 0.5τm 2
2
Ti =
2
2
2
4ξ m Tm1 + 2ξ m Tm1τm + 0.5τm − Tm1
.
2ξm Tm1 + τm
b720_Chapter-03
Rule
Leva et al. (2003).
Model: Method 1
27
28
29
30
Kc =
Kc =
Kc =
Kc =
Comment
Kc
Ti
Tm1
2 K m (Tm 2 + τ m )
4(Tm 2 + τ m )
Hägglund and
Åström (2004).
Model: Method 1;
Tm1 = Tm 2 .
Kc =
FA
Handbook of PI and PID Controller Tuning Rules
202
26
17-Mar-2009
26
Kc
Ti
27
Kc
Ti
28
Kc
Ti
29
Kc
Ti
30
Kc
Ti
2.01 − 0.80φ
( ) 

G jω
1 + (15.45 − 6.30φ) p φ
Km


Tm1 >> Tm 2 + τ m ;
symmetrical optimum
principle.
1.920 < φ < 2.356 ;
M s = 1.2 .
φ = 2.09 radians
(‘good’ choice).
2.094 < φ < 2.443 ;
M s = 1.4 .
φ = 2.269 radians
(‘reasonable’ choice).
2.356 < φ < 2.705 ;
M s = 2. 0 .
6.283(1.72φ − 2.55)
, Ti =
( ) 

G jω
G p jωφ
1 + (3.44φ − 5.20 ) p φ

Km



0.33
6.61
, Ti =
.
2


G p jωφ

G p jωφ 
1 + 2.26
 G p jωφ
 ωφ
1 + 2.01
Km 

Km 





2.50 − 0.92φ
6.283(1.72φ − 3.05)
, Ti =

G p jωφ 

G jω
1 + (10.75 − 4.01φ)
 G p jωφ
1 + (3.44φ − 6.10 ) p φ
Km 

Km




0.41
5.36
, Ti =
.
2


G p jωφ

G p jωφ 
1 + 1.65
 G p jωφ
 ωφ
1 + 1.71
Km 

Km 





3.43 − 1.15φ
6.283(0.86φ − 1.50)
, Ti =

G p jωφ 

G jω
1 + (11.30 − 4.01φ)
 G p jωφ
1 + (1.72φ − 2.90) p φ
Km 

Km




( )
( )
ωφ
( ) 
2
( ) 
2
.
ωφ


( )
( )
( )


( )
( )
( )
( )
( )
.
2
( )


.
ωφ
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Hägglund and
Åström (2004) –
continued.
Desbiens (2007),
p. 90.
31
Kc
32
Kc
Ti
Comment
Ti
φ = 2.53 radians
(‘good’ choice).
Tm1 > Tm 2 ;
Model: Method 1
Ti
Robust
Tm1 + Tm 2 + 0.5τ m
K m τ m (2λ + 1)
Brambilla et al.
(1990).
Model: Method 1;
λ values obtained
from graph.
31
32
Kc =
0.1
0.2
0.5
1.0
2ξ mTm1 + 0.5τm
K m (2λ + τm )
Kc =
0.52
( ) 

G jω
1 + 1.15 p φ
Km


Ti
Km


( )
G p jωφ
, Ti =
3.0
1.8
1.0
0.6
2.0
5.0
10.0
λ
0.4
0.2
0.2
2ξ m Tm1 + 0.5τm
4.25
( ) 

G jω
1 + 1.45 p φ
Km


(Tm1Tm 2 )2 ω6 + (Tm12 + Tm 22 )ω4 + ω2
Ti 2ω2 + 1
τm
≤ 10
Tm1 + Tm 2
Coefficient values:
τm
λ
Tm1 + Tm 2
τm
Tm1 + Tm 2
Marchetti and Scali
(2000).
Model: Method 1
0.1 ≤
Tm1 + Tm 2 + 0.5τ m


.
2
ωφ
, with ω given by solving (for
φm = 58.13 ): 1.015 = 0.5π + tan −1 (Ti ω) − tan −1 (Tm 2ω) − tan −1 (Tm1 ω) − τ m ω .
2

T 
τ
T  τ
Ti = Tm1 1 + 0.175 m + 0.3 m 2  + 0.2 m 2  , m < 2 ;
Tm1
Tm1
Tm1
 Tm1 


2

 Tm 2 
τm
Tm 2  τ m



+ 0.3
+
>2.
0
.
2
Ti = Tm1  0.65 + 0.35
,


Tm1  Tm1
Tm1
 Tm1 


203
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
204
Rule
Kc
Kristiansson (2003).
Model: Method 1
33
Kc
34
Kc
Panda et al. (2004).
Model: Method 1
35
Kc
Ti
37
Kc
Ti
Kuwata (1987);
x 1 , x 2 deduced from
graphs. ξ m = 0.5 .
Kuwata (1987);
x 1 , x 2 deduced from
graphs. ξ m = 0.7 .
Kuwata (1987);
x 1 , x 2 deduced from
graphs. ξ m = 1.5 .
Comment
Ti
2ξ m Tm1
M max = 1.7
36
Ultimate cycle
x 2 Tu
x 1K u
x1
x2
τm Tu
x1
x2
τm Tu
0.08
0.09
0.03
0.32
0.14
0.03
x 1K u
0.05
0.10
0.15
0.07
0.13
0.18
0.04
0.08
0.10
x 2 Tu
0.25
0.30
0.35
x1
x2
τm Tu
x1
x2
τm Tu
0.01
0.04
0.08
0.12
0.29
0.19
0.15
0.14
x 1K u
0.05
0.10
0.15
0.20
0.16
0.18
0.21
0.22
0.14
0.15
0.15
0.15
x 2 Tu
0.25
0.30
0.35
0.40
x1
x2
τm Tu
x1
x2
τm Tu
0.21
0.25
0.26
1.13
0.70
0.43
0.15
0.20
0.25
0.25 0.29
0.23 0.21
0.225 0.17
x1
x2
x1
x2
0.225 0.155
0.225 0.155
Kc =
2ξ m Tm1 
1
 x1 − x12 − 1 +
K m τm 
M max 2
34
Kc =
2ξ m Tm1 
T
T
T 2
1 + 0.1913 m 2 − 0.346 + 0.3826 m 2 + 0.0366 m 22
K m τm 
τm
τm
τm

35
Kc =
2
4ξ 2 T + ξ m τ m + 0.5Tm1
8ξm Tm1 + 2ξ m τm + Tm1
, Ti = m m1
.
2λK m
2ξ m
37
x2
0.225 0.155
0.225 0.155
33
36

T
 , x1 = 1 + m 2
τm

0.30
0.35
0.40
x1
0.21 0.12
0.225 0.14
0.225 0.155

1
1 − 1 −
M max 2

τm Tu
0.40
0.45
0.50
τm Tu
0.45
0.50
τm Tu
0.45
0.50

.


.




T
λ ≥ maximum 0.828 + 0.812 m 2 + 0.172Tm1e − (6.9Tm 2 / Tm1 ) ,1.7 τm  .
T
m1


Tm 2
0
.
558
T
T
m 2 m1
+ 0.172Tm1e − (6.9Tm 2 / Tm1 ) +
0.828 + 0.812
0.5τ m
Tm1
Tm1 + 1.208Tm 2
+
,
Kc =
λK m
λK m
Ti = 0.828 + 0.812
Tm 2
0.558Tm 2Tm1
+ 0.172Tm1e − (6.9Tm 2 / Tm1 ) +
+ 0.5τ m .
Tm1
Tm1 + 1.208Tm 2
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Thyagarajan and Yu
(2003).
38
39
Hägglund and
Åström (2004).
Model: Method 1
38
39
Ti
Comment
Kc
2Tm1
Tm1 = Tm 2 ;
Model: Method 20
Kc
Other tuning
1.95
tan (0.5φ)
ωφ
Kc
0.413 G p ( jω2.27 )
3

T 
 T 
2 0.0200 m1  + 0.2714 m1 

 Tu 
 Tu 
.
Kc = 
2
 Tm1 
 +1
K m 0.2947

 Tu 
2.31
Kc =
.
2
1 + tan (0.5φ) G p ( jωφ )
[
]
4.182 ω 2.27
Tm1 = Tm 2 ;
Tm1 >> τ m .
φ = 2.27 radians
(‘reasonable’ choice).
205
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
206


1
+ Td s 
3.5.2 Ideal PID controller G c (s) = K c 1 +
Ti s


E(s)
R(s)
+
−

 U(s)
1
K c 1 +
+ Td s 
Ti s


Table 21: PID tuning rules – SOSPD model G m (s) =
G m (s ) =
Rule
Auslander et al.
(1975).
1
Kc
1
Tm1 2 s 2 + 2ξ m Tm1s + 1

 Tm 2
 T
 m1
(Tm1 − Tm 2 )2
−
1+
Tm 2
Tm1 − Tm 2

2
 Tm 2
(Tm1 − Tm 2 )  Tm1

1+
Tm 2
Tm1 − Tm 2
Tm 2

 Tm 2
 T
 m1
 Tm 2


 Tm1 
Tm 1

 Tm1 −Tm 2 




Tm 2
 Tm1 −Tm 2  Tm 2

− 

 Tm1
Tm1
 Tm1 −Tm 2
−
Tm1
Tm1 − Tm 2
 Tm1 −Tm 2  Tm 2

− 
 Tm1

Tm1

 Tm1 −Tm 2 




Tm1
 Tm1 −Tm 2
 Tm 2


 Tm1 
Model:
Method 3
 Tm1 −Tm 2  Tm 2

− 

 Tm1
1
2Tm1Tm 2  Tm1 
 −
ln
Tm1 − Tm 2  Tm 2 
Comment
Td
Process reaction
x1 +2τm
0.25Ti

1.2
T T
T 
Kc =
{τm + m1 m 2 ln m1 
K m (Tm1 − Tm 2 ) 
Tm1 − Tm 2 Tm 2 
x1 =
K m e − sτ m
or
(1 + sTm1 )(1 + sTm 2 )
K m e −sτ m
Ti
Kc
Y(s)
SOSPD
model
Tm 2
−
Tm1
Tm1 − Tm 2
Tm1

 Tm1 −Tm 2 




 Tm 2


 Tm1 
Tm 2
 Tm1 −Tm 2
 Tm 2


 Tm1 
Tm 2
 Tm1 −Tm 2
2
.
}−1 ,
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Alfaro Ruiz
(2005a).
2
ξm
τm Tm1 = 0.1
Kc
Ti
Td
Comment
Kc
Ti
Td
Model: Method 1
Tm1 = Tm 2 ; 1.0 ≤ x1 ≤ 1.67 .
Minimum performance index: regulator tuning
Representative
5.8
tuning values;
0.36Tu
0.21Tu
Km
Tm 2 = τ m
5
= 0.1Tm1 .
0.29Tu
0.29Tu
Km
Minimum IAE –
Wills (1962b)
– deduced from
graph.
Model: Method 1
Minimum IAE –
Lopez (1968),
pp. 79–90.
207
x1 K m
x 2Tm1
Model: Method 1
x 3Tm1
Coefficients of K c , Ti and Td deduced from graphs. These are
representative results.
0.5
0.6
0.8
x1
x2
x3
x1
x2
x3
x1
x2
x3
27
-
0.28
28
-
0.26
30
-
0.26
τm Tm1 = 0.2
9.5
0.61
0.48
10.5
0.59
0.45
11.5
0.57
0.41
τm Tm1 = 0.5
2.25
0.97
0.92
2.6
1.00
0.82
3.2
1.05
0.69
τm Tm1 = 1.0
0.78
1.11
1.40
1.0
1.25
1.20
1.4
1.47
0.94
τm Tm1 = 2.0
0.39
1.27
1.40
0.52
1.56
1.30
0.77
1.92
1.15
τm Tm1 = 5.0
0.42
2.7
1.40
0.45
2.9
1.50
0.53
3.0
1.70
τm Tm1 = 10.0
0.38
4.9
1.15
0.41
5.3
1.40
0.47
5.7
1.90
1.0
ξm
τm Tm1 = 0.1
x1
x2
31
-
1.5
x1
x2
0.24 35
x3
-
2.0
x3
4.0
x1
x2
x3
x1
x2
x3
0.22 40
-
-
68
-
-
τm Tm1 = 0.2
13.0 0.59 0.43 16.0 0.56 0.30 19.5 0.56 0.25 47 0.53
τm Tm1 = 0.5
4.0 1.06 0.59 6.0 1.11 0.46 7.8 1.14 0.39 16.5 1.14 0.29
-
τm Tm1 = 1.0
1.85 1.56 0.82 3.0 1.75 0.68 4.3 1.85 0.60 9.0 1.92 0.52
τm Tm1 = 2.0
1.05 2.3 1.10 1.65 2.6 1.05 2.3
2.9 1.00 4.8
τm Tm1 = 5.0
0.62 3.6 1.90 0.85 4.3 2.10 1.1
4.8 2.15 2.15 6.5
τm Tm1 = 10.0
0.52 6.1 2.35 0.60 6.9
3.3 0.98
2.3
2.9 0.70 7.4 3.45 1.2 10.0 4.0
τ

 16.57 + 20.83 m


x
2
.
0
T
T
2

m1
m1
K c = 1 0.70 +
 , Ti = 
τm
Km 
τm 
1
11
.
36
+

Tm1

τ



 2.65 + 3.33 m
T
τ , T = 
m1
 m d 
τm
1
11
.
36
+


Tm1




τ .
 m


b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
208
Rule
Kc
Ti
Td
Comment
Minimum IAE –
Shinskey (1988),
p. 151.
Model: Method 2
0.62 K u
0.38Tu
0.15Tu
Tm 2
= 0.25
Tm 2 + τ m
0.68K u
0.33Tu
0.19Tu
Tm 2
= 0.5
Tm 2 + τ m
0.79 K u
0.26Tu
0.21Tu
Tm 2
= 0.75
Tm 2 + τ m
Ti
Td
Model: Method 1
Minimum IAE –
Kang (1989).
T
0.05 ≤ L ≤ 7.0.
Tm1
Minimum ISE –
Lopez (1968),
pp. 79–90.
ξm
3
Kc
x1 , x 2 …. x14 , x15 values are specified on pp. 127–165 of Kang
(1989), for τm Tm1 = 0.1, 0.2 ….. 0.9, 1.0 with Tm 2 Tm1 = 0.1 ,
0.2…..0.9, 1.0.
x1 K m
Model: Method 1
x 3Tm1
Coefficients of K c , Ti and Td deduced from graphs. These are
representative results.
0.5
0.6
0.8
x1
3
x 2Tm1
x2
x3
x1
x2
x3
x1
x2
x3
τm Tm1 = 0.1
29
0.21
0.34
30
0.21
0.33
31
0.21
0.31
τm Tm1 = 0.2
10.0
0.38
0.56
10.5
0.38
0.54
12.0
0.38
0.48
τm Tm1 = 0.5
2.4
0.67
1.10
2.8
0.69
1.10
3.2
0.74
0.83
τm Tm1 = 1.0
0.86
0.87
1.70
1.1
0.95
1.45
1.55
1.11
1.20
τm Tm1 = 2.0
0.45
1.1
2.00
0.57
1.3
1.80
0.85
1.6
1.60
τm Tm1 = 5.0
0.44
2.4
2.00
0.49
2.6
2.20
0.57
2.9
2.40
τm Tm1 = 10.0
0.45
5.3
1.50
0.48
5.6
1.85
0.53
5.9
2.50
Kc =
Ti =
T


−x2 L 


1 
T 
T 

x1 − e Tm1 x 3 cos x 4 L  − x 5 sin  x 4 L  ,
Km 

 Tm1 
 Tm1 

Tm1
T
T

x8 L 
x 10 m 2


T

Tm1 
+ x 9e TL sin  x11 L + x12 
x 6 + x 7 1 − e

T


m1




1
Load disturbance model =
.
1 + sTL
, Td =
T

x 15 L 
1 
x13 + x14e Tm1  ,

Tm1 


b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
209
Rule
Kc
Ti
Td
Comment
Minimum ISE –
Lopez (1968),
pp. 79–90 –
continued.
ξm
x1 K m
x 2Tm1
x 3Tm1
Model: Method 1
Coefficients of K c , Ti and Td deduced from graphs. These are
representative results.
1.0
x1
τm Tm1 = 0.1
1.5
x2
x3
x1
2.0
x2
x3
x1
x2
4.0
x3
x1
x2
x3
32 0.21 0.30 38 0.21 0.25 42 0.21 0.23 70 0.21
-
τm Tm1 = 0.2
13 0.38 0.44 17 0.38 0.36 21 0.38 0.31 39 0.38 0.20
τm Tm1 = 0.5
4.2 0.77 0.74 6.2 0.80 0.57 8.5 0.82 0.50 17.5 0.83 0.37
τm Tm1 = 1.0
2.0 1.18 1.05 3.2 1.33 0.85 4.4 1.39 0.77 9.5 1.49 0.66
τm Tm1 = 2.0
1.1
2.6
1.2
τm Tm1 = 5.0
0.66 3.1
2.6 0.90 3.7
2.8 1.20 4.2 2.85 2.25 5.3
3.0
τm Tm1 = 10.0
0.58 5.9
3.0 0.67 6.3
4.0 0.80 6.7
Minimum ITAE
– Lopez (1968),
pp. 79–90.
x1 K m
x 2Tm1
x 3Tm1
ξm
1.8
1.5
1.8
2.1 1.35 2.9
2.3 1.30 5.2
-
1.25 8.3
-
Model: Method 1
Coefficients of K c , Ti and Td deduced from graphs. These are
representative results.
0.5
0.6
0.8
x1
x2
x3
x1
x2
x3
x1
x2
x3
τm Tm1 = 0.1
24
-
0.25
24
-
0.25
24
0.50
0.225
τm Tm1 = 0.2
8.8
0.71
0.44
9.2
0.74
0.41
10.0
0.74
0.35
τm Tm1 = 0.5
2.1
1.15
0.84
2.5
1.15
0.74
3.2
1.22
0.60
τm Tm1 = 1.0
0.72
1.22
1.20
0.93
1.39
1.00
1.35
1.61
0.82
τm Tm1 = 2.0
0.36
1.30
1.15
0.48
1.59
1.10
0.75
2.04
1.00
τm Tm1 = 5.0
0.40
2.7
1.20
0.44
2.9
1.30
0.52
3.2
1.45
τm Tm1 = 10.0
0.34
4.3
0.98
0.38
5.0
1.30
0.43
-
1.75
1.0
ξm
x1
τm Tm1 = 0.1
x2
1.5
x1
x2
26 0.53 0.20 31
x3
-
2.0
x1
x2
0.175 36
x3
-
4.0
x1
x2
x3
0.15 68
x3
-
0.1
τm Tm1 = 0.2
11.3 0.77 0.32 14.5 0.71 0.25 18.2 0.68 0.205 36 0.57 0.14
τm Tm1 = 0.5
3.8 1.25 0.52 5.8 1.28 0.40 7.7 1.21 0.34 16.2 1.21 0.26
τm Tm1 = 1.0
1.8 1.75 0.70 2.9 1.92 0.58 4.0 2.00 0.54 8.7 2.13 0.46
τm Tm1 = 2.0
0.36 2.4 0.96 0.48 2.9 0.92 0.73 3.2 0.88 0.97 3.6 0.84
τm Tm1 = 5.0
0.40 3.6
τm Tm1 = 10.0
0.34 5.9
1.6 0.44 4.5
-
0.38 6.9
1.7 0.53 5.3 1.85 0.58 6.9
-
0.43 7.7
-
0.48
-
2.0
-
b720_Chapter-03
Rule
Minimum ITAE
– Lopez et al.
(1969).
Model:
Method 23
Bohl and
McAvoy
(1976b).
Minimum ITAE
– Hassan (1993).
5
FA
Handbook of PI and PID Controller Tuning Rules
210
4
17-Mar-2009
Comment
Kc
Ti
Td
x1 K m
x 2 Tm1
x 3 Tm1
Representative coefficient values (from graphs):
x1
x2
x3
ξm
τ m Tm1
25
0.7
0.35
25
1.8
9.0
0.5
1.3
5
0.5
1.7
2
0.25
1.2
1.0
0.2
0.7
0.45
0.5
0.5
0.5
1.0
1.0
4.0
0.1
1
10
0.1
1
1
4
Kc
Ti
Td
Minimum ITAE;
Model: Method 1
5
Kc
Ti
Td
Model: Method 8
Tm 2 Tm1 = 0.12,0.3,0.5,0.7,0.9 ; τm Tm1 = 0.1,0.2,0.3,0.4,0.5 .
 10 τ m  


−1.2096 + 0.1760 ln 

 Tm1  
10.9507  10τm 


Kc =



K m  Tm1 



 T 
 10 τ m  


0.1044 + 0.1806 ln  10 m 2  − 0.2071 ln 

 10Tm 2 
 Tm1 
 Tm1  



,

 Tm1 



 10 τ m  


0.7750 − 0.1026 ln 

 10τm 
 Tm1  

Ti = 0.2979Tm1 


 Tm1 



 10 Tm 2 
 10 τ m  

 + 0.0081 ln 

0.1701+ 0.0092 ln 

 T 
 10Tm 2 
 Tm1 
 m1 



,

 Tm1 



 10 τ m  


0.6025 − 0.0624 ln 

 10τm 
 Tm1  

Td = 0.1075Tm1 


 Tm1 



 10 Tm 2 
 10 τ m  

 + 0.0128 ln 

0.4531− 0.0479 ln 

 T 
 10Tm 2 
 Tm1 
 m1 



.

 Tm1 



Note: equations continued into the footnote on p. 211.
0.5 ≤ ξ m ≤ 2 ; 0.1 ≤ τm Tm1 ≤ 4 . K c is obtained as follows:
log[K m K c ] = 1.9763370 − 0.6436825ξm − 5.1887660
τm
+ 0.4375558ξ m 2
Tm1
2
τ 

τ
2 τ
+ 2.9005550 m  + 3.1468010ξ m m − 0.1697221ξ m  m 
T
T
T
m1
 m1 
 m1 
2
2
τ 
τ
3
− 0.8161808ξ m  m  − 1.2048220ξ m 2 m − 0.0810373ξ m
T
T
m1
 m1 
3
3
τ 
τ 

2 τ
− 0.4444091 m  + 0.0319431ξ m  m  + 0.1054399ξ m  m 
T
T
T
 m1 
 m1 
 m1 
2
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
2
+ 0.1652807ξ m
3
3


τm
3 τ
3 τ
+ 0.1175991ξ m  m  − 0.0375245ξ m  m  ;
Tm1
T
T
 m1 
 m1 
Ti is obtained as follows:
 T 
τ
log  i  = −0.7865873 + 0.6796885ξm + 2.1891540 m − 0.3471095ξ m 2
T
T
m1
 m1 
2
τ 

τ
2 τ
− 1.9003610 m  − 0.7007801ξ m m + 0.3077857ξ m  m 
T
T
T
m1
 m1 
 m1 
2
2
τ 
τ
3
+ 0.8566974ξ m  m  − 0.2535062ξ m 2 m + 0.0412943ξ m
T
T
m1
 m1 
3
3
τ 
τ 

2 τ
+ 0.3484161 m  − 0.1626562ξ m  m  − 0.0661899ξ m  m 
T
T
T
 m1 
 m1 
 m1 
2
2
+ 0.2247806ξ m
3
3


τm
3 τ
3 τ
− 0.2470783ξ m  m  + 0.0493011ξ m  m  ,
Tm1
T
T
 m1 
 m1 
Td is obtained as follows:
T 
τ
log  d  = −0.6726798 − 0.2072477ξ m + 2.6826330 m + 0.0807474ξm 2
T
T
m1
 m1 
2
τ 

τ
2 τ
− 1.7707830 m  − 1.6685140ξ m m + 0.0845958ξ m  m 
T
T
T
m1
 m1 
 m1 
2
2
τ 
τ
3
+ 0.7159307ξ m  m  + 0.5631447ξ m 2 m − 0.0225269ξ m
T
T
m1
 m1 
3
3
τ 
τ 

2 τ
+ 0.2821874 m  − 0.0616288ξ m  m  − 0.0626626ξ m  m 
T
T
T
 m1 
 m1 
 m1 
2
− 0.0372784ξ m 3
2
3


τm
3 τ
3 τ
− 0.0948097ξ m  m  + 0.0272541ξ m  m  .
Tm1
T
T
 m1 
 m1 
211
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
212
Rule
Minimum ITAE
– Sung et al.
(1996).
6
17-Mar-2009
Kc =
1
Km
6
Kc
Ti
Td
Comment
Kc
Ti
Td
Model:
Method 11
2.001
0.766

 τ
 Tm1 


− 0.67 + 0.297 Tm1 
 , m < 0.9 or


ξ
2
.
189
+
m
τ 
τ 

 Tm1
m 
m 




Kc =
2
0.766
 τ
T 

τ
1 
ξ m  , m ≥ 0. 9 ;
− 0.365 + 0.260 m − 1.4  + 2.189 m1 
Km 
 Tm1
 τm 

 Tm1


0.520

 τ
τ 
− 0.3 , m < 0.4 or
Ti = Tm1 2.212 m 

 Tm1
 Tm1 


2

τ
τ
Ti = Tm1{−0.975 + 0.910 m − 1.845  + x1} , m ≥ 0.4 with
T
T
m1

 m1
ξm

−
τ

0.15 + 0.33 m
Tm1
x1 = 1 − e



Td =
ξm

−
1 − e − 0.15 + 0.939(Tm1



2


 
 5.25 − 0.88 τm − 2.8   ;
T
 

 m1
 


Tm1
;
1.171 
0.530


1.121




T
T
τm )
 1.45 + 0.969 m1 
 − 1.9 + 1.576 m1 
τ 
τ 


m 

 m 

 
0.05 <
τ
τm
≤ 2 (Sung et al., 1996); 0 < m < 2 (Park et al., 1997).
Tm1
Tm1
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Nearly minimum
IAE, ISE, ITAE
– Hwang (1995).
Model:
Method 14
213
Comment
Kc
Ti
Td
7
Kc
Ti
8
Kc
Ti
9
Kc
Ti
3 ≤ ε 2 < 20
10
Kc
Ti
ε 2 ≥ 20
ε 2 < 2.4
2.4 ≤ ε 2 < 3
Td
Decay ratio = 0.2; 0.2 ≤ τ m Tm1 ≤ 2.0 and 0.6 ≤ ξ m ≤ 4.2 .
Minimum performance index: servo tuning
Representative
tuning values;
6.5 K m
1.45Tu
0.14Tu
Tm 2 = τ m
Minimum IAE –
Wills (1962b)
– deduced from
graph.
Model: Method 1
7
7 Km
0.95Tu
[
]
(
)
2
2

0.622 1 − 0.435ωH 2 τm + 0.052ωH 2 τm 
Kc =  KH2 −
,


K m (1 + K H 2 K m )


K c (1 + K H 2 K m )
Ti =
,
ωH 2 K m 0.0697 1 + 0.752ωH 2 τm − 0.145ωH 2 2 τm 2
Td =
8
[
1.45(1 + K u K m )  1.16 
1 −
 1 − 0.612ω u τ m + 0.103ω u 2 τ m 2 .


ε
K m 2ωu
0


2
2

0.724 1 − 0.469ωH 2 τm + 0.0609ωH 2 τm
Kc =  KH2 −

K m (1 + K H 2 K m )

Ti =
9
[
[
(
)
] ,


(
K c (1 + K H 2 K m )
ωH 2 K m 0.0405 1 + 1.93ωH 2 τm − 0.363ωH 2 2 τm 2

1.26(0.506 )ωH 2 τ m 1 − 1.07 ε 2 + 0.616 ε 2 2
Kc =  KH2 −

K m (1 + K H 2 K m )

Ti =
10
= 0.1Tm1 .
0.22Tu
] ,


K c (1 + K H 2 K m )
(
ωH 2 K m 0.0661(1 + 0.824 ln[ωH 2 τm ]) 1 + 1.71 ε 2 − 1.17 ε 2 2
2
2

1.09 1 − 0.497ωH 2 τm + 0.0724ωH 2 τm
Kc =  KH2 −

K m (1 + K H 2 K m )

Ti =
] ,


(
K c (1 + K H 2 K m )
ωH 2 K m 0.054 − 1 + 2.54ωH 2 τ m − 0.457ωH 2 2 τ m 2
).
).
).
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
214
Rule
Minimum IAE –
Gallier and Otto
(1968).
Model:
Method 1
Minimum ITAE
– Wills (1962a) –
deduced from
graph.
Minimum ITAE
– Sung et al.
(1996).
Kc
Ti
x1 K m
τm
2Tm1
11
Ti
Td
Comment
Td
Tm1 = Tm 2
Representative coefficient values – deduced from graphs:
τm
x1
x2
x3
x1
x2
x3
2Tm1
0.053
0.11
0.25
6.7
4.15
2.35
0.89
0.84
0.77
18
Km
12
Kc
0.24
0.24
0.24
0.67
1.50
4.0
1.05
0.70
0.52
≈ Tu
≈ 0.2Tu
Ti
Td
11
Ti = x 2 (Tm1 + Tm 2 + τm ) , Td = x 3 (Tm1 + Tm 2 + τm ) .
12
Kc =
0.64
0.24
0.55
0.26
0.50
0.20
Tm1 = Tm 2 ;
τ m = 0.1Tm1 ;
Model: Method 1
Model:
Method 11
−0.983 



τ 
1 
 ξ m  , ξ m ≤ 0.9 or
− 0.04 + 0.333 + 0.949 m 
Km 

 
 Tm1 

 

Kc =
−0.832

τ 
τ
1 
ξ m  , ξ m > 0.9 .
− 0.544 + 0.308 m + 1.408 m 
Km 
Tm1

 Tm1 


Ti = Tm1 [2.055 + 0.072(τm Tm1 )]ξ m , τ m Tm1 ≤ 1 or
Ti = Tm1[1.768 + 0.329(τm Tm1 )]ξ m , τ m Tm1 > 1 .
Td =
(τ

− m
1 − e


0.05 <
Tm1 )1.060 ξ m
0.870
Tm1
;
1.090 



 0.55 + 1.683 Tm1 

τ 


 m 


τ
τm
≤ 2 (Sung et al., 1996); 0 < m < 2 (Park et al., 1997).
Tm1
Tm1
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Nearly minimum
IAE, ISE, ITAE
– Hwang (1995).
Model:
Method 14
13
Ti
Kc
Ti
15
Kc
Ti
16
Kc
Ti
17
Kc
Ti
13
215
Comment
Td
14
Decay ratio =
0.1;
τ
0.2 ≤ m ≤ 2.0 ;
Tm1
0.471K u
K m ωu
0. 6 ≤ ξ m ≤ 4. 2 .
[
2
2

0.822 1 − 0.549ωH 2 τm + 0.112ωH 2 τm
Kc =  KH2 −

K m (1 + K H 2 K m )

Ti =
] ,


K c (1 + K H 2 K m )
(
ωH 2 K m 0.0142 1 + 6.96ωH 2 τm − 1.77ωH 2 2 τm 2
).
2
14
15
ξ ≤ 0.613 + 0.613
τ 
τm
+ 0.117 m  .
Tm1
 Tm1 
[
2
2

0.786 1 − 0.441ωH 2 τm + 0.0569ωH 2 τm
Kc =  KH2 −

K m (1 + K H 2 K m )

Ti =
16
[


[
(
K c (1 + K H 2 K m )
ωH 2 K m 0.0172 1 + 4.62ωH 2 τm − 0.823ωH 2 2 τm 2
2
ω τ

1.28(0.542) H 2 m 1 − 0.986 ε 2 + 0.558 ε 2
Kc =  K H2 −

K m (1 + K H 2 K m )

Ti =
17
] ,
] ,


K c (1 + K H 2 K m )
(
ωH 2 K m 0.0476(1 + 0.996 ln[ωH 2 τm ]) 1 + 2.13 ε 2 − 1.13 ε 2 2
2
2

1.14 1 − 0.466ωH 2 τm + 0.0647ωH 2 τm
Kc =  KH2 −

K m (1 + K H 2 K m )

Ti =
] ,


(
K c (1 + K H 2 K m )
ωH 2 K m 0.0609 − 1 + 1.97ωH 2 τ m − 0.323ωH 2 2 τ m 2
).
).
).
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
216
Rule
Kc
Ti
Kc
Ti
20
Kc
Ti
21
Kc
Ti
22
Kc
Ti
18
Hwang (1995) –
continued.
18
17-Mar-2009
Comment
Td
19
Decay ratio =
0.1;
τ
0.2 ≤ m ≤ 2.0 ;
Tm1
0.471K u
K m ωu
0. 6 ≤ ξ m ≤ 4. 2 .
[
2
2

0.794 1 − 0.541ωH 2 τm + 0.126ωH 2 τm
Kc =  KH2 −

K m (1 + K H 2 K m )

Ti =
] ,


K c (1 + K H 2 K m )
(
ωH 2 K m 0.0078 1 + 8.38ωH 2 τm − 1.97ωH 2 2 τm 2
),
2
19
20
ξ > 0.649 + 0.58
τ 
τm
− 0.005 m  .
Tm1
 Tm1 
[
2
2

0.738 1 − 0.415ωH 2 τm + 0.0575ωH 2 τm
Kc =  KH2 −

K m (1 + K H 2 K m )

Ti =
21
[


[
K c (1 + K H 2 K m )
(
ωH 2 K m 0.0124 1 + 4.05ωH 2 τm − 0.63ωH 2 2 τm 2

1.15(0.564 )ωH 2 τ m 1 − 0.959 ε 2 + 0.773 ε 2 2
Kc =  KH2 −

K m (1 + K H 2 K m )

Ti =
22
] ,
] ,


K c (1 + K H 2 K m )
(
ωH 2 K m 0.0335(1 + 0.947 ln[ωH 2 τm ]) 1 + 1.9 ε 2 − 1.07 ε 2 2
2
2

1.07 1 − 0.466ωH 2 τm + 0.0667ωH 2 τm
Kc =  KH2 −

K m (1 + K H 2 K m )

Ti =
] ,


(
K c (1 + K H 2 K m )
ωH 2 K m 0.0328 − 1 + 2.21ωH 2 τm − 0.338ωH 2 2 τm 2
).
).
).
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Kc
Ti
25
Kc
Ti
26
Kc
Ti
27
Kc
Ti
23
Hwang (1995) –
continued.
23
Ti
Comment
Td
24
Decay ratio =
0.1;
τ
0.2 ≤ m ≤ 2.0 ;
Tm1
0.471K u
K m ωu
0. 6 ≤ ξ m ≤ 4. 2 .
[
2
2

0.789 1 − 0.527ωH 2 τm + 0.11ωH 2 τm
Kc =  KH2 −

K m (1 + K H 2 K m )

] ,


K c (1 + K H 2 K m )
Ti =
(
ωH 2 K m 0.009 1 + 9.7ωH 2 τm − 2.4ωH 2 2 τm 2
2
24
25
ξ ≤ 0.649 + 0.58
2
[
2
2

0.76 1 − 0.426ωH 2 τm + 0.0551ωH 2 τm
Kc =  KH2 −

K m (1 + K H 2 K m )

[
] ,


[
(
K c (1 + K H 2 K m )
ωH 2 K m 0.0153 1 + 4.37ωH 2 τ m − 0.743ωH 2 2 τ m 2

1.22(0.55)ωH 2 τ m 1 − 0.978 ε 2 + 0.659 ε 2 2
Kc =  KH2 −

K m (1 + K H 2 K m )

Ti =
27
).
τ 
τ 
τm
τ
− 0.005 m  , ξ > 0.613 + 0.613 m + 0.117 m  .
Tm1
T
T
m
1
m
1


 Tm1 
Ti =
26
217
] ,


K c (1 + K H 2 K m )
(
ωH 2 K m 0.0421(1 + 0.969 ln[ωH 2 τm ]) 1 + 2.02 ε 2 − 1.11 ε 2 2
2
2

1.111 − 0.467ωH 2 τm + 0.0657ωH 2 τm
Kc =  KH2 −

K m (1 + K H 2 K m )

Ti =
] ,


(
K c (1 + K H 2 K m )
ωH 2 K m 0.0477 − 1 + 2.07ωH 2 τm − 0.333ωH 2 2 τm 2
).
).
).
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
218
Rule
Kc
Ti
Comment
Td
Minimum performance index: other tuning
Minimum
Wilton (1999).
Model: Method 1
∫ [e (t) + x
∞
2
0
x1
Km
2
2
]
K m 2 (y( t ) − y ∞ )2 dt ; Tm1 > Tm 2 .
Tm1 + Tm 2
Tm 2
1 + Tm 2 Tm1
Coefficient values (obtained from graphs):
Tm 2 Tm1 = 0.25 Tm 2 Tm1 = 0.5 Tm 2 Tm1 = 0.75
x1
x2
x1
x2
x1
x2
1250.1 0.0001
22.946 0.04 20.396 0.04 22.946 0.04
4.8990
0.2
4.8990
0.2
4.8990
0.2
2.1213
1
2.1213
1
2.1213
1
180.01 0.0001 230.01 0.0001 300.01 0.0001
8.1584 0.04 9.1782 0.04 10.708 0.04
2.9394
0.2
3.1843
0.2
3.4293
0.2
1.2728
1
1.3435
1
1.4142
1
90.004 0.0001 100.00 0.0001 112.01 0.0001
4.5891 0.04 4.6911 0.04
5.354
0.04
2.0331
0.2
2.0821
0.2
2.1311
0.2
0.8768
1
1.0324
1
1.0748
1
T
T
28
i
d
K
τ m Tm1 = 0.1
τ m Tm1 = 0.5
τ m Tm1 = 1
Keane et al.
c
(2000).
Model: Method 1 Minimum ITAE servo plus ITAE regulator, with minimum M max and
with good noise attenuation; Tm1 = Tm 2 .
Huang and Jeng
(2003).
28
29
Kc =
29
2ξ m Tm1
Kc
(ln K u )[Tu + Tm1 (1 + ln K u )] ,
Tu K m
Td =
ξ m ≤ 1.1
Model:
Method 17
Td
Tu Tm1
,
Tu + Tm1 (1 + ln K u )
Ti =
(ln K u )[Tu + Tm1 (1 + ln K u )]
.
(ln K u )(1 + ln K u ) + [ln K u + ln(1 + eT − K )][ln(K u + 1.33419τm ) − 1]
Kc =
1.2858Tm1ξ m  τm 


T 
K m τm
 m1 
m
u
0.0544
,
Td =
[(− 0.0349ξm + 1.0064)Tm1 + (0.4196ξm − 0.1100)τm ]2
2Tm1ξm
.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Huang and Jeng
(2003) –
continued.
30
Kc
Ti
Td
Comment
Ti
Td
ξ m > 1.1
Servo or regulator tuning – minimum IAE.
x1 K m
Direct synthesis: time domain criteria
x 2 Tu
x 3 Tu
Wills (1962a).
Representative coefficient values (deduced from graphs):
Model: Method 1
x
x2
x3
x1
x2
x3
x1
x2
x3
1
Servo response;
2.5
2.9 0.035 1.6
1.4 0.143 0.2
0.48 0.143
ξ =1.
2.75
2.9
0.073
16
1.4
0.28
8
0.48 0.36
Tm1 = Tm 2 ;
3.2
2.9 0.143
10
1.4
0.72 0.09 0.28 0.073
τ m = 0.1Tm1 .
20
2.9 0.285 0.42 0.70 0.073 0.1
0.28 0.143
12
2.9
0.57
0.4
0.70 0.143 0.11 0.28 0.28
1.4
1.4 0.035
10
0.70 0.36
1.5
1.4 0.073
2
0.70 0.70
Tm1 + Tm 2 +
31
Step disturbance.
Van der Grinten
Td
Kc
0.5τm
(1963).
Stochastic
Model: Method 1
32
disturbance.
Ti
Td
Kc
Pemberton
(1972b).
30
2(Tm1 + Tm 2 )
3K m τ m
0.5890  τm 


Kc =
K m τm  Tm 2 
0.0030
Tm1 + Tm 2
Tm1Tm 2
Tm1 + Tm 2
Model: Method 1
2


 0.0052 Tm 2 + 0.8980Tm 2 + 0.4877 τm + Tm1  ,


τ
m


2
Ti = 0.0052
Tm 2
+ 0.8980Tm 2 + 0.4877 τm + Tm1 ,
τm
2


 0.0052 Tm 2 + 0.8980Tm 2 + 0.4877 τm 


τm
Td = Tm1 
.
2
T
 0.0052 m 2 + 0.8980T + 0.4877 τ + T 
m2
m
m1 

τm


31
Kc =
(T + T )τ + 2Tm1Tm 2 .
1 
T +T 
 0.5 + m1 m 2  , Td = m1 m 2 m


τm + 2(Tm1 + Tm 2 )
Km 
τm

32
Kc =
T + Tm 2 − ωd τ m
τ
e − ωd τm
; Ti = − ωmτ x1 ,
x 1 , x1 = 1 − 0.5e− ωd τ m + m1
e
Km
τm
e d m
− ωd τ m 

1 − 0.5e − ωd τ m + Tm1Tm 2e
(T + T )

(Tm1 + Tm 2 )τm  m1 m 2

.
Td =
x1
219
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
220
Rule
Kc
Ti
Td
Pemberton
(1972a),
(1972b).
Model: Method 1
(Tm1 + Tm 2 )
K mτm
Tm1 + Tm 2
Tm1Tm 2
Tm1 + Tm 2
Pemberton
(1972b).
Model: Method 1
2(Tm1 + Tm 2 )
3K m τ m
Tm1 + Tm 2
λTm1 + Tm 2
K m (1 + λτ m )
Tm1 + Tm 2
Chiu et al.
(1973).
Model: Method 1
Comment
Tm1 + Tm 2
4
Tm1Tm 2
Tm1 + Tm 2
0.1 ≤
Tm1
≤ 1.0 ,
Tm 2
0.2 ≤
τm
≤ 1.0 .
Tm 2
0.1 ≤
Tm1
≤ 1.0 .
Tm 2
0.2 ≤
τm
≤ 1.0 .
Tm 2
λ variable;
suggested values
are 0.2, 0.4, 0.6
and 1.0.
Appropriate for
“small τ m ”.
Mollenkamp et
2ξ m Tm1
2ξ m Tm1
Tm1
al. (1973).
K m (TCL + τ m )
2ξ m
Model: Method 1
Smith et al.
Tm1 > Tm 2
λTm1
(1975).
T
T
m
1
m
2
(
)
K m 1 + λτ m
Model: Method 1
λ = pole of specified FOLPD closed loop response.
Suyama (1992).
Tm1 + Tm 2
Tm1Tm 2
Model: Method 1
OS = 10%
T
+
T
m1
m2
2K m τ m
Tm1 + Tm 2
Rotach (1995).
Model: Method 9
Wang and
Clements (1995).
33
Kc =
1.06τm
Damping factor for oscillations to a disturbance input = 0.75.
2ξ m Tm1
2λξ m Tm1
Tm1
K m (1 + λτ m )
2ξ m
Gorez and Klàn
(2000).
Model: Method 1
Vítečková
(1999),
Vítečková et al.
(2000a).
Model:
Method 1
1.84τm
8.47 K m
33
Kc
Model: Method 22; underdamped response.
Non-dominant
2ξ m Tm1
Tm1
time delay.
2ξ m
x 1 (Tm1 + Tm 2 )
Kmτm
Tm1 + Tm 2
Tm1Tm 2
Tm1 + Tm 2
x1
OS (%)
x1
OS (%)
x1
OS (%)
0.368
0.514
0.581
0.641
0
5
10
15
0.696
0.748
0.801
0.853
20
25
30
35
0.906
0.957
1.008
40
45
50
2ξ m Tm1
.
K m (2ξ mTm1 + τm )
Closed loop
overshoot
(OS) defined.
Overdamped
process;
Tm1 > Tm 2 .
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Vítečková
(1999),
Vítečková et al.
(2000a) –
continued.
Skogestad
(2004b).
Model: Method 1
35
Kc =
x 1 ξ m Tm1
K mτm
2ξ m Tm1
Tm1
2ξ m
OS (%)
0.74Tm1
K mτm
34
Kc
1.0128ξ m Tm1
K mτm
35
2ξdes
m Tm1
( 2ξ
Comment
x1
OS (%)
x1
OS (%)
1.392
1.496
1.602
1.706
20
25
30
35
1.812
1.914
2.016
40
45
50
2ξ m Tm1
Underdamped
process;
0. 5 < ξ m ≤ 1 .
Recommended
TCL = τ m .
Tm1
2ξ m
A m = 3.14 , φ m = 61.4 0 , M max = 1.59
Chen and Seborg
(2002).
K m τm
Td
x1
Bi et al. (2000).
Kc =
Ti
0.736
0
1.028
5
1.162
10
1.282
15
2ξ m Tm1
K m (TCL + τ m )
Vítečková et al.
(2000b).
Model: Method 5
Arvanitis et al.
(2000a).
34
Kc
des
m
Kc
2Tm1
0.5Tm1
Tm1 = Tm 2
2ξ des
m Tm1
0.5Tm1 ξdes
m
1.9747 K m τ m
0.5064Tm1
K mτm
Model:
Method 1
Model:
Method 10
Ti
Td
2
Model:
Method 1
).
+1
1 [(Tm1 + Tm 2 )τm + Tm1Tm 2 ](3TCL + τm ) − TCL3 − 3TCL 2 τm
,
Km
(TCL + τm )3
[(Tm1 + Tm 2 )τm + Tm1Tm 2 ](3TCL + τm ) − TCL3 − 3TCL 2τm ,
Tm1Tm 2 + (Tm1 + Tm 2 + τm )τm
2
3T T T + Tm1Tm 2 τm (3TCL + τm ) − (Tm1 + Tm 2 + τm )TCL 3
Td = CL m1 m 2
[(Tm1 + Tm 2 )τm + Tm1Tm 2 ](3TCL + τm ) − TCL3 − 3TCL 2τm
Ti =
Tise −sτ m
 y
=
.
 
3
 d desired K c (TCLs + 1)
,
221
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
222
Rule
Kc
Chen and Seborg
(2002) –
continued.
Chidambaram
(2002), p. 156.
Skogestad
(2004c), Manum
(2005).
Model: Method 1
36
Ti
Td
Ti
Td
2ξ m Tm1
0.5Tm1
ξm
4(λ + τm )
Tm 2
Td
Kc
2
1.333ξ m Tm1
K m τm
Tm1
K m (λ + τm )
37
Kc
4(λ + τm ) + Tm 2
38
Kc
Ti
39
Kc
Ti
Comment
Model: Method 1
0.5Tm1 ξ m
τm
x1 (Tm1 + Tm 2 )
Tm1Tm 2
≤2
Trybus (2005).
T
+
T
m
1
m
2
Tm
K m τm
Tm1 + Tm 2
Model: Method 1
x1 OS x1 OS x1 OS x1 OS x1 OS x1 OS
Tm1 > Tm 2
0.34 0% 0.54 5% 0.64 10% 0.74 15% 0.82 20% 0.90 25%
36
Kc =
Ti =
Td =
37
38
39
Kc =
(
)
1 2ξ mTm1τm + Tm12 (3TCL + τm ) − TCL 3 − 3TCL 2 τm
,
Km
(TCL + τm )3
(2ξ
m Tm1τ m
)
+ Tm12 (3TCL + τm ) − TCL 3 − 3TCL 2 τm
Tm12 + (2ξ m Tm1 + τm )τm
Tise −sτ m
 y
,  
=
,
3
 d desired K c (TCLs + 1)
3TCL 2Tm12 + Tm12 τm (3TCL + τm ) − (2Tm1ξm + τm )TCL 3
(2ξ
m Tm1τ m
)
+ Tm12 (3TCL + τm ) − TCL 3 − 3TCL 2 τm
.

Tm1 Tm 2 + 4(λ + τm )
Tm 2 
, Td = Tm 2 1 +
.
2
4K m
(λ + τm )
 4(λ + τm ) 
 2Tm1ξ m
Tm1 [ξ m (Tm1 + 4(λ + τm )ξ m ) + 2Tm1 (1 − ξ m )]
K c = max 
,
 , ξm < 1 ;
2
(
)
K
λ
+
τ
 m

4K m (λ + τ m )
m
Ti = max{2Tm1ξ m , ξ m Tm1 + 4(λ + τ m )(2 − ξ m )} .



Tm1 Tm1 + 4(λ + τm ) ξ m + ξm 2 − 1  
 2T ξ


 

m1 m
K c = max 
,
 , ξm > 1 ;
4K m (λ + τm )2

 K m (λ + τm )


Ti = max{2Tm1ξm , Tm1 + 4(λ + τm )} .
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc

x1  Tm1

+ 0.5 
Trybus (2005) –
Tm1 + 0.5τm
K m  τm

continued.
x1 OS x1 OS x1 OS
Boudreau and
McMillan
(2006).
Td
Comment
0.5Tm1τm
Tm1 + 0.5τm
τm
>2
Tm
Ti
x1
OS
x1
OS
x1
223
OS
0.34 0% 0.54 5% 0.64 10% 0.74 15% 0.82 20% 0.90 25%
Tm1
40
Model: Method 1
Tm 2
Ti
K m (λ + τm )
λ = τm (maximum load rejection capability); otherwise, λ is a
multiple of Tm .
Vítečková
(2006).
Model: Method 5
≥
Sample results:
A m = 2.0 ,
2Tm1
0.5Tm1
1.0472Tm1
K mτm
2Tm1
0.5Tm1
Ti
Td
τ m / Tm < 2ξ m
Tm 2
Tm1 > Tm 2
41
Kc
ω p Tm1
42
Ti
φ m = 45 o .
A m = 3. 0 ,
φ m = 60 o .
AmKm
2
Ti = Tm1 ; for an overdamped response, Ti >
Kc =
Tm1 = Tm 2
0.5Tm1
1.5708Tm1
K mτm
Ho et al. (1994).
Model: Method 1
Ho et al. (1995a).
Model: Method 1
41
2Tm1
Direct synthesis: frequency domain criteria
πTm1
2Tm1
0.5Tm1
Tm1 = Tm 2
AmK mτm
Hang et al.
(1993a).
Model:
Method 10;
τm
> 0.3
Tm1
(Yang and
Clarke, 1996).
40
0.736Tm1
K m τm
4Tm1
λ + τm
1
 T

1 +  m1 
λ
+
τ
m 

2
2
 πξm


2ωp Tm1  πξm
2

+ π − 2ωp τm  , Ti = Tm12 
+ π − 2ωp τm  ,
 Tm1ωp


πA m  ωpTm1
π



Td =
42
.
1
Ti =
2ωp −
4ωp 2 τ m
π
1
+
Tm1
.
πTm1

 πξ
2 m + πTm1 − 2Tm1ωp τm 

 ωp


.
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
224
Rule
Kc
Ho et al. (1995a)
– continued.
0.7854Tm1
K mτm
Tm1
Tm 2
0.5236Tm1
K mτm
Tm1
Tm 2
Ti
Comment
Td
Sample results:
A m = 2.0 ,
φ m = 45 o .
A m = 3. 0 ,
φ m = 60 o .
Ho et al. (1997).
43
Kc
Ti
Td
Model: Method 1
Leva et al.
(1994).
44
Kc
Ti
0.25Ti
Model:
Method 21
43
Kc =
2
πA m K m

τ 
2
 πξm + π − 2 m  , Ti = Tm1 (πξm + π − 2τm ) ,

Tm1 
π

Td =
2
τ
τ
πTm1
; m ≤ 1 , 2ξ m ≥ m ;
Tm
2(πξ mTm1 + πTm1 − 2τm ) Tm

 π + π 2 + 8πτ m ξ m

Tm1
φm < 
44
Kc =
ωcn Ti
Km
(1 + ω T ) + 4ξ ω
(1 − T T ω ) + T ω
2
cn
2 2
m1
2
m
i d
cn
2
cn
2 2
2
i
Tm12
2
(
)
,
cn
ωcn =
Ti =

 A m 2 − 1 − 2 πA m (A m − 1)


4A m
 2ξ m τ m Tm1 (0.5π − φ m ) 
1 
4.07 − φ m + tan −1 
 ;
τm 
 (0.5π − φ m )2 Tm1 2 − τ m 2 

 
 2ξ ω T 
2
tan 0.5ωcn τm + φm − 0.5π − tan −1  m2 cn2 m1  ,
 ω T − 1  
ωcn
 
 cn m1
 

3τ m 
ξ m + ξ m 2 − 1  ); φ m = 70 0 (at least).
ξ m ≤ 1 or ξ m > 1 (with 0.5π − φ m >
Tm1 

b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Leva et al.
(1994) –
continued.
Wang et al.
(1999).
45
ξ m Tm1
K mτm
Wang and Shao
(1999a).
Model:
Method 16
x1
ξ m Tm1
K mτm
2
ω T
K c = cn m1
K m Td
Comment
Ti
Td
Ti
Td
2ξ m Tm1
0.5Tm1
ξm
0. 5
2ξ m Tm1
Model:
Method 16 46
Tm1
ξm
Some coefficient values:
Comment
x1
x1
Comment
1.571
A m = 2 , φ m = 45
0
0.785
A m = 4 , φ m = 67.50
1.047
A m = 3 , φ m = 60 0
0.628
A m = 5 , φ m = 72 0
2
45
Kc
ωcn +
1 
2ξ m ξ m 2 − 1 − 1
2 

Tm1 
2
ωcn + z 2
ωcn =
,
 2ξ m τ m Tm1 (0.5π − φ m ) 
1 
4.07 − φ m + tan −1 
 ,
τm 
 (0.5π − φ m )2 Tm1 2 − τ m 2 

ωcn
z=
,






ωcn Tm1
tan φ m − 0.5π + ωcn τ m + tan −1 


  ξ m + ξ m 2 − 1  

 
 
2
Tm1z +  ξm + ξ m − 1 
Tm1

, T =
Ti =
, ξm > 1
d
z ξm + ξ m 2 − 1 
Tm1z +  ξ m + ξm 2 − 1 




3τ m 
3τ m
2

ξ m − ξ m − 1 < 0.5π − φ m ≤
with

Tm1 
Tm1
46
ξ m > 0.7071 or 0.05 <
or 0.05 <
225
0.7071τ m
2
Tm1 2ξ m − 1
< 0.15 ,
ξm τm
ξ τ
< 0.15 , m m > 1, ξ m < 1 .
Tm1
Tm1
0.7071τ m
Tm1 2ξ m 2 − 1
> 1, ξ m ≥ 1
ξ + ξ 2 − 1  .
m
 m

b720_Chapter-03
Rule
Kc
Wang (2000).
Model:
Method 1.
ξ m Tm1
K mτm
Wang and Shao
(2000b).
Chen et al.
(1999b).
Model: Method 1
0.05 <
Kc
49
Kc
x1
Am
1.00
3.14
1.22
2.58
1.34
2.34
1.40
2.24
50
51
Comment
Ti
Td
2ξ m Tm1
0.5Tm1 ξ m
2ξ m Tm1
0.5Tm1 ξ m
2ξ m τ m
τm
2ξ m
φm
φm
Ms
x1
Am
61.4
1.00
1.44
2.18
48.7
0
1.30
55.0
0
1.10
1.52
2.07
46.50
1.40
51.6
0
1.20
1.60
1.96
0
1.50
0
1.26
Kc
4(Tm 2 + τ m )
Kc
5Tm 2 + 4 τ m
Td
0.7071τm
Model:
Method 16
0
50.0
16(Tm 2 + τ m )
Tm1 (K mξ m − Tm1 )
or 0.05 <
Kc =
48
47
x 1ξ m Tm1
τm K m
Leva et al.
(2003).
Model: Method 1
48
FA
Handbook of PI and PID Controller Tuning Rules
226
47
17-Mar-2009
< 0.15 or
0.7071τm
Tm1 (K m ξ m −Tm1 )
Ms
44.1
> 1 , for ξm > 1 ;
ξm τm
ξ τ
< 0.15 or m m > 1 , for 0.7071 < ξ m < 1 .
Tm1
Tm1
 e − τ m x1 0.368 
2Tm1ξm
,
min 
 , x1 = 2Tm1 (K m ξm − Tm1 ) , ξm > 1 or
τm 
Km
 x1
x1 =
49
Kc =
50
Kc =
51
Kc =
1.508 
2ξ m Tm1 
 , suggested M max = 1.6.
1.451 −
K m τm 
M max 
Tm1Tm 2
8K m (Tm 2 + τm )2
, Tm1 ≥ 4(Tm 2 + τ m ) .
Tm1 (5Tm 2 + 4 τ m )
8K m (Tm 2 + τ m )2
, Td =
4Tm 2 (Tm 2 + τ m )
, Tm1 ≥ 8(Tm 2 + τ m ) .
5Tm 2 + 4τ m
Tm1
, ξm < 1 .
ξm
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Åström and
Hägglund (2006),
pp. 245–246.
Lee et al. (2006).
Model: Method 1
52
Kc
Ti
K m (λ + τm )
Ti
Td
Comment
Ti
Td
Model:
Method 1;
M s = M t = 1.4 .
53
227
Td
Ti
Desired closed loop response =
e − sτ m
(λs + 1)2
; recommended λ = 0.5τm .
Robust
54
Brambilla et al.
(1990).
Model:
Method 1
Kc
Td
Representative coefficient values (deduced from graph):
τm
τm
τm
λ
λ
λ
Tm1 + Tm 2
Tm1 + Tm 2
Tm1 + Tm 2
0.1
0.2
0.5
52
Ti
0.50
0.47
0.39
1.0
2.0
5.0
0.29
0.22
0.16
10.0
0.14
1 
Tm1
T
T T 
+ 0.18 m 2 + 0.02 m1 m 2  ,
0.19 + 0.37
Km 
τm
τm
τm 
Tm1
Tm 2
Tm1Tm 2
+ 0.18
+ 0.02
0.19 + 0.37
τm
τm
τm
,
Ti =
0.48
Tm1
Tm 2
T T
+ 0.03 2 − 0.0007 2 + 0.0012 m1 3m 2
τm
τm
τm
τm
Kc =

T T  T T
 0.29τm + 0.16Tm1 + 0.20Tm 2 + 0.28 m1 m 2  m1 m 2

τm  Tm1 + Tm 2
Td = 
.
T
T
T T
0.19 + 0.37 m1 + 0.18 m 2 + 0.02 m1 m 2
τm
τm
τm
2
2
53
Ti = 2ξm Tm1 +
54
Kc =
2
τm
τm
, Td =
+
2(λ + τm )
2(λ + τm )
Tm1 −
Tm1 + Tm 2 + 0.5τm
, Ti = Tm1 + Tm 2 + 0.5τm ,
K m τm (2λ + 1)
Td =
3
τm
6(λ + τm )
.
Ti
τm
Tm1Tm 2 + 0.5(Tm1 + Tm 2 )τm
, 0.1 ≤
≤ 10 .
Tm1 + Tm 2
Tm1 + Tm 2 + 0.5τm
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
228
Rule
Kc
Lee et al. (1998).
Model: Method 1
desired closed
loop response =
Ti
K m (2λ + τm )
e − τ ms
(λs + 1)2
Comment
Td
55
Ti
Td
56
Ti
Td
57
Ti
Td
58
Ti
Td
λ = maximum [0.25τm ,0.2Tm1 ] (Panda et al., 2004).
Ti
K m (λ + τm )
.
Ti
Huang et al.
2ξ m Tm1
Tm1
(1998), Rivera
2ξ m Tm1
K m (τ m + λ)
2ξ m
and Jun (2000).
Model: Method 1 λ specified graphically for different OS and TR values (Huang et al.,
1998).
59
Marchetti and
Model:
T
Td
Kc
i
Method 1
Scali (2000).
Smith (2003).
Tm1
Tm1
Tm 2
λ ∈ [ τ m , Tm 2 ] ,
Model: Method 1 K (λ + τ )
Tm1 ≥ Tm 2 .
m
m
Kristiansson
(2003).
60
Ti = 2ξ m Tm1 −
56
Ti = Tm1 + Tm 2 −
58
59
60
2ξ m Tm1
Tm1 2ξm
Model: Method 1
2λ2 − τm 2
T 2
τm 3
, Td = Ti − 2ξ mTm1 + m1 −
.
2(2λ + τm )
Ti
6Ti (2λ + τm )
55
57
Kc
2λ2 − τm 2
τm 3
T T
, Td = Ti − Tm1 − Tm 2 + m1 m 2 −
.
2(2λ + τm )
Ti
6Ti (2λ + τm )

τm 3 
2

−
T
m
1

τm 2
6(λ + τm ) 
Ti = 2ξm Tm1 +
, Td = Ti − 2ξ m Tm1 
.
2(λ + τm )
Ti






3



 Tm1Tm 2 − τm

τm
6(λ + τm ) 
Ti = Tm1 + Tm 2 +
, Td = Ti − (Tm1 + Tm 2 )
.
2(λ + τm )
Ti






2ξ T + 0.5τm
T (T + ξ m τm )
K c = m m1
, Ti = 2ξ m Tm1 + 0.5τm , Td = m1 m1
.
K m (2λ + τm )
2ξ m Tm1 + 0.5τm
2
Kc =
2ξmTm1 
1 
1 −
 . Recommended values of M max : 1.4,1.7,2.0.
K m τm  M max 
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Shamsuzzoha et
al. (2006c).
Model: Method 1
Shamsuzzoha
and Lee (2007a).
Model: Method 1
61
Kc =
Ti
Td
Comment
61
Kc
Ti
Td
λ , ξ not
specified.
62
Kc
Ti
Td
λ = x1Tm
Ti
,
K m (4λξ + τm − x1 )
Ti = Tm1 + Tm 2 + x1 −
Td =
Kc
229
4λ2ξ 2 + 2λ2 − x 2 + x1τm − 0.5τm 2
,
4λξ + τm − x1
x 2 + x1 (Tm1 + Tm 2 ) + Tm1Tm 2 −
3
2
0.167 τm − 0.5x1τm + x 2 τm + 4λ3ξ
4λξ + τm − x1
Ti
−
2
τ
4λ2ξ 2 + 2λ2 + x1τm − 0.5τm 2 − x 2
,
4λξ + τm − x1
τ
2
m
2
2
 − m
2λξ  − Tm1
2 λ
2
 λ − 2λξ + 1 e Tm 2 + Tm 2 2 − Tm12
−
+
−
1
e
T
Tm1 
m
2
T 2 T

T 2 T

m1
m2
 m1

 m2

,
x1 =
Tm 2 − Tm1

x 2 = Tm 2
62
Kc =
Td =
2 

λ2
 Tm12

2
τm

2λξ  − Tm1
−
+1 e
− 1 + x1Tm1 .


Tm1


Ti
6λ2 − x 2 + x1τm − 0.5τm 2
,
, Ti = Tm1 + Tm 2 + x1 −
K m (4λ + τm − x1 )
4 λ + τ m − x1
x 2 + x1 (Tm1 + Tm 2 ) + Tm1Tm 2 −
3
2
0.167 τm − 0.5x1τm + x 2 τm + 4λ3
4λ + τm − x1
Ti
2
6λ2 + x1τm − 0.5τm − x 2
,
4λ + τ m − x 1
τm
τm
4
4




λ  − Tm1 
λ  − Tm 2
2 
 e


Tm12 1 −
1
T
1
e
−
− 1
−
−
m2 




 Tm1 
 Tm 2 

,


x1 =
Tm 2 − Tm1
τm
4


λ  − Tm 2
 + x1Tm 2 .
 e
x 2 = Tm 2 2 1 −
1
−

 Tm 2 


b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
230
Rule
Shamsuzzoha
and Lee (2007a)
– continued.
Wills (1962a).
Model: Method 1
Quarter decay
ratio tuning.
Tm1 = Tm 2 ;
τ m = 0.1Tm1 .
Bohl and
McAvoy
(1976b).
63
17-Mar-2009
Kc
Ti
Comment
Td
Coefficients of x1 (deduced from graph):
τm Tm
Ms
1.8
0.27
0.29
0.31
0.33
1.9
0.18
0.20
0.22
0.24
0.25
0.27
1.8
1.9
0.2
0.34
0.28
0.3
0.35
0.29
0.4
0.38
0.32
0.5
0.41
0.34
0.6
0.46
0.38
0.7
0.51
0.41
0.8
0.55
0.43
Ultimate cycle
x 2 Tu
x 3Tu
x1 K m
x1
x2
10
8
14
33
18
2.9
1.4
2.9
2.9
1.4
63
τm Tm
Ms
2.0
0.12
0.15
0.17
0.19
0.21
0.22
0.23
2.0
0.25
0.26
0.28
0.30
0.33
0.36
0.38
0.9
1.0
1.25
1.5
2.0
2.5
3.0
Coefficient values (deduced from graph):
x3
x1
x2
x3
x1
x2
0.036
0.036
0.068
0.143
0.068
28
6
2
0.6
30
Ti
Kc
τ 

2.0302 + 0.9553 ln  m  
 Tu  
2.2689  τm 
Kc =

 
K m  Tu 



1.4
0.67
0.48
0.28
0.67
0.143
0.068
0.068
0.068
0.143
Td
18
0.9
18
18
13
0.48
0.28
0.67
0.48
0.28
x3
0.143
0.143
0.24
0.24
0.28
Model: Method 1
τ 

1.3135+ 0.1809 ln (K u K m )+ 0.5219 ln  m  
(K u K m )
 Tu   ,


τ 

− 0.0517 − 0.0643 ln  m  
 τm 
 Tu  
Ti = 0.2693Tu  


 Tu 


τ 

0.5914 −0.0832 ln (K u K m )+ 0.0687 ln  m  
(K u K m )
 Tu   ,


τ 

− 4.2613−1.5855 ln  m  
 τm 
 Tu  

Td = 0.0068Tu 



 Tu 


τ 

−1.1436 −0.2658 ln (K u K m )−1.1100 ln  m  
(K u K m )
 Tu   .


b720_Chapter-03
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FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
x 1K u
x 2 Tu
Kuwata (1987);
x1 x 2 x 3 τm
x1 x 2
x 1 , x 2 deduced
Tu
from graphs.
0.47
0.23
0.22
0.15
0.29 0.17
ξ m = 0. 5 .
0.45 0.21 0.19 0.18 0.225 0.17
0.35 0.18 0.10 0.30 0.27 0.19
x 1K u
x 2 Tu
Kuwata (1987);
x1 x 2 x 3 τm
x1 x 2
x 1 , x 2 deduced
Tu
from graphs.
0.45
0.30
0.14
0.20
0.34 0.20
ξm = 0.7 .
0.42 0.25 0.11 0.25 0.31 0.19
0.38 0.22 0.09 0.30 0.28 0.18
x 1K u
x 2 Tu
Kuwata (1987);
x1 x 2 x 3 τm
x1 x 2
x 1 , x 2 deduced
Tu
from graphs.
0.45
0.47
0.12
0.20
0.37 0.24
ξm = 1 .
0.43 0.37 0.10 0.25 0.33 0.21
0.40 0.30 0.09 0.30 0.30 0.19
x 1K u
x 2 Tu
Kuwata (1987);
x1 x 2 x 3 τm
x1 x 2
x 1 , x 2 deduced
Tu
from graphs.
0.46
0.80
0.11
0.20
0.38 0.29
ξ m = 1. 5 .
0.44 0.52 0.09 0.25 0.35 0.22
0.42 0.38 0.08 0.30 0.30 0.19
4 + x1 1
64
Landau and Voda
Kc
x1 ω1350
(1992).
3K u
0.51Tu
Model: Method 1
1.9 K u
64
Kc =
4 + x1
x1
.
4 2 2 G p ( jω1350 )
0.64Tu
Comment
Td
x 3 Tu
x3
τm
Tu
x1
x2
x3
τm
Tu
0.08 0.35 0.28 0.19 0.03 0.50
0.01 0.41
0.02 0.47
x 3 Tu
x3
τm
Tu
x1
x2
x3
τm
Tu
0.07 0.35 0.28 0.18 0.03 0.50
0.06 0.40
0.04 0.45
x 3 Tu
x3
τm
Tu
x1
x2
x3
τm
Tu
0.07 0.35 0.28 0.18 0.03 0.50
0.06 0.40
0.04 0.45
x 3 Tu
x3
τm
Tu
x1
x2
x3
τm
Tu
0.07 0.35 0.28 0.18 0.03 0.50
0.06 0.40
0.04 0.45
4
1
1 ≤ x1 ≤ 2
4 + x1 ω1350
0.13Tu
0.16Tu
ωu τ m ≤ 0.16
0.16 < ωu τ m
≤ 0.2
231
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232
3.5.3 Ideal controller in series with a first order lag
 1

1
G c (s) = K c 1 +
+ Td s 
 Tf s + 1
 Ti s
R(s) E(s)
−
+
U(s)
1
Tf s + 1


1
K c 1 +
+ Td s 
Ti s


SOSPD
model
Table 22: PID controller tuning rules – SOSPD model G m (s) =
G m (s) =
Rule
Kc
Lim et al. (1985).
1
Arvanitis et al.
(2000a).
Model: Method 1
Panda et al.
(2004).
Model: Method 1
Huang et al.
(2005a).
Model:
Method 18
Huang and Jeng
(2005).
1
Kc =
2
Kc =
Tm1 2 s 2 + 2ξ m Tm1s + 1
Ti
(
Td
Comment
Direct synthesis: time domain criteria
Model: Method 1
Tm1 + Tm 2
Tm1Tm 2
Tm1 + Tm1
Kc
Kc
ξ m Tm1
K mτm
2ξ m Tm1
Tm1
2ξ des
m
TCL 2 >
τm
2
Tm1
,
2 Nξ m
N not defined.
Direct synthesis: frequency domain criteria
ξ m Tm1
Tm1
0.025Tm1
Tf =
2ξ m Tm1
K mτm
2ξ m
ξm
Tm1
2ξ m
Tf =
A m ≈ 3 , φ m ≈ 60 0
1.1ξ m Tm1
K mτm
2ξ des
m Tm1
+ τm
)
, Tf =
0.5Tm1
ξm
2ξ m Tm1
2
TCL 2
Tm1 + Tm 2
.
, Tf =
K m (2ξTCL 2 + τ m )
2ξTCL 2 + τ m
K m 2ξ des
m TCL 2
K m e − sτ m
or
(1 + sTm1 )(1 + sTm 2 )
K m e − sτ m
2ξ des
m Tm1
2
Y(s)
(
2 2TCL 2 2 − τ m 2
2ξ des
m TCL 2 + τ m
).
Model:
Method 1;
Tf = 0.01τ m .
b720_Chapter-03
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FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Hang et al.
(1993a).
Model:
Method 10
Rivera and Jun
(2000). Model:
Method 1; λ not
specified.
Lee and Edgar
(2002).
2Tm1 + τ m
2(λ + τ m )K m
Zhang et al.
(2002b).
Model:
Method 1;
5% overshoot:
λ = 0.5τ m .
3
Ti
Td
Robust
Tm1 + 0.5τ m
Tm1 τ m
2Tm1 + τ m
Model has a repeated pole (Tm1 ) .
2ξ m Tm1 λ
2ξ m Tm1
0.5Tm1 ξ m
2ξ m Tm1
Tm1
2ξ m
Kc
Ti
Td
Tm1 + Tm 2
K m (τ m + 2λ )
Tm1 + Tm 2
Tm1Tm 2
Tm1 + Tm 2
2ξ m Tm1
K m (2 τ m + λ )
3
233
Comment
λ > 0.25τ m .
Tf =
λτ m
.
2(λ + τ m )
Tf = τ m
Tf =
τmλ
2τ m + λ
Model: Method 1
Tf =
λ2
;
2λ + τ m
H ∞ method.
4
Tm1Tm 2
Tm1 + Tm 2
Kc
Ti
K m (2λ + τm )
5
Tm1 + Tm 2
Td
Ti
Tf =
λτ m
λ + 2τ m
Maclaurin
method.
2
Kc =
1
0.5τm − λ2
+
(2ξ m Tm1 +
K m (2λ + τm )
2λ + τ m
Tm 2 −

τm3
0.5τ m 2 − λ2
+  2ξ m Tm1 +
6(2λ + τ m ) 
2λ + τ m
 0.5τ m 2 − λ2

),
 2λ + τ m

2
Ti = 2ξm Tm1 +
0.5τm − λ2
+
2λ + τ m

τm3
0.5τ m 2 − λ2
+  2ξ m Tm1 +
6(2λ + τ m ) 
2λ + τ m
Tm 2 −
Td =
4
5
Kc =
Tm 2 −

τm3
0.5τ m 2 − λ2
+  2ξ m Tm1 +
6(2λ + τ m ) 
2λ + τ m
 0.5τ m 2 − λ2

.
 2λ + τ m

Tm1Tm 2
, H 2 method.
K m (Tm1 + Tm 2 )(λ + 2τm )
Ti = Tm1 + Tm 2 −
2
2
2λ − τ m
, Td =
2(2λ + τm )
τm 3
2λ2 − τm 2
12λ + 6τm
−
, Tf = 0 .
Ti
2(2λ + τm )
Tm1Tm 2 −
 0.5τ m 2 − λ2

,
 2λ + τ m

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234
Rule
Kc
Ti
Td
Comment
2ξ m Tm1
Tm1
2ξ m
Model: Method 1
Kristiansson
(2003).
6
7
Kc
Panda et al.
(2004).
Model: Method 1
8
Kc
Ti
Td
9
Kc
Ti
Td
6
Kc =
7
Kc =
8
Kc =
Kc
2ξ m Tm1 
1
 x1 − x12 − 1 +
K m τm 
M max 2

T 
1
 , x1 = 1 + f 1 − 1 −
τm 
M max 2

2ξ m Tm1 
T
T
T 2
1 + 0.019 m1 − 0.346 + 0.038 m1 + 0.00036 m12  ;
K m τm 
τm
τm
τm 


for suggested M max = 1.7 , x 2 = 10 .
2
Td =

T
 , Tf = m1 .
x2

2
2ξ m Tm1 + 0.25Tm1 + 0.5ξ m τm
4ξ T + 0.5Tm1 + ξ m τm
, Ti = m m1
,
K m (λ + 2ξ m Tm1 )
2ξ m
(Tm1 + 2ξm τm )Tm1ξm
2
4ξm Tm1 + ξ m τm + 0.5Tm1
, Tf =
λ (Tm1 + 2ξ m τ m )
,
4ξ m λ + 4ξ m τ m + 2Tm1


T

λ = maximum 0.25 m1 + τm ,0.4ξ mTm1  .
 2ξ m



9
1
Kc =
Km
0.558Tm 2Tm1
Tm 2
+ 0.172Tm1e − (6.9 Tm 2 / Tm1 ) +
+ 0.5τm
Tm1 + 1.208Tm 2
Tm1
,
T
λ + 0.828 + 0.812 m 2 + 0.172Tm1e − (6.9Tm 2 / Tm1 )
Tm1
0.828 + 0.812
Ti = 0.828 + 0.812
0.558Tm 2Tm1
Tm 2
+ 0.172Tm1e− (6.9Tm 2 / Tm1 ) +
+ 0.5τm ,
Tm1 + 1.208Tm 2
Tm1


  1.116Tm 2Tm1
Tm 2
+ τm 
+ 0.172Tm1e − (6.9Tm 2 / Tm1 )  
0.828 + 0.812
T
1
.
208
T
+
T
m1
m2
,
  m1
Td = 
Tm 2
1.116Tm 2Tm1
− (6.9 Tm 2 / Tm1 )
+ 0.344Tm1e
+
+ τm
1.656 + 1.624
Tm1
Tm1 + 1.208Tm 2
λ[1.116Tm1Tm 2 + τ m (Tm1 + 1.208Tm 2 )]
,
Tf =
2(λ + τ m )(Tm1 + 1.208Tm 2 ) + 2.232Tm 2 Tm1

 0.279Tm 2Tm1
T
λ = maximum 
+ 0.25τm ,0.1656 + 0.1624 m 2 + 0.0344Tm1e− (6.9Tm 2 / Tm1 )  .
T
+
1
.
208
T
T
m2
m1

 m1
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Naşcu et al.
(2006).
Model:
Method 19
Kc
Ti
Td
2ξ m Tm1
K m (2TCL 2 + τm )
2ξ m Tm1
0.5Tm1
ξm
Desired servo closed loop transfer function =
235
Comment
2
Tf =
TCL 2
2TCL 2 + τm
e − sτ m
(TCL 2s + 1)2
;
suggested TCL 2 values: 0.4ξ m Tm1 , ξm Tm1 .
Ultimate cycle
Huang et al.
(2005a).
10
K c = 0.080
10
Kc
Ti
Td
Tf = 0.05Td
A m ≈ 3 , φ m ≈ 60 0 ; Model: Method 18.
^ ^




^ ^
K u Tu
 6.283τm 
 6.283τm 
=
sin 
,
T
0
.
159
K
T
sin
i
u u


,
^
^
K m τm
 T

 T

u
u







 1 + K^ cos 6.283τm  
u


^
^ 
 T

Tu 
u

 .
Td = 0.159 ^ 




Ku 
 6.283τm 
 sin 


^
 T



u




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236


Td s
1
+
3.5.4 Controller with filtered derivative G c (s) = K c 1 +
sT

Ti s
1+ d

N

E(s)
R(s)
−
+


Td s
1
+
K c 1 +
T

Ti s
1+ d

N


 U(s)


s

SOSPD
model
Table 23: PID controller tuning rules – SOSPD model G m (s) =
G m (s) =
Rule
Kc
Hang et al.
(1994).
Model: Method 1
AmKm
Y(s)
K m e −sτ m
or
(1 + sTm1 )(1 + sTm 2 )
K m e − sτ m
Tm1 2 s 2 + 2ξ m Tm1s + 1
Ti
ω p Tm1






Td
Comment
Direct synthesis: frequency domain criteria
1
N = 20;
Tm 2
Ti
Tm1 > Tm 2 .
0.7854Tm1
K mτm
Sample result:
Tm1
Tm 2
A m = 2.0 ,
φ m = 45 o .
Robust
Hang et al.
Tm1
(1994).
K m (TCL + τ m )
Model: Method 1
2
Zhong and Li
Kc
(2002).
1
1
Ti =
2ωp −
2
2
4ωp τm
π
1
+
Tm1
Tm1
Tm 2
N = 20;
Tm1 > Tm 2 .
Ti
Td
Model: Method 1
.
2
Kc =
2 x1 + τm
2ξ m Tm1 (2 x1 + τm ) − x1
, Ti =
,
K m ( 2 x1 + τ m ) 2
2ξm Tm1 (2 x1 + τm ) − x12
 2 2ξ T x 2 ( 2 x 1 + τ m ) − x 1 4 
2x + τ
Td = Ti Tm1 − m m1 1
 , N = 1 2 m Td .
(2x1 + τ m ) 2
x1


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FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Wang (2003),
pp. 50–54.
Lavanya et al.
(2006a).
Model:
Method 13;
N = 10.
3
Kc
Ti
Td
Comment
Kc
Ti
Td
Model: Method 1
2ξm Tm1
K m (λ + τm )
2ξ m Tm1
0.5ξm
Tm1
3
Ti
K m (2λ + τm )
4
λ = max [0.25τm ,0.2Tm1 ]
2
2 ξ T ( 2 x + τ ) − x1
2ξ m Tm1 (2 x1 + τm ) − x1
, Ti = m m1 1 m
,
2 x1 + τ m
K m ( 2 x1 + τ m ) 2
Td =
4
τm
< 1;
Tm1
λ not specified.
τm
>1
Tm1
Td
Ti
2
Kc =
0.1 ≤
237
Ti = 2ξm Tm1 −
2
2
Tm12 (2 x1 + τm )
2ξ m Tm1 (2x1 + τm ) − x12
2λ − τ m
, Td = Ti − 2ξ mTm1 +
2(2λ + τm )
Tm12 −
τm 3
6(2λ + τm )
.
Ti
−
2x + τ
x12
, N = 1 2 m Td .
2x1 + τm
x1
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238



1  1 + Td s 


3.5.5 Classical controller G c (s) = K c 1 +
 Ti s  1 + Td s 
N 

E(s)
R(s)
+
−





1  1 + Td s  U(s)

K c 1 +
Td 
Ti s 

s
1 +
N 

SOSPD
model
Table 24: PID controller tuning rules – SOSPD model G m (s) =
G m (s ) =
Rule
Kc
Y(s)
K m e − sτ m
or
(1 + sTm1 )(1 + sTm 2 )
K m e −sτ m
2 2
Tm1 s + 2ξ m Tm1s + 1
Ti
Td
Comment
Minimum performance index: regulator tuning
1
0.954
0.780
0.982
Approximately
Tm1 > Tm 2 ;
 x2 
 x2 
 
minimum IAE – 0.817  x1 
 
 
0
.
602
x
0
.
903
x
1
1
x 
x 
x 
N = 8.
K
 1
Huang and Chao
 1
m  2 
(1982).
Also given by Chao et al. (1989); Model: Method 1.
Minimum IAE –
Shinskey (1988),
p. 149.
Model:
Method 1;
N not specified.
1
0.80Tm1
K mτm
1.5(Tm 2 + τ m )
Tm 2
0.60(Tm 2 + τ m ) T + τ = 0.25
m2
m
0.77Tm1
K mτm
1.2(Tm 2 + τ m )
0.70(Tm 2 + τ m )
0.83Tm1
K mτm
0.75(Tm 2 + τ m )
Tm 2
0.60(Tm 2 + τ m ) T + τ = 0.75
m2
m

τ
1.398
x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1  m + 1.039 −
.
T
1
.
349
(
T
T
)
+
m1
m2 
 m2
Tm 2
= 0.5
Tm 2 + τ m
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
239
Comment
Td
2
Minimum IAE –
N=8
Ti
Td
Kc
Chao et al.
0.8 ≤ ξ m ≤ 3; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1.
(1989).
3
Minimum IAE –
Ti
Td
Tm 2 τm ≤ 3
Kc
Shinskey (1994),
2.5Tm1 K m τm
τ m + 0.2Tm 2
τ m + 0.2Tm 2
Tm 2 τm > 3
p. 159.
Model: Method 1; N not specified.
Minimum IAE –
τm Tm1 = 0.2 ,
Shinskey (1996),
0.59 K u
0.36Tu
0.26Tu
Tm 2 Tm1 = 0.2 .
p. 121.
Model: Method 1; N not specified.
0.85Tm1 K m τm
1.98τ m
0.86τ m
Tm 2 Tm1 = 0.1
Minimum IAE –
0.87Tm1 K m τm
2.30τ m
1.65τ m
Tm 2 Tm1 = 0.2
Shinskey (1996),
1.00Tm1 K m τm
2.50τ m
2.00τ m
Tm 2 Tm1 = 0.5
p. 119.
Model: Method 1 1.25T K τ
2.75τ
2.75τ
T T = 1. 0
m1
m m
m
m
m2
m1
τm Tm1 = 0.2 ; N not specified.
Minimum ISE – 0.667Tm1 K m τm
Model: Method 1
Tm1
Tm 2
Haalman (1965).
0
Tm1 > Tm 2 . M s = 1.9, A m = 2.36, φ m = 50 ; N = ∞ .
2
− 0.01133 + 0.5017ξm − 0.07813ξ m
Kc =
Km
2
(
 τm 


 2ξ T 
 m m1 
Ti = 2ξ m Tm1 1.235 − 0.4126ξ m + 0.09873ξ m
(
Td = 2ξ m Tm1 1.214 − 0.6250ξ m + 0.1358ξm
3
Kc =
2
2
)
−1.890 + 0.7902 ξ m − 0.1554 ξ m 2
)
,
 τm 


 2ξ T 
 m m1 
 τm 


 2ξ T 
 m m1 
−0.03793 + 0.3975ξ m − 0.05354 ξ m 2
0.5696 − 0.8484 ξ m + 0.0505ξ m 2
100




 48 + 57 1 − e



1.2 Tm1
−
τm

  K m τm
  Tm1

2

1 + 0.34 Tm 2 − 0.2  Tm 2  

 

τm
 τm  

T

− m1

Ti = τ m 1.5 − e 1.5τ m







,
.
,
1.2 Tm1 
T



−
− m2  

1 − e τm   , Td = 0.56τm 1 − e τ m  + 0.6Tm 2 .
1
0
.
9
+




 






b720_Chapter-03
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Handbook of PI and PID Controller Tuning Rules
240
Rule
Minimum ISE –
McAvoy and
Johnson (1967).
ξm
Kc
Ti
Td
Comment
x1 K m
x 2τm
x 3τ m
Model: Method 1
Representative coefficient values (deduced from graph):
x1
x2
x3
ξm
x1
x2
Tm1
Tm1
τm
τm
N = 20
1
1
1
0.5
4.0
10.0
0.7
7.6
34.3
0.97
3.33
5.00
0.75
2.03
2.7
N = 10
1
1
1
0.5
4.0
10.0
0.9
8.0
33.5
1.10
4.00
6.25
0.64
1.83
2.43
4
4
7
7
4
4
7
7
0.5
4.0
0.5
4.0
0.5
4.0
0.5
4.0
3.0
22.7
5.4
40.0
3.2
23.9
5.9
42.9
1.16
1.89
1.19
1.64
1.33
2.17
1.39
1.89
x3
0.85
1.28
0.85
1.14
0.78
1.17
0.78
1.04
0.881
4
0.883
0.881
Approximately
Tm1 > Tm 2 ;
 x2 
1.101  x1 
 x2 
 
minimum ISE –




1.134x1  
0.563x1  


N = 8.
Huang and Chao K m  x 2 
 x1 
 x1 
(1982).
Also given by Chao et al. (1989); Model: Method 1.
5
Minimum ISE –
N=8
Ti
Td
Kc
Chao et al.
0.8 ≤ ξ m ≤ 3; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1.
(1989).
1.036
6
0.595
1.006
Approximately
Tm1 > Tm 2 ;
 x2 
 x2 
0.745  x1 
 
minimum ITAE




0
.
771
x
0
.
597
x
1
1




N = 8.
Km  x2 
– Huang and
 x1 
 x1 
Chao (1982).
Also given by Chao et al. (1989); Model: Method 1.
4
5
τ

1.398
x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1  m + 1.039 −
.
T
1
.
349
(
T
T
)
+
m1
m2 
 m2
0.2538 + 0.3875ξ m − 0.04283ξ m 2  τm 


Kc =
 2ξ T 
Km
 m m1 
(
Ti = 2ξ m Tm1 1.8283 − 0.8083ξm + 0.1852ξ m
(
Td = 2ξ m Tm1 1.077 − 0.4952ξ m + 0.1062ξ m
6
2
2
)
−1.652 + 0.5416 ξ m − 0.09544 ξ m 2
)
,
 τm 


 2ξ T 
 m m1 
 τm 


 2ξ T 
 m m1 
0.1363+ 0.2342 ξ m − 0.01445ξ m 2
,
0.4772 + 0.0442 ξ m + 0.01694 ξ m 2
τ

1.398
x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1  m + 1.039 −
.
T
1
.
349
(
T
T
)
+
m1
m2 
 m2
.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Bohl and
McAvoy
(1976b).
Minimum ITAE
– Chao et al.
(1989).
7
Kc
Ti
Td
Comment
7
Kc
Ti
Td = Ti
Minimum ITAE;
Model: Method 1
8
Kc
Ti
Td
0.8 ≤ ξ m ≤ 3
0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1.
τm Tm1 = 0.1,0.2, 0.3,0.4,0.5; Tm 2 Tm1 = 0.12,0.3 ,0.5,0.7,0.9. N = ∞ .
 τ 

−1.1445 + 0.1100 ln  10 m  
 Tm1  
5.5030  τm 

Kc =

10

K m  Tm1 



 τ 

0.6212 − 0.0760 ln  10 m  
 τm 
 Tm1  

Ti = 1.8681Tm1 10



 Tm1 


8
241
− 0.1033 + 0.5347ξm − 0.07995ξ m
Kc =
Km
2

 Tm 2
10
 Tm1





 Tm 2
10
 Tm1




 τm 


 2ξ T 
 m m1 
(
Ti = 2ξ m Tm1 0.9096 − 0.1299ξ m + 0.04179ξm
(
)
2
 τ 
 T 
0.1648+ 0.1709 ln  10 m 2  − 0.2142 ln  10 m  
 Tm 1  
 Tm1 



 τ 
 T 
0.3144 − 0.0062 ln  10 m 2  + 0.0085 ln  10 m  
 Tm 1  
 Tm1 



−1.798 + 0.7191ξ m − 0.1416 ξ m 2
)
,
 τm 


 2ξ T 
 m m1 
τm 
2 

Td = 2ξ m Tm1 1.2226 − 0.6456ξ m + 0.1373ξ m 

ξ
2
 m Tm1 
−0.1277 + 0.5220 ξ m − 0.08629 ξ m 2
,
0.4780 − 0.03653ξ m + 0.04523ξ m 2
; N = 8.
b720_Chapter-03
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Handbook of PI and PID Controller Tuning Rules
242
Rule
Kc
Ti
Comment
Td
9
Chao et al.
Ti
Td
0.3 < ξ m < 0.8
Kc
(1989) –
N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 .
continued.
10
0.884
0.907
0.869
Approximately
Tm1 > Tm 2 ;
x 
x 
 
minimum ITSE – 0.994  x1 
0.570x1  2 
1.032x1  2 
N = 8.


 x1 
Huang and Chao Km  x2 
 x1 
(1982).
Also given by Chao et al. (1989); Model: Method 1.
11
Minimum ITSE
Ti
Td
0.8 ≤ ξ m ≤ 3
Kc
– Chao et al.
N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1.
(1989).
9
1.3292 − 2.8527ξm + 2.0365ξ m 2  τm 


Kc =
 2ξ T 
Km
 m m1 
−1.6159 − 0.06616 ξ m + 0.5351ξ m 2
,
(
)
 τm 

Ti = 2ξm Tm1 exp(x1 ) + 1.4094 − 4.4581ξ m + 3.4857ξ m 2 ln
 2ξ m Tm1 
(
+ 0.1160 − 0.8142ξ m + 0.7402ξ m
2
2
  τ m 
 ,
ln
  2ξ m Tm1 
)
(
x 1 = 3.3368 − 7.7919ξ m + 4.3556ξ m
(
)
2
);
2
).
 τm 

Td = 2ξ m Tm1 exp(x 2 ) + 1.8243 − 5.6458ξ m + 4.6238ξ m 2 ln
 2ξ m Tm1 
2
τ m 
2  
 ,
+ 0.2866 − 1.4727ξ m + 1.2893ξ m ln
  2ξ m Tm1 
(
)
(
x 2 = 2.3054 − 6.4328ξ m + 3.9743ξ m
10
11
τ

1.398
x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1  m + 1.039 −
.
1.349 + (Tm1 Tm 2 ) 
 Tm 2
0.1034 + 0.4857ξ m − 0.0709ξ m 2  τm 


Kc =
 2ξ T 
Km
 m m1 
(
2
(
)
Ti = 2ξ m Tm1 1.5144 − 0.6060ξ m + 0.1414ξm
−1.713 + 0.6238ξ m − 0.1178ξ m 2
)
,
 τm 


 2ξ T 
 m m1 
τm 
2 

Td = 2ξ m Tm1 1.0783 − 0.4886ξ m + 0.1038ξm 

ξ
2
 m Tm1 
0.1008 + 0.2625ξ m − 0.02203ξ m 2
,
0.4316 + 0.08163ξ m + 0.009173ξ m 2
.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Chao et al.
(1989) –
continued.
Approximately
minimum IAE –
Huang and Chao
(1982).
12
Kc
12
Kc
Ti
Td
Comment
Ti
Td
0. 3 < ξ m < 0. 8
243
N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 .
Minimum performance index: servo tuning
1.00
1.032
Tm1 > Tm 2 ;
0.673  x1 
 x2 
13
 
 
0
.
502
x
T
1
N = 8.


x 
i
Km  x2 
 1
Also given by Chao et al. (1989); Model: Method 1.
1.8342 − 3.8984ξ m + 2.7315ξm 2  τm 


Kc =
 2ξ T 
Km
 m m1 
(
−1.5641+ 0.1628ξ m + 0.09337 ξ m 2
,
)
 τm 

Ti = 2ξm Tm1 exp(x1 ) + 0.9475 − 3.6318ξ m + 2.9969ξ m 2 ln
 2ξ m Tm1 
(
+ − 0.06927 − 0.3875ξ m + 0.4220ξ m
2
2
  τ m 
 ,
ln
  2ξ m Tm1 
)
(
2
)
2
).
x 1 = 3.2026 − 7.4812ξ m + 4.3686ξ m ;
(
)
 τm 

Td = 2ξm Tm1 exp(x 2 ) + 1.0983 − 1.5794ξ m + 1.0744ξ m 2 ln
 2ξ m Tm1 
(
+ 0.01553 − 0.01414ξ m − 0.009083ξ m
(
2
2
  τ m 
 ,
ln
  2ξ m Tm1 
)
x 2 = 1.7393 − 3.7971ξ m + 1.7608ξ m
13

τ
1.398
x1
Ti =
, x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1  m + 1.039 −
T
x 
 Tm 2
1.349 + m1
0.148 2  + 0.979

T
m2

 x1 


.



b720_Chapter-03
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Handbook of PI and PID Controller Tuning Rules
244
Rule
Kc
Ti
Comment
Td
14
Minimum IAE –
N=8
Ti
Td
Kc
Chao et al.
0.8 ≤ ξ m ≤ 3; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1.
(1989).
0.964
15
0.971
1.028
Approximately
Tm1 > Tm 2 ;
 x2 
0.787  x1 
 x2 
 
minimum ISE –




0
.
594
x
1
.
678
x
1
1
x 



N = 8.
K
m  2 
Huang and Chao
 x1 
 x1 
(1982).
Also given by Chao et al. (1989); Model: Method 1.
16
Minimum ISE –
N=8
Ti
Td
Kc
Chao et al.
0.8 ≤ ξ m ≤ 3; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1.
(1989).
14
− 0.1779 + 0.6763ξ m − 0.1264ξ m
Kc =
Km
2
 τm 


 2ξ T 
 m m1 
(
−0.9941+ 0.1669 ξ m − 0.04381ξ m 2
,
x
)
 τm  1
 ,
Ti = 2ξ m Tm1 0.2857 + 0.4671ξm − 0.08353ξ m 2 

 2ξ m Tm1 
2
3
x 1 = 0.1665 − 0.3324ξ m + 0.1776ξ m − 0.02897ξ m ;
(
)
 τm 

Td = 2ξm Tm1 exp(x 2 ) + − 2.0552 + 2.786ξ m − 0.5687ξ m 2 ln
 2ξ m Tm1 
2
  τ m 
 ,
+ − 0.5008 + 0.6232ξ m − 0.1430ξ m 2 ln
  2ξ m Tm1 
(
)
(
x 2 = − 0.8776 + 0.4353ξ m − 0.1237ξ m
15
16
2
).
2
);
τ

1.398
x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1  m + 1.039 −
.
T
1
.
349
(
T
T
)
+
m1
m2 
 m2
0.1446 + 0.4302ξ m − 0.07501ξm 2  τm 


Kc =
 2ξ T 
Km
 m m1 
−1.214 + 0.3123ξ m − 0.06889 ξ m 2
;
(
)
 τm 

Ti = 2ξm Tm1 exp(x1 ) + − 2.2502 + 3.7015ξ m − 1.7699ξ m 2 + 0.2680ξ m 3 ln
 2ξ m Tm1 
(
)
+ − 0.4823 + 0.9479ξm − 0.4913ξ m + 0.07792ξ m [ln (0.5τm ξTm1 )] ,
(
2
)
3
(
x 1 = − 0.1256 + 0.1434ξ m − 0.03277ξ m
Td = 2ξm Tm1 0.9105 − 0.3874ξ m + 0.08662ξ m (0.5τm ξm Tm1 ) ,
2
2
x2
2
x 2 = 0.0779 + 0.2855ξm − 0.01985ξm .
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
245
Comment
Td
0.995
1.049
Approximately
Tm1 > Tm 2 ;
 
 x2 
17
minimum ITAE 0.684  x1 


0
.
491
x
T
N = 8.
1
i

K m  x 2 
– Huang and
 x1 
Chao (1982).
Also given by Chao et al. (1989); Model: Method 1.
Minimum ITAE
– Chao et al.
(1989).
18
Kc
Ti
Td
0.8 ≤ ξ m ≤ 3
19
Kc
Ti
Td
0.3 < ξ m < 0.8
N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1.
17
18

τ
1.398
x1
Ti =
, x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1  m + 1.039 −
T
x 
 Tm 2
1.349 + m1
0.114 2  + 0.986

T
m2

 x1 
− 0.2641 + 0.7736ξ m − 0.1486ξ m
Kc =
Km
2
 τm 


 2ξ T 
 m m1 
(


.



−0.7225 − 0.04344 ξ m − 0.001247 ξ m 2
;
x
)
 τm  1
 ,
Ti = 2ξ m Tm1 0.22257 + 0.7452ξ m − 0.1451ξm 2 

 2ξ m Tm1 
2
x 1 = 0.03138 − 0.04430ξ m + 0.01120ξ m ;
(
)
 τm 

Td = 2ξm Tm1 exp(x 2 ) + − 1.2256 + 1.8544ξ m − 0.3366ξ m 2 ln
 2ξ m Tm1 
2
  τ m 
 ,
+ − 0.2315 + 0.3402ξ m − 0.06757ξ m 2 ln
  2ξ m Tm1 
(
)
(
x 2 = 0.05515 − 0.7031ξ m + 0.1433ξ m
19
Kc =
1.1295 − 2.8584ξ m + 2.1176ξ m 2  τm 


 2ξ T 
Km
 m m1 
(
Ti = 2ξ m Tm1 8.6286 − 20.70ξm + 13.203ξm
2
).
2
−0.9904 − 2.5311ξ m + 2.8008ξ m 2
)(0.5τ
;
ξ m Tm1 ) 1 ,
x
m
2
(
)
x 1 = 0.7962 − 2.4809ξ m + 1.7611ξ m ;
Td = 2ξm Tm1 exp(x 2 ) + 3.2228 − 12.034ξ m + 9.2547ξ m 2 ln (0.5τm ξm Tm1 )
(
)
+ 0.6366 − 3.0208ξ m + 2.4603ξ m [ln (0.5τm ξ m Tm1 )] ,
2
(
2
2
)
x 2 = 3.9285 − 12.874ξ m + 8.6434ξ m .
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246
Rule
Kc
Ti
Td
Comment
1.00
1.028
Approximately
Tm1 > Tm 2 ;
 x2 
0.718  x1 
20
minimum ITSE –
 
 
0
.
596
x
T
1
N = 8.
 
x 
i
Huang and Chao K m  x 2 
 1
(1982).
Also given by Chao et al. (1989); Model: Method 1.
21
Minimum ITSE
Ti
Td
0.8 ≤ ξ m ≤ 3
Kc
– Chao et al.
N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1.
(1989).
20
21

τ
1.398
x1
Ti =
, x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1  m + 1.039 −
T
 x2 
 Tm 2
1.349 + m1
0.063  + 1.061

T
m2

 x1 
0.0820 + 0.4471ξm − 0.07797ξ m 2  τm 


Kc =
 2ξ T 
Km
 m m1 
(
)


.



−0.9246 + 0.07955ξ m − 0.02436 ξ m 2
;
x
 τm  1
 ,
Ti = 2ξ m Tm1 0.5930 + 0.1457ξ m − 0.02062ξ m 2 

 2ξ mTm1 
2
x 1 = −0.2201 + 0.1576ξ m − 0.03202ξ m ;
(
)
 τm 

Td = 2ξm Tm1 exp(x 2 ) + − 1.0018 + 1.5914ξ m − 0.2831ξ m 2 ln
 2ξ m Tm1 
(
+ − 0.2291 + 0.3064ξ m − 0.0634ξ m
(
2
2
  τ m 
 ,
ln
  2ξ m Tm1 
)
x 2 = − 0.6534 − 0.1770ξ m − 0.0240ξ m
2
).
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FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Chao et al.
(1989) –
continued.
22
Kc
Td
Comment
Ti
Td
0.3 < ξ m < 0.8
N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 .
Smith et al.
(1975).
Model: Method 1
Direct synthesis: time domain criteria
Tm1
Tm 2
λ = 1 TCL ;
λTm1
K m (λτ m + 1)
N not specified.
x 1Tm1
K mτm
Smith et al.
(1975).
Model:
Method 1;
N = 10.
ξm
ρ
6
1.75
1.75
1.5
1.0
≥ 0.02
0.27
0.07
0.11
0.25
Tm1
ρ=
Tm 2
τm
Tm1 + Tm 2
Coefficient values (deduced from graph):
ρ
ρ
x1
ξm
x1
ξm
0.7
0.3
0.8Tm1 Tm 2
Hougen (1979),
τm
pp. 348–349.
Model: Method 1 0.84Tm1 0.8 Tm 2 0.2
τm
22
Ti
247
0.51
0.46
0.36
0.33
0.46
3
1.75
1.5
1.5
1.0
0.5Tm1 + Tm 2
23
0.50
0.42
0.44
0.28
0.48
≥ 0.04
0.13
0.33
0.08
0.17
3
(
Ti = 2ξ m Tm1 6.3891 − 15.773ξ m + 10.916ξm
2
≥ 0.06
0.09
0.17
0.50
0.13
x1
0.45
0.39
0.38
0.40
0.49
τ m Tm1Tm 2
N = 10
Td
N = 30
Ti
0.9666 − 2.3853ξm + 2.4096ξm 2  τm 


Kc =
 2ξ T 
Km
 m m1 
2
1.75
1.5
1.0
1.0
−1.533 − 0.05442 ξ m +10.916 ξ m 2
;
x
)
 τm  1


 2ξ T  ,
 m m1 
2
x 1 = 0.04175 − 1.3860ξ m + 1.4053ξ m ;
(
)
 τm 

Td = 2ξm Tm1 exp(x 2 ) + 2.277 − 8.9402ξ m + 7.7176ξ m 2 ln
 2ξ m Tm1 
(
+ 0.3465 − 1.9274ξ m + 1.8030ξ m
(
2
2
  τ m 
 ,
ln
  2ξ m Tm1 
)
x 2 = 3.2620 − 10.789ξ m + 7.6420ξ m
23
Ti = 0.53Tm1 + 1.3Tm 2 , Td = 2.4(τm Tm1Tm 2 )
0.28
.
2
).
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248
Rule
Kc
Hougen (1988).
Model: Method 1
Sklaroff (1992).
Ti
Td
Comment
N = 10;
τ m < Tm1 .
24
Kc
0.5Tm1 + Tm 2
Td
25
Kc
Tm1 + Tm 2
Tm1Tm 2
Tm1 + Tm 2
Also given by Åström and Hägglund (1995), pp. 250–251.
Model: Method 8. N = 8 – Honeywell UDC6000 controller.
26
N not defined.
Ti
Td
K
Schaedel (1997).
c
Model: Method 1
Skogestad
(2003).
0.5Tm1 K m τm
N =∞.

24

1
Kc =
10 
Km
T
x 1 − x 2 log10  m 2
 Tm1



 
min(Tm1 ,8τ m )
Model:
Method 24
Tm 2
. Coefficients of K c deduced from graphs:
τ m Tm 2
0.1
0.3
0.5
1
2
x1
0.9
0.42
0.24
-0.10
-0.39
x2
0.7
0.7
0.68
0.70
0.69
Td = (Tm1 + Tm 2 )10
Td = (Tm1 + Tm 2 )10
25
Kc =
26
Kc =


 τ 
 0.33 log10  m  − 0.40 


 Tm1 


 τ 
 0.31 log10  m  − 0.63 


 Tm1 
,
Tm 2
= 0.1 ;
Tm1
, 0.2 ≤
Tm 2
≤1.
Tm1
3
.


3τ m


K m 1 +
Tm1 + Tm 2 

0.375
Km
2
2
2
2
4ξ m Tm1 + 2ξ m Tm1τm + 0.5τm − Tm1
,
(2ξmTm1 + τm )2 +  1 − 1(2ξmTm1 + τm )Tm1 2(2ξm 2 − 1) − x
N 
2
2
2
2
x = 4ξ m Tm1 + 2ξ m Tm1 τ m + 0.5τ m − Tm1 ;
2
Ti =
Td =
2
2
2
4ξ m Tm1 + 2ξ m Tm1τm + 0.5τm − Tm1
;
2ξm Tm1 + τm
(2ξmTm1 + τm )2 − 2(Tm12 + 2ξmTm1τm + 0.5τm 2 ) .
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
27
Panda et al.
(2004).
Kc
Tm1
2K m τ m
Ti
Td
Comment
2ξ m Tm1
Tm1 2ξm
Tm1
Tm 2
N = 8;
Model: Method 8
N not defined;
Model: Method 1
Tm 2
N =∞;
Tm1 > Tm 2 .
Skoczowski
(2004).
Tm1
28
Kc
Model: Method 1
Huang and Jeng
T
0.495 m1 + 0.2
(2005).
0.4τ m + 0.9Tm1
τm
Model: Method 1
Km
Skogestad
(2006b).
Model: Method 1
27
Kc =
28
Kc ≤
29
Tm 2
N =∞;

4T 
min Tm1 , m1 
Tm 2
T
m1 > Tm 2 .
Km 

Slow control; ‘acceptable’ disturbance rejection performance.
Direct synthesis: frequency domain criteria
Tm1
Tm1
Tm 2
K m (Tm1 + τ m )
N = 5; Tm1 ≤ 5Tm 2 ; minimum φ m = 550 .
ωp Tm1
KmAm
x 1Tm1
K mτm
Ti
Tm1
Tm1
Tm 2
29
N =∞.
A m = 1.571 x1 ;
φ m = 0.5π − x 1 .
Sample result ( A m = 2.0; φ m = 45 ):
0.785Tm1 K m τm
Tm1
2Tm1 
1 + 2 x12 − 2 x1 1 + x12  , with x1 ≈
τm 

1
2ωp −
Tm1 = Tm 2 ;
0
Tm 2
3
.

1.5τm 

K m 1 +

 ξm Tm1 
Ti =
Tm 2
τm
1
Poulin et al.
(1996).
Model: Method 1
Yang and Clarke
(1996).
Model:
Method 10
O’Dwyer
(2001a).
Model:
Method 1;
N =∞.
N = 100
4ωp 2 τm
π
+
1
Tm1
.
ln (1 OS)
π + [ln (1 OS)]2
2
.
249
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250
Rule
Kc
Majhi and Litz
(2003).
30
Kc
Ti
Td
Comment
Ti
Tm1
N=∞
Model: Method 10; Tm1 = Tm 2 .
Ultimate cycle
Bohl and
McAvoy
(1976b).
30
Kc =
31
2φm + π(A m − 1) Tm1
,
2
K m τm
2 Am − 1
(
)
Ti =
1+
31
N =∞;
Model: Method 1
Td = Ti
Ti
Kc
Tm1
τm
Tm1
.


Am
Am
[
]
(
)
(
(
)
)
2
φ
+
π
A
−
1
1
−
2
φ
+
π
A
−
1


m
m
m
m
2
Am2 − 1

 π A m − 1
(
τ 

1.2264 + 0.4568 ln  m  
 Tu  
0.5458  τm 
Kc =
 

K m  Tu 



)
(
)
τ 

1.2190 − 0.0958 ln (K u K m )−0.1007 ln  m  
(K u K m )
 Tu   ,


τ 

−1.9314 − 0.6221 ln  m  
 τm 
 Tu  
Ti = 0.0393Tu  


 Tu 


τ 

−0.0703−0.1125 ln (K u K m )−0.2944 ln  m  
(K u K m )
 Tu   .


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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
251
3.5.6 Generalised classical controller


Td s
1
+
G c (s) = K c 1 +
T

Ti s
1+ d

N

R(s)
E(s)
+
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N



2
 b f 0 + b f 1s + b f 2 s
 1 + a f 1s + a f 2 s 2
s



2
 b f 0 + b f 1s + b f 2 s
 1 + a f 1s + a f 2 s 2
s 

U(s)
SOSPD
model
Table 25: PID controller tuning rules – SOSPD model G m (s) =
G m (s) =
Rule
Huang et al.
(2003).
Model:
Method 12
Kc
0.8ξm Tm1
K m τm
Y(s)
K m e − sτ m
or
(1 + sTm1 )(1 + sTm 2 )
K m e − sτ m
2
Tm1 s 2 + 2ξ m Tm1s + 1
Ti
Comment
Td
Direct synthesis: time domain criteria
2ξ m Tm1
0.5Tm1
ξm
b f 0 = 1 ; b f 1 not defined; b f 2 = 0 ; N = ∞ ; a f 1 = a f 2 = 0 .
Robust
Jahanmiri and
Fallahi (1997).
Model: Method 7
2ξ m Tm1
K m (τ m + λ)
2ξ m Tm1
b f 1 = 0.5τ m ,
Tm1
2ξ m
a f1 =
λτ m
.
2( τ m + λ )
N = ∞ , b f 0 = 1 , b f 2 = 0 , a f 2 = 0 ; λ = 0.25τ m + 0.1ξ m Tm1 .
Kristiansson
(2003).
Model: Method 1
1
Kc =
1
Kc
2ξ m Tm1
M max = 1.7
Tm1
2ξ m
2
b f 0 = 1 , b f 1 = b f 2 = 0 , a f 1 = 0.08Tm1 , a f 2 = 0.01Tm1 , N = ∞ .
2ξ m Tm1 
T
T
T 2
1 + 0.019 m1 − 0.346 + 0.038 m1 + 0.00036 m12
K m τm 
τm
τm
τm


.






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252
Rule
Shi and Lee
(2004).
Kc
Ti
Td
Comment
Kc
Ti
Td
Model: Method 1
0.5τm
K m (λ + τm )
0.5τm
0.167 τm
N=∞
2
Shamsuzzoha
and Lee (2006a).
Model: Method 1
af1 =
2
τmλ
τm λ
, af 2 =
, b f 0 = 1 , b f 1 = 2ξ m Tm1 ,
2(τm + λ )
12(τm + λ )
2
b f 2 = Tm1 . λ = x1τm ; x1 = 1.360 , M s = 1.4 ; x1 = 0.677 , M s = 1.6 ;
x1 = 0.372 , M s = 1.8 ; x1 = 0.337 , M s = 2.0 .
2
Kc =

1 + ω−6dB τ m
ω− 6dB
0.5 
 Tm1 + Tm 2 −
 , bf 0 = 1 , b f 1 =
,


2ω−6dB
ω− 6 dB 
K m (2 + ω− 6dBτm ) 
bf 2 =
Ti = Tm1 + Tm 2 −
τm
τm
1
, a f1 =
, af 2 =
;
4ω −6dB
2ω−6dB
2ω−6dB (1 + ω −6dB τ m )
2
0.5
T T ω
− 0.5[(Tm1 + Tm 2 )ω− 6dB − 0.5]
, Td = m1 m 2 − 6dB
,
ω− 6dB
ω− 6 dB [(Tm1 + Tm 2 )ω− 6dB − 0.5]
N=
0.5[(Tm1 + Tm 2 )ω−6dB − 0.5]
Tm1Tm 2 ω−6dB 2 − 0.5[(Tm1 + Tm 2 )ω −6dB − 0.5]
.
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FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
3.5.7 Two degree of freedom controller 1


Td s
1
+
U (s ) = K c  1 +
 Ti s 1 + Td

N






βTd s 
 E (s ) − K c  α +
 R (s )
Td 


s
1+ s 

N 






T
s
β
d
α +
K
Td  c

1
s
+


N 

E(s)
R(s)
−
+


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N





s

−
U(s)
+
SOSPD
model
Table 26: PID controller tuning rules – SOSPD model G m (s) =
Y(s)
K m e − sτ m
or
(1 + sTm ) 2
K m e − sτ
K m e − sτ m
or
(1 + sTm1 )(1 + sTm 2 ) Tm1 2 s 2 + 2ξ m Tm1s + 1
m
Rule
Kc
Ti
Td
Comment
Minimum performance index: regulator tuning
1.18Tm1 K m τm
2.20τ m
0.72τ m
Tm 2 Tm1 = 0.1
Minimum IAE –
1.25Tm1 K m τm
Shinskey (1996),
1.67Tm1 K m τm
p. 119.
Model: Method 1 2.5T K τ
m1
m m
253
2.20τ m
1.10τ m
Tm 2 Tm1 = 0.2
2.40τ m
1.65τ m
Tm 2 Tm1 = 0.5
2.15τ m
2.15τ m
Tm 2 Tm1 = 1.0
τm Tm1 = 0.2 ; α = 0 ; β = 1 ; N = ∞ .
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254
Rule
Minimum IAE –
Shinskey (1996),
p. 121.
Model: Method 1
Minimum IE –
Shen (1999).
Model: Method 1
Td
Comment
Ti
Td
Tm 2 τm ≤ 3
τ m + 0.2Tm 2
τ m + 0.2Tm 2
Tm 2 τm > 3
α = 0 ;β =1; N = ∞ .
0.85K u
0.35Tu
2
Ti
Kc
− 1.5 Tm1 τ m
m m
Km = 1
Td
m1
− Tm1 1.5 τ m
m
m
Tm 2 Tm1 = 0.2 .
α = 0 ;β =1; N = ∞ .
)]
(38 + 40[1 − e (
)(K τ T )(1 + 0.34[T τ
(
)
]),
T = τ (0.5 + 1.4[1 − e
]) (1 + 0.48[1 − e
(
)
) + 0.6T .
T = 0.42τ (1 − e
d
τm Tm1 = 0.2 ,
0.17Tu
100
Kc =
i
2
Ti
Kc
1
Minimum IAE –
Kc
Shinskey (1994),
3.33Tm1 K m τm
p. 159.
Model: Method 1
1
FA
m2
m
] − 0.2[Tm 2
τm ]2
),
− Tm 2 τ m
− 1.2 Tm1 τ m
m2
x1 , x 2 , x 3 and x 4 are given by the following equation:
2
3
τ  τ 
τ 
ξ τ
exp[a + b m  + c m  + d m  + e m m + fξ m + gξ m 2
Tm1
T
T
T
 m1   m1 
 m1 
3
2
3
2
τ 
τ 
τ 
ξm τm
2
2
+ i m  ξ m + j m  ξ m + k  m  ξ m ] ;
T
T
T
Tm1
 m1 
 m1 
 m1 
K c = Tm1x1 τm , Ti = τm x 2 , Td = τm x 3 , α = 1 − x 4 ; coefficients of x1 , x 2 , x 3 and x 4
+h
are given below. 1.4 ≤ M s ≤ 2 ; β = 0 ; N = ∞ .
a
b
c
d
e
f
g
h
i
j
k
x1
x2
x3
x4
1.40
-11.60
14.17
-6.44
8.28
-0.07
0.13
-4.66
-0.99
-3.03
2.89
2.43
-2.50
-4.16
2.85
-0.30
0.1
-1.72
15.51
-7.98
-8.45
4.66
1.32
-0.51
0.70
-1.06
0.02
-0.09
-7.43
1.77
2.64
4.77
-3.74
-1.44
1.27
1.34
-0.37
1
-5.97
-0.78
-10.63
18.85
-10.97
6.44
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Minimum IE –
Shen (2000).
Model: Method 1
3
3
Kc
Ti
Td
Comment
Kc
Ti
Td
Km = 1
255
x1 , x 2 , x 3 and x 4 are given by the following equation:
2
3
 τm   τ m 
 τm 
 + c


 + eξ m + f ξ m 2
exp[a + b
  τ + T  + d τ + T 
τ
+
T
m1 
m1 
m1 
 m
 m
 m
2
3
 τm

 τm

 τm

ξ m + h 
 ξ m + i
 ξ m
+ g
τ
+
T
τ
+
T
τ
+
T
m1 
m1 
m1 
 m
 m
 m
2
3
 τm
 2
 τm

 τm

ξ m + k 
 ξ m 2 + l
 ξ m 2 ] ;
+ j
τ
+
T
τ
+
T
τ
+
T
m1 
m1 
m1 
 m
 m
 m
K c = Tm1x1 τm , Ti = τm x 2 , Td = τm x 3 , α = 1 − x 4 ; coefficients of x1 , x 2 , x 3 and x 4
are given below. M s = 2 ; β = 0 ; N = ∞ ; 0 < ξ m < 1 ; 0 ≤ τm Tm1 ≤ 1 .
a
b
c
d
e
f
g
h
i
j
k
l
x1
x2
x3
x4
1.73
-17.51
30.73
-24.69
0.15
-0.12
3.30
33.07
-37.64
2.63
-34.57
36.88
2.28
0.87
-24.00
15.09
-0.01
-0.02
-8.03
45.21
-22.48
3.89
-22.39
8.32
1.43
-1.94
14.82
-21.44
-0.25
0.16
-1.06
-43.39
51.16
-3.25
36.04
-34.95
-1.60
4.30
-12.16
26.54
1.33
-1.08
-17.56
38.13
-56.45
14.80
-39.33
51.18
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
256
Rule
Minimum IE –
Shen (2000).
Model: Method 1
Madhuranthakam
et al. (2008).
Model: Method 1
4
17-Mar-2009
4
5
Kc
Ti
Td
Comment
Kc
Ti
Td
Km = 1
Kc
Ti
Td
Tm1 > Tm 2 ;
τm < Tm1 + Tm 2 .
x1 , x 2 , x 3 and x 4 are given by the following equation:
2
3
 τm   τ m 
 τm 
 + c
 + d
 + eξ m + f ξ m 2
exp[a + b



τ +T 
m1 
 τm + Tm1   τm + Tm1 
 m
2
3
 τm

 τm

 τm

ξ m + h 
 ξ m + i
 ξ m
+ g
τ
+
T
τ
+
T
τ
+
T
m1 
m1 
m1 
 m
 m
 m
2
3
 τm
 2
 τm

 τm

ξ m + k 
 ξ m 2 + l
 ξ m 2 ] ;
+ j
τ
+
T
τ
+
T
τ
+
T
m1 
m1 
m1 
 m
 m
 m
K c = Tm1x1 τm , Ti = τm x 2 , Td = τm x 3 , α = 1 − x 4 ; coefficients of x1 , x 2 , x 3 and x 4
are given below. M s = 2 ; β = 0 ; N = ∞ ; 0.4 ≤ ξ m ≤ 1 ; 1 < τm Tm1 ≤ 15 .
a
b
c
d
e
f
g
h
i
j
k
l
5
Kc =
x1
x2
x3
x4
128.9
-560.3
769.2
-338.9
-324.8
194.3
1421
-1981.6
895.0
-848.1
1186.1
-538.1
92.8
-391
521.6
-225.5
-224.5
137.5
963.9
-1310.6
574.4
-590.8
808.0
-356.7
-3.1
46.3
-107.4
63.1
5.2
-2.5
-79.6
176.0
-103.6
36.2
-78.4
45.8
-119.9
539.5
-785.1
371.8
233.9
-115.7
-1068
1560.5
-741.5
529.2
-774.6
369.2
0.6202  Tm1 + Tm 2 
1 +


K m 
τm

0.9931
,
2
 




τm
τm
τ 
 ,
 + 13.81
Ti = τm 1 + m  4.566 − 14.906
τ +T +T  

m1
m2 
 m
 τm + Tm1 + Tm 2 
 Tm1  

Td = 0.0921(τ m + Tm 2 )
1.4849
τm 
T + Tm 2 
1 + m1


Tm1 
τm

, N = 10, α = 0 , β = 1 .
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Madhuranthakam
et al. (2008).
Model: Method 1
6
Ti
Kc
Comment
Td
Minimum performance index: servo tuning
Tm1 > Tm 2 ;
Ti
Td
τm < Tm1 + Tm 2 .
Minimum performance index: servo/regulator tuning
x1 K m
x 2Tm1
x 3Tm1
N=∞
Taguchi et al.
(1987).
x
Model: Method 1 1
11.47
3.39
1.70
1.06
0.75
0.58
0.48
Minimum ITAE
– Pecharromán
(2000a),
Pecharromán and
Pagola (2000).
Model:
Method 15;
Km = 1;
Tm1 = 1 ;
ξm = 1 .
x2
x3
α
β
0.74
1.17
1.35
1.39
1.37
1.34
1.32
x1K u
0.40
0.67
0.89
1.06
1.18
1.25
1.28
0.64
0.57
0.47
0.35
0.21
0.08
-0.05
0.84
0.80
0.75
0.69
0.62
0.55
0.46
x 2 Tu
x1
x2
α
0.147
1.150
0.170
1.013
τm
Tm1
x1
x2
x3
α
β
τm
Tm1
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.41
0.38
0.35
0.33
0.33
0.35
0.37
1.32
1.34
1.38
1.53
1.75
2.26
2.82
0
1.28
1.25
1.21
1.12
1.06
1.08
1.19
-0.15
-0.24
-0.30
-0.37
-0.37
-0.32
-0.24
0.37
0.27
0.18
-0.04
-0.19
-0.30
-0.18
1.6
1.8
2.0
2.5
3.0
4.0
5.0
Coefficient values:
φc
x1
x2
α
φc
0.280
0.512
0.281
− 730
− 140 0
0.291
0.503
0.270
− 630
0.1713 1.0059 0.4002 − 139.7 0
0.195 0.880 0.386 − 1330
0.297
0.483
0.260
− 52 0
0.303
0.462
0.246
− 410
0.210
0.720
0.342
− 1250
0.307
0.431
0.229
− 30 0
0.234
0.672
0.345
− 115 0
0.317
0.386
0.171
− 19 0
0.249
0.610
0.323
− 105
0
0.324
0.302
0.004
− 10 0
0.262
0.568
0.308
− 94 0
0.320
0.223
-0.204
− 60
0.274
0.545
0.291
− 84 0
0.411
− 146
0
0.401
1.0409
6
Kc =
0.5723  Tm1 + Tm 2 

1 +

τm
K m 

,
1.6501
Ti = 0.2476


τm
(τm + Tm1 + Tm 2 )1 + Tm1 + Tm 2 
τm
Tm1


Td = 0.0943(τ m + Tm 2 )
,
1.4636
T + Tm 2 
τm 
1 + m1


Tm1 
τm

, N = 10, α = 0 , β = 1 .
257
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
258
Rule
Kc
Ti
Td
Comment
Minimum ITAE
– Pecharromán
and Pagola
(2000).
0.7236K u
0.5247Tu
0.1650Tu
N = 10;
Model:
Method 15
Minimum ITAE
– Pecharromán
(2000a).
Model:
Method 15;
Km = 1;
Tm1 = 1 ;
β = 1 ; N = 10;
ξm = 1 ;
0.1 < τ m < 10 .
Minimum ITAE
– Pecharromán
(2000b), p. 145.
Model:
Method 15;
Km = 1;
Tm1 = 1 ;
β = 1 ; N = 10;
ξm = 1 .
α = 0.5840 , β = 1 , φc = −139.650 , K m = 1 ; Tm1 = 1 ; ξ m = 1 .
x1K u
x1
x 2 Tu
x 3Tu
Coefficient values and other data:
α
x2
x3
φc
0.803
0.509
0.167
0.585
− 146 0
0.727
0.524
0.165
0.584
− 140 0
0.672
0.532
0.161
0.577
− 134 0
0.669
0.486
0.170
0.550
− 1250
0.600
0.498
0.157
0.543
− 1150
0.578
0.481
0.154
0.528
− 1050
0.557
0.467
0.149
0.504
− 930
0.544
0.466
0.141
0.495
− 84 0
0.537
0.444
0.144
0.484
− 730
0.527
0.450
0.131
0.477
− 630
0.521
0.440
0.129
0.454
− 52 0
0.515
0.429
0.126
0.445
0.509
0.399
0.132
0.433
− 410
0.496
0.374
0.123
0.385
− 19 0
0.480
0.315
0.112
0.286
− 10 0
0.430
0.242
0.084
0.158
− 60
x1K u
x1
x 2 Tu
− 30 0
x 3Tu
Coefficient values and other data:
α
x2
x3
φc
0.7965
0.5139
0.1638
0.5854
− 146.10
0.7139
0.5275
0.1625
0.5844
− 140.00
0.6675
0.5109
0.1600
0.5771
− 134.00
0.6540
0.4240
0.1704
0.5498
− 125.00
0.5996
0.4736
0.1629
0.5428
− 115.00
0.5801
0.4660
0.1573
0.5276
− 105.00
0.5604
0.4545
0.1512
0.5042
− 93.00
0.5500
0.4520
0.1470
0.4951
− 84.00
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Minimum ITAE
– Pecharromán
(2000b), p. 145 –
continued.
Kuwata (1987).
Model: Method 1
7
Kc =
Ti
Td
x1K u
x 2 Tu
x 3Tu
Coefficient values and other data:
α
x2
x3
x1
Kuwata (1987).
Model: Method 1
0.4321
0.1491
0.4844
− 73.00
0.5312
0.4431
0.1361
0.4769
− 63.00
0.5247
0.4356
0.1325
0.4544
− 52.0 0
0.5189
0.4260
0.1281
0.4452
− 41.00
0.5153
0.3971
0.1324
0.4327
− 30.00
0.4987
0.3680
0.1227
0.3855
− 19.00
0.4846
0.3131
0.1146
0.2860
− 10.00
0.4610
0.2471
0.0966
0.1577
− 6.00
Direct synthesis: time domain criteria
Ti
0
Ti
7
Kc
8
Kc
9
Kc
Ti
Td
10
Kc
Ti
Td
0.8(2Tm + τm )
1
−
, x1 =
K m (2Tm + τm − x1 ) K m
2
+ 4Tm τm + τm
2Tm + τm
m


α = 1; β = 1;
N =∞.
2
].
(2ξmTm1 + τm )2 − 0.267τm [6Tm12 + 6ξmTm1τm + τm 2 ] ;
[
 0.833 2Tm12 + 4ξm Tm1τm + τm 2
Ti = 1 −

(2ξmTm1 + τm )2

9
α =1
(2Tm + τm )2 − 0.267τm [6Tm 2 + 6Tm τm + τm 2 ] ;
] 1.667[2T
 0.833 2Tm 2 + 4Tm τm + τm 2
Ti = 1 −

(2Tm + τm )2

0.8(2ξ m Tm1 + τm )
1
Kc =
−
,
K m (2ξ m Tm1 + τm − x1 ) K m
x1 =
φc
0.5451
[
8
Comment
Kc
] 1.667[2T
2
m1


+ 4ξ mTm1τm + τm
2ξm Tm1 + τm
2
].
2
2
1.20 x1
1
x
x
x − (2Tm + τm )x 2
−
; Ti = 1.667 2 − 4.167 23 ; Td = 0.833x 2 1 3
,
2
Km x2
Km
x1
x1
x1 − 0.833x 2 2
3
Kc =
2
2
2
2
3
x1 = 2Tm + 4Tm τm + τm , x 2 = 6Tm τm + 6Tm τm + τm .
10
(
3
Kc =
)
2
2
1.20 x1
1
2.778 x1 − 2.5x 2 x 2
=
−
;
; x1 = 2Tm12 + 4ξm Tm1τm + τm 2 ;
T
i
K m x 22 K m
x14
x1 − (2ξm Tm1 + τm )x 2
2
Td = 0.833x 2
x13 − 0.833x 2 2
, x 2 = 6Tm12 τm + 6ξ m Tm1τm 2 + τm 3 .
259
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
260
Rule
Taguchi and
Araki (2000).
Model: Method 1
Kc
Ti
Td
Comment
11
Kc
Ti
0
ξm = 1
12
Kc
Ti
0
ξ m = 0.5
τ m Tm ≤ 1.0 ; OS ≤ 20% .
Taguchi and
Araki (2000).
Model: Method 1
14
Kc
Ti
Td
ξm = 1
15
Kc
Ti
Td
ξ m = 0.5
τ m Tm1 ≤ 1.0 ; OS ≤ 20% ; N not specified.
11
Kc =

1 
0.5316
,
 0.3717 +

[τm Tm ] + 0.0003414 
Km 
(
)
Ti = Tm 2.069 − 0.3692[τm Tm ] + 1.081[τm Tm ] − 0.5524[τm Tm ] ,
2
3
α = 0.6438 − 0.5056(τm Tm ) + 0.3087(τm Tm ) − 0.1201(τm Tm ) .
2
12
Kc =
3

0.05627
1 
 0.1000 +
,
2 

Km 
[(τm Tm ) + 0.06041] 
(
)
Ti = Tm 4.340 − 16.39[τm Tm ] + 30.04[τm Tm ] − 25.85[τm Tm ] + 8.567[τm Tm ] ,
2
3
4
α = 0.6178 − 0.4439(τm Tm ) − 7.575(τm Tm ) + 9.317(τm Tm ) − 3.182(τm Tm ) .
2
14
Kc =
3
4

0.6978
1 
1.389 +
,
K m 
[(τm Tm1 ) + 0.02295]2 
(
)
Ti = Tm1 0.02453 + 4.104[τm Tm1 ] − 3.434[τm Tm1 ] + 1.231[τm Tm1 ] ,
2
3
4

τ 
τ 
τ  
τ
Td = Tm1  0.03459 + 1.852 m − 2.741 m  + 2.359  m  − 0.7962  m   ,
Tm1
 Tm1 
 Tm1 
 Tm1  

2
3
α = 0.6726 − 0.1285(τm Tm1 ) − 0.1371(τm Tm1 ) + 0.07345(τm Tm1 ) ,
2
3
β = 0.8665 − 0.2679(τm Tm1 ) + 0.02724(τm Tm1 ) .
2
15
Kc =

0.5013
1 
 0.3363 +
,
K m 
[(τm Tm1 ) + 0.01147]2 
(
(0.03392 + 2.023[τ
)
] ),
Ti = Tm1 − 0.02337 + 4.858[τm Tm1 ] − 5.522[τm Tm1 ] + 2.054[τm Tm1 ] ,
Td = Tm1
2
Tm1 ] − 1.161[τm Tm1 ] + 0.2826[τm Tm1
2
m
3
3
α = 0.6678 − 0.05413(τm Tm1 ) − 0.5680(τm Tm1 ) + 0.1699(τm Tm1 ) ,
2
β = 0.8646 − 0.1205(τm Tm1 ) − 0.1212(τm Tm1 ) .
2
3
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Arvanitis et al.
(2000a).
Model: Method 1
Huang et al.
(2000).
16
ξ
τm
Kc = λ
α=
17
Kc
Comment
Direct synthesis: frequency domain criteria
0 < x1 < 1 ;
Tm1
Ti
suggested


T
x1  2ξm + m1 
x1 = 0.5 .
τm 

α = 1, β = 1, N = ∞ .
Model: Method 1
Ti
Td
(
where
17
Kc
Td
x1Tm1 
T 
(x1Tm1 τm )(2ξm + (Tm1 τm ))
 2ξm + m1  , Ti =
,

K m τm 
τm 
1  Tm1 (2ξ m τm + Tm1 ) 4ξ 2 (1 − x1 ) + x1 
 − x2
1 +
τm 
τm 2

Kc =
x2 =
16
Ti
8(1 − x1 )
2
Tm1
Km
)


Tm1
(2ξm τm + Tm1 ) 2ξ2 (1 − x1 ) + x1 Tm1 (2ξm τm + Tm1 ) + τm  ,
τm
τm


(2ξ m τ m + Tm1 ) +
2
(
)
2

1 
2  Tm1 
2

 (2ξ m τ m + Tm1 )2  ≥ 0 .
x
2
1
x
+
ξ
(
−
)
1
1
2
K 

τm 
 m


2
2Tm1ξ m + 0.4τm
T + 0.8Tm1ξ m τm
, Ti = 2Tm1ξ m + 0.4τm , Td = m1
,
K m τm
0.8Tm1ξm τm
2Tm1ξ m
0.5Tm1 2
−1, β =
− 1 , N = min[20,20Td ] ;
2
2Tm1ξ m + 0.4τ m
Tm1 + 0.8Tm1ξ m τ m
λ = 0.6 , τmξ m Tm1 ≤ 1 , A m = 2.83 ; λ = 0.7 , 1 < τmξ m Tm1 ≤ 2 , A m = 2.43 ;
λ = 0.8 , τmξ m Tm1 > 2 , A m = 2.13 .
261
b720_Chapter-03
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Handbook of PI and PID Controller Tuning Rules
262
Rule
Kc
18
Shamsuzzoha
and Lee (2007a).
Model: Method 1
Kuwata (1987);
x 1 , x 2 deduced
from graphs.
ξ m = 0. 5 .
Kuwata (1987);
x 1 , x 2 deduced
from graphs.
ξ m = 0. 7 .
Kuwata (1987);
x 1 , x 2 deduced
from graphs.
ξ m = 1. 0 .
Kc =
Td =
Td
Comment
Ti
Td
λ = x1Tm
τm Tm
Ms
1.9
0.18
0.20
0.22
0.24
0.25
0.27
2.0
0.12
0.15
0.17
0.19
0.21
0.22
0.23
1.8
0.2
0.34
0.3
0.35
0.4
0.38
0.5
0.41
0.6
0.46
0.7
0.51
0.8
0.55
Ultimate cycle
0
x 2 Tu
x 1K u
x1
0.05
0.11
x2
0.01
0.03
0.07
0.11
x2
1.9
0.28
0.29
0.32
0.34
0.38
0.41
0.43
2.0
0.25
0.26
0.28
0.30
0.33
0.36
0.38
0.9
1.0
1.25
1.5
2.0
2.5
3.0
α =1
τm Tu
x1
x2
τm Tu
x1
0.25
0.30
0.16
0.18
x 2 Tu
0.23
0.18
0.35
0.40
0
0.20
0.20
τm Tu
x1
x2
τm Tu
x1
x2
τm Tu
0.05
0.10
0.15
0.20
0.14
0.17
0.19
0.20
x 2 Tu
0.14
0.14
0.14
0.14
0.25
0.30
0.35
0.40
0
0.20
0.20
0.14
0.14
0.45
0.50
0.83
0.39
x 1K u
x1
τm Tm
Ms
0.27
0.17
0.14
0.14
x 1K u
x2
τm Tu
0.15 0.45
0.14 0.50
α =1
α =1
x1
x2
τm Tu
x1
x2
τm Tu
x1
x2
τm Tu
0.04
0.11
0.17
0.20
0.95
0.53
0.38
0.30
0.05
0.10
0.15
0.20
0.21
0.21
0.21
0.20
0.23
0.20
0.17
0.15
0.25
0.30
0.35
0.40
0.20
0.20
0.14
0.14
0.45
0.50
2
Ti
6λ2 − x 3 + x1τm − 0.5τm
, Ti = Tm1 + Tm 2 + x1 − x 2 , x 2 =
,
K m (4λ + τm − x1 )
4 λ + τ m − x1
x 3 + x1 (Tm1 + Tm 2 ) + Tm1Tm 2 −
[
Tm1 (1 − (λ Tm1 )) e − τ m
2
x1 =
Kc
Ti
N = ∞ , α = 0.6 , β = 0 . Coefficients of x1 (deduced from graph):
1.8
0.27
0.29
0.31
0.33
18
17-Mar-2009
[
4
Tm1
x 3 = Tm 2 (1 − (λ Tm 2 )) e − (τ m
2
4
]
3
2
0.167 τm − 0.5x1τm + x 3τm + 4λ3
4λ + τm − x1
− x2 ,
Ti
[
− 1 − Tm 2 (1 − (λ Tm 2 )) e − τ m
Tm 2 − Tm1
Tm 2 )
2
]
− 1 + x1Tm 2 .
4
Tm 2
],
−1
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Td
Comment
Kuwata (1987);
x 1 , x 2 deduced
from graphs.
ξ m = 1. 5 .
x 1K u
x 2 Tu
0
α =1
Kuwata (1987);
x 1 , x 2 deduced
from graphs.
ξ m = 0. 5 ;
x1
0.15
0.23
0.27
x1
x2
τm Tu
x1
x2
τm Tu
x1
x2
τm Tu
0.10
0.15
0.20
0.27
0.24
0.22
x 2 Tu
0.38
0.27
0.20
0.25
0.30
0.35
0.20
0.20
0.20
0.17
0.15
0.14
0.40
0.45
0.50
0.80
0.60
0.47
x 1K u
x2
x3
0.42 0.24 0.19
α = 1; β = 1;
0.27 0.15 0.17
N =∞.
Kuwata (1987);
x 1K u
x 1 , x 2 deduced
x1 x 2 x 3
from graphs.
ξm = 0.7 ;
0.54 0.32 0.14
α = 1; β = 1;
0.37 0.25 0.12
N =∞.
0.29 0.19 0.08
Kuwata (1987);
x 1K u
x 1 , x 2 deduced
x1 x 2 x 3
from graphs.
ξm = 1 ; α = 1 ;
0.61 0.42 0.12
β =1; N = ∞ .
0.47 0.34 0.10
0.38 0.27 0.08
Kuwata (1987);
x 1K u
x 1 , x 2 deduced
x1 x 2 x 3
from graphs.
ξ m = 1. 5 ; α = 1 ;
0.67 0.48 0.11
β =1; N = ∞ .
0.54 0.43 0.09
0.43 0.34 0.08
Oubrahim and
19
Leonard (1998).
0. 6 K u
Model: Method 1
19
α = 1 − 1.3
α = 1−
τm
Tu
x1
x 3 Tu
x2
x3
τm
Tu
x1
x2
x3
τm
Tu
x1
x2
x3
τm
Tu
0
0.50
x3
τm
Tu
0
0.50
x3
τm
Tu
0.18 0.19 0.11 0.07 0.30
0.23 0.17 0.10 0 0.36
x 2 Tu
τm
Tu
x1
x 3 Tu
x2
x3
τm
Tu
0.18 0.22 0.16 0.03 0.36 0.20 0.14
0.23 0.20 0.14 0.01 0.42
0.30 0.20 0.14 0 0.47
x 2 Tu
x 3 Tu
τm
Tu
x1
x2
x3
τm
Tu
x1
x2
0.18 0.30 0.20 0.06 0.36 0.20 0.14
0.23 0.21 0.17 0.025 0.42
0.30 0.20 0.14 0.01 0.47
x 2 Tu
x 3 Tu
τm
Tu
x1
x2
x3
τm
Tu
x1
x2
0.18 0.33 0.25 0.07 0.36 0.20 0.14 0 0.50
0.23 0.25 0.18 0.03 0.42
0.30 0.20 0.15 0.02 0.47
Tm1 = Tm 2 ;
0.5Tu
0.125Tu
0.1 < τm Tm1 < 3
 16 − K m K u
16 − K m K u
, β = 1 − 1.69
17 + K m K u
 17 + K m K u
2

 , 10% overshoot.


38
1.717
, 20% overshoot.
, β =1−
29 + 3.5K m K u
(1 + 0.121K m K u )2
263
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FA
Handbook of PI and PID Controller Tuning Rules
264
3.5.8 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s)




(
)
1
−
β
T
s
1 + b f 1s
[
]
1
−
χ
δ
d

,
F1 (s) = K c  [1 − α ] +
+
+


T
Ti s
1 + Ti s
1 + a f 1s + a f 2 s 2
d
1
+
s


N 





[
]
K (b + b f 4 s )
1
−
ϕ
T
s
1+ bf 2s
[
]
−
φ
1
d 
F2 (s) = K c  [1 − ε] +
+
− 0 f3
2

 1 + a f 3s + a f 4 s
T
Ti s
1+ a f 5s
1+ d s 

N 

R(s)
+
F1 (s)
U(s)
Y(s)
Model
−
F2 (s)
Table 27: PID controller tuning rules – SOSPD model G m (s) =
G m (s) =
Rule
Huang et al.
(1996).
1
Kc
1
Kc
K m e − sτ m
or
(1 + sTm1 )(1 + sTm 2 )
K m e − sτ m
2 2
Tm1 s + 2ξ m Tm1s + 1
Ti
Td
Comment
Minimum performance index: regulator tuning
0
Model: Method 1
Ti
Minimum IAE. Equations continued into the footnote on pp. 265 and 266;
0 < Tm 2 Tm1 ≤ 1 ; 0.1 ≤ τ m Tm1 ≤ 1 .
Kc =
1
Km

τm
τ T 
T
+ 76.6760 m 2 − 3.3397 m m2 2 
0.1098 − 8.6290
Tm1
Tm1
Tm1 

1
+
Km
−0.8058
0.6642
2.1482 

 τm 
 τm 
τ 

1.1863 m 




+ 20.3519
+ 23.1098
T 
Tm1 
Tm1 


m1 





1
+
Km

T
 − 52.0778 m 2
T

 m1




0.8405
T
− 12.1033 m 2
 Tm1



2.1123 



b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
+
τ
1 
9.4709 m
Km 
Tm1

+
T
1 
 − 19.4944 m 2
Km 
Tm1

 Tm 2

 Tm1



0.5306
Tm 2
Tm1
+ 13.6581
 τm 


 Tm1 
−0.4500
− 28.2766
 τm 


 Tm1 
τm
Tm1
−1.0781 



 Tm 2

 Tm1



1.1427 



τ m Tm 2
τm
Tm 2


2
T
1 
Tm1
Tm 1
+
− 19.1463e
+ 8.8420e
+ 7.4298e Tm1 − 11.4753 m 2  ;
Km 
τm 


2

τ 
T
τ T 
τ
Ti = Tm1  − 0.0145 + 2.0555 m + 0.7435 m 2 − 4.4805 m  + 1.2069 m m2 2 

Tm1
Tm1
Tm1 
 Tm1 



T
+ Tm1 0.2584 m 2

 Tm1


τ 
T
 + 7.7916 m  − 6.0330 m 2
Tm1

 Tm1 

τ
+ Tm1 3.9585 m
T

m1

 Tm 2

 Tm1

T
+ Tm1 7.0004 m 2
Tm1


T
 τm 
 − 2.7755 m 2

T
 Tm1
 m1 

T
+ Tm1 3.1663 m 2

 Tm1

T
τ 

 + 2.4311 m  − 0.9439 m 2
T
 Tm1
 m1 

2
3
2
T

 − 3.0626 m 2
 Tm1

3

T
+ Tm1  − 2.4506 m 2
Tm1


4
 τm 


 Tm1 
3
τ 

 − 7.0263 m 
 Tm1 




2
4
T
 τm 
 − 0.2227 m 2

T
 Tm1
 m1 
3

τ
 − 0.5494 m
T
m1




4



2
 τ  T
 τm 
 − 1.5769 m  m 2

T
 Tm1  Tm1
 m1 
5
2

 τ  T
+ Tm1 1.9228 m   m 2

 Tm1   Tm1

2



2
 τm 


 Tm1 
 Tm 2

 Tm1






5



3



4
;


2

τ 
τ T 
τ
T
x1 = Tm1 − 0.0206 + 0.9385 m + 0.7759 m 2 − 2.3820 m  + 2.9230 m m2 2 

Tm1
Tm1
Tm1 
 Tm1 



T
+ Tm1 − 3.2336 m 2

 Tm1


τ
+ Tm1 2.7095 m
T

m1

2
3

τ 
T
 + 7.2774 m  − 9.9017 m 2
Tm1
T

 m1 
 Tm 2

 Tm1
2
T

 + 6.1539 m 2
 Tm1

3
 τm 


 Tm1 
τ 

 − 11.1018 m 
 Tm1 

2



4






3



265
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
266

T
+ Tm1 10.6303 m 2
Tm1


3
τ 
 τm 
 + 5.7105 m 

T
 Tm1 
 m1 
2

τ
+ Tm1  − 7.9490 m
Tm1


 Tm 2

 Tm1

T
+ Tm1  − 4.4897 m 2
Tm1


T
 τm 
 − 7.6469 m 2

T
 Tm1
 m1 
4
2

 τ m   Tm 2




+ Tm1 2.2155
Tm1   Tm1




T
+ Tm1 0.519 m 2
Tm1


3
T

 − 6.6597 m 2
 Tm1

4
τ 

 + 8.0849 m 
 Tm1 




2
 τm 


 Tm1 
3
5
2
2
τ 

 − 2.274 m 
 Tm1 

 τ  T

 + 5.0694 m  m 2
 Tm1  Tm1

T
 τm 
 − 1.1295 m 2

T
 Tm1
 m1 
4

 τ m   Tm 2




+ Tm1 2.2875
Tm1   Tm1



 Tm 2

 Tm1
5






4
T

 + 4.1225 m 2
 Tm1

6
3



3
τ 

 − 1.6307 m 
 Tm1 

τ 

 + 0.9524 m 
 Tm1 

6
 Tm 2

 Tm1
3
2
 Tm 2

 Tm1






5



4




τ
 − 0.9321 m
T
m1

 Tm 2

 Tm1
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3
a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ; a f 5 = x1 10 .



5
;


= 0;
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Huang et al.
(1996).
2
2
Kc
Ti
Td
Comment
Kc
Ti
0
Model: Method 1
267
Minimum IAE. Equations continued into the footnote on p. 268.
0.4 ≤ ξ m ≤ 1 , 0.05 ≤ τ m Tm1 ≤ 1 .
Kc =
 τ 
1 
τ
− 35.7307 + 14.19 m + 1.4023ξ m + 6.8618ξ m  m  
K m 
Tm1
 Tm1  
+
3
τ 
τ
1 
 − 0.9773 m  + 55.5898ξ m  m
T 
T
Km 
 m1 
 m

+
−1.6624 
τ 
τ
1 

53.8651 m ξ m 2 + 11.4911ξ m 3 + 0.8778 m 
T 
Km 
Tm1

 m1 


+
−0.6951
−0.4762

τ 
 τm 
1 
− 29.8822 m 


+
53
.
535
− 16.9807ξ m 1.1197 




Km 

 Tm1 
 Tm1 


+
−2.1208 
τ 
1 

 − 25.4293ξ m 1.4622 − 0.1671ξ m 58981 + 0.0034ξ m  m 
T 
Km 

 m1 


1
+
Km
+
1
Km



0.086
τ 
− 3.3093ξ m  m 
 Tm1 
2



τm 

τ
τ
 − 25.0355 m ξ m 3.0836 − 54.9617 m ξ m 1.2103 − 0.1398e Tm1 
Tm1
Tm1




τ m ξm

ξ T 
 − 8.2721e ξm + 6.3542e Tm1 + 1.0479 m m1  ;
τm 



2

τ 
τ 
τ
Ti = Tm1 0.2563 + 11.8737 m − 1.6547ξ m − 16.1913 m  − 9.7061ξ m m 

Tm1
Tm1 
 Tm1 


3
2

τ 
τ 
 τ 
+ Tm1 3.5927ξ m 2 + 19.5201 m  − 14.5581ξ m  m  + 2.939ξ m 2  m 

 Tm1 
 Tm1 
 Tm1 

4
3

 τm  
 τm 
3






+ 50.5163ξ m 
+ Tm1 − 0.4592ξ m − 34.6273
Tm1 
Tm1  





2


τm
2  τm 
3
4




+
ξ
−
ξ
8
.
6966
6
.
9436
+ Tm1 8.9259ξ m 
m
m
Tm1 
Tm1





b720_Chapter-03
268
17-Mar-2009
Handbook of PI and PID Controller Tuning Rules
5
4
3

τ  
τ 
τ 
+ Tm1 27.2386 m  − 20.0697ξ m  m  − 42.2833ξ m 2  m  

 Tm1  
 Tm1 
 Tm1 

2


τ 
τ 
+ Tm1 8.5019 m  ξ m 3 − 12.2957 m ξ m 4 + 8.0694ξ m 5 − 2.7691ξ m 6 


 Tm1 
 Tm1 


6
5


τ 
τ 
τ
+ Tm1  − 7.7887 m  + 2.3012ξ m  m  + 4.6594 m ξ m 5 
Tm1


 Tm1 
 Tm1 


4
3
2


τ 
τ 
τ 
+ Tm1 8.8984 m  ξ m 2 + 10.2494 m  ξ m 3 − 5.4906 m  ξ m 4  ;


 Tm1 
 Tm1 
 Tm1 


2

τ 
τ 
τ
x1 = Tm1 − 0.021 + 3.3385 m + 0.185ξ m − 0.5164 m  − 0.9643ξ m m 

Tm1
Tm1 
 Tm1 


3
2

τ 
τ 
 τ 
+ Tm1  − 0.8815ξ m 2 + 0.584 m  − 12.513ξ m  m  + 1.3468ξ m 2  m 

 Tm1 
 Tm1 
 Tm1 

4
3


 τm 
 τm 
3
4






−
ξ
15
.
3014
3
.
1988
+
ξ
+ Tm1 2.3181ξ m − 5.2368
m
m
T 
Tm1 


m1 




2
5

 τm  
τm
2  τm 
3






− 2.0411
ξ m + 3.4675
+ Tm1 11.9607ξ m 
Tm1 
Tm1
Tm1  





4
3
2


 τm 
 τm 
2  τm 
3








+ Tm1 − 0.8219ξ m 
−
ξ
−
ξ
15
.
0718
1
.
8859
m
m
T 
T 
Tm1 


m1 
m1 





6
5

 τm  
 τm 
 τm  4
5








+ 0.529ξ m 
ξ m + 2.2821ξ m − 0.9315
+ Tm1 0.4841
Tm1 
Tm1 
Tm1  






4
3


 τm 
 τm 
6
2
3






ξ
7
.
1176
ξ
+
+ Tm1 − 0.6772ξ m − 1.4212
m
m
T 
Tm1 


m1 




2


 τm 
τm
4
5

;


ξ
+
ξ
0
.
5497
+ Tm1 − 2.3636
m
m
Tm1 
Tm1





α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 = 0 ;
a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ; a f 5 = x1 10 .
FA
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Madhuranthakam
et al. (2008).
Model: Method 1
3
Kc
Ti
Td
Kc
Ti
0
269
Comment
Tm1 > Tm 2 ;
τm < Tm1 + Tm 2 .
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = K c ; b f 3 = 0 , b f 4 = x1 , a f 5 = 0.1x1 .
Madhuranthakam
et al. (2008).
Model: Method 1
4
Minimum performance index: servo tuning
Tm1 > Tm 2 ;
0
Ti
τm < Tm1 + Tm 2 .
Kc
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = K c ; b f 3 = 0 , b f 4 = x1 , a f 5 = 0.1x1 .
3
0.6202  Tm1 + Tm 2 

1 +
Kc =

K m 
τm

0.9931
1.4849
τ 
T + Tm 2 

, x1 = 0.0921(τ m + Tm 2 ) m 1 + m1

Tm1 
τm

,
2
 




τm
τm
τ 
 .
 + 13.81
Ti = τm 1 + m  4.566 − 14.906
τ +T +T  

m1
m2 
 m
 τm + Tm1 + Tm 2 
 Tm1  

1.0409
4
Kc =
0.5723  Tm1 + Tm 2 

1 +

K m 
τm

, x1 = 0.0943(τm + Tm 2 )
1.6501
Ti = 0.2476


τm
(τm + Tm1 + Tm 2 )1 + Tm1 + Tm 2 
Tm1
τ
m


.
1.4636
τm  Tm1 + Tm 2 

1 +

Tm1 
τm

,
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
270
Rule
Huang et al.
(1996).
5
17-Mar-2009
Kc
Ti
Td
Comment
Kc
Ti
0
Model: Method 1
5
Minimum IAE. Equations continued into the footnote on p. 271; 0 < Tm 2 Tm1 ≤ 1 ,
0.1 ≤ τ m Tm1 ≤ 1 .
Kc =
1 
T
τm
τ T 
− 137.8937 m 2 − 122.7832 m m2 2 
7.0636 + 66.6512
K m 
Tm1
Tm1
Tm1 
+
0.0865
2.6405
T
 τm 
τ 
1 
26.1928 m 


33
.
6578
+ 3.0098 m 2
+
T 
T 
Km 
m
1
m
1
 Tm1





+
T
1 
 − 10.9347 m 2
T
Km 
 m1

+
T
1 
34.3156 m 2
Km 
Tm1

+
τ
1 
152.6392 m
Km 
Tm1

+
τ m Tm 2
τm
Tm 2

2
τ
1 
− 57.9370e Tm1 + 10.4002e Tm1 + 6.7646e Tm1 + 7.3453 m
Km 
 Tm




2.345
 τm 


 Tm1 
 Tm 2

 Tm1
T
+ 141.511 m 2
 Tm1
−0.9450






1.0570
+ 29.4068
− 70.1035
Tm 2
Tm1
 τm 


 Tm1 
− 47.9791
τm
Tm1
 Tm 2

 Tm1
0.8866






1.0309 



Tm 2 

τm 

−0.9282 



0.8148 






−0.4062 
;


2

τ 
τ T 
τ
T
Ti = Tm1 0.9923 + 0.2819 m − 0.2679 m 2 − 1.4510 m  − 0.6712 m m2 2 

Tm1
Tm1
Tm1 
 Tm1 



T
+ Tm1 0.6424 m 2

 Tm1

2
3

τ 
T
 + 2.504 m  + 2.5324 m 2
Tm1
T

 m1 

τ
+ Tm1 2.3641 m
Tm1


 Tm 2

 Tm1
T

 + 2.0500 m 2
 Tm1


τ
+ Tm1 0.8519 m
Tm1


 Tm 2

 Tm1

T
 − 1.3496 m 2
Tm1


τ
+ Tm1  − 2.4216 m
Tm1


2
 Tm 2

 Tm1
4
3
3
 τm 


 Tm1 
2



τ 

 − 1.8759 m 
 Tm1 

4



3
τ 
 τm 

 − 3.4972 m 
T
 Tm1 
 m1 
T

 − 3.1142 m 2
 Tm1

4
τ 

 + 0.5862 m 
 Tm1 

5



2
 Tm 2

 Tm1



2



b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models

T
+ Tm1 0.0797 m 2
Tm1


4
T
 τm 

 + 0.985 m 2
T
 Tm1
 m1 
2

 τ  T
+ Tm1 1.2892 m   m 2

 Tm1   Tm1




2
 τm 


 Tm1 
3
T

 + 1.2108 m 2
 Tm1




271
3



5
;


2

τ 
τ T 
τ
T
x1 = Tm1 0.0075 + 0.3449 m + 0.3924 m 2 − 0.0793 m  + 2.7495 m m2 2 

Tm1
Tm1
Tm1 
 Tm1 



T
+ Tm1 0.6485 m 2

 Tm1


τ 
T
 + 0.8089 m  − 9.7483 m 2
Tm1
T

 m1 

τ
+ Tm1 3.4679 m
Tm1


 Tm 2

 Tm1
2
3

T
+ Tm1 12.0049 m 2
Tm1



T
+ Tm1 10.0045 m 2

 Tm1

2
T

 − 5.8194 m 2
 Tm1

3
2
 Tm 2

 Tm1
5



2
3
τ 
 τm 

 + 7.0404 m 
T
 Tm1 
 m1 

 τ  T
+ Tm1 1.4294 m  m 2

 Tm1  Tm1


T
+ Tm1 1.1839 m 2
Tm1


3
τ 

T
 + 0.3520 m  − 6.3603 m 2
T
Tm1
 m1 


T
+ Tm1  − 3.2980 m 2

 Tm1

4

T
 − 6.9064 m 2

 Tm1
5
 τm 
T

 + 1.7087 m 2
T
 m1 
 Tm1
2
τ 

 − 1.0884 m 
 Tm1 

τ 
 τm 

 − 1.4056 m 
T
 Tm1 
 m1 
4
 τm 


 Tm1 
2
 Tm 2

 Tm1



4



2

τ
 − 3.7055 m
T
m1

 τm 


 Tm1 






3



3



5

τ 
 + 1.7444 m 

 Tm1 
4
6



 Tm 2

 Tm1



2



3
3
2
4

 τ m   Tm 2 
 τ m   Tm 2 
τm









+ Tm1 − 1.2817
  T  − 2.1281 T   T  + 1.5121 T
T

m1
 m1   m1 
 m1   m1 

α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 = 0 ;
a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ; a f 5 = x1 10 .



4

τ 
 + 0.0471 m 

 Tm1 
6
 Tm 2

 Tm1
 Tm 2

 Tm1



5
;


b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
272
Rule
Huang et al.
(1996).
6
17-Mar-2009
6
Kc
Ti
Td
Comment
Kc
Ti
0
Model: Method 1
Minimum IAE. Equations continued into the footnote on p. 273. 0.4 ≤ ξ m ≤ 1 ;
0.05 ≤ τ m Tm1 ≤ 1 .
Kc =
1
Km

 τ 
τ
− 8.1727 − 32.9042 m + 31.9179ξm + 38.3405ξm  m 
Tm1

 Tm1 
−1.9009
0.1571
1.2234 
 τm 
 τm 
τ 
1 
 + 0.2079 m 





29
.
3215
35
.
9456
+
+
T 
T 
T 
Km 

m1 
m1 
m1 





1
0.1311
1.9814
1.737
+
− 21.4045ξ m
+ 5.1159ξ m
− 21.9381ξ m
Km
+
[
]
+
−0.1303
1.2008 
 τm 
τ 
1 
 − 17.7448ξ m  m 



26
.
8655
+
ξ
m

T 
Km 

 Tm1 
 m1 


+
τm 
τ
τ
1 
− 52.9156 m ξ m 1.1207 − 22.4297 m ξ m 0.3626 − 3.3331e Tm1 
Km 
Tm1
Tm1



1
+
Km
τ m ξm

ξ T 
8.5175e ξm − 1.5312e Tm1 + 0.8906 m m1  [Huang (2002)];
τm 



2

τ 
τ 
τ
Ti = Tm1 1.1731 + 6.3082 m − 0.6937ξ m + 8.5271 m  − 24.7291ξ m m 

Tm1
Tm1 
 Tm1 


3
2


τ 
τ 
+ Tm1  − 6.7123ξ m  m  + 7.9559ξ m 2 − 32.3937 m  + 5.5268ξ m 6 


 Tm1 
 Tm1 


4
3


 τm 
 τm 
2  τm 








166
.
9272
36
.
3954
+
ξ
+
ξ
+ Tm1 − 27.1372
m
m
 T 
T 
Tm1 

m1 
m1 





2


τm
2  τm 
3
3
4




−
ξ
−
ξ
+
ξ
22
.
6065
1
.
6084
29
.
9159
+ Tm1 − 94.8879ξ m 
m
m
m
Tm1 
Tm1





5
4
3

 τm 
 τm 
2  τm 








− 93.8912ξ m 
− 84.3776ξ m 
+ Tm1 49.6314
Tm1 
Tm1 
Tm1  






2


 τm  4
 τm 
3
5






ξ
−
ξ
25
.
1896
19
.
7569
ξ
−
+ Tm1 110.1706
m
m
m
T 
Tm1 


m1 




b720_Chapter-03
17-Mar-2009
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
6
5
4


τ 
τ 
τ 
+ Tm1  − 12.4348 m  − 11.7589ξ m  m  + 68.3097 m  ξ m 2 


 Tm1 
 Tm1 
 Tm1 


3
2


τ 
τ 
τ
+ Tm1  − 17.8663 m  ξ m 3 − 22.5926 m  ξ m 4 + 9.5061 m ξ m 5  ;
Tm1


 Tm1 
 Tm1 


2

τ 
τ 
τ
x1 = Tm1 0.0904 + 0.8637 m − 0.1301ξ m + 4.9601 m  + 14.3899ξ m m 

Tm1
Tm1 
 Tm1 


3
2


τ 
τ 
+ Tm1 0.7170ξ m 2 − 12.5311 m  − 42.5012ξ m  m  + 17.0952ξ m 4 


 Tm1 
 Tm1 


4

 τm  
2  τm 
3






− 6.9555ξ m − 12.3016
+ Tm1 − 21.4907ξ m 
Tm1 
Tm1  





3
2


 τm 
τm
2  τm 
3






+
ξ
7
.
5855
19
.
1257
+
ξ
+ Tm1 102.9447ξ m 
m
m
T 
Tm1 
Tm1


m1 




5
4
3

 τm 
 τm 
2  τm 








− 110.0342ξ m 
− 17.2130ξ m 
+ Tm1 10.8688
Tm1 
Tm1 
Tm1  






2


 τm  4
 τm 
3
5
6






ξ
−
ξ
+
ξ
16
.
7073
16
.
2013
5
.
4409
ξ
−
+ Tm1 50.6455
m
m
m
m
T 
Tm1 


m1 




6
5
4


 τm 
 τm 
 τm 
2








ξ
10
.
9260
29
.
4445
+
−
ξ
+ Tm1 − 0.0979
m
m
T 
T 
Tm1 


m1 
m1 





3
2


 τm 
 τm 
τm
3
4
5

;




ξ
+
ξ
24
.
1917
6
.
2798
ξ
−
+ Tm1 21.6061
m
m
m
T 
Tm1 
Tm1


m1 




α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 = 0 ;
a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ; a f 5 = x1 10 .
FA
273
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
274
Rule
Majhi and
Atherton (1999),
Atherton and Boz
(1998).
Model: Method 1
Kc
7
Ti
Comment
Td
Direct synthesis: time domain criteria
Coefficient
x3
0
values deduced
1
1
1
+
+
from graphs.
τm Tm 2 Tm1
Kc
α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ;
φ = 0 ; ϕ = 1 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = x1 ; b f 3 = 1 ,
bf 4 = x 2 , a f 5 = 0 .
x3
Minimum ISE –
servo – Atherton
and Boz (1998).
1.2
2.6
4.5
6.4
1.8
4.2
6.9
10.0
Minimum ITSE – 1.8
servo – Majhi
4.2
and Atherton
6.9
(1999).
10.0
Minimum ISTSE 2.3
– servo –
5.2
Atherton and Boz 8.4
(1998).
12.4
Minimum ISTSE 2.3
– servo – Majhi
5.2
and Atherton
8.7
(1999).
12.4
7
x4
x5
x6
x3
x4
x5
x6
0.58
0.91
0.89
0.98
0.17
0.22
0.25
0.26
0.17
0.22
0.25
0.26
0.08
0.11
0.14
0.13
0.08
0.11
0.12
0.13
-0.81
-0.55
-0.30
-0.22
-0.31
-0.16
-0.11
-0.08
-0.29
-0.16
-0.11
-0.08
-0.16
-0.10
-0.07
-0.04
-0.16
-0.10
-0.06
-0.04
2.36
2.49
2.18
2.23
1.08
0.96
0.95
0.92
1.05
0.96
0.95
0.92
0.71
0.65
0.68
0.63
0.69
0.65
0.62
0.64
8.5
10.8
13.3
16.0
13.0
16.2
20.3
24.0
13.0
16.8
20.3
1.02
1.03
1.02
1.00
0.28
0.31
0.29
0.30
0.28
0.27
0.29
-0.16
-0.14
-0.12
-0.09
-0.07
-0.06
-0.04
-0.04
-0.06
-0.05
-0.04
2.18
2.16
2.13
2.10
0.94
0.99
0.93
0.95
0.94
0.90
0.97
16.5
20.4
25.2
31.2
16.5
20.4
25.2
0.14
0.15
0.15
0.14
0.14
0.15
0.15
-0.04
-0.03
-0.02
-0.02
-0.03
-0.03
-0.02
0.64
0.65
0.64
0.61
0.64
0.67
0.65
Equations developed from information provided by Majhi and Atherton (1999).
(T T + τmTm1 + τmTm 2 )3 , x = 1 1 − (Tm1Tm 2 + τmTm1 + τmTm 2 )3 x  ,
K c = x 4 m1 m 2

1
5
2
2
2
K m 
τm Tm1 Tm 2
K m τm 2Tm12Tm 2 2

 (T T + τ T + τ T )2

m m1
m m2
 m1 m 2
x 6 − (Tm1 + Tm 2 − τ m ) 
τ m Tm1Tm 2

 τm + Tm 2
≤ 0. 8 .
x2 = 
,
3
Tm1
(T T + τm Tm1 + τ m Tm 2 ) x 5


1 − m1 m 2


τ m 2 Tm12 Tm 2 2


b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
8
Hansen (1998).
Model: Method 1
Kc
Ti
Td
Kc
Ti
0
275
Comment
χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 ,
a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = K c − (1 K m ) , b f 4 = x 2 , a f 5 = 0 .
9
Ti
Td
Kc
Arvanitis et al.
10
Ti
Td
Kc
(2000a).
Model: Method 1 α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; b = 0 , a = T , a = T T ;
f1
f1
i
f2
i d
ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 0 .
8
Kc =
[
[
2 Tm1Tm 2 + (Tm1 + Tm 2 )τm + 0.5τm 2
]
3
9K m Tm1Tm 2 τm + 0.5(Tm1 + Tm 2 )τm 2 + 0.167 τm 3
]
2
,
Ti = 1.54 x1 , α = 0.802 , nearly zero overshoot; Ti = 1.27 x1 , α = 0.711 , nearly
minimum IAE; Ti = 1.75x1 , α = 0.857 , conservative tuning;
x1 =
x2 =
9
10
[
3 Tm1Tm 2 τm + 0.5(Tm1 + Tm 2 )τm + 0.167 τm
[T
2
m1Tm 2 + (Tm1 + Tm 2 )τ m + 0.5τ m
[
[
2
]
2 Tm1Tm 2 + (Tm1 + Tm 2 )τm + 0.5τm 2
3
]
],
2
3K m Tm1Tm 2 τm + 0.5(Tm1 + Tm 2 )τm 2 + 0.167 τm 3
]−
Tm1 + Tm 2 + τm
.
Km
Kc =
x1Tm1 
T 
Tm1τm
 2ξm + m1  , 0 < x1 < 1 ; suggested x1 = 0.4 ; Td =
;
τm 
K m τm 
x1 ( 2ξ m τm + Tm1 )
Ti =

 1

τm 
x + 2ξ 2  − 1 + 2ξ
2  2
x
x2
 1 

Kc =
1 (2ξTCL 2 + τm )(2ξm τm + Tm1 )Tm1 − τmTCL 2 2
,
K m τm
TCL 2 2 + τm (2ξTCL 2 + τm )

 1

 1

 − 1 x 2 + ξ2  − 1   , with
x

x
 
 1 
 1  
2
2

 1
τm
τm
1+
+ ξ2  − 1 ≥ 0 , x 2 = 1 +
.
x1Tm1 (2ξ m τm + Tm1 )
x1Tm1 (2ξm τm + Tm1 )
 x1 
(2ξTCL 2 + τm )(2ξm τm + Tm1 )Tm1 − τmTCL 2 2 ,
τm 2 + τm 2Tm1 (2ξ m τm + Tm1 )
2
2
Tm1 [TCL 2 + τm (2ξTCL 2 + τm )]
Td =
.
(2ξTCL 2 + τm )(2ξm τm + Tm1 )Tm1 − τmTCL 22
Ti =
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
276
Rule
Kc
Ti
Td
Kc
Ti
0
11
Huang et al.
(2000).
Model: Method 1
Comment
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ;
a f 5 = x1 x 2 , x 2 not specified.
Åström and
Hägglund (2006),
pp. 252, 265.
Model:
Method 24
11
Kc = λ
12
Kc
Direct synthesis: frequency domain criteria
Ti
Td
M s = M t = 1.4
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ;
ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 ; a f 3 = x1 , Td 30 ≤ x1 ≤ Td 3 ;
2
a f 4 = 0 . 5 x1 ; K 0 = 0 .
2
2Tm1ξ m + 0.4τm
T + 0.8Tm1ξ m τm
, Ti = 2Tm1ξ m + 0.4τm , x1 = m1
;
K m τm
0.8Tm1ξm τm
λ = 0.5, τm ξ m Tm1 ≤ 1 , λ = 0.6,1 < τm ξm Tm1 ≤ 2 , λ = 0.7, τmξ m Tm1 > 2 , servo;
λ = 0.6, τm ξ m Tm1 ≤ 1 , λ = 0.7,1 < τm ξ m Tm1 ≤ 2 , λ = 0.8, τm ξm Tm1 > 2 , regulator.
12
1 
Tm1
T + 0.5x1
T (T + 0.5x1 ) 
+ 0.18 m 2
+ 0.02 m1 m 2
0.19 + 0.37
,
Km 
τm + 0.5x1
τm + 0.5x1
τm + 0.5x1 
Tm1
T + 0.5x1
T (T + 0.5x1 )
0.19 + 0.37
+ 0.18 m 2
+ 0.02 m1 m 2
τm + 0.5x1
τm + 0.5x1
τm + 0.5x1
Ti =
,
Tm1 (Tm 2 + 0.5x1 )
Tm 2 + 0.5x1
0.48
Tm1
0
.
0012
0
.
0007
+
+ 0.03
−
τm + 0.5x1
(τm + 0.5x1 )3
(τm + 0.5x1 )2
(τm + 0.5x1 )2
Kc =
Td =
Tm1 (Tm 2 + 0.5x1 )
τm + 0.5x1
Tm1
Tm 2 + 0.5x1
Tm1 (Tm 2 + 0.5x1 )
0.19 + 0.37
+ 0.18
+ 0.02
τ m + 0. 5 x 1
τm + 0.5x1
τ m + 0. 5 x 1
0.29(τm + 0.5x1 ) + 0.16Tm1 + 0.2(Tm 2 + 0.5x1 ) + 0.28
 Tm1 + Tm 2 + 0.5x1 


T +T +τ +x .
m2
m
1
 m1
b720_Chapter-03
17-Mar-2009
FA
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277
3.6 SOSPD Model with a Zero
K (1 + sTm 3 )e −sτ m
K m (1 + Tm 3 s )e − sτ m
G m (s) = m
or G m (s) =
2
(1 + sTm1 )(1 + sTm 2 )
Tm1 s 2 + 2ξ m Tm1s + 1
or G m (s) =
K m (1 − Tm 4 s )e − sτ m
K m (1 − Tm 4 s )e −sτ m
or G m (s) =
2
(1 + sTm1 )(1 + sTm 2 )
Tm1 s 2 + 2ξ m Tm1s + 1

1 

3.6.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
R(s)
+

1 

K c 1 +
Ti s 

E(s)
−
U(s)
Model
Y(s)
Table 28: PI controller tuning rules – SOSPD model with a zero
K m (1 − Tm 4s )e −sτ m
K m (1 + Tm3s )e −sτ m
K m (1 − Tm 4s )e −sτ m
G m (s ) =
or
or
2 2
(1 + sTm1 )(1 + sTm 2 ) Tm12s 2 + 2ξmTm1s + 1
Tm1 s + 2ξ m Tm1s + 1
Rule
Chien et al. (2003).
Model: Method 1
Kc
1
Kc
2
Kc
Ti
Direct synthesis: time domain criteria
Tm1
Tm1 > Tm 2
Ti
Pomerleau and Poulin 1
Tm1
(2004).
K m Tm1 + Tm 4 + τm
Model: Method 3
1
Kc =
Comment
1.5Tm1
OS < 10%;
Tm1 = Tm 2 .
Tm1
,
K m (1.414x1 + Tm 4 + τm )
x1 = 0.707Tm 2 + 0.5 4(Tm 4 τm + Tm 2Tm 4 + Tm 2 τm ) + 2Tm 2 .
2
2
Kc =
Ti
,
K m (2.414x1 + Tm 4 + τm − Ti )
4.828ξ m Tm1x1 + 2ξ mTm1 (Tm 4 + τm ) + Tm 4 τm − 2.414 x1
,
2ξ m Tm1 + Tm 4 + τm
2
Ti =
x1 obtained numerically from the solution of a cubic equation.
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Rule
Kc
Direct synthesis: frequency domain criteria
Ti
Tm1 > Tm 2
3
Kc
Khodabakhshian and
Golbon (2004).
Marchetti and Scali
(2000).
Model: Method 1
Comment
Ti
Also Desbiens (2007), p. 90. Model: Method 1
Robust
Negative zero.
2ξ m Tm1 + 0.5τm
2ξ mTm1 + 0.5τm
K m (2λ + τm )
2ξm Tm1 + 0.5τm
K m (2λ + τm + 2Tm 4 )
Positive zero.
2ξ m Tm1 + 0.5τm
Ultimate cycle
Luyben (2000).
Tm 4 Tm1 values.
x 1 values.
3
Kc =
Ti
Km
K u x1 K m
Model: Method 1
Tu x 2
0.2
0.4
0.8
1.6
3.9
3.3
2.8
3.0
τm Tm 4 = 0.2
3.8
3.4
3.0
2.9
τm Tm 4 = 0.4
4.1
3.6
3.2
3.1
τm Tm 4 = 0.6
4.3
3.8
3.4
3.2
τm Tm 4 = 0.8
4.8
4.3
3.7
3.3
τm Tm 4 = 1.0
5.2
4.6
3.9
3.4
τm Tm 4 = 1.2
5.5
5.0
4.4
3.7
τm Tm 4 = 1.4
6.0
5.4
4.7
3.8
τm Tm 4 = 1.6
(Tm1Tm 2 )2 ω6 + (Tm12 + Tm 22 )ω4 + ω2
(TiTm 4 )2 ω4 + (Ti 2 + Tm 4 2 )ω2 + 1
 1 − 10−0.1M max
cos −1 

2

4
, with ω given by solving

 = 0.5π + tan −1 (Ti ω) − tan −1 (Tm 4ω) − tan −1 (Tm 2ω)


− tan −1 (Tm1 ω) − τ m ω with recommended M max = 0dB (Khodabakhshian
and Golbon, 2004); ω given by solving, for φm = 58.13 (Desbiens, 2007):
1.015 = 0.5π + tan −1 (Ti ω) − tan −1 (Tm 4ω) − tan −1 (Tm 2ω) − tan −1 (Tm1 ω) − τ m ω ;
2

T 
T  τ
τ
Ti = Tm1 1 + 0.175 m + 0.3 m 2  + 0.2 m 2  , m < 2 ;
Tm1
Tm1
Tm1
 Tm1 


2

 Tm 2 
T  τ
τm



Ti = Tm1  0.65 + 0.35
+ 0.2 m 2  , m > 2 .
+ 0.3

Tm1
Tm1
Tm1
 Tm1 


4
x 2 = (0.5 + 1.56[Tm 4 Tm1 ]) + (3.44 − 1.56[Tm 4 Tm1 ])(τm Tm 4 ) ; Tm1 = Tm 2 .
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

1
+ Td s 
3.6.2 Ideal PID controller G c (s) = K c 1 +
 Ti s

R(s)
E(s)
−
+


1
K c 1 +
+ Td s 
T
s
i


U(s)
Y(s)
Model
Table 29: PID controller tuning rules – SOSPD model with a zero
K m (1 + sTm3 )e −sτ m
K m (1 − sTm 4 )e −sτ m
K m (1 − sTm 4 )e −sτ m
G m (s ) =
or
or
(1 + sTm1 )(1 + sTm 2 ) Tm12s 2 + 2ξmTm1s + 1
Tm1 2 s 2 + 2ξ m Tm1s + 1
Rule
Kc
Wang et al.
(1995a).
1
Ti
Td
Comment
Minimum performance index: servo tuning
p 0 q 2 − p 2 p1 Model: Method 1
p
1 p 0 q 1 − p1
q1 − 1
−
1
r
p0
p 0 q 1 − p1 p 0
Km
p0 2
p 0 = Tm1 + Tm 2 + Tm 4 + τm , p1 = Tm1Tm 2 + 0.5τm (Tm1 + Tm 2 ) − 0.5τm Tm 4 ,
p 2 = 0.5Tm1Tm 2 τ m , q 1 = Tm1 + Tm 2 + 0.5τ m , q 2 = Tm1Tm 2 + 0.5τ m (Tm1 + Tm 2 ) ,
r = δ0 + δ1
T 
T    τ 
T 
 T  τ
τm
+ δ 2  m 2  + δ3  m 4  + δ4  m  + δ7  m 2  + δ9  m 4  m
Tm1
 Tm1 
 Tm1    Tm1 
 Tm1 
 Tm1  Tm1
 τ 
 τ 
T 
T 
 T  T
 T  T
+ δ7  m  + δ5  m 2  + δ8  m 4  m 2 + δ9  m  + δ8  m 2  + δ6  m 4  m 4 ;
  Tm1 
 Tm1 
 Tm1 
 Tm1  Tm1   Tm1 
 Tm1  Tm1
Minimum IAE: δ 0 = 3.5550 , δ1 = −3.6167 , δ 2 = 2.1781 , δ 3 = −5.5203 , δ 4 = 1.4704 ,
δ 5 = −0.4918 , δ 6 = 2.5356 , δ 7 = −0.4685 , δ 8 = −0.3318 , δ 9 = 1.4746 .
Minimum ISE: δ 0 = 3.9395 , δ1 = −3.2164 , δ 2 = 1.6185 , δ 3 = −5.8240 , δ 4 = 1.0933 ,
δ 5 = −0.6679 , δ 6 = 2.5648 , δ 7 = −0.2383 , δ 8 = −0.0564 , δ 9 = 1.3508 .
Minimum ITAE: δ 0 = 3.2950 , δ1 = −3.4779 , δ 2 = 2.5336 , δ 3 = −5.5929 ,
δ 4 = 1.4407 , δ 5 = −0.3268 , δ 6 = 2.7129 , δ 7 = −0.5712 , δ 8 = −0.5790 , δ 9 = 1.5340 .
279
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Rule
Kc
Chidambaram
(2002), p. 157.
Model: Method 1
2
Kc
3
Kc
4
Kc
5
Ti
Comment
Td
Direct synthesis: time domain criteria
‘Smaller’
Tm1Tm 2
Tm1 + Tm 2
τm Tm 4 .
Tm1 + Tm 2
‘Larger’
τm Tm 4 .
0.5
2ξ m Tm1
Tm1
ξm
Kc
‘Smaller’
τm Tm 4 .
‘Larger’
τm Tm 4 .
Direct synthesis: frequency domain criteria
Wang et al.
(2000b), (2001a).
Model: Method 2
6
2ξ m Tm1
Kc
Tm1
2
M s = 1.5
Robust
Marchetti and 2ξ mTm1 + 0.5τm
Scali (2000).
K m (2λ + τm )
Model: Method 1
7
Kc
2
3
4
5
6
Kc =
Kc =
1.571(Tm1 + Tm 2 )
2ξ m Tm1 + 0.5τm Tm1 (Tm1 + ξm τm )
2ξm Tm1 + 0.5τm
(
)
.
(
)
.
)
.
)
.
K m τm 1 + 3.142 Tm 4 τm 2
2.358(Tm1 + Tm 2 )
K m τm 1 + 4.712 Tm 4 τm 2
Negative zero.
Positive zero.
2
Kc =
3.142ξ mTm1
(
K m τm 1 + 3.142 Tm 4 τm 2
2
Kc =
Kc =
4.716ξ m Tm1
(
K m τm 1 + 4.712 Tm 4 τm 2
1
cos θ
2ξ m Tm1ω
; ω is determined by solving the
M s cos ωτ m − tan −1 (x1ω) K x 2ω2 + 1
m
1
[
]
 cos θ 
 = tan −1 (x1ω) − ωτ m − 0.5π , with
equation: tan −1 

 M s − sin θ 
(
(
)
)
 x τ ω2 − 1 cos(ωτ m ) − ωτ m sin (ωτ m ) 
θ = tan −1  1 m 2
 ; x1 = Tm3 or −Tm 4 , as appropriate.
 x1τm ω − 1 sin (ωτ m ) + ωτ m sin (ωτ m ) 
7
Kc =
2ξm Tm1 + 0.5τm
.
K m (2λ + τm + 2Tm 4 )
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Lee et al. (2006). K (T + τ )
m CL
m
Model: Method 1
9
Kc
8
8
Kc =
Ti
Td
Recommended
τm ≤ TCL ≤ 2τm .
Ti
Td
Recommended
τ m ≤ λ ≤ 2τm .
2
Desired closed loop transfer function =
9
Comment
0 . 5τ m
T
0. 5 τ m
τ (0.1667 τm − 0.5Tm 3 )
, Td =
+ m1 + m
− Tm3 ;
TCL + τm
Ti
Ti (TCL + τm )
TCL + τm
2
Ti = 2ξm Tm1 − Tm3 +
Td
Ti
281
2
2
e − sτ m
.
TCLs + 1
2
T (τ − λ ) + 0.5τm
Ti
, Ti = 2ξm Tm1 + m 4 m
,
K m (λ + τm + 2Tm 4 )
λ + τm + 2Tm 4
Tm 4 (τm − λ ) + 0.5τm
T
τ (0.1667 τm + 0.5Tm 4 )
+ m1 − m
;
λ + τm + 2Tm 4
Ti
Ti (λ + τm + 2Tm 4 )
2
Td =
Desired closed loop transfer function =
(1 − sTm 4 )e −sτ .
(1 + sTm 4 )(1 + sλ )
m
2
2
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3.6.3 Ideal controller in series with a first order lag

 1
1
+ Td s 
G c (s) = K c 1 +
Ti s

 Tf s + 1
R(s)
+
E(s)
U(s)

 1
1
K c 1 +
+ Td s 
 Ti s
 Tf s + 1
−
Y(s)
Model
Table 30: PID controller tuning rules – SOSPD model with a zero
K m (1 + sTm3 )e −sτ m
K m (1 + Tm3s )e −sτ m
K m (1 − sTm 4 )e −sτ m
G m (s ) =
or
or
(1 + sTm1 )(1 + sTm 2 ) (1 + sTm1 )(1 + sTm 2 )
Tm1 2 s 2 + 2ξ m Tm1s + 1
Rule
Chien et al.
(2003).
Model:
Method 1;
Tf = Tm3 .
Kc
Ti
0.829ξm Tm1
K m τm
1
Kc
2
Kc
Pomerleau and
Tm1
1
Poulin (2004).
K m Tm1 + τ m
Model: Method 3
Tm1
Tm1
0
Ti
0
1.5Tm1
0
(
Tm1 > Tm 2
Tm1 = Tm 2 ,
Tf = Tm 3 ;
OS < 10%.
)
2
2
, x 1 = 0.707T m 2 + 0.5 2 T m 2 + τ m + 4T m 2 τ m .
1
Kc =
2
Kc =
Ti
,
K m (2.414 x1 + τm − Ti )
Ti =
4.828ξm Tm1x1 + 2ξ m Tm1τm + 0.5τm − 2.414 x1
,
2ξm Tm1 + τm
K m (1.414 x1 + Tm3 + τm )
Comment
Td
Direct synthesis: time domain criteria
2ξ m Tm1
Tm1
2ξ m
2
2
x1 is obtained numerically from the solution of a cubic equation.
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Li et al. (2004).
Model: Method 1
Kc
Ti
Td
Kc
Tm1 + Tm 2
Tm1Tm 2
Tm1 + Tm 2
2ξm Tm1
K m (λ + τm )
2ξ m Tm1
3
283
Comment
Robust
Bialkowski
(1996).
Model: Method 1
Chen et al.
(2006).
Model: Method 1
Chen et al.
(2006).
3
Kc =
4
Kc
0.6K u
Tm1 + Tm 2
Tm1
2ξ m
Tm1Tm 2
Tm1 + Tm 2
Ultimate cycle
0.5Tu
0.125Tu
Tf = Tm3 or
Tm 4 ; λ not
defined.
τm ≤ λ ≤ 10τm
Tf = 0.0125Tu
2
T
− Tm 4 τm
Tm1 + Tm 2
;
, Tf = CL 2
K m (2TCL 2 + Tm 4 + τm )
2TCL 2 + Tm 4 + τm
desired closed loop transfer function = −
4
Kc =
(λ − τ m )Tm 4
Tm1 + Tm 2
.
, Tf =
K m (λ + 2Tm 4 + τ m )
λ + 2Tm 4 + τ m
(1 − sTm 4 )e−sτ
(TCL 2s + 1)2
m
.
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


Td s 
1

+
3.6.4 Controller with filtered derivative G c (s) = K c 1 +
 Ti s 1 + Td s 


N 

R(s)
+


Td s
1
+
K c 1 +
T

Ti s
1+ d

N

E(s)
−

 U(s)


s

Y(s)
Model
Table 31: PID controller tuning rules – SOSPD model with a zero
K (1 + sTm 3 )e − sτ m
K (1 − sTm 4 )e −sτ m
K m (1 + sTm3 )e −sτ m
or
or m
or
G m (s) = m
2 2
(1 + sTm1 )(1 + sTm 2 ) Tm1 s + 2ξmTm1s + 1 (1 + sTm1 )(1 + sTm 2 )
K m (1 − sTm 4 )e −sτ m
Tm12s 2 + 2ξ mTm1s + 1
Rule
Kc
Chien et al.
(2003).
Model: Method 1
0.829ξm Tm1
K m τm
2
Kc
Ti
Td
Comment
Direct synthesis: time domain criteria
1
2ξ m Tm1 − Tm 3
Td
2ξ m Tm1 − 0.1Td
Td
N = 10
2
1
Td =
Tm1
Tm1
− Tm3 , N =
−1 .
2ξ m Tm1 − Tm3
Tm 3 (2ξ mTm1 − Tm3 )
2
Kc =
Tm1
2
, x1 = 0.1Td + 0.5 4Tm 4 τm + 0.4Td Tm 4 + 0.4Td τm + 0.04Td ;
K m (2 x1 + Tm 4 + τm )
2
Td = Tm1 10ξ m − 3.02 11ξm − 1  , ξm > 0.302 .


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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Td
Comment
Td
Tm1 > Tm 2 > Tm 3
285
Robust
3
Kc
Chien (1988).
ξ
2
T
m m1 − Tm 3
Model: Method 1
K m (λ + τ m )
Ti
2ξ m Tm1 − Tm 3
4
Td
5
Kc
Ti
Td
6
Kc
Ti
Td
Tm1 > Tm 2 > Tm 4
N = 10; λ = [Tm1 , τ m ] (Chien and Fruehauf, 1990).
3
Kc =
4
Td =
5
Tm1 + Tm 2 − Tm3
T T − (Tm1 + Tm 2 − Tm3 )Tm3
.
, Ti = Tm1 + Tm 2 − Tm3 , Td = m1 m 2
K m (λ + τm )
Tm1 + Tm 2 − Tm 3
Tm2 1 − (2ξ m Tm1 − Tm 3 )Tm3
.
2ξ m Tm1 − Tm3
Tm 4 τm
Tm 4 τm
λ + Tm 4 + τm
, Ti = Tm1 + Tm 2 +
,
K m (λ + Tm 4 + τm )
λ + Tm 4 + τm
Tm1 + Tm 2 +
Kc =
Td =
6
Tm 4 τm
Tm1Tm 2
+
.
λ + Tm 4 + τm Tm1 + Tm 2 + (Tm 4 τm (λ + Tm 4 + τm ))
Tm 4 τm
Tm 4 τm
λ + Tm 4 + τm
, Ti = 2ξm Tm1 +
,
K m (λ + Tm 4 + τm )
λ + Tm 4 + τm
2ξ m Tm1 +
Kc =
Td =
2
Tm 4 τm
Tm1
+
.
λ + Tm 4 + τm 2ξ m Tm1 + (Tm 4 τm (λ + Tm 4 + τm ))
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


1  1 + Td s 


3.6.5 Classical controller G c (s) = K c 1 +
 Ti s  1 + Td s 
N 

R(s)
+





1  1 + Td s  U(s)

K c 1 +
Td 
Ti s 

s
1 +
N 

E(s)
−
Y(s)
Model
Table 32: PID controller tuning rules – SOSPD model with a zero
K (1 + sTm 3 )e − sτ m
K (1 − sTm 4 )e −sτ m
or m
G m (s) = m
(1 + sTm1 )(1 + sTm 2 ) (1 + sTm1 )(1 + sTm 2 )
Rule
Kc
Zhang (1994).
Model: Method 1
Tm1 K m τm
Chien et al.
(2003).
Model:
Method 1;
Tm1 > Tm 2 .
0.414Tm1
K m τm
Poulin et al.
(1996).
Model: Method 1
Kc =
Comment
Td
Direct synthesis: time domain criteria
Tm1
Tm 2
‘Small’ τ m ; ‘good’ servo response. N = Tm 2 Tm 3 or N = ∞ .
1
Tm1
Tm 2
N = Tm 2 Tm3
N = 10
Kc
Direct synthesis: frequency domain criteria
Tm1
Tm1
Tm 2
N = Tm 2 Tm 3
K m (Tm1 + τ m )
Minimum φ m = 90 0 ; φ m = 60 0 , τ m = Tm1 .
2
1
Ti
Kc
Tm1
Tm 2
Minimum
φ m = 650 .
Tm1
,
K m (2 x1 + Tm 4 + τm )
2
x1 = 0.1Tm 2 + 0.5 4Tm 4 τm + 0.4Tm 2Tm 4 + 0.4Tm 2 τm + 0.04Tm 2 .
2
Kc =
T (T + 2Tm 4 )
Tm1
, φ m = 56 0 when Tm 4 = Tm1 = τ m .
, N = m 2 m1
K m (Tm1 + 2Tm 4 + τ m )
Tm1Tm 4
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
O’Dwyer
(2001a).
Model: Method 1
Kc
Ti
Td
x1Tm1 K m τm
Tm1
Tm 2
Comment
A m = 0.5π x 1 ; φ m = 0.5π − x 1 ; N = Tm 2 Tm 4 ; sample result:
0.785Tm1
K mτm
Tm1
Tm 2
A m = 2.0;
φ m = 450 .
Robust
Chien (1988).
Model: Method 1
Chien (1988).
Model: Method 1
Chien (1988).
Model:
Method 1;
Tm1 > Tm 2 > Tm 4 .
Chien (1988).
Model: Method 1
Bialkowski
(1996).
Model: Method 1
Tm1 > Tm 2 ;
N = 10;
[Tm1 , τ m ]
λ
∈
Tm1
(Chien
and
Tm1
Tm 2
K m (λ + τ m )
Fruehauf, 1990).
Tm 2
1.1(Tm1 − Tm 3 ) Tm1 > Tm 2 > Tm 3
Tm 2
K m (λ + τ m )
N = 11;
[Tm1 , τ m ]
λ
∈
Tm1
1.1(Tm 2 − Tm 3 )
Tm1
(Chien
and
K m (λ + τ m )
Fruehauf, 1990).
Tuning rules converted to this equivalent controller architecture.
N = 10;
Tm 2
Tm1
Tm 2
[Tm1 , τ m ]
λ
∈
K m (λ + τ m )
(Chien and
Tm1
Tm 2
Tm1
Fruehauf, 1990).
K m (λ + τ m )
Tm 2
K m (λ + τ m )
Tm 2
Tm1
Tm 2
K m (λ + τ m )
Tm 2
3
Td
Tm1
K m (λ + τ m )
Tm1
4
Td
N = 11; λ ∈ [T, τ m ] , T = dominant time constant (Chien and Fruehauf,
1990). Tuning rules converted to this equivalent controller
architecture.
Tm1
Tm 2
Tm1 > Tm 2
Tm1
K m (λ + τm )
3
Td = 1.1Tm1 +
1.1Tm 4 τm
.
λ + Tm 4 + τm
4
Td = 1.1Tm 2 +
1.1Tm 4 τ m
.
λ + Tm 4 + τ m
N = Tm 2 Tm3 or N = Tm 2 Tm 4 .
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3.6.6 Generalised classical controller


Td s
1
+
G c (s) = K c 1 +
T

Ti s
1+ d

N

R(s) +
E(s)
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N



2
 b f 0 + b f 1s + b f 2 s
 1 + a f 1s + a f 2 s 2
s



2
 b f 0 + b f 1s + b f 2 s
 1 + a f 1s + a f 2 s 2
s 

U(s)
Model
Y(s)
Table 33: PID controller tuning rules – SOSPD model with a zero
G m (s) =
Rule
K m (1 + sTm 3 )e −sτ m
2
Tm1 s 2 + 2ξ m Tm1s + 1
Kc
or
K m (1 − sTm 4 )e−sτ m
Tm12s 2 + 2ξ mTm1s + 1
Ti
Td
Comment
0.167 τm
Negative zero.
Robust
Shamsuzzoha
and Lee (2006a).
Model: Method 1
0.5τm
K m (λ + τm )
1
0.5τm
Positive zero.
Kc
0.5(λ + Tm3 )τm + Tm3λ − 0.083τm + 0.5Tm3τm
or
λ + τm
2
af1 =
0.5(λ + Tm 4 )τm + Tm 4λ − 0.083τm − 0.5Tm 4 τm
;
λ + τm + 2Tm 4
2
af1 =
af 2 =
0.5τm Tm3 (λ − 0.167 τm )
0.5τm Tm 4 (λ + 0.167 τm )
or a f 2 =
;
λ + τm
λ + τm + 2Tm 4
2
b f 0 = 1 , b f 1 = 2ξ m Tm1 , b f 2 = Tm1 , N = ∞ .
λ = x1τm ; x1 = 1.360 , M s = 1.4 ; x1 = 0.677 , M s = 1.6 ;
x1 = 0.372 , M s = 1.8 ; x1 = 0.337 , M s = 2.0 .
1
Kc =
0.5τm
.
K m (λ + τm + 2Tm 4 )




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3.6.7 Two degree of freedom controller 1


Td s
1
+
U (s ) = K c  1 +
 Ti s 1 + Td

N

289





βTd s 
 E (s ) − K c  α +
 R (s )
Td 


s
1+ s 

N 






T
s
β
d
α +
K
Td  c

1+
s

N 

E(s)
R(s)
+
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N





s

−
U(s)
Y(s)
Model
+
Table 34: PID controller tuning rules – SOSPD model with a zero
K m (1 + sTm 3 )e −sτ m
K m (1 − sTm 4 )e −sτ m
K m (1 + sTm3 )e−sτ m
G m (s) =
or
or
or
(1 + sTm1 )(1 + sTm 2 ) (1 + sTm1 )(1 + sTm 2 )
Tm12s 2 + 2ξmTm1s + 1
K m (1 − sTm 4 )e −sτ m
Tm12s 2 + 2ξ m Tm1s + 1
Rule
Kc
Madhuranthakam
et al. (2008).
Model: Method 1
1
Ti
Kc =
α = 0 , β = 1 , N = 10.
0.4938  Tm1 + Tm 2 − Tm 3 
1 +


K m 
τm

Ti = 12.585
Comment
Minimum performance index: regulator tuning
Ti
Td
Tm1 > Tm3
Kc
1.3594
1
Td
, 0.1 ≤
τm
≤ 0.66 ;
τm + Tm1 + Tm 2 − Tm3
Tm 2 (Tm1 + Tm 2 − Tm3 )  Tm1 + Tm 2 − Tm3 
1 +



τm
τm


2.4858
,
Td = 6.465(Tm1 + Tm 2 − Tm 3 )
Tm 2  Tm1 + Tm 2 − Tm3 
1 +


τm 
τm

2.9185
.
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290
Rule
Madhuranthakam
et al. (2008).
Model: Method 1
Chien et al.
(2003).
Model: Method 1
Huang et al.
(2000).
Model: Method 1
Kc
2
Kc
Ti
Kc =
Comment
Td
Minimum performance index: servo tuning
Ti
Td
Tm1 > Tm3
α = 0 , β = 1 , N = 10.
Direct synthesis: time domain criteria
3
4
Ti
Kc
Kc
α = 0 ; β =1;
N = 10.
Td
Direct synthesis: frequency domain criteria
N = min[20,
2Tm1 ξ m + 0.4τ m
Td
20Td ]
1.2206
2
FA
0.5254  Tm1 + Tm 2 − Tm 3 
1 +


K m 
τm

, 0.1 ≤
τm
≤ 0.66 ;
τm + Tm1 + Tm 2 − Tm3
1.4569
 T + Tm 2 − Tm3 

Ti = 0.5657 τm 1 + m1

τm


,
Td = 6.7572(Tm1 + Tm 2 − Tm3 )
3
Kc =
Tm 2
τm

T + Tm 2 − Tm 3 
1 + m1



τm


Tm1
, x1 = 0.1Td + 0.5 4Tm 4 τm + 0.4Td Tm 4 + 0.4Td τm + 0.04Td 2 ;
K m (2 x1 + Tm 4 + τm )
2
Ti = 2ξ m Tm1 − 0.1Td ; Td = Tm1 10ξ m − 3.02 11ξm − 1  , ξm > 0.302 .


4
Kc = λ
α=
2
T + 0.8Tm1ξ m τm
2Tm1ξ m + 0.4τm
2T ξ + 0.4τm
;
or K c = λ m1 m
, Td = m1
K m (τm + 2Tm 3 )
K m (τm + 2Tm 4 )
0.8Tm1ξm τm
2Tm1ξ m
0.5Tm1 2
−1 ; β =
−1 ;
2Tm1ξ m + 0.4 τ m
Tm1 2 + 0.8Tm1ξ m τ m
λ = 0.6,
τm
τ
ξ m ≤ 1 , A m = 2.83 ; λ = 0.7,1 < m ξ m ≤ 2 , A m = 2.43 ;
Tm1
Tm1
λ = 0.8,
τm
ξ m > 2 , A m = 2.13 .
Tm1
2.9057
.
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Jyothi et al.
(2001).
Model:
Method 1;
α = 0 , β =1,
N =∞.
Kc
Ti
Td
Comment
5
Kc
Tm1 + Tm 2
τm Tm3 < 1
6
Kc
Tm1Tm 2
Tm1 + Tm 2
7
Kc
τm Tm3 < 1
8
Kc
Tm1
2ξ m
0.5(Tm1 + Tm 2 )
0.5Tm1Tm 2
Tm1 + Tm 2
τm Tm 4 < 1
Tm1 + Tm 2
Tm1Tm 2
Tm1 + Tm 2
2ξ m Tm1
Tm1
2ξ m
2ξ m Tm1
Tm1 + Tm 2
K m Tm 4
9
0.5
Kc
Tm1 + Tm 2
K m Tm 4
10
Kc
ξm Tm1 K m Tm 4
11
5
6
7
8
9
10
11
Kc =
Kc =
Kc =
Kc =
Kc =
Kc =
Kc =
Kc
1.57(Tm1 + Tm 2 )
K m τm 1 + 9.870(Tm3 τm )2
0.79(Tm1 + Tm 2 )
K m τm 1 + 2.467(Tm3 τm )2
3.14ξ m Tm1
K m τm 1 + 9.870(Tm3 τm )2
1.57ξ m Tm1
K m τm 1 + 2.467(Tm3 τm )2
1.57(Tm1 + Tm 2 )
K m τm 1 + 2.467(Tm 4 τm )2
0.79(Tm1 + Tm 2 )
K m τm 1 + 2.467(Tm 4 τm )2
1.57ξ m Tm1
K m τm 1 + 2.467(Tm 4 τm )2
.
.
.
.
.
.
.
τm Tm3 > 1
τm Tm3 > 1
τm Tm 4 > 1
τm Tm 4 < 1
τm Tm 4 > 1
τm Tm 4 < 1
τm Tm 4 > 1
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3.6.8 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s)




(
)
1
−
β
T
s
1 + b f 1s
[
]
1
−
χ
δ
d

,
F1 (s) = K c  [1 − α ] +
+
+


T
Ti s
1 + Ti s
1 + a f 1s + a f 2 s 2
d
1
+
s


N 





[
]
K (b + b f 4 s )
1
−
ϕ
T
s
1+ bf 2s
[
]
−
φ
1
d 
F2 (s) = K c  [1 − ε] +
+
− 0 f3
2

 1 + a f 3s + a f 4 s
T
Ti s
1+ a f 5s
1+ d s 

N 

R(s)
+
F1 (s)
U(s)
−
Y(s)
Model
F2 (s)
Table 35: PID controller tuning rules – SOSPD model with a zero
K m (1 + sTm 3 )e −sτ m
K m (1 − sTm 4 )e −sτ m
G
(
s
)
=
or
G m (s) =
m
Tm12s 2 + 2ξ mTm1s + 1
Tm1 2 s 2 + 2ξ m Tm1s + 1
Rule
Kc
1
Huang et al.
(2000).
Model: Method 1
Kc
Ti
Td
Comment
Direct synthesis: time domain criteria
2Tm1ξ m + 0.4τ m
0
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ;
a f 5 = x1 x 2 , x 2 not specified.
Chien et al.
(2003).
Model: Method 1
2
Kc
Tm1
0
Tm1 > Tm 2
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 1 ; b f 3 = 1 , b f 4 = Tm 2 , a f 4 = 0.1Tm 2 .
1
Kc = λ
2
2Tm1ξ m + 0.4τm
2T ξ + 0.4τm
T + 0.8Tm1ξ m τm
;
or K c = λ m1 m
, x1 = m1
K m ( τm + 2Tm3 )
K m (τm + 2Tm 4 )
0.8Tm1ξ m τm
λ = 0.5, τ m ξ m Tm1 ≤ 1 ; λ = 0.6,1 < τm ξ m Tm1 ≤ 2 ; λ = 0.7, τmξ m Tm1 > 2 ; servo.
λ = 0.6, τmξ m Tm1 ≤ 1 ; λ = 0.7,1 < τm ξ m Tm1 ≤ 2 ; λ = 0.8, τm ξm Tm1 > 2 ; regulator.
2
Kc =
Tm1
, x1 = 0.1Tm 2 + 0.5 4Tm 4 τm + 0.4Tm 2 (Tm 4 + τm ) + 0.04Tm 2 2 .
K m (2 x1 + Tm 4 + τm )
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3.7 TOSPD Model G m (s) = K m
G m (s) =
G m (s) =
1 + b 1s + b 2 s 2 + b 3 s 3
1 + a 1s + a 2 s 3 + a 3 s 3
e −sτ m or
K m e −sτ m
or
(1 + sTm1 )(1 + sTm 2 )(1 + sTm3 )
(T
K m (Tm 3 s + 1)e −sτ m
m1
)
s + 2ξ m Tm1s + 1 (Tm 2 s + 1)
2 2

1 

3.7.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
E(s)
R(s)
+
−

1 

K c 1 +

T
is 

U(s)
TOSPD
model
Y(s)
Table 36: PI controller tuning rules – third order system plus time delay model
K m e − sτ m
1 + b1s + b 2s 2 + b3s3 −sτ m
G m (s ) = K m
e
or
or
3
3
(1 + sTm1 )(1 + sTm 2 )(1 + sTm3 )
1 + a1s + a 2s + a 3s
K m (Tm3s + 1)e −sτ m
(T
2 2
m1 s
Rule
Minimum IAE –
Murata and Sagara
(1977). Model:
Method 2; K c , Ti
deduced from graphs.
Minimum IAE –
Murata and Sagara
(1977). Model:
Method 2; K c , Ti
deduced from graphs.
)
+ 2ξ m Tm1s + 1 (Tm 2s + 1)
Kc
Comment
Ti
Minimum performance index: regulator tuning
x1 K m
x 2Tm1
Tm1 = Tm 2 = Tm3
x1
1.3
0.9
0.7
x2
τm Tm1
x1
x2
τm Tm1
3.4
0.2
0.6
3.6
0.8
3.4
0.4
0.6
3.8
1.0
3.5
0.6
Minimum performance index: servo tuning
x1 K m
x 2Tm1
Tm1 = Tm 2 = Tm3
x1
x2
τm Tm1
x1
x2
τm Tm1
0.9
0.7
0.6
3.0
3.0
3.1
0.2
0.4
0.6
0.5
0.5
3.3
3.5
0.8
1.0
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Rule
Kc
Kuwata (1987).
Model: Method 1
1
Comment
Ti
Direct synthesis: time domain criteria
Ti
Tm1 = Tm 2 = Tm3
Kc
Direct synthesis: frequency domain criteria
Hougen (1979),
pp. 350–351.
Model: Method 1;
Tm1 ≥ Tm 2 ≥ Tm 3 .
0.7
Km
2
Vrančić et al. (1996),
Vrančić (1996),
p. 121.
Model: Method 1
Vrančić et al.
(2004a).
Model: Method 1
Vrančić et al.
(2004b).
1
Kc =
 Tm1 


 τm 
3
0.333
τm Tm1 ≤ 0.04
Kc
0.5A 3
( A1A 2 − K m A 3 )
A3
A2
0.5A 3
( A1A 2 − K m A 3 )
4
undefined
5
τ m Tm1 > 0.04
Ti
A m ≥ 2, φ m ≥ 60 0
(Vrančić et al., 1996).
Ti
A1 A 3
A1A 2 − K m A 3
G c = 1 (Tis )
Ti
Model: Method 1
Kc
0.4(3Tm1 + τm )
0.5
−
, Ti = 1.25x1 − 0.25(3Tm1 + τm ) , with
K m (3Tm1 + τm − x1 ) K m
x1 =
(3Tm1 + τm )2 − 0.267[6Tm13 + 18Tm12 τm + 9Tm1τm 2 + τm3 ] .
0.333
1   Tm1 
T + Tm 2 + Tm 3 

+ 0.8 m1
0.7
, Ti = 1.5τm 0.08 Tm1 (Tm 2 + Tm3 ) .

2K m   τm 
(
Tm1Tm 2Tm 3 )0.333 


2
Kc =
3
A1 = K m (a 1 − b1 + τ m ) , A 2 = K m b 2 − a 2 + A1a 1 − b1 τ m + 0.5τ m
(
(
2
),
2
3
)
A 3 = K m a 3 − b 3 + A 2 a 1 − A1a 2 + b 2 τ m − 0.5b1 τ m + 0.167 τ m .
A1A3 (A1A 2 − K m A 3 ) )
(A1A 2 − K m A3 )2 + K m A3 (A1A 2 − K m A3 ) + 0.25K m 2A32
4
Ti =
5
Kc =
A1A 2 − A3K m − [sign (A1A 2 − A 3K m )]A1 A 2 − A1A 3
.
2
A 3K m 2 − 2A1A 2 K m + A13
,
A1A 2 − A3K m − [sign (A1A 2 − A 3K m )]A1 A 2 − A1A 3
2
Ti = 2A1
2
(A3K m − 2A1A 2 K m +
A13 )(1 +
K c K m )2
.
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Comment
Robust
Marchetti and Scali
(2000).
Model: Method 1
a1 + 0.5τm
(3λ + τm )K m
a1 + 0.5τm
(3λ + τm + 2Tm3 )K m
b1 = b 2 = b3 = 0 or
a 1 + 0. 5 τ m
b 2 = b3 = 0; b1 = Tm 3 .
b 2 = b3 = 0 ;
b1 = −Tm3 .
Ultimate cycle
x 2 Tu
Non-oscillatory
x 1K u
TOSPD model –
x1
x 2 τm Tu x 1
x 2 τm Tu x 1
x 2 τm Tu
Kuwata (1987).
x 1 , x 2 deduced from 0.13 0.40 0.05 0.22 0.21 0.25 0.225 0.155 0.45
0.17 0.33 0.10 0.225 0.19 0.30 0.225 0.155 0.50
graphs.
0.19 0.30 0.15 0.225 0.17 0.35
0.21 0.24 0.20 0.225 0.16 0.40
Yu (2006),
Ziegler–Nichols.
0.45K u
0.83Tu
pp. 107–112.
0.33K u
2Tu
Model: Method 3
Oscillatory TOSPD model.
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296


1
+ Td s 
3.7.2 Ideal PID controller G c (s) = K c 1 +
 Ti s

R(s)
+
E(s)
−

 U(s)
1
K c 1 +
+ Td s 
Ti s


Table 37: PID controller tuning rules – TOSPD model G m (s) =
G m (s ) =
Rule
Kc
1
Polonyi (1989).
Model: Method 1
Jones and Tham
(2006).
Marchetti and
Scali (2000).
Kuwata (1987);
x 1 , x 2 deduced
from graphs.
1
2
3
Kc
Y(s)
TOSPD
model
K m e − sτ m
or
(1 + sTm1 )(1 + sTm 2 )(1 + sTm 3 )
K m e − sτ m
1 + a1s + a 2s3 + a 3s3
Ti
Comment
Td
Minimum performance index: other tuning
Ti
τm
Tm1 >
10(Tm 2 + τ m )
Standard form optimisation – binomial.
2
Ti
τm
Kc
Standard form optimisation – minimum ITAE.
Robust
3
Tm1 + Tm 2 + Tm3
Td
Kc
a1 + 0.5τm
(3λ + τm )K m
a 1 + 0. 5 τ m
a 2 + 0.5a1τm
a1 + 0.5τm
Model: Method 1
Model: Method 1
Ultimate cycle
x 2 Tu
x 3 Tu
x 1K u
x1
x2
x3
τm
Tu
x1
x2
x3
0.45
0.46
0.45
0.43
0.57
0.47
0.40
0.33
0.16
0.13
0.11
0.10
0.05
0.10
0.15
0.20
0.41
0.38
0.35
0.32
0.29
0.25
0.22
0.20
0.08
0.08
0.07
0.06
τm
Tu
x1
x2
x3
τm
Tu
0.25 0.29 0.19 0.04 0.45
0.30 0.28 0.18 0.03 0.50
0.35
0.40

T
T T
K c = 1 − 4 m 2 + m 2  m1 , Ti = 4 6Tm 2 τm + τm .

6τm Tm3  τm


Tm 2
T T
+ m 2  m1 , Ti = 2.7 3.4Tm 2 τm + τm .
K c = 1 − 2.1

3.4τm Tm3  τm

T + Tm 2 + Tm3
T T + Tm 2Tm 3 + Tm1Tm3
K c = m1
, Td = m1 m 2
, λ not specified.
K m (λ + τm )
Tm1 + Tm 2 + Tm3
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3.7.3 Ideal controller in series with a first order lag

 1
1
+ Td s 
G c (s) = K c 1 +
 Ti s
 Tf s + 1
R(s)
E(s)
−
+
U(s)
Y(s)

 1
1
K m e − sτ
K c 1 +
+ Td s 
2
3
1 + a 1s + a 2 s + a 3 s
 Ti s
 Tf s + 1
m
Table 38: PID controller tuning rules – TOSPD model G m (s) =
Rule
Schaedel (1997).
Model: Method 1
1
Kc =
Kc
1
Td
1 + a 1s + a 2 s 2 + a 3 s 3
Comment
Direct synthesis: frequency domain criteria
Ti
Td
Tf = Td N ;
5 ≤ N ≤ 20 .
2
2
0.375Ti
a − a 2 + a1τm + 0.5τm
, Ti = 1
,
T


a1 + τm − Td
K m  a1 + τm + d − Ti 
N


2
Td =
Kc
Ti
K m e − sτ m
2
3
a 2 + a1τm + 0.5τm
a + a 2τm + 0.5a1τm + 0.167 τm
.
− 3
a1 + τ m
a 2 + a1τm + 0.5τm 2
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

Td s
1
+
3.7.4 Controller with filtered derivative G c (s) = K c 1 +
 Ti s 1 + Td

N

R(s)
E(s)
−
+


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N





s

U(s)
K m e − sτ
2
Rule
Schaedel (1997).
1
Kc =
Td =
N=
Kc
1
Kc
Y(s)
m
1 + a 1s + a 2 s + a 3 s
Table 39: PID controller tuning rules – TOSPD model G m (s) =
Ti
Td



s 

3
K m e − sτ m
1 + a 1s + a 2 s 2 + a 3 s 3
Comment
Direct synthesis: frequency domain criteria
Model: Method 1
Ti
Td
2
2
0.375Ti
a − a 2 + a1τm + 0.5τm
, Ti = 1
,
T


a1 + τm − Td
K m  a1 + τm + d − Ti 
N


a 2 + a1τm + 0.5τm 2 a 3 + a 2 τm + 0.5a1τm 2 + 0.167 τm 3
,
−
a1 + τ m
a 2 + a1τm + 0.5τm 2
Td
 x + τ − Ti 
− 1 m
+
2


(x1 + τm − Ti )2 + 0.375 TiTd
4
,
Kv
0.05 ≤ x1 ≤ 0.2 , K v = prescribed overshoot in the manipulated variable for a step
response in the command variable.
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3.7.5 Two degree of freedom controller 1


Td s
1
+
U (s ) = K c  1 +
 Ti s 1 + Td

N






βTd s 
 E (s ) − K c  α +
 R (s )
Td 


s
1+ s 

N 






T
s
β
d
α +
K
Td  c

1+
s

N 

E(s)
R(s)
+
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N





s

−
U(s)
+
TOSPD
model
Y(s)
Table 40: PID controller tuning rules – third order system plus time delay model
1 + b 1s + b 2 s 2 + b 3 s 3 −sτ m
K m e − sτ m
G m (s) = K m
e
or G m (s) =
3
3
(1 + sTm1 )(1 + sTm 2 )(1 + sTm3 )
1 + a 1s + a 2 s + a 3 s
Rule
Taguchi and
Araki (2000).
1
Kc =
Kc
1
Ti
Td
Comment
Minimum performance index: servo/regulator tuning
0
Model: Method 1
Ti
K
c
OS ≤ 20% ; Tm1 = Tm 2 = Tm3 ; τ m Tm1 ≤ 1.0 .

1 
0.7399
 0.2713 +
,

Km 
(τm Tm1 ) + 0.5009 
(
Ti = Tm1 2.759 − 0.003899[τm Tm1 ] + 0.1354[τm Tm1 ]
2
α = 0.4908 − 0.2648(τm Tm1 ) + 0.05159(τm Tm1 ) .
2
),
299
b720_Chapter-03
Rule
Taguchi and
Araki (2000).
2
Kc
Ti
Td
Comment
Kc
Ti
Td
Model: Method 1
OS ≤ 20% ; N not specified; Tm1 = Tm 2 = Tm3 ; τ m Tm1 ≤ 1.0 .
Kuwata (1987).
Tm1 = Tm 2 = Tm3 .
Kuwata (1987).
Model: Method 1
Kc =
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2
17-Mar-2009
3
Kc
4
Kc
Direct synthesis: time domain criteria
0
Ti
α = 1;
Model: Method 1
Ti
Tm1 = Tm 2 = Tm3
Td
α = 1, β = 1, N = ∞ .

1 
1.275
 0.4020 +
,

[τm Tm1 ] + 0.003273 
Km 
(
(0.8335 + 0.2910[τ
)
Ti = Tm1 0.3572 + 7.467[τm Tm1 ] − 12.86[τm Tm1 ] + 11.77[τm Tm1 ] − 4.146[τm Tm1 ] ,
Td = Tm1
2
3
m
)
Tm1 ] − 0.04000[τm Tm1 ] ,
2
α = 0.6661 − 0.2509[τm Tm1 ] + 0.04773[τm Tm1 ] ,
4
2
β = 0.8131 − 0.2303[τm Tm1 ] + 0.03621[τm Tm1 ] .
2
3
Kc =
x1 =
0.8(3Tm1 + τm )
1
−
, with
K m (3Tm1 + τm − x1 ) K m
(3Tm1 + τm )2 − 0.267[6Tm13 + 18Tm12τm + 9Tm1τm 2 + τm3 ] ;
[
 0.833 6Tm12 + 6Tm1τm + τm 2
Ti = 1 −

(3Tm1 + τm )2

4
Kc =
] 1.667[6T
2
m1


+ 6Tm1τm + τm
3Tm1 + τm
2
].
x
1.20 x13
1
x 2
x 2 − (3T + τ )x
−
, Ti = 1.667 2 − 4.167 23 , Td = 0.833x 2 1 3 m1 m 2 2 ;
2
x1
Km x2
Km
x1
x1 − 0.833x 2
2
2
3
2
2
3
x1 = 6Tm1 + 6Tm1τm + τm , x 2 = 6Tm1 + 18Tm1 τm + 9Tm1τm + τm .
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Rule
Kc
Vrančić (1996),
pp. 136–137.
5
Ti
Kc
x1
Kc =
x1
x2
τm Tu
x1
x2
τm Tu
0.05
0.10
0.15
0.20
0.21
0.21
x 2 Tu
0.22
0.20
0.18
0.20
0.25
0.30
0.21
0.20
0.20
0.16
0.15
0.14
Tm1 = Tm 2
0.35
0.40
0.45
= Tm3
(1 − [1 − α] )(K
2
m
τm
Tu
x1
x2
x3
0.13
0.12
0.11
0.10
0.05
0.10
0.15
0.20
0.38
0.32
0.28
0.23
0.26
0.22
0.18
0.16
0.08
0.07
0.04
0.02
A 3 + A13 − 2K m A1A 2
x1 =
Kc =
), K
(1 − [1 − α] )(K
2
m
A 3 + A13 − 2K m A1A 2
A1
Ti =
2
Km +
mA3
x1
τm
Tu
x2
x3
τm
Tu
0.25 0.21 0.15 0.01 0.45
0.30 0.20 0.14 0 0.50
0.35
0.40
− A1A 2 < 0 ;
(K m A3 − A1A 2 )2 − (1 − [1 − α]2 )A3 (K m 2A3 + A13 − 2K m A1A 2 ) ;
A1A 2 − K m A 3 + x1
2
x 3 Tu
x3
A1A 2 − K m A 3 − x1
2
Tm1 = Tm 2 = Tm3
τm Tu
0.33
0.29
0.26
x 1K u
Kuwata (1987);
x1 x 2
x 1 , x 2 deduced
from graphs.
α = 1 , β = 1 , 0.69 0.41
0.60 0.38
N =∞.
0.51 0.36
0.43 0.31
5
Ultimate cycle
0
x 2 Tu
x2
0.14
0.17
0.19
Comment
Td
Direct synthesis: frequency domain criteria
Model: Method 1
0
Ti
x 1K u
Kuwata (1987);
x 1 , x 2 deduced
from graphs.
α = 1.
301
(
1
KK
2
+ c m 1 − [1 − α ]
2K c
2
(
)
), K
mA3
− A1A 2 > 0 .
with
)
A1 = K m (a 1 − b1 + τ m ) , A 2 = K m b 2 − a 2 + A1a1 − b1τm + 0.5τm ,
2
(
2
3
)
A 3 = K m a 3 − b 3 + A 2 a 1 − A1a 2 + b 2 τ m − 0.5b1 τ m + 0.167 τ m ;
0.2 ≤ α ≤ 0.5 – good servo and regulator response (p. 139).
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3.7.6 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s)




(
)
1
−
β
T
s
1 + b f 1s
[
]
1
−
χ
δ
d

,
F1 (s) = K c  [1 − α ] +
+
+


T
Ti s
1 + Ti s
1 + a f 1s + a f 2 s 2
d
1
+
s


N 





[
]
K (b + b f 4 s )
1
−
ϕ
T
s
1+ bf 2s
[
]
−
φ
1
d 
F2 (s) = K c  [1 − ε] +
+
− 0 f3
2

 1 + a f 3s + a f 4 s
T
Ti s
1+ a f 5s
1+ d s 

N 

R(s)
+
F1 (s)
U(s)
−
Y(s)
Model
F2 (s)
Table 41: PID controller tuning rules – third order system plus time delay model
K m e − sτ m
G m (s ) =
(1 + sTm1 )(1 + sTm 2 )(1 + sTm3 )
Rule
Kc
Ti
Kc
Ti
Td
Comment
Td
0.5τm ≤ λ ≤ 3τm
Robust
1
Jones and Tham
(2006).
Model: Method 1
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ;
ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ;
K0 =
1
Tm1 + Tm 2 + Tm3 −
Kc =
0.4
; bf 3 = 1 , bf 4 = 0 , a f 5 = 0 .
K m (τm + 0.75)
0.4τm
τm + 0.75
K m (λ + τm )
Tm1 + Tm 2 + Tm3 −
, Ti =
1+
0.4τm
τm + 0.75
0.4τm
τm + 0.75
Td =
,
Tm1Tm 2 + Tm1Tm 3 + Tm 2Tm3
.
0.4τm
Tm1 + Tm 2 + Tm 3 −
τm + 0.75
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303
3.8 Fifth Order System Plus Delay Model
K (1 + b1s + b 2 s 2 + b 3s 3 + b 4 s 4 + b s s 5 )e − sτm
G m (s ) = m
1 + a 1s + a 2 s 2 + a 3s 3 + a 4 s 4 + a 5s 5
(
)


1
+ Td s 
3.8.1 Ideal PID controller G c (s) = K c 1 +
 Ti s

U(s)
E(s)
R(s)
−
+
(
Y(s)
)
)
K m 1 + b1s + b 2s 2 + b 3s 3 + b 4s 4 + b 5s 5 e − sτ
1 + a 1s + a 2s 2 + a 3s 3 + a 4s 4 + a 5s 5


1
K c 1 +
+ Td s 
 Ti s

(
m
Table 42: PID controller tuning rules – fifth order model with delay
K (1 + b 1s + b 2 s 2 + b 3 s 3 + b 4 s 4 + b s s 5 )e − sτ m
G m (s) = m
1 + a 1s + a 2 s 2 + a 3 s 3 + a 4 s 4 + a 5 s 5
(
Rule
Kc
Vrančić et al.
(1999).
1
1
Kc
)
Ti
Comment
Td
Direct synthesis: frequency domain criteria
Model: Method 1
Ti
Td
Note: equations continued into the footnote on p. 304.
Kc =
a13 − a12 b1 + a1b 2 − 2a1a 2 + a 2 b1 + a 3 − b3 + y1
[
2K m − a12 b1 + a1a 2 + a1b12 − a 3 − b1b 2 + b3 + y 2
(
)
] , with
y1 = τ m a 1 − a 1 b1 − a 2 + b 2 + 0.5(a 1 − b1 )τ m + 0.167τ m and
2
2
3
y 2 = (a1 − b1 ) τ m + (a1 − b1 )τm + 0.333τm − (a1 − b1 + τm ) Td ;
2
Ti =
[a
2
3
2
a13 − a12 b1 + a1b 2 − 2a1a 2 + a 2 b1 + a 3 − b3 + y1
2
1
− a1b1 − a 2 + b 2 + (a1 − b1 )τm + 0.5τm 2 − (a1 − b1 + τm )Td
Td is found by solving the following equation:
2
3
];
4
Td x1 + 360 x 2 − 360τm x 3 + 180τm x 4 − 60τm x 5 − 15τm x 6
− 3τ m x 7 + 7τ m (b1 − a 1 ) − τ m = 0 , with
5
4
6
3
x 1 = 360[a 1 b 2 − a 1 (a 2 b1 + a 3 + b1b 2 + b 3 )
2
2
2
2
+ a 1 (a 2 + a 2 ( b1 − 2 b 2 ) + 3a 3 b1 + 2a 4 + b1b 3 + b 2 − b 4 )
2
− a 1 (a 2 ( 2a 3 − 3b 3 ) + a 3 ( 2 b1 − b 2 ) + 3a 4 b1 + a 5 − b1 b 4 + 2 b 2 b 3 − b 5 )
2
2
2
− a 2 b1 b 3 + a 3 + a 3 ( b1 b 2 − 2 b 3 ) + a 4 b1 + a 5 b1 − b1 b 5 + b 3 ]
4
3
2
2
− 360τ m [a 1 b1 − a 1 (a 2 + b1 + 2 b 2 ) + 3a 1 (a 3 + b1 b 2 )
2
+ a 1 (3a 2 b 2 − 3a 3 b1 − 3a 4 − b1 b 3 − 2 b 2 + 2 b 4 )
7
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2
− a 2 ( b 1 b 2 + b 3 ) + a 3 ( b 1 − b 2 ) + 2a 4 b 1 + a 5 − b 1 b 4 + 2 b 2 b 3 − b 5 ]
2
4
3
2
2
+ 180τ m [a 1 − 4a 1 b1 + 3a 1 ( b1 + b 2 ) + a 1 (3a 2 b1 − 3a 3 − 5b1 b 2 + b 3 )
2
2
− a 2 ( b1 + 2 b 2 ) + a 3 b1 + 2a 4 + b1 b 3 + 2( b 2 − b 4 )]
3
3
2
2
+ 60τ m [4a 1 − 9a 1 b1 − a 1 (3a 2 − 5b1 − 4b 2 ) +4a 2 b1 − a 3 − 5b1 b 2 + b 3 ]
4
2
5
2
6
+ 15τ m [ 9a 1 −14a 1 b1 − 4a 2 + 5b1 + 4 b 2 ] − 42 τ m ( b1 − a 1 ) + 7τ m ,
4
3
x 2 = a 1 b 3 − a 1 ( a 2 b 2 + a 1 b 3 + a 4 + b1 b 3 )
2
2
2
+ a 1 (a 2 b1 + a 2 ( 2a 3 + b1 b 2 − 3b 3 ) + a 3 ( b1 + b 2 ) + 2a 4 b1 + a 5 + b 2 b 3 − b 5 )
3
2
2
2
− a 1 ( a 2 + a 2 ( b 1 − 2 b 2 ) + a 2 ( 2a 3 b 1 − 2 b 1 b 3 + b 2 − b 4 )
2
2
2
+ 2a 3 + a 3 ( 2 b1 b 2 − 3b 3 ) + a 4 ( b1 + b 2 ) + a 5 b1 − b1 b 5 + b 3 )
2
2
2
+ a 2 a 3 + a 2 [a 3 ( b1 − 2 b 2 ) − a 5 − b1 b 4 + b 5 ] + a 3 b1
2
+ a 3 ( a 4 − b1 b 3 + b 2 − b 4 ) + a 4 ( b1 b 2 − b 3 ) + a 5 b 2 − b 2 b 5 + b 3 b 4 ,
4
3
x 3 = a 1 b 2 − a 1 ( a 2 b1 + a 3 + b1 b 2 + b 3 )
2
2
2
2
+ a 1 (a 2 + a 2 ( b1 − 2 b 2 ) + 3a 3 b1 + 2a 4 + b1b 3 + b 2 − b 4 )
2
−a 1 (a 2 ( 2a 3 − 3b 3 ) + a 3 ( 2 b1 − b 2 ) + 3a 4 b1 + a 5 − b1 b 4 + 2 b 2 b 3 − b 5 )
2
2
2
− a 2 b1 b 3 + a 3 + a 3 ( b1 b 2 − 2 b 3 ) + a 4 b1 + a 5 b1 − b1 b 5 + b 3 ,
4
3
2
x 4 = a 1 b1 − a 1 ( a 2 + b1 + 2 b 2 )
2
2
+ 3a 1 (a 3 + b1 b 2 ) + a 1 (3a 2 b 2 − 3a 3 b1 − 3a 4 − b1 b 3 − 2 b 2 + 2b 4 )
2
− a 2 ( b1 b 2 + b 3 ) + a 3 ( b 1 − b 2 ) + 2a 4 b 1 + a 5 − b 1 b 4 + 2 b 2 b 3 − b 5 ) ,
4
3
2
2
x 5 = a 1 − 4a 1 b1 + 3a 1 ( b1 + b 2 ) + a 1 (3a 2 b1 − 3a 3 − 5b1b 2 + b 3 )
2
2
− a 2 ( b 1 + 2 b 2 ) + a 3 b 1 + 2a 4 + b 1 b 3 + 2( b 2 − b 4 ) ,
3
2
2
x 6 = 4a 1 − 9a 1 b1 − a 1 (3a 2 − 5b1 − 4 b 2 ) +4a 2 b1 −a 3 − 5b1 b 2 + b 3 ,
2
2
x 7 = 9a 1 − 14a 1 b1 − 4a 2 + 5b1 + 4b 2 .
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305
3.8.2 Controller with filtered derivative


Td s
1

G c (s) = K c 1 +
+
 Ti s 1 + (Td N )s 
E(s)
R(s)
+
−


Td s
1
+
K c 1 +
T

Ti s
1+ d

N

Y(s)
 U(s)

2
3
4
5 − sτ m
 K m 1 + b1s + b 2s + b 3s + b 4s + b 5s e

1 + a 1s + a 2s 2 + a 3s 3 + a 4s 4 + a 5s 5
s

(
)
)
(
Table 43: PID controller tuning rules – fifth order model with delay
K (1 + b 1s + b 2 s 2 + b 3 s 3 + b 4 s 4 + b s s 5 )e − sτ m
G m (s) = m
1 + a 1s + a 2 s 2 + a 3 s 3 + a 4 s 4 + a 5 s 5
(
Rule
)
Kc
Ti
Comment
Td
Direct synthesis: frequency domain criteria
Vrančić (1996),
pp. 118–119.
Model: Method 1
1
A3
Ti =
2
A 2 − Td A1 −
Td =
Ti
2(A1 − Ti )
Td
N
1
N < 10
Td
Ti
,
− (A 32 − A 5A1 ) +
(A
2
3
)
4
(A3A 2 − A5 )(A5A 2 − A 4A3 )
N
,
2
(A3A 2 − A5 )
N
2
− A 5A1 −
(
A1 = K m (a 1 − b1 + τ m ) , A 2 = K m b 2 − a 2 + A1a 1 − b1 τ m + 0.5τ m
2
),
(
)
(b − a + A a − A a + A a − b τ + 0.5b τ + 0.167b τ + 0.042τ )
(a − b + A a − A a + A a − A a + b τ − 0.5b τ )
+ K (0.167 b τ − 0.042b τ + 0.008τ ) .
2
3
A 3 = K m a 3 − b 3 + A 2 a 1 − A1a 2 + b 2 τ m − 0.5b1 τ m + 0.167 τ m ,
A4 = Km
A5 = K m
2
4
4
3 1
2 2
1 3
3 m
3
2 m
4
1 m
m
2
5
5
4 1
3 2
2 3
1 4
4 m
3 m
3
m
2 m
4
1 m
5
m
b720_Chapter-03
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306
Rule
Vrančić et al.
(1999).
2
17-Mar-2009
Kc =
[
2
Kc
Ti
Td
Comment
Kc
Ti
Td
8 ≤ N ≤ 20 ;
Model: Method 1
a13 − a12 b1 + a1b 2 − 2a1a 2 + a 2 b1 + a 3 − b3 + y1
2K m − a12 b1 + a1a 2 + a1b12 − a 3 − b1b 2 + b3 + (a1 − b1 ) 2 τ m + y 2
(
)
] , with
y1 = τ m a 1 − a 1 b1 − a 2 + b 2 + 0.5(a 1 − b1 )τ m + 0.167τ m and
2
2
3
2
3
2
y 2 = (a1 − b1 )τm + 0.333τm − (a1 − b1 + τm ) Td −
3
Td 2
(a1 − b1 + τm ) ;
N
2
a1 − a1 b1 + a1b 2 − 2a1a 2 + a 2 b1 + a 3 − b3 + y1
Ti =
2
 2
T 
2
a1 − a1b1 − a 2 + b 2 + (a1 − b1 )τ m + 0.5τ m − (a1 − b1 + τ m )Td − d 
N 

Td is found by solving the following equation:
4
3
2
Td N −3z1 + Td N −2 z 2 + Td N −1z3 + Td x1 + 360 x 2 − 360τm x 3
+ 180τm x 4 − 60τm x 5 − 15τm x 6 − 3τ m x 7 + 7τ m (b1 − a 1 ) − τ m = 0
2
2
5
4
6
7
with x 1 , x 2 , x 3 , x 4 , x 5 , x 6 and x 7 as defined in Table 42; z 1 , z 2 and z 3 are defined
as follows:
3
2
z 1 = 360[a 1 − a 1 b1 + a 1 ( b 2 − 2a 2 ) + a 2 b1 + a 3 − b 3 ]
2
2
3
+ 360τ m [a 1 − a 1 b1 − a 2 + b 2 ] + 180τ m (a 1 − b1 ) + 60τ m ,
3
2
z 2 = 360(a 1 − b1 )[a 1 − a 1 b1 + a 1 ( b 2 − 2a 2 ) + a 2 b1 + a 3 − b 3 ]
3
2
2
+ 360τ m [2a 1 − 3a 1 b1 − a 1 (3a 2 − b1 − 2b 2 ) + 2a 2 b1 + a 3 − b1 b 2 − b 3 ]
2
2
2
3
4
+ 180τ m (3a 1 − 4a 1 b1 − 2a 2 + b1 + 2 b 2 ) + 240τ m (a 1 − b1 ) + 60τ m ,
2
3
4
5
z 3 = z 4 + z 5 τ m + z 6 τ m + z 7 τ m + 135(a 1 − b1 ) τ m + 27τ m , with
4
3
2
2
z 4 = −360[a 1 b1 − a 1 (a 2 + b1 + b 2 ) + a 1 ( 2(a 3 + b1 b 2 ) − a 2 b1 )
2
2
2
+ a 1 ( a 2 + a 2 ( b 1 + b 2 ) − a 3 b 1 − 2a 4 − b 1 b 3 − b 2 + b 4 )
− a 2 ( a 3 + b1 b 2 ) + a 4 b1 + a 5 + b 2 b 3 − b 5 ] ,
4
3
2
2
z 5 = 360[a 1 − 3a 1 b1 + a 1 ( 2( b1 + b 2 ) − a 2 ) + a 1 (3a 2 b1 − a 3 − 3b1 b 2 )
2
2
− a 2 ( b1 + b 2 ) + a 4 + b1 b 3 + b 2 − b 4 ] ,
3
2
2
z 6 = 540[a 1 − 2a 1 b1 − a 1 (a 2 − b1 − b 2 ) + b1 (a 2 − b 2 )] and
2
2
z 7 = 180[ 2a 1 − 3a 1 b1 − a 2 + b1 + b 2 ] .
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Vrančić et al.
(2004a), Vrančić
and Lumbar
(2004).
3
Kc
Ti
Td
Comment
Kc
Ti
Td
8 ≤ N ≤ 20 ;
Model: Method 1
3
307
K c , Ti and Td are determined by solving the following equations:
0.5
K c . Td =
A2
(A 4 A1 − A 5 K m ) − 0.5 A 4 (A1A 2 − A 3 K m )
A1
Km
2
2
K c 1 + 2K m K c + K m K c
=
Ti
2 A1 + K m 2 K cTd
(
)
and
2
A 3K m − A1 K m K cTd − A 2 A1 + A 2 K m K c Td + x
2A1A 2 K m −
[
,
2
2
Kc =
2
A 5 A 1 K m + A 1 A 2 A 3 − A 4 A1 − A 3 K m
A13
− A 3K m
2
(
, with
][
)
]
x 2 = (2A1K m K cTd + A 2 )A 2 − A1A 3 − A 3 K m + A1 K c Td A1 + K m K cTd ,
and with
2
2
2
(
A1 = K m (a 1 − b1 + τ m ) , A 2 = K m b 2 − a 2 + A1a 1 − b1 τ m + 0.5τ m
(
(b
(a
2
3
)
2
),
A 3 = K m a 3 − b 3 + A 2 a 1 − A1a 2 + b 2 τ m − 0.5b1 τ m + 0.167 τ m ,
A4 = K m
A5 = K m
2
4
5
− a 4 + A3a1 − A 2a 2 + A1a 3 − b3τm + 0.5b 2 τm + 0.167 b1τm3 + 0.042τm 4
− b5 + A 4a1 − A3a 2 + A 2a 3 − A1a 4 + b 4 τm − 0.5b3τm
(
2
)
3
)
4
5
)
+ K m 0.167 b 2 τm − 0.042b1τm + 0.008τm .
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308
3.8.3 Two degree of freedom controller 1


Td s
1
+
U (s ) = K c  1 +
 Ti s 1 + Td

N






βTd s 
 E (s ) − K c  α +
 R (s )
Td 


s
1+ s 

N 






T
s
β
d
α +
K
Td  c

1+
s

N 

E(s)
R(s)
−
+


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N





s

−
U(s)
Y(s)
Model
+
Table 44: PID controller tuning rules – fifth order model with delay
K (1 + b 1s + b 2 s 2 + b 3 s 3 + b 4 s 4 + b s s 5 )e − sτ m
G m (s) = m
1 + a 1s + a 2 s 2 + a 3 s 3 + a 4 s 4 + a 5 s 5
(
Rule
)
Kc
Ti
Comment
Td
Direct synthesis: frequency domain criteria
Vrančić et al.
(2004a), Vrančić
and Lumbar
(2004).
1
1
Ti
Kc
Model:
Method 1;
8 ≤ N ≤ 20 .
Td
Note: equations continued into the footnote on the next page. K c , Ti and Td are determined by
solving the following equations:
A
A
0.5 2 (A 4 A1 − A 5 K m ) − 0.5 4 (A1A 2 − A 3 K m )
A1
Km
K c . Td =
,
2
2
A 5 A 1 K m + A 1 A 2 A 3 − A 4 A1 − A 3 K m
K c 1 + 2K m K c + K m 2 K c 2
=
and
Ti
2 A1 + K m 2 K cTd
(
Kc =
A 3K m −
)
A12 K m K cTd
− A 2 A1 + A 2 K m 2 K c Td + x
2A1A 2 K m − A13 − A 3K m 2
[
(
)
, with
][
]
x 2 = (2A1K m K cTd + A 2 )A 2 − A1A 3 − A 3 K m + A1 K c Td A1 + K m K cTd ,
and with
2
2
2
(
A1 = K m (a 1 − b1 + τ m ) , A 2 = K m b 2 − a 2 + A1a 1 − b1 τ m + 0.5τ m
(
2
3
)
A 3 = K m a 3 − b 3 + A 2 a 1 − A1a 2 + b 2 τ m − 0.5b1 τ m + 0.167 τ m ,
2
),
b720_Chapter-03
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FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
(
(a
A 4 = K m b 4 − a 4 + A 3 a 1 − A 2 a 2 + A 1a 3 − b 3 τ m + 0.5b 2 τ m 2 + 0.167b1 τ m 3 + 0.042 τ m 4
A5 = K m
5
− b 5 + A 4 a 1 − A 3a 2 + A 2 a 3 − A1a 4 + b 4 τ m − 0.5b 3 τ m
(
2
)
3
4
309
)
5
)
+ K m 0.167 b 2 τ m − 0.042b1 τ m + 0.008τ m ;
α = 1, β = 1.
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Handbook of PI and PID Controller Tuning Rules
310
3.9 General Model G m (s) =
K m e − sτ m
(1 + sTm1 ) n

1 

3.9.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
E(s)
R(s)
−
+

1 

K c 1 +
Ti s 

U(s)
K m e − sτ
Y(s)
m
(1 + sTm1 ) n
Table 45: PI controller tuning rules – general model G m (s) =
Rule
Minimum IAE –
Murata and Sagara
(1977). n = 4.
Model: Method 3;
K c , Ti deduced from
graphs.
Minimum IAE –
Murata and Sagara
(1977). n = 5.
Model: Method 3;
K c , Ti deduced from
graphs.
Minimum IAE –
Murata and Sagara
(1977). n = 4. Model:
Method 3; K c , Ti
deduced from graphs.
Minimum IAE –
Murata and Sagara
(1977). n = 5. Model:
Method 3; K c , Ti
deduced from graphs.
Kc
K m e − sτ m
(1 + sTm1 ) n
Comment
Ti
Minimum performance index: regulator tuning
x1 K m
x 2Tm1
x1
x2
τm Tm1
x1
x2
τm Tm1
0.9
0.7
0.6
4.0
4.0
4.2
0.2
0.4
0.6
0.5
0.5
4.4
4.7
0.8
1.0
x1 K m
x 2Tm1
x1
x2
τm Tm1
x1
x2
τm Tm1
0.7
0.6
0.5
4.5
4.7
4.9
0.2
0.4
0.6
0.5
0.4
5.2
5.5
0.8
1.0
Minimum performance index: servo tuning
x1 K m
x 2Tm1
x1
x2
τm Tm1
x1
x2
τm Tm1
0.7
0.6
0.5
4.5
4.7
4.9
0.2
0.4
0.6
0.5
0.4
5.2
5.5
0.8
1.0
x1 K m
x 2Tm1
x1
x2
τm Tm1
x1
x2
τm Tm1
0.6
0.5
0.5
4.1
4.3
4.6
0.2
0.4
0.6
0.5
0.4
5.0
5.3
0.8
1.0
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Kuwata (1987).
1
Davydov et al.
(1995).
Kc =
x1 =
2
Kc =
Comment
Direct synthesis: time domain criteria
Model: Method 1
Ti
Kc
Model: Method 1;
ξ = 0.9.
Direct synthesis: frequency domain criteria
τ m + Tar
0.5
‘Normal’ design.
0.5
Km n −1
n
( n + 1)( τ m + Tar )
0.375  n + 1 
‘Sharp’ design.


2n
K m  n − 1
2
[
Schaedel (1997).
Model: Method 2
1
Ti
311
Ti
Kc
]
0.4(nTm1 + τm )
0.5
−
, Ti = 1.25x1 - 0.25(nTm1 + τm ) , with
K m (nTm1 + τm − x1 ) K m
(nTm1 + τm )2 − 0.267[n (n − 1)(n − 2)Tm13 + 3n (n − 1)Tm12τm + 3nTm1τm 2 + τm3 ] .


τ*m
, Ti = (nTm1 + τ m )0.624 − 0.467




 nTm1 + τ m
K m  4.345
− 0.151
nT
+
τ
m1
m


1
τ*m




n


(n − 1)!
1
(
)
n
1
!
with τ*m = τ m + Tm1 (n − 1) −
.
+
−
n −1 − (n −1)
n −i
(n − 1) e
( ) (i − 1)!

i =1 n − 1
∑
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312


1
+ Td s 
3.9.2 Ideal PID controller G c (s) = K c 1 +
 Ti s

E(s)
R(s)
+
−
U(s)
Y(s)
Model
Table 46: PID controller tuning rules – general model
Comment
Kc
Ti
Td
Rule
Gorez and Klàn
(2000), Klán
(2000).
Model: Method 4


1
K c 1 +
+ Td s 
T
s
i


Direct synthesis: time domain criteria
Td
1
Ti
K m (Ti + τm )

τ 
1 − m 
 T 
ar 

0.25Ti
Ti τm
Tar
Ti
< 0.25Ti
2
Alternative
tuning.
Alternative
tuning.
Klán (2000).
τ 
τ
1 + 1 + 2 m  − 2 m
Tar
 Tar 
1
General delayed model. Ti = Tar
,
2
2
T 
T
K  2τ   
τ 
Td = ar (Ti + Tcr ) cr + Ti − Taa − m 1 − c 1 + m   ,

Ti 
Tar
2Tar  K m 
3Ti  


Tar = average residence time of the process (which equals Tm + τ m for a FOLPD
process, for example); Taa = additional apparent time constant;

K 
τ 
Tcr = 1 − c 1 + m  τm .
 K m  2Ti 
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Gorez and Klàn
(2000).
Lee and Teng
(2003).
313
Kc
Ti
Td
Comment
2
Kc
Ti
Td
Model: Method 4
3
Kc
Ti
Td
Model: Method 1
2 x1 K m
2Tm1
0.5Tm1
0.2 ≤ τm Tm1 ≤ 2
Trybus (2005).
Model: Method 1 τm
n
Tm1 0.20 0.22 0.25 0.29 0.33 0.40 0.50 0.67 1.0 2.0
2
x1
0.23 0.23 0.22 0.22 0.21 0.21 0.20 0.18 0.16 0.12
3
x1
0.14 0.14 0.14 0.14 0.14 0.13 0.13 0.12 0.11 0.09
4
x1
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.09 0.07
5
General delayed model has a positive zero. K c =
Ti = Tar
τ
1 + 1 + 2 m
 Tar
Ti
,
K m [Ti + (1 + x1 )τm ]
2

τ 
τ 
 − 2x1 m 1 − (3 + 2.5x1 ) m 

Tar 
Tar  τm (1 + 1.5x1 )

−
,
2
Tar

Tar 
Tcr
1
+ Ti − Taa − Tca 
,
(Ti + Tcr )
Ti 
Tar
 1 + x1 K c τ m
K m Tar
with Tar = average residence time of the process, Taa = additional apparent time
Td =
constant and x1 is a normalised area parameter characterising the inverse response of
the process.


K 
K  2τ   τ 2
τ 
Tcr = 1 − (1 + x1 ) c 1 + m  τm ; Tca = 1 − (1 + x1 ) c 1 + m   m .
K m  2Ti 
Km 
3Ti   2Tar


3
General delayed model. The three dominant desired poles of the closed loop response
are − ξωCL ± jωCL 1 − ξ 2 and − x1ωCL . G p ( jω) evaluated at − ξωCL + jωCL 1 − ξ 2
equals x 2∠φ2 , with G p ( jω) evaluated at − x1ωCL equalling − x 3 . Then,
[
2
Kc = −
Ti =
[
x 2 x 3 x12
[
]
−1
2
2
2
1
3
−1
− 2 x1ξ + 1 sin cos ξ
−1
−1
3
−1
3
−1
1 2
ωCL
] [
−1
[
]
];
[cos ξ + φ ]+ x sin[cos ξ − φ ]+ x x sin[2 cos ξ] ;
x ω {x sin [cos ξ] + x [sin (cos ξ − φ ) + x sin φ ]}
x x sin [cos ξ] + x [x sin (cos ξ + φ ) − sin φ ]
[x x sin(cos ξ + φ ) + x sin(cos ξ − φ ) + x x sin(2 cos ξ)] .
2
x1 x 3 sin
1 CL
Td =
]
x1 x 3 sin cos −1 ξ + φ2 + x 3 sin cos −1 ξ − φ2 + x1x 2 sin 2 cos −1 ξ
3
2
3
2
2
1
−1
1
−1
−1
1 2
2
2
2
2
1 2
−1
b720_Chapter-03
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Handbook of PI and PID Controller Tuning Rules
314
Rule
Trybus (2005) –
continued.
Kc
Ti
Td
Comment
(1 + τm ) x1
Km
Tm1 + 0.5τm
0.5Tm1τm
Tm1 + 0.5τm
τm
>2
Tm1
τm
Tm1
2.2
2.5
2.9
3.3
4.0
5.0
6.7
10
20
n
x1
0.12 0.13 0.15 0.16 0.19 0.21 0.26 0.32 0.43
3
x1
0.09 0.10 0.11 0.13 0.15 0.17 0.21 0.26 0.37
4
x1
0.07 0.08 0.09 0.10 0.12 0.14 0.17 0.22 0.33
5
G m (s) = K m e −sτ m (1 + sTm1 ) .
n
Direct synthesis: frequency domain criteria
Skoczowski and
Tarasiejski
(1996).
4
Td
Model: Method 1
0.5(n − 1)Tm1
λ ≥ nTm1
Tm1
Kc
Robust
Larionescu
(2002).
Model: Method 1
4
nTm1
K m (λ + τm )
ωg Tm1
K m e − sτ m
G m (s ) =
. Kc ≤
n
Km
(1 + sTm1 )
nTm1
G m (s) = K m e −sτ m (1 + sTm1 ) .
n
(1 + ω
g
2
Tm12
2
1 + ωg Td
)
n −1
, Td =
2
n −1
Tm1 , n ≥ 2 with
n+2

2n + 4 τm 4n + 2 
− Tm1  n 2 − 2n − 2 +
+
φm  ± b
π Tm1
π


ωg =
, and with


+
τ
−
4
n
2
2
n
2
2
2
m
+
φm 
2Tm1 n − 4n + 3 +
π Tm1
π


2
b = Tm1
 2
2n + 4 τ m 4 n + 2 
+
φm  + c , with
 n − 2n − 2 +
π Tm1
π


2 
4n + 2  τ m  2n − 2 


+
φm  ,
c = 4(n + 2)1 + φm  n 2 − 4n + 3 +
π
π  Tm1 
π

 

φ m = specified phase margin.
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
315
3.9.3 Ideal controller in series with a first order lag
 1

1
G c (s) = K c 1 +
+ Td s 
 Tf s + 1
 Ti s
E(s)
R(s)
U(s)

 1
1
K m e − sτ
K c 1 +
+ Td s 
(1 + sTm1 ) n
 Ti s
 Tf s + 1
m
+
−
Y(s)
Table 47: PID controller tuning rules – general model G m (s) =
Rule
Schaedel (1997).
Model: Method 2
1
Ti =
Kc
Ti
Td
K m e − sτ m
(1 + sTm1 ) n
Comment
Direct synthesis: frequency domain criteria
1
Td
Tf = Td N ,
0.563  n + 1 
Ti


5 ≤ N ≤ 20 .
Km  n − 2
3(n + 1)
(τm + Tar ) , Td = n + 1 (τm + Tar ) .
5n − 1
6n
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316


Td s
1
+
3.9.4 Controller with filtered derivative G c (s) = K c 1 +
 Ti s 1 + s Td

N

R(s)


Td s
1
+
K c 1 +
T

Ti s
1+ s d

N

E(s)
−
+






U(s)
K m e − sτ
(1 + sTm1 ) n
m
Table 48: PID controller tuning rules – general model G m (s) =
Rule
Kc
Ti






Y(s)
K m e − sτ m
(1 + sTm1 ) n
Comment
Td
Direct synthesis: time domain criteria
Davydov et al.
(1995).
ξ = 0.9;
N = Km ;
Model: Method 1
1
2
Kc
Ti
0.25Ti
τ*m
> 0.5
nTm1 + τm
Kc
Ti
Td
τ*m
< 0. 5
nTm1 + τm
0.5(n − 1)Tm1
λ ≥ nTm1 ; N = 5.
Robust
Larionescu
(2002).
Model: Method 1
1
Kc =
nTm1
K m (λ + τm )
nTm1



τ*m
 ,
, Ti = (nTm1 + τm )0.911 − 0.716
nTm1 + τm  






K m  3.540
− 0.718 
nTm1 + τm


1
τ*m


(n − 1)! + (n − 1)! n
1
with τ*m = τm + Tm1 (n − 1) −
.
n −1 − (n −1)
n −i
(n − 1) e
(n − 1) (i − 1)!

i =1
∑
2
Kc =
1

τ*m
K m  2.766

nTm1 + τ m


 0.150τ*m
, Ti = (nTm1 + τ m )0.559 − 
 nT + τ


 m1 m
− 0.521



 .


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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
3.9.5 Two degree of freedom structure 1


Td s
1
+
U (s ) = K c  1 +
 Ti s 1 + Td

N






βTd s 
 E (s ) − K c  α +
 R (s )
Td 


s
1+ s 

N 






T
s
β
d
α +
K
Td  c

1+
s

N 

E(s)
R(s)
−
+


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N





s

−
U(s)
+
Y(s)
K m e − sτ
(1 + sTm1 )n
Table 49: PID controller tuning rules – general model G m (s) =
Rule
Kc
Kuwata (1987).
Model: Method 1
1
Kc =
x1 =
1
Kc
2
Kc
Ti
Kc =
m
K m e − sτ m
(1 + sTm1 ) n
Comment
Direct synthesis: time domain criteria
0
Ti
Td
0.8(nTm1 + τm )
1
−
, α = 1,
K m (nTm1 + τm − x1 ) K m
(nTm1 + τm )2 − 0.267[n (n − 1)(n − 2)Tm13 + 3n (n − 1)Tm12τm + 3nTm1τm 2 + τm3 ] ,
[
 0.833 n ( n − 1)Tm12 + 2nTm1τm + τm 2
Ti = 1 −

(nTm1 + τm )2

2
Td
Ti
] 1.667[n(n − 1)T
2
m1
2
]
+ 2nTm1τm + τm
.
nTm1 + τm



x 2 − (nT + τ m )x 2
x x
1.20 x13
1
−
, Ti = 1.6671 − 2.5 22  2 , Td = 0.833x 2 1 3 m1
,
2

Km x2
Km
x1  x1
x 1 − 0.833x 2 2

α = 1 , β = 1 , N = ∞ ; x1 = n (n − 1)Tm12 + 2nTm1τm + τm 2 ,
x 2 = n (n − 1)(n − 2 )Tm1 + 3n (n − 1)Tm1 τm + 3nTm1τm + τm .
3
2
2
3
317
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318
3.10 Non-Model Specific

1 

3.10.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
R(s)
E(s)
+
−

1 

K c 1 +
Ti s 

U(s)
Non-model
specific
Y(s)
Table 50: PI controller tuning rules – non-model specific
Rule
Kc
Ti
Comment
Ziegler and Nichols
(1942).
Idzerda et al. (1955).
0.45K u
Ultimate cycle
0.83Tu
Quarter decay ratio.
Johnson (1956).
0.5K u
Hwang and Chang
(1987).
0.83K u
1
3TuK c
Tu
 5.22 5.22 

Tu
−
T1 
 Tu
p1 , T1 = decay rate, period measured under proportional control
0.45K u
1
p1
when K c = 0.5K u .
1
Parr (1989).
0. 5K u
0.43Tu
p. 191.
p. 192.
Parr (1989).
0.33K u
2Tu
Chen and Yang
(1993).
Hang et al. (1993b),
p. 59.
Pessen (1994).
0.36K u
5.25Tu
0.25K u
0.25Tu
0.25K u
0.167Tu
McMillan (1994).
0.3571K u
Tu
Dominant time delay
process.
Dominant time delay
process.
p. 90.
TuK c = period of the continuous oscillation signal (of the measured variable) generated
when K c = 0.83K u , with Ti being decreased until continuous oscillations exist.
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Åström and Hägglund
(1995), pp. 141–142.
Kc
Ti
Comment
0.4698K u
0.4373Tu
A m = 2, φ m = 20 0 .
0.1988K u
0.0882Tu
A m = 2.44,
φ m = 610 .
0.2015K u
0.1537Tu
A m = 3.45,
φ m = 46 0 .
Calcev and Gorez
(1995).
Edgar et al. (1997),
p. 8-15.
Klán and Gorez
(2000).
ABB (2001).
0.3536 K u
0.1592Tu
0.59 K u
0.81Tu
0.5 K m
0.5Tu
0. 5K u
0.8Tu
Robbins (2002).
Prokop et al. (2005).
0.3K u
2
Ti
0.3K u
3
Ti
0.5 K m
Wang et al. (1995b).
Vrančić et al. (1996),
Vrančić (1996),
p. 121.
Vrančić (1996),
p. 140.
Friman and Waller
(1997).
4
φm = 150 , large τ m .
Minimum IAE –
regulator.
K m assumed known.
Minimum IAE –
servo.
Good response –
regulator.
K m assumed known.
Direct synthesis: frequency domain criteria
Ti
Kc
0.5A 3
( A1A 2 − K m A 3 )
A3
A2
A m ≥ 2, φ m ≥ 60 0
(Vrančić et al., 1996).
0.45K u
A3
A2
Modified Ziegler–
Nichols method.
0.4830
3.7321
ω1500
A m > 2,
G p ( jω1500 )
2
Ti = (0.24 + 0.111K u K m )Tu .
3
Ti = (0.27 + 0.054K u K m )Tu .
4
0.4Tu
φm = 450 , small τ m


1
jωCL G CL ( jωCL )
1
Kc =
Im 
 , Ti =
ωCL
ωCL  G p ( jωCL ){1 − G CL ( jωCL )}
φ m > 150 .


jωCLG CL ( jωCL )
Im

(
)
{
(
)
}
ω
−
ω
G
j
1
G
j

CL
CL 
 p CL
.


jωCLG CL ( jωCL )
Rl 

 G p ( jωCL ){1 − G CL ( jωCL )}
319
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320
Rule
Kc
Kristiansson and
Lennartson (2000).
KuKm
K 1350 K m
K 1350 K m
6
Ti =
0.50K u K m − 0.36
.
(0.33K u K m − 0.17)ωu
7
Ti =
9
10
Ti =
K c = K135 0
K u K m > 0. 5
7
Ti
K u K m < 0.1 ;
K 1350 K m ≤ 0.1 .
K u K m < 0.1 ;
Ti
K 1350 K m > 0.1 .
KuKm
ω u (0.18K u K m + 1)
K u K m ≤ 10
Kc
Ti
K u K m > 10
10
Kc
Ti
1.6 ≤ M s ≤ 1.9
(0.315K1350 K m − 0.175)ωu
5.4K1350 K m − 13.6
Ti
9
20K135 0 K m − 160
(1.32K1350 K m − 3.2)ωu
6
8
2
Kristiansson and
0.33K u K m − 0.15
Lennartson (2002a). K (0.18K K + 1)
m
u m
1.18K u K m − 1.72
.
(0.33K u K m − 0.17)ωu
0.1 ≤ K u K m ≤ 0.5
2
5.4 K 1350 K m − 13.6
Ti =
Ti
2
20 K 1350 K m − 160
5
5
2
0.50K u K m − 0.36
Kristiansson (2003).
Comment
Ti
1.18K u K m − 1.72
KuKm
8
FA
.
.
0.09K1350 K m + 0.25
0.09K1350 K m + 1.2
, Ti =
(
K1350 K m
ω1350 0.09K1350 K m + 1.2
).





K 0
ω150 0 
K 0
0.075

Kc =
0
.
2
+
, Ti = ω150 0 0.06 + 1.6 150 − 0.06 150


K150 0
Km

Km
 Km

+ 0.05 


K
m






2
.


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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Kristiansson and
Lennartson (2006).
11
Kristiansson and
Lennartson (2003).
12
Kristiansson and
Lennartson (2006).
13
Kc
1.6(Tar − τm )
0.15
Ku
0.8Tu
1 + 3.7(K u K m )
0.16
Ku
Tu
1 + 4.5(K u K m )
0.81K 37%
Other tuning rules
1.29T37%
0.05 ≤
K 1500
Km
≤ 0. 9 ;
K 1500
Km
≤ 0. 9 ;
K u > 0.2K m (close to
optimal tuning);
M s = M t = 1.4 .
37% decay ratio.
[0.63K 33% ,0.97 K 33% ] [1.1T33% ,1.3T33% ]
(
33% decay ratio.
0.90K 37%
T37%
37% decay ratio.
1.19 K 37%
0.92T37%
37% decay ratio.
1.02 K 37%
0.92T37%
37% decay ratio.


0.075
2 0.2 +

K150 0 K m + 0.05 

Kc =
K m 0.06 + 1.6 K150 0 K m − 0.06 K150 0 K m
(
K 0

K c = 2Tar  0.7 − 0.45 150
Km

K c = 1.6(Tar − τm )
0.05 ≤
Ti
Kc
)
)
(
Ti =
13
≤ 0. 9 ;
K u ≥ 0.2 K m (robust); K u ≥ 0.5K m (close to optimal tuning).
MacLellan (1950).
12
Km
1.60 ≤ M max ≤ 1.85 .
Rutherford (1950).
[
K 1500
1.60 ≤ M max ≤ 1.85 .
Åström and Hägglund
(2006), p. 239.
11
0.05 ≤
Ti
1.65 ≤ M max ≤ 1.85 .
Hägglund and Åström
(2004).
Aikman (1950).
Comment
Ti
Kc
321
)2 ]
,
[
(
2
)
(
ω150 0 0.06 + 1.6 K150 0 K m − 0.06 K150 0 K m



K
 ω150 0 
0.075
 , T = 2T  0.7 − 0.45 150 0

+
0
.
2
i
ar 
 K 
K150 0

Km
 m

+ 0.05 

K
m



ω150 0 
0.075
0.2 +
.
K m 
K150 0 K m + 0.05 
(
)

.


)2 ]
.
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322
Rule
Kc
Ti
Comment
Young (1955).
K 37%
< 1.1Tu
37% decay ratio.
K x%
0.5 + 0.361 ln x
Tx %
1.2 1 + 0.025 ln 2 x
0.999 K 25%
0.814T25%
Parr (1989), p. 191.
0.667 K 25%
T25%
Example: x = 25 i.e.
quarter decay ratio.
Quarter decay ratio.
McMillan (1994),
pp. 42–43.
0.42 K 25%
T25%
‘Fast’ tuning.
0.33K 25%
T25%
‘Slow’ tuning.
Wade (1994), p. 114.
0.7515K 25%
0.7470T25%
ECOSSE team
(1996b).
Hay (1998), p. 189.
0.65K 50%
0.7917T50%
K 25%
T25%
Bateson (2002),
p. 616.
K 25%
Tu
McMillan (2005),
p. 62.
0.25K 25%
Chesmond (1982),
p. 400.
sin φ m
Åström (1982).
G p ( jω900 )
Leva (1993).
Åström and Hägglund
(1995), p. 248.
 τ
0.1 + 65.31 m
 T25%
tan φ m
ω900
Ti
0.5
4
ω1350
G p ( jω1350 )
0.25
1. 6
ω1350
G p ( jω1350 )
Hägglund and Åström
(1991).
14
Kc =
^
^
0.25 K u
tan (φm − φω − 0.5π )
G p ( jω) 1 + tan (φm − φω − 0.5π )
2
‘Modified’ ultimate
cycle method.
T25%
Kc
14
x = 100(decay ratio).
0.2546 Tu
, Ti =



2
tan( φ m − φ ω − 0.5π)
>0
Alfa-Laval
Automation ECA400
controller.
Alfa-Laval
Automation ECA400
controller – large
delay.
Model: Method 2
tan (φm − φω − 0.5π )
.
ω
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FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
NI Labview (2001).
Model: Method 2
0.4 K u
^
0.18 K u
Jones et al. (1997).
Model: Method 2
^
^
‘Little’ overshoot.
∧
Model: Method 5
∧
0.333 A p
2 Tu
0.2 hTu sin φ m
Ap
0.16Tu tan φ m
^
h
Ap
x1 Tu
0.4614  K m h 
K m  A p 
Majhi (1999),
pp. 132–137.
0.5380  K m h 
K m  A p 
2.5465h sin φ m
Lee et al. (2005).
‘Some’ overshoot.
1.6 Tu
Majhi (1999),
pp. 132–137.
Tang et al. (2002).
Model: Method 4
^
0.8 Tu
0.3 K u
0.5787
Quarter decay ratio.
0.8 Tu
^
Cox et al. (1994).
^
0.8 Tu
0.13 K u
Pagola and
Pecharromán (2002).
Parr (1989).
Comment
Ti
^
323
0.5554
0.9395
^
^
Tu  K m h 
3.7736  A p 
^
0.7958 ≤ x1 ≤ 1.5915.
Tu  K m h 
3.2541  A p 
0.5752
tan φ m
^
A p x 1 ω900 A m
ω900
0.45
1 + Tm 2 x12
Km
5.215
x1
Minimum ISTE –
servo.
0.5468
Minimum ISTE –
regulator.
0.81 ≤ x 1 ≤ 1 ;
x 1 = 1 , ‘small’ τ m ;
x 1 = 0.91 , ‘large’ τ m .
Equivalent to Ziegler
and Nichols (1942). 15
15
Representative result for FOLPD process model. Other formulae may be deduced, equivalent to
other ultimate cycle tuning rules, for this process model. In addition, tuning rules may also be
deduced for process models in SOSPD form, FOLIPD form, FOLIPD form (with a negative zero)
and SOSIPD form, as well as processes of 4th, 5th and 6th order without an explicit time delay term.
x1 =
2
τm
16(τm Tm ) + 66 − 181(τm Tm )2 + 762(τm Tm ) + 3081
(τm
Tm ) + 17
, 0.1 ≤
τm
≤ 2.0 .
Tm
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324


1
3.10.2 Ideal PID controller G c (s) = K c 1 +
+ Td s 

 Ti s
E(s)
R(s)
+
−

 U(s)
1
K c 1 +
+ Td s 
 Ti s

Non-model
specific
Y(s)
Table 51: PID controller tuning rules – non-model specific
Rule
Kc
Ziegler and
Nichols (1942).
[ 0. 6 K u , K u ]
Ti
Td
Ultimate cycle
0.5Tu
0.125Tu
Farrington
Tu
[ 0.1Tu , 0.25Tu ]
[ 0.33K u ,0.5K u ]
(1950).
McAvoy and
0.54 K u
Tu
0.2Tu
Johnson (1967).
Atkinson and
0.25K u
0.75Tu
0.25Tu
Davey (1968).
20% overshoot – servo response.
Ku
0.5Tu
0.125Tu
Carr (1986);
0.6667 K u
Tu
0.167Tu
Pettit and Carr
(1987).
0. 5K u
1.5Tu
0.167Tu
Tu
0.2Tu
0. 5K u
Tu
0.25Tu
0. 5K u
0.34Tu
0.08Tu
Tinham (1989).
0.4444 K u
0.6Tu
0.19Tu
Blickley (1990).
0. 5K u
Tu
Corripio (1990),
p. 27.
0.75K u
0.63Tu
Parr (1989),
pp. 190,191,193.
0. 5K u
[0.125Tu ,
0.167Tu ]
0.1Tu
Comment
Quarter decay
ratio.
Underdamped.
Critically
damped.
Overdamped.
ξ ≈ 0.45
Less than quarter
decay ratio
response.
Quarter decay
ratio.
Quarter decay
ratio.
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Åström (1982).
Kc
Td
Ti
K u cos φ m
1
Ti
Comment
x1Ti
0.87 K u
Representative results:
0.55Tu
0.14Tu
φ m = 30 0
0.71K u
0.77Tu
0.19Tu
φ m = 450
0.50K u
1.19Tu
0.30Tu
φ m = 60 0
Ku
Am
Arbitrary
4 π 2 Ti
K u cos φ m
x1Td
Hang and Åström
(1988b).
K u sin φ m
Tu (1 − cos φ m )
π sin φ m
Tu (1 − cos φ m )
4π sin φ m
Åström and
Hägglund (1988),
p. 60.
0.35K u
0.77Tu
0.19Tu
0.4698K u
0.4546Tu
0.1136Tu
0.1988K u
1.2308Tu
0.3077Tu
0.2015K u
0.7878Tu
0.1970Tu
0.27 K u
2.40Tu
1.32Tu
0.906K u
0.5Tu
0.125Tu
φ m = 250
0.866K u
0.5Tu
0.125Tu
φ m = 30 0
0. 5K u
0.5Tu
0.125Tu
p. 90.
0.5Tu
0.125Tu
p. 21.
0.1592Tu
0.0398Tu
Åström and
Hägglund (1984).
Åström and
Hägglund (1995),
pp. 141–142.
Chen and Yang
(1993).
De Paor (1993).
McMillan (1994)
McMillan (2005) [0.3K u ,0.5K u ]
1
2
Calcev and
Gorez (1995).
0.3536K u
ABB (1996).
0.625K u
325
Tu
2
Td
Specify A m .
Specify φ m .
Am ≥ 2 ,
φ m ≥ 450 .
A m = 2,
φ m = 20 0 .
A m = 2.44,
φ m = 610 .
A m = 3.45,
φ m = 46 0 .
0
φ m = 45 , ‘small’ τ m ; φ m = 150 , ‘large’ τ m .
0.5Tu
0.083Tu
‘P+D’ tuning
rule.
T
Ti =  tan φm + 4 x1 + tan 2 φm  u . x1 = 0.25 (Åström, 1982);

 4απ
x1 = 0.5 (Seki et al., 2000); 0.1 ≤ x1 ≤ 0.3 (Seki et al., 2000).
T
Td =  tan φ m + 1 + tan 2 φ m  u , x1 not specified.

 π
b720_Chapter-03
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Handbook of PI and PID Controller Tuning Rules
326
Rule
Kc
Ti
Td
Luo et al. (1996).
0.48K u
0.5Tu
0.125Tu
[0.213Tu ,
[0.133Tu ,
1.406Tu ]
0.469Tu ]
0.46K u
2.20Tu
0.16Tu
0. 4 K u
0.5Tu
0.125Tu
Based on work
by Tyreus and
Luyben (1992).
p. 26.
0. 5K u
Tu
0.125Tu
p. 27.
[0.32 K u ,0.6K u ]
Karaboga and
Kalinli (1996).
Luyben and
Luyben (1997),
p. 97.
Tan et al.
(1999a).
Tan et al.
(1999a).
Tighter damping than quarter decay ratio tuning.
0. 4 K u
0.333Tu
0.083Tu
Wojsznis et al.
(1999).
Yu (1999), p. 11.
Tan et al. (2001).
Robbins (2002).
Smith (2003).
0.125Tu
Some overshoot.
0. 2 K u
0.5Tu
0.125Tu
No overshoot.
Kc
Ti
0.25Ti
0.33K u
0.5Tu
0.333Tu
0. 2 K u
0.5Tu
0.333Tu
0.45K u
4
Ti
0.25Tu
0.45K u
5
Ti
0.25Tu
0.625Tu
0.1Tu
0.625Tu
0.1Tu
0.6Tu
1 + 2(K u K m )
Td
0.75K u
Åström and
Hägglund (2006),
pp. 240–241.
Ti ωu G 'c ( jωu )
2
0.5Tu
[K u ,1.67K u ]
Alfaro Ruiz
(2005a).
Kc =
0.33K u
3
Chau (2002),
p. 115.
3
Comment
2
0.25Ti ωu + 1
6
Kc
, Ti = 0.3183Tu tan
‘Just a bit of’
overshoot.
No overshoot.
Minimum IAE –
servo.
‘Good’ response
– regulator.
Quarter decay
ratio.
K u > 0.2K m
(close to optimal
tuning);
M s = M t = 1. 4 .
0.5π + ∠G 'c ( jωu )
,
2
G 'c ( jωu ) = desired frequency response of controller G c ( jω) at ωu .
4
Ti = (0.24 + 0.111K m K u )Tu .
5
Ti = (0.27 + 0.054K m K u )Tu .
6
1
Kc =
Ku
4

 

0.3 − 0.1 K u   , Td = 0.15Tu 1 − (K u K m ) .
K  
1 − 0.95(K u K m )

 m 

b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
Rutherford
(1950).
1.25K 37%
T37%
Other rules
0.125T37%
1.12 K 37%
T37%
0.25T37%
1.77 K 37%
1.29T37%
0.16T37%
1.67 K 37%
1.29T37%
0.16T37%
MacLellan
(1950).
Harriott (1964),
pp. 179–180.
Lipták (1970),
pp. 846–847.
1.49 K 37%
0.98T37%
0.24T37%
K 25%
0.167T25%
0.667T25%
K 25%
0.667T25%
0.167T25%
Kc
Ti
0.25Ti
0.999 K 25%
0.488T25%
0.122T25%
K 25%
0.67T25%
0.17T25%
37% decay ratio.
0.83K 25%
0.5T25%
0.1T25%
0.67 K 25%
0.5T25%
0.1T25%
‘Slow’ tuning.
Wade (1994), [ 1.002 K 25% ,
p. 114.
1.67 K
0.45T25%
0.1125T25%
0.8497 K 50%
0.475T50%
0.1188T50%
K 25%
0.667T25%
0.167T25%
K 25%
0.6667Tu
0.1667Tu
7
Parr (1989),
p. 191.
McMillan
(1994), p. 43.
ECOSSE team
(1996b).
Hay (1998),
p. 189.
Bateson (2002),
p. 616.
Rotach (1994).
8
37% decay ratio.
Quarter decay
ratio.
Quarter decay
ratio.
x = 100(decay
ratio).
Example: x = 25
i.e. quarter decay
ratio.
Quarter decay
ratio.
‘Fast’ tuning.
Chesmond
(1982), p. 400.
7
Comment
Td
8
Kc
25%
]
Ti
− 0.5
Tx %
K x%
, Ti =
.
0.5 + 0.361ln x
2 1 + 0.025 ln 2 x
M max
2
Kc =
G p ( jωM max ) , Ti = −
2

dφωM
M max − 1
max
ωM max 2 
 dωM max
Kc =



.
dφ ωM
max
dω M max
‘Modified’
ultimate cycle.
327
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Handbook of PI and PID Controller Tuning Rules
328
Rule
Rotach (1994) –
continued.
García and
Castelo (2000).
Ti
Td
9
Kc
Ti
Td
10
Kc
Ti
0.25Ti
^
0. 4 K u
Parr (1989).
0.5 A p
^
M max 2 − 1
Td = −
11
Kc =
Kc =
^
Td =
^
0.159 Tu
∧
∧
Tu
0.25 Tu
Ti
Td
G p ( jωr ) ,
dφωr 
 


0.5
1
1
−1
2
[ ωr
+ φ ωr  ].
+ φωr   + tan  π − sin −1
1 + tan  π − sin
M
dωr 
M
ωr

max
max




(
cos 1800 + φm − ∠G p ( jω1 )
) , T = 2[sin(180 + φ − ∠G ( jω )) + 1] .
ω cos(180 + φ − ∠G ( jω ) )
0
i
G p ( jω1 )
1 + x12 x 2 2 + sin 2 φm
, with x1 =
m
0
1
1
x3 =
Ti =
Self-regulating
processes.
Non selfregulating
processes.
^
0.083 Tu
Tu
Kc
ω1 < ωu
2
,
 dφω 



1
1
−1
−1
2
r
+ φω r 
ωr ωr
+ φω r  − tan π − sin
1 + tan  π − sin
M max
M max
 dωr 




Ti = −
10
11
M max
Kc =
^
0.3333 Tu
0. 4 K u
Zhang et al.
(1996a).
Comment
Kc
Lloyd (1994).
Model: Method 3
9
FA
x3
1 + x 32
p
m
p
1
π
2
2
, x2 =
Ap − ε4 ,
4h
x 2 − cos φm
2
1
2
x 2 + sin φm
sin φm or x 3 =
cos φm − x 2
x 2 2 + sin 2 φm
A x −x
Am2 − 1
Am2 − 1
or Ti =
; Td = m 4 2 1 or
A mωg (x 4 − A m x1 )
A mωg (x 4 + A m x1 )
ωg A m − 1
(
A m x 4 + x1
(
2
) , with x
ωg A m − 1
4
)
∈ [A m x1 + 1.0, A m x1 + 1.2] or
∧ 
∧ 
∧ 
∧ 
 
 ∧

x 4 ∈   ωpc − ωu  ωu  − 0.4,   ωpc − ωu  ωu  + 0.4 , ωu obtained from an ideal


 




∧ 
∧ 

relay experiment. x 4 = A m x1 + 1.2 or x 4 =   ωpc − ωu  ωu  + 0.4 (minimum IAE,



regulator).
sin φm ;
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Jones et al.
(1997).
Shin et al.
(1997).
Majhi (1999),
pp. 132–137.
Majhi (1999),
pp. 132–137.
NI Labview
(2001).
Model: Method 2
12
Kc =
0.7490h A p
^
Model: Method 2
^
x1 Tu
0.125 Tu
0.2526 ≤ x1 ≤ 0.5053 .
12
Kc
Ti
x 3Ti
Model: Method 2
13
Kc
Ti
Td
14
Kc
Ti
Td
Minimum ISTE –
servo.
Minimum ISTE –
regulator.
Quarter decay
ratio.
‘Some’
overshoot.
∧
∧
0.6 K u
∧
0.5 Tu
∧
0.12 Tu
∧
0.25 K u
∧
0.5 Tu
0.12 Tu
ρ
,
 ∧


1 
1 

 − x 2  x 3 ωu Ti − ∧
x1  x 3ωu Ti −
 − y 2 − ρx 2

ωu Ti 



T
ω
u i 


(x 2 − x1 ) + (x1 − x 2 )2 + 4 x 3ρx1ωu + x 3 y2 ωu  ρx1 +
∧
 ωu

Ti =
Comment
Td
Ti
329
y 2 
∧ 

ωu 
,
∧ 

2 x 3ρx1ωu + x 3 y 2 ωu 


∧ 

 ωu − ωu 
2
1
1


 1− ξ ,
x1 = −
, x 2 = ∧ cos ∠G p ( j ωu )  , ρ = 
Ku
ωu
ξ


Ku
∧
1


y 2 = ∧ sin ∠G p ( j ωu )  . Typical x 3 : 0.1; typically 0.3 ≤ ξ ≤ 0.7 .


Ku
∧
13
Kc =

∧ K h
K h
1 
0.5585 m  + 0.03835 , Ti = 0.259 Tu  m 
 Ap 
 Ap 
Km 






0.516
,
∧
(
)
.
∧
(
)
.
Td = 0.124 Tu K m h A p
14
Kc =
1
Km


∧ K h
K h
0.7837 m  − 0.0526 , Ti = 0.257 Tu  m 
 Ap 
 Ap 






0.403
0.197
,
Td = 0.131Tu K m h A p
1.109
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Handbook of PI and PID Controller Tuning Rules
330
Rule
Kc
NI Labview
(2001) –
continued.
Chen and Yang
(1993).
Ti
∧
15
0.5 Tu
0.12 Tu
‘Little’
overshoot.
4Td
Td
Model: Method 6
∧
0.15 K u
0.25 + 0.87 x1
0.25 + x12
Comment
Td
∧
2.5465h sin φ m
16
^
Tang et al.
x 2Td
0.81 ≤ x 1 ≤ 1
Td
A p x 1 ω900 A m
(2002).
Model: Method 4
x 1 = 1 , ‘small’ τ m ; x 1 = 0.91 , ‘large’ τ m ; 1.5 ≤ x 2 ≤ 4 .
Dutton et al.
(1997).
17
Tue
0.5K eu
0.25Tue
pp. 576–578.
Direct synthesis: time domain criteria
Ramasamy and
Ti
18
Sundaramoorthy K (τ + T )
Td
Ti
m CL
CL
(2007).
2
15
x1 =
π Ap − εr
, 2ε r = relay deadband, d r = relay output amplitude;
4d r
− x 2 ± x 22 + 1
Td =
16
2
^
2 ωu
, x2 =
0.43 − 0.5x1
.
0.25 + 0.87 x1
− cot φ m + cot 2 φ m + 4 x 2
Td =
^
.
2 ω90 0
17
K *c
K eu =
(8 − r )2
1−
, Tue =
6.283  (8 − r )3.5 
1 −

ωd 
1110 
0.286
with r = ratio of the height of the first peak
55
to the height of the first trough of the closed loop response, when controller gain K *c is applied;
ωd = damped natural frequency.
18
Desired closed loop transfer function = e −sτ CL 1 + sTCL .
2
2
−
Ti = t − τCL +
−

τCL
, Td =  Ti + τCL
2(τCL + TCL )

−

 τCL − t  − σ 2
_
τCL 3

,
−
− t  + 
2Ti
6Ti (τCL + TCL )

with t = mean and σ 2 = variance of the process impulse response.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ramasamy and
Sundaramoorthy
(2007) –
continued.
19
Kc
21
Td
Ti
Td
Kc
max
22
Åström and
Hägglund (1995),
p. 217.
Streeter et al.
(2003), Keane et
al. (2005).
19
23
24
Comment
Direct synthesis: frequency domain criteria
0 < KmKu < ∞ ;
Ti
Td
M
= 1.4.
20
Åström and
Hägglund (1995),
p. 217.
Ti
Kc
Ti
Td
Kc
Ti
Td
(
0 < KmKu < ∞ ;
M max = 2.0.
0 < KmKu < ∞ ;
M max = 2.0.
)
Desired closed loop transfer function = e −sτ CL 1 + 2ξCL TCL1s + TCL12s 2 .
Kc =
2
2
− 2TCL1
τCL
Ti
, Ti = t − τCL +
,
K m (τCL + 2ξCL TCL1 )
2(τCL + 2ξCL TCL1 )
−
2

Td =  Ti + τCL

−

 τCL − t  − σ 2
_
−
τCL 3

− t  + 
, with t = mean
−
2Ti
6Ti (τCL + 2ξCL TCL1 )

and σ 2 = variance of the process impulse response.
20
21
22
23
24


1
Tuning rules converted from formulae developed for G c (s) = K c  b +
+ Td s  .
Ti s


(
2
)
(
)
(
2
)
(
2
)
K c = 0.19e −1.61κ + 2.5 κ K u , Ti = 0.44e −2.9 κ + 3.14 κ Tu , Td = 0.29e0.84 κ −5.6 κ Tu


1
Tuning rules converted from formulae developed for G c (s) = K c  b +
+ Td s  .
T
s
i


(
2
2
)
(
2
)
K c = 0.18e −1.04 κ +1.08 κ K u , Ti = 0.15e −0.74 κ + 0.26 κ Tu , Td = 0.60e −1.96 κ + 0.68 κ Tu .
(
K c x1 = 0.72e
(
Ti
0.59e
=
x1
−1.6 κ +1.2 κ 2
−1.3 κ + 0.38 κ 2
)K
)([ 0.72e
[0.72e
− 0.0012340Tu − 6.1173.10 ,
−1.6 κ +1.2 κ 2
−1.6 κ +1.2 κ
2
Td x1 =
u
−6
2
)K
u
− 0.040430e
( )
0.108e− 3κ +1.76 κ K u Tu − 0.0026640 eTu
2
]
− 0.0012340Tu − 6.1173.10−6 Tu
−1.3 κ + 0.38 κ 2
(
log 1.6342 log K u
0.72K u e −1.6 κ +1.2 κ − 0.0012340Tu − 6.1173.10− 6
)
]K
,
u
2
, x1 = 0.25e0.56 κ − 0.12 κ +
Ku
.
eK u
331
b720_Chapter-03
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FA
Handbook of PI and PID Controller Tuning Rules
332
3.10.3 Ideal controller in series with a first order lag
 1

1
G c (s) = K c 1 +
+ Td s 
 Tf s + 1
 Ti s
R(s) E(s)
−
+


1
K c 1 +
+ Td s 
Ti s


Rule
Kc =
K 0
K m  150
 Km
U(s)
Non-model
specific
Y(s)
Table 52: PID controller tuning rules – non-model specific
Comment
Kc
Ti
Td
Kristiansson
(2003).
1
1
Tf s + 1
1
Kc
Minimum performance index: other tuning
Ti
Td
‘Almost’ optimal tuning rule; K 1500 ≥ 0.03 ; 1.6 ≤ M s ≤ 1.9 .
0.60

K 0
K 0

+ 0.07  0.32 + 1.6 150 − 0.8 150
K

m

 Km




2
1.5
,
2

K150 0
 K150 0  
 
ω150 0 0.32 + 1.6
− 0.8
Km
K m  




0.6667
,
Td =
2

K150 0
 K150 0  
 
ω150 0 0.32 + 1.6
− 0.8
Km
K m  




0.4
Tf =
K 0
K 0
K 0

ω150 0  150 + 0.07  0.32 + 1.6 150 − 0.8 150
Km
 Km
 Km
 
,



Ti =
Tf =
 K 0
ω150 0 20 150
Km

1

K 0

K 0
+ 0.5 0.32 + 1.6 150 − 0.8 150

K
m

 Km
2
2



K
  min 6 + m ,25
 
K150 0


 




2



.
or
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ramasamy and
Sundaramoorthy
(2007).
2
2
Kc
3
Kc
Ti
333
Comment
Td
Direct synthesis: time domain criteria
Applies to
x1 + Tf
Td
processes with
x 3 + Tf
dominant
Tf
numerator
x 2 + Tf
1+
dynamics.
x2
Desired closed loop transfer function = e −sτ CL 1 + sTCL .
Kc =
2
−
 Tf 
τCL
, Tf not specified;
1 +  , x1 = t − τCL +
K m (τCL + TCL )  x1 
2(τCL + TCL )
Ti
2
−

 τ − t  − σ 2
3
−  CL
τCL

x1 + τCL − t + 
+ Tf −
2 x1
6 x1 (τCL + TCL )
Td =
,
Tf
1+
2
_
τCL
t − τCL +
2(τCL + TCL )
−
with t = mean and σ 2 = variance of the process impulse response (treated as a statistical
distribution).
3
(
)
Desired closed loop transfer function = e −sτ CL 1 + 2ξCL TCL1s + TCL12s 2 .
Kc =
 Tf 
τCL
x2
,
1 +  , x 2 = t − τCL +
K m (τCL + 2ξCLTCL1 )  x 2 
2(τCL + 2ξCL TCL1 )
−
2
2
− 2TCL1
2

x 3 =  x 2 + τCL

−

 τCL − t  − σ 2
_
−
τCL3

, with t = mean
−
− t  + 
2x 2
6 x 2 (τCL + 2ξCLTCL1 )

and σ 2 = variance of the process impulse response (treated as a statistical distribution).
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
334
Rule
Kc
Wang et al
(1995b).
Kristiansson and
Lennartson
(2002b).
4
17-Mar-2009
Ti
4
Kc
5
Kc
Td
Comment
Direct synthesis: frequency domain criteria
Ti
Td
Ti
0.4444Ti
Plant does not contain an integrator; 1.6 ≤ M max ≤ 1.9 .
For a stable process, K m is assumed known; for an integrating process, K m and τ m are assumed
known. K c =


1
jωCL G CL ( jωCL )
Im 
,
ωCL  G p ( jωCL ){1 − G CL ( jωCL )}


jωCL G CL ( jωCL )
Im

G
j
1
G
j
ω
−
ω
(
)
{
(
)
}

1
CL
CL 
 p CL
Ti =
ωCL 

jωCL G CL ( jωCL )
Rl 

 G p ( jωCL ){1 − G CL ( jωCL )}
+



jωCL G CL ( jωCL )
j2ωCL G CL ( j2ωCL )
1  
 Rl 
 − Rl 
 ,
3   G p ( jωCL )[1 − G CL ( jωCL )] 
 G p ( j2ωCL )[1 − G CL ( j2ωCL )] 

 


 
j2ωCLG CL ( j2ωCL )
jωCLG CL ( jωCL )
 Rl
 − Rl

 G p ( j2ωCL )[1 − G CL ( j2ωCL )] 
1   G p ( jωCL )[1 − G CL ( jωCL )]
Td = 
.
3


jωCL G CL ( jωCL )

Im



 G p ( jωCL )[1 − G CL ( jωCL )]


5
Kc =
Ti =
Tf =


0.48
1.5
− 0.15
 K150 0 K m + 0.07

(
[
)
(
)
(
K m 0.34 + 1.5 K150 0 K m − 0.7 K150 0 K m
[
1.5
(
)
(
)2 ]
ω150 0 0.34 + 1.5 K150 0 K m − 0.7 K150 0 K m
[
(
,
)2 ]
,


0.48
− 0.15


 K150 0 K m + 0.07
(
)
(
)
ω150 0 0.34 + 1.5 K 150 0 K m − 0.7 K 150 0 K m
)2 ]2 min{3 + 2(K m
) }
K 150 0 ,25
.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
6
Kristiansson and
Lennartson
(2006).
Kc
Ti
Td
7
Kc
Ti
Td
8
Kc
Ti
Td
Comment
M max ≈ 1.7
Plant contains an integrator.
Ultimate cycle
0.5Tu
0.125Tu
[0.3K u ,0.5K u ]
(
[ (
Tf =
Kc =
Ti =
8
Td


0.45
1. 5 
− 0.1
1.5
 K150 0 K m + 0.07
 , T =
,
Kc =
i
K m 0.86 K150 0 K m + 0.44
ω150 0 0.86 K150 0 K m + 0.44
Td =
7
Ti
Plant does not contain an integrator; 1.65 ≤ M max ≤ 1.85 .
McMillan
(2005).
6
Kc
Kc =
Td =
ω1500
Km
)
[ (
]
)
]
)
0.6667
1
, Tf =
or
K150 0
K 1500
K 1500 




ω150 0 0.86
+ 0.44
ω1500 0.86
+ 0.44 2 + 14

Km
Km
Km 






0.45
− 0.1

K
K
0
.
07
+
 150 0 m

(
[ (
)
)
]
{
(
) }
ω150 0 0.86 K150 0 K m + 0.44 2 min 3 + 2 K m K150 0 ,25
3.333ω150 0
e
Km
3.333
ω150 0
3.5 −
3.5 −
2.94ω150 0
Km
2.94
ω150 0
(
6.4 −
9ω
e
4.4 −
3.5 −
G p jω
3ω150 0
150 0
(
(
G p jω150 0
6.5ω
150 0
Km
3ω150 0
Km
)
G p jω150 0
Km
1
3ω150 0
Km
(
150 0
Km
(
)
, Tf =
, Td =
)
G p jω
150 0
(
)
)
Km
≤ 0.91 .
0.083 ω1500
,
G p jω1500 ≥ 0.7 ;
ω1500
Km
(
)
, Ti =
(
2.94
ω150 0
3.5 −
3ω150 0


0.11 4.4−
G p jω150 0  , Tf =
e
3.5 −
ω1500
Km


(
K 1500
3ω150 0

0.834 
G p jω150 0  .
3.5 −
ω150 0 
Km

)
G p jω150 0
; 0.04 ≤
)
G p jω1500 < 0.7 , 1.7 ≤ M max ≤ 1.8 .
1
3ω150 0
Km
6.5ω
150 0
Km
(
)
(
G p jω150 0
G p jω
1500
)
,
)
,
335
b720_Chapter-03
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
336



Td s 
1

+
3.10.4 Controller with filtered derivative G c (s) = K c 1 +
 Ti s 1 + Td s 


N 



Td s
1
+
K c 1 +
T

Ti s
1+ d

N

E(s)
R(s)
−
+
 U(s)

Non-model


specific
s

Y(s)
Table 53: PID controller tuning rules – non-model specific
Comment
Td
Kc
Ti
Rule
Direct synthesis: frequency domain criteria
Ti
6.283 x1ω
φω > φm − π
Leva (1993).
N not specified.
1
Kc
2
Kc
x1Td
Td
6 ≤ x1 ≤ 10
Yi and De Moor
(1994).
Vrančić (1996),
p. 130.
3
Kc
4Td
Td
N not defined.
4
Td
N < 10
1
2
Ti
2(A1 − Ti )
ωTi
Kc =
2

2π 
 + ω2Ti 2
G p ( jω) 1 − ωTi
x1 

ωTi
Kc =
,
2
G p ( jω) 1 − ω2Ti Td + ω2Ti 2
(
4
Kc =
Ti =
0.5 cos θ
G p ( jωφ )
, Td =
A3
, Ti =
tan (φm − φω − 0.5π)
2π
ω 1+
tan (φm − φω − 0.5π )
x1
, x1 = 10 .
)
Td =
3
Ti
(
A 2 − Td A1 − Td
2
− x1ω + x12ω2 + 4x1ω2 tan 2 (φm − φω − 0.5π )
2 x1ω2 tan (φm − φω − 0.5π )
, φω < φm − π .
tan θ + 4 + tan 2 θ
, θ = 2250 − ∠G p ( jωφ ) .
2ωφ
N
),
2
Td =
− (A 3 − A 5A1 ) +
(A
2
3
)
2
− A 5A1 − (4 N )(A 3A 2 − A 5 )(A 5A 2 − A 4 A 3 )
(2 N )(A3A 2 − A5 )
.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Ti
5
Kc
Ti
Vrančić (1996),
p. 134.
6
Kc
Vrančić et al.
(1999).
Lennartson and
Kristiansson
(1997).
Kristiansson and
Lennartson
(1998a).
7
Vrančić (1996),
pp. 120–121.
5
Kc =
(A
2
3
)
Kc =
7
Kc =
Comment
Td
A3A4 − A 2 A5
A 3 2 − A1 A 5
N ≥ 10
Ti
x1Ti
N = 10 ;
0.2 ≤ x1 ≤ 0.5 .
Kc
Ti
Td
8 ≤ N ≤ 20
8
Kc
Ti
0.4Ti
K u K m ≥ 1.67
9
Kc
Ti
0.4Ti
(
0.5A3 A32 − A1A5
)
− A1A5 (A1A 2 − A3K m ) − (A3A 4 − A 2 A5 )A12
,
Ti =
6
337
(A
(
A3 A 32 − A1A 5
2
3
)
)
− A1A5 A 2 − (A3A 4 − A 2 A 5 )A1
A − A 2 2 − 4x1A1A 3
0.5Ti
.
, Ti = 2
A1 − K m Ti
2 x1A1
A3
2


T
2
2 A1A 2 − A 0 A 3 − Td A1 − d A 0 A1 


N


Td is obtained by solving:
A3
, Ti =
2
A 2 − Td A1 −
(
Td
Km
N
,
)
A 0 A 3 4 A1A 3 3 A 0 A 5 − A 2 A 3 2
2
Td +
Td −
Td + A 3 − A1A 5 Td + (A 2 A 5 − A 3A 4 ) = 0 .
N
N3
N2
8
Kc = KuKm
Ti =
9
N=
[
K u K m + 2.5 12K u 2 K m 2 − 35K u K m + 30
Kc
3
2
− 0.053ωu + 0.47ωu − 0.14ωu + 0.11
Kc = KuKm
Ti =
12K u 2 K m 2 − 35K u K m + 30
,N=
K u K m + x1 12K u 2 K m 2 − 35K u K m + 30
Kc
− 0.525ωu 3 + 0.473ωu 2 − 0.143ωu + 0.113
x1
KmKu
2.5 
35
30 
+
12 −
.
K m K u 
K m K u K m 2 K u 2 
12K u 2 K m 2 − 35K u K m + 30
[
,
],
];
ωu
≤ 0.4 ;
KuKm

35
30 
+
12 −
 ; x 1 = 3 , K u K m > 10 ; x 1 = 2.5 , K u K m < 10 .
K m K u K m 2 K u 2 

.
b720_Chapter-03
Rule
Kc
Kristiansson and
Lennartson
(1998a) –
continued.
Kristiansson and
Lennartson
(1998b).
Kristiansson and
Lennartson
(2000).
Kc
11
Kc =
2
Km Ku
2
0.4Ti
Ti
11
Kc
Ti
0.4Ti
12
Kc
Ti
0.333Ti
13
Kc
Ti
Td
K c as in footnote 9; Ti =
7.71
10
Kc =
0. 1 ≤ K u K m
≤ 0.5
Kc
3
− 0.185ωu + 1.052ωu
12K u 3K m 2 − 35K u 2 K m + 30K u
[
3
K m K u 3 + 2.5 12K u 2 K m 2 − 35K u K m + 30
3
2
Kc =
N = 2.5
2
ωu
≥ 0.4 .
− 0.854ωu + 0.309 K u K m
,
ωu
9.14
> 0.45;
−
+ 3.14 , K u K m > 1.67 ,
KuKm
K mK u
Ti =
13
Comment
Td
Ti
Ti =
12
FA
Handbook of PI and PID Controller Tuning Rules
338
10
17-Mar-2009
[
KcKm Ku
2
2
2
(1.1K u K m − 2.3K u K m + 1.6)
− 0.63ωu + 0.39ωu 2 + 0.15ωu + 0.0082
.
],
ωu 0.95K m K u − 2K m K u + 1.4
2
Kc
3
]
,N =
2.5
KmKu

35
30 
+
12 −
.
K m K u K m 2 K u 2 

2
2
K m K u (−20 + 13K m K u )(1 + 0.37 K m K u ) 2
 1.6( −20 + 13K m K u )(1 + 0.37 K m K u ) 
− 1 ,

2
2

 1.1K m K u − 2.3K m K u + 1.6
Ti =
2
2
K m K u (1.1K u K m − 2.3K u K m + 1.6)  1.6( −20 + 13K m K u )(1 + 0.37 K m K u ) 
− 1 ,

ωu (−20 + 13K m K u )(1 + 0.37 K m K u ) 2  1.1K m 2 K u 2 − 2.3K m K u + 1.6

Td =
K m K u (1.1K u K m − 2.3K u K m + 1.6)
ωu (−20 + 13K m K u )(1 + 0.37 K m K u ) 2
2
2

 ( −20 + 13K m K u ) 2 (1 + 0.37 K m K u ) 2


2
2
2
(1.1K m K u − 2.3K m K u + 1.6)

− 1 ,

 1.6( −20 + 13K m K u )(1 + 0.37 K m K u )
−1 

2
2
1.1K m K u − 2.3K m K u + 1.6


N=
2
2
K K (1.1K u K m − 2.3K u K m + 1.6)
Td
.
, Tf = m u
Tf
ωu (−20 + 13K m K u )(1 + 0.37 K m K u ) 2
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Kristiansson and
Lennartson
(2000) –
continued.
14
Kc =
Ti =
Ti
Td
Comment
14
Kc
Ti
Td
K u K m > 0.5
15
Kc
Ti
Td
K u K m < 0.1
2
2

(1.1K u K m − 2.3K u K m + 1.6) 2  4.8K m K u (1 + 0.37 K m K u )
− 1 ,
2
2
2
2
2 
3ωu K m K u (1 + 0.37 K m K u )  1.1K m K u − 2.3K m K u + 1.6 
2
2

(1.1K u K m − 2.3K u K m + 1.6)  4.8K m K u (1 + 0.37 K m K u )
− 1 ,

2
2
2
3ωu (1 + 0.37 K m K u )
 1.1K m K u − 2.3K m K u + 1.6 

 3K m 2 K u 2 (1 + 0.37 K m K u ) 2


2
2
2
2
2
(1.1K u K m − 2.3K u K m + 1.6)  (1.1K m K u − 2.3K m K u + 1.6)
,
Td =
1
−

 4.8K m K u (1 + 0.37 K m K u )
3ωu (1 + 0.37 K m K u ) 2
−1 

2
2
1.1K m K u − 2.3K m K u + 1.6


N=
15
2
2
Td
1.1K u K m − 2.3K u K m + 1.6
.
, Tf =
Tf
3ωu (1 + 0.37 K m K u ) 2
Kc =
 20.8(1.8 + 0.3K m K135 0 ) 
− 1 ,

13K m (1.8 + 0.3K m K1350 )  ( −6 + 3.7 K m K1350 )

Ti =
(−6 + 3.7 K1350 K m )K1350 K m  20.8(1.8 + 0.3K m K135 0 ) 
− 1 ,

13ω1350 (1.8 + 0.3K m K135 0 ) 2  (−6 + 3.7 K m K1350 )

(−6 + 3.7 K1350 K m ) 2
2

 169(1.8 + 0.3K m K 0 ) 2
135


2
(−6 + 3.7 K m K135 0 )K m K1350 
(−6 + 3.7 K m K1350 )

Td =
−
1
,

13ω1350 (1.8 + 0.3K m K135 0 ) 2  20.8(1.8 + 0.3K m K1350 )
−1 

 (−6 + 3.7 K m K1350 )


(−6 + 3.7 K m K1350 ) K m K1350
T
.
N = d , Tf =
Tf
13ω1350 (1.8 + 0.3K m K1350 ) 2
339
b720_Chapter-03
Rule
Kc
Kristiansson and
Lennartson
(2002a).
Ti
Td
Comment
16
Kc
Ti
Td
K u K m ≤ 10
17
Kc
Ti
Td
K u K m > 10
K u [1.5(K m K u + 4)(0.4K m K u + 0.75) − K m K u x1 ]x1
,
Km
(0.4K m K u + 0.75) 2 ( 4 + K m K u )
Kc =
Ti =
K m K u [1.5(K m K u + 4)(0.4K m K u + 0.75) − K m K u x1 ]
,
ωu
(0.4K m K u + 0.75) 2 (4 + K m K u )
Td =
K mK u
(4 + K m K u )
,
ωu [1.5(4 + K m K u )(0.4K m K u + 0.75) − K m K u x1 ] 1 + N −1
(
2
N=
K K
x1
Td
,
, Tf = m u
Tf
ωu
(0.4K m K u + 0.75) 2 (4 + K m K u )
2
2
[
Km
(0.44K m K1350 + 1.4) 2 x 2
[
K m K135 0 1.5(0.44K m K135 0 + 1.4) x 2 − K m K u x 3
Td =
(0.44K m K135 0 + 1.4) 2 x 2
ω135 0
K m K1350
ω1350
[1.5x (0.44K
2
2
N=
]
K1350 1.5(0.44K m K135 0 + 1.4) x 2 − K m K1350 x 3 x 3
Kc =
Ti =
)
2
x 1 = 0.13 + 0.16K m K u − 0.007 K m K u .
17
FA
Handbook of PI and PID Controller Tuning Rules
340
16
17-Mar-2009
m K135 0
,
],
x2
,
+ 1.4) − K m K1350 x 3 1 + N −1
](
)
2
K m K1350
x3
Td
,
, Tf =
ω135 0
Tf
(0.44K m K1350 + 1.4) 2 x 2
x 2 = min(14 + 0.5K m K 1350 ,25) , x 3 = 1.35 + 0.35K m K 1350 − 0.006K m 2 K 1350 2 .
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models

1  1 + sTd

3.10.5 Classical controller G c (s) = K c 1 +
 Ti s  1 + s Td
N
R(s)
+
E(s)
−
Rule



1  1 + sTd

K c 1 +
Td
Ti s 

1 + s
N

 U(s)


Non-model

specific


Y(s)
Table 54: PID controller tuning rules – non-model specific
Comment
Td
Kc
Ti
Minimum performance index: regulator tuning
Minimum IAE –
Edgar et al.
(1997), p. 8-15.
Pessen (1953).
N =∞.
Pessen (1954).
N =∞.
Atkinson (1968).
N =∞.
0.56K u
0.39Tu
0.25K u
Ultimate cycle
0.33Tu
0.5Tu
0. 2 K u
0.25Tu
N = 10
0.14Tu
‘Optimum’ servo
response.
0.5Tu
‘Optimum’ regulator response – step changes.
0.33K u
0.5Tu
0.33Tu
0.750K u
0.75Tu
‘Some’
overshoot.
0.25Tu
0.375K u
‘Rather underdamped response’ – p. 396.
0.75Tu
0.25Tu
Kinney (1983).
0.25K u
‘Good results’ – p. 396.
0.5Tu
0.12Tu
Corripio (1990),
p. 27.
Hang et al.
(1993b), p. 58.
0. 6 K u
0.5Tu
0.125Tu
10 ≤ N ≤ 20
0.35K u
1.13Tu
0.20Tu
N not specified.
0. 5K u
1.2Tu
0.125Tu
0.25K u
1.2Tu
0.125Tu
‘Aggressive’
tuning.
‘Conservative’
tuning.
St. Clair (1997),
p. 17. N ≥ 1 .
N not specified.
341
b720_Chapter-03
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Handbook of PI and PID Controller Tuning Rules
342
Rule
Kc
Ti
Td
Comment
Harrold (1999).
0.67 K u
0.5Tu
0.125Tu
N not specified.
Tan et al.
(1999a), p. 26.
0.3K u
0.15Tu
0.25Tu
N=∞
O’Dwyer
(2001b) –
representative
results – Åström
and Hägglund
(1995)
equivalent,
p. 142.
N =∞.
Tan et al. (2001).
1
17-Mar-2009
Kc =
Ti =
Ziegler–Nichols ultimate cycle equivalent.
A m = 2,
0.2349 K u
0.2273Tu
0.2273Tu
0.0944 K u
0.6154Tu
0.6154Tu
0.1008K u
0.3939Tu
0.3939Tu
φ m = 46 0 .
Ti
0.25Ti
N=∞
1
Kc
Ti ωu G 'c ( jωu )
Ti 2ωu 2 + 1 0.0625Ti 2ωu 2 + 1
,
 2∠G 'c ( jωu ) + π 
2
2
6.25ωu + 4ωu tan 2 
 − 2.5ωu
2


 2∠G 'c ( jωu ) + π 
ωu 2 tan 

2


,
G 'c ( jωu ) = desired controller frequency response at ω = ωu .
φ m = 20 0 .
A m = 2.44,
φ m = 610 .
A m = 3.45,
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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
343
3.10.6 Generalised classical controller


Td s
1
+
G c (s) = K c 1 +
T

Ti s
1+ d

N

R(s) E(s)
+
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N

U(s)


2
b
+
b
s
+
b
s
Non-model
f1
f2
 f0
 1 + a f 1s + a f 2 s 2
specific
s

Kristiansson et
al. (2000).
1
Kc
‘Almost’ optimal tuning rule.
0.625

G p ( jωu )
ωu 0.37 +
Km


Tf = x1 ,
Y(s)
Minimum performance index: other tuning
Ti
Td
M s ≈ 1.7
2


G p ( jωu )
G p ( j ωu )


−
+
1.6 1.6
2
.
3
1
.
1
2


Km
Km
1.6


Kc =
, Ti =



G p ( jωu )
G p ( jωu )

ωu 0.37 +
K m 0.37 +
Km 
Km





Td =




Table 55: PID controller tuning rules – non-model specific
Comment
Kc
Ti
Td
Rule
1


2
 b f 0 + b f 1s + b f 2 s
 1 + a f 1s + a f 2 s 2
s 

G p ( jω u )
Km




, x1 =
≤ 0.5 ; Tf =




,
2


G (j )
1.6 G p ( jωu ) − 2.3 p ωu + 1.1
2


Km
Km



G p ( jωu )
ωu 0.37 +

Km





2

G ( j ωu )
13 − 20 p

Km





,
x1 G p ( j ω u )
,
> 0. 5 ;
Km
3
2
b f 0 = 1 , b f 1 = 0 , b f 2 = 0 , a f 1 = Tf , a f 2 = Tf , N = ∞ .
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344
Rule
Kristiansson
(2003).
2
Kc
Ti
Td
Kc
Ti
Td
Comment
‘Almost’ optimal tuning rule; K 1500 ≥ 0.03 ; 1.6 ≤ M s ≤ 1.9 .
Direct synthesis: time domain criteria
Ramasamy and
Sundaramoorthy
(2007).
2
Kc =
3
x1 + Tf
Kc
Td
0.60
K 0
K 0

K 0
K m  150 + 0.07  0.32 + 1.6 150 − 0.8 150
Km
 
 Km
 Km




,
2



1.5
Ti =
,
2

K150 0
 K150 0  
 
ω150 0 0.32 + 1.6
− 0.8

Km

 K m  

0.6667
, bf 0 = 1 , bf 1 = 0 , bf 2 = 0 , N = ∞ ,
Td =
2

K150 0
 K150 0  
 
ω150 0 0.32 + 1.6
− 0.8

Km

 K m  

af1 =
 K 0
ω150 0 20 150
Km

0.8

K 0

K 0
+ 0.5 0.32 + 1.6 150 − 0.8 150

Km

 Km
K 0

2
ω150 0 20 150 + 0.5
K
m


Kc =
2
,



1
af 2 =
3




x1
K m (τCL
2
K150 0
0.32 + 1.6

Km

K 0
− 0.8 150
 Km




2
2
.



2
−
 af1 
a f 2 + (a f 1 + x 2 )x1
τCL
,
, x1 = t − τCL +
 , Td =
1 +
x1 
x1 + a f 1
2(τCL + TCL )
+ TCL ) 
2

with x 2 =  x 1 + τ CL

−

2
3
 τ CL − t  − σ
_
τ CL


− t  +
,
−
2x 1
6x 1 (τ CL + TCL )

−
a f 1 , a f 2 not specified; b f 0 = 1 , b f 1 = 0 , b f 2 = 0 , N = ∞ ; t = mean and σ2 =
variance of the process impulse response; desired closed loop transfer function =
e −sτ CL 1 + sTCL .
b720_Chapter-03
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FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Ramasamy and
Sundaramoorthy
(2007).
4
4
Kc
Ti
Td
Kc
x3 + af1
Td
Comment
Applies to processes with dominant numerator dynamics.
(
)
Desired closed loop transfer function = e −sτ CL 1 + 2ξCL TCL 2s + TCL 2 2s 2 .
Kc =
 af1 
a f 2 + (a f 1 + x 4 )x 3
x3
with
1 +
 , Td =
K m (τCL + 2ξCL TCL 2 ) 
x3 
x3 + a f1
−
x 3 = t − τCL +
2
2
τCL − 2TCL 2
,
2(τCL + 2ξCL TCL 2 )
2

x 4 =  x 3 + τCL

−

 τCL − t  − σ 2
_
τCL 3

,
−
− t  + 
2x 3
6x 3 (τCL + 2ξCL TCL 2 )

−
a f 1 , a f 2 not specified; b f 0 = 1 , b f 1 = 0 , b f 2 = 0 , N = ∞ ; t = mean and σ 2 =
variance of the process impulse response.
345
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Handbook of PI and PID Controller Tuning Rules
346
3.10.7 Two degree of freedom controller 1


Td s
1
+
U (s ) = K c  1 +
 Ti s 1 + Td

N






βTd s 
 E (s ) − K c  α +
 R (s )
Td 


s
1+ s 

N 






T
s
β
d
K
α +
Td  c

1+
s

N 

E(s)
R(s) +
−
Rule
Edgar et al.
(1997), p. 8–15.
Majhi (1999),
pp. 132–137.
1
Kc =


Td s
1
+
K c 1 +
T

Ti s
1+ d

N





s

−
+
U(s)
Non-model
specific
Y(s)
Table 56: PID controller tuning rules – non-model specific
Comment
Td
Kc
Ti
Minimum performance index: regulator tuning
0.77 K u
0.48Tu
0.11Tu
1
Kc
Minimum IAE; α = 0 , β = 1 , N = ∞ .
Minimum ISTE;
Ti
Td
N = 10.
∧  K h + 4.267 A 
17.08  K m h − 0.4079A p 
m
p

 , Ti = 0.654 Tu 
,
K m  K m h + 14.9A p 
 K m h + 12.15A p 
∧  K h − 0.6498A 
m
p
Td = 2.891Tu 
 , α = 0 , β =1.
 K m h + 10.52A p 
b720_Chapter-03
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FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
Rule
Kc
Majhi (1999),
pp. 132–137.
Shen (2002).
2
Kc
3
Kc
Minimum performance index: servo tuning
Minimum ISTE;
Ti
Td
N = 10.
Minimum performance index: other tuning
∧
Ti
Td
Ku > 1 Km
Kc
Direct synthesis: frequency domain criteria
2
0
M s = 1.4
0.90T e −4.4 κ + 2.7 κ
4
Åström and
Hägglund (1995),
p. 215.
Comment
Td
Ti
347
u
2
α = 1 − 1.1e −0.0061κ +1.8κ ; 0 < K m K u < ∞ .
2
0.13K u e1.9 κ −1.3κ 0.90Tu e −4.4 κ + 2.7 κ
0
2
M s = 2.0
2
α = 1 − 0.48e 0.40 κ −0.17 κ ; 0 < K m K u < ∞ .
2
Kc =
∧  K h + 0.1205A 
15.75  K m h − 0.3287A p 
m
p

 , Ti = 4.602 Tu 
,
K m  K m h + 18.04A p 
 K m h + 16.07 A p 
∧  K h − 0.5917 A 
m
p
Td = 2.93 Tu 
 , α = 0 , β =1.
 K m h + 18.22A p 
3
4
Minimise
∞
∞
0
0
∫ r( t) − y(t) dt + ∫ d( t) − y(t) dt ; N = ∞ , β = 0 , M

∧
^  


K c = K u exp 0.17 −  2.62  K m K u   + 1.79


  


 2 ^ 2  
 K m K u   ,
 

 


∧

Ti = T u exp  − 0.02 −  2.62



^  

 K m K u   + 1.34

  
 2 ^ 2  
 K m K u   ,
 

 


∧

Td = T u exp  − 1.70 −  0.59



^  

 K m K u   −  0.25

  
 2 ^ 2  
 K m K u   ,
 

 



α = 1 − exp − 0.30 −  0.48



^  

 K m K u   +  0.93

  
2
K c = 0.053K u e 2.9 κ − 2.6 κ .
 2 ^ 2  
 K m K u   .
 

 

s
≤2;
b720_Chapter-03
FA
Handbook of PI and PID Controller Tuning Rules
348
Rule
Vrančić (1996),
pp. 136–137.
5
17-Mar-2009
5
Kc
Ti
Td
Kc
Ti
0
Ultimate cycle
0.5Tu
0.125Tu
Mantz and
Tacconi (1989).
0.6K u
VanDoren
(1998).
0.75K u
0.625Tu
0.1Tu
OMEGA Books
(2005).
0.61K u
0.625Tu
0.1Tu
Kc =
Kc =
x1 =
Ti =
Comment
Quarter decay ratio; α = 0.83 , β = 0.346 , N = ∞ .
A1A 2 − K m A 3 − x1
(1 − [1 − α] )(K
2
2
m
A 3 + A13 − 2K m A1A 2
A1A 2 − K m A 3 + x1
(1 − [1 − α] )(K
2
2
m
A 3 + A13 − 2K m A1A 2
), K
mA3
), K
mA3
− A1A 2 < 0 ;
− A1A 2 > 0 ;
(K m A3 − A1A 2 )2 − (1 − [1 − α]2 )A3 (K m 2A3 + A13 − 2K m A1A 2 ) ;
A1
2
(
1
K K
2
Km +
+ c m 1 − [1 − α ]
2K c
2
)
.
0.2 ≤ α ≤ 0.5 – good servo & regulator action (Vrančić, 1996, p. 139).
α = 0 ,β =1,
N =∞.
α = 0 ,β =1,
N =∞.
b720_Chapter-03
17-Mar-2009
FA
Chapter 3: Controller Tuning Rules for Self-Regulating Process Models
3.10.8 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s)




(
)
1
−
β
T
s
1 + b f 1s
[
]
1
−
χ
δ
d

,
F1 (s) = K c  [1 − α ] +
+
+


T
Ti s
1 + Ti s
1 + a f 1s + a f 2 s 2
d
1
+
s


N 





[
]
K (b + b f 4 s )
1
−
ϕ
T
s
1+ bf 2s
[
]
−
φ
1
d 
F2 (s) = K c  [1 − ε] +
+
− 0 f3
2

 1 + a f 3s + a f 4 s
T
Ti s
1+ a f 5s
1+ d s 

N 

R(s)
+
F1 (s)
U(s)
Y(s)
Model
−
F2 (s)
Table 57: PID controller tuning rules – non-model specific
Comment
Td
Kc
Ti
Rule
1
Bateson (2002),
pp. 629–637.
Direct synthesis: frequency domain criteria
Model: Method 1
Ti
2 ω180 0
Kc
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ;
ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 1 ;
b f 3 = 0 ; b f 4 = 0.1Td ; a f 5 = 0 .
1
K c = 10
{ (
min  − G p jω 0

140
[
) } or {− G (jω ) − 6}
p
]
[
180 0
(
, G p jω1400
]
Ti = min ω82 0 ,0.2ω170 0 . min ω82 0 ,0.2ω170 0 .
)
(
and G p jω1800
)
are given in dB;
349
b720_Chapter-04
04-Apr-2009
FA
Chapter 4
Controller Tuning Rules for Non-Self-Regulating
Process Models
4.1 IPD Model G m (s) =
K m e −sτ m
s

1 

4.1.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
R(s)
E(s)
+
−

1 

K c 1 +
Ti s 

U(s)
K m e − sτ
s
m
Y(s)
Table 58: PI controller tuning rules – IPD model G m (s) =
K m e − sτ m
s
Rule
Kc
Ti
Comment
Ziegler and Nichols
(1942).
Model: Method 2
0.9 K m τm
Process reaction
3.33τ m
Quarter decay ratio.
1.0 K m τm
∞
Quarter decay ratio.
Two constraints
method – Wolfe
(1951).
Model: Method 2
Also given by Coon (1956), (1964) and NI Labview (2001).
Decay ratio = 0.4.
0.6 K m τm
2.78τ m
Decay ratio is as
small as possible.
Minimum error integral (regulator mode).
0.87 K m τm
4.35τ m
350
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Ti
Comment
Åström and Hägglund
(1995), p. 13.
Model: Method 1
Hay (1998), p. 188.
0.63
K mτm
3.2 τ m
0.42 K m τm
5. 8τ m
Ultimate cycle
Ziegler–Nichols
equivalent.
Model: Method 3
Hay (1998), p. 199.
Model: Method 1;
K c and Ti
deduced from graphs.
Kp
s(1 + sTp ) n
NI Labview (2001).
Model: Method 12
Minimum IAE –
Shinskey (1988),
p. 123.
Minimum IAE –
Shinskey (1994),
p. 74.
Minimum IAE –
Shinskey (1988),
p. 148.
Model: Method 1
K m τ m = 0. 1
3.5
K m τ m = 0. 2
2.5
2.2
K m τ m = 0. 4
2.0
K m τ m = 0.5
1.7
K m τ m = 0.6
0% overshoot (servo).
x 2τm
x 3 K m τm
x1
K m τ m = 0.3
4.5K m τ m
x1 K m τ m
Bunzemeier (1998).
Model: Method 5
Gp =
7.5
x2
0.18 1.350
0.24 3.050
0.22 1.910
0.21 1.690
0.20 1.584
0.20 1.557
0.44 K m τm
x3
0.23
0.40
0.30
0.28
0.26
0.26
10% overshoot
(servo).
Coefficient values:
n
x1
x2
0
1
2
3
4
5
0.20
0.19
0.19
0.18
0.18
1.477
1.463
1.453
1.385
1.378
x3
n
0.25
0.24
0.24
0.24
0.24
6
7
8
9
10
‘Some’ overshoot.
∞
‘No’ overshoot.
0.9 K m τm
3.33τm
Quarter decay ratio.
0.4 K m τm
5.33τm
‘Some’ overshoot.
0.24 K m τm
5.33τm
‘No’ overshoot.
0.26 K m τm
Minimum performance index: regulator tuning
0.9524
Model: Method 1
4τ m
K τ
m m
0.9259
K m τm
4τ m
Model: Method 1
0.61K u
Tu
Also specified by
Shinskey (1996),
p. 121.
351
b720_Chapter-04
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FA
Handbook of PI and PID Controller Tuning Rules
352
Rule
Kc
Ti
Comment
Minimum ISE –
Hazebroek and Van
der Waerden (1950).
Minimum ISE –
Haalman (1965).
Model: Method 1
Minimum ITAE –
Poulin and Pomerleau
(1996).
Model: Method 1
1.5
K mτm
5.56τ m
Model: Method 2
0.6667 K m τm
∞
Skogestad (2001).
Model: Method 1
Skogestad (2003).
Model: Method 1
Skogestad (2001).
Model: Method 1
Kuwata (1987).
Model: Method 1
Regulator – Gorecki
et al. (1989).
Model: Method 1
Tyreus and Luyben
(1992).
Model:
Method 1
or Method 10.
Fruehauf et al.
(1993).
Rotach (1995).
Model: Method 6
Wang and Cluett
(1997).
1
Kc =
M s = 1.9 , A m = 2.36 , φ m = 50 0 .
Process output step
load disturbance.
Process input step
0.5327 K m τm
3.8853τ m
load disturbance.
Minimum performance index: other tuning
0.28 K m τm
7τ m
M max = 1.4
0.5264 K m τm
4.5804 τ m
0.404 K m τm
7τ m
M max = 1.7
0.49 K m τm
3.77τ m
M max = 2.0
Direct synthesis: time domain criteria
Also given by Hay
∞
0.5 K m τm
(1998), p. 188.
e
 τ +T 
−  m m 
 Tm 
∞
K m τm
0.828 K m τm
5.828τm
0.487
K mτm
8.75τ m
0.31K u
2.2Tu
0.5 K m τm
5τ m
0.75 K m τm
2.41τ m
Pole is real and has
maximum attainable
multiplicity.
Maximum closed
loop log modulus
= 2dB ; TCL =
τ m 10 .
Model: Method 2
Damping factor for oscillations to a disturbance input = 0.75.
1
Model: Method 1;
Ti
Kc
ξ = 0.707 .
1
, Ti = (1.4020TCL + 1.2076)τm , τm ≤ TCL ≤ 16τm .
K m τm (0.7138TCL + 0.3904)
b720_Chapter-04
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FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Wang and Cluett
(1997) – continued.
Cluett and Wang
(1997).
Model: Method 1
Kc
Ti
Comment
Kc
Ti
Model: Method 1;
ξ =1 .
0.9588 K m τ m
3.0425τ m
TCL = τ m
0.6232 K m τ m
5.2586τ m
TCL = 2τ m
2
0.4668 K m τ m
7.2291τ m
TCL = 3τ m
0.3752 K m τ m
9.1925τ m
TCL = 4τ m
0.3144 K m τ m
11.1637 τ m
TCL = 5τ m
0.2709 K m τ m
13.1416τ m
TCL = 6τ m
1.04Tu
M max = 5 dB
Poulin and Pomerleau
2.13
0.34 K u or
(1999).
K m Tu
Model: Method 1
x1 K m τ m
Vítečková (1999),
Vítečková et al.
OS
x1
(2000a).
0.368
0%
Model: Method 1
0.514
5%
0.581
10%
0.641
15%
Chidambaram and
1.1111 K m τm
Sree (2003).
Huba and Žáková
(2003).
Skogestad (2003),
(2004b).
Model: Method 1
0.696
20%
0.748
25%
0.801
30%
0.853
35%
4.5τ m
0.23 K m τm
2.914 τ m
3.555τ m
1
K m (TCL + τ m )
4ξ 2 (TCL + τ m )
0.5
K mτm
0.4689 K m τm
Sree and
Chidambaram (2006).
Model: Method 1
2x1
(1 + x1 )K m τm
Kc =
Coefficient values:
OS
x1
0.281 K m τ m
Benjanarasuth et al.
(2005).
Model: Method 1
2
∞
0.30K u
x1
OS
0.906
0.957
1.008
40%
45%
50%
Model: Method 1
Model: Method 1
Suggested
ξ = 0.7 or 1.
8τ m
‘Good’ robustness –
TCL = τ m , ξ = 1 .
∞
‘Stability index’ =
2.5.
0.5(1 + x1 )τm
x1 − 1
p. 47. x1 > 1 ;
x1 = 1.25
(Chidambaram and
Sree, 2003).
1
, Ti = (1.9885TCL + 1.2235)τm , τm ≤ TCL ≤ 16τm .
K m τm (0.5080TCL + 0.6208)
353
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Handbook of PI and PID Controller Tuning Rules
354
Rule
Kc
Comment
Ti
Direct synthesis: frequency domain criteria
Maximise crossover
3
frequency.
∞
Kc
Hougen (1979),
p. 333.
Model: Method 1
Buckley et al. (1985),
pp. 389–390.
Model: Method 1
Chidambaram (1994),
Srividya and
Chidambaram (1997).
≤
0.67075
Kmτm
ωp
Gain and phase
margin – Kookos et
al. (1999).
Model: Method 1
≥ 20τm
‘Well-damped’
response.
3.6547τ m
Model: Method 7;
recommended
A m = 2.
0.45
K m τm
AmKm
1
ω p 0.5π − ω p τ m
0.942 K m τ m
Representative results:
4.510τ m
(
)
A m = 1.5;
φ m = 22.5 0 .
Chen et al. (1999a).
Model: Method 11
0.698 K m τ m
4.098τ m
A m = 2; φ m = 30 0 .
0.491 K m τ m
6.942 τ m
A m = 3; φ m = 450 .
0.384 K m τ m
18.710τ m
A m = 4; φ m = 600 .
π


φ m + 2 (A m − 1)


K mτm Am2 − 1
(
3
)
Ti
0.5236 K m τ m
8τ m
Values recorded deduced from a graph:
0.1
0.2
0.3
0.4
K mτm
7.0
Kc
4
Ti =
5
Kc =
φm ≥
Ti
Kc
5
Cheng and Yu
(2000).
Model: Method 1
4
3.5
2.2
1.8
A m = 2.83;
φ m = 46.10 .
0.5
0.6
0.7
0.8
0.9
1.0
1.4
1.1
1.0
0.8
0.75
0.7
2

π
τm
Am − 1
.


4  φm + 0.5π(A m − 1)  
φm + 0.5π(A m − 1) 
0.5π − φm −

2
Am − 1


r1π(2r1A m − 1)
(
)
4K m τm r12 A m 2 − 1
, Ti =
(
2
2
)
4τm r1 A m − 1
2
πr1 A m (r1A m − 2 )(2r1A m − 1)
2
π  1.0607 
1 −

2 
A m 
.
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
O’Dwyer (2001a).
Model: Method 1
Kc
Ti
Comment
x1 K m τ m
x2τm
6
Representative coefficient values:
Am
φm
x1
x2
Am
x2
0.558
1.4
1.5
46.2
0
0.357
4.3
4.0
60.0 0
0.484
1.55
2.0
45.50
0.305
12.15
5.0
75.0 0
0.458
3.35
3.0
0
O’Dwyer (2001a).
Model: Method 1
59.9
x1K u
x 2Tu
7
2.01 − 0.80φ
6.283(1.72φ − 2.55)
ωφ
1.920 < φ < 2.356 ;
M s = 1.2
0.33
6.61
ω 2.09
φ = 2.09 radians
(‘good’ choice).
2.50 − 0.92φ
6.283(1.72φ − 3.05)
ωφ
2.094 < φ < 2.443 ;
M s = 1.4 .
5.36
ω 2.27
φ = 2.27 radians
(‘reasonable’ choice).
( )
Hägglund and Åström
(2004).
Model: Method 1
G p jωφ
G p ( jω2.09 )
( )
G p jωφ
0.41
G p ( jω2.27 )
6
Am =
π
 +
2

x2
4x 1
π2
π 

−
4
x2 

x π
1+ 2  +
4 2

2
π
π 

−
4
x2 

2
2
2
, x 2 > 1.273 ;


φ m = tan −1  0.5x 1 2 x 2 2 + 0.5x 1 x 2 x 1 2 x 2 2 + 4  − 0.5x 1 2 + 0.5

7
Am =

2x 2
πx1
1+
π
 +
2

π2
π 

−
4 4x 2 

π
 +
x12 π2  2

1
φm
x1
x12 x 2 2 + 4 .
2
π2
π 

−
4 4x 2 

φ m = tan −1 x 3 − 0.125π2 x12 +
x1
x2
2
; x 3 = 2π2 x12 x 2 2 + 2πx1x 2 4π2 x12 x 2 2 + 4 ;
πx1
4π2 x12 x 2 2 + 4 .
16 x 2
355
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Handbook of PI and PID Controller Tuning Rules
356
Rule
Kc
Ti
Comment
6.283(0.86φ − 1.50)
ωφ
2.356 < φ < 2.705 ;
M s = 2. 0 .
G p ( jω2.53 )
0.52
4.25
ω 2.53
φ = 2.53 radians
(‘good’ choice).
Hägglund and Åström
(2004).
0.35 K m τ m
7τm
Model: Method 2
Åström and Hägglund
(2006), pp. 228–229.
0.35 K m τ m
13.4τm
Model: Method 2;
‘AMIGO’ tuning rule;
M s = M t = 1. 4 .
Clarke (2006).
Model: Method 4
 0.222 0.444 
,


 K m τm K m τm 
[1.592τm ,3.183τm ]
Hägglund and Åström
(2004) – continued.
Model: Method 1
Litrico and Fromion
(2006).
Litrico et al. (2006).
8
Kc =
3.43 − 1.15φ
( )
G p jωφ
Kc
Ti
0.333 K m τm
6τm
8
A m = 12.7dB , φm = 44.50 .
Model: Method 1
Model: Method 1
A

 − Am  
1.571 − 20m
10
sin  0.0175φm + 1.57110 20   ,
K m τm

 



Am

 − Am  
Ti = (0.637 τm )10 20 tan  0.0175φm + 1.57110 20   ; A m in dB, φm in degrees.

 



b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
357
Comment
Ti
Robust
Chien (1988).
Model: Method 1
1  2λ + τ m
K m  [λ + τ m ]2




Boudreau and
McMillan (2006).
Model: Method 1
1
Km
 2λ + τ m

 [λ + τ ]2
m

x1 K m τ m




Ogawa (1995).
Model: Method 1;
coefficients of K c
and Ti deduced from
graphs.
Alvarez-Ramirez et
al. (1998).
Model: Method 1
9
x1
x2
0.45
0.39
0.34
0.30
0.27
11
12
13
14
15
1
Km
 1
1 


+
T
T
m 
 CL
2λ + τ m
9
4(λ + τm )
2λ + τ m
2
>
Overdamped
response.
x 2τm
20% uncertainty in process parameters.
30% uncertainty in process parameters.
40% uncertainty in process parameters.
50% uncertainty in process parameters.
60% uncertainty in process parameters.
Tm + τ m
TCL > τ m
λ ∈ [1 K m , τm ] (Chien and Fruehauf, 1990);
λ = 3τ m (Bialkowski, 1996);
λ > τ m + Tm (Thomasson, 1997);
1.5τm ≤ λ ≤ 4.5τm (Zhang et al., 1999b).
From a graph, λ = 1.5τ m ….overshoot = 58%, settling time = 6τ m
λ = 2.5τ m ….overshoot = 35%, settling time = 11τ m
λ = 3.5τ m ….overshoot = 26%, settling time = 16τ m
λ = 4.5τ m ….overshoot = 22%, settling time = 20τ m .
λ = e max / 100 K m ; e max = maximum output error after a step load disturbance
(Andersson, 2000, p. 12);
Ti se − τ ms
y
=
(Chen and Seborg, 2002);
 
2
 d  desired K c (λs + 1)
λ = τ m (Smith, 2002);
λ = 1.65τm , minimum IAE, servo (Panda et al., 2006);
λ = 1.15τm , minimum IAE, regulator (Panda et al., 2006);
λ = 2.65τm , overshoot (servo) ≈ 20% (Panda et al., 2006);
λ = τm , Maximum load rejection capability (Boudreau and McMillan, 2006);
λ > τm (Yu, 2006, p. 19).
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358
Rule
Kc
Zhong and Li (2002).
Model: Method 2
2 x1 + τm
Smith (2002).
Skogestad (2004a).
Model: Method 1
Ti
Comment
K m (x1 + τm )2
2 x1 + τ m
x1 not specified.
1 K m τm
τm
Model: Method 1
10
4
K cK m
Kc
Ultimate cycle
Litrico et al. (2007).


1.234 K u 10


∧
11
Kc
^
0.37 K u
−
Am
20
A

− m 
φm = 900 1 − 10 20 




Model: Method 10
∞
Ti
^
1.5 Tu
Representative result; A m = 10 dB; φm = 430 .
Other methods
Penner (1988).
Model: Method 1
10
0.50
K m τm
∞
Maximum closed
loop gain = 1.0.
0.58
K mτm
10τ m
Maximum closed
loop gain = 1.26.
0.8
K mτm
5.9 τ m
Maximum closed
loop gain = 2.0.
K c ≥ ∆d 0 ∆y max , ∆d 0 = maximum magnitude of a sinusoidal disturbance over all
frequencies, ∆y max = maximum controlled variable change, corresponding to the
sinusoidal disturbance.
11
A

 − Am  
∧  − m

K c = 1.234 K u 10 20 sin  0.0175φm + 1.57110 20   ,

 





Am

 − Am
∧ 

Ti =  0.159 Tu 10 20 tan  0.0175φ m + 1.57110 20







  ; A m in dB, φm in degrees.
 

b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models


1
4.1.2 Ideal PID controller G c (s) = K c 1 +
+ Td s 

 Ti s
E(s)
R(s)
+
−

 U(s)
1
K c 1 +
+ Td s 
 Ti s

K m e − sτ
s
Y(s)
m
K m e − sτ m
s
Table 59: PID controller tuning rules – IPD model G m (s) =
Rule
Kc
Callender (1934).
τm = 1 ;
Model: Method 1
Ziegler and
Nichols (1942).
0.2155 K m
x1 K m τ m
2τm
0.5τm
Ford (1953).
Model: Method 3
1.48 K m τm
2τ m
0.37τ m
1.36 K m τm
∞
0.265τm
Åström and
Hägglund (1995),
p. 139.
Hay (1998),
p. 188.
0.94 K m τm
2τ m
0.5τ m
Hay (1998),
p. 199.
Model:
Method 1;
K c , Td deduced
from graphs.
10.0
NI Labview
(2001).
Model:
Method 12
1.1 K m τm
2τm
0.53 K m τm
4τm
0.8τm
0.32 K m τm
4τm
0.8τm
x1 K m τ m
2. 5τ m
0.4τm
Alfaro Ruiz
(2005a).
Ti
0.1 K m
Comment
Td
Process reaction
2.529
∞
5.0
Decay ratio =
0.11.
Decay ratio = 0.
1.2 ≤ x1 ≤ 2.0 ;
Model: Method 2
Decay ratio
2.7:1.
Model: Method 1
Ultimate cycle Ziegler–Nichols equivalent.
0.4 K m τm
3.2 τ m
0.8τ m
Model: Method 3
0.5 K m τm
∞
0.5τm
Model: Method 3
0.55τ m
K m τ m = 0. 1
0.30τ m
K m τ m = 0. 2
0.25τ m
K m τ m = 0.3
0.25τ m
K m τ m = 0. 4
0.25τ m
K m τm = 0.5, 0.6
0.5τm
Quarter decay
ratio.
‘Some’
overshoot.
‘No’ overshoot.
4.0
2.5
2.0
3. 2 K m τ m
1.8
2
2.0 ≤ x1 ≤ 3.33 ;
Model: Method 1
359
b720_Chapter-04
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Handbook of PI and PID Controller Tuning Rules
360
Rule
Kc
Visioli (2001).
Model: Method 1
Paz Ramos et al.
(2005).
Åström and
Hägglund (2004).
Model: Method 1
Alenany et al.
(2005).
Model: Method 1
Regulator –
Gorecki et al.
(1989).
Model: Method 1
Leonard (1994).
Model: Method 1
Ti
Comment
Td
Minimum performance index: regulator tuning
x1 K m τ m
x 2τm
x 3τ m
Visioli (2001).
Model: Method 1
1
04-Apr-2009
Coefficient values:
1.49
0.59
Minimum ISE.
1.66
0.53
Minimum ITSE.
1.83
0.49
Minimum ISTSE
Minimum performance index: servo tuning
∞
x1 K m τ m
x 3τ m
1.37
1.36
1.34
Coefficient values:
0.49
0.45
0.45
∞
2.1883τm
1.03
0.96
0.90
0.5992 K m τm
Minimum ISE.
Minimum ITSE.
Minimum ISTSE
Minimum IAE;
Model: Method 1
Minimum performance index: other tuning
x1 K m τ m
x 2τm
x 3τ m
x1
x2
0.139
76.9
0.261
23.3
0.367
12.2
0.460
7.85
0.543
5.78
0.645 K m τm
1.104 K m τm
x3
Coefficient values:
M max
x1
0.346
0.365
0.378
0.389
0.400
1.1
1.2
1.3
1.4
1.5
x2
x3
M max
∞
0.616
4.58
0.681
3.82
0.740
3.28
0.793
2.89
0.841
2.61
0.357 τm
0.410
1.6
0.418
1.7
0.426
1.8
0.434
1.9
0.440
2.0
Optimum servo.
2.405τm
0.417 τm
Optimum
regulator.
Direct synthesis: time domain criteria
Pole is real and
Tm
has maximum
∞
1
1 + 4(Tm τm )
Kc
attainable
2.785 K m τm
3.731τm
0.263τm
multiplicity.
OS (step input) <
0.74 K m τm
12.2 τ m
0.41τ m
10%; Minimum
IAE (disturbance
0.47 K u
3.05Tu
0.10Tu
ramp).
τ + 4Tm −
Kc = m
e
K m Tm τm
τ m + 2 Tm
Tm
.
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Ti
Td
2
Kc
Ti
Td
3
Kc
Ti
Td
x1 K m τ m
x 2τm
x 3τ m
Wang and Cluett
(1997).
Cluett and Wang
(1997).
Model: Method 1
Rotach (1995).
Model: Method 6
x1
x2
x3
Coefficient values:
x4
x1
0.9588 3.0425 0.3912
1
0.6232 5.2586 0.2632
2
0.4668 7.2291 0.2058
3
1.21 K m τm
1.60τ m
361
Comment
Model: Method 1
TCL = x 4 τ m
x2
x3
x4
0.3752 9.1925 0.1702
0.3144 11.1637 0.1453
0.2709 13.1416 0.1269
0.48τ m
4
5
6
Damping factor for oscillations to a disturbance input = 0.75.
Chidambaram
Model: Method 1
1.2346 K m τm
4.5τ m
0.45τm
and Sree (2003).
Sree and
Chidambaram
Model: Method 1
0.896 K m τm
2.5τ m
0.55τm
(2005b).
4
Chen and Seborg
Model: Method 1
Ti
Td
Kc
(2002).
2
Kc =
1
, Ti = (1.4020TCL + 1.2076)τm ,
K m τm (0.7138TCL + 0.3904)
Td =
3
Kc =
τm
; τm ≤ TCL ≤ 16τm , ξ = 0.707 .
1.4167TCL + 1.6999
1
, Ti = (1.9885TCL + 1.2235)τm ,
K m τm (0.5080TCL + 0.6208)
Td =
4
τm
; τm ≤ TCL ≤ 16τm , ξ = 1 .
1.0043TCL + 1.8194
τ (3TCL + 0.5τm )
T s(1 + 0.5τms )e −sτ m
 y
, Ti = 3TCL + 0.5τm ,
= i
, Kc = m
 
3
3
K
d
K c (TCLs + 1)
 desired
m (TCL + 0.5τ m )
2
Td =
2
3
3
1.5TCL τm + 0.75TCL τm + 0.125τm − TCL
.
τm (3TCL + 0.5τm )
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Rule
Sree and
Chidambaram
(2006).
Model: Method 1
Kc
Ti
Td
Comment
1 K m τm
∞
0.5τm
p. 46.
4x12
x1 2 K m τm
(1 + )
0.8956 K m τm
Regulator –
Gorecki et al.
(1989).
Chen et al.
(1999a).
Åström and
Hägglund (2006),
p. 244.
Åström and
Hägglund (2006),
pp. 263–264.
0.5(1 + x1 )
τm
x1 − 1
0.25(1 + x1 )
τm
x1
2.5τm
0.5τm
p. 47. x1 > 1 ;
x1 = 1.25
(Chidambaram
and Sree, 2003).
p. 48.
Direct synthesis: frequency domain criteria
Model: Method 1
∞
0.75 K m τm
0.333τm
Low frequency part of the magnitude Bode diagram is flat.
1.661
AmK mτm
∞
0.4603
K m τm
7.880τm
0.390τm
M s = M t = 1.4 ;
Model: Method 1
0.45
K m τm
8τm
0.5τm
M s = M t = 1.4 ;
Model: Method 2
5
Td
6
Td
Model: Method 1
Robust
λ + 0.25τ m
2
τm
Zou et al. (1997), K (λ + 0.5τ )
2λ + τ m
2λ + τ m
m
m
Zou and Brigham
(1998).
Model: Method 8 or Method 9; 0.5τ m ≤ λ ≤ 3τ m .
Rice (2004).
2λ + τm
Model: Method 1 K (λ + 0.5τ )2
m
m
2λ + τ m
0.25τm + λ
τm
2λ + τm
7
2
Recommended
1
0.5τm
Lee et al. (2006). K (T + τ )
τm ≤ TCL ≤ 2τm .
∞
TCL + τm
m CL
m
Model: Method 1
Desired closed loop response = e −sτ m (1 + sTCL ) .
5
6
7
Td = 1.273τm [1 − 0.6002A m (1.571 − φm )] .
rA 

Td = τm 1.2732 − 1 m  .
1.6661 

λ = 2.149τm , ‘very aggressive’; λ = 2.444τm , ‘aggressive’;
λ = 2.780τm , ‘moderately aggressive’; λ = 3.162τm , ‘moderate’;
λ = 4.091τm , ‘moderately conservative’; λ = 5.293τm , ‘conservative’;
λ = 6.847 τm , ‘very conservative’.
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Shamsuzzoha
and Lee (2006a).
Shamsuzzoha
and Lee (2007a).
Kc
Ti
Td
Comment
Kc
Ti
Td
Model: Method 1
8
Also given by Shamsuzzoha et al. (2006c); λ , ξ not specified.
9
Model: Method 1
Ti
Td
K
c
Ultimate cycle
Litrico and
Fromion (2006).
8
Kc =
10
Ti
Kc
Ti
, Td =
x1K m (2λξ + τm − x 2 )
Ti = x1 + x 2 −
x1x 2 −
Model:
Method 10
Td
0.167 τm 3 − 0.5x 2 τm 2
2
λ2 + x 2 τm − 0.5τm
2λξ + τm − x 2
,
Ti
2λξ + τm − x 2
(
)
2
2

λ2 − 2λξx1 + x1 e− τ m
λ2 + x 2 τm − 0.5τm
, x 2 = x1 1 −
2
2λξ + τm − x 2
x1

x1

,

x1 is a constant of a ‘sufficiently large value’.
9
Kc =
2
2
3λ2 + 2x 2 τm − 0.5τm − x 2
Ti
,
, Ti = x1 + 2 x 2 −
x1K m (3λ + τm − 2x 2 )
3λ + τm − 2 x 2
2 x1x 2 + x 2 2 −
Td =
λ3 + 0.167 τm 3 − x 2 τm 2 + x 2 2 τm
2
2
3λ2 + 2x 2 τm − 0.5τm − x 2
3λ + τm − 2 x 2
,
Ti
3λ + τm − 2 x 2

 , x is a constant of a ‘sufficiently large value’;
 1

From graphs: λ = 2.3τm , M s = 1.6 ; λ = 1.7 τm , M s = 1.8 ; λ = 1.4τm , M s = 2.0 .

x 2 = x1 1 −


10
3

λ
1 −  e − τ m
 x 
1

Tm
A
 − Am 
∧  − m

K c = 1.234 K u 10 20 sin θ , θ = 0.0175φm + 1.57110 20  ;




A
Am
m
∧  x
∧  10 20


,
Ti =  0.159 Tu  1 10 20 tan θ , Td =  0.159 Tu 


 x1 tan θ
 1 + x1
A

− m
with 2 ≤ x1 ≤ 6 , φm < 901 − 10 20



 , A in dB, φ in degrees.
m
m


363
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364
4.1.3 Ideal controller in series with a first order lag
 1

1
G c (s) = K c 1 +
+ Td s 
 Tf s + 1
 Ti s
R(s)
E(s)
−
+


1
K c 1 +
+ Td s 
Ti s


U(s)
1
1 + Tf s
K m e − sτ
s
m
Table 60: PID controller tuning rules – IPD model G m (s) =
Rule
Kc
Ti
Td
Y(s)
K m e − sτ m
s
Comment
Robust
Tan et al.
τm
0.463λ + 0.277
(1998b).
0
λ = 0.5
K mτm
0.238λ + 0.123
Model: Method 1
Tf = τm (5.750λ + 0.590) ; labelled H ∞ optimal.
Zhang et al.
(1999a).
Rivera and Jun
(2000).
Rice and Cooper
(2002).
Model: Method 1
Ou et al. (2005a).
Model: Method 1
1
2
3
4
Kc =
Kc =
Kc =
Kc =
(
1
Kc
2λ + τ m
0
2
Kc
2(τ m + λ )
0
3
Kc
2λ + 1.5τ m
0.5τ m 2 + λτ m
2λ + 1.5τ m
4
Kc
3λ + τm
2λ + τm
K m λ2 + 2λτ m + 0.5τm 2
2(τm + λ )
2
2
K m ( 2τ m + 4 τ m λ + λ )
(
2λ + 1.5τm
K m λ2 + 2λτ m + 0.5τm 2
)
, Tf =
, Tf =
)
λ2 τm
.
2λ2 + 4λτ m + τm 2
τ m λ2
2
2 τ m + 4 τ m λ + λ2
, Tf =
(6λ + τm )τm
4(3λ + τm )
.
0.5λ2 τ m
λ2 + 2λτ m + 0.5τ m 2
1
4(3λ + τm )
4λ3
, Tf =
.
2
2
2
K m 12λ + 6λτ m + τm
12λ + 6λτ m + τm 2
.
λ not specified;
Model: Method 1
λ not specified;
Model: Method 1
Suggested
λ = 3.162τ m .
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Ou et al.
(2005b).
Model: Method 1
Wang et al.
(2005).
Arbogast and
Cooper (2007).
Model: Method 1
Kc
Ti
5
Kc
2λ + 1.5τm
6
Kc
7
365
Comment
Td
(2λ + τm )τm
λ > 0.3034τm
4λ + 3τm
(for stability).
2λ + 1.3866τm
Td
Kc
2TCL + 1.5τm
(TCL + 0.5τm )τm
λ not specified;
Model: Method 1
0.465
K m τm
7.82τm
2TCL + 1.5τm
0.468τm
Tf = 0.297 τm
Recommended TCL = 3.16τm .
5
Kc =
6
Kc =
λ2 τ m
1
2λ + 1.5τm
, Tf =
.
2
2
2
K m λ + 2λτ m + 0.5τm
2λ + 4λτ m + τ m 2
1
2λ + 1.3866τm
0.3866τm (2λ + τm )
, Td =
,
K m λ2 + 3.2286λτ m + 0.4896τm 2
2λ + 1.3866τm
Tf = τm
7
Kc =
(
2(2TCL + 1.5τm )
2
K m 2TCL + 4τm TCL + τm
2
)
, Tf =
0.3866λ2 − 0.2494λτ m − 0.1247 τm 2
TCL 2 τm
2
2TCL + 4TCL τm + τm 2
λ2 + 3.2286λτ m + 0.4896τm 2
.
.
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

Td s
1
+
4.1.4 Controller with filtered derivative G c (s) = K c 1 +
 Ti s 1 + sTd

N

R(s)


Td s
1
+
K c 1 +
T

Ti s
1+ d

N

E(s)
−
+




s

U(s)
K m e − sτ
s
Table 61: PID controller tuning rules – IPD model G m (s) =
Rule
Kc
Ti
m






Y(s)
K m e − sτ m
s
Comment
Td
Direct synthesis: frequency domain criteria
Kristiansson and
Lennartson
(2002a).
1
Chien (1988).
Model: Method 1
2
Kc
Ti
Kc
2λ + τ m
Yu (2006), p. 19.
2
Model: Method 1 K (λ + 0.5τ )
m
m
1
Kc =
Ti =
2.5ωu
Km
2λ + τ m


 − 0.08 + 0.055 + 0.059 
2

K1
K1 

(
)
(
)
Model: Method 1
Td
Robust
τ m (λ + 0.25τ m )
2λ + τ m
τ m (λ + 0.25τ m )
2λ + τ m
(
)
2
2.5 
2
0.059 K1 + (0.055 K1 ) − 0.08 
−
,

ωu  2.3 − 2K1
0.4(10 + 1 K1 )

2
Kc =
(
−1
−1
)
2
Td
0.059 K1 + (0.055 K1 ) − 0.08
, Tf =
, 0.5 ≤ K1 ≤ 1 .
Tf
0.16ωu (10 + 1 K1 )
2λ + τm
K m (0.5λ + τm )2
N = 10;
λ > τm .

0.059 K 1 2 + (0.055 K 1 ) − 0.08 
2

−
,
 2.3 − 2 K 1

0.4(10 + 1 K 1 )


2.5 
2
0.059 K12 + (0.055 K1 ) − 0.08   1

Td =
−
 + 1 ,


0.4(10 + 1 K1 )
ωu  2.3 − 2K1
 N 
N=
N = 10
; λ ∈ [1 K m , τ m ] (Chien and Fruehauf, 1990).
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FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Ti
Td
Kc
Ti
Td
0.447
K m τm
7.528τm
0.189τm
3
Arbogast and
Cooper (2007).
Model: Method 1
Comment
N = 0.636
Recommended TCL = 3.16τm .
3
Kc =
(2TCL + 1.5τm )(2TCL 2 + 4τmTCL + τm 2 ) − TCL 2τm
(
2
2
0.5K m 2TCL + 4τmTCL + τm
Ti = 2TCL + 1.5τm −
Td =
TCL 2 τm
2
2TCL + 4τm TCL + τm 2
τm (TCL + 0.5τm )
2
Td
TCL 2 τm
.
, Tf =
2
Tf
2TCL + 4τm TCL + τm 2
,
,
2
TCL τm
2TCL + 4τm TCL + τm
2
−
2
2TCL + 1.5τm −
N=
)
2
TCL τm
2
2TCL + 4τm TCL + τm
2
,
367
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368



1  1 + Td s 


4.1.5 Classical controller G c (s) = K c 1 +
 Ti s  1 + Td s 
N 

R(s)



1  1 + sTd

K c 1 +
Td
Ti s 

1 + s
N

E(s)
+
−






U(s)
K m e − sτ
s
m
Table 62: PID controller tuning rules – IPD model G m (s) =
Rule
Kc
Bunzemeier
(1998).
Model:
Method 5;
Gp =
x1 K m τ m
Ti
Y(s)
K m e − sτ m
s
Comment
Td
Process reaction
Kp
s(1 + sTp ) n
.
Minimum IAE –
Shinskey (1988),
p. 143.
Minimum IAE –
Shinskey (1994),
p. 74.
Model: Method 1
Minimum IAE –
Shinskey (1996),
p. 117.
x 2τm
x 4 K m τm
x1
x2
0% overshoot
(servo).
10% overshoot
(servo).
x 3τ m
Coefficient values:
x3
x4
N
n
0.21
0.68
0.47
0.35
0.30
0.27
0.25
0.23
0.22
0.22
0.730
0.280
0.24
5.60
0
1.570
0.796
0.94
5.99
2
1.200
0.669
0.62
6.03
3
0.981
0.587
0.46
5.99
4
0.866
0.530
0.38
6.02
5
0.828
0.487
0.34
6.01
6
0.801
0.452
0.32
6.03
7
0.782
0.424
0.30
5.97
8
0.767
0.401
0.29
5.99
9
0.756
0.381
0.27
5.95
10
Minimum performance index: regulator tuning
Model:
0.93
Method
2;
1
.
57
τ
0
.
58
τ
m
m
K mτm
N not specified.
0.93
K mτm
0.93
K mτm
1.60τ m
0.58τ m
N = 10
1.48τ m
0.63τ m
N = 20
1.57 τ m
0.56τ m
Model:
Method 2;
N not specified.
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FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Minimum IAE –
Shinskey (1996),
p. 121.
0.56K u
Ti
Comment
Td
Model:
Method 1;
N not specified.
Direct synthesis – frequency domain criteria
1
∞
0.45τm
Kc
Hougen (1979),
pp. 333–334.
0.39Tu
0.15Tu
Model: Method 1; maximise crossover frequency; 10 ≤ N ≤ 30 .
2
∞
10[log10 τ m +1.65] Five criteria are
Kc
fulfilled; N = 10.
Robust
λ ∈ [1 K m , τ m ]
0.5τm
Chien (1988). K (λ + 0.5τ )2
0.5τ m
2λ + 0.5τ m
(Chien and
m
Model: Method 1 m
Fruehauf, 1990);
2λ + 0.5τm
N = 10.
2λ + 0.5τ m
0.5τ m
K m (λ + 0.5τm )2
Hougen (1988).
Model: Method 1
2λ + 0.5τm
Chien (1988). K (λ + 0.5τ )2
m
Model: Method 1 m
0.5τm
2λ + 0.5τ m
0.55τm
0.5τ m
2.2λ + 0.55τm
K m (λ + 0.5τm )2
Tuning rules converted to this equivalent controller architecture.
3
0.5τ m
3λ + 0.5τm
λ not specified;
Kc
Model: Method 1
Suggested
4
2λ + τ m
0.5τ m
λ = 3.162τ m .
Kc
Zhang et al.
(1999a).
Rice and Cooper
(2002).
Model: Method 1
1
Values deduced from graph:
0.1
0.2
K mτm
0.3
0.4
0.5
Kc
9
4.2
2.8
2.1
1.7
K mτm
0.6
0.7
0.8
0.9
1.0
Kc
1.45
1.2
1.1
0.95
1
2
Equations deduced from graph; K c ≈ ' 10
Km
3
4
Kc =
Kc =
0.5τm
(
K m 3λ2 + 1.5λτ m + 0.25τm 2
(
2λ + τm
2
λ ∈ [1 K m , τ m ]
(Chien and
Fruehauf, 1990);
N = 11.
K m λ + 2λτ m + 0.5τm
2
)
)
, N=
, N=
0.85

τ
−  log10  m'
T

 m




, Km =
K 'm
Tm'
.
(3λ + 0.5τm )(3λ2 + 1.5λτm + 0.25τm 2 ) .
λ3
λ2 + 2λτ m + 0.5τ m 2
λ2
.
369
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Rule
Kc
Ti
Td
2λ + 0.5τm
0.5τm
Kc
2TCL + τm
0.5τm
0.435 K m τm
7.324τm
0.5τm
Rice (2004).
2λ + 0.5τm
Model: Method 1 K (λ + 0.5τ )2
m
m
Arbogast and
Cooper (2007).
Model: Method 1
6
Comment
N=∞
5
N = 1.684
Recommended TCL = 3.16τm .
Ultimate cycle
Luyben (1996).
Model:
Method 10
0.46K u
2.2Tu
0.16Tu
Maximum closed
loop log modulus
of +2dB ;
N = 10.
Belanger and
Luyben (1997).
Model: Method 1
3.11K u
2.2Tu
3.64Tu
N = 0.1
5
λ = 2.149τm , ‘very aggressive’; λ = 2.444τm , ‘aggressive’;
λ = 2.780τm , ‘moderately aggressive’; λ = 3.162τm , ‘moderate’;
λ = 4.091τm , ‘moderately conservative’; λ = 5.293τm , ‘conservative’;
λ = 6.847 τm , ‘very conservative’.
6
Kc =
(
2(2TCL + τm )
2
K m 2TCL + 4τm TCL + τm
2
)
, N=
TCL 2 + 2TCL τm + 0.5τm 2
TCL 2
.
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371
4.1.6 Generalised classical controller


Td s
1
+
G c (s) = K c 1 +
T

Ti s
1+ d

N

R(s) E(s)
+
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N



2
 b f 0 + b f 1s + b f 2 s
 1 + a f 1s + a f 2 s 2
s 



2 U(s)
K m e − sτ m
 b f 0 + b f 1s + b f 2 s
2
 1 + a f 1s + a f 2 s
s
s

Table 63: PID controller tuning rules – IPD model G m (s) =
Rule
Kc
Ti
Kc
0.667 τm
Y(s)
K m e − sτ m
s
Td
Comment
0.25τm
Model: Method 1
Robust
Shamsuzzoha et
al. (2007).
Yu (2006), p. 18.
Model: Method 1
1
Kc = −
1
0.46K u
Ultimate cycle
2.22Tu
0.16Tu
b f 0 = 0 , b f 1 = 0.16Tu , b f 2 = 0 , a f 1 = 0.016Tu , a f 2 = 0 .
3

4τm
λ
, b f 0 = 1 , b f 1 = x1 1 +  e τ m
K m x1 (6τm + 18λ − 6b f 1 )
x1 


2
N = ∞ , af1 =
af 2 = 0 .
x1

− 1 , b f 2 = 0 ,


2b f 1τm + τm + 12λτ m + 18λ2
+ x1 ; x1 , λ not specified explicitly;
6τm + 18λ − 6 x1




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4.1.7 Two degree of freedom controller 1


Td s
1
+
U (s ) = K c  1 +
 Ti s 1 + Td

N






βTd s 
 E (s ) − K c  α +
 R (s )
Td 


s
1+ s 

N 






T
s
β
d
α +
K
Td  c

1+ s

N 

E(s)
−
R(s) +


Td s
1
+
K c 1 +
T

Ti s
1+ d

N





s

U(s)
−
+
K m e − sτ
s
Table 64: PID controller tuning rules – IPD model G m (s) =
Rule
Minimum IAE –
Shinskey (1988),
p. 143.
Minimum IAE –
Shinskey (1994),
p. 74.
Minimum IAE –
Shinskey (1988),
p. 148.
Minimum IAE –
Shinskey (1996),
p. 121.
Kc
Ti
Td
m
Y(s)
K m e − sτ m
s
Comment
Minimum performance index: regulator tuning
α = 0 , β =1,
1.28
1.90τ m
0.48τ m
N =∞.
K mτm
Model: Method 2
α = 0 , β =1,
1.28
1.90τ m
0.46τ m
N =∞.
K mτm
Model: Method 1
α = 0 , β =1,
0.82 K u
0.48Tu
0.12Tu
N =∞.
Model: Method 1
α = 0 , β =1,
0.77 K u
0.48Tu
0.12Tu
N =∞.
Model: Method 1
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Ti
Td
Comment
K m τm
2.3800τ m
0
α =1
K m τm
2.4569 τ m
0.3982τ m
α = 1, β = 1,
N =∞.
Kc
Minimum ISE – 0.6330
Arvanitis et al.
(2003a).
1.4394
Model: Method 1
0.6114
Arvanitis et al.
1.2986
(2003a).
Model: Method 1
K m τm
2.7442 τ m
0
α =1
K m τm
3.2616τ m
0.4234τ m
α = 1, β = 1,
N =∞.
∞
∫ [e ( t) + K
2
Minimise performance index
2
m
]
u 2 ( t ) dt .
0
0.6146 K m τm
Arvanitis et al.
1.1259 K m τm
(2003a).
Model: Method 1
2.6816τ m
0
α =1
6.7092τ m
0.4627τ m
α = 1, β = 1,
N =∞.
∞
2

 du  
Minimise performance index e 2 ( t ) + K m 2   dt .
 dt  

0 
Minimum performance index: servo tuning
0
0.6270 K m τm
2.4688τ m
α =1
∫
Minimum ISE –
Arvanitis et al.
(2003a).
1.5221 K m τm
Model: Method 1
0.6064 K m τm
Arvanitis et al.
1.4057 K m τm
(2003a).
Model: Method 1
2.1084 τ m
0.3814τ m
α = 1, β = 1,
N =∞.
2.8489 τ m
0
α =1
2.5986τ m
0.4038τ m
α = 1, β = 1,
N =∞.
∞
∫ [e ( t) + K
2
Minimise performance index
2
m
]
u 2 ( t ) dt .
0
0.6130 K m τm
Arvanitis et al.
1.3332 K m τm
(2003a).
Model: Method 1
2.7126τ m
0
α =1
3.0010τ m
0.4167τ m
α = 1, β = 1,
N =∞.
∞
Minimise performance index
2
 2
2  du 
e ( t ) + K m   dt .
 dt  

0 
∫
373
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Rule
Minimum ITAE
– Pecharromán
(2000a).
Model:
Method 11
Minimum ITAE
– Pecharromán
and Pagola
(2000).
Model:
Method 11
Kc
Ti
x1K u
Comment
Td
Minimum performance index: other tuning
0
x 2 Tu
Km = 1
Coefficient values:
φc
x1
x1
x2
α
x2
α
φc
0.049
2.826
0.506
− 164 0
0.066
2.402
0.512
− 160
0
0.218
1.279
0.564
− 1350
0.250
1.216
0.573
− 130 0
0.099
1.962
0.522
− 1550
0.286
1.127
0.578
− 1250
0.129
1.716
0.532
− 150 0
0.330
1.114
0.579
− 120 0
1.506
0.544
0
0.159
0.351
1.093
0.577
− 118 0
0.189
1.392
0.555
− 145
− 140 0
x 2 Tu
x1K u
x 3Tu
x1
x2
Coefficient values:
x3
α
φc
1.672
0.366
0.136
0.601
− 164 0
1.236
0.427
0.149
0.607
− 160 0
0.994
0.486
0.155
0.610
− 1550
0.842
0.538
0.154
0.616
− 150 0
0.752
0.567
0.157
0.605
− 1450
0.679
0.610
0.149
0.610
− 140 0
0.635
0.637
0.142
0.612
− 1350
0.590
0.669
0.133
0.610
− 130 0
0.551
0.690
0.114
0.616
− 1250
0.520
0.776
0.087
0.609
− 120 0
0.509
0.810
0.068
0.611
− 118 0
Km = 1
0
x1K u
x 2 Tu
Minimum ITAE
Coefficient values:
– Pecharromán
α
α
x1
x2
φc
x1
x2
φc
(2000b), p. 142.
0
0.0469 2.8052 0.5061 − 164.1 0.2174 1.2566 0.5642 − 135.00
Model:
Method 11
0.0662 2.3876 0.5119 − 160.50 0.2511 1.1948 0.5730 − 130.00
0.0988 1.9581 0.5220 − 155.00 0.2870 1.1281 0.5777 − 125.00
0.1271 1.6757 0.5319 − 150.00 0.3280 1.0860 0.5794 − 120.00
0.1589 1.5052 0.5439 − 145.00 0.3501 1.0822 0.5767 − 118.00
0.1866 1.3619 0.5546 − 140.00
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Minimum ITAE
– Pecharromán
(2000b), p. 148.
Model:
Method 11
Comment
Kc
Ti
Td
x1K u
x 2 Tu
x 3Tu
x1
x2
Coefficient values:
x3
α
φc
1.6695
0.3684
0.1323
0.6007
− 164.10
1.2434
0.4389
0.1507
0.6073
− 160.00
0.9901
0.4846
0.1587
0.6102
− 155.00
0.8378
0.5393
0.1562
0.6163
− 150.00
0.7383
0.5789
0.1548
0.6052
− 145.00
0.6728
0.6283
0.1488
0.6105
− 140.00
0.6196
0.6221
0.1376
0.6120
− 135.00
0.5948
0.6939
0.1326
0.6096
− 130.00
0.5513
0.7050
0.1190
0.6157
− 125.00
0.5271
0.7471
0.0992
0.6088
− 120.00
0.5229
0.8886
0.0862
0.6115
− 118.0 0
α = 0.6810
Taguchi and
0
0.7662 K m τm
4.091τ m
Araki (2000).
1.253 K m τm
2.388τ m
0.4137τ m
OS ≤ 20% ;
α = 0.6642 , β = 0.6797 , N fixed but not specified.
Model: Method 1
∞
0.82 K m τm
0.40τm
β = 0.41 ;
Taguchi et al.
N = 10.
(2002).
N = 10;
∞
0
.
4046
τ
1
m
0.8203 K m τm
Model:
0.1 ≤ τm ≤ 1.0 .
Method 1;
∞
0.87 K m τm
0.44τm
β = 0.44 ;
α=0.
N= ∞.
Minimum IAE – 0.571 K m τm
0
6.139τm
Tavakoli et al.
M max = 2 ; α = 0.417 ; Model: Method 1.
(2005).
Direct synthesis: time domain criteria
Kuwata (1987).
0
1 K m τm
3.33τm
α =1
Model: Method 1
2
1.067 K m τm
0
.
313
τ
α
=
1, β = 1,
m
6.25τm
N =∞.
1
β = 0.3623 +
0.02488
.
τm + 0.1035
375
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Rule
Chien et al.
(1999).
Model: Method 2
Hansen (2000).
Model: Method 1
Chidambaram
(2000b).
Model: Method 1
Kc
Ti
Td
Comment
2
Kc
1.414TCL 2 + τ m
0
α =1
3
Kc
α = 1, β = 1,
N =∞.
Underdamped system response: ξ = 0.707 .
1.414TCL 2 + τ m
Td
0.938K m τ m
2.7 τ m
0.313τ m
0.64 K m τ m
3.333τ m
0.487 K m τ m
8.75τ m
0.9425 K m τm
2τ m
α = 0.833 ,
β =1, N = ∞ .
α = 0.65
0
α = 0.557
In general, α = 0.6 (on average) or α = 0.5[1 − τ m Ti ] .
0.5τ m
N = ∞ ; β = 1 ; α = 0.707 or α = 0.65 .
Hägglund and
Åström (2002).
Model: Method 2
0.35
K mτm
Arvanitis et al.
(2003a).
Model: Method 1
(8 − x1 )K m τm
4
(
)
16 2ξ2 + 1
32ξ2 + 4 K m τ m
Chidambaram
and Sree (2003).
Model: Method 1
Sree and
Chidambaram
(2006).
Model: Method 1
Klán (2001).
Model: Method 1
2
3
Kc =
Kc =
(
(
)
0
4τ m
x1
0
α = 1,
4
.
x1 = 2
4ξ + 1
16ξ 2 + 4
τm
16 2ξ 2 + 1
α = 1, β = 1,
N =∞.
2
(
)
4.5τ m
0.45τ m
0.9 K m τm
3.33τm
0
1.2 ≤ x1 ≤ 2
2τ m
0.5τ m
0.8956
K m τm
2.5τm
0.5τm
x1 K m τ m ,
α = 0.6 , β = 0 ,
N =∞.
α = 0.65 .
pp. 37–38.
α = 0.65 , β = 1 ,
N =∞.
p. 48. N = ∞ ,
β = 0 , α = 0.6 .
Direct synthesis: frequency domain criteria
0
α = 0.19
0.29 K m τm
8.9τm
1.414TCL 2 + τm
).
2
1.414TCL 2 + τm
2
)
+ 1 τm
0.3 < α < 0.5 .
1.2346
K mτm
K m TCL 2 2 + 1.414TCL 2τm + τm 2
(
(2ξ
M s = 1.4;
7τ m
K m TCL 2 + 0.707TCL 2 τm + 0.25τm
2
)
, Td =
0.25τm + 0.707TCL 2 τm
.
1.414TCL 2 + τm
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Td
Comment
0
α =1
0.25τm + λ
τm
2λ + τm
α = 0 ,β =1;
N =∞,
λ = 3τm .
Ti
377
Robust
Arvanitis et al.
(2003a).
Model: Method 1
2
4
4π τ m
x 1 π 2 − x1
(
Kc
2λ + τm
Rice (2006).
K m (λ + 0.5τm )2
Model: Method 1
Kc =
[(8 − x )π
1
Ti
Kc
Td
equations for λ deduced from graphs.
4π2
2
2λ + τ m
α = 0.6 , β = 0 ,
N =∞.
λ = 2.3τm , M s = 1.6 ; λ = 1.7 τm , M s = 1.8 ; λ = 1.4τm , M s = 2.0 ;
5
Shamsuzzoha
and Lee (2007a).
Model: Method 1
4
)
]
+ x12 K m τm
with 0 < x1 < π2 ;
recommended x1 =
5
Kc =
2
π2 
16
1− 1− 2 2

2 
π 4ξ + 1
(
2
)

 , ξ > 0.3941 .


2
Ti
3λ + 2x 2 τm − 0.5τm − x 2
,
, Ti = x1 + 2 x 2 −
x1K m (3λ + τm − 2x 2 )
3λ + τm − 2 x 2
2
2 x1x 2 + x 2 −
Td =

x 2 = x1 1 −


3
2
2
λ3 + 0.167 τm − x 2 τm + x 2 τm
2
2
3λ2 + 2x 2 τm − 0.5τm − x 2
3λ + τm − 2 x 2
−
,
Ti
3λ + τm − 2 x 2
3

λ
1 −  e − τ m
 x 
1

Tm

 , x is a constant with a ‘sufficiently large value’.
 1

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4.1.8 Two degree of freedom controller 2




T
s
1
d
 1 + b f 1s (F(s) R (s) − Y(s) ) − K Y(s) ,
U(s) = K c 1 +
+
0
 Ti s
T  1 + a f 1s
1+ d s 

N 

1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4
F(s) =
1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5 s 4
R(s)
F(s)
+
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N



 1 + b f 1s
 1 + a f 1s
s

+ U(s)
−
K m e − sτ
s
K0
Table 65: PID controller tuning rules – IPD model G m (s) =
Rule
Alenany et al.
(2005).
Model: Method 1
Kc
Ti
Y(s)
m
Td
K m e − sτ m
s
Comment
Minimum performance index: servo tuning
1.104 K m τm
2.405τm
0.417 τm
N=∞
a f 1 = 0 , bf 1 = 0 ; bf 2 = 0 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , bf 3 = 0 ,
bf 4 = 0 , bf 5 = 0 , K 0 = 0 .
a f 2 = 1.866τm : OS ≈ 7% ; a f 2 = 2.38τm : OS ≈ 0% .
Normey-Rico et
al. (2000).
Model: Method 1
0.563 K m τm
Direct synthesis: time domain criteria
1.5τ m
0.667 τ m
N=∞
a f 1 = 0.13τm , b f 1 = 0 ; a f 2 = 0.75τm , b f 2 = 0.30τm ;
a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , bf 3 = 0 , bf 4 = 0 , bf 5 = 0 ,
K 0 = 0. 5 K m τ m .
Normey-Rico et
al. (2001).
Model: Method 1
1
Kc
(1.5 − x1 )τm
Td
a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , a f 4 = 0 ,
a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0. 5 K m τ m .
1
Kc =
1.5 − x1 
τm
1 − x1 (1.5 − x1 )
4 x1 + 1 − 4 x12 + 0.5x1  , Td =
− x1τm , N =
;
(1.5 − x1 )x1
K m τm 
1.5 − x1

Recommended x1 = 0.5 ; non-oscillatory response.
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Benjanarasuth et
al. (2005).
Model: Method 1
Kc
Ti
Td
0.37 K u
Tu
0
Comment
a f 1 = 0 , b f 1 = 0 ; a f 2 = Tu , b f 2 = 0 ; a f 3 = 0 , b f 3 = 0 ; a f 4 = 0 ,
a f 5 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 .
0.63K u
0.76Tu
0.078Tu
N=∞
a f 1 = 0 , b f 1 = 0 ; a f 2 = 0.76Tu , b f 2 = 0 ;
2
a f 3 = 0.06Tu , b f 3 = 0 ; a f 4 = 0 , a f 5 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0 .
2
Huang et al.
(2005b).
Model: Method 1
Kc
Ti
[
0.6325 τm
]
N=∞
a f 1 = max 0.0063 τm ,0.01 , b f 1 = 0 ; a f 2 = Ti , b f 2 = 1.0630τm ;
1.5
a f 3 = 0.6723τm
+ 0.6988K m τm
2.5
, b f 3 = 0.2652τm 2 ; a f 4 = 0 ,
a f 5 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 .
Robust
Lee and Edgar
(2002).
Model: Method 1
3
Ti
Kc
0
a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , a f 4 = 0 ,
a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0.25K u .
4
Zhang et al.
(2002a).
Model: Method 1
2λ 2 + 1.5τm
Kc
Td
N=∞
2
a f1 =
0.5λ 2 τ m
2
λ 2 + 2λ 2 τ m + 0.5τ m 2
, b f 1 = 0 ; a f 2 = 2λ 2 + τ m ,
b f 2 = 2λ1 + τm ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 ,
bf 5 = 0 , K 0 = 0 .
2
K c = 0.3221 + (0.3099 K m τm ) , Ti = τm (1.0630 + 1.1048K m τm ) .
3
Kc =
4
Kc =
2
0.25K u Ti
4
τm
− τm +
.
, Ti =
(λ + τm )
KuKm
2(λ + τm )
1
2λ 2 + 1.5τm
(2λ 2 + τm )0.5τm , λ ,λ not specified.
, Td =
1 2
K m λ 2 2 + 2λ 2 τm + 0.5τm 2
2λ 2 + 1.5τm
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Rule
Kc
Ti
Td
Comment
5
2λ 2 + (1 + δ)τm
Td
N=∞
Kc
Zhang et al.
λ 2 2δτm − r2 τm 2 (2λ + τm )
(2006).
af1 = 2
, b f 1 = 0 ; a f 2 = 2λ 2 + τ m ,
Model: Method 1
λ 2 + 2λ 2δτm − r2 τm 2 − r1τm (2λ + τm )
b f 2 = 2λ1 + τm ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 ,
bf 5 = 0 , K 0 = 0 .
5
Kc =
2
1
2λ 2 + τm + δτm
2λ 2 + δτm
T
=
,
,
d
K m λ 2 2 + 2λ 2δτm − r2 τm 2 − r1τm (2λ + τm )
2λ 2 + (1 + δ)τm
λ1 , λ 2 , λ , r1 , r2 , δ not specified.
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
4.1.9 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s)




(
)
1
−
β
T
s
1 + b f 1s
[
]
1
−
χ
δ
d

,
F1 (s) = K c  [1 − α ] +
+
+


T
Ti s
1 + Ti s
1 + a f 1s + a f 2 s 2
d
1
+
s


N 





[
]
K (b + b f 4 s )
1
−
ϕ
T
s
1+ bf 2s
[
]
−
φ
1
d 
F2 (s) = K c  [1 − ε] +
+
− 0 f3
2

 1 + a f 3s + a f 4 s
T
Ti s
1+ a f 5s
1+ d s 

N 

R(s)
+
F1 (s)
U(s)
Y(s)
Model
−
F2 (s)
Table 66: PID controller tuning rules – IPD model G m (s) =
Rule
Kc
Ti
K m e − sτ m
s
Comment
Td
Minimum performance index: other tuning
0.45 K m
8τ m
0.5τ m
Åström and
α = 1 , τm Tm ≤ 1 ; α = 0 , τm Tm > 1 ; β = 1 ; χ = 0 ; δ = 0 ; N = ∞ ;
Hägglund (2004).
bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 ,
Model: Method 1
2
a f 3 = 0.2τm , a f 4 = 0.01τm ; K 0 = 0 .
Kaya and
Atherton (1999),
Kaya (2003).
Model:
Method 10
1
Direct synthesis: frequency domain criteria
∧
0
1
0.313 K u
0.733 T u
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
∧

0.567  K m K u
K0 =
∧
K m 
 τm T u
∧
bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; bf 3 = 1 ; a f 5 = 0 .





0.667
∧
− 0.313 K u , b f 4

∧
 K K u
= 0.753 m ∧
 τm T u






0.333
−

1  τm Tm
.

τm  K m

381
b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
382
Rule
Kc
Ti
Kc
Ti
Td
Comment
Robust
2
Lee and Edgar
(2002).
Model: Method 1
Td
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ;
ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0.25K u ;
bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 .
3
Kc
Ti
x1
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; b f 1 = 0 , a f 1 = Td , a f 2 = 0 ;
ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = Td , a f 4 = 0 ; K 0 = 0.25K u ;
bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 .
2
3
Kc =
2
2

0.25K u  4
4
τm
τm
− τm +
,
− τm +
 , Ti =

(λ + τm )  K u K m
2(λ + τm ) 
KuKm
2(λ + τm )
Td =
2
4
0.25K u 
τm 3
τm
τ (1 − 0.25K u K m τm ) 
2
+
+ m
.
0.25τm −
2
(λ + τm ) 
6(λ + τm ) 4(λ + τm )
0.5K u K m (λ + τm ) 
Kc =
2
2

0.25K u  4
τm
4
τm
− τm +
+ x1  , Ti =
− τm +
+ x1

(λ + τm )  K u K m
2(λ + τm )
KuKm
2(λ + τm )

2

τm
τm
(1 − 0.25K u K m τm ) 
x1 = τm 0.5 −
+
+

2
6( λ + τ m ) 4( λ + τ m )
0.5K u K m (λ + τm ) 

0.5
.
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
4.2 IPD Model with a Zero
G m (s) =
K m (1 + sTm 3 )e −sτ m
K (1 − sTm 4 )e −sτ m
or G m (s) = m
s
s

1 

4.2.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
E(s)
R(s)
−
+

1 

K c 1 +
Ti s 

U(s)
Model
Y(s)
Table 67: PI controller tuning rules – IPD model with a negative zero
K (1 + sTm3 )e −sτ m
K (1 − sTm 4 )e −sτ m
G m (s) = m
or m
s
s
Rule
Kc
2
3
4
Kc =
Kc =
Kc =
Kc =
1.571
3
Kc
(
)
.
(
)
.
K m τm 1 + 3.142 Tm 4 τm 2
2.358
Comment
Direct synthesis: frequency domain criteria
τm Tm3 < 1
∞
4
τm Tm3 > 1
Kc
Jyothi et al. (2001).
Model: Method 1
1
Ti
Direct synthesis: time domain criteria
1
‘Smaller’ τm Tm 4 .
Kc
∞
2
‘Larger’ τm Tm 4 .
Kc
Chidambaram (2002),
p. 157.
Model: Method 1
K m τm 1 + 4.712 Tm 4 τm 2
1.57
K m τm 1 + 9.870(Tm3 τm )2
0.79
K m τm 1 + 2.467(Tm3 τm )2
.
.
383
b720_Chapter-04
Rule
Kc
Jyothi et al. (2001).
Model: Method 1
0.5 K m Tm 4
Kc =
FA
Handbook of PI and PID Controller Tuning Rules
384
5
04-Apr-2009
5
Ti
Kc
0.79
K m τm 1 + 2.467(Tm 4 τm )2
.
∞
Comment
τm Tm 4 < 1
τm Tm 4 > 1
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
4.3 FOLIPD Model G m (s) =
K m e − sτm
s(1 + sTm )

1 

4.3.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
R(s)
E(s)
+
−

1 

K c 1 +
Ti s 

U(s)
K m e − sτ
s(1 + sTm )
m
Y(s)
Table 68: PI controller tuning rules – FOLIPD model G m (s) =
Rule
Kc
Ti
K m e − sτ m
s(1 + sTm )
Comment
Process reaction
Quarter decay ratio
Coon (1956), (1964).
criterion.
∞
K m (τ m + Tm )
Model: Method 1;
Coefficient values:
coefficients of K c
τ m Tm
x1
τ m Tm
x1
τ m Tm
x1
deduced from graphs.
0.020
5.0
0.25
2.2
4.0
1.1
0.053
4.0
0.43
1.7
0.11
3.0
1.0
1.3
Minimum performance index: regulator tuning
Minimum IAE –
0.556
Shinskey (1994),
Model: Method 1
3.7(τ m + Tm )
K m (τ m + Tm )
p. 75.
Minimum IAE –
0.952
Shinskey (1994),
Model: Method 1
4(Tm + τ m )
K m (Tm + τ m )
p. 158.
x1
385
b720_Chapter-04
Rule
Minimum ITAE –
Poulin and Pomerleau
(1996).
Model: Method 1
Process output step
load disturbance.
Process input step
load disturbance.
Velázquez-Figueroa
(1997).
Model: Method 1
Velázquez-Figueroa
(1997).
Model: Method 1
Kuwata (1987).
Model: Method 1
Kc =
FA
Handbook of PI and PID Controller Tuning Rules
386
1
04-Apr-2009
1
Kc
Ti
Comment
Kc
x 1 (τ m + Tm )
K c and Ti
coefficients deduced
from graph.
τ m Tm
x1
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
5.0728
4.9688
4.8983
4.8218
4.7839
3.9465
3.9981
4.0397
4.0397
4.0397
0.029  Tm 


K m  τm 
Coefficient values:
x2
τ m Tm
0.5231
0.5237
0.5241
0.5245
0.5249
0.5320
0.5315
0.5311
0.5311
0.5311
1.2
1.4
1.6
1.8
2.0
1.2
1.4
1.6
1.8
2.0
0.157
∞
x1
x2
4.7565
4.7293
4.7107
4.6837
4.6669
4.0337
4.0278
4.0278
4.0218
4.0099
0.5250
0.5252
0.5254
0.5256
0.5257
0.5312
0.5312
0.5312
0.5313
0.5314
0.1 ≤
τm
≤ 0.33
Tm
0.195
τ
τ 
0.1 ≤ m ≤ 0.5
6.824Tm  m 
Tm
 Tm 
Minimum performance index: servo tuning
0.528
τ
0.031  Tm 
0.1 ≤ m ≤ 0.33
∞


Tm


K m  τm 
0.406
0.119
τ
 τm 
0.042  τm 
0.1 ≤ m ≤ 0.5




8
.
807
T
T
m
T 
m
K m  Tm 
 m
Direct synthesis: time domain criteria
0.5
∞
K (τ + T )
0.044  τm 


K m  Tm 
m
m
0.122
m
x2
Tm 2
+1 .
K m (τm + Tm ) x1 (τm + Tm )2
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
2
Huba and Žáková
(2003).
Model: Method 1
Kc
Ti
Kc
∞
387
Comment
Representative tuning rules; tuning rules for other τ m Tm values
may be obtained from a graph.
0.193 K m τ m
3.717 τ m
τ m Tm = 0.5
0.207 K m τ m
3.386τ m
τ m Tm = 1
0.219 K m τ m
3.127 τ m
τ m Tm = 2
Direct synthesis: frequency domain criteria
Maximise crossover
3 1.26x 1
frequency.
∞
Kmτm
Hougen (1979),
p. 338.
Model: Method 1
Poulin and Pomerleau
2.13
0.34 K u or
(1999).
K m Tu
Robust
∞
Velázquez-Figueroa 0.833(2λ + Tm + τm )
(1997).
K m (λ + τm )2
McMillan (1984).
Model: Method 1
4
Kc
Tuning rules developed from K u , Tu .
^
K u tan 2 φ m
Perić et al. (1997).
Model: Method 8
− 2Tm + τm + 4Tm
3
4
2
2
2 Tm + τ m − τ m + 4 Tm
K m τ m 2e
0.1592 Tu
tan φ m
1 + tan 2 φ m
2
Kc =
Model: Method 1; λ
not specified.
Ultimate cycle
Ti
^
2
Model: Method 1;
M max = 5 dB .
1.04Tu
2
.
2 Tm
x 1 values recorded deduced from a graph:
τ m Tm
1
2
3
4
5
6
7
8
9
10
x1
0.31
0.19
0.14
0.11
0.09
0.078
0.068
0.06
0.053
0.05
2
  T 0.65 

1.477 Tm 
1


m 
Kc =

 , Ti = 3.33τm 1 + 
 .
τ
K m τ m 2 1 + (Tm τ m )0.65 
  m  
b720_Chapter-04
04-Apr-2009
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Handbook of PI and PID Controller Tuning Rules
388


1
+ Td s 
4.3.2 Ideal PID controller G c (s) = K c 1 +
Ti s


E(s)
R(s)
+
−

 U(s)
1
K c 1 +
+ Td s 
 Ti s

K m e − sτ
s(1 + sTm )
m
Y(s)
Table 69: PID controller tuning rules – FOLIPD model G m (s) =
Rule
Kc
Ti
Td
K m e − sτ m
s(1 + sTm )
Comment
Minimum performance index: regulator tuning
Minimum ISE – 0.6667 K m τm
Model: Method 1
∞
Tm
Haalman (1965).
0
A m = 2.36 ; φ m = 50 ; M s = 1.9 .
0.717
Velázquez

1
Figueroa (1997). 0.062  Tm 
Td
Ti


Model: Method 1 K m  τm 
Minimum performance index: servo tuning
0.75
0.73
Velázquez

 τm 
2
Figueroa (1997). 0.064  Tm 


1
.
2
T
T
m
i


T 
Model: Method 1 K m  τm 
 m
Direct synthesis: time domain criteria
Van der Grinten
∞
1 K m τm
Tm + 0.5τ m Step disturbance.
(1963).
−2 ω d τ m
Stochastic
e
Model: Method 1
3
disturbance.
∞
Td
K τ
m m
Tachibana
(1984).
1
2
λ
K m (1 + λτ m )
τ 
Ti = 3.537Tm  m 
 Tm 
0.951
∞
τ 
, Td = 1.458Tm  m 
 Tm 
Tm
0.754
.
Ti = 8.071τm , 0 < τm Tm ≤ 0.0333 ;
Ti = 22.183τm − 0.4701Tm , 0.0333 < τm Tm ≤ 0.2121 ;
Ti = 8.2076Tm − 18.724τm , 0.2121 < τm Tm ≤ 0.3256 ;
Ti = 1.7571Tm + 1.089τm , τm Tm > 0.3256 .
3


T
Td = 1 − 0.5e − ωd τ m + m e − ωd τ m τme − ωd τ m .
τ
m


Model: Method 5
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Ti
389
Comment
Td
Vítečková
∞
x1 K m τ m
Tm
(1999),
Coefficient values:
Vítečková et al.
OS
OS
OS
OS
x
x
x1
x1
1
1
(2000a).
0%
0.641
15%
0.801
30%
0.957
45%
Model: Method 1 0.368
0.514
5%
0.696
20%
0.853
35%
1.008
50%
0.581
10%
0.748
25%
0.906
40%
4
Chen and Seborg
Model: Method 1
3TCL + τ m
Td
Kc
(2002).
Sree and
Model:
Chidambaram
Method 1;
T
T
5
i
d
Kc
(2006).
p. 58.
Direct synthesis: frequency domain criteria
6
2
Åström and
0.67e−1.9 τ + 0.2 τ
Hägglund (1995), K (T + τ )
m m
m
pp. 212–213.
−9.3τ + 6.6 τ 2
Model: Method 1 4.8e
or Method 3.
K m (Tm + τm )
0.56τm e1.1τ −1.1τ
ωpTm
Chen et al.
(1999a).
4
2
0.13τm e13.6 τ −11.0 τ 14.2τm e −13.3τ + 8.6 τ
K m A mTd
r1ωx Tm
K mTd
2
0.68τm e 2.26 τ −1.8τ
2
2
M max = 1.4
M max = 2.0
∞
7
Td
Model: Method 1
∞
8
Td
Model: Method 1
(3TCL + τm )se−sτ m , K = (3TCL + τm )(Tm + τm ) ,
 y
=
 
c
(TCL + τm )3
K c (TCLs + 1)3
 d desired
2
Td =
0.895 − 0.100
5
6
7
8
Kc =
K m τm
Tm
τm
Tm
T
0.450 + 0.960 m
τm
τm
τ , T =
τ .
, Ti =
Tm m d
Tm m
0.358 − 0.040
0.895 − 0.100
τm
τm
0.895 − 0.100


1
Tuning rules converted from formulae developed for G c (s) = K c  [1 − α ] +
+ Tds  .
Tis


1
.
Td =
2
−1
2ωp − 1.273ωp τm + Tm
Td =
1
ωx − 1.273ωx 2 τm + Tm −1
, ωx =
0.25π(2r1A m − 1)
(r A
2
1
2
m
)
− 1 τm
.
3
2
3TCL Tm + 3TCL Tm τm − TCL + Tm τm
.
(3TCL + τm )(Tm + τm )
b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
390
Rule
Kc
Ti
Klán (2001).
0.67e−1.9 τ + 0.2 τ
Model: Method 1
K m (Tm + τm )
Comment
Td
2
9
x 1Tm
K mτm
O’Dwyer
(2001a).
Model: Method 1
Td
∞
Tm
A m = 0. 5 x 1 ;
φ m = 0.5π − x 1 .
Representative coefficient values:
Am
φm
x1
Am
x1
0.785
Åström and
Hägglund (2006),
p. 246.
Ti
10
2
45
0.524
0
Ti
Kc
Td
Robust
Tm (τ m + 2λ )
τ m + Tm + 2λ
τ m + Tm + 2λ
Rivera and Jun τ m + Tm + 2λ
(2000).
K m (τ m + λ )2
Model: Method 1
φm
3
60 0
M s = M t = 1. 4 ;
Model: Method 1
λ not specified.
2
Recommended
1
0.5τm
Tm +
Lee et al. (2006). K (T + τ )
τ
m ≤ TCL ≤ 2τ m .
∞
TCL + τm
m CL
m
Model: Method 1
Desired closed loop response = e −sτ m (1 + sTCL ) .
Shamsuzzoha et
al. (2006c).
Model: Method 1
9
11
Ti
Kc
Td
λ , ξ not
specified.


1
Tuning rules converted from formulae developed for G c (s) = K c  [1 − α ] +
+ Tds  .
Tis


Ti = 0.13(τm + Tm )e13.6 τ −11.0 τ ; Td = 14.2(τm + Tm )e−13.3τ +8.6 τ .
2
10
11
1
Kc =
K m τm
Kc =
Td =
2
T
T
0.37 + 0.02 m
0.16 + 0.28 m

Tm 
τm
τm
τm , Td =
τm .
0.37 + 0.02  , Ti =
T
T
τ
m
m 

0.03 + 0.0012
0.37 + 0.02 m
τm
τm
Ti
, Ti = x1 + Tm + x 2 − x 3 ,
x1K m (4λξ + τm − x 2 )
x 4 + x 2 (Tm + x1 ) + x1Tm −
0.167 τm 3 − 0.5x 2 τm 2 + x 4 τm + 4λ3ξ
4λξ + τm − x 2
− x3 ;
Ti
x1 is a constant with ‘a sufficiently large value’,
2
x2 =
Tm x 5e − τ m
2
[
x 4 = Tm x 5e − τ m
x1
Tm
2
− x1 x 5 e − τ m
x1 − Tm
x1
2
+ x1 − Tm
2
2
, x3 =
4λ2ξ 2 + 2λ2 + x 2 τm − 0.5τm − x 4
,
4λξ + τm − x 2
2
 λ2
2λξ 
− 1 + x 2Tm , x 5 =  2 −
+1 .

T
Tm

 m
]
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Shamsuzzoha
and Lee (2007a).
12
Ti
Td
Comment
Ti
Td
Model: Method 1
0.25Ti
Tuning rules
developed from
K u , Tu .
Kc
Ultimate cycle
McMillan
(1984).
Model: Method 1
13
Perić et al.
(1997).
Model: Method 8
12
Kc =
Td =
Ti
Kc
∧
K u cos φ m
x1Td
14
Td
Recommended
x1 = 4 .
Ti
, Ti = x1 + Tm + x 2 − x 3 ,
x1K m (4λ + τm − x 2 )
x 4 + x 2 (Tm + x1 ) + x1Tm −
3
2
0.167 τm − 0.5x 2 τm + x 4 τm + 4λ3
4λ + τm − x 2
− x3 ;
Ti
x1 is a constant with ‘a sufficiently large value’,
2
x2 =
[
x1 x 5e − τ m
[
x1
]
2
[
− 1 − Tm x 5e − τ m
Tm − x1
Tm
],x
−1
2
=
3
6λ2 − x 4 + x 2 τm − 0.5τm
,
4λ + τ m − x 2
4
]

λ
x 4 = Tm x 5e
− 1 + x 2Tm ; x 5 = 1 −  ;
x
1

From graphs: λ = 0.26τm , M s = 1.6 , 0.05 ≤ τm Tm ≤ 1.1 ;
2
− τ m Tm
λ = 0.21τm , M s = 1.8 , 0.05 ≤ τm Tm ≤ 1.75 ;
λ = 0.19τ m , M s = 2.0 , 0.05 ≤ τm Tm ≤ 2.5 .
13
14
2
  T 0.65 

1.111 Tm 
1


m 
Kc =

 , Ti = 2τm 1 + 
 .
τ
K m τ m 2 1 + (Tm τ m )0.65 
  m  

Td =  tan φ m +


4
+ tan 2 φ m
x1
^
 Tu

.
 4π

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4.3.3 Ideal controller in series with a first order lag
 1

1
G c (s) = K c 1 +
+ Td s 
 1 + Tf s
 Ti s
E(s)
R(s)
−
+
U(s) K e −sτ m
1
m
Tf s + 1
s(1 + sTm )


1
K c 1 +
+ Td s 
Ti s


Y(s)
Table 70: PID controller tuning rules – FOLIPD model G m (s) =
Rule
Kc
Kristiansson and
Lennartson
(2006).
Model:
Method 1;
1.7 < M max
< 1.8 .
1
Kc =
10e
−8 τ m T m
K m Tm
2
Ti
Kc
2
Kc
Ti
Kc
Ti
(1 − 0.35e
4e
−4.5 τ m T m
K m Tm
2
(1.7 − e
(
− 4 τ m Tm
)(2.4τ
m
(
− τ m 1.3Tm
)(1.7τ
m
+ 0.9Tm ) , Td =
Tm
)(2.4τ
m
Kc =
1.3Tm
)
)
0.25 ≤
1≤
τm
<1
Tm
τm
< 50
Tm
2.4τm + 0.9Tm
,
2 1 − 0.35e − 4 τ m Tm
(
)
+ 0.9Tm ) , Tf =
−8 τ m T m
0.25e
Tm
(2.4τm + 0.9Tm )2 .
+ 1.3Tm ) ,
(
Ti = 2 1.7 − e − τ m
3
1.3Tm
1.7 τ m + 1.3Tm
2 1.7 − e − τ m
(
Kc =
1.7 τ m + 1.3Tm
2 1.7 − e − τ m
Ti = 2 1 − 0.35e −4 τ m
2
Comment
Td
Direct synthesis: frequency domain criteria
Ti
Td
τ
0 ≤ m < 0.25
Tm
1
3
K m e − sτ m
s(1 + sTm )
(
τ 
2 
 0.06 + 0.004 m  1.7 − e− τ m
Tm 
K m Tm 2 
(
1.3Tm
1.3Tm
)(1.7τ
Ti = 2 1.7 − e − τ m
m
)(1.7τ
m
+ 1.3Tm ) , Tf = 0.85τm + 0.65Tm .
+ 1.3Tm ) ,
1.3Tm
)(1.7τ
m
+ 1.3Tm ) , Tf = 0.85τm + 0.65Tm .
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Rule
Kc
Ti
Kc
Ti
Td
Comment
Td
Model: Method 1
Robust
H ∞ optimal –
Tan et al.
(1998b).
4
λ = 0.5 (performance), λ = 0.1 (robustness), λ = 0.25 (acceptable).
5
3λ + Tm + τ m
1.5τ m ≤ λ
(3λ + τ m )Tm
Kc
Zhang et al.
3λ + τ m + Tm
≤ 4.5τ m
(1999b).
Model: Method 1 λ = 1.5τ m , OS = 58%, TS = 6τ m ; λ = 2.5τ m , OS = 35%, TS = 11τ m ;
λ = 3.5τ m , OS = 26%, TS = 16τ m ; λ = 4.5τ m , OS = 22%,
TS = 20τ m ; λ > 0.2291τm for stability (Ou et al., 2005b).
Tan et al.
(1998a).
Model:
Method 1;
λ not specified.
Rivera and Jun
(2000).
Model: Method 1
4
Kc =
Kc =
Kc
Tm + 8.1633τ m
Tm τ m
0.1225Tm + τ m
Tf = 0.5549τ m
7
Kc
Tm + 5.3677 τ m
Tm τ m
0.1863Tm + τ m
Tf = 0.4482τ m
8
Kc
Tm + 3.9635τ m
Tm τ m
0.2523Tm + τ m
Tf = 0.2863τ m
9
Kc
2( τ m + λ ) + Tm
(0.463λ + 0.277)([0.238λ + 0.123]Tm + τm )
K m τm 2
Ti = Tm +
5
6
K m 3λ2 + 3λτ m + τm
2
)
, Tf =
6
Kc =

0.0337Tm 
τm
1 +
.
2 
K m τm  0.1225Tm 
7
Kc =

0.0754Tm 
τm
1 +
.
2 

0
.
1863
T
K m τm 
m 
8
9
, Tf =
λ3
3λ2 + 3λτ m + τ m 2
.

0.1344Tm 
τm
1 +
.
2 
K m τm  0.2523Tm 
2(τ m + λ ) + Tm
τ m λ2
Kc =
, Tf =
.
2
2
2
K m 2τ m + 4τ m λ + λ
2τ m + 4 τ m λ + λ2
Kc =
(
)
λ not specified.
τm
,
5.750λ + 0.590
Tm τm
τm
, Td =
.
(0.238λ + 0.123)Tm + τm
0.238λ + 0.123
3λ + Tm + τm
(
2Tm (τ m + λ )
2( τ m + λ ) + Tm
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394


Td s
1
+
4.3.4 Controller with filtered derivative G c (s) = K c 1 +
 Ti s 1 + s Td

N

R(s)


Td s
1
+
K c 1 +
T

Ti s
1+ d

N

E(s)
−
+




s

U(s)
K m e − sτ
s(1 + sTm )
m
Table 71: PID controller tuning rules – FOLIPD model G m (s) =
Rule
Kc
Ti






Y(s)
K m e − sτ m
s(1 + sTm )
Comment
Td
Direct synthesis: frequency domain criteria
Kristiansson and
Lennartson
(2002a).
1
Td
Model:
Method 1;
0.5 ≤ K1 ≤ 1 .
Tm (2λ + τ m )
2λ + Tm + τ m
λ ∈ [τ m , Tm ]
(Chien and
Fruehauf, 1990);
N = 10.
Ti
Kc
Robust
Chien (1988).
2λ + τ m + Tm
Model: Method 1
K m (λ + τ m )2
1
Kc =
Ti =
2.5ωu
Km
2λ + Tm + τ m


 − 0.08 + 0.055 + 0.059 
2

K1
K1 

(
)
(
)
(
)

0.059 K 1 2 + (0.055 K 1 ) − 0.08 
2

−
,
 2.3 − 2 K 1

0.4(10 + 1 K 1 )


2
2.5 
2
0.059 K1 + (0.055 K1 ) − 0.08 
−
,

ωu  2.3 − 2K1
0.4(10 + 1 K1 )

−1
2.5 
2
0.059 K12 + (0.055 K1 ) − 0.08   1

Td =
−
 + 1 ,


0.4(10 + 1 K1 )
ωu  2.3 − 2K1
 N 
N=
(
)
2
Td
0.059 K1 + (0.055 K1 ) − 0.08
, Tf =
.
Tf
0.16ωu (10 + 1 K1 )
−1
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models





1  1 + Td s 

4.3.5 Classical controller G c (s) = K c 1 +
Td s 
Ti s 

1 +

N 

R(s)



1  1 + sTd

K c 1 +
Td
Ti s 

1 + s
N

E(s)
+
−






Y(s)
K m e − sτ
s(1 + sTm )
U(s)
m
Table 72: PID controller tuning rules – FOLIPD model G m (s) =
Rule
Kc
Ti
K m e − sτ m
s(1 + sTm )
Comment
Td
Minimum performance index: regulator tuning
Minimum IAE –
τ m = Tm ;
0.78
Shinskey (1994), K ( τ + T ) 1.38( τ m + Tm ) 0.66( τ m + Tm )
N = 10.
m
m
m
p. 75.
Model: Method 1
Minimum IAE –
Shinskey (1994),
pp. 158–159.
1
Minimum ITAE
– Poulin and
Pomerleau
(1996), (1997).
Model:
Method 1.
Process output
step load
disturbance.
3
2
Kc
Kc
Kc
τm
Tm N
0.2
0.4
0.6
0.8
1.0
τm Tm < 2
0.56τ m + 0.75Tm
Ti
Model: Method 1; N not specified.
x 2 (τ m + Tm )
Tm
τm Tm ≥ 2
3.33 ≤ N ≤ 10
Coefficient values (deduced from graphs):
τm
x1
x2
x1
T N
m
5.0728
4.9688
4.8983
4.8218
4.7839
0.5231
0.5237
0.5241
0.5245
0.5249
(
1.2
1.4
1.6
1.8
2.0
[
1
Kc =
0.926
, Ti = 1.57 τm 1 + 1.2 1 − e − Tm
K m τm (1.22 − 0.03(Tm τm ))
2
Kc =
0.926
.
K m τm (1 + 0.4(Tm τm ))
3
Kc =
τm
x2
Tm 2
+1 ; 0 ≤
≤ 2.
(Tm N )
K m (τm + Tm ) x1 (τm + Tm )2
4.7565
4.7293
4.7107
4.6837
4.6669
τm
]) .
x2
0.5250
0.5252
0.5254
0.5256
0.5257
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Rule
Kc
Poulin and
Pomerleau
(1996), (1997) –
continued.
Process input
step load
disturbance.
Ti
Comment
Td
Coefficient values (continued):
3.9465
0.5320
1.2
4.0337
3.9981
0.5315
1.4
4.0278
4.0397
0.5311
1.6
4.0278
4.0397
0.5311
1.8
4.0218
4.0397
0.5311
2.0
4.0099
0.2
0.4
0.6
0.8
1.0
0.5312
0.5312
0.5312
0.5313
0.5314
Direct synthesis: time domain criteria
Skogestad
(2003).
Model: Method 1
Hougen (1979),
p. 338.
0.5
K mτm
8τ m
N=∞
Tm
Direct synthesis: frequency domain criteria
0.7
0.3
∞
0
.
106
x
4
1
0.135 NTm τm
Kmτm
Model: Method 1; maximise crossover frequency; 10 ≤ N ≤ 30 .
N = 5.
Poulin et al.
0.5455
(1996).
Tm
M max = 1 dB
K m (τm + 0.2Tm ) 21(0.2Tm + τ m )
Model: Method 1
Robust
Tm
λ ∈ [τ m , Tm ]
Chien (1988).
Tm
2λ + τ m
(Chien and
K m (λ + τ m )2
Model: Method 1
Fruehauf, 1990);
2λ + τ m
N = 10.
2λ + τ m
Tm
K m (λ + τ m )2
Chien (1988).
Model: Method 1
2λ + τ m
K m (λ + τ m )
2
Tm
K m (λ + τ m )2
2λ + τ m
1.1Tm
Tm
2.2λ + 1.1τm
λ ∈ [τ m , Tm ]
(Chien and
Fruehauf, 1990);
N = 11.
Tuning rules converted to this equivalent controller architecture.
4
x 1 values deduced from graph:
τ m Tm
1
2
3
4
5
6
x1
6
3
2.1
1.8
1.6
1.3
τ m Tm
7
8
9
10
11
12
x1
1.1
1.0
0.9
0.8
0.7
0.7
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4.3.6 Generalised classical controller


Td s
1
+
G c (s) = K c 1 +
T

Ti s
1+ d

N

R(s) E(s)
+
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N



2
 b f 0 + b f 1s + b f 2 s
 1 + a f 1s + a f 2 s 2
s 



2 U(s)
K m e − sτ
 b f 0 + b f 1s + b f 2 s
 1 + a f 1s + a f 2 s 2
s(1 + sTm )
s

m
K m e − sτ m
s(1 + sTm )
Table 73: PID controller tuning rules – FOLIPD model G m (s) =
Rule
Tsang et al.
(1993).
Model: Method 2
Kc
x1 K m τ m
Direct synthesis: time domain criteria
0
∞
bf 0 = 1
2
ξ
1.6818
0.0
1.3829
0.1
1.1610
0.2
x1 K m τ m
x1
Coefficient values:
ξ
x1
0.9916
0.8594
0.7542
0.3
0.4
0.5
ξ
0.6693
0.6
0.6000
0.7
0.5429
0.8
0.35τm
∞
x1
ξ
0.4957
0.4569
0.9
1.0
bf 0 = 1
2
2
b f 1 = 0.5τm , b f 2 = 0.0833τm , a f 1 = 0.2τm , a f 2 = 0.01τm .
x1
Tsang and Rad
(1995).
Model: Method 2
Comment
Td
b f 1 = Tm + 0.25τm , b f 2 = 0.25Tm τm , a f 1 = 0.2τm , a f 2 = 0.01τm .
x1
Tsang et al.
(1993).
Model: Method 2
Ti
Y(s)
ξ
1.8512
0.0
1.5520
0.1
1.3293
0.2
0.809 K m τm
x1
Coefficient values:
ξ
x1
1.1595
1.0280
0.9246
0.3
0.4
0.5
0.8411
0.7680
0.6953
∞
b f 0 = 1 , b f 1 = Tm + 0.5τm , b f 2
0
ξ
x1
ξ
0.6
0.7
0.8
0.6219
0.5527
0.9
1.0
Maximum
overshoot = 16%.
= 0.5Tm τm , a f 1 = 0.12τm ,
2
a f 2 = 0.0036τm .




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Rule
Kc
Ti
Td
Comment
Direct synthesis: frequency domain criteria
τ
1
0 ≤ m ≤ 2.5
∞
0.333τm
Tm
K m (λ + τm )
Shamsuzzoha et
al. (2006a).
Model: Method 1 b = 1 , b = T , b = 0 , a = 0.167 τ 2λ − τm , a = 0 , N = ∞ .
f0
f1
m
f2
f2
f1
m
λ + τm
λ values deduced from graph: λ = 1.4τm , M s = 1.4 ;
λ = 0.7 τm , M s = 1.6 ; λ = 0.4τm , M s = 1.8 ; λ = 0.35τm , M s = 2.0 .
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
4.3.7 Two degree of freedom controller 1


Td s
1
+
U (s ) = K c  1 +
 Ti s 1 + Td

N






βTd s 
 E (s ) − K c  α +
 R (s )
Td 


s
1+ s 

N 






T
s
 α + β d K c
T 

1+ d s

N 

E(s)
−
R(s) +


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N





s

−
U(s)
+
Kc
Ti
Y(s)
K m e − sτ
s(1 + sTm )
m
Table 74: PID controller tuning rules – FOLIPD model G m (s) =
Rule
Td
K m e − sτ m
s(1 + sTm )
Comment
Minimum performance index: regulator tuning
Minimum IAE –
α = 0 ,β =1,
1.16
Shinskey (1994), K ( τ + T )
1.38( τ m + Tm ) 0.55( τ m + Tm )
N =∞.
m
m
m
p. 75.
Model: Method 1
Minimum IAE –
Shinskey (1994),
p. 159.
1
Kc =
(
1
Kc
1.28
0.48τ m + 0.7Tm
Ti
K m τm 1 + 0.24(Tm τm ) − 0.14(Tm τm )2
), T
i
399
(
α = 0 ,β =1,
N =∞.
Model: Method 1
= 1.9τ m 1 + 0.75[1 − e − (T m
τm
)
)
] .
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Rule
Kc
4
(8 − x1 )K m τm
Arvanitis et al.
(2003b).
Model: Method 1
Ti
Td
Comment
4τm
x1
0
α =1
Minimum ISE. 2
Minimum performance index. 3
2
2
3
τ 
τ 
τ 
τ 
x1 = 2.6302 − 2.1248 m  + 2.5776 m  − 1.8511 m  + 0.82557 m 
T
T
T
 m
 m
 m
 Tm 
τ
− 0.23641 m
 Tm
5

τ
 + 0.044128 m

 Tm
τ
− 1.7164.10 −5  m
 Tm
6

τ
 − 0.0053335 m

 Tm
9

τ
 + 3.1685.10 −7  m

 Tm
4
7

τ
 + 0.00040201 m

 Tm
10

τ
 , 0 < m < 4.1 ;
T
m

τ
 τ
x1 = 1.7110073 − 0.003554 m − 4.1 , m ≥ 4.1 .
 Tm
 Tm
∞
3
Performance index =
∫ [e ( t) + K
2
2
m
]
u 2 ( t ) dt .
0
2
3
τ 
τ 
τ 
τ 
x1 = 2.1054 − 0.76462 m  + 0.79445 m  − 0.51826 m  + 0.21771 m 
 Tm 
 Tm 
 Tm 
 Tm 
τ
− 0.06 m
 Tm
5

τ
 + 0.010927 m

 Tm
τ
− 4.132.10 −6  m
 Tm
9
6

τ
 − 0.0013006 m

 Tm

τ
 + 7.6235.10 −8  m

 Tm
7

τ
 + 9.717.10 −5  m

 Tm
10

τ
 , 0 < m < 4.6 ;
Tm

τ
 τ
x1 = 1.69667833406 − 0.00148768 m − 4.6  , m ≥ 4.6 .
T
 m
 Tm



4
8



8
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Arvanitis et al.
(2003b) –
continued.
Arvanitis et al.
(2003b).
Model: Method 1
Ti
401
Comment
Td
Minimum performance index. 4
5
0
Ti
Kc
α =1
Minimum ISE. 6
∞
4
Performance index =
2
 2
2  du 
e ( t ) + K m   dt .
 dt  

0 
∫
τ
 τ
x1 = 1.6505623 − 0.0034985 m − 8  , m ≥ 2 ;
T
 m
 Tm
2
3
τ 
τ 
τ 
τ 
x1 = 2.1806 − 0.60273 m  + 0.3875 m  − 0.14955 m  + 0.034535 m 
 Tm 
 Tm 
 Tm 
 Tm 
τ
− 0.0042243 m
 Tm
5

τ
 + 8.2395.10 −5  m

 Tm
τ
+ 4.2603.10 −7  m
 Tm
5
Kc =
[
9
6

τ
 + 5.0912.10 −5  m

 Tm
7

τ
 − 7.283.10 −6  m

 Tm



8
10

τ 
τ
 − 9.5824.10 −9  m  , m range not defined.
Tm

 Tm 
12.5664(τm + Tm ) − 4
) ]
(
K m 2(τm + Tm )(12.5664(τm + Tm ) − 4 ) − 1.5τm 2 + 3τm Tm + 2Tm 2 x1
3.4674τm + 3.1416Tm − 1
12.5664(τm + Tm ) − 4
]x 2 ; Ti =
.
x1
τm
with x1 = [0.5708 +
2
6
4
3
τ 
τ 
τ 
τ 
x 2 = 0.99315 + 1.9536 m  − 2.5157 m  + 1.9163 m  − 0.8893 m 
 Tm 
 Tm 
 Tm 
 Tm 
τ
+ 0.26167 m
 Tm
5

τ
 − 0.049809 m

 Tm
τ
+ 2.0083.10 −5  m
 Tm
9
6

τ
 + 0.0061098 m

 Tm

τ
 − 3.7368.10 −7  m

 Tm
10
7

τ
 − 0.00046593 m

 Tm

τ
 , 0 < m < 6.0 ;
Tm

τ
 τ
x 2 = 1.96705757 + 0.01308189 m − 6  , m ≥ 6.0 .
T
 m
 Tm
4



8
b720_Chapter-04
04-Apr-2009
Handbook of PI and PID Controller Tuning Rules
402
Rule
Kc
Ti
Arvanitis et al.
(2003b) –
continued.
Td
Comment
Minimum performance index. 7
Minimum performance index. 8
∞
7
FA
Performance index =
∫ [e ( t) + K
2
2
m
]
u 2 ( t ) dt .
0
2
3
τ 
τ 
τ 
τ 
x 2 = 0.73114 + 2.3461 m  − 2.8509 m  + 2.0965 m  − 0.95252 m 
T
T
T
 m
 m
 m
 Tm 
τ
+ 0.27639 m
 Tm
5

τ
 − 0.052089 m

 Tm
τ
+ 2.0611.10 −5  m
 Tm
6

τ
 + 0.0063413 m

 Tm
9

τ
 − 3.8178.10 −7  m

 Tm
4
7

τ
 − 0.00048064 m

 Tm



8
10

τ
 , 0 < m < 4.5 ;
T
m

τ
 τ
x 2 = 1.916423116 + 0.018175439 m − 4.5  , m ≥ 4.5 .
 Tm
 Tm
∞
8
Performance index =
2
 2
2  du 
e ( t ) + K m   dt .
 dt  

0 
∫
2
3
τ 
τ 
τ 
τ 
x 2 = 0.60896 + 3.0593 m  − 3.9339 m  + 2.933 m  − 1.3353 m 
 Tm 
 Tm 
 Tm 
 Tm 
τ
+ 0.38696 m
 Tm
5

τ
 − 0.072771 m

 Tm
τ
+ 2.8623.10 −5  m
 Tm
9
6

τ
 + 0.0088395 m

 Tm

τ
 − 5.2941.10 −7  m

 Tm
10
7

τ
 − 0.00066864 m

 Tm

τ
 , 0 < m < 4.5 ;
Tm

τ
 τ
x 2 = 1.9283751 + 0.016846129 m − 4.5  , m ≥ 4.5 .
T
 m
 Tm
4



8
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Ti
Td
Comment
4
τm
x1
(8 − x1 )
τm
16
α = 1, β = 1,
N =∞.
Kc
16
Arvanitis et al. (16 − 3x )K τ
1
m m
(2003b).
Model: Method 1
Minimum ISE. 9
Minimum performance index. 10
2
9
3
τ 
τ 
τ 
τ 
x1 = 1.5841 + 0.3101 m  + 0.38841 m  − 0.4622 m  + 0.21918 m 
 Tm 
 Tm 
 Tm 
 Tm 
τ
− 0.060691 m
 Tm
5

τ
 + 0.010771 m

 Tm
τ
− 3.8302.10 −6  m
 Tm
9

τ
 + 7.0316.10 −8  m

 Tm
∞
10
Performance index =
6

τ
 − 0.0012466 m

 Tm
∫ [e ( t) + K
2
2
m
4
7

τ
 + 9.1201.10 −5  m

 Tm



10

 .

]
u 2 ( t ) dt .
0
2
3
τ 
τ 
τ 
τ 
x1 = 0.066485 + 2.0757 m  + 1.0972 m  − 2.7957 m  + 1.9019 m 
 Tm 
 Tm 
 Tm 
 Tm 
τ
− 0.68418 m
 Tm
5

τ
 + 0.14763 m

 Tm
τ
− 7.2368.10 −5  m
 Tm
9
6

τ
 − 0.019733 m

 Tm

τ
 + 1.397.10 −6  m

 Tm
10
7

τ
 + 0.0016016 m

 Tm

τ
 , 0 < m < 4.0 ;
Tm

τ
 τ
x1 = 2.128638 − 0.00560812 m − 4.0  , m ≥ 4.0 .
T
 m
 Tm
4



8
8
403
b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
404
Rule
Kc
11
Arvanitis et al.
(2003b).
Model: Method 1
Kc
Ti
Td
Comment
Ti
Td
α = 1, β = 1,
N =∞.
Minimum ISE. 12
Minimum performance index. 13
11
Kc =
12.5664(τm + Tm ) − 4
K m τm [25.1328(τm + Tm ) − 8 − (τm + 2Tm )x1 ]
with x1 = [0.5708 +
Ti =
12.5664(τm + Tm ) − 4
x1
2
.
, Td = Tm − Tm
x1
12.5664(τm + Tm ) − 4
2
12
3.4674τm + 3.1416Tm − 1
]x 2 ,
τm
3
τ 
τ 
τ 
τ 
x 2 = 1.178 + 4.5663 m  − 7.0371 m  + 5.5784 m  − 2.6103 m 
 Tm 
 Tm 
 Tm 
 Tm 
τ
+ 0.7679 m
 Tm
5

τ
 − 0.1458 m

 Tm
τ
+ 5.83.10 −5  m
 Tm
6

τ
 + 0.017833 m

 Tm
7
4

τ
 − 0.0013561 m

 Tm



8
10
9

τ
 − 1.0823.10 −6  m

 Tm

τ
 , 0 < m < 3.5 ,
Tm

τ
 τ
x 2 = 2.2036 − 0.0033077 m − 6.5  , m ≥ 3.5 .
T
 m
 Tm
∞
13
Performance index =
∫ [e ( t) + K
2
2
m
]
u 2 ( t ) dt .
0
2
3
τ 
τ 
τ 
τ 
x 2 = −0.1185 + 4.781 m  − 3.9977 m  + 1.733 m  − 0.42245 m 
T
T
T
 m
 m
 m
 Tm 
τ
+ 0.057708 m
 Tm
5

τ
 − 0.0038421 m

 Tm
6

τ
 + 2.6386.10 −5  m

 Tm
τ
− 4.2486.10 −7  m
 Tm
9

 .

7
4

τ
 + 1.078.10 −5  m

 Tm



8
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Arvanitis et al.
(2003b).
Model: Method 1
Ti
Comment
Td
Minimum performance index: servo tuning
4
4τm
(8 − x1 )K m τm
0
α =1
x
1
Minimum ISE. 14
Minimum performance index. 15
Minimum performance index. 16
2
14
3
τ 
τ 
τ 
τ 
x1 = 1.997 − 0.82781 m  + 1.0002 m  − 0.72106 m  + 0.32269 m 
 Tm 
 Tm 
 Tm 
 Tm 
τ
− 0.092452 m
 Tm
5

τ
 + 0.017215 m

 Tm
τ
− 6.5691.10 −6  m
 Tm
6

τ
 − 0.0020706 m

 Tm
9

τ
 + 1.2025.10 −7  m

 Tm
4
7

τ
 + 0.00015505 m

 Tm



8
10

τ
 , 0 < m < 5.5 ;
T
m

τ
 τ
x1 = 1.6281745 − 0.001171 m − 5.5  , m ≥ 5.5 .
T
 m
 Tm
∞
15
∫ [e ( t) + K
2
Performance index =
2
m
]
u 2 ( t ) dt .
0
2
3
τ 
τ 
τ 
τ 
x1 = 1.5693 + 0.1066 m  − 0.10101 m  + 0.05217 m  − 0.016147 m 
T
T
T
 m
 m
 m
 Tm 
τ
+ 0.0030412 m
 Tm
5

τ
 − 0.00032187 m

 Tm
−7 
τ
− 1.2274.10  m
T
 m
9
6

τ
 + 1.2181.10 −5  m

 Tm

τ
 + 3.7365.10 −9  m

 Tm
10

τ
 , 0 < m < 4.6 .
T
m

∞
16
Performance index =
2
 2
2  du 
e ( t ) + K m   dt
 dt  

0 
∫
τ

τ
x1 = 0.780651417 + 4.7464131 m − 0.1 , 0 < m < 0.3 ;
Tm
 Tm

τ

τ
x1 = 1.729934 − 0.051535 m − 0.3  , 0.3 ≤ m < 2 .
T
T
m
 m

7
4

τ
 + 9.9698.10 −7  m

 Tm



8
405
b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
406
Rule
Kc
Arvanitis et al.
(2003b).
Model: Method 1
17
Kc
Ti
Td
Comment
Ti
0
α =1
Minimum ISE. 18
Minimum performance index. 19
17
Kc =
12.5664(τm + Tm ) − 4
[
3.4674τm + 3.1416Tm − 1
12.5664(τm + Tm ) − 4
]x 2 , Ti =
.
x1
τm
with x1 = [0.5708 +
2
18
) ]
(
K m 2(τm + Tm )(12.5664(τm + Tm ) − 4 ) − 1.5τm 2 + 3τm Tm + 2Tm 2 x1
3
τ 
τ 
τ 
τ 
x 2 = 0.79001 + 2.114 m  − 2.5601 m  + 1.8971 m  − 0.86871 m 
 Tm 
 Tm 
 Tm 
 Tm 
τ
+ 0.25371 m
 Tm
5

τ
 − 0.048068 m

 Tm
6

τ
 + 0.0058771 m

 Tm
7

τ
 − 0.00044707 m

 Tm



8
10
9
τ
+ 1.9231.10 −5  m
 Tm
4

τ
 − 3.5718.10 −7  m

 Tm

τ
 , 0 < m < 4.5 ;
T
m

τ
 τ
x 2 = 1.903177576 + 0.0181584 m − 4.5  , m ≥ 4.5 .
T
 m
 Tm
∞
19
Performance index =
∫ [e ( t) + K
2
2
m
]
u 2 ( t ) dt .
0
2
3
τ 
τ 
τ 
τ 
x 2 = 0.48124 + 2.5549 m  − 2.896 m  + 2.0495 m  − 0.91115 m 
T
T
T
 m
 m
 m
 Tm 
τ
+ 0.26094 m
 Tm
−5 
5

τ
 − 0.048775 m

 Tm
τ
+ 1.9095.10  m
T
 m
9
6

τ
 + 0.0059064 m

 Tm

τ
 − 3.5335.10 −7  m

 Tm
10
7

τ
 − 0.00044618 m

 Tm

τ
 , 0 < m < 4.0 ;
T
m

τ
 τ
x 2 = 1.87054524 + 0.021802 m − 4  , m ≥ 4.0 .
T
 m
 Tm
4



8
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Arvanitis et al.
(2003b) –
continued.
Arvanitis et al.
16
(2003b).
(16 − 3x1 )K m τm
Model: Method 1
Ti
407
Comment
Td
Minimum performance index. 20
α = 1, β = 1,
N =∞.
(8 − x1 )
τm
16
4
τm
x1
Minimum ISE. 21
∞
20
Performance index =
2
 2
2  du 
e ( t ) + K m   dt .
 dt  

0 
∫
2
3
τ 
τ 
τ 
τ 
x 2 = 0.31373 + 3.7796 m  − 5.0294 m  + 3.8589 m  − 1.7956 m 
T
T
T
 Tm 
 m
 m
 m
τ
+ 0.52906 m
 Tm
5

τ
 − 0.10079 m

 Tm
τ
+ 4.0678.10 −5  m
 Tm
6

τ
 + 0.012369 m

 Tm
9

τ
 − 7.5694.10 −7  m

 Tm
7
4

τ
 − 0.00094357 m

 Tm



8
10

τ
 , 0 < m < 3.5 ;
T
m

 τ
τ
x 2 = 1.86144723 + 0.021989 m − 3.5  , m ≥ 3.5 .
 Tm
 Tm
2
21
3
τ 
τ 
τ 
τ 
x1 = 1.2171 + 0.7339 m  + 0.083449 m  − 0.14905 m  + 0.030257 m 
T
T
T
 Tm 
 m
 m
 m
τ
+ 0.0045567 m
 Tm
5

τ
 − 0.0029943 m

 Tm
τ
+ 2.7505.10 −6  m
 Tm
6

τ
 + 0.00056821 m

 Tm
9
7
4

τ
 − 5.4957.10 −5  m

 Tm

τ
 − 5.6642.10 −8  m

 Tm
10

 .




8
b720_Chapter-04
04-Apr-2009
Handbook of PI and PID Controller Tuning Rules
408
Rule
Kc
Arvanitis et al.
(2003b) –
continued.
Arvanitis et al.
(2003b).
Model: Method 1
Ti
Comment
Td
Minimum performance index. 22
23
Ti
Kc
Td
α = 1, β = 1,
N =∞.
Minimum ISE. 24
∞
22
FA
∫ [e ( t) + K
2
Performance index =
2
m
]
u 2 ( t ) dt .
0
2
3
τ 
τ 
τ 
τ 
x1 = 0.083709 + 2.2875 m  − 0.18739 m  − 1.0059 m  + 0.79718 m 
 Tm 
 Tm 
 Tm 
 Tm 
τ
− 0.3036 m
 Tm
5

τ
 + 0.067859 m

 Tm
τ
− 3.5473.10 −5  m
 Tm
6

τ
 − 0.0093104 m

 Tm
9

τ
 + 6.9487.10 −7  m

 Tm
7

τ
 + 0.00077161 m

 Tm



4
8
10

τ
 , 0 < m < 4.5
Tm

τ
 τ
x 1 = 2.444547 − 0.0146274714 m − 4.5  , m ≥ 4.5 .
T
 m
 Tm
12.5664(τm + Tm ) − 4
23
Kc =
K m τm [25.1328(τm + Tm ) − 8 − (τm + 2Tm )x1 ]
with x1 = [0.5708 +
Ti =
12.5664(τm + Tm ) − 4
x1
2
.
, Td = Tm − Tm
x1
12.5664(τm + Tm ) − 4
2
24
3
3.4674τm + 3.1416Tm − 1
]x 2
τm
τ 
τ 
τ 
τ 
x 2 = 1.1571 + 5.0775 m  − 8.2021 m  + 6.6353 m  − 3.1355 m 
 Tm 
 Tm 
 Tm 
 Tm 
τ
+ 0.92683 m
 Tm
5

τ
 − 0.17635 m

 Tm
τ
+ 7.0439.10 −5  m
 Tm
9
6

τ
 + 0.02158 m

 Tm

τ
 − 1.3056.10 −6  m

 Tm
10
7

τ
 − 0.0016402 m

 Tm

τ
 , 0 < m < 4.5
Tm

 τ
τ
x 2 = 2.124307 − 0.004844182 m − 4.5  , m ≥ 4.5 .
T
 Tm
 m
4



8
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Ti
Arvanitis et al.
(2003b) –
continued.
Comment
Minimum performance index. 25
Minimum performance index. 26
∞
25
Td
Performance index =
∫ [e ( t) + K
2
2
m
]
u 2 ( t ) dt .
0
2
3
τ 
τ 
τ 
τ 
x 2 = −0.70861 + 7.4212 m  − 8.1214 m  + 4.8168 m  − 1.7217 m 
T
T
T
 Tm 
 m
 m
 m
τ
+ 0.38998 m
 Tm
5

τ
 − 0.056975 m

 Tm
6

τ
 + 0.0053141 m

 Tm
7

τ
 − 0.00030233 m

 Tm



8
10
9
τ
+ 9.3969.10 −6  m
 Tm
4

τ
 − 1.1861.10 −7  m

 Tm

τ
 , 0 < m < 4.5
T
m

τ
 τ
x 2 = 2.1069751341 − 0.0017098 m − 4.5  , m ≥ 4.5 .
 Tm
 Tm
∞
26
Performance index =
2
 2
2  du 
e ( t ) + K m   dt .
 dt  

0 
∫
2
3
τ 
τ 
τ 
τ 
x 2 = 0.05486 + 0.81662 m  − 1.4989 m  + 1.1955 m  − 0.48464 m 
 Tm 
 Tm 
 Tm 
 Tm 
τ
+ 0.11011 m
 Tm
5

τ
 − 0.013386 m

 Tm
τ
− 5.307.10 −6  m
 Tm
9
6

τ
 + 0.00059665 m

 Tm

τ
 + 1.6437.10 −7  m

 Tm
10
7

τ
 + 3.8633.10 −5  m

 Tm

τ
 , 0 < m < 6.0 ;
Tm

 τ
τ
x 2 = 2.1111 − 0.001025 m − 6  , m ≥ 6 .
T
 Tm
 m
4



8
409
b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
410
Rule
Kc
Ti
x1 K m
Taguchi et al.
(1987).
x1
Model: Method 1
12.06
3.93
2.19
1.50
1.14
0.92
0.77
Minimum ITAE
– Pecharromán
and Pagola
(2000).
Model:
Method 4;
Km = 1;
Comment
Td
Minimum performance index: other tuning
x 2Tm
x 3Tm
N=∞
x2
x3
α
β
0.81
1.55
2.23
2.88
3.50
4.10
4.68
x1K u
0.38
0.62
0.78
0.90
1.00
1.09
1.18
0.67
0.67
0.67
0.66
0.66
0.66
0.66
0.84
0.82
0.80
0.79
0.77
0.76
0.75
x 2 Tu
τm
Tm
x1
x2
x3
α
β
τm
Tm
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.67
0.59
0.53
0.42
0.36
0.27
0.22
5.25
5.82
6.36
7.74
9.06
11.38
14.76
x 3Tu
1.26
1.34
1.42
1.61
1.82
2.26
2.59
0.66
0.66
0.66
0.66
0.66
0.66
0.68
0.74
0.74
0.73
0.72
0.72
0.70
0.76
1.6
1.8
2.0
2.5
3.0
4.0
5.0
x1
x2
Coefficient values:
x3
α
φc
1.672
0.366
0.136
0.601
− 164 0
1.236
0.427
0.149
0.607
− 160 0
0.994
0.486
0.155
0.610
− 1550
Tm = 1 ;
0.842
0.538
0.154
0.616
− 150 0
β = 1 , N = 10;
0.05 < τ m < 0.8 .
0.752
0.567
0.157
0.605
− 1450
0.679
0.610
0.149
0.610
− 140 0
0.635
0.637
0.142
0.612
− 1350
0.590
0.669
0.133
0.610
− 130 0
0.551
0.690
0.114
0.616
− 1250
0.520
0.776
0.087
0.609
− 120 0
0.509
0.810
0.068
0.611
− 118 0
Minimum ITAE
– Pecharromán
(2000a).
Model:
Method 4;
Km = 1;
Tm = 1 ;
0.05 < τ m < 0.8 .
x1K u
0
x 2 Tu
x1
x2
α
0.049
2.826
0.066
Coefficient values:
φc
x1
x2
α
φc
0.218
1.279
0.564
− 1350
− 160 0
0.250
1.216
0.573
− 130 0
0.522
− 1550
0.286
1.127
0.578
− 1250
1.716
0.532
− 150
0
0.330
1.114
0.579
− 120 0
0.159
1.506
0.544
− 1450
0.351
1.093
0.577
− 118 0
0.189
1.392
0.555
− 140 0
0.506
− 164
0
2.402
0.512
0.099
1.962
0.129
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
x1K u
Minimum ITAE
– Pecharromán
x1
x2
(2000b), p. 142.
0.0469 2.8052
Model:
Method 4;
0.0662 2.3876
Km = 1;
0.0988 1.9581
Tm = 1 ;
0.05 < τ < 0.8 . 0.1271 1.6757
m
Minimum ITAE
– Pecharromán
(2000b), p. 148.
Model:
Method 4;
Km = 1;
Tm = 1 ;
β = 1 , N = 10;
0.05 < τm < 0.8 .
Taguchi and
Araki (2000).
Model: Method 1
27
Kc =
Ti
Td
x 2 Tu
0
Comment
Coefficient values:
φc
x1
α
411
x2
α
φc
0.5061 − 164.10 0.2174 1.2566 0.5642 − 135.00
0.5119 − 160.50 0.2511 1.1948 0.5730 − 130.00
0.5220 − 155.00 0.2870 1.1281 0.5777 − 125.00
0.5319 − 150.00 0.3280 1.0860 0.5794 − 120.00
0.1589 1.5052 0.5439 − 145.00 0.3501 1.0822 0.5767 − 118.00
0.1866 1.3619 0.5546 − 140.00
x1K u
x 2 Tu
x 3Tu
x1
x2
Coefficient values:
x3
α
φc
1.6695
0.3684
0.1323
0.6007
− 164.10
1.2434
0.4389
0.1507
0.6073
− 160.00
0.9901
0.4846
0.1587
0.6102
− 155.00
0.8378
0.5393
0.1562
0.6163
− 150.00
0.7383
0.5789
0.1548
0.6052
− 145.00
0.6728
0.6283
0.1488
0.6105
− 140.00
0.6196
0.6221
0.1376
0.6120
− 135.00
0.5948
0.6939
0.1326
0.6096
− 130.00
0.5513
0.7050
0.1190
0.6157
− 125.00
0.5271
0.7471
0.0992
0.6088
− 120.00
0.5229
0.8886
0.0862
0.6115
27
Kc
Ti
0
− 118.0 0
OS ≤ 20% ;
τm Tm ≤ 1.0 .

0.2839
1 
,
 0.1787 +

Km 
(τm Tm ) + 0.001723 
Ti = 4.296 + 3.794(τm Tm ) + 0.2591(τm Tm ) , α = 0.6551 + 0.01877(τ m Tm ) .
2
b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
412
Rule
Kc
Taguchi and
Araki (2000) –
continued.
28
Kc
Ti
Td
Ti
Td
Comment
N=∞
x1 K m
x 2Tm
x 3Tm
Taguchi and
α
α
x1
x 2 x3
β
x4
x1
x 2 x3
β
x4
Araki (2002).
8.91
1.73
0.46
0.32
0.72
0.1
12.36
0.68
0.39
0.66
0.80
1.5
Model: Method 1
11.13 1.40 0.39 0.45 0.91 0.5 13.00 0.54 0.40 0.67 0.76 2.0
12.04 0.81 0.38 0.67 0.84 1.0
Load disturbance model = K m e −sτ m (1 + sx 4Tm ) ; τm Tm = 0.2 .
Taguchi et al.
(2002).
Model:
Method 1;
α=0.
Taguchi et al.
(2002).
Model:
Method 1;
α=0.
Taguchi et al.
(2002).
28
Kc =
∞
x1Tm K m
x1
x2
16.54
0.32
5.33
0.52
1.96
0.76
1.17
0.91
x1Tm K m
N = 10
x 2Tm
β
τm Tm
0.89
0.85
0.78
0.73
0.1
0.2
0.4
0.6
x1
∞
x1
x2
β
τm Tm
x1
x2
26.56
7.75
2.54
1.42
0.29
0.50
0.78
0.95
0.81
0.79
0.74
0.70
0.1
0.2
0.4
0.6
0.98
0.74
0.35
0.15
1.08
1.19
1.59
2.77
29
Kc
∞
β
Td
m
)
)
Tm ) − 1.774(τm Tm ) + 0.6878(τm Tm ) , α = 0.6647 ,
2
m
τm Tm
0.66
0.8
0.62
1.0
0.50
2.0
0.37
5.0
N = 10;
Model: Method 1
Ti = Tm 0.03330 + 3.997(τm Tm ) − 0.5517(τm Tm ) ,
2
τm Tm
0.68
0.8
0.64
1.0
0.50
2.0
0.37
5.0
N= ∞
Td

1 
0.5184
,
0.7608 +
2 
K m 
((τm Tm ) + 0.01308) 
(
= T (0.03432 + 2.058(τ
β
x2
0.83
1.02
0.65
1.10
0.32
1.47
0.14
1.47
x 2Tm
β = 0.8653 − 0.1277(τ m Tm ) + 0.03330(τ m Tm ) , N not specified.
3
2
29
Kc =

τ
τ
Tm 
0.2938
, β = 0.9305 − 0.4147 m + 0.1234 m
0.4073 +
2
K m 
Tm
((τm Tm ) + 0.03829) 
 Tm
2

 ,


2
3

 τm  
 τm 
τm
τm






Td = Tm 0.08289 + 2.682
− 2.959
 + 1.309 T   , α = 0 , 0.1 ≤ T ≤ 1.0 .
Tm
T

m
 m 
 m

b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Kuwata (1987).
Model: Method 1
Comment
Td
Direct synthesis: time domain criteria
3.33(Tm + τm )
0
K m (Tm + τm )
1
30
Arvanitis et al.
(2003b).
Model: Method 1
Ti
Ti
Kc
4
(8 − x1 )K m τm
Åström and
Hägglund (1995),
pp. 210–212.
31
Kc
32
Kc
33
Kc
34
Kc
35
Kc
0
Ti
(2ξ
2
α = 1, β = 1 ,
N =∞.
α = 1;
4
x1 = 2
.
4ξ + 1
Td
4τm
x1
)
+ 1 τm
Td
Ti
Td
α =1
α = 1, β = 1,
N =∞.
Direct synthesis: frequency domain criteria
2
M s = 1.4
5.7τ m e1.7 τ−0.69 τ
0
Ti
M s = 2.0
0.14 ≤ τm Tm ≤ 5.5 ; Model: Method 3.
Klán (2001).
30
31
1.067(Tm + τm )
33
2
; Ti =
2
2
2
m
m
(Tm + τm )
m
2
m
m
2
m m
2
m m
2.5τm (Tm + 0.5τm )
3
; Td =
(Tm + τm )
3
2
2
2
i
m
m
).
m
2
2
Kc
2
2
Model: Method 1
(1 + 4ξ )(τ + T )
, T = (1 + 4ξ )(τ + T
K [τ (0.5 + 8ξ ) + τ T (1 + 16ξ ) + 8ξ T ]
16(2ξ + 1)
16ξ + 4
=
(32ξ + 4)K τ , T = 16(2ξ + 1) τ .
τ T (1 + 4ξ )
τ + T + 4τ ξ
.
=
, T = τ + T + 4τ ξ , T =
τ + T + 4τ ξ
K τ (1 + 8ξ )
Kc =
Kc
6.25τm (2Tm + τm )
2
K m τm (2Tm + τm )
2
0
Ti
Kc
3
Kc =
m
32
36
d
m
2
m
2
m m
2
i
m
m
m
2
m m
d
m
m
2
m
2
2
34
Kc =
2
0.41e−0.23τ + 0.019 τ
, α = 1 − 0.33e 2.5τ−1.9 τ .
K m (Tm + τm )
35
Kc =
2
2
0.81e−1.1τ + 0.76 τ
, Ti = 3.4τm e0.28τ − 0.0089 τ , α = 1 − 0.78e −1.9 τ+1.2 τ .
K m (Tm + τm )
2
2
36
2
2
0.41e−0.23τ + 0.019 τ
Kc =
, Ti = 5.7(τm + Tm )e1.7 τ − 0.69 τ , α = 1 − 0.33e 2.5τ−1.9 τ .
K m (Tm + τm )
.
413
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414
Rule
Kc
Wang and Cai
37
Kc
(2001).
Model: Method 1 0.6068 K m τm
Wang and Cai
(2002).
Xu and Shao
(2003a).
0.6189 K m τm
Xu and Shao
(2003c).
1.5τ m + 0.5Tm
Xu and Shao
(2004a).
Model: Method 1
40
Td
Ti
Td
5.9784 τ m
38
Td
5.9112 τ m
39
Td
2(Tm + 1.5τ m )
Kc
2K m τ m
Ti
2
(2λ − 1)τm + Tm
K m λ2 τm 2
Tm + τ m
(Tm + 0.5τm )2
(Tm + 1.5τ m )
1.5Tm τ m
1.5τ m + 0.5Tm
(2λ − 1)Tm τm
(2λ − 1)τm + Tm (2λ − 1)τm + Tm
N =∞; α =
Comment
β = 0, N = ∞.
Model: Method 1
Model: Method 1
41
Model: Method 1
λ not defined.
λτ m
λ
; β=
.
Tm + (2λ − 1)τm
2λ − 1
τm (1.3608M s − 1.2064)
1 
1.2064 
,
1.3608 −
 , Ti =
K m τm 
Ms 
0.29M s − 0.3016
0. 2 M s
(T + 0.1τm )(1.450Ms − 1.508) , α =
, 1.3 ≤ M s ≤ 2 .
Td = m
1.3608M s − 1.2064
1.3608M s − 1.2064
37
Kc =
38
Td = 0.8363(Tm + 0.1τm ) , α = 0.3296 , M s = 1.6 ( A m > 2.66 , φ m > 36.4 0 ) .
39
Td = 0.8460(Tm + 0.1τm ) , N = ∞ , β = 0 , α = 0.3232 , A m = 3 , φ m = 60 0 .
40
Kc =
41
α=
τm
0.5(Tm + 1.5τm )
, N= ∞ , β = 0 , A m > 2 , φ m > 60 0 .
, α=
Tm + 1.5τ m
K m τm (Tm + τm )
τm
, β = 0.6667 , N= ∞ , A m > 2 , φ m > 60 0 .
1.5Tm + 0.5τ m
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Ti
Td
Comment
Td
α = 0. 6 , β = 0 .
415
Robust
42
Shamsuzzoha
and Lee (2007a).
Model: Method 1
Ti
Kc
λ = 0.26τm , M s = 1.6 , 0.05 ≤ τm Tm ≤ 1.1 ;
λ = 0.21τm , M s = 1.8 , 0.05 ≤ τm Tm ≤ 1.75 ;
λ = 0.19τ m , M s = 2.0 , 0.05 ≤ τm Tm ≤ 2.5 ;
Kuwata (1987);
x 1 , x 2 deduced
from graphs.
x 1K u
x1
0.12
0.19
0.25
x1
Oubrahim and
Leonard (1998).
Model: Method 1
42
Kc =
Td =
equations for λ deduced from graphs.
Ultimate cycle
0
x 2 Tu
x2
1.14
1.09
0.91
x 1K u
x2
0.80
0.54
0.73
0.57
0. 6 K u
α =1
τm Tu
x1
x2
τm Tu
x1
x2
τm Tu
0.08
0.10
0.13
0.30
0.34
0.36
x 2 Tu
0.89
0.86
0.84
0.15
0.18
0.20
0.38
0.83
0.25
x3
τm Tu
0.11
0.15
0.09
0.17
0.5Tu
x 3 Tu
x1
x2
0.71
0.61
0.69
0.62
0.125Tu
α = 1, β = 1,
N =∞.
τm Tu
x3
0.08
0.20
0.08
0.22
20% overshoot.
α = 0.9 ; β = 0.99 ; N = ∞ ; 0.05 < τm Tm < 0.8 .
2
Ti
6λ2 − x 4 + x 2 τm − 0.5τm
, Ti = x1 + Tm + x 2 − x 3 , x 3 =
,
x1K m (4λ + τm − x 2 )
4λ + τm − x 2
x 4 + x 2 (Tm + x1 ) + x1Tm −
0.167 τm 3 − 0.5x 2 τm 2 + x 4 τm + 4λ3
4λ + τ m − x 2
− x3 ,
Ti
x1 is a constant with a ‘sufficiently large value’;

2 
x1 1 −
x2 =


τm
τm
4
4



λ  − x1
λ  − Tm
2 


 e


−
−
−
−
1
T
1
e
1
m

 Tm 

x1 
,


Tm − x1
τm
4


λ  − Tm
2 
 e
− 1 + x 2Tm , N = ∞ .
x 4 = Tm 1 −

 Tm 



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416
4.3.8 Two degree of freedom controller 2




T
s
1
d
 1 + b f 1s (F(s) R (s) − Y(s) ) − K Y(s) ,
U(s) = K c 1 +
+
0
 Ti s
T  1 + a f 1s
1+ d s 

N 

1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4
F(s) =
1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5 s 4
R(s)
F(s)
+
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N



 1 + b f 1s
 1 + a f 1s
s

+ U(s)
−
K m e − sτ
s(1 + sTm )
m
K0
Table 75: PID controller tuning rules – FOLIPD model G m (s) =
Rule
Kc
1
Huang et al.
(2005b).
Model: Method 1
Kc
Ti
[
Td
Direct synthesis: time domain criteria
Ti
T
m
]
Y(s)
K m e − sτ m
s(1 + sTm )
Comment
N=∞
a f 1 = max 0.01 τm ,0.01 , b f 1 = 0 ; a f 2 = Ti , b f 2 = 1.452τm + Tm ;
a f 3 = Ti Tm , b f 3 = 1.452τmTm ; a f 4 = 0 , a f 5 = 0 , b f 4 = 0 , b f 5 = 0 ,
K0 = 0 .
Robust
Lee and Edgar
(2002).
Model: Method 1
2
Kc
Ti
0
a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , a f 4 = 0 ,
a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0.25K u .
1
Kc =

2.2221K m τ m 
0.2328 
T 
.
1 + 0.6887 m  + 0.5173 , Ti = (1.452τ m + Tm )1 +




K m τm 
τm 
 1 + 0.6887(Tm τ m ) 
2
Kc =
0.25K u Ti
4
τm
, T =
.
−τ +
(λ + τm ) i K u K m m 2(λ + τm )
2
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
3
Liu et al. (2003).
Model: Method 1
Kc
Ti
Td
Kc
Ti
Td
417
Comment
a f 1 = 0 , b f 1 = 0 ; a f 2 = 3λ + τm , b f 2 = λ ; a f 3 = 0 , a f 4 = 0 ,
a f 5 = 0 , bf 3 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 .
Representative λ values (deduced from graph):
OS
Rise time
OS
Rise time
λ
λ
0.3τ m
≈ 57%
≈ 3τ m
2τ m
≈ 0%
≈ 8τ m
0.5τ m
≈ 33%
≈ 3.5τ m
2.5τ m
≈ 0%
≈ 10.5τ m
τm
≈ 7%
≈ 5τ m
3τ m
≈ 0%
≈ 12.5τ m
1.5τ m
≈ 0%
≈ 8.5τ m
3.5τ m
≈ 0%
≈ 14τ m
a f 1 = 0 , b f 1 = 0 ; a f 2 = 7λ + τm , b f 2 = 3λ ; a f 3 = λ(16λ + 4τm ) ,
b f 3 = 3λ2 ; a f 4 = 4λ2 (3λ + τm ) , b f 4 = λ3 , a f 5 = 0 , b f 5 = 0 , K 0 = 0 .
Representative λ values (deduced from graph):
OS
Rise time
OS
Rise time
λ
λ
0.3τ m
≈ 43%
≈ 3.5τ m
2τ m
≈ 0%
≈ 16.5τ m
0.5τ m
≈ 15%
≈ 4.5τ m
2.5τ m
≈ 0%
≈ 20τ m
τm
≈ 0%
≈ 8τ m
3τ m
≈ 0%
≈ 24τ m
1.5τ m
≈ 0%
≈ 12.5τ m
3.5τ m
≈ 0%
≈ 28τ m
a f 1 = 0 , b f 1 = 0 ; a f 2 = 4λ + τm , b f 2 = 3λ ; a f 3 = λ (3.25λ + τm ) ,
b f 3 = 3λ2 ; a f 4 = 0.25λ2 (3λ + τm ) , b f 4 = λ3 , a f 5 = 0 , b f 5 = 0 ,
K0 = 0 .
Representative λ values (deduced from graph):
OS
Rise time
OS
Rise time
λ
λ
0.3τ m
3
Kc =
Td =
(
≈ 61%
2τ m
≈ 4%
≈ 6.5τ m
0.5τ m
≈ 40%
≈ 3τ m
2.5τ m
≈ 4%
≈ 8τ m
τm
≈ 15%
≈ 3.5τ m
3τ m
≈ 1%
≈ 10τ m
1.5τ m
≈ 7%
≈ 5τ m
3.5τ m
≈ 0%
≈ 12τ m
Ti
2
K m 3λ + 3λτ m + τm
(
≈ 3τ m
2
)
, Ti =
Tm (3λ + τm ) 3λ + 3λτ m + τm
x1
2
x1
(3λ + 3λτ
)−
2
2
m
+ τm
2
)
,N=
Tm (3λ + τm )
−1
λ3 x1
3
λ
3λ2 + 3λτ m + Tm 2
,
(
)
x1 = (3λ + τm + Tm ) 3λ2 + 3λτ m + τm − λ3 .
2
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418
4.3.9 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s)




(
)
1
−
β
T
s
1 + b f 1s
[
]
1
−
χ
δ
d

,
F1 (s) = K c  [1 − α ] +
+
+


T
Ti s
1 + Ti s
1 + a f 1s + a f 2 s 2
d
1
+
s


N 





[
]
K (b + b f 4 s )
1
−
ϕ
T
s
1+ bf 2s
[
]
−
φ
1
d 
F2 (s) = K c  [1 − ε] +
+
− 0 f3
2

 1 + a f 3s + a f 4 s
T
Ti s
1+ a f 5s
1+ d s 

N 

R(s)
+
F1 (s)
U(s)
−
Y(s)
Model
F2 (s)
Table 76: PID controller tuning rules – FOLIPD model G m (s) =
Rule
Kc
Ti
Td
K m e − sτ m
s(1 + sTm )
Comment
Direct synthesis: time domain criteria
1
2
0
Skogestad
K m (TCL + τ m ) 4ξ (TCL + τ m )
(2004b).
Model: Method 1 For ‘good’ robustness: TCL = τ m , ξ = 0.7 or 1. α = 0 ; β = 1 ; χ = 0 ;
δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 1 ; bf 2 = 0 ,
a f 3 = 0 , a f 4 = 0 ; K 0 = 1 ; b f 3 = 1 ; b f 4 = Tm ,
Xu and Shao
(2004a).
Model: Method 1
(λ − 1)τm + Tm
2
K m λ τm
2
a f 5 = Tm x1 , 2 ≤ x1 ≤ 10 .
(λ − 1)τm + Tm
(λ − 1)Tm τm
(λ − 1)τm + Tm
λ not defined.
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ;
ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 1 λK m τ m ;
b f 3 = 1 ; b f 4 = Tm ; a f 5 = 0 .
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Majhi and
Mahanta (2001).
Model: Method 6
0.5236
Kmτm
Ti
Td
419
Comment
Direct synthesis: frequency domain criteria
A m = 3;
0
2τ m e τ ms
φ = 60 0 .
m
α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; b f 1 = Tm , a f 1 = 0 , a f 2 = 0 ; ε = 0 ;
φ = 0 ; ϕ = 1 ; b f 2 = Tm , a f 3 = 0 , a f 4 = 0 ; K 0 = 0.5 K m τm ; b f 3 = 1 ;
b f 4 = Tm ; a f 5 = 0 .
Wang et al.
(2001b).
Model: Method 7
0.4189
K m τm
4τ m
1.25(Tm + 0.1τ m )
A m = 3;
φ m = 60 0 .
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ;
ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0. 2 K m τ m ;
bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 .
Robust
1
Lee and Edgar
(2002).
Model: Method 1
Kc
Ti
Td
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ;
ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0.25K u ;
bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 .
2
Kc
Ti
x1
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; b f 1 = 0 , a f 1 = Td , a f 2 = 0 ;
ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = Td , a f 4 = 0 ; K 0 = 0.25K u ;
bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 .
1
2
Kc =
2
2

0.25K u  4
4
τm
τm
− τm +
,
− τm +
 , Ti =

(λ + τm )  K u K m
2(λ + τm ) 
KuKm
2(λ + τm )
Td =
2
3
4
0.25K u  4Tm
τm
τm
τ (1 − 0.25K u K m τm ) 
2
+ 0.5τm −
+
+ m
.

2
(λ + τm )  K u K m
6(λ + τm ) 4(λ + τm )
0.5K u K m (λ + τm ) 
Kc =
0.25K u
λ + τm
2
2

 4
4
τm
τm
− τm +
+ x1
− τm +
+ x1  , Ti =

2(λ + τm )
KuKm
2(λ + τm )

 K u K m
3
4
 4Tm
τm
τm
(1 − 0.25K u K m τm )τm 2 
2
x1 = 
+ 0.5τm −
+
+

2
6( λ + τ m ) 4( λ + τ m )
0.5K u K m (λ + τm ) 
 K u K m
0.5
.
b720_Chapter-04
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420
4.4 FOLIPD Model with a Zero
K (1 + sTm 3 )e −sτ m
K (1 − sTm 4 )e −sτm
G m (s) = m
or G m (s) = m
s(1 + sTm1 )
s(1 + sTm1 )


1
+ Td s 
4.4.1 Ideal PID controller G c (s) = K c 1 +
 Ti s

E(s)
R(s)
−
+

 U(s)
1
K c 1 +
+ Td s 
Ti s


Model
Y(s)
Table 77: PID controller tuning rules – FOLIPD model with a zero
K (1 − sTm 4 )e −sτ m
K (1 + sTm3 )e−sτ m
G m (s ) = m
or m
s(1 + sTm1 )
s(1 + sTm1 )
Rule
Kc
Ti
Td
Comment
Direct synthesis: time domain criteria
1
Chidambaram
(2002), p. 157.
Model: Method 1
Jyothi et al.
(2001).
Model: Method 1
1
2
3
4
Kc =
Kc =
Kc =
Kc =
1.571
Kc
2
Kc
3
Kc
4
Kc
∞
‘Smaller’
τm Tm 4 .
‘Larger’
τm Tm 4 .
Direct synthesis: frequency domain criteria
τm Tm3 < 1
∞
Tm1
τm Tm3 > 1
(
)
.
(
)
.
K m τm 1 + 3.142 Tm 4 τm 2
2.358
Tm1
K m τm 1 + 4.172 Tm 4 τm 2
1.57
K m τm 1 + 9.870(Tm3 τm )2
0.79
K m τm 1 + 2.467(Tm3 τm )2
.
.
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Jyothi et al.
(2001).
Model: Method 1
0.5 K m Tm 4
5
Kc
Comment
Ti
Td
∞
Tm1
τm Tm 4 > 1
Td
Recommended
τm ≤ λ ≤ 2τm .
τm Tm 4 < 1
Robust
Lee et al. (2006).
Model: Method 1
5
6
Kc =
6
∞
Kc
0.79
K m τm 1 + 2.467(Tm 4 τm )2
.
T (τ − λ ) + 0.5τm
1
;
, Td = Tm1 + m 4 m
K m (λ + τm + 2Tm 4 )
λ + τm + 2Tm 4
2
Kc =
(1 − sTm 4 )e −sτ
.
(1 + sTm 4 )(1 + sλ )
m
Desired closed loop transfer function =
421
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Handbook of PI and PID Controller Tuning Rules
422
4.4.2 Ideal controller in series with a first order lag
E(s)
R(s)
−
+
 1

1
G c (s) = K c 1 +
+ Td s 
 Tf s + 1
 Ti s
U(s)

 1
1
K c 1 +
+ Tds 
 Tis
 Tf s + 1
Model
Y(s)
Table 78: PID controller tuning rules – FOLIPD model with a zero
K (1 + sTm 3 )e − sτ m
K (1 − sTm 4 )e −sτ m
G m ( s) = m
or m
s(1 + sTm1 )
s(1 + sTm1 )
Rule
Td
Comment
Ti
Td
Model: Method 1
Ti
Td
Model: Method 1
Kc
Ti
1
Kc
2
Kc
Robust
Anil and Sree
(2005).
Shamsuzzoha et
al. (2006c).
1
2
Kc =
Km
x1 is a constant of a ‘sufficiently large value’.
[(λ + T
m4
Tm1 + Tm 4 + τm + 3λ
+ τm ) + 2λ2 + λ (Tm 4 + τm ) − Tm 4 τm
2
], T =T
i
m1
+ Tm 4 + τm + 3λ ,
Td =
Tm1 (Tm 4 + τm + 3λ )
λ3 − Tm 4 τm (3λ + Tm 4 + τm )
.
, Tf =
Tm1 + Tm 4 + τm + 3λ
(λ + Tm 4 + τm )2 + 2λ2 + λ(Tm 4 + τm ) − Tm 4τm
Kc =
Ti
, λ , ξ not specified; Ti = x1 + Tm1 + x 2 − x 3 , Tf = Tm3 ,
x1K m (4λξ + τm − x 2 )
Td =
x2 =
x3 =
x 4 + x 2 (Tm1 + x1 ) + x1Tm1 −
Tm12 x 5e − τ m
Tm1
− x12 x 5e − τ m
x1 − Tm1
0.167 τm3 − 0.5x 2 τm 2 + x 4 τm + 4λ3ξ
4λξ + τm − x 2
− x3 ,
Ti
x1
+ x12 − Tm12
,
[
4λ2ξ2 + 2λ2 − x 4 + x 2 τm − 0.5τm 2
2
, x 4 = Tm1 x 5e− τ m
4λξ + τm − x 2

 λ2
2λξ
x5 = 
−
+ 1

T 2 T
m1

 m1
2
.
Tm1
]
− 1 + x 2Tm1 ,
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423





1  1 + Td s 

4.4.3 Classical controller G c (s) = K c 1 +
Td s 
Ti s 

1 +

N 

E(s)
R(s)
−
+



1  1 + sTd

K c 1 +
 Ti s  1 + s Td

N







U(s)
Y(s)
Model
Table 79: PID controller tuning rules – FOLIPD model with a zero
K (1 + sTm 3 )e −sτ m
K (1 − sTm 4 )e −sτ m
G m (s) = m
or m
s(1 + sTm1 )
s(1 + sTm1 )
Rule
Kc
Ti
Comment
Td
Direct synthesis: time domain criteria
Anil and Sree
(2005).
Model: Method 1
1
0.5τmTm1
λ3K m
2
Kc
Tm1
2.414λ + 0.5τm
N=
2.414λ + 0.5τm
Tm3
Tm1
Td
N=∞
1
λ is determined by solving λ3 − 1.207 τm λ2 − 0.603τm 2λ − 0.125τm 3 = 0 .
2
Kc =
0.5τm Tm1
, Td = 2.414λ + 0.5τm + Tm 4 ;
K m λ3 − 0.5τmTm 4 [2.414λ + 0.5τm + Tm 4 ]
(
)
3
2
λ is determined by solving λ − 1.207τm λ − (0.6035τm + 2.414Tm 4 )τm λ
2
2
− τm (0.125τm + 0.5Tm 4 τm + Tm 4 ) = 0 .
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424
4.5 I 2 PD Model G m (s) =
K m e −sτm
s2


1
+ Td s 
4.5.1 Ideal PID controller G c (s) = K c 1 +
Ti s


R(s)
+
E(s)
−
U(s)


1
K c 1 +
+ Td s 
Ti s


K m e − sτ
s
Table 80: PID tuning rules – I 2 PD model G m (s) =
Rule
Åström and
Hägglund (2006),
pp. 244–245.
Kc
Ti
Td
m
Y(s)
2
K m e − sτ m
s2
Comment
Direct synthesis: frequency domain criteria
M s = M t = 1. 4 ;
0.02140
17.570τm
14.019τm
Model: Method 1
K m τm 2
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models



1  1 + Td s 


4.5.2 Classical controller G c (s) = K c 1 +
 Ti s  1 + Td s 
N 

E(s)
R(s)
+
−





1  1 + Td s  U(s)

K c 1 +
Td 
Ti s 

s
1 +
N 

K m e − sτ
m
s2
Table 81: PID tuning rules – I 2 PD model G m (s) =
Rule
Kc
Skogestad
(2003).
Model: Method 1
0.0625
Ti
Y(s)
K m e − sτ m
Td
s2
Comment
Direct synthesis: time domain criteria
K mτm2
8τ m
8τ m
N=∞
425
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426
4.5.3 Two degree of freedom controller 1


Td s
1
+
U (s ) = K c  1 +
 Ti s 1 + Td

N






βTd s 
 E (s ) − K c  α +
 R (s )
Td 


s
1+ s 

N 






T
s
β
d
α +
K
Td  c

1+ s

N 

E(s)
R(s) +
−


Td s
1
+
K c 1 +
T

Ti s
1+ d

N





s

+
U(s)
Hansen (2000).
Model: Method 1
Kc
3.75K m τ m
Ti
2
K me
s2
+
Table 82: PID tuning rules – I 2 PD model G m (s) =
Rule
Y(s)
− sτ m
Td
K m e − sτ m
s2
Comment
Direct synthesis: time domain criteria
5.4 τ m
2.5τ m
N = ∞ ; β =1;
α = 0.833 .
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427
4.5.4 Two degree of freedom controller 2




T
s
1
d
 1 + b f 1s (F(s) R (s) − Y(s) ) − K Y(s) ,
U(s) = K c 1 +
+
0
 Ti s
T  1 + a f 1s
1+ d s 

N 

1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4
F(s) =
1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5 s 4
R(s)


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N

+
F(s)
−


 1 + b f 1s
 1 + a f 1s
s

+ U(s)
−
Kc
Ti
Kc
Ti
Y(s)
m
s2
K0
Table 83: PID tuning rules – I 2 PD model G m (s) =
Rule
K m e − sτ
K m e − sτ m
s2
Comment
Td
Robust
1
Liu et al. (2003).
Model: Method 1
Td
a f 1 = 0 , b f 1 = 0 ; a f 2 = 4λ + τm , b f 2 = 2λ ;
2
a f 3 = 6λ + 4λτ m + τm , b f 3 = λ2 ; a f 4 = 0 , a f 5 = 0 , b f 4 = 0 ,
2
bf 5 = 0 , K 0 = 0 .
1
Kc =
Td =
(
Ti
3
2
2
K m 4λ + 6λ τm + 4λτ m + τm
(6λ + 4λτ
2
2
m
+ τm 4λ3 + 6λ2 τm
x1
2
N=
)(
3
)
, Ti =
(4λ + 6λ τ
+ 4λτ + τ )
−
3
2
2
m
(
x1
m
+ 4λτ m 2 + τm 3
3
),
λ4
m
4λ + 6λ τ m + 4λτ m 2 + τ m 3
3
2
)
6λ2 + 4λτ m + τm
2
3
− 1 , x1 = (4λ + τm ) 4λ3 + 6λ2 τm + 4λτ m + τm − λ4 .
λ3x1
;
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Rule
Liu et al. (2003)
– continued.
Kc
Ti
Comment
Td
Representative λ values (deduced from graph):
OS
Rise time
OS
Rise time
λ
λ
0.8τ m
≈ 33%
≈ 4τ m
2.5τ m
≈ 0%
≈ 10τ m
τm
≈ 20%
≈ 5τ m
3τ m
≈ 0%
≈ 12.5τ m
1.5τ m
≈ 3%
≈ 7τ m
3.5τ m
≈ 0%
≈ 14τ m
2τ m
≈ 0%
≈ 8τ m
4τ m
≈ 0%
≈ 16τ m
2
3
a f 1 = 0 , b f 1 = 0 ; b f 2 = 4λ , b f 3 = 6λ , b f 4 = 4λ , b f 5 = λ4 ;
a f 2 = 8λ + τm , a f 3 = 26λ2 + 8λτ m + τm
(
2
)
2
(
2
)
a f 4 = 4λ 10λ2 + 20λτ m + 4τm , a f 5 = 4λ2 6λ2 + 4λτ m + τm , K 0 = 0 .
λ
0.8τ m
Representative λ values (deduced from graph):
OS
Rise time
OS
Rise time
λ
≈ 14%
≈ 3τ m
2.5τ m
≈ 0%
≈ 7τ m
≈ 4%
≈ 3.5τ m
3τ m
≈ 0%
≈ 8.5τ m
1.5τ m
≈ 0%
≈ 4.5τ m
3.5τ m
≈ 0%
≈ 10.5τ m
2τ m
≈ 0%
≈ 6τ m
4τ m
≈ 0%
≈ 13τ m
τm
a f 1 = 0 , b f 1 = 0 ; b f 2 = 4λ , b f 3 = 6λ2 , b f 4 = 4λ3 , b f 5 = λ4 ;
2
(
a f 2 = 5λ + τm , a f 3 = 10.25λ2 + 5λτ m + τm ,
2
)
(
2
)
a f 4 = λ 7λ2 + 4.25λτ m + τm , a f 5 = 0.25λ2 6λ2 + 4λτ m + τm ,
λ
0.8τ m
K0 = 0 .
Representative λ values (deduced from graph):
OS
Rise time
OS
Rise time
λ
≈ 46%
≈ 6τ m
2.5τ m
≈ 4%
≈ 20.5τ m
τm
≈ 34%
≈ 13.5τ m
3τ m
≈ 2%
≈ 24τ m
1.5τ m
≈ 16%
≈ 13τ m
3.5τ m
≈ 1%
≈ 28τ m
2τ m
≈ 8%
≈ 16.5τ m
4τ m
≈ 0%
≈ 32τ m
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
4.5.5 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s)




(
)
1
−
β
T
s
1 + b f 1s
[
]
1
−
χ
δ
d

,
F1 (s) = K c  [1 − α ] +
+
+


T
Ti s
1 + Ti s
1 + a f 1s + a f 2 s 2
d
1
+
s


N 





[
]
K (b + b f 4 s )
1
−
ϕ
T
s
1+ bf 2s
[
]
−
φ
1
d 
F2 (s) = K c  [1 − ε] +
+
− 0 f3
2

 1 + a f 3s + a f 4 s
T
Ti s
1+ a f 5s
1+ d s 

N 

R(s)
+
F1 (s)
U(s)
−
Y(s)
Model
F2 (s)
Table 84: PID tuning rules – I 2 PD model G m (s) =
Rule
Kc
Ti
K m e −sτm
Td
s2
Comment
Direct synthesis: time domain criteria
0.25
0
Skogestad
2
4ξ 2 (TCL + τ m )
K m (TCL + τ m )
(2004b).
Model: Method 1 For ‘good’ robustness: TCL = τ m , ξ = 0.7 or 1; α = 0 ; β = 1 ; χ = 0 ;
δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 1 ; bf 2 = 0 ,
a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = 1 , b f 4 = 4(TCL + τm ) ,
a f 5 = b f 4 x 2 , 2 ≤ x 2 ≤ 10 .
Huba (2005).
Model: Method 1
−
0.079
K m τm 2
∞
5.835τm
α = 1; β = 1; χ = 1 ; δ = 0 ; ε = 0 ; φ = 1; ϕ = 0 ; N = ∞ ;
bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 0 .
429
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430
4.6 SOSIPD Model
G m (s) =
K m e −sτ m
s(1 + Tm1s) 2
or G m (s) =
K m e − sτ m
2
s(Tm1 s 2 + 2ξ m Tm1s + 1)

1 

4.6.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
E(s)
R(s)
+
−

1 

K c 1 +
Ti s 

U(s)
K m e − sτ
Y(s)
m
s(1 + sTm1 ) 2
Table 85: PI controller tuning rules – SOSIPD model G m (s) =
Rule
Kc
Ti
s(1 + sTm1 )2
Comment
Direct synthesis: time domain criteria
Kuwata (1987).
Model: Method 1
0.5
K m (2Tm1 + τm )
∞
K m e − sτ m
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
4.6.2 Two degree of freedom controller 1


Td s
1
+
U (s ) = K c  1 +
 Ti s 1 + Td

N






βTd s 
 E (s ) − K c  α +
 R (s )
Td 


s
1+ s 

N 






T
s
β
d
α +
K
Td  c

1+
s

N 

R(s)
+
E(s)
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N





s

−
U(s)
Rule
+
Minimum ITAE
– Pecharromán
(2000a).
Model:
Method 2;
Km = 1;
Tm1 = 1 ;
0.1 < τ m < 10 .
K m e − sτ m
or
s(1 + Tm1s) 2
K m e − sτ m
s(Tm12s 2
Kc
+ 2ξm Tm1s + 1)
Ti
x1K u
Y(s)
SOSIPD
model
Table 86: PID controller tuning rules – SOSIPD model G m (s) =
G m (s) =
Comment
Td
Minimum performance index: other tuning
0
x 2 Tu
x1
x2
α
0.049
2.826
0.066
Coefficient values:
φc
x1
x2
α
φc
0.218
1.279
0.564
− 1350
− 160 0
0.250
1.216
0.573
− 130 0
0.522
− 1550
0.286
1.127
0.578
− 1250
1.716
0.532
− 150
0
0.330
1.114
0.579
− 120 0
0.159
1.506
0.544
− 1450
0.351
1.093
0.577
− 118 0
0.189
1.392
0.555
− 140 0
0.506
− 164
0
2.402
0.512
0.099
1.962
0.129
431
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432
Rule
Kc
x1K u
Minimum ITAE
– Pecharromán
x1
x2
(2000b), p. 142.
0.0469 2.8052
Model:
Method 2;
0.0662 2.3876
Km = 1;
0.0988 1.9581
Tm1 = 1 ;
0.05 < τ < 10 . 0.1271 1.6757
m
Minimum ITAE
– Pecharromán
and Pagola
(2000).
Model:
Method 2;
Km = 1;
Ti
Td
x 2 Tu
0
Comment
Coefficient values:
φc
x1
α
x2
α
φc
0.5061 − 164.10 0.2174 1.2566 0.5642 − 135.00
0.5119 − 160.50 0.2511 1.1948 0.5730 − 130.00
0.5220 − 155.00 0.2870 1.1281 0.5777 − 125.00
0.5319 − 150.00 0.3280 1.0860 0.5794 − 120.00
0.1589 1.5052 0.5439 − 145.00 0.3501 1.0822 0.5767 − 118.00
0.1866 1.3619 0.5546 − 140.00
x1K u
x 2 Tu
x 3Tu
x1
x2
Coefficient values:
x3
α
φc
1.672
0.366
0.136
0.601
− 164 0
1.236
0.427
0.149
0.607
− 160 0
0.994
0.486
0.155
0.610
− 1550
Tm1 = 1 ;
0.842
0.538
0.154
0.616
− 150 0
0.1 < τ m < 10 ;
0.752
0.567
0.157
0.605
− 1450
β = 1 , N = 10.
0.679
0.610
0.149
0.610
− 140 0
0.635
0.637
0.142
0.612
− 1350
0.590
0.669
0.133
0.610
− 130 0
0.551
0.690
0.114
0.616
− 1250
0.520
0.776
0.087
0.609
− 120 0
0.509
0.810
0.068
0.611
− 118 0
α
φc
Minimum ITAE
– Pecharromán
(2000b), p. 148.
Model:
Method 2;
Km = 1;
Tm1 = 1 ;
β = 1 , N = 10;
0.05 < τm < 0.8 .
x1K u
x 2 Tu
x 3Tu
x1
x2
Coefficient values:
x3
1.6695
0.3684
0.1323
0.6007
− 164.10
1.2434
0.4389
0.1507
0.6073
− 160.00
0.9901
0.4846
0.1587
0.6102
− 155.00
0.8378
0.5393
0.1562
0.6163
− 150.00
0.7383
0.5789
0.1548
0.6052
− 145.00
0.6728
0.6283
0.1488
0.6105
− 140.00
0.6196
0.6221
0.1376
0.6120
− 135.00
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04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Pecharromán
(2000b) –
continued.
Ti
Td
x1K u
x 2 Tu
x 3Tu
x1
x2
Coefficient values:
x3
α
φc
0.5948
0.6939
0.1326
0.6096
− 130.00
0.5513
0.7050
0.1190
0.6157
− 125.00
0.5271
0.7471
0.0992
0.6088
− 120.00
0.5229
0.8886
0.0862
0.6115
− 118.0 0
Tm1 ≤ 1.0 ;
Taguchi and
Araki (2000).
Model: Method 1
Taguchi et al.
(2002).
Model:
Method 1;
N = 10; α = 0 .
1
Kc
Ti
0
2
Kc
Ti
Td
x1Tm1 K m
∞
Kc =
τm
OS ≤ 20% .
x 2Tm1
x1
x2
β
τm Tm1
x1
x2
β
τm Tm1
1.48
1.04
0.71
0.56
1.35
1.41
1.49
1.57
0.89
0.81
0.72
0.67
0.1
0.2
0.4
0.6
0.46
0.40
0.24
0.12
1.63
1.70
2.02
3.08
0.63
0.60
0.50
0.38
0.8
1.0
2.0
5.0
3
1
Comment
Kc
Kc
∞
Td

1 
0.3840
 0.07368 +
 , T = 8.549 + 4.029[τm Tm1 ] ,

[τm Tm1 ] + 0.7640  i
Km 
α = 0.6691 + 0.006606[τm Tm1 ] .
2
Kc =

1 
0.5667
 0.1778 +
 , T = T (1.262 + 0.3620[τm Tm1 ]) ,

[τm Tm1 ] + 0.002325  d m1
Km 
(
)
Ti = Tm1 0.2011 + 11.16[τm Tm1 ] − 14.98[τm Tm1 ] + 13.70[τm Tm1 ] − 4.835[τm Tm1 ] ,
2
3
α = 0.6666 , β = 0.8206 − 0.09750[τm Tm1 ] + 0.03845[τm Tm1 ] ; N not specified.
2
3
Kc =

Tm1 
0.3084
,
0.1309 +
Km 
([τm Tm1 ] + 0.1315) 
[
]
Td = Tm 1.308 + 0.5000[τm Tm1 ] − 0.1119[τm Tm1 ] ,
2
β = 0.9231 − 0.6049[τm Tm1 ] + 0.2906[τm Tm1 ] ; 0.1 ≤ τm Tm1 ≤ 1.0 .
2
4
433
b720_Chapter-04
Rule
Taguchi et al.
(2002).
Model:
Method 1;
N = 10; α = 0 .
Kc
Ti
Td
Comment
x1Tm1 K m
∞
x 2Tm1
ξm = 0.5
x1
x2
β
τm Tm1
x1
x2
β
τm Tm1
0.52
0.43
0.34
0.30
1.11
0.89
0.61
0.50
0.82
0.67
0.26
-0.13
0.1
0.2
0.4
0.6
0.28
0.26
0.21
0.13
0.49
0.53
0.91
2.26
-0.34
-0.39
-0.08
0.23
0.8
1.0
2.0
5.0
Kc
∞
Td
x1Tm1 K m
∞
x 2Tm1
4
Taguchi et al.
(2002).
Model:
Method 1;
N = 10; α = 0 .
Taguchi et al.
(2002).
Model:
Method 1;
N = 10; α = 0 .
Taguchi et al.
(2002).
Model:
Method 1;
N = 10; α = 0 .
Taguchi et al.
(2002).
Model:
Method 1;
N = 10; α = 0 .
Kc =
FA
Handbook of PI and PID Controller Tuning Rules
434
4
04-Apr-2009
x1
1.06
0.80
0.57
0.46
x1Tm1
x1
2.18
1.48
0.93
0.69
x1Tm1
x1
2.93
1.90
1.13
0.81
x1Tm1
x2
β
τm Tm1
1.34
1.32
1.30
1.31
Km
0.89
0.81
0.70
0.62
0.1
0.2
0.4
0.6
x2
β
τm Tm1
1.34
1.46
1.64
1.77
Km
0.92
0.87
0.81
0.77
0.1
0.2
0.4
0.6
∞
∞
x2
β
τm Tm1
1.29
1.46
1.71
1.90
Km
0.93
0.89
0.83
0.80
0.1
0.2
0.4
0.6
∞
x1
ξm = 0.8
x2
0.40
1.33
0.35
1.37
0.23
1.64
0.12
2.74
x 2Tm1
x1
x2
0.55
1.88
0.46
1.98
0.26
2.37
0.12
3.42
x 2Tm1
x1
x2
0.63
2.06
0.51
2.19
0.27
2.68
0.12
3.75
x 2Tm1
β
τm Tm1
0.56
0.8
0.52
1.0
0.40
2.0
0.33
5.0
ξ m = 1. 2
β
τm Tm1
0.73
0.8
0.70
1.0
0.59
2.0
0.43
5.0
ξ m = 1. 4
β
τm Tm1
0.77
0.8
0.74
1.0
0.64
2.0
0.47
5.0
ξ m = 1. 6
x1
x2
β
τm Tm1
x1
x2
β
τm Tm1
3.80
2.37
1.34
0.93
1.24
1.45
1.76
2.00
0.94
0.90
0.85
0.82
0.1
0.2
0.4
0.6
0.70
0.56
0.28
0.12
2.20
2.36
2.95
4.08
0.79
0.77
0.68
0.51
0.8
1.0
2.0
5.0

Tm1 
0.08343
,
0.1904 +
Km 
((τm Tm1 ) + 0.1496) 
[
]
Td = Tm1 1.237 − 2.031(τm Tm1 ) + 1.344(τm Tm1 ) ,
2
β = 1.048 − 1.637(τm Tm1 ) − 1.676(τm Tm1 ) + 1.898(τm Tm1 ) ; 0.1 ≤ τm Tm1 ≤ 1.0 .
2
3
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Kuwata (1987).
Model: Method 1
Ti
Kc
Ultimate cycle
0
x 2 Tu
x 1K u
x1
0.16
0.19
0.23
Kc =
x2
Td = 1.25
(2.5T
2
m1
α =1
τm Tu
x1
x2
τm Tu
x1
0.01
0.03
0.05
0.26
0.30
0.32
x 2 Tu
0.89
0.88
0.86
0.08
0.10
0.13
0.35
0.36
0.38
1.05
0.97
0.95
x 1K u
1.067
(2Tm1 + τm )3
K m 2T 2 + 4T τ + τ 2
m1
m1 m
m
[
α =1
α = 1,
β =1, N = ∞ .
Td
Kuwata (1987);
x 1 x 2 x 3 τm
x1 x 2 x 3
x 1 , x 2 deduced
Tu
from graphs;
β = 1 , N = ∞ . 0.98 0.45 0.12 0.03 0.79 0.54 0.11
0.90 0.47 0.12 0.05 0.76 0.56 0.10
0.82 0.51 0.11 0.08 0.72 0.58 0.09
5
Comment
Td
Direct synthesis: time domain criteria
1
0
K m (2Tm1 + τm ) 3.33(2Tm1 + τm )
5
Kuwata (1987);
x 1 , x 2 deduced
from graphs.
Ti
]
2
; Ti =
)(
[
x 3 Tu
τm
Tu
x1
x2
x3
τm
Tu
0.10 0.70 0.60 0.09 0.17
0.12 0.70 0.62 0.08 0.20
0.15
6.25 2Tm12 + 4Tm1τm + τm 2
(2Tm1 + τm )2
+ Tm1τm + 0.25τm 2 2Tm12 + 4Tm1τm + τm 2
(2Tm1 + τm )3
τm Tu
x2
0.85 0.15
0.84 0.18
0.83 0.25
α =1
).
]
2
;
435
b720_Chapter-04
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FA
Handbook of PI and PID Controller Tuning Rules
436
4.7 SOSIPD Model with a Zero G m (s) =
K m (1 + sTm 3 )e − sτ m
s(1 + sTm1 )(1 + sTm 2 )



1  1 + Td s 


4.7.1 Classical controller G c (s) = K c 1 +
 Ti s  1 + Td s 
N 

R(s)
E(s)
+
−





1  1 + Td s 

K c 1 +
Td 
Ti s 

s
1 +
N 

Y(s)
U(s)
G m (s)
Table 87: PID controller tuning rules – SOSIPD model with a zero
K (1 + sTm3 )e −sτ m
G m (s ) = m
s(1 + sTm1 )(1 + sTm 2 )
Rule
Kc
Bialkowski
(1996).
Model: Method 1
Tm1
Ti
Td
Comment
Robust
K m (λ + τm )2
Tm1
2λ + τ m
N=
2λ + τm
;
Tm3
Tm1 > Tm 2 .
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
437
4.8 TOSIPD Model
G m (s) =
K m e − sτ m
K m e −sτ m
or G m (s) =
3
s(1 + sTm1 )(1 + sTm 2 )(1 + sTm 3 )
s(1 + sTm1 )
4.8.1 Two degree of freedom controller 1


Td s
1
+
U (s ) = K c  1 +
 Ti s 1 + Td

N






βTd s 
 E (s ) − K c  α +
 R (s )
Td 


s
1+ s 

N 






T
s
β
d
α +
K
Td  c

1+
s

N 

E(s)
R(s)
+
−
Rule
Kuwata (1987);
x 1 , x 2 deduced
from graphs.


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N





s

−
Y(s)
U(s)
Model
+
Table 88: PID controller tuning rules – TOSIPD model
Comment
Kc
Ti
Td
Ultimate cycle
0
x 2 Tu
x 1K u
x1
0.23
0.26
0.28
x2
0.91
0.89
0.88
x 1K u
α =1
τm Tu
x1
x2
τm Tu
x1
x2
τm Tu
0.01
0.03
0.05
0.31
0.33
0.35
x 2 Tu
0.86
0.86
0.85
0.08
0.10
0.13
0.36
0.37
0.38
0.84
0.84
0.83
0.15
0.18
0.25
x 3 Tu
Kuwata (1987);
x1 x 2 x 3 τm
x1 x 2 x 3 τm
x1 x 2 x 3 τm
x 1 , x 2 deduced
Tu
Tu
Tu
from graphs.
0.85
0.50
0.11
0.03
0.74
0.56
0.09
0.10
0.70
0.62
0.08
0.18
α = 1, β = 1,
0.80 0.52 0.11 0.05 0.72 0.58 0.09 0.12 0.70 0.63 0.075 0.22
N =∞.
0.77 0.55 0.10 0.08 0.70 0.61 0.08 0.16
b720_Chapter-04
04-Apr-2009
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Handbook of PI and PID Controller Tuning Rules
438
4.9 General Model with Integrator G m (s) =
K
G m (s) = m
s
∏ (T
∏ (T
s(1 + sTm1 )
)∏ (T
s + 1)∏ (T
m1i s
i
K m e − sτ m
+1
m 2i
n
or
2 2
m 3i
i
m 4i
)
e
s + 1)
s + 2ξ m n i Tm 2i s + 1
i
2 2
s + 2ξ m di Tm 4i
i

1 

4.9.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
R(s)
−
+
U(s)

1 

K c 1 +

T
is 

E(s)
G m (s)
Y(s)
Table 89: PI controller tuning rules – general model with integrator
Comment
Kc
Ti
Rule
Direct synthesis: time domain criteria
0.5
∞
K m (nTm1 + τm )
Kuwata (1987).
Model: Method 1
Direct synthesis: frequency domain criteria
(1)
Symmetrical
Kc
optimum method.
α1TQ
( 2)
Non-symmetrical
Kc
optimum method.
1
Loron (1997).
Model: Method 1
1
Kc
(1)
=

TQ = 

Kc
( 2)
 1 + sin φ m
, α1 = 
α1 TQ
 cos φ m
1
Km
∑T
=
m 3i
i
+2
∑ξ
i
λ
K m α 2 TQ
 
−
 
m d i Tm 4i 
, λ=



2
∑T
m1i
i
Mc2
Mc2
+2
∑ξ

m n i Tm 2 i 

i
2
+ τm .
 1 + sin ∆φ 
−1
, α 2 = 
 , ∆φ = cos (1 λ ) ,
−1
 cos ∆φ 
M c = desired closed loop resonant peak.
−sτ m
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
4.9.2 Two degree of freedom controller 1


Td s
1
+
U (s ) = K c  1 +
 Ti s 1 + Td

N

439





βTd s 
 E (s ) − K c  α +
 R (s )
Td 


s
1+ s 

N 






T
s
β
d
α +
K
Td  c

1+
s

N 

E(s)
R(s)
+
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N





s

U(s)
−
K m e − sτ
n
s(1 + sTm1 )
m
+
Table 90: PID controller tuning rules – general model with integrator
Comment
Kc
Ti
Td
Rule
Direct synthesis: time domain criteria
Kuwata (1987).
1
Model: Method 1 K (nT + τ ) 3.33(nTm1 + τm )
0
m
m1
m
Kuwata (1987).
Model: Method 1
1
Kc =
Td
Y(s)
1
Ti
Kc
1.067
(nTm1 + τm )3
K m n (n − 1)T 2 + 2nT τ + τ 2
m1
m1 m
m
[
[(nT
= 1.25x
1
m1
(
]
2
, Ti =
Td
[
α = 1; β = 1;
N =∞.
6.25 n (n − 1)Tm12 + 2nTm1τm + τm 2
+ τm )2 − 0.75 n (n − 1)Tm12 + 2nTm1τm + τm 2
(nTm1 + τm )3
α =1
(nTm1 + τm )2
]
2
,
)] ,
x1 = [n (n − 1)Tm1 + 2nTm1τm + τm ] .
2
2
b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
440
4.10 Unstable FOLPD Model G m (s) =
K m e −sτ m
Tm s − 1

1 

4.10.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
R(s)
E(s)
+
−

1 

K c 1 +
Ti s 

U(s)
Y(s)
K m e − sτ
Tm s − 1
m
Table 91: PI controller tuning rules – unstable FOLPD model G m (s) =
Rule
Minimum ITSE –
Majhi and Atherton
(2000).
0 < τm Tm < 0.693 .
Jacob and
Chidambaram (1996).
Model: Method 1
Kc
1
2
K m e − sτ m
Tm s − 1
Comment
Ti
Minimum performance index: servo tuning
Model: Method 1
Ti
K
c
10
TS
Model: Method 3
Ti
Kc
Direct synthesis: time domain criteria
TS is the ±1%
0.4ξ 2TS
settling time.
Chidambaram (1997).
1.678  Tm
ln
K m  τ m
Valentine and
Chidambaram (1998).
Model: Method 1
2.695 −1.277 τ m
e
Km



Tm
0.4015Tm e 5.8τ m
Tm
0.4015Tm e 5.8τ m
Tm
Model: Method 1
3
(
ξ = 0.333 ;
τm Tm < 0.7 .
)
1
Kc =
e − τ m Tm − 0.064 
2.6316Tm e τ m Tm − 0.966
1 
, Ti =
.
0.889 + τ T


m
m
Km 
e
e − τ m Tm − 0.377
− 0.990 
2
Kc =

1 
1
Tm K (1 + K )
 0.889 +
 , Ti =
, K = − Ap Kmh .
K m 
K (1 + K ) 
0.5(1 − 0.6382K ) tanh −1 K
3
±1% Ts = 7.5Tm , τ m Tm = 0.1 ; ±1% Ts = 8.75Tm , τ m Tm = 0.2 ;
(
)
±1% Ts = 11.25Tm , τ m Tm = 0.3 ; ±1% Ts = 15.0Tm , τ m Tm = 0.4 ;
±1% Ts = 17.5Tm , τ m Tm = 0.5 ; ±1% Ts = 20.0Tm , τ m Tm = 0.6 .
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Ti
Model: Method 1;
‘small’ τm Tm .
Chandrashekar
et al. (2002).
5
Kc
Ti
Model: Method 1
49.535 − 21.644 τm
e
Km
0.8668  Tm 


K m  τm 
Chidambaram (2002),
p. 156.
Model: Method 1
Kc =
Tm Ti
2 2
0.04TS ξ + Ti τm
τm
< 0.1
Tm
Tm
0.1523Tm e
0.8288
7.9425τ m Tm
0. 1 ≤
τm
≤ 0. 7
Tm
6.0000 K m
Alternative tuning rules:
0.6000Tm
τ m Tm = 0.1
3.2222 K m
1.4500Tm
τ m Tm = 0.2
2.2963 K m
2.6571Tm
τ m Tm = 0.3
1.8333 K m
4.4000Tm
τ m Tm = 0.4
1.5556 K m
7.0000Tm
τ m Tm = 0.5
1.3265 K m
14.6250Tm
τ m Tm = 0.6
1.1875 K m
31.0330Tm
τ m Tm = 0.7
τm
2.695 −1.277 Tm
e
K m,
0.4015Tme
5.8
τm
Tm
0<
τm
≤ 0.6
Tm
2
, Ti =
0.04TS ξ2 + Tm τm + 0.4TSξ2
.
Tm − τm
Desired closed loop transfer function =
Kc =
6
Comment
4
Chandrashekar et
al. (2002).
Model: Method 1
5
Ti
Chidambaram
(2000c).
6
4
Kc
441
(
(1)
TCL
2s
)(
+1
m
( 2)
TCL
2s
1
(1)
K m Ti − TCL
2
Desired closed loop transfer function =

τ
TCL 2 =  0.025 + 1.75 m
Tm

(
(Tis + 1)e−sτ
( 2)
− TCL
2
− τm
(ηs + 1)e −sτ
(TCL2 s + 1)2
m
);
+1
)
, Ti =
(
)
(1)
( 2)
(1)
( 2)
TCL
2 TCL 2 + Tm TCL 2 + TCL 2 + τ m
.
Tm − τm
;

τ
τ
Tm , m ≤ 0.1 ; TCL 2 = 2 τ m , 0.1 ≤ m ≤ 0.5 ;
Tm
Tm

τ

τ
TCL 2 = 2 τ m + 5τ m  m − 0.5  , 0.5 ≤ m ≤ 0.7 .
T
T
m
 m

b720_Chapter-04
04-Apr-2009
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Handbook of PI and PID Controller Tuning Rules
442
Rule
Minimum ITAE –
Chidambaram (2002),
p. 156.
Model: Method 1
Sree et al. (2004).
Model: Method 1
Kc
7
0.898
0.8624  Tm 


K m  τm 
Manum (2005).
Model: Method 1
Sree and
Chidambaram (2006),
pp. 19–22.
Minimum ITAE –
Sree and
Chidambaram (2006),
pp. 22–23.
Model: Method 1
Kc
T 
0.876 m 
 τm 
Comment
Ti
0.324Tm e
6.46
8
Ti
0.2 ≤
τm
≤ 0.6
Tm
0.01 ≤
τm
≤ 0.6
Tm
4τm Tm (2Tm − 3τm )
τm
< 0.25
Tm
(Tm − 2τm )
2
τm
2.695 −1.277 Tm
e
Km
0.4015Tme
Kc
T 
0.876 m 
 τm 
Tm
0.9744
0.5Tm
K m τm
9
τm Tm ≤ 0.2
τm
0.898
0.324Tm e
5.8
τm
Tm
Model: Method 1
τm Tm ≤ 0.2
6.461τ m
Tm
0.2 ≤
τm
≤ 0.6
Tm
Direct synthesis: frequency domain criteria
De Paor and
O’Malley (1989).
Model: Method 1
7
8
9
10
Kc =
10
A m = 2;
Ti
Kc
τm
<1.
Tm
τm 
1 
10.7572 − 35.1  .
Km 
Tm 
3
2


τ 
τ 
τ
Ti = Tm 143.34 m  − 73.912 m  + 19.039 m − 0.2276 .
Tm


 Tm 
 Tm 


1 
τm 
Kc =
10.7572 − 35.1  .
Km 
Tm 
Kc =

1 
Tm 
τ 
τ τ
1 − m  sin
cos 1 − m  m +

Km 
τm  Tm 
 Tm  Tm

Ti =
Tm
 T 

 m 1 − τm   tan (0.5φ )

 τm  Tm  



τ τ
1 − m  m
 T T
m  m

, φ = tan −1

,


Tm
τm


τ
τ 
1 − m  − 1 − m
 Tm
 Tm 
 τm

.
 Tm
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Venkatashankar and
Chidambaram (1994).
11
Comment
Ti
25(Tm − τ m )
Kc
443
Model: Method 1;
τ m Tm < 0.67 .
Chidambaram
τm
T 
1 
< 0.6
1 + 0.26 m 
(1995b),
25Tm − 27τ m
Tm
K m 
τm 
Chidambaram (1998),
p. 91.
2
τ  
1 
τ
Model: Method 1
4.656 − 13.05 m + 13.436 m   .
Alternatively, K c =
Km 
Tm
 Tm  

12
τm
ω p Tm
Ti
< 0.62
Ho and Xu (1998).
Tm
AmKm
Model: Method 1
Representative results:
A m = 2.3 ,
0.5775Tm
τ m Tm
K mτm
0.3212Tm − τ m
φ = 250 .
m
0.6918Tm
Kmτm
τ m Tm
0.3359Tm − τ m
x 1Tm K m τ m
−Tm
1.0472
1.5
30
0
0.5236
3
0.7854
2
450
0.3927
4
Luyben (1998).
Model: Method 1
0.31K u
12
Ti =
13
1 
0.04Tm2
0.98 1 +

Km
(Tm − τm )2

1
2
1.57ωp − ωp τm − Tm
2.2Tu
TCL
φm
60 0
67.50
= 3.16τ m
Maximum closed loop log modulus = 2 dB.
Model: Method 1;
13
Ti
0.1 ≤ τ m Tm ≤ 0.6 .
Kc
Gaikward and
Chidambaram (2000).
Kc =
φ m = 22.50 .
Other representative results: coefficient values:
Am
φm
x1
Am
x1
11
A m = 1.9 ,
−1
2
2
2
 25 
τm + x1 Tm
  x (T − τ )
,
m
2
2
2
 τ  1 m
τm + 625x1 (Tm − τm )
m 


x1 = 1.373, τm Tm < 0.25 ; x1 = 0.953, 0.25 ≤ τ m Tm < 0.67 .
.
2

 τm  
τm
1 
Tm 

.


Kc =
+ 8.214
0.2619 + 0.7266  , Ti = Tm 0.193 + 4.19
Tm
Tm  
Km 
τm 




b720_Chapter-04
Rule
Kc
Sree and
Chidambaram (2006),
pp. 15–16.
Paraskevopoulos et
al. (2006).
Model: Method 1
15
Kc =
x1 =
FA
Handbook of PI and PID Controller Tuning Rules
444
14
04-Apr-2009
1
Km
14
15
Kc
Ti
Comment
Ti
Model: Method 1;
0 ≤ τm Tm < 0.6 .
Ti
τm
< 0.9
Tm
2
1 + x1
Ti x1
1 + Ti 2 x12
2


 τm  
τm
0.7266Tm 




T
T
0
.
193
4
.
19
8
.
214
0
.
2619
,
+
=
+
+

 i
m
T  .
Tm
τm

m 





τ 
τ
2
Ti 1 − m  + Ti − m + x 2
T
T
m 
m

2Ti 2 (τm Tm )
,
2
 
τ 
τ 
τ
τ 
2
x 2 = Ti 2 1 − m  + Ti − m  + 4 m Ti 1 + Ti − m  ;
Tm 
Tm 
Tm 
  Tm 
φm
 T 
 φmax

τ
τ
, with φm > 0.2φmax
Ti = x 3 1 + m 0.632 m + 0.436 m − 0.0153  m
m ;


φm
τm 
Tm
Tm


1
−


max
φm
0.0029 − 0.0682
x3 =
τm
τ
+ 1.4941 m
Tm
Tm

τ 
1.003 − m 


T
m 

2
=−
, φ max
m
τm
Tm
 T

Tm
− 1 + tan −1  m − 1  .
 τm

τm


b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Paraskevopoulos et
al. (2006) –
continued.
16
K min = Ti ωmin
16
K max
or K min A m
Am
1 + ωmin
2
2
1 + Ti ωmin
2
0.03(τm Tm )
1.14 − (τm Tm )
= 1+


0.05
 0.973 +
Ti − x1


(
)
−
τ
1
T
m
m 

ωmin
Comment
Ti
τm Tm < 0.9
1 + ωmax
and K max = Ti ωmax
0.006 +
Ti
2
Ti −
2
1 + Ti ωmax 2
x1
τm
(1 + Ti )
Tm
x1 =
445
, with
,
0.0029 − 0.0682 τm Tm + 1.4941(τm Tm )
(1.003 − (τm
Tm ))2
,
4
2

 τ   
τ  x   1.571Tm
ωmax = 1 + 0.22 m   1 +  0.1 − 0.3 m  1  
Tm  Ti   τm

 Tm    



 

Ti
Ti
− 0.9463 
− 0.5609 ;

 (Ti + 1)(τm Tm )
  (Ti + 1)(τm Tm )

1+ (τ m Tm ) 


 max  


 A m − 1  A m − 1 


+
Ti = x1 1 + 0.65 


 
max


1 −  Am − 1
A
−1  

 
 m



[
with A max
m
[
]
]
2
3

τ 
 τ  3(τm Tm )4 − 2.3(τm Tm )2 
τ
τ
,
0.01 − 0.18 + 5 m − 32 m + 75 m  − 51 m  +
Tm
Tm
Tm 
Tm 


(
1 − (τm Tm ))3




1.571Tm
max
=
; A m > 0.2(A m − 1) +1.

0.4085(τm Tm ) 


τ m 1 +

 1 − 0.2864(τm Tm ) 
b720_Chapter-04
04-Apr-2009
Handbook of PI and PID Controller Tuning Rules
446
Rule
Kc
Rotstein and Lewin
(1991).
Model: Method 1
 λ

Tm λ 
+ 2 
 Tm

λ2 K m
Ti
Comment
Robust
 λ

λ 
+ 2 
 Tm

λ obtained
graphically – sample
values below.
τ m Tm = 0.2, 0.6Tm ≤ λ ≤ 1.9Tm
τ m Tm = 0.2, 0.5Tm ≤ λ ≤ 4.5Tm
τ m Tm = 0.4, 1.5Tm ≤ λ ≤ 4.5Tm
τ m Tm = 0.6, 3.9Tm ≤ λ ≤ 4.1Tm
Tan et al. (1999b).
Model: Method 1
Ou et al. (2005a).
17
18
17
Kc
18
Kc
λ2 + 2λTm + Tm τm
K m (λ + τm )
2
, Ti =
λ2 + 2λTm + Tm τm
.
Tm − τm
K m uncertainty =
50%.
K m uncertainty =
30%.
Ti
τm Tm < 0.5
Ti
Model: Method 1


τ
Tm 12.7708 m + 3.8019 
Tm
1

 .
Kc =
, Ti =
τm


τm
0.1486
+ 0.6564
K m  0.1486
+ 0.6564 
Tm
Tm


Kc =
FA
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models


1
+ Td s 
4.10.2 Ideal PID controller G c (s) = K c 1 +
 Ti s

E(s)
R(s)

 U(s)
1
K c 1 +
+ Td s 
 Ti s

−
+
K m e − sτ
Tm s − 1
m
Y(s)
Table 92: PID controller tuning rules – unstable FOLPD model G m (s) =
Rule
Kc
Minimum ISE –
Visioli (2001).
Ti
Td
K m e − sτ m
Tm s − 1
Comment
Minimum performance index: regulator tuning
1.18
1.37  τ m 
τ 


Model: Method 1
2.42Tm  m 
0.60τ m
K m  Tm 
 Tm 
Minimum ITSE –
Visioli (2001).
1.37  τ m

K m  Tm



τ
4.12Tm  m
 Tm



0.90
Minimum ISTSE
– Visioli (2001).
1.70  τ m

K m  Tm



τ
4.52Tm  m
 Tm



1.13
0.55τ m
Model: Method 1
0.50τ m
Model: Method 1
Minimum performance index: servo tuning
Minimum ISE –
Visioli (2001).
1.32  τ m

K m  Tm



0.92
Minimum ITSE –
1.38  τ m

Visioli (2001).
K m  Tm



0.90
Minimum ISTSE
1.35  τ m

– Visioli (2001).
K m  Tm



0.95
[
= 3.62T [1 − 0.85(T
= 3.70T [1 − 0.86(T
1
Td = 3.78Tm 1 − 0.84(Tm τm )
2
m
τm )
m
τm )
3
Td
Td
m
m
0.02
0.02
0.02
](τ
](τ
](τ
τ
4.00Tm  m
 Tm



0.47
τ
4.12Tm  m
 Tm



0.90
τ
4.52Tm  m
 Tm



m
Tm )
.
m
Tm )
.
m
Tm )
.
0.95
0.93
0.97
1
Td
Model: Method 1
2
Td
Model: Method 1
3
Td
Model: Method 1
1.13
447
b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
448
Rule
Kc
Ti
Comment
Td
Minimum performance index: servo/regulator tuning
Jhunjhunwala
and
Chidambaram
(2001).
4
Ti
Kc
Direct synthesis: time domain criteria
Ti
1.165  Tm 


5
T
x1
i
Km  τm 
Chidambaram
(1997).
Model: Method 1
6
Valentine and
Chidambaram
1.165  Tm
(1997b).

Model: Method 1 K m  τ m



0.245
Kc =
τm Tm < 0.6
Ti
4.4Tm + 9.0τm
7
4
0.176Tm +
0.36τm
2.044 τ m
Tm
0.6 ≤
0.8 <
Ti
1 
τ  τ
1.397  Tm 
13.0 − 39.712 m  , m < 0.2 ; K c =


K m 
Tm  Tm
K m  τm 
Ti = 0.856Tm e
Model: Method 1
Td
τ
, 0 ≤ m ≤ 1 ; Ti = 0.3444Tme
Tm
0.769
2.9595 τ m
Tm
, 1≤
, 0.2 ≤
τm
≤ 0.8
Tm
τm
≤1
Tm
τm
≤ 1.4 ;
Tm
τm
≤ 1.4 ;
Tm


τ
τ
Td = Tm  0.5643 m + 0.0075  , 0 ≤ m ≤ 1.4 .
Tm
Tm


2
5

τ  τ
τ 
τ
Ti =  0.176 + 0.360 m Tm ; x1 = 0.179 − 0.324 m + 0.161 m  , m < 0.6 ;
Tm 
Tm

 Tm  Tm
x1 = 0.04 , 0.6 < τm Tm < 0.8 ; x1 = 0.12 − 0.1(τm Tm ) , 0.8 < τm Tm < 1.0 .
6
Ti =
7
Ti =
0.176Tm + 0.36τm
0.179 − 0.324(τm Tm ) + 0.161(τm Tm )2
0.176Tm + 0.36τm
.
0.12 − 0.1(τm Tm )
.
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Pramod and
Chidambaram
(2000),
Chidambaram
(2002), p. 156.
Model: Method 1
8
Chidambaram
(2002), p. 156.
Sree et al.
(2004).
Model: Method 1
8
9
1
Kc =
Km
Ti
Ti
Kc
9
Ti
10
Ti
449
Comment
Td
τm Tm < 0.6
0.176Tm
+0.36τm 0.6 <
0.8 <
11
Kc
Ti
Td
12
Kc
Ti
0.4917 τ m
τm
< 0.8
Tm
τm
≤1
Tm
Model: Method 1
0.01 ≤
τm
≤ 0.9
Tm
2

 

0.176Tm + 0.36τm
2.121 − 1.906 τm + 0.945 τm   , Ti =
.
2
T  
Tm

 τm 
 m 
τm



−
+
0.179 0.324
0.161

Tm
 Tm 
Ti = 0.44Tm + 0.90τm (Pramod and Chidambaram, 2000);
Ti = 4.4Tm + 9.0τm (Chidambaram, 2002).
10
Ti =
0.176Tm + 0.36τm
(Chidambaram, 2002).
0.12 − 0.1(τm Tm )
11
Kc =
1
Km

1.397Tm
τ
τm  τm
, 0.2 < m ≤ 1.4 ;
≤ 0.2 ; K c =
13.0 − 39.712  ,
T
K
τ
T
T
m
m m
m
m

Ti = 0.856Tm e
2.044
τm
Tm
τm
2.9595
τ
τ
Tm
, 0 < m < 1 ; Ti = 0.344Tm e
, 1 < m ≤ 1.4 ;
Tm
Tm

τ
τ 
Td = Tm 0.0075 + 0.5643 m  , 0 < m ≤ 1.4 .
Tm 
Tm

12
Kc =
1.4183  Tm 


K m  τm 
0.9147
2


τ 
τ
, Ti = Tm 16.327 m  + 5.5778 m + 0.8158 ,
Tm


 Tm 


  τ 2

τm
τ
τ
0.01 ≤
≤ 0.6 ; Ti = Tm 196 m  − 247.28 m + 87.72 , 0.6 ≤ m ≤ 0.9 .
Tm
Tm
Tm
  Tm 



b720_Chapter-04
FA
Handbook of PI and PID Controller Tuning Rules
450
Rule
Kc
Sree et al. (2004)
– continued;
Vivek and
Chidambaram
(2005).
Sree and
Chidambaram
(2006),
pp. 117,118.
13
04-Apr-2009
Kc =
1.2824  Tm 


K m  τm 
Ti = 0.483Tme
Td
13
Kc
Ti
Td
14
Kc
Ti
Td
Comment
0.01 ≤
τm
≤ 1.2
Tm
Model: Method 1
Alternative tuning rules:
Ti
Td
Kc
TCL 2
τ m Tm
0.02Tm
0.0
101.0000 K m
0.0404Tm
0.0
10.3662 K m
0.3874Tm
0.0435Tm
0.10Tm
0.1
5.3750 K m
0.8600Tm
0.0884Tm
0.20Tm
0.2
3.2655 K m
1.8018Tm
0.1375Tm
0.40Tm
0.3
2.4516 K m
3.0400Tm
0.1868Tm
0.60Tm
0.4
2.0217 K m
4.6500Tm
0.2366Tm
0.80Tm
0.5
1.7525 K m
6.7286Tm
0.2866Tm
1.00Tm
0.6
1.5458 K m
10.337Tm
0.3380Tm
1.30Tm
0.7
1.3723 K m
17.693Tm
0.3910Tm
1.80Tm
0.8
1.2420 K m
33.610Tm
0.4440Tm
2.60Tm
0.9
1.1400 K m
80.620Tm
0.4969Tm
4.20Tm
1.0
1.0895 K m
143.6Tm
0.5975Tm
5.00Tm
1.2
1.0536 K m
277.37Tm
0.6982Tm
6.00Tm
1.4
0.8325
3.3739
Ti


τ
τ
, Ti = Tm 5.5734 m − 0.0063 , 0.01 ≤ m ≤ 0.5 ;
Tm
Tm


τm
Tm
, (Sree et al., 2004); Ti = 0.483Tme
− 3.3739
τm
Tm
(Vivek and
Chidambaram, 2005; Model: Method 5); 0.5 ≤ τm Tm ≤ 1.2 ;
Td = 0.507 τm + 0.0028Tm .
14
Desired closed loop transfer function =
Kc = −
([Ti − 0.5τm ]s + 1)e−sτ
(
(1)
TCL
2s
m
)
+1
;
0.5(Ti − 0.5τ m )τ m
Ti
, Td =
,
(1)
( 2)
Ti
K m TCL
+
T
+
1
.
5
τ
−
T
2
CL 2
m
i
(
(
)
)
(1)
( 2)
(1)
( 2)
(1)
( 2)
TCL
2TCL 2 + 0.5τ m TCL 2 + TCL 2 + 0.5TCL 2TCL 2
Ti =
)(
+1
( 2)
TCL
2s
Tm − 0.5τm
(
τm
(1)
( 2)
+ Tm TCL
2 + TCL 2 + τ m
Tm
)
.
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Sree and
Chidambaram
(2006),
pp. 117,118.
Sree and
Chidambaram
(2006).
Model: Method 1
15
Kc =
Ti
Td
Comment
451
15
Kc
Ti
0.5τ m
Model: Method 1
16
Kc
Ti
Td
pp. 15–16.
Kc
Ti
Td
p. 248;
τ
0.01 ≤ m ≤ 1.1 .
Tm
17
2

τ 
τ
τ
1 
4282 m  − 1334.6 m + 101 , 0 < m < 0.2 ;
Tm
Km 
Tm 
Tm




Kc =
1.1161  Tm 


K m  τm 
0.9427
, 0.2 ≤
τm
≤ 1.0 .
Tm
, 0.8 ≤
τm
≤ 1.0 .
Tm
2


 τm 
 τm 
τ


 + 0.8288 , 0 ≤ m ≤ 0.8 ;


Ti = Tm 36.842
−
10
.
3

T 
T
T


m
 m
 m


τ 
Ti = 76.241Tm  m 
 Tm 
Desired closed loop transfer function =
TCL 2 = 4.5115Tm (τm Tm )
5.661
(
(ηs + 1)e −sτ
(TCL2 s + 1)2
m
6.77
;
, 0.8 ≤ τm Tm ≤ 1.0 ,
)
TCL 2 = 0.0326 + 0.4958(τm Tm ) + 2.03(τm Tm )2 Tm , 0 ≤ τm Tm ≤ 0.8 .
16
Kc =
1
Km
2


 τm  
τm
0.726Tm 




+
T
T
0
.
096
2
.
099
4
.
104
0
.
229
,
=
+
+

 i
m
T   ,
τm 
Tm


 m 

2

τ  
τ
τ
Td = Tm 0.019 + 0.419 m + 0.822 m   , 0 ≤ m < 0.6 .
Tm
T
T


m
 m 

17
Kc =
1.214  Tm 


K m  τm 
0.8449
1.028
τ 
, Ti = 3.05Tm  m 
 Tm 
1.0145
τ 
, Td = 0.672Tm  m 
 Tm 
.
b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
452
Rule
Kc
Sree and
Chidambaram
(2006) –
continued.
18
Kc
Ti
Td
Comment
Ti
Td
p. 248;
τ
0.01 ≤ m ≤ 1.1 .
Tm
Direct synthesis: frequency domain criteria
De Paor and
O’Malley (1989).
Model: Method 1
Chidambaram
(1995b).
19
1.166  Tm 


Kc =
K m  τm 
Kc =
Ti =
Kc
Ti
Td
τm
<1
Tm
20
Kc
25Tm − 27 τm
0.46τ m
τ m Tm < 0.6
Also Chidambaram (1998), p. 93. Model: Method 1
τ
0.1 ≤ m ≤ 0.6 ;
21
Ti
Td
Kc
Tm
Model: Method 1
Gaikward and
Chidambaram
(2000).
18
19
0.6365
1.105
, Ti = 1.0553Tme
Tm
τm
τ 
, Td = 0.7542Tm  m 
 Tm 


1 
1 − τm Tm
τ τ
τ τ
cos 1 − m  m +
sin 1 − m  m
Km 
τm Tm
 Tm  Tm
 Tm  Tm

0.789
.

,


τm Tm
Tm
, Td = Tm
tan (0.75φ) ,
 1 − τm Tm 
1 − τm Tm

 tan (0.75φ )
 τm Tm 
φ = tan −1
(Tm
τm − 1) −
(τm
Tm ) − (τm Tm ) .
2
20
Kc =
2
 τm  
τm
1 
1 
T 
 .

1.3 + 0.3 m  or K c =
5
.
332
15
.
02
16
.
214
−
+
T  
τm 
Km 
Tm
K m 
 m 

21
Kc =
1
Km
2



 τm 
τ
Tm 



+
+ 2.099 m + 0.096 ,
0
.
299
0
.
726
,
T
=
T
4
.
104

 i
m


τm 
Tm



 Tm 


2


 τm 
τm

.


+
+
Td = Tm 0.822
0
.
419
0
.
019
Tm 
Tm





b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Ti
Td
453
Comment
Robust
Ti
Td
Kc
Rotstein and
Lewin (1991), λ determined graphically – sample values provided:
Marchetti et al.
τm Tm = 0.2 , 0.5Tm ≤ λ ≤ 1.9Tm .
K m uncertainty
(2001).
= 50%.
τm Tm = 0.4 , 1.3Tm ≤ λ ≤ 1.9Tm .
Model: Method 1
τm Tm = 0.2 , 0.4Tm ≤ λ ≤ 4.3Tm .
K m uncertainty
τm Tm = 0.4 , 1.1Tm ≤ λ ≤ 4.3Tm .
= 30%.
τm Tm = 0.6 , 2.2Tm ≤ λ ≤ 4.3Tm .
22
Tan et al.
(1999b).
22
23
23
Ti
Kc
Td
τm Tm < 1.0 ;
Model: Method 1
  λ



 λ
Tm λ
+ 2  + 0.5τm 
0.5λ
+ 2 τ m
T


 Tm
  m
 , T = λ λ + 2  + 0.5τ , T =
Kc =
.
d
i
m

T


λ2 K m
λ

 m
λ
+ 2  + 0.5τ m

 Tm
1
,
Kc =
K m {[− (0.0373 λ ) + 0.1862](τm Tm ) + [0.6508 − 0.1498 log λ ]}
Ti =
{[(10.5045 λ ) + 0.8032](τm
{[− (0.0373 λ ) + 0.1862](τm
Tm ) + [(3.2312 λ ) − 0.3445]}
Tm ,
Tm ) + [0.6508 − 0.1498 log λ ]}


τ
1
Td = Tm 0.9065λ0.1632 m −

Tm 0.2075λ + 2.0335 


 − 0.0373
τ
+ 0.1862  m + (0.6508 − 0.1498 log λ ) , λ not specified.

λ
T


m


b720_Chapter-04
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FA
Handbook of PI and PID Controller Tuning Rules
454
Rule
Kc
Lee et al. (2000).
Model: Method 1
Shamsuzzoha
and Lee (2006b).
Ti
Td
Comment
24
Kc
Ti
Td
0 ≤ τm Tm < 2
25
Kc
Ti
Td
Model: Method 1
26
Ti
Td
Kc
Shamsuzzoha et
If τm Tm >> 1 , λ = τm (Shamsuzzoha et al., 2005); λ = x1Tm , with
al. (2005),
Shamsuzzoha
x1 values given below (Shamsuzzoha and Lee, 2007a).
and Lee (2007a).
Ms
τm Tm
Ms
τm Tm
Model: Method 1
3.0
3.5
4.0
3.0
3.5
4.0
0.01
0.01
0.01
0.01
0.40
0.34
0.31
0.3
0.05
0.05
0.04
0.05
0.65
0.55
0.48
0.4
0.10
0.09
0.08
0.1
1.10
0.90
0.75
0.5
0.23
0.20
0.18
0.2
24
Kc =
 T

, x1 = Tm  CL + 1 eτ m
− K m (2TCL + τm − x1 )
 Tm


2
Ti
Tm

− 1 , Ti = −Tm + x 1 − x 2 ,


τm (0.167 τm − 0.5x1 )
2TCL + τm − x1
− x2 .
Ti
2
2
x2 =
25
2
TCL + x1τm − 0.5τm
, Td =
2TCL + τm − x1
Kc = −
− Tm x1 −
λ2 − 0.5τm 2 + x1τm
Ti
’
, Ti = −Tm + x1 − x 3 , x 3 =
K m (τm − x1 + 2λx 2 )
2λx 2 + τm − x1
3
− Tm x1 −
Td =
2
0.167 τm − 0.5x1τm
 λ2 + 2λx 2Tm + Tm 2 e τ m
τm − x1 + 2λx 2
− x 3 , x1 = Tm 
Ti
Tm 2

(
)
Tm

− 1 ;

x 2 not explicitly specified ( x 2 ≥ 0.707 normally); λ is ‘slightly more’ than τm .
26
Kc = −
3λ2 + 2x1τm − 0.5τm 2 − x12
Ti
,
, Ti = −Tm + 2 x1 −
K m (3λ + τm − 2x1 )
3λ + τm − 2x1
2
− 2Tm x1 + x1 −
Td =

x 1 = Tm 



λ
1 +
 T
m

λ3 + 0.167 τm 3 − x1τm 2 + x12 τm
3λ2 + 2x1τm − 0.5τm 2 − x12
3λ + τm − 2x1
,
Ti
3λ + τm − 2x1
3
 τm
 e


Tm

− 1 .

b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
455
4.10.3 Ideal controller in series with a first order lag
U(s)
E(s)
R(s)
+
−

 1
1
K c 1 +
+ Td s 
 Ti s
 Tf s + 1
 1

1
G c (s) = K c 1 +
+ Td s 
 Tf s + 1
 Ti s
K m e − sτ
Tm s − 1
m
Y(s)
Table 93: PID controller tuning rules – unstable FOLPD model G m (s) =
Rule
Kc
Chandrashekar et
al. (2002).
1
Kc =
1
Km
1
Kc
Ti
Td
K m e − sτ m
Tm s − 1
Comment
Direct synthesis: time domain criteria
Model: Method 1
Ti
0.5τ m
2




τ
 4282 τm  − 1334.6 τm + 101 , 0 < m < 0.2 ;


Tm
Tm


 Tm 


Kc =
1.1161  Tm 


K m  τm 
0.9427
, 0. 2 ≤
τm
≤ 1. 0 .
Tm
, 0. 8 ≤
τm
≤ 1.0 .
Tm
2


τ 
τ 
τ
Ti = Tm 36.842 m  − 10.3 m  + 0.8288 , 0 ≤ m ≤ 0.8 ;
T
T
T


m
 m
 m


τ 
Ti = 76.241Tm  m 
 Tm 


τ
τ
Tf =  0.1233 m + 0.0033Tm , 0 ≤ m ≤ 1.0 .
Tm
Tm


Desired closed loop transfer function =
τ
TCL 2 = 4.5115Tm  m
 Tm
TCL 2



5.661
, 0.8 ≤
(ηs + 1)e −sτ
(TCL2 s + 1)2
m
;
τm
≤ 1.0 ,
Tm

τ
τ
=  0.0326 + 0.4958 m + 2.03 m

Tm
 Tm




2

 T , 0 ≤ τ m ≤ 0. 8 .
 m
Tm

6.77
b720_Chapter-04
04-Apr-2009
Handbook of PI and PID Controller Tuning Rules
456
Rule
2
Chandrashekar et
al. (2002).
Model: Method 1
Kc
Ti
Td
Kc
Ti
Td
TCL 2
τ m Tm
0.02Tm
0.0
0.0435Tm 0.0134Tm
0.10Tm
0.1
0.0884Tm 0.0250Tm
0.20Tm
0.2
3.2655 K m 1.8018Tm
0.1375Tm 0.0435Tm
0.40Tm
0.3
2.4516 K m 3.0400Tm
0.1868Tm
0.0581Tm
0.60Tm
0.4
2.0217 K m 4.6500Tm
0.2366Tm 0.0696Tm
0.80Tm
0.5
1.7525 K m 6.7286Tm
0.2866Tm 0.0784Tm
1.00Tm
0.6
1.5458 K m 10.337Tm
0.3380Tm
0.0885Tm
1.30Tm
0.7
1.3723 K m 17.693Tm
0.3910Tm
0.1005Tm
1.80Tm
0.8
1.2420 K m 33.610Tm
0.4440Tm 0.1124Tm
2.60Tm
0.9
0.1247Tm
4.20Tm
1.0
0.5975Tm
0.1138Tm
5.00Tm
1.2
0.6982Tm
0.0957Tm
6.00Tm
1.4
Ti
101.0000
Km
0.0404Tm
10.3662
Km
0.3874Tm
5.3750 K m 0.8600Tm
0.0
0.0
1.1400 K m 80.620Tm 0.4969Tm
Desired closed loop transfer function =
Kc = −
Model: Method 1; p. 118.
([Ti − 0.5τm ]s + 1)e−sτ
(
(1)
TCL
2s
Td =
)(
+1
( 2)
TCL
2s
m
)
+1
;
Ti
,
(1)
( 2)
K m TCL
+
T
2
CL 2 + 1.5τ m − Ti
(
(
)
)
(1)
( 2)
(1)
( 2)
(1)
( 2)
TCL
2TCL 2 + 0.5τ m TCL 2 + TCL 2 + 0.5TCL 2TCL 2
Ti =
Comment
Alternative tuning rules:
Td
Tf
Kc
Sree and
1.0895 K m 143.6Tm
Chidambaram
1.0536 K m 277.37Tm
(2006).
2
FA
(
τm
(1)
( 2)
+ Tm TCL
2 + TCL 2 + τ m
Tm
Tm − 0.5τm
(1)
( 2)
0.5TCL
0.5(Ti − 0.5τm )τm
2 TCL 2 τ m
, Tf = −
.
(1)
( 2)
Ti
Tm Ti − 1.5τ m − TCL
2 − TCL 2
(
)
)
,
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Ti
Kc
Ti
457
Comment
Td
Robust
Zhang et al.
(2000).
Model: Method 1
Lee et al. (2006).
Model: Method 1
3
4
Kc
Ti
Td
Recommended
τm ≤ λ ≤ 2τm .
Zhang (2006).
Model: Method 1
5
Kc
Ti
Td
Recommended
τm ≤ λ ≤ 2τm .
)
2
3
Kc =
Kc =
2τm ≤ λ ≤ 5τm , τm Tm < 1 ; λ large, τm Tm > 1 .
(
λ2 τm + λ2 + 2λτ m Tm + 2(λ + τm )Tm
λ2 τm λ (λ + 2τm )
, Ti =
+
+ 2(λ + τm ) ,
2
K mλ[λτ m + (λ + 2τm )Tm ]
Tm
Tm
λ2 τm + λ(λ + 2τm )Tm + (2λ + τm )Tm
Td = τm
4
Td
2
λ2 τm + λ(λ + 2τm )Tm + 2(λ + τm )Tm 2
, Tf =
λ2 τm Tm
.
λ2 τm + λ (λ + 2τm )Tm
2
Ti
λ2 + x1τm − 0.5τm
, Ti = −Tm + x1 −
,
− K m ( 2λ + τ m − x 1 )
2λ + τm − x1
τm (0.167 τm − 0.5x1 )
2
λ2 + x1τm − 0.5τm
2λ + τm − x1
−
;
Ti
2λ + τm − x1
2
− Tm x1 −
Td =
Desired closed loop transfer function =
λ3
5
Kc =
Tm
2
+
3λ2
+ 3λ + τm
Tm
 λ3
3λ2 
Km  2 +
T
Tm 
 m
, Ti =
λ3
Tm
2
+
 λ

, x1 = Tm 
+ 1 e τ m
2
 Tm
(λs + 1)


2
e − sτ m
Tm

− 1 .


3λ2
+ 3λ + τm ,
Tm
λ2
Td =
Tm
λ3
Tm
2
2
+
3λ
+3
Tm
3λ2
+
+ 3λ + τm
Tm
λτ m , Tf =
λTm
.
λ + 3Tm
b720_Chapter-04
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458





1  1 + Td s 

4.10.4 Classical controller G c (s) = K c 1 +
Td s 
Ti s 

1 +

N 

E(s)
R(s)
+
−



 U(s)

1  1 + Td s 

K c 1 +
Td 
Ti s 

s
1 +
N 

K m e − sτ
Tm s − 1
m
Y(s)
Table 94: PID controller tuning rules – unstable FOLPD model G m (s) =
Rule
Minimum IAE –
Shinskey (1988),
p. 381.
Method:
Model 1;
N not specified.
Huang and Chen
(1997), (1999).
1
Kc
0.91Tm
Ti
Td
K m e − sτ m
Tm s − 1
Comment
Minimum performance index: regulator tuning
K m τm
1.70τ m
0.60τ m
τ m Tm = 0.1
1.00Tm K m τm
1.90τ m
0.60τ m
τ m Tm = 0.2
0.89Tm K m τm
2.00τ m
0.80τ m
τ m Tm = 0.5
0.86Tm K m τm
2.25τ m
0.90τ m
τ m Tm = 0.67
0.83Tm K m τm
2.40τ m
1.00τ m
τ m Tm = 0.8
1
Kc
Direct synthesis: time domain criteria
Ti
Td
N =∞;
Model: Method 4
0 < τm Tm < 1 ; ‘Good’ servo and regulator response.
Kc =
1.02379 


T 
0.5 
 , Ti = K c K m Tm  2 x1 + τm  ,
1 + 1.61519 m 
T

K
K
1
T
−
Km 
τ

c m
m 
 m
 m



τ
τ
with recommended x 1 = Tm  − 1.3735.10 −3 + 0.22091 m + 1.6653 m
Tm

 Tm

2

 τm  
τm
−4

.


Td = Tm 1.5006.10 + 0.23549
+ 0.16970
Tm
Tm  







2
;


b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Ti
Comment
Td
Direct synthesis: frequency domain criteria
Paraskevopoulos
x1τm Tm
N =∞;
2
2
Ti
et al. (2006). Ti x 2 1 + x 2
τ
> 0.15τm Tm 0.02 < m < 0.9.
3
1 + Ti 2 x 2 2
Model: Method 1
Ti
Tm
2
0 ≤ x1 ≤ 0.55 +
τm 
0.0098 
 0.31 −
 ; x1 < 0.15 .

Tm 
1 − (τm Tm ) 
Ti 2 (1 − (τm Tm ) + Td ) + Ti − (τm Tm ) + Td + x 3
x2 =
2Ti 2 ((τm Tm ) − Td )
,
2
 


τ
 2

τ
τ
τ
x 3 = Ti 2 1 − m + Td  + Ti − m + Td  + 4 m − Td Ti 1 + Ti − m + Td  ;
Tm
Tm
  Tm


 Tm
 

(
)

0.632((τ m Tm ) − Td ) + 0.436 (τ m Tm ) − Td − 0.0153 φ m φ max
m
Ti = x 4 1 +
(τm Tm ) − Td
1 − φ m φ max

m
x4 =
0.0029 − 0.0682 (τm Tm ) − Td + 1.4941((τm Tm ) − Td )
(1.003 − (τm
Tm ) + Td )
2
(
;
 T

τ
 T
max
φm > 0.2φmax
= − m − Td  m − Td − 1 + tan −1  m − Td − 1  .
m , φm
 τm

 Tm
 τm


3
x 2 and x 3 given in footnote 2;
 T 
 
Tτ
Ti = x 4 1 + m 1 + 0.4 d m − 0.22Td  x 5  , x 4 given in footnote 2;
Tm
 
 τm 
(
)1 −(φ(φ φ φ ) ) ,
τ ) − 1).
x 5 = − 0.0153 − 0.436 τm Tm + 0.632(τm Tm )
φmax
= −(τm Tm ) (Tm τm ) − 1 + tan −1
m
( (T
m
m
m
max
m
max
m
m
)

,

459
b720_Chapter-04
FA
Handbook of PI and PID Controller Tuning Rules
460
Rule
Paraskevopoulos
et al. (2006).
Model: Method 1
4
04-Apr-2009
0 ≤ x1 ≤ 0.55 +
Kc =
4
Kc
Ti
Td
Comment
Kc
Ti
x1τm Tm
N=∞
τm 
0.0098 
 0.31 −
 ; x1 < 0.15 . 0.02 < τm Tm < 0.9 .

Tm 
1 − (τm Tm ) 
2
2
K max
1 + ωmin
1 + ωmax
or K min A m , K min = Ti ωmin
, K max = Ti ωmax
;
2
2
2
Am
1 + Ti ωmin
1 + Ti ωmax 2
0.03([τm Tm ] − Td )
1.14 − [τm Tm ] + Td
= 1+


0.05
 0.973 +
Ti − x 2


[
]
−
τ
+
1
T
T
m
m
d 

0.006 +
ωmin
x2
T i −([τm Tm ] − Td )(1 + Ti )
,
4
2

 x  
τ
  
τ
1.571
ωmax = 1 + 0.22 m − Td   1 +  0.1 − 0.3 m − Td  2  


Tm

 Tm
   
 Ti   (τm Tm ) − Td


 

Ti
Ti
− 0.9463 
− 0.5609 .

 (Ti + 1)([τm Tm ] − Td )
  (Ti + 1)([τm Tm ] − Td )

1+ (τ m Tm )− Td


 max  

 A m − 1  A m − 1 


Ti = x 2 1 + 0.65 




1 −  Am − 1
A max − 1 

 
 m


[
[
]
]
[




 + 0.01 − 0.18 + 5 τm − T 
d

Tm




2
3
+ 0.01 − 32([τm Tm ] − Td ) + 75([τm Tm ] − Td ) − 51([τm Tm ] − Td )
x2 =
0.0029 − 0.0682 [τm Tm ] − Td + 1.4941([τm Tm ] − Td )
(1.003 − [τm
max
A m > 0.2(A max
=
m − 1) +1, A m
]
 3([τm Tm ] − Td )4 − 2.3([τm Tm ] − Td )2 
+ 0.01
,
(1 − [τm Tm ] + Td )3


Tm ] + Td )
2
;
1.571
 τm

0.4085[[τm Tm ] − Td ] 



 T − Td 1 + 1 − 0.2864[[τ T ] − T ] 
m
m
d 
 m

1
(7.41 − 3.52[τm Tm ])(TdTm τm )3
1+
3
1 − (0.47 + 0.49[τm Tm ])(Td Tm τm )
.
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Paraskevopoulos
et al. (2006) –
continued.
5
5
Kc
Ti
Kc
Ti
Comment
Td
>
461
0.15τm
Tm
N=∞
K c and x 2 as given in footnote 4, with
0.03([τm Tm ] − Td )
1.14 − [τm Tm ] + Td
= 1+


0.05
 0.973 +
Ti − x 2


[
]
−
τ
+
1
T
T
m
m
d 

0.006 +
ωmin
x2
τ

T i − m − Td (1 + Ti )
 Tm

 (0.14 + 1.15[τm Tm ])(Td Tm τm )3  x 2  
  ,
1 +
2
T 
1 + 2(Td Tm τm )
 i  

ωmax
2
4

 x 2  
 τm
   
τm
1.571








= 1 + 0.22
− Td  1 + 0.1 − 0.3
− Td   


[
T
T
T
τ




m
 m
  
 i   m Tm ] − Td




Ti
Ti
− 0.9463 
− 0.5609

 ;
 (Ti + 1)([τm Tm ] − Td )
  (Ti + 1)([τm Tm ] − Td )
(
5.53 − 0.41[τm Tm ])(Td Tm τm )4
1+
3
2
1 − 0.65(Td Tm τm ) 1.27 − 0.4(Ti x 2 )
[
1+ (τ m Tm )− Td


 max  

 A m − 1  A m − 1 


Ti = x 3 1 + 0.65 




1 −  Am − 1
A max − 1 

 
 m


[
[
]
]
[
][
]




 + 0.01 − 0.18 + 5 τm − T 
d

Tm




2
3
+ 0.01 − 32([τm Tm ] − Td ) + 75([τm Tm ] − Td ) − 51([τm Tm ] − Td )
]
2

 3([τm Tm ] − Td )4 − 2.3([τm Tm ] − Td )2    Td Tm 



+
.
1
x4  ,
+ 0.01

3



(1 − [τm Tm ] + Td )
   τm 


  T 3 0.367 + 1.78[τ T ]
m
m 
x 3 = x 2 1 +  Td m 
,
  τm 
1 − (Td Tm τm )2 


3τ m

  Td Tm  Tm

x 4 =  

  τm 



2
 (A − 1)3 
  τ 3
 τm 
τ
m
m
m

 2
 − 3.32

,
3 
 T  + 1.2 T − 0.27  − max
  Tm 
m
 m
 A m − 1 


A max
as given in footnote 4. 0.02 < τm Tm < 0.9 .
m
(
)
b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
462
4.10.5 Generalised classical controller


Td s
1
+
G c (s) = K c 1 +
T

Ti s
1+ d

N

R(s) E(s)
+
−


Td s
1
K c 1 +
+
 Ti s 1 + Td

N


 b + b s + b s 2 U(s)
f1
f2
 f0
+
+
1
a
s
a
s2

f
1
f
2
s



2
 b f 0 + b f 1s + b f 2 s
 1 + a f 1s + a f 2 s 2
s 

K m e − sτ
Tm s − 1
Y(s)
m
Table 95: PID controller tuning rules – unstable FOLPD model G m (s) =
Rule
Kc
K m e − sτ m
Tms − 1
Td
Comment
0.25τm
Model: Method 1
Ti
Robust
1
Shamsuzzoha et
al. (2007).
0.667 τm
Kc
For τm Tm > 0.6 , λ = τm . Otherwise, λ = x1Tm ; x1 given for
various τm Tm and M s values:
1
Kc = −
τm Tm
M s = 3. 0
M s = 3. 6
τm Tm
M s = 3. 0
M s = 3. 6
0.05
0.1
0.2
0.3
0.015
0.04
0.09
0.14
0.015
0.04
0.08
0.13
0.4
0.5
0.6
0.21
0.32
0.41
0.18
0.26
0.36
3

4τm
λ  τm
 e
, b f 0 = 1 , b f 1 = Tm 1 +
K m (6τm + 18λ − 6b f 1 )
 Tm 

2
N = ∞ , af1 =
2b f 1τm + τm + 12λτ m + 18λ2
+ Tm , a f 2 = 0 .
6τ m + 18λ − 6b f 1
Tm

− 1 , b f 2 = 0 ,






b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
4.10.6 Two degree of freedom controller 1


Td s
1
+
U (s ) = K c  1 +
 Ti s 1 + Td

N






βTd s 
 E (s ) − K c  α +
 R (s )
Td 


s
1+ s 

N 






T
s
 α + β d K c
T 

1+ d s

N 

E(s)
R(s) +
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N





s

−
U(s)
+
K m e − sτ
Tm s − 1
m
Table 96: PID controller tuning rules – unstable FOLPD model G m (s) =
Rule
Minimum IAE –
Huang and Lin
(1995).
1
Kc = −
Kc
1
Kc
Ti
Td
Y(s)
K m e − sτ m
Tm s − 1
Comment
Minimum performance index: regulator tuning
Model: Method 2
Ti
Td
α = 0 , β = 1 , N = 10; 0.01 ≤ τm Tm ≤ 0.8 .
T 
1 
τ
0.2675 + 0.1226 m + 0.8781 m 
Km 
Tm
 τm 

1.004 
(
,


Ti = Tm 0.0005 + 2.4631(τm Tm ) + 9.5795(τm Tm )
2.9123
), T
d
463
= 0.0011Tm + 0.4759τm .
b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
464
Rule
Kc
Minimum ISE –
Paraskevopoulos
et al. (2004).
Minimum IAE –
Huang and Lin
(1995).
Minimum ISE –
Paraskevopoulos
et al. (2004).
2
Kc
4
Ti
3
(1)
Ti
Td
Comment
0
Model: Method 1
Minimum performance index: servo tuning
Model: Method 2
Ti
Td
Kc
α = 0 , β = 1 , N = 10; 0.01 ≤ τm Tm ≤ 0.8 .
5
Kc
0
Ti
(1)
Model: Method 1
Minimum performance index: other tuning
Paraskevopoulos
et al. (2004).
Model: Method 1
2
Kc
(1)
Kc
= Kc
(max)
=
(min)
Kc
ωmax Ti
Km
6
(max)
Kc
, Kc
0
Ti
(1)
(min)
=
1 + ω max 2 Tm 2
2
1 + ωmax Ti
2
ωmin Ti
Km
, ωmin =
1 + ω min 2 Tm 2
α =1
,
1 + ωmin 2 Ti 2
τm 
1  Ti
−

,
Tm  Tm Tm + Ti 
[4.935 τm][(Ti (Ti + τm )) − (τm Tm )] , T ω 2 >> 1 .
(τm Tm ) + π[(Ti (Ti + τm )) − (τm Tm )] i max
1.916 + 4.78(τ m Tm )
3
Ti =
τm , α = 1 , 0.05 ≤ τm Tm ≤ 0.85 .
0.88 − (τm Tm )
1.0251
4
),
K c = −(1 K m )(− 0.433 + 0.2056(τm Tm ) + 0.3135(Tm τm )
6.6423
),
Ti = Tm (− 0.0018 + 0.8193(τm Tm ) + 7.7853(τm Tm )
7.6983(τ T )
).
Td = Tm (− 0.0312 + 1.6333(τm Tm ) + 0.0399e
2.725 + 1.71(τ m Tm )
(1)
5
K c given in footnote 2. Ti =
τm , α = 1 , 0.05 ≤ τm
0.88 − (τm Tm )
ωmax =
m
6
Kc
(1)
m
∫ [e (t) + (u( t) − u ) ]dt .
given in footnote 2. Minimise performance index =
[
]
∞
2
0
Ti = Tm 0.747 + 3.66(τm Tm ) + 18.64(τm Tm ) , 0.01 ≤ τm Tm ≤ 0.3 .
3
 4.52 + 0.476(τm Tm )2 
τm
Ti = τm 
≤ 0.85 .
 , 0.3 ≤
Tm
0.88 − (τm Tm ) 

Tm ≤ 0.85 .
∞
2
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Paraskevopoulos
et al. (2004) –
continued.
Chidambaram
(2000c).
Model: Method 1
7
Kc
(1)
8
Kc
9
Kc
Ti
Td
Comment
Ti
0
α =1
465
Direct synthesis: time domain criteria
Ti
‘Small’ τ m Tm .
0
τ
0.1 ≤ m ≤ 0.6
Ti
Tm
10 (1)
Chidambaram
τm Tm < 0.6
Ti
0.245
(2000c), Srinivas
1.165  Tm 
4.4Tm + 9 τ m 0.176T
τ
and


m
0.6 ≤ m ≤ 0.8
K m  τ m 
Chidambaram
T
+0.36τm
m
(2001), Sree and
( 2)
τ
Chidambaram
Ti
0.8 ≤ m ≤ 0.9
(2006),
Tm
pp. 182–184.
Model: Method 1 ξ = 0.333 ; N = ∞ , β = 1 , α = 0.74 + 0.42(τm Tm ) − 0.23(τm Tm )2
(Chidambaram, 2000c);
2
τ
1
1 
Tm
τ 
− m  (Srinivas
; β=
N =∞, α = m +
τm −
Ti K c K m
Td 
K c K m 2Ti 
and Chidambaram, 2001);
ξ ≥ 1 ; N = ∞ , β = 1 ; α = 1 (Sree and Chidambaram, 2006).
7
Kc
(1)
∞
∫ [e (t) + (du dt) ] dt.
given in footnote 2. Minimise performance index =
2
2
0
 2.58 + 3.60(τm Tm )2 
τm
Ti = τm 
≤ 0.85 .
 , 0.03 ≤
T
(
)
0
.
88
−
τ
T
m
m
m


8
Kc =
2
Tm Ti
2 2
0.04TS ξ + Ti τm
α = 1−
, Ti =
0.04TS ξ2 + Tm τm + 0.4TSξ 2
,
Tm − τm
0.2TS ξ 2 (Tm − τ m )
0.04TS 2 ξ 2 + Tm τ m + 0.4TS ξ 2
(Chidambaram, 2000c); α = 1 −
0.2TSξ 2
or
Ti
α = 0.733 + 0.6(τm Tm ) − 0.36(τm Tm ) (Sree and Chidambaram, 2006).
5.8 τ m Tm
Tm
, Ti = 0.4015Tm e
,
2
9
K c = (2.695 K m )e −1.277 τ m
α = 0.733 + 0.6(τ m Tm ) − 0.36(τ m Tm ) ; ξ = 0.333 .
2
10
Ti (1) =
0.176Tm + 0.36τm
0.179 − 0.324(τm Tm ) + 0.161(τm Tm )2
, Ti
( 2)
=
0.176Tm + 0.36τm
.
0.12 − 0.1(τm Tm )
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Rule
Kc
Ti
Comment
Td
τm Tm ≤ 0.2
11
Kc
6.461τ m
Jhunjhunwala
0.898
and
0.324Tm e Tm
 Tm 

Chidambaram 0.876

 τm 
(2001).
12
Ti
Model: Method 1
Kc
τm
≤ 0.6
Tm
0
0.2 ≤
Td
β =1, N = ∞ .
Minimum ISE (Chidambaram, 2002, pp. 157–158).
Chandrashekar et
al. (2002).
Model: Method 1
13
0
Ti
Kc

K c K m (2 − Tm ) + (K c K m − 1)Ti
τm 
(Jhunjhunwala and
10.7572 − 35.1  ; α =
T
2K c K m
m

K K (2 + τm ) − (K c K m + 1)Ti
Chidambaram, 2001); α = c m
(Chidambaram, 2002).
2K c K m
11
Kc =
1
Km
12
Kc =
1 
τ  τ
1.397  Tm 
13.0 − 39.712 m  , m < 0.2 ; K c =


K m 
Tm  Tm
K m  τm 
2.044 τ m
Ti = 0.856Tm e
Tm
τ
, 0 ≤ m ≤ 1 ; Ti = 0.3444Tme
Tm
0.769
2.9595 τ m
Tm
, 1≤
, 0.2 ≤
τm
≤ 1.4 ;
Tm
τm
≤ 1.4 ;
Tm
Td = 0.5643τm + 0.0075Tm , 0 ≤ τm Tm ≤ 1.4 ;
α = 0.3137 , τm Tm = 0.1 ; α = 1 − 0.4772e − (1.58τ m
13
49.535 − 21.644 τ m
e
Km
Ti = 0.1523Tm e7.9425τ m
Kc =
Tm
Tm
,
Tm )
, 0.2 ≤ τm Tm ≤ 1 .
τm
0.8668  τm 


< 0.1 ; K c =
K m  Tm 
Tm
−0.8288
, 0.1 ≤
τm
≤ 0.7 ;
Tm
;
Desired closed loop transfer function =
(ηs + 1)e −sτ
(TCL2 s + 1)2
m
;
TCL 2 = 0.025Tm + 1.75τm , τm Tm ≤ 0.1 ; TCL 2 = 2 τ m , 0.1 ≤ τm Tm ≤ 0.5 ;
TCL 2 = τm (5(τm Tm ) − 0.5) , 0.5 ≤ τm Tm ≤ 0.7 .
α = 0.505 + 3.426(τm Tm ) − 10.57(τm Tm ) , τm Tm ≤ 0.2 ;
2
α = 0.713 + 0.232(τm Tm ) , 0.2 ≤ τm Tm ≤ 0.7 (Chandrashekar et al, 2002);
α = 0.67 , τm Tm ≤ 0.1 ; α = 0.6354 + 0.437(τm Tm ) , 0.1 ≤ τ m Tm ≤ 0.7
(Sree and Chidambaram, 2006).
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Chandrashekar et
al. (2002).
Model: Method 1
14
Kc =
14
Kc
467
Comment
Ti
Td
Ti
0.5τ m
2

τ 
τ
τ
1 
4282 m  − 1334.6 m + 101 , 0 < m < 0.2 ;
T
Km 
T
T

m
m
 m


Kc =
[
]
1.1161  Tm 


K m  τm 
0.9427
, 0.2 ≤
τm
≤ 1.0 ;
Tm
Ti = Tm 36.842(τm Tm ) − 10.3(τm Tm ) + 0.8288 , 0 ≤ τm Tm ≤ 0.8 ;
2
Ti = 76.241Tm (τm Tm )
6.77
, 0.8 ≤ τm Tm ≤ 1.0 ;
α = 0.83 , τm Tm ≤ 0.1 ; α = 0.8053 − 0.1935(τm Tm ) , 0.1 ≤ τm Tm ≤ 0.7
(Chandrashekar et al., 2002);
α = 0.505 + 3.426(τm Tm ) − 10.17(τm Tm ) , τm Tm ≤ 0.2 ;
2
α = 0.713 + 0.232(τm Tm ) , 0.2 ≤ τm Tm ≤ 1.0 (Sree and Chidambaram, 2006).
(ηs + 1)e −sτ ;
(TCL2 s + 1)2
2
Tm ) + 2.03(τm Tm ) )Tm , 0 ≤ τm
m
Desired closed loop transfer function =
(
TCL 2 = 0.0326 + 0.4958(τm
TCL 2 = 4.5115Tm (τ m Tm )
Tm ≤ 0.8 ;
5.661
, 0.8 ≤ τm Tm ≤ 1.0 , N = ∞ ; β = 0
(Chandrashekar et al., 2002; Sree and Chidambaram, 2006);
[(1 − α )Tis + 1]e−sτm .
Desired closed loop transfer function =
TCL 2s 2 + 2ξTCLs + 1
If it is desired to minimise the overshoot (servo response), for ξ ≥ 1 , then α = 1 .
Otherwise, the optimum value of α is chosen by minimising the ISE (servo
response). Then, α = 1 − 2(TCL Ti ) , ξ = 1 ; α = 1 − 2(TCL ξ Ti ) , ξ < 1 ;
T
α = 1 − CL
Ti
2
3


2
2
2

 

  − ξ + ξ − 1  − ξ − ξ − 1  −  − ξ + ξ − 1  + x1 

 

 , ξ > 1 with

3


 − ξ + ξ2 − 1  − ξ − ξ2 − 1  + x




2







3
2
x 1 = − − ξ − ξ 2 − 1  +  − ξ + ξ 2 − 1   − ξ − ξ 2 − 1  and

 
 

3
2
2
x 2 =  − ξ − ξ 2 − 1  − ξ + ξ 2 − 1  − 2 − ξ + ξ 2 − 1   − ξ − ξ 2 − 1  ;




 

N= ∞ ; β = 1 (Sree and Chidambaram, 2005a).
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Rule
Chandrashekar et
al. (2002).
Model: Method 1
Kc
Ti
Td
Kc
Ti
0
Comment
6.0000 K m
Alternative tuning rules:
α
Ti
0.6000Tm
0.6667
τ m Tm = 0.1
3.2222 K m
1.4500Tm
0.7246
τ m Tm = 0.2
2.2963 K m
2.6571Tm
0.7742
τ m Tm = 0.3
1.8333 K m
4.4000Tm
0.8182
τ m Tm = 0.4
1.5556 K m
7.0000Tm
0.8571
τ m Tm = 0.5
1.3265 K m
14.6250Tm
0.8974
τ m Tm = 0.6
1.1875 K m
31.0330Tm
0.9323
τ m Tm = 0.7
Kc
Alternative tuning rules; TCL 2 = x 5 Tm :
x1 K m
x 2Tm
N = x3 x 4 ,
x 3Tm
β = 0.
Sree and
Chidambaram
(2006).
x1
x2
10.3662
5.3750
3.2655
2.4516
2.0217
1.7525
1.5458
1.3723
1.2420
1.1400
1.0895
1.0536
0.3874
0.8600
1.8018
3.0400
4.6500
6.7286
10.337
17.693
33.610
80.620
143.6
277.37
x3
Coefficient values:
x4
x5
0.0435 0.0134
0.10
0.0884 0.0250
0.20
0.1375 0.0435
0.40
0.1868 0.0581
0.60
0.2366 0.0696
0.80
0.2866 0.0784
1.00
0.3380 0.0885
1.30
0.3910 0.1005
1.80
0.4440 0.1124
2.60
0.4969 0.1247
4.20
0.5975 0.1138
5.00
0.6982 0.0957
6.00
pp. 118, 125. Model: Method 1
α
τ m Tm
0.742
0.767
0.778
0.803
0.828
0.851
0.874
0.898
0.923
0.948
0.965
0.978
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.2
1.4
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Ti
> Ti
Paraskevopoulos
et al. (2004).
Model:
Method 1;
α = 1.
15
Kc
(1)
>
Comment
Td
(min)
1.35τm
(1 − τm
0
Tm )
2
x1τ m +
x2
Coefficient values:
x1
x2
ξ
ξ
0.7269 2.5362 0.5 3.0696 3.7509
1.7016 3.0585 0.75 6.975 5.7053
(1)
x 1 τ m + x 2 Tm +
K
1
1.5
c
6
x 3 τ m Tm
x2
x3
4.59
6.16
8.21
-0.19
0.028
0.37
8.28
9.90
16.88
Kc
16
(1)
x1
x2
x1
x2
1.518 10.54
3.34 16.89
15
Kc
(1)
TCL 2
0.5
0.75
Ti
1
1.5
0
Coefficient values:
x1
x2
TCL 2
5.81
13
τm
< 0.7
Tm
0.1 <
τm
< 0.9
Tm
29.17
49.66
1
1.5
ξ
x2
12.452 8.4358
0.2 <
2
τm
< 0.7
Tm
x2
x3
ξ
1.89
3.44
51.95
95
1.5
2
0
Ti
1.142 1.177
2.353 2.67
17
(1)
12.41
19.91
Coefficient values:
x1
x2
TCL 2
TCL 2
0.409 0.2817 0.5
0.717 0.6542 0.75
Kc
0.5
0.75
1
x1
0
−5
Coefficient values:
ξ
x1
x1
0.02 <
τm
< 0.3
Tm
x 2 τ m Tm
x1
469
τm Tm < 0.2
x1
4.07
8.88
x1
x2
TCL 2
4.76
2
10.73
3
τm Tm < 0.2
x2
TCL 2
23.08 82.87
51.94 176.8
2
3
given in footnote 2. Ti (min) = Tm [1.3146(τm Tm ) − 2.3805(τm Tm )2 + x1 ],
x1 = 57.622(τm Tm ) − 158.99(τm Tm ) + 173.41(τm Tm ) .
3
16
17


x2
.
Ti = Tm  x1 −

(τm Tm ) − τm Tm 
[
]
Ti = Tm x1 + x 3 (τm Tm ) .
3
4
5
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Rule
Kc
18
Prashanti and
Chidambaram
(2000).
Model: Method 1
Ti
Comment
Td
Direct synthesis: frequency domain criteria
Ti
Td
Kc
Modified method of De Paor and O’Malley (1989);
τ m Tm < 1 ; α = 0.83 ; β = 0.346 ; N = ∞ .
Alternatively,
2
α = 0.6873 + 0.3369
3
τ 
τ 
τm
− 0.0378 m  + 0.045 m  ,
Tm
T
 m
 Tm 
2
3
τ 
τ 
τ
τm
− 5 m  + 4.81 m  , 0.05 ≤ m ≤ 0.90 .
Tm
T
T
T
m
 m
 m
Representative results:
α
α
Tm
β
τm Tm
β
τm Tm
β = −1.779 + 2.44
α
β
0.691
0.719
0.756
0.787
-1.834
-1.605
-1.413
-1.346
τm
0.05
0.10
0.20
0.30
0.818
0.850
0.884
0.920
-1.336
-1.207
-1.093
-0.884
0.40 0.960 -0.508
0.50 0.9909 -0.159
0.60
0.70
0.80
0.90
−6.405 τ m
Alternatively, α = 1 − 0.9726e Tm
α
α
τm Tm
τm Tm
0.223
0.543
0.773
18
Kc =
Ti =
1
Km

cos


0.05
0.10
0.20
0.879
0.922
0.954
0.30
0.40
0.50
, β = 1 . Representative results:
α
α
τm Tm
τm Tm
0.974
0.987
0.9943


τ τ
τ τ
1 − τm Tm
1 − m  m +
sin 1 − m  m
 T T
τm Tm
m  m

 Tm  Tm
0.60
0.70
0.80

,


τm Tm
Tm
, Td = Tm
tan (0.75φ) ,
 1 − τm Tm 
1 − τm Tm

 tan (0.75φ )
 τm Tm 
φ = tan −1
(Tm
τm − 1) −
(τm
Tm ) − (τm Tm ) .
2
0.9978
0.90
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
19
Prashanti and
Chidambaram
(2000).
Model: Method 1
Kc
Ti
Td
Comment
Ti
x1Ti
N=∞
2
α = 0.4835 + 0.8482
3
τ 
τ 
τm
− 0.5133 m  + 0.071 m  ,
Tm
T
 m
 Tm 
3
2
β = −2.206 + 3.461
α
β
τm Tm
0.458
0.563
0.664
0.697
-1.916
-1.878
-1.525
-1.307
0.05
0.10
0.20
0.30
τm
τ 
τ 
− 0.984 m  − 0.089 m  .
Tm
 Tm 
 Tm 
Representative results:
α
α
β
τm Tm
0.737
0.786
0.836
0.849
-0.948
-0.729
-0.530
-0.297
0.40
0.50
0.60
0.70
β
0.861 -0.093
0.887 0.038
τm Tm
0.80
0.90
2
Alternatively, α = 0.2243 + 1.289
τ 
τm
− 0.6267 m  , β = 1 ,
Tm
 Tm 
τm
≤ 0.90 . Representative results:
Tm
α
α
α
τm Tm
τm Tm
0.05 ≤
Hägglund and
Åström (2002).
Model:
Method 1;
M s = 1.4;
0.3 < α < 0.5 .
19
Kc =
1
Km
α
τm Tm
0.395
0.336
0.450
0.05
0.10
0.20
0.28 K m Tm
0.544
0.30
0.605
0.40
0.678
0.50
8τ m
0.752
0.772
0.791
0.60
0.70
0.80
0.831
0.2 K m Tm
11τ m
10τ m
0.13 K m Tm
7τ m
0.90
τm
= 0.02
Tm
7.8τ m
0.28 K m Tm
τm Tm
τm
= 0.05
Tm
0
0.28 K m Tm
19τ m
τm
= 0.10
Tm
0.28 K m Tm
135τ m
τ m Tm = 0.15
2


 


2.121 − 1.906 τm + 0.945 τm   , Ti = Tm 0.176 + 0.36 τm  ;


Tm
x1 
Tm 

 Tm  

x1 = 0.179 − 0.324(τm Tm ) + 0.161(τm Tm ) , τm Tm < 0.6 ;
2
x1 = 0.04 , 0.6 < τm Tm ≤ 0.8 ; x 1 = 0.12 − 0.1(τ m Tm ) , 0.8 < τ m Tm ≤ 1.0 .
471
b720_Chapter-04
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FA
Handbook of PI and PID Controller Tuning Rules
472
Rule
Kc
Ti
Td
Comment
Td
N = ∞; β = 0;
Model: Method 1
Td
N = ∞; β = 0;
Model: Method 1
0
τm < Tm
Wang and Cai
(2002).
20
Kc
Ti
Xu and Shao
(2003b).
21
Kc
Ti
Kc
Ti
Robust
Zhang and Xu
(2002).
Model: Method 1
20
Kc =
Td =
21
Kc =
α=
22
For τm Tm ≤ 0.5 , λ = 2 − 5τ m ; α = 1 .
0.5236Tm − 0.4764 Tm τm
0.5236Tm 0.4764 Tm
,
−
, Ti =
K m τm
Km
τm
0.5236 Tm τm − 1
(
0.2618τm Tm τm
0.5236Tm − 0.4764 Tm τm
, α=
(
)
1.9099 τm Tm
1 + 0.9099 τm Tm
)
( A m = 3 , φ m = 60 0 ).
τ (T τ ) + Tm τm
0.5 Tm τm
0.5  Tm
Tm 
, Td =
,
+
, Ti = m m m
τm 
K m  τm
Tm τm − 1
(Tm τm ) + Tm τm
2 Tm τ m
Tm + Tm τ m
Kc =
22
( A m > 2 , φ m > 60 0 ).
λ2 + 2λTm + Tm τm
K m (λ + τm )
2
, Ti =
λ2 + 2λTm + Tm τm
.
Tm − τm
(
)
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
473
4.10.7 Two degree of freedom controller 2




T
s
1
d
 1 + b f 1s (F(s)R (s) − Y(s) ) − K Y(s) ,
+
U(s) = K c 1 +
0

T  1 + a f 1s
Ti s
1+ d s

N 

1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4
F(s) =
1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5s 4
R(s)
F(s)
+
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N



 1 + b f 1s
 1 + a f 1s
s

+ U(s)
−
K0
Table 97: PID controller tuning rules – unstable FOLPD model G m (s) =
Rule
Kc
Ti
Y(s)
K m e − sτ m
Tms − 1
K m e − sτ m
Tm s − 1
Comment
Td
Direct synthesis: time domain criteria
Jacob and
Chidambaram
(1996).
Model: Method 1
Tm − Tm τm
K m (Tm + τm )
Tm − Tm τm
0
Tm τm − 1
K0 =
1
Km
Tm
τm
a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , bf 3 = 0 ; a f 4 = 0 ,
a f 5 = 0 , bf 4 = 0 , bf 5 = 0 .
Arvanitis et al.
(2000b).
Model: Method 1
1
Kc =
1
a f 1 = 0 , b f 1 = 0 ; a f 2 = Ti , b f 2 = 0 ; a f 3 = 0 , b f 3 = 0 ; a f 4 = 0 ,
a f 5 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 .
Tm [2φm + π(A m − 1)]
(
Direct synthesis: frequency domain criteria
0
Ti
Kc
)
2K m τm A m 2 − 1
,
Ti =
2πTm
.
 2φm A m + πA m (A m − 1)  
2φm A m + πA m (A m − 1) 
−
π
π
−
3Tm 
2



2τm A m 2 − 1
Am2 − 1

 

(
)
b720_Chapter-04
04-Apr-2009
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474
Rule
Kc
Ti
Comment
Td
Robust
Tm − Tm τm
Tm − τ m
λK m
Chidambaram
(1997).
Model: Method 1
Tm τm − 1
+
λ > 1.7 ;
0
0.5τm
τm
<1.
Tm
a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , bf 3 = 0 ; a f 4 = 0 ,
a f 5 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = (Tm τm )
0.5
2
Lee et al. (2000).
Model: Method 1
a f 5 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 .
3
a f 5 = 0 , bf 4 = 0 , bf 5 = 0 .
4
0.5
2
 T

Ti
, a f 2 = Tm  CL + 1 e τ m
− K m ( 2TCL + τm − a f 2 )
 Tm


− Tm a f 2 −
Kc =
Kc =
Tm
Km .

− 1 ;


2
TCL + a f 2 τm − 0.5τm
,
2TCL + τm − a f 2
τm 2 (0.167 τm − 0.5a f 2 )
T 2 + a f 2 τm − 0.5τm 2
2TCL + τm − a f 2
− CL
.
Ti
2TCL + τm − a f 2
(K 0K m − 1)Ti ,
K m (λ + τm )
2
Ti =
Tm − K 0 K m τm
τm
,
+
K 0K m − 1
2(λ + τm )
K0 =
4
λ > 1.7
a f 5 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = (Tm τm )
2
3
0
Ti
Kc
a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , bf 3 = 0 ; a f 4 = 0 ,
Ti = −Tm + a f 2 −
Td =
0
Ti
Kc
a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , bf 3 = 0 ; a f 4 = 0 ,
Sree and
Chidambaram
(2006),
pp. 27–28.
Model: Method 1
Kc =
0 ≤ τm Tm < 2
Td
a f 1 = 0 , bf 1 = 0 ; N = ∞ ; bf 2 = 0 ; a f 3 = 0 , bf 3 = 0 ; a f 4 = 0 ,
Lee and Edgar
(2002).
Model: Method 1
2
Ti
Kc
Km .
2Tm − Tm τm − τm
2λK m
, Ti =

1 
Tm 
τ 
τ τ
1 − m  sin
cos 1 − m  m +

Km 
τm  Tm 
 Tm  Tm

Tm − Tm τ m
Tm τ m − 1
+ 0.5τm .

τ τ
1 − m  m
 T T
m  m


.


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FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
4.10.8 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s)




(
)
1
−
β
T
s
1 + b f 1s
[
]
1
−
χ
δ
d

,
F1 (s) = K c  [1 − α ] +
+
+


T
Ti s
1 + Ti s
1 + a f 1s + a f 2 s 2
d
1
+
s


N 





[
]
K (b + b f 4 s )
1
−
ϕ
T
s
1+ bf 2s
[
]
−
φ
1
d 
F2 (s) = K c  [1 − ε] +
+
− 0 f3
2

 1 + a f 3s + a f 4 s
T
Ti s
1+ a f 5s
1+ d s 

N 

R(s)
+
F1 (s)
U(s)
Y(s)
Model
−
F2 (s)
Table 98: PID controller tuning rules – unstable FOLPD model G m (s) =
Rule
Kc
1
Minimum ITSE –
Majhi and
Atherton (2000).
Model: Method 1
1
Kc =
(
Kc
Ti
K m e − sτ m
Tms − 1
Comment
Td
Minimum performance index: servo tuning
0
Ti
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 2Tm τm K m ; b f 3 = 1 ;
b f 4 = 0.5τm Tm ; a f 5 = 0 .
0.8011Tm 1 − 0.9358 τm Tm
K m τm
) , T = T  0.1227 + 1.4550 τ
i
m

m
Tm
− 1.2711
τm 2 
.
2
Tm 
475
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Rule
Minimum ITSE –
Majhi and
Atherton (2000).
Model:
Method 3;
Ap
K=
.
K mh
Kc
Ti
Td
2
Kc
Ti
0
3
Kc
Ti
0
Comment
0<
bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; bf 3 = 1 ; a f 5 = 0 .
m m m
Unit impulse
noise rejection
response.
x1
1.8
4.2
6.9
10.0
2.4
5.2
8.4
11.6
bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; bf 3 = 1 ; a f 5 = 0 .
Coefficient values (deduced from graphs):
x2
x3
x4
x1
x2
x3
0.17
0.22
0.25
0.26
0.07
0.11
0.14
0.16
(
Kc =
)
1.08
0.98
0.95
0.94
0.82
0.86
0.93
1.00
13.5
16.8
21.0
24.0
15.0
18.6
23.1
27.2
0.25
0.27
0.26
0.27
0.19
0.20
0.20
0.20
-0.06
-0.05
-0.04
-0.03
-0.09
-0.09
-0.07
-0.06
0.1227 + 1.4550 ln (1 + K ) − 1.2711[ln (1 + K )]2
ln (1 + K )
Tm , b f 4 =
.
4 tanh −1 K
2 tanh −1 K
2(K + 0.9946 )
,
K + 0.0682
0.0237(K + 34.5338)
0.0145K 3 + 0.5773K 2 + 2.6554K + 0.3488
Tm , b f 4 =
.
K + 4.1530
K 3 + 5.2158K 2 + 4.4816K + 0.2817
Equations developed from information provided by Atherton and Majhi (1998).
3

 (2Tm − τm )2 x 4 Tm − 2(Tm − τm ) 
1  (2Tm − τm )
,
K
1
x3  .
b f 4 = 0. 5 
=
−


0
2
3
2
K m 
2Tm τm

 1 − (2Tm − τm ) x 3 2Tm τm 
(
(
x4
0.89
0.91
0.88
0.89
1.10
1.14
1.14
1.13
2
,
ln (1 + K )
1
1  0.8011(K + 0.9946 ) 0.7497 K + 0.9946 
−

 , K0 =
K
K m 
K + 0.0682
K + 0.0682

Ti =
4
-0.31
-0.16
-0.11
-0.08
-0.22
-0.16
-0.12
-0.10
0.8011 1 − 0.9358 ln (1 + K )
1
, K0 =
K m ln (1 + K )
Km
Ti =
3
τm
< 0.5
Tm
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
Unit step setpoint
response.
Kc =
τm
<1
Tm
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
Direct synthesis: time domain criteria
(2Tm − τm ) x1 τmTm
4
x2
0
2Tm − τm
2K τ T 2
Atherton and
Majhi (1998).
Model: Method 1
τm
< 0.693
Tm
0.693 <
3
2
FA
)
(
))
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Ti
Td
477
Comment
Atherton and Boz
Parameters as specified by Atherton and Majhi (1998), with the
(1998), Majhi following coefficient values (deduced from graphs). Model: Method 1.
and Atherton
τm Tm < 0.8 .
(1999).
x1
x2
x3
x4
x1
x2
x3
x4
Minimum ISE –
servo – Atherton
and Boz (1998).
1.2
0.58
-0.81
2.36
8.5
1.02
-0.16
2.18
2.6
0.91
-0.55
2.49
10.8
1.03
-0.14
2.16
4.5
0.89
-0.30
2.18
13.3
1.02
-0.12
2.13
6.4
0.98
-0.22
2.23
16.0
1.00
-0.09
2.10
Minimum ISTE – 1.8
0.17
-0.31
1.08
13.0
0.28
-0.07
0.94
servo – Atherton 4.2
0.22
-0.16
0.96
16.2
0.31
-0.06
0.99
and Boz (1998).
6.9
0.25
-0.11
0.95
20.3
0.29
-0.04
0.93
10.0
0.26
-0.08
0.92
24.0
0.30
-0.04
0.95
Minimum ITSE – 1.8
0.17
-0.29
1.05
13.0
0.28
-0.06
0.94
servo – Majhi
4.2
0.22
-0.16
0.96
16.8
0.27
-0.05
0.90
and Atherton
6.9
0.25
-0.11
0.95
20.3
0.29
-0.04
0.97
(1999).
10.0
0.26
-0.08
0.92
Minimum ISTSE 2.3
0.08
-0.16
0.71
16.5
0.14
-0.04
0.64
– servo –
5.2
0.11
-0.10
0.65
20.4
0.15
-0.03
0.65
Atherton and Boz 8.4
0.14
-0.07
0.68
25.2
0.15
-0.02
0.64
(1998).
12.4
0.13
-0.04
0.63
31.2
0.14
-0.02
0.61
Minimum ISTSE 2.3
0.08
-0.16
0.69
16.5
0.14
-0.03
0.64
– servo – Majhi
5.2
0.11
-0.10
0.65
20.4
0.15
-0.03
0.67
and Atherton
8.7
0.12
-0.06
0.62
25.2
0.15
-0.02
0.65
(1999).
12.4
0.13
-0.04
0.64
Sree and
0
10Tm K m TS
0.4ξ 2TS
Chidambaram
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = τm , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
(2006),
pp. 24–26.
b f 2 = τm , a f 3 = 0 , a f 4 = 0 ; K 0 = 1 K m ; b f 3 = 1 ; b f 4 = τm ;
Model: Method 1
a =0.
f5
Direct synthesis: frequency domain criteria
0
Ti
Arvanitis et al.
5
Kc
(2000b).
Model: Method 1 α = 0 ; χ = 0 ; δ = 0 ; b f 1 = 0.5τm , a f 1 = Ti , a f 2 = 0 ; ε = 0 ; φ = 0 ;
b f 2 = 0.5τm , a f 3 = 0 , a f 4 = 0 ; K 0 = 0 .
5
Kc =
Tm [2φm + π(A m − 1)]
(
)
2K m τm A m 2 − 1
,
Ti =
2πTm
.
 2φm A m + πA m (A m − 1)  
2φm A m + πA m (A m − 1) 
−
π
π
−
3Tm 
2



2τ m A m 2 − 1
Am2 − 1

 

(
)
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478
Rule
Padhy and Majhi
(2006a).
Model: Method 6
6
Kc
Ti
Td
Kc
x1
1 + (x1x 2 τm )
0
Comment
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = (2Tm τm )
0.5
K m ; bf 3 = 1 ;
b f 4 = 0. 5 τ m ; a f 5 = 0 .
Robust
7
Park et al.
(1998).
Model: Method 1
Ti
Kc
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ;
ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ;
K 0 = (Tm τm )
0.5
6
Kc =
Td
) [T
2φm + π(A m − 1)
(
2K m τm A m 2 − 1
m
]
K m ; bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 .
− 0.7071 Tm τm , x1 =
Tm − 0.7071 Tm τm
1.4142 Tm τm − 1
,
 2φ + π(A m − 1)   2A m  2φm + π(A m − 1) 
x 2 = 0.7854A m  m
 1 −

 .
2
π  2 A m 2 − 1 
 2 A m − 1  
(
7
)
(
)
Apply minimum ITAE – regulator tuning rule or minimum ITAE – servo tuning rule defined by
Sung et al. (1996) for SOSPD model, ideal PID controller (Table 21). For these formulae, K m ,
Tm1 and ξ m are replaced by the following parameters from the unstable FOLPD model:
Km
Tm τm − 1
0.71Tm
[
0.71 Tm
(Tm
0.5
0.25
0.75
τm )0.5 − 1
− τm
(Tm
τm
(Unstable FOLPD) → K m (SOSPD)
0.5
]T
0.25
m
τm )0.5 − 1
(Unstable FOLPD) → Tm1 (SOSPD)
τm
−0.75
(Unstable FOLPD) → ξ m (SOSPD)
b720_Chapter-04
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Ti
479
Comment
Td
8
Ti
Td
Kc
Lee and Edgar
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ;
(2002).
Model: Method 1 ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; b f 3 = 1 ; b f 4 = 0 ;
af 5 = 0 .
9
10
Kc
Ti
x1
Kc
Ti
x1
K 0 = 0.25K u
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; b f 1 = 0 , a f 1 = Td , a f 2 = 0 ;
ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = Td , a f 4 = 0 ; b f 3 = 1 ; b f 4 = 0 ;
af 5 = 0 .
8
Kc =
2
2

K 0 K m − 1  Tm − K 0 K m τm
τm
T − K 0 K m τm
τm
,
+
+

 , Ti = m
K m (λ + τm )  K 0 K m − 1
2(λ + τm ) 
K 0K m − 1
2(λ + τm )
Td =
4
2
3
(Tm − K 0K m τm )τm 2  ,
K 0 K m − 1  K 0 K m τm
τm
τm
−
+
+


K m (λ + τm )  2(K 0 K m − 1) 6(λ + τm ) 4(λ + τm )2 2(K 0 K m − 1)(λ + τm ) 
K0 =
9
Kc =
1
Km

cos



Tm 
τ 
τ τ
1 − m  sin
1 − m  m +
 T 
 T T
τ
m  m
m 
m 

(K 0K m − 1)Ti ,
K m (λ + τm )

τ τ
1 − m  m
 T T
m  m


.


2
Ti =
Tm − K 0 K m τm
τm
+
+ x1 ,
K 0K m − 1
2(λ + τm )
 K 0K m
τm
τm 2
(Tm − K 0 K m τm ) 
x1 = τ m 
−
+
+

2
2(K 0 K m − 1)(λ + τm ) 
 2(K 0 K m − 1) 6(λ + τm ) 4(λ + τm )
K0 =
10
Kc =
1
Km

cos



Tm 
τ 
τ τ
1 − m  sin
1 − m  m +
 T 
 T T
τ
m  m
m 
m 

(1 + 0.25K u K m )Ti
K m (λ + τ m )

τ τ
1 − m  m
 T T
m  m


.


2
, Ti =
Tm − 0.25K u K m τm
τm
+
+ x1 ,
1 + 0.25K u K m
2(λ + τm )
2
 0.25K u K m
(Tm − 0.25K u K m τ m )τ m 
τm
τm 2
−
+
+
x1 = τ m 

2
2(1 + 0.25K u K m )(λ + τ m ) 
 2(1 + 0.25K u K m 6(λ + τ m ) 4(λ + τ m )
0.5
.
0.5
;
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480
4.11 Unstable FOLPD Model with a Zero
K (1 + sTm 3 )e −sτ m
K (1 − sTm 4 )e −sτ m
G m (s) = m
or G m (s) = m
Tm1s − 1
Tm1s − 1

1 

4.11.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
E(s)
R(s)
+
−

1 

K c 1 +
Ti s 

U(s)
Model
Y(s)
Table 99: PI controller tuning rules – unstable FOLPD model with a zero
K (1 − sTm 4 )e −sτ m
G m (s ) = m
Tm1s − 1
Rule
Sree and
Chidambaram
(2003c).
1
Kc
1
Ti
Comment
Direct synthesis: time domain criteria
x1Tm1
Model: Method 1
Ti
K mTm 4
x1 = the negative of the value of the ‘inverse jump’ of the closed loop system;
x1Tm1
[Tm 4 (1 − x 2 ) − 0.5τm (1 + x 2 )]
Tm 4
T
.
x1 ≥
; Ti = m 4
x1Tm1
Tm1
(1 − x 2 ) + x 2
Tm 4
x2 >
x1Tm1
(Sree and Chidambaram, 2003c);
x1Tm1 − Tm 4
Tm 4 + 0.5τm
x1Tm1
< x2 <
(Sree and Chidambaram, 2006, p. 106).
Tm 4 − 0.5τm
x1Tm1 − Tm 4
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
481
4.11.2 Ideal controller in series with a first order lag
 1

1
G c (s) = K c 1 +
+ Td s 
 Tf s + 1
 Ti s
E(s)
R(s)
+
U(s)

 1
1
K c 1 +
+ Td s 
 Ti s
 Tf s + 1
−
Y(s)
Model
Table 100: PID controller tuning rules – unstable FOLPD model with a zero
K (1 + sTm3 )e −sτ m
K (1 − sTm 4 )e −sτ m
G m (s ) = m
or G m (s) = m
Tm1s − 1
Tm1s − 1
Rule
Kc
Ti
Comment
Td
Direct synthesis: time domain criteria
Sree and
Chidambaram
(2003b).
Sree and
Chidambaram
(2006).
1
2
Kc
Ti
0
Model: Method 1
Kc
Ti
0
Model:
Method 1;
pp. 128–130.
(1 − sTm 4 )(1 + Tis )e−sτ
Desired closed loop transfer function =
Kc = −
Ti =
2
1
(
(1)
1 + TCL
2s
(
(
− Tm 4 τm +
Tm12
[
(1)
TCL
2
)
(
( 2)
+ TCL
2
] ),
+ Tm 4 + τm Tm1
+ Tm 4 τm − (τm + Tm 4 )Tm1
)
;
)
τm
τ
− m < 0.62 .
Tm1 Tm 4
(1 + sTm3 )(1 + Tis )e−sτ
Desired closed loop transfer function =
Ti =
m
Ti
Tm 4 τm Ti
, Tf =
,
(1)
( 2)
(1)
( 2)
K m TCL
+
T
+
T
+
τ
−
T
T
T
+
T
2
CL 2
m4
m
i
m1 CL 2
CL 2 + Tm 4 + τ m − Ti
(1)
( 2)
Tm1 TCL
2TCL 2
Kc = −
)(
( 2)
1 + TCL
2s
(
(1)
1 + TCL
2s
)(
( 2)
1 + TCL
2s
m
)
.
Ti
Tm3τm Ti
, Tf = −
,
(1)
( 2)
(1)
( 2)
K m TCL
Tm1 TCL
2 + TCL 2 − Tm 3 + τ m − Ti
2 + TCL 2 − Tm 3 + τ m − Ti
(
(
(1)
( 2)
Tm1 TCL
2TCL 2
+ Tm3τm +
[
(1)
TCL
2
)
+
( 2)
TCL
2
− Tm 3
Tm12 − Tm3τm − (τm − Tm 3 )Tm1
(
+ τ ]T )
.
m
m1
)
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Rule
Sree and
Chidambaram
(2006).
Sree and
Chidambaram
(2006).
Kc
Ti
Td
3
Kc
Ti
Td
4
Kc
Ti
0
Kc
Ti
Kc
Ti
Comment
Model:
Method 1;
pp. 128–130.
Model:
Method 1; pp.
137–138, 143.
Robust
Sree and
Chidambaram
(2006).
5
6
Tf = Tm3
Td
Model: Method 1; λ not specified; pp. 163–165.
3
(1 + sTm3 )(1 + Tis )e−sτ
Desired closed loop transfer function =
Kc = −
Ti =
(
(1)
1 + TCL
2s
Ti
;
(1)
( 2)
K m TCL
+
T
−
Tm 3 + 1.5τm − Ti
2
CL 2
(
(1)
( 2)
0.5τmTCL
2TCL 2
+
)
(
(1)
0.5τm Tm1 TCL
2
m
.
)
)
(1)
( 2)
+ Tm3 + TCL
2TCL 2 Tm1 + x1
+ 0.5τ m ,
(Tm − 0.5τm )(Tm3 + Tm1 )
+
( 2)
TCL
2
)(
( 2)
1 + TCL
2s
2
Td =
4
0.5τm (Ti − 0.5τm )
, Tf = −
Ti
(
(1)
( 2)
0.5τ m TCL
2 TCL 2
(1)
( 2)
Tm1 TCL 2 + TCL 2
(
Desired closed loop transfer function =
(
).
)
+ Tm3 [Ti − 0.5τ m ]
− Tm3 + 1.5τ m − Ti
(1 − sTm 4 )(1 + Tis )e−sτ
(1 + TCL3s )3
m
;
3
Kc = −
Ti =
Tm 4 τm Ti − TCL 3
Ti
,
, Tf =
K m (3TCL3 + Tm 4 + τm − Ti )
Tm1 (3TCL3 + Tm 4 + τm − Ti )
(
)
TCL33 + Tm1 3TCL 32 − τm Tm 4 + Tm12 (3TCL3 + Tm 4 + τm )
Tm12 + Tm 4 τm − (τm + Tm 4 )Tm1
,
τm
τ
− m < 0.62 .
Tm1 Tm 4
5
Kc = −
λ2 + 2λTm1 + 0.5τm Tm1
Ti
0.5(Ti − 0.5τm )τm
, Td =
, Ti = 0.5τm +
K m (2λ + τm − Ti )
Tm1 − 0.5τm
Ti
6
Kc = −
Ti
0.5τm Tm 4 (Ti − 0.5τm ) − λ3
,
, Tf =
Tm1 (3λ + Tm 4 + τm − Ti )
K m (3λ + Tm 4 + τ m − Ti )
Ti = 0.5τm +
)
(1)
( 2)
x 1 = Tm1 TCL
2 + TCL 2 − Tm 3 + τ m ;
(
)
λ3 + (3λ + Tm 4 + 0.5τm )Tm12 + 3λ2 − 0.5Tm 4 τm Tm1
0.5τmTm 4 + Tm12 − Tm1 (Tm 4 + 0.5τm )
.
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
483
4.11.3 Generalised classical controller


Td s
1
+
G c (s) = K c 1 +
T

Ti s
1+ d

N

R(s) E(s)
+
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N



2
 b f 0 + b f 1s + b f 2 s
 1 + a f 1s + a f 2 s 2
s 





U(s)


2
 b f 0 + b f 1s + b f 2 s
 1 + a f 1s + a f 2 s 2
s

K m (1 − sTm 4 )e − sτ
Tm1s − 1
m
Y(s)
Table 101: PID controller tuning rules – unstable FOLPD model with a positive zero
K (1 − sTm 4 )e −sτ m
G m (s) = m
Tm1s − 1
Rule
Kc
Ti
Comment
Td
Direct synthesis: time domain criteria
Sree and
Chidambaram
(2006).
1
1
Ti
Kc
Desired closed loop transfer function =
Kc = −
Ti
,
K m (3TCL3 + Tm 4 + 1.5τm − Ti )
3
Ti =
0.5τmTCL3 +
2
Tm1
(3T
2
CL 3
Td
Model:
Method 1;
pp. 143–145.
(1 − sTm 4 )(1 + Tis )e−sτ .
(TCL3s + 1)3
0.5τm (Ti − 0.5τm )
T =
,
m
d
Ti
)
+ 1.5τm TCL 3 − 0.5τm Tm 4 + x1
(Tm1 − 0.5τm )(Tm 4 + Tm1 )
+ 0.5τ m ,
x1 = Tm1TCL 3 (TCL3 + 1.5τm ) + Tm1 (3TCL3 + Tm 4 + τm ) ;
2
3
3
af 2 = −
0.5τm TCL3
, bf 0 = 1 , bf 1 = 0 , bf 2 = 0 , N = ∞ ,
(
Tm1 3TCL 3 + Tm 4 + 1.5τm − Ti )
3TCL3 + 1.5τm Tm1 + Tm 4 (Ti − 0.5τm ) + 0.5τm (Ti − 0.5τm − Tm 4 )
.
3TCL3 + Tm 4 + 1.5τm − Ti
2
a f 1 = Tm1 +
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4.11.4 Two degree of freedom controller 1


Td s
1
+
U (s ) = K c  1 +
 Ti s 1 + Td

N






βTd s 
 E (s ) − K c  α +
 R (s )
Td 


s
1+ s 

N 






T
s
 α + β d K c
T 

1+ d s

N 

R(s)
E(s)
+
FA
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N





s

−
Y(s)
U(s)
Model
+
Table 102: PID controller tuning rules – unstable FOLPD model with a zero
G m (s) =
Rule
K m (1 + sTm3 )e −sτ m
K (1 − sTm 4 )e −sτ m
or m
Tm1s − 1
Tm1s − 1
Kc
Ti
Comment
Td
Direct synthesis: time domain criteria
Sree and
Chidambaram
(2003b).
1
1
(1 − sTm 4 )(1 + Tis )e−sτ
Desired closed loop transfer function =
Kc = −
Ti =
0
Ti
Kc
(
(1)
1 + TCL
2s
)(
( 2)
1 + TCL
2s
m
)
Model: Method 1
;
τm
τ
− m < 0.62 .
Tm1 Tm 4
(1)
( 2) 
 Tm 4 + TCL
Ti
2 + TCL 2 

α
=
−
1
0
.
5
,
,
(1)
( 2)


Ti
K m TCL
2 + TCL 2 + Tm 4 + τ m − Ti


(
(
(1)
( 2)
Tm1 TCL
2TCL 2
− Tm 4 τm +
[
(1)
TCL
2
)
( 2)
+ TCL
2
] ).
+ Tm 4 + τm Tm1
Tm12 + Tm 4 τm − (τm + Tm 4 )Tm1
b720_Chapter-04
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FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Sree and
Chidambaram
(2003c).
2
Sree and
Chidambaram
(2006).
Sree and
Chidambaram
(2006).
2
Kc
Ti
Td
Comment
x1Tm1
K mTm 4
Ti
0
Model:
Method 1;
x1 ≥ Tm 4 Tm1 .
3
Kc
Ti
0
4
Kc
Ti
0
Model:
Method 1;
p. 130.
Model:
Method 1;
p. 138.
x1 = the negative of the value of the ‘inverse jump’ of the closed loop system;
Ti =
Tm 4 )[Tm 4 (1 − x 2 ) − 0.5τm (1 + x 2 )]
x1Tm1
;
, x2 >
(x1Tm1 Tm 4 )(1 − x 2 ) + x 2
x1Tm1 − Tm 4
(x1Tm1
Desired closed loop transfer function =
(1 − sTm 4 )(1 + Tis )e−sτ
(
(1)
1 + TCL
2s
)(
( 2)
1 + TCL
2s
m
)
(Sree and Chidambaram,
(1)
( 2) 
 T + TCL
2 + TCL 2 
2003b); α = 1 − 0.5 m 4
.
Ti


3
Kc = −
Ti =
4
(1 + sTm3 )(1 + Tis )e−sτ
Desired closed loop transfer function =
(
(
+ Tm3τm +
Tm12
)
[
(1)
TCL
2
( 2)
+ TCL
2
m
)
.
] ).
− Tm 3 + τm Tm1
− Tm3τm − (τm − Tm 3 )Tm1
Desired closed loop transfer function =
Ti =
)(
( 2)
1 + TCL
2s
(1)
( 2)
TCL
Ti
2 TCL 2 + Tm 3
α
=
1
−
,
(1)
( 2)
Ti
K m TCL
2 + TCL 2 − Tm 3 + τ m − Ti
(1)
( 2)
Tm1 TCL
2TCL 2
Kc = −
(
(1)
1 + TCL
2s
(1 − sTm 4 )(1 + Tis )e−sτ
(1 + TCL3s )3
m
.
Ti
T
, α = 1 − CL3 (pp. 138, 143),
K m (3TCL3 + Tm 4 + τm − Ti )
Ti
(
)
TCL33 + Tm1 3TCL 32 − τm Tm 4 + Tm12 (3TCL3 + Tm 4 + τm )
Tm12 + Tm 4 τm − (τm + Tm 4 )Tm1
.
485
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486
4.12 Unstable SOSPD Model (one unstable pole)
G m (s ) =
K m e − sτm
(Tm1s − 1)(1 + sTm 2 )

1 

4.12.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
E(s)
R(s)
+

1 

K c 1 +
Ti s 

−
U(s)
K m e −sτ m
(Tm1s − 1)(1 + Tm 2 s )
Y(s)
Table 103: PI controller tuning rules – unstable SOSPD model (one unstable pole)
K m e − sτ m
G m (s) =
(Tm1s − 1)(1 + sTm 2 )
Rule
Kc
Ti
Comment
Minimum performance index: regulator tuning
Minimum ITAE –
4Tm1 (τ m + Tm 2 ) Coefficients of K c , Ti
1
Poulin and Pomerleau
Kc
x 1 Tm1 − 4(τ m + Tm 2 ) deduced from graph.
(1996).
τ m Tm
x1
x2
τ m Tm
x1
x2
Model: Method 1
0.05
0.9479
2.3546
0.30
1.6163
2.6612
0.10
1.0799
2.4111
0.35
1.7650
2.7368
0.15
1.2013
2.4646
0.40
1.9139
2.8161
Process output step
0.20
1.3485
2.5318
0.45
2.0658
2.9004
load disturbance.
0.25
1.4905
2.5992
0.50
2.2080
2.9826
0.05
1.1075
2.4230
0.25
1.6943
2.7007
Process input step
0.10
1.2013
2.4646
0.35
1.8161
2.7637
load disturbance.
0.15
1.3132
2.5154
0.40
1.9658
2.8445
0.20
1.4384
2.5742
0.45
2.1022
2.9210
0.25
1.5698
2.6381
0.50
2.2379
3.0003
x 2Tm1 1 +
1
Kc =
x1Tm 2
2
4(τm + Tm 2 )2
K m (x1Tm1 − 4[τm + Tm 2 ])
.
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Poulin et al. (1996).
Model: Method 1
Kc
487
Comment
Ti
Direct synthesis: frequency domain criteria
Tm 2 + τ m < 0.16Tm1 ;
3.53
Tm1
Km
250 ≤ φ ≤ 900 .
m
Ultimate cycle
McMillan (1984).
Model: Method 1
2
Kc
Tuning rules
developed from
K u , Tu .
Ti
2
2
Kc =
1.477 Tm1Tm 2
Km
τm 2






1
,

0.65 
  (Tm1 + Tm 2 )Tm1Tm 2  
 
1 + 
  (Tm1 − Tm 2 )(Tm1 − τm )τm  
0.65
  (T + T )T T
 

m1
m 2 m1 m 2
Ti = 3.33τm 1 + 
 .
  (Tm1 − Tm 2 )(Tm1 − τm )τm  
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488


1
+ Td s 
4.12.2 Ideal PID controller G c (s) = K c 1 +
Ti s


E(s)
R(s)
−
+
U(s)


1
K c 1 +
+ Td s 
Ti s


K m e − sτ m
(Tm1s − 1)(1 + sTm 2 )
Y(s)
Table 104: PID controller tuning rules – unstable SOSPD model (one unstable pole)
K m e −sτ m
G m (s) =
(Tm1s − 1)(1 + sTm 2 )
Rule
Kc
Wang and Jin
(2004).
1
Kc
2
Kc
Ti
Comment
Td
Direct synthesis: time domain criteria
Model: Method 1
Ti
Td
Robust
Rotstein and
Lewin (1991).
Model:
Method 1;
Tm 2 = 0.5τm ;
Ti
Td
λ determined graphically – sample values provided:
τm Tm1 = 0.2 , 0.5Tm1 ≤ λ ≤ 1.9Tm1
K m uncertainty
= 50%.
τm Tm1 = 0.4 , 1.3Tm1 ≤ λ ≤ 1.9Tm1
Tm 2 >> τm
(Marchetti et al.,
2001).
τm Tm1 = 0.2 , 0.4Tm1 ≤ λ ≤ 4.3Tm1
K m uncertainty
= 30%.
τm Tm1 = 0.4 , 1.1Tm1 ≤ λ ≤ 4.3Tm1
τm Tm1 = 0.6 , 2.2Tm1 ≤ λ ≤ 4.3Tm1
1
Kc =
K m Ti (Tm1 − τm )(Tm 2 + τm )
(TCL3 + τm )3
Ti =
TCL3 + 3τm TCL3 + 3(Tm1Tm 2 + τm Tm1 − τm Tm 2 )TCL 3 + τm Tm 2 (Tm1 − τm )
,
(Tm1 − τm )(Tm 2 + τm )
Td =
(Tm 2 + τm − Tm1 )TCL33 + 3Tm1Tm 2TCL3 (TCL3 + τm ) + Tm1Tm 2τm 2
Ti (Tm1 − τm )(Tm 2 + τm )
3
2
,
2
.
  λ



 λ
Tm1 λ
+ 2  + Tm 2 
λ
+ 2 Tm 2
T


λ
  Tm1


 m1

 , T = λ

Kc =
.
i
 T + 2  + Tm 2 , Td =  λ

λ2 K m
 m1

λ
+ 2  + Tm 2
 Tm1

b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Lee et al. (2000).
Shamsuzzoha et
al. (2005),
Shamsuzzoha
and Lee (2007a).
3
4
Kc
Ti
Td
Comment
Kc
Ti
Td
Model: Method 1
Kc
Ti
Td
Model:
Method 1;
λ values not
defined.
0.25Ti
Tuning rules
developed from
K u , Tu .
489
Ultimate cycle
McMillan
(1984).
Model: Method 1
3
Kc =
5
Ti
Kc
2
Ti
λ2 + x1τm − 0.5τm
,
, Ti = −Tm1 + Tm 2 + x1 −
− K m (2λ + τm − x1 )
2λ + τm − x1
τm (0.167 τm − 0.5x1 )
2λ + τm − x1
2
− Tm1x1 + Tm 2 x1 − Tm1Tm 2 −
Td =
4
Ti
2
−
λ2 + x1τm − 0.5τm
;
2λ + τm − x1
2
 λ

λ = desired closed loop dynamic parameter, x1 = Tm1 
+ 1 e τ m
 Tm1 

Ti
Kc = −
, Ti = Tm 2 − Tm1 + 2 x1 − x 2 ,
K m (4λ + τm − 2 x1 )

− 1 .


0.167 τm 3 − x1τm 2 + x12 τm + 4λ3
4λ + τm − 2 x1
− x2 ,
Ti
x1 − 2x1 (Tm1 − Tm 2 ) − Tm1Tm 2 −
2
Td =
Tm1


x1 = Tm1 


4 τm
2
2
 Tm
 λ
6λ2 − x1 + 2x1τm − 0.5τm
 e − 1 , x 2 =

.
1
+

T

4 λ + τ m − 2 x1
 m1 

2
5
Kc =
1.111 Tm1Tm 2
K m τm 2






1
,

0.65 
  (Tm1 + Tm 2 )Tm1Tm 2  
 
1 + 
  (Tm1 − Tm 2 )(Tm1 − τm )τm  
0.65
  (T + T )T T
 

m1
m 2 m1 m 2
Ti = 2τm 1 + 
 .
  (Tm1 − Tm 2 )(Tm1 − τm )τm  
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490
4.12.3 Ideal controller in series with a first order lag
 1

1
G c (s) = K c 1 +
+ Td s 
 Tf s + 1
 Ti s
E(s)
R(s)
−
+


1
K c 1 +
+ Td s 
Ti s


U(s)
1
(Tf s + 1)
Y(s)
K m e − sτ m
(Tm1s − 1)(1 + sTm 2 )
Table 105: PID controller tuning rules – unstable SOSPD model (one unstable pole)
K m e − sτ m
G m (s) =
(Tm1s − 1)(1 + sTm 2 )
Rule
Kc
Ti
Kc
Ti
Td
Comment
Td
τ m < Tm1
Robust
Zhang and Xu
(2002).
Model: Method 1
1
Kc =
1
For τm Tm1 ≤ 0.5 , λ = 2 − 5(τ m + Tm 2 ) .
Tm 2Tm1 (Tm1 − τm ) + λ3 + 3λ2Tm1 + 3λTm12 + τmTm12
(
K m λ3 + 3λ2Tm1 + 3λTm1τm + τm 2Tm1
2
Ti = Tm 2 +
Td =
Tf =
)
2
λ3 + 3λ2Tm1 + 3λTm1 + τmTm1
,
Tm1 (Tm1 − τm )
Tm 2 (λ3 + 3λ2Tm1 + 3λTm12 + τm Tm12 )
Tm1Tm 2 (Tm1 − τ m ) + λ3 + 3λ2Tm1 + 3λTm12 + τmTm12
(
,
λ3Tm1 (Tm1 − τm )
Tm1 λ + 3λ2Tm1 + 3λTm1τm + τm 2Tm1
3
).
,
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
491



1  1 + Td s 


4.12.4 Classical controller G c (s) = K c 1 +
 Ti s  1 + Td s 
N 

R(s)
E(s)
−
+



 U(s)
Y(s)

1  1 + Td s 
K m e − sτ m

K c 1 +

T


T
s
i  1+ d s

(Tm1s − 1)(1 + sTm 2 )


N 

Table 106: PID controller tuning rules – unstable SOSPD model (one unstable pole)
K m e − sτ m
G m (s) =
(Tm1s − 1)(1 + sTm 2 )
Rule
Kc
1
Minimum ITAE
Kc
– Poulin and
Pomerleau
(1996), (1997). τ m + Tm 2
Tm1
Process output
step load
disturbance.
Process input
step load
disturbance.
x 2Tm1 1 +
1
Kc =
Ti
Minimum performance index: regulator tuning
Model: Method 1
Ti
Tm 2
0.05
0.10
0.15
0.20
0.25
0.05
0.10
0.15
0.20
0.25
x1Tm 2
Comment
Td
Coefficient values (deduced from graph):
τ m + Tm 2
x1
x2
x1
Tm1
0.9479
1.0799
1.2013
1.3485
1.4905
1.1075
1.2013
1.3132
1.4384
1.5698
2.3546
2.4111
2.4646
2.5318
2.5992
2.4230
2.4646
2.5154
2.5742
2.6381
0.30
0.35
0.40
0.45
0.50
0.30
0.35
0.40
0.45
0.50
1.6163
1.7650
1.9139
2.0658
2.2080
1.6943
1.8161
1.9658
2.1022
2.2379
x2
2.6612
2.7368
2.8161
2.9004
2.9826
2.7007
2.7637
2.8445
2.9210
3.0003
2
4(τm + Tm 2 )2
K m (x1Tm1 − 4[τm + Tm 2 ])
, Ti =
4Tm1 (τm + Tm 2 )
;
x1Tm1 − 4(τm + Tm 2 )
0≤
T
τm
≤ 2 ; 0.1Tm 2 ≤ d ≤ 0.33Tm 2 .
N)
N
(Td
b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
492
Rule
Kc
Huang and Chen
(1997), (1999).
Model: Method 4
2
Ti
Comment
Td
Direct synthesis: time domain criteria
Ti
Tm 2
Kc
‘Good’ servo and regulator response; N = ∞ .
Direct synthesis: frequency domain criteria
Chidambaram
and Kalyan
(1996).
Ho and Xu
(1998).
Model:
Method 1;
N =∞.
3
ωpTm1 A m K m
Kc =
4
Ti
Tm 2
Ti
Tm 2
N =∞;
Model: Method 1
Representative result:
−Tm1
Tm 2
0.7854Tm1
K m τm
Paraskevopoulos
et al. (2006). Ti x1
2
Kc
Am = 2 ,
φ m = 450 .
N =∞;
Model: Method 1
2
1 + x1
5
1 + Ti 2 x12
Tm 2
Ti
1.13076 


T 
τ
0.5 
 , Ti = K c K m Tm  2 x1 + τm  , 0 < m < 0.8 ;
1 + 1.11416 m 


T
K
K
1
T
T
−
Km 
τ

m1
m
c
m
m
m






Recommended x 1 = 0.160367Tm e
3
6.06426
τm
Tm 1
.
K c and Ti are defined by De Paor and O’Malley (1989), Rotstein and Lewin (1991) or
Chidambaram (1995b), for the ideal PI control of an unstable FOLPD model [see Table 91].
1
4
.
Ti =
2
1.57ωp − ωp τm − (1 Tm1 )
5
τm Tm1 < 0.9 ; Tm 2 Tm1 > 0.3 .
Ti (1 − (τm Tm1 )) + Ti − (τm Tm1 ) + x 2
2
x1 =
2Ti 2 (τm Tm1 )
{T (1 − (τ T )) + T − (τ
T = x (1 + (T τ )[0.632(τ T
i
x3 =
i
3
m
m1
m
m1
i
m
}
2
2
x2 =
,
m
m1
Tm1 ) + 4(τm Tm1 )Ti 2 (1 + Ti − (τm Tm1 )) ;
) + 0.436
0.0029 − 0.0682 τm Tm1 + 1.4941(τm Tm1 )
(1.003 − (τm Tm1 ))2
max
φm > 0.2φmax
= −(τm Tm1 ) (Tm1
m , φm
])1 −(φ(φ φ φ ) ) ,
τm Tm1 − 0.0153
m
max
m
max
m
m
,
τm ) − 1 + tan −1
( (T
m1
)
τm ) − 1 .
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Paraskevopoulos
et al. (2006).
Model: Method 1
6
Kc
Ti
Td
Comment
Kc
Ti
Tm 2
N=∞
6
A m > 0.2(A max
m − 1) +1;
493
τm
T
< 0. 9 ; m 2 > 0. 3 .
Tm1
Tm1
K c = K max A m or K min A m ,
K min = Ti ωmin
1 + ωmin 2
1 + Ti 2ωmin 2
K max = Ti ωmax
0.03[τm Tm1 ]
1.14 − [τm Tm1 ]
= 1+


0.05
 0.973 +
Ti − x1

1 − [τm Tm1 ] 

1 + ωmax 2
1 + Ti 2ωmax 2
0.006 +
ωmin
x1
,
Ti −[τm Tm1 ](1 + Ti )
,
4
2

 τ   
τ  x   1.571Tm1
ωmax = 1 + 0.22 m   1 +  0.1 − 0.3 m  1  
Tm1  Ti   τm

 Tm1    



 

Ti
Ti
− 0.9463 
− 0.5609 ,

T
+
1
τ
T
T
+
1
τ
T
(
)(
)
(
)(
)
m
m1
m
m1
 i
  i

x1 =
0.0029 − 0.0682 τm Tm1 + 1.4941(τm Tm1 )
(1.003 − [τm
Tm1 ])2
;
1+ (τ m Tm1 ) 


 max  


 A m − 1  A m − 1 


+
Ti = x 2 1 + 0.65 


 A max − 1  
1
A
1
−
−


m
  
 m



[
with A max
m
[
]
]
2
3

τ 
 τ  3(τm Tm1 )4 − 2.3(τm Tm1 )2 
τ
τ
,
0.01− 0.18 + 5 m − 32 m + 75 m  − 51 m  +
Tm1
Tm1


(1 − (τm Tm1 ))3
 Tm1 
 Tm1 


1.571Tm1
=
.

0.4085[τm Tm1 ] 

τm 1 +

 1 − 0.2864[τm Tm1 ] 
b720_Chapter-04
04-Apr-2009
Handbook of PI and PID Controller Tuning Rules
494
Rule
Kc
Ti
Paraskevopoulos
2
et al. (2006). Ti x 2 1 + x 2
2 2
1 + Ti x 2
Model: Method 1
7
0 ≤ x1 ≤ 0.55 +
(τm + Tm 2 )  0.31 −
Tm1
Comment
Td
x1 ( τm + Tm 2 )
Tm1
Ti
N=∞
>
8
7
FA


Ti
0.0098
0.15(τm + Tm 2 )
Tm1

 ; x1 < 0.15 .
1 − ([τm + Tm 2 ] Tm1 ) 
Ti (1 − ([τm + Tm 2 ] Tm1 ) + Td ) + Ti − ([τm + Tm 2 ] Tm1 ) + Td + x 3
2
x2 =
x3 =
2Ti 2 (([τm + Tm 2 ] Tm1 ) − Td )
{T
i
2
(1 − ([τm + Tm 2 ] Tm1 ) + Td ) + Ti − ([τm + Tm 2 ] Tm1 ) + Td }
2
,
+ x4 ,
x 4 = 4(([τm + Tm 2 ] Tm1 ) − Td )Ti (1 + Ti − ([τm + Tm 2 ] Tm1 ) + Td ) ;
2

 τ + Tm 2

τ + Tm 2
φm
− Td  + 0.436 m
− Td − 0.0153
 0.632 m
max
T
T
φ

m1
m1


m
Ti = x 5 1 +
[
(
τm + Tm 2 ] Tm1 ) − Td
1 − φm φmax
m


(
x5 =
0.0029 − 0.0682
)


,



([τm + Tm 2 ] Tm1 ) − Td + 1.4941(([τm + Tm 2 ] Tm1 ) − Td )
.
(1.003 − ([τm + Tm 2 ] Tm1 ) + Td )2
0.02 < τm Tm1 < 0.9 ; Tm 2 Tm1 < 0.3 . φm > 0.2φmax
m ,



 τm + Tm 2
Tm1
Tm1

− Td − 1 + tan −1 
− Td − 1  .
φmax
− Td 
m = −
 τm + Tm 2

 τ m + Tm 2
 Tm1


8
x 2 , x 3 and x 4 given in footnote 7;

 
Tm1 
T (τ + Tm 2 )
1 + 0.4 d m
− 0.22Td  x 6  ; x 5 given in footnote 7;
Ti = x 5 1 +

Tm1
 
 τm + Tm 2 
(
)

τ + Tm 2
τ + Tm 2  φm φmax
m
x 6 =  − 0.0153 − 0.436 m
+ 0.632 m
,

Tm1
Tm1  1 − φm φmax
m

 τ + Tm 2
φ max
= − m
m
 Tm1
(



Tm1
Tm1
−1 


 τ + T − 1 + tan  τ + T − 1  .
m2
m
m
2
 m


)
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
9
Ti
K max A m or
Paraskevopoulos
et al. (2006).
K min A m
Comment
Td
x3
Ti
495
τm + Tm 2
Tm1
N =∞;
Model: Method 1
A m > 0.2(A max
m − 1) +1; 0.02 < τ m Tm1 < 0.9 .
9
K min = Ti ωmin
1 + ωmin 2
2
1 + Ti ωmin
2
, K max = Ti ωmax
0.03(x1 − Td )
1.14 − x1 + Td
= 1+

0.05 
Ti − x 2
 0.973 +

1
x1 + Td 
−

1 + ωmax 2
0.006 +
ωmin
with
1 + Ti 2ωmax 2
x2
Ti −(x1 − Td )(1 + Ti )
, x1 =
τm + Tm 2
,
Tm1
2

 x   1.571
ωmax = 1 + 0.22(x1 − Td )4 1 + 0.1 − 0.3 x1 − Td  2  

 Ti   x1 − Td


 

Ti
Ti
− 0.9463 
− 0.5609 ;

 (Ti + 1)(x1 − Td )
  (Ti + 1)(x1 − Td )

[
] (
[
] 



 Am − 1

Ti = x 2 1 + 0.65 


1 −  Am



)
x 1 − Td +1 

− 1 
A max
m
 

− 1  A max
− 1 
 m
 

 + 0.01 5 x − T − 0.18
1
d



 3(x − T )4 − 2.3(x1 − Td )2 
+ 0.01 − 32(x1 − Td ) + 75(x1 − Td )2 − 51(x1 − Td )3 + 0.01 1 d
,
(1 − x1 + Td )3


[
[
]
[
x2 =
0.0029 − 0.0682 x1 − Td + 1.4941(x1 − Td )
(1.003 − x1 + Td )2
]
]
;

T
0.0098 
 ; x 3 < 0.15 ; m 2 < 0.3 .
0 ≤ x 3 ≤ 0.55 + x1  0.31 −

Tm1
1 − x1 

b720_Chapter-04
FA
Handbook of PI and PID Controller Tuning Rules
496
Rule
Kc
Paraskevopoulos
et al. (2006) –
continued.
N =∞.
10
04-Apr-2009
K min = Ti ωmin
ωmin
10
Ti
K max A m or
>
Tm 2 Tm1 < 0.3 ;
0.15(τm + Tm 2 )
τ
0.02 < m < 0.9.
Tm
T
Ti
K min A m
m1
1 + ωmin 2
2
1 + Ti ωmin
, K max = Ti ωmax
2
1 + ωmax 2
2
 (0.14 + 1.15x1 )(Td x1 )
1 +
2
1 + 2(Td x1 )

3
][ (
2
1 + Ti ωmax
0.006 + [0.03(x1 − Td ) (1.14 − x1 + Td )]
= 1+
(0.973 + 0.05 [1 − x1 + Td ])Ti − x 2
[
Comment
Td
, x1 =
τm + Tm 2
, with
Tm1
x2
T i −(x1 − Td )(1 + Ti )
0.0029 − 0.0682 x1 − Td + 1.4941(x1 − Td )
 x 2 
  , x 2 =
;
T 
(1.003 − x1 + Td )2
 i 
]
)
ωmax = 1 + 0.22(x1 − Td )4 1 + 0.1 − 0.3 x1 − Td (x 2 T1 )2 [1.571 (x1 − Td )]



Ti
Ti
− 0.9463 
− 0.5609

(
)(
)
(
)(
)
T
1
x
T
T
1
x
T
+
−
+
−
1
d
1
d
 i
 i
 ;
(
5.53 − 0.41x1 )(Td x1 )4
1+
1 − 0.65(Td x1 )3 1.27 − 0.4(Ti x 2 )2
[
] 



 Am − 1

Ti = x 3 1 + 0.65 


1 −  Am



[
[
][
]
x 1 − Td +1 

− 1 
A max
m
 

− 1  A max
− 1 
 m
 
]
 3(x − T )4 − 2.3(x1 − Td )2 
+ 0.01 1 d

(1 − x1 + Td )3



 + 0.01 5 x − T − 0.18
1
d



[
  T 2 
1 +  d  x 4 
  x1 



[
]
]
+ 0.01 − 32(x1 − Td ) + 75(x1 − Td )2 − 51(x1 − Td )3 ,
  T  0.367 + 1.78x 
1
x 3 = x 2 1 +  d 
,
  x1  1 − (Td x1 )2 


3

(A m − 1)3  ,
 2x13 − 3.32 x12 + 1.2x1 − 0.27 −
3


A max
m −1 
1
1.571
=
.
3


(x1 − Td )1 + 0.4085[x1 − Td ]  1 + (7.41 − 3.52x1 )(Td x1 ) 3
1 − (0.47 + 0.49x1 )(Td x1 )
 1 − 0.2864[x1 − Td ] 
T 
x 4 =  d 
 x1 
A max
m
3x1
(
)
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
4.12.5 Two degree of freedom controller 1


Td s
1
+
U (s ) = K c  1 +
 Ti s 1 + Td

N

E(s)
+





βTd s 
 E (s ) − K c  α +
 R (s )
Td 


s
1+ s 

N 






T
s
β
d
α +
K
Td  c

1+
s

N 

R(s)
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N





s

−
+
U(s)
Y(s)
K m e − sτ
(Tm1 s − 1)(1 + sTm 2 )
m
Table 107: PID controller tuning rules – unstable SOSPD model (one unstable pole)
K m e − sτ m
G m (s ) =
(Tm1s − 1)(1 + sTm 2 )
Rule
Kc
Huang and Lin
(1995) –
minimum IAE.
1
Ti
Td
Comment
Minimum performance index: regulator tuning
α = 0 , β =1;
1
Ti
Td
N not specified;
Kc
Model: Method 2
Note: equations continued onto p. 498. 0.05 ≤ τ m Tm1 ≤ 0.4 ; Tm 2 ≤ Tm1 ;
Kc = −
−1.164
τ 
τ
1 
T 
− 174.167 − 31.364 m + 0.4642 m 
− 103.069 m 2 
Km 
Tm1
Tm1 
 Tm1 


2
.
54
1
.
065
T 
T 
 Tm 2
T τ
1 
 − 83.916 m 2 

−
− 66.962 m 2 2m + 59.496 m1 
T 
τ
Km 
T
m
1
m




 Tm1
m1

−
497
Tm 2 τ m
Tm 2
τm

2
T
1 
− 70.79 m 2 + 23.121e Tm1 + 126.924e Tm1 + 26.944e Tm1
Km 
τm


;





1.014 



b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
498
Rule
Huang and Lin
(1995) –
continued.
2
Kc
Ti
Td
Comment
Kc
Ti
Td
α = 0 ,β =1,
N not specified;
Model: Method 2
2
2

T  
τ 
T
τ
Ti = Tm1 0.008 + 2.0718 m + 6.431 m  + 0.4556 m 2 + 0.7503 m 2  
Tm1
Tm1

 Tm1  
 Tm1 

3
3

T τ 2
τ 
T 
T τ
+ Tm1 2.4484 m 2 2m − 18.686 m  − 2.9978 m 2  − 21.135 m 2 m
 T 3

Tm1
 Tm1 
 Tm1 
 m1

4
3 




 + 39.001 τ m  + 22.848 Tm 2 τ m 
T 

 T 4 
 m1 

 m1 

τ T 2
+ Tm1 12.822 m m32
 T

 m1


 T 3τ
+ Tm1  − 4.754 m 2 4 m
 T

 m1

2
2


 − 0.527 Tm 2 τ m

 T 4
m1




 + 1.64 Tm 2
T

 m1




4
;


2
2

τ 
T  
T
τ
Td = Tm1 1.1766 m − 4.4623 m  + 0.5284 m 2 − 1.4281 m 2  
Tm1
Tm1

 Tm1 
 Tm1  

3
3
 T τ
 τ 2T
τ 
T 
+ Tm1 4.6 m 2 2m + 11.176 m  + 1.0886 m 2  − 5.0229 m 3m 2
 T

Tm1
 Tm1 
 Tm1 
 m1


τ T 2
+ Tm1  − 0.0301 − 0.5039 m m32
 T

 m1





4
3 




 − 9.8564 τ m  − 7.528 Tm 2 τ m 
T 

 T 4 
 m1 

 m1 

τ T 3
 τ 2T 2
+ Tm1 − 2.3542 m m42  + 9.3804 m m4 2

 T
 T

m1
 m1 


2






 − 0.1457 Tm 2
T

 m1




4
.


Note: equations continued onto p. 499. 0.05 ≤ τ m Tm1 ≤ 0.25 , Tm1 < Tm 2 ≤ 10Tm1 ;
Kc = −
2.1984
τ 
1 
T 
τ
1750.08 + 1637.76 m + 1533.91 m 
− 7.917 m 2 
Km 
Tm1
Tm1 
 Tm1 


−
T
1 
6.187 m 2
T
Km 
 m1

−
Tm 2 τ m
Tm 2
τm

2
T
1 
1.3729 m 2 − 1739.77e Tm1 − 0.000296e Tm1 + 2.311e Tm1
Km 
τm




0.791
− 6.451
Tm 2 τ m
Tm1 2
T 
+ 0.002452 m1 
 τm 
3.2927
 Tm 2

 Tm1

;





1.0757 



b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
2
2

T  
τ 
T
τ
Ti = Tm1 51.678 − 57.043 m + 1337.29 m  + 0.1742 m 2 − 0.1524 m 2  

Tm1
Tm1
 Tm1  
 Tm1 

3

τ 
T
T τ
+ Tm1 7.7266 m 2 2m − 6011.57 m  + 0.0213 m 2
T

Tm1
 m1 
 Tm1


T τ 2
+ Tm1  − 65.283 m 2 m
 T 3

 m1

 T 3τ
+ Tm1 0.003926 m 2 4 m
 T

 m1

2


 + 0.0645 τ m Tm 2

 T 3

 m1
3

τ 
 + 9135.52 m 

 Tm1 
3 


 + 274.851 Tm 2 τ m 

 T 4 

 m1 
2
2


 − 2.0997 Tm 2 τ m

 T 4
m1




 − 0.001077 Tm 2
T

 m1

Tm 2 τ m
τm
Tm 2

2
+ Tm1  − 49.007e Tm1 + 0.000026e Tm1 + 0.2977e Tm1





4




;


2

 τm 
Tm 2 
τm



+
Td = Tm1 − 0.0605 + 4.6998
0
.
0117
− 29.478

Tm1
Tm1 

 Tm1 



T
+ Tm1  − 0.0129 m 2

 Tm1

τ 

T τ
 + 0.6874 m 2 2m + 140.135 m 
Tm1
 Tm1 


T
+ Tm1 0.002455 m 2

 Tm1

 τ 2T

 + 1.4712 m 3m 2
 T

 m1
2
3
3



2


 − 0.1289 τ m Tm 2
 T 3

 m1





4
3 


 Tm 2 τ m 3 
 τm 

 + 0.007725 τ m Tm 2 



0
.
6884
+ Tm1 − 238.864
+

 T 4 
 T 4 

 Tm1 
 m1 
 m1 


 τ 2T 2
+ Tm1 − 0.1222 m m4 2
 T

m1




 − 0.000135 Tm 2
T

 m1




4
.


4



499
b720_Chapter-04
FA
Handbook of PI and PID Controller Tuning Rules
500
Rule
Kc
Minimum IAE –
Huang and Lin
(1995).
3
04-Apr-2009
3
Kc
Ti
Comment
Td
Minimum performance index: servo tuning
α = 0 , β =1;
Ti
Td
N not specified;
Model: Method 2
Note: equations continued onto p. 501. 0.05 ≤ τ m Tm1 ≤ 0.4 , Tm 2 ≤ Tm1 ;
1
Kc = −
Km
−1.344



Tm 2 
10.741 − 13.363 τm + 0.099 τm 
+
727
.
914
T 
Tm1
Tm1 

 m1 


1
−
Km

T
 − 708.481 m 2
T

 m1




0.995
+ 9.915
Tm 2 τ m
Tm1 2
T 
+ 84.273 m1 
 τm 
Tm 2 τ m
τm
Tm 2

2
Tm 2
1 
Tm 1
Tm 1
−
− 90.959
+ 9.034e
− 2.386e
− 16.304e Tm1

Km
τm

1.031
 Tm 2

 Tm1




2.12

τ 
T 
τ
Ti = Tm1 − 149.685 − 141.418 m − 88.717 m  − 17.29 m 2 
Tm1
Tm1 

 Tm1 



T
+ Tm1 20.518 m 2

 Tm1

0.985



− 12.82
Tm 2 τ m
Tm1 2
τ 
+ 3.611 m 
 Tm1 
0.286
 Tm 2

 Tm1
Tm 2 τ m
τm
Tm 2

2
Tm 2
Tm1
Tm1

+ Tm1 0.000805
+ 141.702e
− 2.032e
+ 10.006e Tm1

τm




1.988 




;


2

τ 
T 
τ
Td = Tm1  − 0.4144 + 15.805 m − 142.327 m  + 0.7287 m 2 
Tm1
Tm1 

 Tm1 



T
+ Tm1 0.1123 m 2

 Tm1

2
3
T
τ 

T τ
 − 18.317 m 2 2m + 486.95 m  − 10.542 m 2
Tm1
 Tm1
 Tm1 


 τ 2T
+ Tm1 204.009 m 3m 2
 T

 m1

2


 + 47.26 τ m Tm 2

 T 3

 m1

 τ 3T
+ Tm1  − 138.038 m 4m 2
 T

 m1

T
+ Tm1 19.302 m 2

 Tm1

4



 + 396.349 τ m 
T 

 m1 

4



3 
2


 2
 + 52.155 τ m Tm 2  − 646.848 τ m Tm 2
 T 4

 T 4 
m1

 m1 

5
 τ 4T
τ 

 − 4731.72 m  + 425.789 m 5m 2
 T
 Tm1 

 m1











3






0.997 



b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Minimum IAE –
Huang and Lin
(1995) –
continued.
Kc
Ti
Td
Comment
Kc
Ti
Td
α = 0 , β =1;
N not specified;
Model: Method 2
4

τ T 4
+ Tm1  − 289.476 m m52
 T
 m1


T
+ Tm1  − 3.7688 m 2

 Tm1

2

 3
 − 841.807 τ m Tm 2
5

 T
m1


5
3 

 2
 + 1313.72 τ m Tm 2 
5

 T

m1



6
 τ 5T
τ 

 + 6264.79 m  − 161.469 m 6m 2
 T
 Tm1 

 m1

τ T 5
 τ 4T 2
+ Tm1 204.689 m m62  + 25.706 m m6 2
 T

 T

m1
 m1 


τ T
+ Tm1 648.217 m m2 2
 T

 m1

4
501
3


 − 5.71 Tm 2
T

 m1








4

 2
 − 791.857 τ m Tm 2
6

 T
m1






6
.


Note: equations continued onto p. 502. 0.05 ≤ τ m Tm1 ≤ 0.25 , Tm1 < Tm 2 ≤ 10Tm1 ;
Kc = −
−0.3055
τ 
1 
T 
τ
+ 10.442 m 2 
− 130.2 + 85.914 m + 34.82 m 
Km 
Tm1
Tm1 
 Tm1 


−
T
1 
 − 22.547 m 2
T
Km 
 m1

−

T
1 
− 51.47 m 2
Km 
 τm




0.5174
− 14.698
Tm 2 τ m
Tm1 2
τm
T 
+ 52.408 m1 
 τm 
Tm 2

 + 53.378e Tm1 − 0.000001e Tm1 + 0.286e

1.0077
Tm 2 τ m
Tm 12
 Tm 2

 Tm1




 ;


2

τ  
T 
τ
Ti = Tm1 72.806 − 268.746 m − 4.9221 m 2  + 2468.19 m  
Tm1

 Tm1  
 Tm1 


T
+ Tm1 0.6724 m 2

 Tm1

2
3
T
τ 

T τ
 + 151.351 m 2 2m − 6914.46 m  − 0.0092 m 2
Tm1
 Tm1
 Tm1 


 τ 2T
+ Tm1  − 795.465 m 3m 2
 T

 m1


 τ 3T
+ Tm1 1417.65 m 4m 2
 T

 m1
2


 − 14.27 τ m Tm 2

 T 3

 m1
3


 + 0.4057 τ m Tm 2

 T 4

 m1



 + 5580.17 τ m 
T 

 m1 

4
2

 2
 + 55.536 τ m Tm 2

 T 4
m1












3



0.9879 



b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
502
Rule
Kc
Ti
Comment
Td
Direct synthesis: time domain criteria
Sree and
Chidambaram
(2006).
5
Ti
Kc

T
+ Tm1  − 0.001119 m 2

 Tm1

τm
4
Model:
Method 1;
p. 189;
β =1; N = ∞ .
Td
Tm 2

 − 44.903e Tm1 + 0.000034e Tm1 − 15.694e

19.056

τ 
 Tm 2

+ Tm1 678778 m 
T

m1 

 Tm1




Tm 2 τ m
Tm 12
7.3464 
;


1.1798

 τm 
Tm 2 
τm



−
Td = Tm1 175.515 − 86.2
0
.
008207
+ 348.727

Tm1
Tm1 

 Tm1 


0.1064
0.0355

T 
τ 
 Tm 2
T τ

+ Tm1 − 55.619 m 2 
+ 0.0418 m 2 2m + 78.959 m 
T
T

Tm1
m1 
m1 


 Tm1


T
+ Tm1 0.005048 m 2

 τm

5
Kc =




τm
Tm 2

 − 187.01e Tm1 + 0.000001e Tm1 − 0.0149e

Tm 2 τ m
Tm 12



0.0827 




.


1.13076 

T 

0.5 
 , Ti = K c K m Tm1  2 x1 + τm  ; α = 1 + τm ;
1 + 1.11416 m1 

T
K
K
1
T
K c K m Ti
Km 
τ
−

c m
m1 
 m 
 m1


Recommended x 1 = 0.160367Tm1e
6.06426
τm
T m1
; 0<
τm
< 0.8 (Huang and Chen, 1999).
Tm1
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
4.12.6 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s)




(
)
1
−
β
T
s
1 + b f 1s
[
]
1
−
χ
δ
d

,
F1 (s) = K c  [1 − α ] +
+
+


T
Ti s
1 + Ti s
1 + a f 1s + a f 2 s 2
d
1
+
s


N 





[
]
K (b + b f 4 s )
1
−
ϕ
T
s
1+ bf 2s
[
]
−
φ
1
d 
F2 (s) = K c  [1 − ε] +
+
− 0 f3
2

 1 + a f 3s + a f 4 s
T
Ti s
1+ a f 5s
1+ d s 

N 

R(s)
+
F1 (s)
U(s)
−
Y(s)
Model
F2 (s)
Table 108: PID controller tuning rules – unstable SOSPD model (one unstable pole)
K m e − sτ m
G m (s) =
(Tm1s − 1)(1 + sTm 2 )
Rule
Kc
Majhi and
Atherton (1999),
Atherton and Boz
(1998).
Model: Method 1
Minimum ISE –
servo – Atherton
and Boz (1998).
1
Ti =
x5 =
−1
x1
τm + Tm 2
−1
1
Ti
x 2x5 K m
Direct synthesis: time domain criteria
0
Ti
α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;
b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = [1 − x 5 x 3 ] K m , b f 3 = 1 ; a f 5 = 0 .
Coefficient values deduced from graphs:
x1
x2
x3
x4
x1
x2
x3
x4
1.2
2.6
4.5
6.4
− Tm1−1
0.58
0.91
0.89
0.98
τm Tm12Tm 2 2
-0.81
-0.55
-0.30
-0.22
2.36
2.49
2.18
2.23
8.5
10.8
13.3
16.0
1.02
1.03
1.02
1.00
-0.16
-0.14
-0.12
-0.09
 x x − (Tm1 − Tm 2 − τm )  τm − Tm 2
≤ 0.8 ;
, bf 4 =  6 4
,
1 − x 5x 3
Tm1


(Tm1Tm 2 + τmTm1 − τmTm 2 )3 ,
2
Comment
Td
x6 =
(Tm1Tm 2 + τ m Tm1 − τ m Tm 2 )2
τ m Tm1Tm 2
.
2.18
2.16
2.13
2.10
503
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504
Rule
Kc
Minimum ISTE –
servo – Atherton
and Boz (1998).
Ti
Comment
Td
Parameters as specified by Majhi and Atherton (1999) and Atherton
and Boz (1998).
x1
x2
x3
x4
x1
x2
x3
x4
1.8
4.2
6.9
10.0
Minimum ITSE – 1.8
servo – Majhi
4.2
and Atherton
6.9
(1999).
10.0
Minimum ISTSE 2.3
– servo –
5.2
Atherton and Boz 8.4
(1998).
12.4
Minimum ISTSE 2.3
– servo – Majhi
5.2
and Atherton
8.7
(1999).
12.4
Kwak et al.
(2000).
Model: Method 3
2
0.17
-0.31
1.08
13.0
0.28
-0.07
0.94
0.22
-0.16
0.96
16.2
0.31
-0.06
0.99
0.25
-0.11
0.95
20.3
0.29
-0.04
0.93
0.26
-0.08
0.92
24.0
0.30
-0.04
0.95
0.17
-0.29
1.05
13.0
0.28
-0.06
0.94
0.22
-0.16
0.96
16.8
0.27
-0.05
0.90
0.25
-0.11
0.95
20.3
0.29
-0.04
0.97
0.26
-0.08
0.92
0.08
-0.16
0.71
16.5
0.14
-0.04
0.64
0.11
-0.10
0.65
20.4
0.15
-0.03
0.65
0.14
-0.07
0.68
25.2
0.15
-0.02
0.64
0.13
-0.04
0.63
31.2
0.14
-0.02
0.61
0.08
-0.16
0.69
16.5
0.14
-0.03
0.64
0.11
-0.10
0.65
20.4
0.15
-0.03
0.67
0.12
-0.06
0.62
25.2
0.15
-0.02
0.65
0.13
-0.04
0.64
Direct synthesis: frequency domain criteria
Continued on
2
next page.
T
T
Kc
i
d
Optimal gain margin: regulator response. 0 ≤ Tm1 τm ≤ 1 − (Tm 2 Tm1 ) .
[
)[− 0.365 + 0.260((τ
K c = (1 K m ) − 0.67 + 0.297(Tm1 τm )
K c = (1 K m
2.001
+ 2.189(Tm1 τm )
0.766
m
Tm1 ) − 1.4 ) + 2.189(Tm1 τm )
0.766
[
− 0.3] , τ T < 0.4 or
[− 0.975 + 0.910((τ T ) − 1.845) + x [5.25 − 0.88((τ
Ti = Tm1 2.212(τm Tm1 )
]
ξ m , τm Tm1 < 0.9 or
2
]
ξm , τm Tm1 ≥ 0.9 ;
0.520
Ti = Tm1
m
m1
2
m
m1
1
Tm1 ) − 2.8)
2
m
],
τm Tm1 ≥ 0.4 with x1 = 1 − e − (ξ m [0.15 + 0.33(τ m
Td =
(
x 2 1.45 + 0.969(Tm1 τm )
Tm1
1.171
)− 1.9 + 1.576(T
m1
τm )
0.530
Tm1 )])
;
,
with x 2 = 1 − e − (ξ m
[−0.15+ 0.939(T
m1
τ m )1.121
]) .
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FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kwak et al.
(2000) –
continued.
3
Kc
Ti
Td
Kc
Ti
Td
505
Comment
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ;
ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; b f 3 = 1 ; a f 5 = 0 .4
Robust
5
Lee et al. (2000).
Model: Method 1
Ti
Kc
α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 ,
[
a f 1 = Tm1 ((λ Tm1 ) + 1) e τ m
3
0 ≤ τm Tm1 < 2
Td
2
]
Tm1
− 1 , af 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ;
bf 2 = 0 , a f 3 = a f 1 , a f 4 = 0 ; K 0 = 0 .
Optimal gain margin: servo response. 0 ≤ Tm1 τm ≤ 1 − (Tm 2 Tm1 ) .
[

)[− 0.544 + 0.308(τ
K c = (1 K m ) − 0.04 + 0.333 + 0.949(Tm1 τm )
K c = (1 K m
0.983
ξ ], ξ
m
≤ 0.9 or
m
]
Tm1 ) + 1.408(Tm1 τm )
0.832
m
ξ m , ξ m > 0.9 .
Ti = Tm1 [2.055 + 0.072(τm Tm1 )]ξ m , τ m Tm1 ≤ 1 or
Ti = Tm1 [1.768 + 0.329(τm Tm1 )]ξ m , τ m Tm1 > 1 ;
Td =
4
K0 =
[
Tm1
x1 0.55 + 1.683(Tm1 τm )
1.090
1
G m ( jωu )(1 + jb f 4ωu ) K m
, x1 = 1 − e −1.149ξ m (τ m
]
Tm1 )1.060
.
(
)
, b f 4 = x 3 + x 4 (τm Tm1 ) + x 5 (τm Tm1 )2 Tm1 , with
2
3
4
x 3 = −0.0030 + 0.6482x 6 − 2.2841x 6 + 2.6221x 6 − 0.9611x 6 ,
2
3
4
x 4 = 0.2446 − 1.0410 x 6 − 13.6723x 6 − 16.7622 x 6 + 5.1471x 6 ,
2
3
4
x 5 = 0.1685 + 0.8289 x 6 − 9.3630 x 6 + 2.9855x 6 + 7.3803x 6 , x 6 = Tm 2 Tm1 .
5
Kc =
Ti
, recommended τm ≤ λ ≤ 2τm ;
− K m (2λ + τm − a f 1 )
2
Ti = −Tm1 + Tm 2 + a f 1 −
λ2 + a f 1τm − 0.5τm
,
2λ + τm − a f 1
− Tm1a f 1 + Tm 2a f 1 − Tm1Tm 2 −
Td =
2
τm (0.167 τm − 0.5a f 1 )
2λ + τm − a f 1
Ti
Desired closed loop transfer function =
e − sτ m
(λs + 1)2
2
−
λ2 + a f 1τm − 0.5τm
;
2λ + τm − a f 1
(Lee et al., 2006).
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506
4.13 Unstable SOSPD Model (two unstable poles)
G m (s ) =
K m e − sτm
(Tm1s − 1)(Tm 2s − 1)


1
+ Td s 
4.13.1 Ideal PID controller G c (s) = K c 1 +
 Ti s

R(s)
E(s)
−
+


1
K c 1 +
+ Td s 
Ti s


U(s)
K m e − sτ
(Tm1s − 1)(Tm 2 s − 1)
m
Y(s)
Table 109: PID controller tuning rules – unstable SOSPD model (two unstable poles)
K m e − sτ m
G m (s) =
(Tm1s − 1)(Tm 2 s − 1)
Rule
Kc
Wang and Jin
(2004).
1
Kc =
1
Kc
Ti
K m Ti (Tm1 − τm )(Tm 2 − τm )
(TCL3 + τm )3
Comment
Td
Direct synthesis: time domain criteria
Model: Method 1
Ti
Td
,
Ti =
TCL3 + 3τm TCL3 + 3(Tm1Tm 2 + τm Tm1 − τm Tm 2 )TCL 3 + τm Tm 2 (Tm1 − τm )
,
(Tm1 − τm )(τm − Tm 2 )
Td =
(Tm 2 + τm − Tm1 )TCL33 + 3Tm1Tm 2TCL3 (TCL3 + τm ) + Tm1Tm 2τm 2
Ti (Tm1 − τm )(Tm 2 − τm )
3
2
.
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Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Ti
Kc
Ti
Td
Comment
Td
Model: Method 1
Robust
Lee et al. (2000).
2
Kc =
2
Ti
, λ = desired closed loop dynamic parameter;
K m (4λ + τm − x1 )
2
Ti = −Tm1 − Tm 2 + x1 − x 2 , x 2 =
6λ2 − x 3 + x1τm − 0.5τm
;
4 λ + τ m − x1
x 3 − (Tm1 + Tm 2 ) x1 + Tm1Tm 2 −
Td =
3
2
4λ3 + x 3τm + 0.167 τm − 0.5x1τm
4λ + τm − x1
− x2 ;
Ti
x1 , x 3 values determined by solving 1 −
(x s
3
2
)
+ x1s + 1 e − τ m s
(λs + 1)4
s=
1
1
,
Tm1 Tm 2
=0.
507
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508
4.13.2 Generalised classical controller


Td s
1
+
G c (s) = K c 1 +
T

Ti s
1+ d

N

R(s)
+
E(s)
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N



2
 b f 0 + b f 1s + b f 2 s
 1 + a f 1s + a f 2 s 2
s 



U(s)
2
 b f 0 + b f 1s + b f 2 s
Model
 1 + a f 1s + a f 2 s 2
s

Y(s)
Table 110: PID controller tuning rules – unstable SOSPD model (two unstable poles)
K m e − sτ m
G m (s) =
(Tm1s − 1)(Tm 2 s − 1)
Rule
Kc
Ti
Kc
Ti
Td
Comment
Td
Model: Method 1
Robust
Shamsuzzoha
and Lee (2007d).
1
Recommended λ = τm .
4
τm
4
τm
 λ

 λ

Tm12 
+ 1 e Tm1 − Tm 2 2 
+ 1 e Tm 2 + Tm 2 2 − Tm12
Ti
 Tm1 
 Tm 2

1
Kc =
, Ti =
K m (4λ + τm − Ti )
Tm1 − Tm 2

4 τm
 Tm1 
λ

+ 1 e − 1 − Ti Tm
 Tm1 



, b f 0 = 1 , b f 1 = 0.5τm , b f 2 = 0 , N = ∞ ,
Ti
2 
Tm1 
Td =

 0.5τm Ti − Ti Td + 2λτ m + 6λ2
a f 2 = 0 , a f 1 = 0.1
+ Tm1 + Tm 2  , for large parametric
4
T
τ
+
λ
−
m
i



 0.5τm Ti − Ti Td + 2λτ m + 6λ2
+ Tm1 + Tm 2  .
uncertainties; otherwise, a f 1 = 
τm + 4λ − Ti






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509
4.13.3 Two degree of freedom controller 2




T
s
1
d
 1 + b f 1s (F(s) R (s) − Y(s) ) − K Y(s) ,
U(s) = K c 1 +
+
0
 Ti s
T  1 + a f 1s
1+ d s 

N 

1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4
F(s) =
1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5 s 4
R(s)
F(s)
+
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N



 1 + b f 1s
 1 + a f 1s
s

+ U(s)
−
Kc
Ti
Kc
Ti
Y(s)
K0
Table 111: PID tuning rules – unstable SOSPD model G m (s) =
Rule
Model
K m e − sτ m
(Tm1s − 1)(Tm 2 s − 1)
Td
Comment
Td
N=∞
Robust
1
Lee et al. (2000).
Model: Method 1
a f 1 = 0 , bf 1 = 0 ; bf 2 = 0 ; bf 3 = 0 ; a f 4 = 0 , a f 5 = 0 , bf 4 = 0 ,
b f 5 = 0 , K 0 = 0 ; recommended τm ≤ λ ≤ 2τm .
1
Desired closed loop transfer function = e −sτ m (λs + 1)4 (Lee et al., 2006).
2
Kc =
6λ2 − a f 3 + a f 2 τm − 0.5τm
Ti
,
, Ti = −Tm1 − Tm 2 + a f 2 − x1 , x1 =
K m (4λ + τm − a f 2 )
4λ + τ m − a f 2
a f 3 − (Tm1 + Tm 2 )a f 2 + Tm1Tm 2 −
Td =
a f 2 , a f 3 determined by 1 −
(a
f 3s
2
4λ3 + a f 3τm + 0.167 τm 3 − 0.5a f 2 τm 2
4λ + τm − a f 2
− x1 ;
Ti
)
+ a f 2s + 1 e − τ m s
(λs + 1)4
s=
1
1
,
Tm1 Tm 2
=0; 0≤
τm
<2.
Tm
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510
Rule
2
Shamsuzzoha et
al. (2006b).
Model: Method 1
Kc
Ti
Td
Comment
Kc
Ti
Td
N=∞
a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , bf 3 = 0 ; a f 4 = 0 ,
a f 5 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 .
λ , ξ , a f 1 not specified.
2
Kc =
Td =
Ti
, Ti = x1 − Tm1 − Tm 2 − x 2 ,
K m (4λξ + τm − x1 )
x 3 − x1 (Tm1 + Tm 2 ) + Tm1Tm 2 −
2
τ
0.167 τm 3 − 0.5x1τm 2 + x 3τm + 4λ3ξ
4λξ + τm − x1
− x2 ,
Ti
2
τ
m
m
2
λ2
2λξ  Tm1
2λξ  Tm1
2 λ
2

+
+
−
+
+
1
e
+ Tm 2 2 − Tm12
1
e
T
Tm1 
m2 
2
T 2 T


T
T
m1
m2
 m1

 m2

,
x1 =
Tm1 − Tm 2
2
x2 =
τ
m
2
2λξ  Tm1
x
4λ2ξ 2 + 2λ2 − x 3 + x1τm − 0.5τm 2
2 λ
, x 3 = Tm1 
−1− 1 .
1
e
+
+

T 2 T
T
4λξ + τm − x1
m
1
m1

 m1
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511
4.14 Unstable SOSPD Model with a Zero
K m (1 + Tm3 s )e −sτ m
K m (1 + Tm 3s )e −sτ m
G m (s) =
or
G
(
s
)
=
or
m
2
2
− Tm1 s 2 + 2ξ mTm1s + 1
Tm1 s 2 − 2ξ m Tm1s + 1
G m (s) =
K m (1 − Tm 4 s )e −sτm
2
Tm1 s 2 − 2ξ m Tm1s + 1

1 

4.14.1 Ideal PI controller G c (s) = K c 1 +
 Ti s 
R(s)

1  U(s)

K c 1 +
Model
Ti s 

E(s)
−
+
Y(s)
Table 112: PI controller tuning rules – unstable SOSPD model with a zero
K m (1 − Tm 4s )e −sτ m
G m (s ) =
2
Tm1 s 2 − 2ξ m Tm1s + 1
Rule
Kc
Sree and
Chidambaram (2004).
1
1
Kc
Direct synthesis: time domain criteria
Model: Method 1
Ti
Desired closed loop transfer function =
Kc =
Ti = −
x1 =
(
(1)
K m TCL
Tm12
(
( 2)
+ TCL
(1)
TCL
(1 − sTm 4 )(1 + Tis)e−sτ .
(1 + sTCL(1) )(1 + sTCL(2) )(1 + sx1 )
m
Ti
;
+ x1 + Tm 4 + τm − Ti
( 2)
+ TCL
(
Comment
Ti
)
)
(1) ( 2 )
+ x1 + Tm 4 + τm − TCL
TCL x1
Tm 4 τm − Tm12
)
;
(1)
( 2)
+ TCL
+ Tm 4 + τm (− 2ξ m Tm 4 τm + Tm1 (Tm 4 + τm ) ) − x 2
Tm1 TCL
(1) ( 2 )
(1)
( 2)
(− 2ξmTm1 + Tm 4 + τm )TCL
+ TCL
+ 2ξ m Tm1 )
TCL − Tm12 + (Tm 4 τm − Tm12 )(TCL
−1
2
(1) ( 2 )
x 2 = (Tm 4 τm − Tm1 )(TCL
TCL − τm Tm 4 ) ; − τm ([2ξ m Tm1 ] + Tm 4 ) < 0.62 .
,
b720_Chapter-04
Rule
Kc
Sree and
Chidambaram (2006).
Kc
K c given in footnote 1; Ti = −
x1 =
FA
Handbook of PI and PID Controller Tuning Rules
512
2
04-Apr-2009
(
)
Comment
Ti
2
(
Model: Method 1;
pp. 147–148.
Ti
)
(1)
( 2)
(1)
Tm12 TCL
+ TCL
+ x1 + Tm 4 + τm − Tm1TCL
Tm 4
Tm 4 τm − Tm12
(
, with
)(
(1)
( 2)
(1) ( 2 )
Tm1 TCL
+ TCL
+ Tm 4 + τm (− 2ξ mTm 4 τm + Tm1 (Tm 4 + τm ) ) − Tm 4 τm − Tm12 TCL
TCL − τmTm 4
(1) ( 2 )
(1)
( 2)
(− 2ξmTm1 + Tm 4 + τm )(TCL
TCL − Tm12 ) + (Tm 4 τm − Tm12 )(TCL
+ TCL
− Tm12 )
)
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
513
4.14.2 Ideal controller in series with a first order lag

 1
1
+ Td s 
G c (s) = K c 1 +
 Ti s
 Tf s + 1
R(s)
E(s)
U(s)

 1
1
K c 1 +
+ Td s 
 Ti s
 Tf s + 1
−
+
Y(s)
Model
Table 113: PID controller tuning rules – unstable SOSPD model with a zero
K m (1 − Tm 4s )e −sτ m
K m (1 + Tm3s )e−sτ m
G m (s ) =
or
2 2
2
Tm1 s − 2ξmTm1s + 1
Tm1 s 2 − 2ξ mTm1s + 1
Rule
Kc
Ti
Comment
Td
Direct synthesis: time domain criteria
Sree and
Chidambaram
(2004).
1
1
Ti
Kc
Desired closed loop transfer function =
Kc =
Td =
Ti =
Km
(T
(1)
CL
( 2)
+ TCL
Model: Method 1
Td
(1 − sTm 4 )(1 + Tis)e−sτ .
(1 + sTCL(1) )(1 + sTCL(2) )(1 + sx1 )
m
Ti
,
+ x 2 + Tm 4 + 0.5τm − Ti
)
(1) ( 2 )
0.5(Ti − 0.5τm )τm
0.5τmTCL
TCL x 2
,
, Tf =
2 (1)
( 2)
Ti
Tm1 TCL + TCL + x 2 + Tm 4 + 0.5τm − Ti
x1 − Tm12
(
(1)
TCL
(
+
( 2)
TCL
+ x 2 + Tm 4 + τm
0.5Tm12Tm 4 τm − Tm14
(
)(T
)
) with
)
2
(1) ( 2 )
(1) ( 2 )
( 2 ) ( 3)
( 3) (1)
(1) ( 2 )
x1 = TCL
TCL x 2 + 0.5τmTm1 TCL
TCL + TCL
TCL + TCL
TCL + ξmTm1τmTCL
TCL x 2 ,
x2 =
4
Tm1
(
2
0.5Tm 4 τm − Tm1
(1) ( 2 )
CL TCL
+
[
(1)
0.5τm TCL
x4
+
( 2)
TCL
]
( 2)
− Tm 4 TCL
+ 2ξ m Tm 4 x 3
(
(1)
( 2)
x 3 = TCL
+ TCL
+ Tm 4 + τm ,
2
(
)( [
− 0.5τ ) (T [T
2
2
m
2
m1
]
)]+ ξ
(1)
( 2)
(1) ( 2 )
x 4 = −Tm1 0.5Tm 4 τm − Tm1 Tm1 TCL
+ TCL
+ 0.5τm + 2ξ m Tm1 − 0.5τmTCL
TCL
+ (2ξTm1 − Tm 4
(1) ( 2 )
CLTCL
)
(1) ( 2 )
Tm1 (− 2ξ m Tm1 + Tm 4 + 0.5τm ) 0.5τm TCL
TCL − Tm 4 x 3
,
x4
4
+
)
(
(1)
+ 0.5τm TCL
( 2)
+ TCL
2
)
(1) ( 2 )
m Tm1τm TCL TCL
4
)
− Tm1 .
b720_Chapter-04
FA
Handbook of PI and PID Controller Tuning Rules
514
Rule
Sree and
Chidambaram
(2006).
2
04-Apr-2009
2
Kc
Ti
Td
Comment
Kc
Ti
Td
p. 165;
Model: Method 1
(1 + sTm3 )(1 + Tis)e−sτ .
(1 + sTCL(1) )(1 + sTCL(2) )(1 + sx 2 )
m
Desired closed loop transfer function =
Kc =
Ti
,
(1)
( 2)
K m TCL
+ TCL
+ x 2 − Tm3 + 1.5τm − Ti
(
x1 − Tm14
)
(
( 2)
+ TCL
+ x 2 − Tm 3
2
− 0.5Tm1 Tm3τm − Tm14
Ti =
Tf =
Tm12
(
(1)
TCL
+ τm
) + 0.5τ
(1) ( 2 )
0.5τm TCL
TCL x 2
(1)
TCL
( 2)
+ TCL
+ x 2 − Tm3 + 1.5τm − Ti
2
(
)(T
m
, Td =
);
0.5(Ti − 0.5τm )τm
,
Ti
)
(1) ( 2 )
(1) ( 2 )
( 2)
(1)
(1) ( 2 )
x1 = TCL
TCL x 2 + 0.5τ m Tm1 TCL
TCL + TCL
x 2 + x 2 TCL
− ξ m Tm1τ m TCL
TCL Tm3 ,
x2 =
4
Tm1
(
2
− 0.5Tm3τ m − Tm1
(1) ( 2 )
CL TCL
+
[
(1)
0.5τm TCL
x4
+
( 2)
TCL
]
( 2)
+ Tm 3 TCL
− 2ξ m Tm3 x 3
(
(1)
( 2)
x 3 = TCL
+ TCL
− Tm 3 + τm ,
2
(
2
)( [
]
2
(1)
( 2)
(1) ( 2 )
x 4 = Tm1 0.5Tm3τm + Tm1 Tm1 TCL
+ TCL
+ 0.5τm + 2ξm Tm1 − 0.5τm TCL
TCL
2
+ Tm1
(2ξTm1 + Tm3 − 0.5τm )
(T [T
2
m1
)
(1) ( 2 )
Tm1 (− 2ξ m Tm1 − Tm3 + 0.5τm ) 0.5τm TCL
TCL + Tm3 x 3
,
x4
4
+
)
(1) ( 2 )
CL TCL
(
)]
2
)
4
)
(1)
(2)
(1) ( 2 )
+ 0.5τ m TCL
+ TCL
+ ξ m Tm1 τ m TCL
TCL − Tm1 .
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Sree and
Chidambaram
(2006).
3
3
Kc
Ti
Td
Comment
Kc
Ti
Td
p. 148;
Model: Method 1
Desired closed loop transfer function =
Kc =
Tf =
Ti =
(1 − sTm 4 )(1 + Tis)e−sτ .
(1 + sTCL(1) )(1 + sTCL(2) )(1 + sx 2 )
0.5(Ti − 0.5τm )τm
, T =
m
Ti
(1)
( 2)
K m TCL
+ TCL
+ x 2 + Tm 4 + 1.5τm − Ti
(
Tm12
)
(1) ( 2 )
0.5τm TCL
TCL x 2
(T
(1)
CL
( 2)
+ TCL
+ x 2 + Tm 4 + 1.5τm − Ti
(
(1)
( 2)
x1 − Tm14 TCL
+ TCL
+ x 2 + Tm 4 + τm
0.5Tm12Tm 4 τm − Tm14
(
)(T
d
Ti
,
);
) + 0.5τ
m
, with
)
2
(1) ( 2 )
(1) ( 2 )
( 2)
(1)
(1) ( 2 )
x1 = TCL
TCL x 2 + 0.5τm Tm1 TCL
TCL + TCL
x 2 + x 2TCL
+ ξ m Tm1τmTCL
TCL Tm 4 ,
x2 =
4
Tm1
(
2
0.5Tm 4 τm − Tm1
(1) ( 2 )
CL TCL
+
[
(1)
0.5τm TCL
x4
+
( 2)
TCL
]
( 2)
− Tm 4 TCL
+ 2ξ m Tm 4 x 3
2
(
2
)
(
)( [
]
2
(1)
( 2)
(1) ( 2 )
x 4 = −Tm1 0.5Tm 4 τm − Tm1 Tm1 TCL
+ TCL
+ 0.5τm + 2ξ m Tm1 − 0.5τmTCL
TCL
2
+ Tm1
(2ξTm1 − Tm 4 − 0.5τm )
(T [T
2
m1
)
(1) ( 2 )
Tm1 (− 2ξ m Tm1 + Tm 4 + 0.5τm ) 0.5τm TCL
TCL − Tm 4 x 3
,
x4
4
+
(1)
( 2)
x 3 = TCL
+ TCL
+ Tm 4 + τm ,
515
(1) ( 2 )
CLTCL
(
)]
2
)
4
)
(1)
( 2)
(1) ( 2 )
+ 0.5τm TCL
+ TCL
+ ξmTm1τmTCL
TCL − Tm1 .
b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
516
4.14.3 Generalised classical controller


Td s
1
+
G c (s) = K c 1 +
T

Ti s
1+ d

N

E(s)
R(s) +
−


Td s
1
K c 1 +
+
 Ti s
T
1+ d

N



2
 b f 0 + b f 1s + b f 2 s
 1 + a f 1s + a f 2 s 2
s 


U(s)

2
b
+
b
s
+
b
s
f1
f2
 f0
Model
 1 + a f 1s + a f 2 s 2
s





Y(s)
Table 114: PID controller tuning rules – unstable SOSPD model with a zero
K m (1 + Tm 3s )e −sτ m
K m (1 − Tm 4s )e −sτ m
G m (s ) =
or
2 2
2
− Tm1 s + 2ξ m Tm1s + 1
Tm1 s 2 − 2ξ mTm1s + 1
Rule
Kc
Rao and
Chidambaram
(2006a), (2006b).
1
1
Ti
Kc
Comment
Td
Direct synthesis: time domain criteria
x1x 2 − x 3x 4
x 5x 3 − x 6 x 2
Model: Method 1
x x −x x
x x −x x
1 5
Desired closed loop transfer function =
6 4
(T T s
i d
2
1 2
3 4
)
+ Tis + 1 (1 − sTm 4 )e −sτ m
(λs + 1)3
,
recommended λ = 1.2τm ; b f 0 = 1 , b f 1 = 0.5τ m , b f 2 = 0 , N = ∞ .
Kc =
1
Ti
2
, x1 = − τmTm1Tm 4ξ m − Tm1 (Tm 4 + 0.5τm ) ,
K m 3λ + τm + Tm 4 − Ti
(
)
x 2 = 0.5Tm1 6λ2 + 3τm λ − Tm 4 τ m − 2ξm Tm1 (3λ + τm + Tm 4 ) − 0.5τm λ3 ,
2
3
x 3 = λ2Tm1 (λ + 1.5τm ) − λ3τm Tm1ξ m − Tm1 (3λ + τm + Tm 4 ) ,
2
4
x 4 = Tm1 − 0.5τm Tm 4 , x 5 = −Tm1 (2ξm Tm1 + Tm 4 + 0.5τm ) ,
2
2
2
(
2
)
x 6 = Tm1 0.5τm Tm 4 − Tm1 ;
af1 =
0.5τ m
Tm1
2
3
λ − Tm 4 Ti Td
λ3
0.05τm
, Tm 4 = 0 .
, Tm 4 ≠ 0 ; a f 1 =
2
3λ + τ m + Tm 4 − Ti
Tm1 3λ + τm − Ti
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Kc
Td
Comment
x 5 − x 2Ti
Ti x1
Model:
Method 1;
λ not specified.
x 3 − x 2Ti
Ti x1
Model:
Method 1;
λ not specified.
Ti
517
Robust
Sree and
Chidambaram
(2006),
pp. 168–169.
Sree and
Chidambaram
(2006),
pp. 167–168.
2
Kc =
2
3
x 5 x 3 − x 6 x1
x 2 x 3 − x 4 x1
Kc
x 3 x 4 − x 6 x1
x 4 x 2 − x1x 5
Kc
Ti
2
2
, x1 = Tm1 , x 2 = −Tm1 (0.5τm − 2ξm Tm1 ) ,
K m (4λ + 0.5τm − Ti )
x 3 = −0.5τm Tm1 , x 4 = −Tm1 , x 5 = λ4 + 6λ2Tm1 + 2ξ m Tm1 (4λ + 0.5τm ) ,
2
4
2
3
x 6 = −2ξ m Tm1λ4 + 4Tm1 λ3 − Tm1 (4λ + 0.5τm ) ; b f 0 = 1 , b f 1 = 0.5τm , b f 2 = 0 , N = ∞ ,
2
af 2 = −
3
Kc =
(4λ
λ4Tm3
4
)
+ 0.5τm − Ti Tm12
, a f 1 = Tm3 −
λ4
(4λ + 0.5τm − Ti )Tm12
.
Ti
λ5
, bf 2 = 0 , N = ∞ ,
, af 2 =
2
K m (5λ + Tm 4 + 0.5τm − Ti )
Tm1 (5λ + Tm 4 + 0.5τm − Ti )
x 3 − x 2Ti
+ Ti (Tm 4 + 0.5τm )
x1
, b f 0 = 1 , b f 1 = 0.5τm ,
5λ + Tm 4 + 0.5τm − Ti
10λ2 − 0.5Tm 4 τm −
a f 1 = 2ξ mTm1 +
(
)
x1 = Tm1 0.5Tm 4 τm − Tm1 , x 2 = Tm1 (Tm 4 + 0.5τm − 2ξ mTm1 ) ,
2
2
(
4
)
x 3 = 2ξm Tm1λ5 − Tm1 10λ2 − 0.5Tm 4 τm + 2ξ mTm1[5λ + Tm 4 + 0.5τm ] + 5Tm1 λ4 ,
4
x 4 = Tm1 (Tm 4 + 0.5τm − 2ξ mTm1 ) ,
2
2
[
]
x 5 = Tm1 2ξm Tm1 (Tm 4 + 0.5τm ) − 4ξ m Tm1 + Tm1 − 0.5Tm 4 τm ,
2
5
x6 = λ
3
− 2ξ m Tm1
2
(10λ
2
)
− 0.5Tm 4 τm −
2
2
4
4ξm Tm1
2
(5λ + Tm 4 + 0.5τm ) − 10Tm12λ3
+ Tm1 (5λ + Tm 4 + 0.5τm ) .
4
b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
518
4.14.4 Two degree of freedom controller 1
U (s) = G c (s)E (s) − F1 (s)R (s) ,


Td s
1
+
G c (s) = K c 1 +
T

Ti s
1+ d

N







β
T
s
d
E (s) , F1 (s) = K c  α +

Td 


s
s
1+

N 


F1 (s)
+
R(s)
E(s)
−
−
G c (s)
U(s)
Model
+
Y(s)
Table 115: PID controller tuning rules –– unstable SOSPD model with a zero
K m (1 − Tm 4 s )e −sτ
G m (s) =
Rule
m
2 2
Tm1 s − 2ξ m Tm1s + 1
Kc
Ti
Comment
Td
Direct synthesis: time domain criteria
Sree and
Chidambaram
(2004).
1
1
Desired closed loop transfer function =
Kc =
(
(1)
K m TCL
Ti = −
x1 =
Tm12
(
( 2)
+ TCL
(1)
TCL
Model: Method 1
(1 − sTm 4 )(1 + Tis)e−sτ .
(1 + sTCL(1) )(1 + sTCL(2) )(1 + sx1 )
m
 2ξ
Ti
1 
 < 0.62 ;
, − τm  m +

+ x1 + Tm 4 + τm − Ti
 Tm1 Tm 4 
( 2)
+ TCL
(
0
Ti
Kc
)
)
(1) ( 2 )
+ x1 + Tm 4 + τm − TCL
TCL x1
Tm 4 τm − Tm12
,
)
(1)
( 2)
+ TCL
+ Tm 4 + τm (− 2ξ m Tm 4 τm + Tm1 (Tm 4 + τm ) ) − x 2
Tm1 TCL
(1) ( 2 )
(1)
( 2)
(− 2ξmTm1 + Tm 4 + τm )TCL
+ TCL
+ 2ξ m Tm1 )
TCL − Tm12 + (Tm 4 τm − Tm12 )(TCL
(
)(
)
(1) ( 2 )
and with x 2 = Tm 4 τm − Tm12 TCL
TCL − τm Tm 4 , α = 1 −
(
)
(1)
( 2)
Tm 4 + 0.5 TCL
+ TCL
.
Ti
,
b720_Chapter-04
04-Apr-2009
FA
Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models
Rule
Sree and
Chidambaram
(2004) –
continued.
Sree and
Chidambaram
(2006).
2
Kc
Ti
Td
Comment
2
Kc
Ti
Td
Model:
Method 1;
β = 0, N = ∞.
3
Kc
Ti
0
Model:
Method 1;
p. 154.
Ti
(1)
( 2)
K m TCL
+ TCL
+ x 2 + Tm 4 + 0.5τ m −
(1)
( 2)
+ TCL
+ x 2 + Tm 4 + τm
x1 − Tm12 TCL
2
0.5Tm1 Tm 4 τm − Tm14
Kc =
(
Ti =
(
Ti
)
(
d
=
)
(1)
( 2)
Tm 4 + 0.5 TCL
+ TCL
,
Ti
, α = 1−
), T
2
(
0.5(Ti − 0.5τm )τm
,
Ti
)
(1) ( 2 )
(1) ( 2 )
( 2 ) ( 3)
( 3) (1)
(1) ( 2 )
x1 = TCL
TCL x 2 + 0.5τmTm1 TCL
TCL + TCL
TCL + TCL
TCL + ξmTm1τmTCL
TCL x 2 ,
(
)(
4
Tm1
x2 =
2
x 4 0.5Tm 4 τ m − Tm1
)(
(1) ( 2 )
TCL
TCL
+
x3 =
(1)
TCL
( 2)
+ TCL
x4 =
2
−Tm1
(0.5T
− x (T [T
m 4 τm
2
m1
5
3
+ Tm 4 + τm ,
2
Ti = −
x1 =
Tm12
(
(1)
TCL
(1) ( 2 )
CL TCL
( 2)
+ TCL
(
4
Tm1
)( [
( 2)
CL
4
m m1
]
( 2)
− Tm 4 TCL
+ 2ξ m Tm 4 x 3
(
(
(1)
+ 0.5τ m TCL
( 2)
+ TCL
)
)]
]
(1) ( 2 )
+ ξ m Tm1τm TCL
TCL
Tm 4 τm − Tm12
4
− Tm1
),
x 5 = −2ξTm1 + Tm 4 + 0.5τm .
(1 − sTm 4 )(1 + Tis)e
(1 + sTCL(1) )(1 + sTCL(2) )(1 + sx1 ) .
)
,
)
(1)
( 2)
Tm1 TCL
+ TCL
+ Tm 4 + τm (− 2ξ m Tm 4 τm + Tm1 (Tm 4 + τm ) ) − x 2
(1) ( 2 )
(1)
( 2)
(− 2ξmTm1 + Tm 4 + τm )(TCL
TCL − Tm12 ) + (Tm 4 τm − Tm12 )(TCL
+ TCL
− Tm12 )
(
)(
)
(1) ( 2 )
and with x 2 = Tm 4 τm − Tm12 TCL
TCL − τm Tm 4 , α = 1 −
)
)
− sτ m
(1)
+ x1 + Tm 4 + τm − Tm1TCL
Tm 4
)
2
(1) ( 2 )
+ Tm 4 + 0.5τ m ) 0.5τm TCL
TCL − Tm 4 x 3 ,
2
Ti
(1)
( 2)
K m TCL
+ TCL
+ x1 + Tm 4 + τm − Ti
(
(
[ +T
x )(− 2ξ T
(1)
0.5τm TCL
(1)
( 2)
(1) ( 2 )
− Tm1 Tm1 TCL
+ TCL
+ 0.5τm + 2ξ m Tm1 − 0.5τmTCL
TCL
Desired closed loop transfer function =
Kc =
+
519
(1) ( 2 )
TCL
TCL − Tm 4
Ti
.
,
b720_Chapter-04
04-Apr-2009
FA
Handbook of PI and PID Controller Tuning Rules
520
4.14.5 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s)




(
)
1
−
β
T
s
1 + b f 1s
[
]
1
−
χ
δ
d

,
F1 (s) = K c  [1 − α ] +
+
+


T
Ti s
1 + Ti s
1 + a f 1s + a f 2 s 2
d
1
+
s


N 





[
]
K (b + b f 4 s )
1
−
ϕ
T
s
1+ bf 2s
[
]
−
φ
1
d 
F2 (s) = K c  [1 − ε] +
+
− 0 f3
2

 1 + a f 3s + a f 4 s
T
Ti s
1+ a f 5s
1+ d s 

N 

R(s)
+
F1 (s)
U(s)
Y(s)
Model
−
F2 (s)
Table 116: PID controller tuning rules –– unstable SOSPD model with a zero
Rule
Kc
Ti
Comment
Td
Direct synthesis: time domain criteria
x1x 2 − x 3x 4
x 5x 3 − x 6 x 2
x1x 5 − x 6 x 4
x1x 2 − x 3 x 4
Rao and
1
Kc
Chidambaram
(2006b).
Model: Method 1 β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; b f 1 = 0.5τm ; a f 2 = 0 ; ε = 0 ; φ = χ ;
ϕ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 0 .
1
(
)
Desired closed loop transfer function = TiTds 2 + Tis + 1 (1 − sTm 4 )e−sτ m
K c = Ti (K m (3λ + τm + Tm 4 − Ti )) , recommended λ = 1.2τm ;
af1 =
0.5τm
Tm12
(λs + 1)3 ;
λ3 − Tm 4Ti Td
λ3
0.05τm
, Tm 4 ≠ 0 ; a f 1 =
, Tm 4 = 0 with
3λ + τm + Tm 4 − Ti
Tm12 3λ + τm − Ti
x1 = − τmTm1Tm 4ξ m − Tm1 (Tm 4 + 0.5τm ) , x 4 = Tm1 − 0.5τm Tm 4 ,
2
(
2
)
x 2 = 0.5Tm1 6λ2 + 3τm λ − Tm 4 τm − 2ξm Tm1 (3λ + τm + Tm 4 ) − 0.5τm λ3 ,
2
3
x 3 = λ2Tm1 (λ + 1.5τm ) − λ3τm Tm1ξ m − Tm1 (3λ + τm + Tm 4 ) ,
2
4
(
)
x 5 = −Tm1 (2ξm Tm1 + Tm 4 + 0.5τm ) , x 6 = Tm1 0.5τm Tm 4 − Tm1 ;
2
2
2
Formulate φ = (Ti − Tm 4 + b f 1 ) (Ti + K m K cTi − 0.5τm K m K c − Tm 4 K m K c ) ;
α = 1 − ((Tm 4 − b f 1 ) Ti ) − (x 7 Ti )[(Ti K m K c ) + Ti − Tm 4 − 0.5τm ] ;
if φ > 1 , x 7 = 1 ; if φ ≤ 1 , 0.6 ≤ x 7 ≤ 0.7 .
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Chapter 5
Performance and Robustness Issues in the
Compensation of FOLPD Processes with PI
and PID Controllers
5.1 Introduction
This chapter will discuss the compensation of FOLPD processes using PI and
PID controllers whose parameters are specified using appropriate tuning rules.
The gain margin, phase margin and maximum sensitivity of the compensated
system as the ratio of time delay to time constant of the process varies, are used
as ways of judging the performance and robustness of the system. This work
follows the method of Ho et al. (1995b), (1996), in which good approximations
for the gain and phase margins of the PI or PID controlled system are analytically
calculated. The method will be extended to determine analytically the maximum
sensitivity of the compensated system. Insight will be obtained into the range of
time delay to time constant ratios over which it is sensible to apply various tuning
rules, to compensate a FOLPD process.
The chapter is organised as follows. Formulae for calculating analytically the
gain margin, phase margin and maximum sensitivity are outlined in Sections 5.2
and 5.3. The performance and robustness of FOLPD processes compensated with
a variety of PI and PID tuning rules are evaluated in Section 5.4. In Section 5.5,
tuning rules are designed to achieve constant gain and phase margins for all
values of delay, for a number of process models and controller structures.
Conclusions of the work are drawn in Section 5.6. This work was previously
published by O’Dwyer (1998), (2001a).
521
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5.2 The Analytical Determination of Gain and Phase Margin
5.2.1 PI tuning formulae
The controller and process model are respectively given by

1 

G c (s) = K c 1 +
 Ti s 
G m (s ) =
(5.1)
K m e −sτ m
1 + sTm
(5.2)
K m e − jωτ m K c ( jωTi + 1)
1 + jωTm
jωTi
(5.3)
Then
G m ( jω)G c ( jω) =
From the definition of gain and phase margin, the following sets of equations are
obtained:
[
]
φ m = arg G c ( jωg )G m ( jωg ) + π
(5.4)
1
G c ( jω p )G m ( jω p )
(5.5)
Am =
where ωg and ωp are given by
G c ( j ω g ) G m ( jω g ) = 1
[
]
arg G c ( jωp )G m ( jωp ) = − π
(5.6)
(5.7)
From Equation (5.3),
G m ( jω)G c ( jω) =
K m K c 1 + ω2 Ti
ωTi 1 + ω2 Tm
2
2
∠ − 0.5π + tan −1 ωTi − tan −1 ωTm − ωτ m (5.8)
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523
Therefore, from Equation (5.4)
φ m = π − 0.5π + tan −1 ωg Ti − tan −1 ωg Tm − ωg τ m
(5.9)
with ωg given by the solution of Equation (5.6), i.e.
2
K m K c 1 + ωg Ti
2
ωg Ti 1 + ωg Tm
2
2
=1
(5.10)
Also, from Equations (5.5) and (5.8),
Am =
1
G m ( jω p ) G c ( jω p )
=
2
2
2
2
ωp Ti
1 + ωp Tm
KmKc
1 + ωp Ti
(5.11)
with ωp given by the solution of Equation (5.7), i.e.
− 0.5π + tan −1 ωp Ti − tan −1 ωp Tm − ωp τ m = − π
(5.12)
From Equation (5.10), ωg may be determined analytically to be
ωg =
(
2
2
) (K
Ti K c K m − 1 +
2
c
)
2
2
2
2
2
K m − 1 Ti + 4 K c K m Tm
2Ti Tm
2
2
(5.13)
An analytical solution of Equation (5.12) (to determine ωp ) is not possible. An
approximate analytical solution may be obtained if the following approximation
for the arctan function is made:
tan −1 x ≈
π
π π
x , x < 1 and tan −1 x ≈ −
, x >1
4
2 4x
(5.14)
This is quite an accurate approximation, as is shown in Figures 1a and 1b
(next page). Considering Equation (5.12), four possibilities present themselves if
the approximation in Equation (5.14) is to be used. These possibilities, together
with the formula for ωp that may be determined analytically for each of these
cases, are
(i)
1
1 

π ± π 2 − 4πτ m  −
 Ti Tm 
ωp Ti > 1, ωp Tm > 1 : ω p =
4τ m
(5.15)
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π ± π2 −
ωp Ti > 1, ωp Tm < 1 : ωp =
(ii)
π
(0.25πTm + τ m )
Ti
2(0.25πTm + τ m )
(5.16)
π
(iii)
ωp Ti < 1, ωp Tm > 1 : ωp =
(iv)
ωp Ti < 1, ωp Tm < 1 : ωp =
Tm
4τ m − πTi
2π
4τ m + π(Tm − Ti )
(5.17)
(5.18)
The gain and phase margin of the compensated system for each of the tuning
rules, as a function of τ m Tm , may be calculated by applying Equations (5.9),
(5.11), (5.13) and the relevant approximation for ωp from Equations (5.15) to
(5.18).
Figure 1a: Arctan x and its approximation (Equation 5.14)
arctan x
approximation
Figure 1b: % error in taking approximation (Equation 5.14) to arctan(x)
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525
5.2.2 PID tuning formulae
The classical PID controller structure is considered, whose transfer function is
given as

1  1 + sTd

G c (s) = K c 1 +
 Ti s  1 + sαTd



(5.19)
and the process model is given by Equation (5.2). Substituting Equations (5.2)
and (5.19) into Equations (5.4) and (5.5) gives
φ m = 0.5π + tan −1 ωg Ti + tan −1 ωg Td − tan −1 ωg Tm − tan −1 ωg αTd − ωg τ m (5.20)
and
Am =
(1 + ω T )(1 + ω α T )
(1 + ω T )(1 + ω T )
2
ωp Ti
p
KcKm
2
m
2 2
p
i
2
2
p
2
p
2
d
2
(5.21)
d
From Equation (5.6), ωg is given by the solution of
2
K m K c 1 + ωg Ti
2
ωg Ti 1 + ωg Tm
2
2
2
1 + ωg Td
2
2
1 + α 2 ωg Td
2
=1
(5.22)
The equation for ωg may be determined to be
6
4
2
ωg + a 1ω g + a 2 ωg + a 3 = 0
2
with
a1 =
2
a2 =
(
2
2
)
2
2
2
Ti Tm + α 2 Td − K c K m Ti Td
2
2
Ti Tm α 2 Td
2
2
2
2
2
2
Ti Tm α 2 Td
2
,
2
2
Ti − K c K m (Ti + Td )
2
(5.23)
and a 3 = −
Kc Km
2
2
2
Ti Tm α 2 Td
2
.
Following the procedure outlined by Ho et al. (1996), the analytical solution of
Equation (5.23) is given as
ωg =
3
R + Q3 + R 2 + 3 R − Q3 + R 2 −
3
with Q =
2
9a a − 27a 3 − 2a 1
3a 2 − a 1
and R = 1 2
.
9
54
a1
3
(5.24)
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From Equation (5.7), ω p is given by the solution of
0.5π + tan −1 ωp Td + tan −1 ωp Ti − tan −1 ωp Tm − tan −1 ωp αTd − ωp τ m = 0 (5.25)
As with Equation (5.12), an analytical solution of this equation is not possible.
An approximate analytical solution may be obtained if the approximation
detailed in Equation (5.14) is made. Looking at Equation (5.25), twelve
possibilities present themselves if the approximation in Equation (5.14) is to be
used; these possibilities are
(1) ωp Ti > 1, ωp Tm > 1
(2) ωp Ti > 1, ωp Tm < 1
(a) ω p Td ≤ 1, ω p αTd < 1
(a) ω p Td ≤ 1, ωp αTd < 1
(b) ω p Td > 1, ωp αTd < 1
(b) ω p Td > 1, ωp αTd < 1
(c) ω p Td > 1, ω p αTd > 1
(c) ω p Td > 1, ωp αTd > 1
(3) ωp Ti < 1, ωp Tm > 1
(4) ωp Ti < 1, ωp Tm < 1
(a) ω p Td ≤ 1, ωp αTd < 1
(a) ω p Td ≤ 1, ωp αTd < 1
(b) ω p Td > 1, ωp αTd < 1
(b) ω p Td > 1, ωp αTd < 1
(c) ω p Td > 1, ωp αTd > 1
(c) ω p Td > 1, ωp αTd > 1
The formulae for ωp that may be determined for each of these cases are as
follows:
(1) ωp Ti > 1, ωp Tm > 1
(a) ωp Td ≤ 1, ωp αTd < 1 :
1
1 

π ± π 2 − π(4τ m − πTd (1 − α )) −
 Ti Tm 
ωp =
(5.26)
4τ m − πTd (1 − α )
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(b) ωp Td > 1, ωp αTd < 1 :
 1
1
1 

+ −
2π ± 4π 2 − π(4τ m + πTd α )
 Td Ti Tm 
ωp =
(5.27)
4τ m + πTd α
(c) ωp Td > 1, ωp αTd > 1 :
 1
1
1
1 

π ± π 2 − 4πτ m 
+ −
−
 Td Ti αTd Tm 
ωp =
(5.28)
4τ m
(2) ωp Ti > 1, ωp Tm < 1 :
(a) ωp Td ≤ 1, ωp αTd < 1 :
2π ± 4π 2 +
ωp =
π
(πTd − πTm − παTd − 4τ m )
Ti
πTm + 4 τ m + παTd − πTd
(5.29)
(b) ωp Td > 1, ωp αTd < 1 :
 1
1
3π ± 9π 2 − π
+ (πTm + παTd + 4τ m )
 Td Ti 
ωp =
(5.30)
πTm + 4τ m + παTd
(c) ωp Td > 1, ωp αTd > 1 :
 1
1
1
2π ± 4π 2 + π
−
−
 αTd Td Ti
ωp =
πTm + 4τ m

(πTm + 4τ m )

(5.31)
(3) ωp Ti < 1, ωp Tm > 1 :
(a) ωp Td ≤ 1, ωp αTd < 1 : ωp =
π
Tm (4τ m + π[αTd − Td − Ti ])
(5.32)
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(b) ωp Td > 1, ωp αTd < 1 :
 1
1
− π ± π 2 − π
−
 Tm Td
ωp =
πTi − 4τ m
(c) ωp Td > 1, ωp αTd > 1 : ωp =

(πTi − παTd − 4 τ m )

(5.33)
− παTd
π
π
π
−
−
Td Tm αTd
πTi − 4τ m
(5.34)
(4) ωp Ti < 1, ωp Tm < 1 :
(a) ωp Td ≤ 1, ωp αTd < 1 : ωp =
2π
π(αTd − Td ) + π(Tm − Ti ) + 4τ m
(5.35)
(b) ωp Td > 1, ωp αTd < 1 :
2π ± 4π 2 −
ωp =
π
(πTm − πTi + παTd + 4τ m )
Td
πTm − πTi + 4τ m − παTd
(5.36)
(c) ωp Td > 1, ωp αTd > 1 :
 1
1 
π ± π 2 − π
− (πTi − πTm − 4τ m )
 αTd Td 
ωp =
(5.37)
4τ m − πTm − πTi
The gain and phase margin of the compensated system for each of the
tuning rules, as a function of τ m Tm , may now be calculated by applying
Equations (5.20), (5.21), (5.24) and the relevant approximation for ωp from
Equations (5.26) to (5.37). As Equation (5.19) shows, the procedure applies to
tuning rules defined for the classical PID controller structure only; however, Ho
et al. (1996) suggest that the method may be applied to tuning rules defined for
the ideal PID controller structure, by using equations that approximately convert
these tuning rules into their equivalent in the classical controller structure.
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529
5.3 The Analytical Determination of Maximum Sensitivity
The maximum sensitivity is the reciprocal of the shortest distance from the
Nyquist curve to the (-1,0) point on the Rl-Im axis. It is defined as follows:
M max = Max
all ω
1
1 + G m ( jω)G c ( jω)
(5.38)
For a FOLPD process model controlled by a PI controller,
G m ( jω)G c ( jω) =
1 + ω2 Ti
2
1 + ω2 Tm
2
KcKm
ωTi
(5.39)
and
arg[G m ( jω)G c ( jω)] = −0.5π − tan −1 ωTm + tan −1 ωTi − ωτ m
(5.40)
For a FOLPD process model controlled by a PID controller,
2
G m ( jω)G c ( jω) =
2
(1 + ω2 Ti )(1 + ω2 Td )
KcKm
2
2
(1 + ω2 Tm )(1 + α 2 ω2 Td ) ωTi
(5.41)
and arg[G m ( jω)G c ( jω)] =
− 0.5π − tan −1 ωTm − tan −1 ωαTd + tan −1 ωTi + tan −1 ωTd − ωτ m
(5.42)
The maximum sensitivity may be calculated over an appropriate range of
frequencies corresponding to phase lags of 100 0 to 260 0 .
5.4 Simulation Results
Space considerations dictate that only representative simulation results may be
provided; an extensive set of simulation results covering 103 PI controller tuning
rules and 125 PID controller tuning rules (for the ideal and classical controller
structures) are available (O’Dwyer, 2000). The MATLAB package has been used in
the simulations. Figures 2 to 7 show how gain margin, phase margin and maximum
sensitivity vary as the ratio of time delay to time constant varies, if some PI tuning
rules are used (Figures 2 to 4) and corresponding PID tuning rules for the classical
controller structure (with α = 0.1) are used (Figures 5 to 7). In these results, Z-N
refers to the process reaction curve method of Ziegler and Nichols (1942); W-W
refers to the process reaction curve method of Witt and Waggoner (1990); IAE reg,
ISE reg and ITAE reg refer to the tuning rules for regulator applications that
minimise the integral of absolute error criterion, the integral of squared error criterion
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and the integral of time multiplied by absolute error criterion, respectively, as defined
by Murrill (1967) for PI tuning rules and Kaya and Scheib (1988) for PID tuning
rules based on the classical controller structure. Figures 8 to 15 show gain and phase
margin comparisons between corresponding PI and PID controller tuning rules.
It is clear that the gain margin is generally less when the PID rather than the PI
tuning rules are considered, over the ratios of time delay to time constant taken; the
difference between the phase margins is less clear cut. This suggests that these PID
tuning rules should provide a greater degree of performance than the corresponding
PI tuning rules, but may be less robust. Comparing the individual tuning rules, it is
striking that the ISE based tuning rules have generally the smallest gain margin and
have also a small phase margin, suggesting that this is a less robust tuning strategy.
The results in Figures 4 and 7 confirm these comments.
Figure 2: Gain margin
Figure 3: Phase margin
Figure 4: Max. sensitivity
120
4
- = Z-N
* = IAE reg
+ = ITAE reg
o = ISE reg
3.5
3
8
7
100
6
80
5
2.5
60
2
40
4
3
20
1.5
1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2
0
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1
0
Ratio of τ m to Tm
Ratio of τ m to Tm
Figure 5: Gain margin
- = W-W
* = IAE reg
+ = ITAE reg
o = ISE reg
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Ratio of τ m to Tm
Figure 7: Max. sensitivity
Figure 6: Phase margin
2.5
2
0.2
100
6
90
5.5
80
5
70
4.5
60
4
50
3.5
40
3
30
2.5
1.5
1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Ratio of τ m to Tm
1.8
2
20
2
10
1.5
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Ratio of τ m to Tm
1.8
2
1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Ratio of τ m to Tm
2
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531
No general conclusion can be reached as to the best tuning rule (as expected);
it is interesting, though, that a full panorama of simulation results (O’Dwyer,
2000) show that many tuning rules may be applied at ratios of time delay to time
constant greater than that normally recommended. One example may be seen in
Figures 5 to 7, where the gain margin, phase margin and maximum sensitivity
(associated with the use of the PID tuning rule for obtaining minimum IAE in the
regulator mode) tends to level out when the ratio of time delay to time constant is
greater than 1; normally, the tuning rule is used when the ratio is less than 1
(Murrill, 1967). On the other hand, it is clear from Figures 8 and 9 that there is a
significant degradation of performance when using the PID tuning rule of Witt
and Waggoner (1990) and the PI tuning rule of Ziegler and Nichols (1942) for
large ratios of time delay to time constant, which is compatible with application
experience.
The decision between the use of a PI and PID controller to compensate the
process depends on the ratio of time delay to time constant in the FOLPD model,
together with the desired trade-off between performance and robustness, as
expected. It turns out, however, that the analytical method explored allows the
calculation of a far wider range of gain and phase margins for PI controllers; it is
also true that stability tends to be assured when a PI controller is used (O’Dwyer,
2000). Thus, a cautious design approach is to use a PI controller, particularly at
larger ratios of time delay to time constant.
Finally, the volume of tuning rules and data generated means that the use of
an expert system to recommend a tuning rule based on user defined requirements
is indicated; work is ongoing on such an implementation (Feeney and O’Dwyer,
2002).
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Figure 8: Gain margin comparison
Figure 9: Phase margin comparison
4
120
3.5
100
- = W-W PID
o = Z-N PI
3
80
2.5
60
2
40
1.5
20
1
FA
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
Ratio of τ m to Tm
0.8
1
1.2
1.4
1.6
1.8
2
Ratio of τ m to Tm
Figure 11: Phase margin comparison
Figure 10: Gain margin comparison
100
3
90
2.8
- = IAE reg PID
o = IAE reg PI
2.6
80
2.4
70
2.2
60
2
50
1.8
40
1.6
30
1.4
20
1.2
10
1
0.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Ratio of τ m to Tm
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Ratio of τ m to Tm
1.8
2
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Figure 12: Gain margin comparison
533
Figure 13: Phase margin comparison
2.5
100
- = ISE reg PID
o = ISE reg PI
90
80
70
2
60
50
40
1.5
30
20
10
1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0
2
0.2
0.4
Figure 14: Gain margin comparison
100
2.8
90
- = ITAE reg PID
o = ITAE reg PI
70
2.2
60
2
50
1.8
40
1.6
30
1.4
20
1.2
10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
80
2.4
0
1
Figure 15: Phase margin comparison
3
1
0.8
Ratio of τ m to Tm
Ratio of τ m to Tm
2.6
0.6
1.2
1.4
Ratio of τ m to Tm
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Ratio of τ m to Tm
1.8
2
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5.5 Design of Tuning Rules to Achieve Constant Gain and Phase Margins,
for All Values of Delay
Normally, the gain and phase margins of the compensated systems tend to
increase as the time delay increases, supporting the common view that PI and
PID controllers are less suitable for the control of dominant time delay processes.
However, for the PI control of a FOLPD process model, O’Dwyer (2000) shows
that the tuning rules proposed by Chien et al. (1952), Haalman (1965) and
Pemberton (1972a), among others, facilitate the achievement of a constant gain
and phase margin as the time delay of the process model varies. All of these
tuning rules have the following structure: K c = aTm K m τ m , Ti = Tm . Following
this observation, original approaches to the design of tuning rules for PI, PD and
PID controllers are proposed, for a wide variety of process models, which allow
constant gain and phase margins for the compensated system.
This work is organised as follows. In Section 5.5.1, PI controller tuning
rules are specified for processes modelled in FOLPD form and IPD form. In
Section 5.5.2, PID controller tuning rules are described for processes modelled in
FOLPD form, SOSPD form and SOSPD form with a negative zero. Section 5.5.3
deals with the design of PD controller tuning rules for the control of processes
modelled in FOLIPD form.
5.5.1 PI controller design
5.5.1.1 Processes modelled in FOLPD form
For such processes and controllers, Equations (5.1) and (5.2) apply, i.e.,
G m (s) =
K m e − sτ m
1 + sTm

1 

G c (s) = K c 1 +
 Ti s 
with
From Section 5.2.1,
φ m = π − 0.5π + tan −1 ω g Ti − tan −1 ω g Tm − ω g τ m
(Equation 5.9)
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535
with ω g given by the solution of
2
K m K c 1 + ωg Ti
2
ωg Ti 1 + ωg Tm
2
2
=1
(Equation 5.10)
and
Am =
2
2
2
2
ωpTi
1 + ωp Tm
K mK c
1 + ωp Ti
(Equation 5.11)
with ωp given by the solution of
− 0.5π + tan −1 ω p Ti − tan −1 ω p Tm − ω p τ m = −π
(Equation 5.12)
If K c and Ti are designed as follows:
Kc =
aTm
K mτm
(5.43)
and
Ti = Tm
(5.44)
Then Equation (5.12) becomes
− 0.5π − ωp τm = −π
i.e.
ω p = π 2τ m
(5.45)
(5.46)
Substituting Equation (5.46) into Equation (5.11) gives
A m = πTm 2K m K c τ m
(5.47)
Substituting Equation (5.44) into Equation (5.10) gives
i.e.
K m K c ω g Tm = 1
(5.48)
ω g = K m K c Tm
(5.49)
Substituting Equations (5.44) and (5.49) into Equation (5.9) gives
φ m = 0 . 5 π − K m K c τ m Tm
(5.50)
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Finally, substituting Equation (5.43) into Equation (5.47) gives
A m = π 2a
(5.51)
and substituting Equation (5.43) into Equation (5.50) gives
φ m = 0 .5 π − a
(5.52)
Some typical tuning rules are shown in Table 117.
Table 117: Typical PI controller tuning rules – FOLPD process model
a
Kc
Ti
Am
φm
π6
π3
1.047Tm K m τ m
Tm
1.5
π 4
0.785Tm K m τm
Tm
2.0
π 4
π6
0.524Tm K m τm
Tm
3.0
π3
These rules are also provided in Table 9. It may also be demonstrated that, if K c
and Ti are designed as follows:
Kc =
aTm
τm
Tu K u
2
Tu + 4π 2 Tm
2
(5.53)
and
Ti = Tm
(5.54)
where K u and Tu are the ultimate gain and ultimate period, respectively, then
the constant gain and phase margins provided in Equations (5.51) and (5.52) are
obtained.
5.5.1.2 Processes modelled in IPD form
A similar analysis to that of Section 5.5.1.1 may be done for the design of PI
controllers for processes modelled in IPD form. The process is modelled as
follows:
G m (s) = K m e −sτ m s
(5.55)
Corresponding to Equations (5.4) and (5.9), the phase margin is
φ m = π − 0.5π + tan −1 ω g Ti − 0.5π − ω g τ m
(5.56)
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537
with ωg given by the solution of
2
K m K c 1 + ω g Ti
2
=1
2
ω g Ti
(5.57)
If K c and Ti are designed as follows:
Kc =
a
K m τm
(5.58)
and
Ti = bτ m
(5.59)
then it may be shown that
 a2
a
ωg = 
±
2
2
2bτ m
 2τ m

a b + 4

2
0.5
2
(5.60)
and, substituting Equations (5.59) and (5.60) into Equation (5.56),
φ m = tan −1 0.5a 2 b 2 + 0.5ab a 2 b 2 + 4 


0.5
− 0.5a 2 + 0.5
a
a 2b2 + 4
b
(5.61)
Corresponding to Equations (5.5) and (5.11), the gain margin,
2
Am =
ω p Ti
2
K m K c 1 + ω p Ti
2
(5.62)
with ω p given by the solution of
− 0.5π + tan −1 ω p Ti − 0.5π − ω p τ m = −π
(5.63)
i.e. ω p is given by the solution of
tan −1 ω p Ti = ω p τ m
(5.64)
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An analytical solution to this equation is not possible, though if the
π
(when ω p Ti > 1 ) is used, simple
approximation tan −1 ω p Ti ≈ 0.5π −
4ω p Ti
calculations show that the following analytical solution for ω p may be obtained:
0.5 
π
2
0.5π + 0.25π − 
τ m 
b 
ωp =
(5.65)
The inequality ω p Ti > 1 may be shown to be equivalent to b > 1.273.
Substituting Equations (5.58), (5.59) and (5.65) into Equation (5.62), calculations
show that:
b
4a
Am =
1+
π
π2 π 
 +
− 
4 b
 2

2
b π
π
π
 +
− 
4 2
4 b


2
2
(5.66)
2
Some typical tuning rules are shown in Table 118.
Table 118: Typical PI controller tuning rules – IPD process model
Kc
Ti
Am
φm
0.558 K m τm
1.4τ m
1.5
46.2 0
0.484 K m τm
1.55τm
2.0
45.5 0
0.458 K m τm
3.35τ m
3.0
59.9 0
0.357 K m τm
4.3τ m
4.0
60.0 0
0.305 K m τm
12.15τ m
5.0
75.0 0
It may also be shown that the maximum sensitivity is a constant if K c and Ti are
specified according to Equations (5.58) and (5.59). The maximum sensitivity
may be calculated to be:
M max

a2 
= 1 − 2a cos(ω r τ m ) − 2 2 
ω r τ m 

−0.5
(5.67)
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539
with ω r τ m being a constant obtained numerically from the solution of the
2
2
equation (a 2 ω r τ m ) + cos ω r τ m = sin(ω r τ m ) / ω r τ m .
It may also be shown that, if K c and Ti are designed as follows:
K c = aK u
(5.68)
Ti = bTu
(5.69)
and
then the following constant gain and phase margins are determined:
2b  π
π2 π 
 +
− 
4 4b 
πa  2


Am =
1+
2
π
π2 π 

+
− 
4 4b 
a 2 π 2  2

1
[
and φ m = tan −1 2π 2 a 2 b 2 + 2πab 4π 2 a 2 b 2 + 4
− 0.125π 2 a 2 +
(5.70)
2
]
0 .5
πa
4π 2 a 2 b 2 + 4
16b
(5.71)
5.5.2 PID controller design
5.5.2.1 Processes modelled in FOLPD form – classical controller
The classical PID controller is given by

1  1 + sTd

G c (s) = K c 1 +
 Ti s  1 + sαTd



(Equation 5.19)
and the process model is given by
G m (s) =
K m e − sτ m
1 + sTm
(Equation 5.2)
From Section 5.2.2,
φm = 0.5π + tan −1 ωg Ti + tan −1 ωg Td − tan −1 ωg Tm − tan −1 ωg αTd − ωg τm
(Equation 5.20)
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with ωg given by the solution of
2
K m K c 1 + ω g Ti
2
ω g Ti 1 + ω g Tm
Also
Am =
ω p Ti
KcK m
2
2
2
1 + ω g Td
2
2
1 + α 2 ω g Td
=1
2
(Equation 5.22)
(1 + ω T )(1 + ω α T )
(1 + ω T )(1 + ω T )
2
p
2
2
m
2
p
2
p
2
i
2
p
2
d
2
(Equation 5.21)
d
with ω p given by
0.5π + tan −1 ω p Td + tan −1 ω p Ti − tan −1 ω p Tm − tan −1 ω p αTd − ω p τ m = 0
(Equation 5.25)
If K c , Ti and Td are designed as follows:
aTm
K mτm
Ti = αTm
Kc =
(5.72)
(5.73)
and
Td = Tm
(5.74)
then, Equation (5.25) becomes
0.5π − ω p τ m = 0
(5.75)
ω p = π 2τ m
(5.76)
i.e.
and Equation (5.21) becomes
A m = α πTm 2K m K c τ m
(5.77)
K m K c ω g Tm = 1
(5.78)
ω g = K m K c Tm
(5.79)
Equation (5.22) becomes
i.e.
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541
Finally, substituting Equations (5.73), (5.74) and (5.79) into Equation (5.20)
gives
φ m = 0 .5 π − K m K c τ m α T m
(5.80)
Substituting Equations (5.72), (5.73), (5.74) and (5.76) into Equation (5.21) gives
A m = πα 2a
(5.81)
and substituting Equation (5.72) into Equation (5.80) gives
φ m = 0.5π − a α
(5.82)
This design reduces to the PI controller design when α = 1 .
5.5.2.2 Processes modelled in SOSPD form – series controller
For such processes and controllers,
K m e − sτ m
G m (s) =
(1 + sTm1 )(1 + sTm 2 )
(5.83)

1 
(1 + sTd )
G c (s) = K c 1 +
 Ti s 
(5.84)
with
Therefore,
G m (s)G c (s) =
K m K c e −sτ m (1 + sTi )(1 + sTd )
Ti s(1 + sTm1 )(1 + sTm 2 )
(5.85)
If Ti and Td are designed as follows:
Ti = Tm1
(5.86)
Td = Tm 2
(5.87)
and
then, from Equation (5.85)
G m (s)G c (s) =
K m K c e −sτ m
Ti s
(5.88)
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which is equal to G m (s)G c (s) in Section 5.5.1.1 when Ti = Tm . Therefore,
designing
aTm
Kc =
(5.89)
K mτm
will allow A m = π 2a and φ m = 0.5π − a , as before.
5.5.2.3 Processes modelled in SOSPD form with a negative zero –
classical controller
For such processes,
K m e −sτ m (1 + sTm 3 )
G m (s) =
(1 + sTm1 )(1 + sTm 2 )
(5.90)
with G c (s) given by Equation (5.19). Therefore,
G m (s)G c (s) =
K m K c e −sτ m (1 + sTm 3 )(1 + sTi )(1 + sTd )
Ti s(1 + sTm1 )(1 + sTm 2 )(1 + αsTd )
(5.91)
If Ti , Td and α are designed as follows:
Ti = Tm1
(5.92)
Td = Tm 2
(5.93)
α = Tm 3 Tm 2
(5.94)
and
therefore, designing
Kc =
aTm
K mτm
(5.95)
will allow A m = π 2a and φ m = 0.5π − a , as in Sections 5.5.1.1 and 5.5.2.2.
5.5.3 PD controller design
In this case, the process is modelled in FOLIPD form, i.e.
G m (s) =
K m e −sτ m
s(1 + sTm )
(5.96)
with
G c (s) = K c (1 + Td s )
(5.97)
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Therefore,
G m (s)G c (s) =
K m K c e −sτ m (1 + sTd )
s(1 + sTm )
(5.98)
Therefore, designing
Kc =
aTm
K mτm
(5.99)
and
Td = Tm
(5.100)
will allow A m = π 2a and φ m = 0.5π − a , as in Sections 5.5.1.1, 5.5.2.2 and
5.5.2.3.
5.6 Conclusions
This chapter has considered the performance and robustness of a PI and PID
controlled FOLPD process, with the parameters of the controllers determined by
a variety of tuning rules. The original contributions of this work are as follows:
(a) an expansion of the analytical approach of Ho et al. (1995b), (1996) to
determine the approximate gain and phase margin analytically under all operating
conditions, (b) the analytical determination of the approximate maximum
sensitivity of the compensated system and (c) the application of the algorithms
using a wide variety of PI and PID tuning rules. The implementation of
autotuning algorithms in commercial controllers means that choice of a suitable
tuning rule is an important issue; the techniques discussed allow an analytical
evaluation to be performed of candidate tuning rules.
Finally, the chapter discusses an original approach to design tuning rules for
both PI and PID controllers, for a variety of delayed process models, with the
objective of achieving constant gain and phase margins for all values of delay. In
one of the cases discussed (PI control of an IPD process model), an analytical
approximation is used in the development; this approximation may also be used
to determine further tuning rules for other process models with delay, and for
other PID controller structures.
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Appendix 1
Glossary of Symbols and Abbreviations
a f 1 , a f 2 , a f 3 , a f 4 , a f 5 , a f 6 = Denominator parameters of a filter associated with
some PID controller structures
a 1 , a 2 , a 3 , b1 , b 2 , b 3 = Parameters of a third order process model
a 1 , a 2 , a 3 , a 4 , a 5 , b1 , b 2 , b 3 , b 4 , b 5 = Parameters of a fifth order process model
A 1 , A 2 , A 3 , A 4 , A 5 = Areas calculated from the process open loop step
response (see, for example, Vrančić et al., 1996)
A m = Gain margin
A p = Peak output amplitude of limit cycle determined from relay autotuning
b f 1 , b f 2 , b f 3 , b f 4 , b f 5 , b f 6 = Numerator parameters of a filter associated with
some PID controller structures
D R = Desired closed loop damping ratio
d(t) = Disturbance variable (time domain)
du dt = Time derivative of the manipulated variable (time domain)
e(t) = Desired variable, r(t), minus controlled variable, y(t) (time domain)
E(s) = Desired variable, R(s), minus controlled variable, Y(s)
FOLPD model = First Order Lag Plus time Delay model
FOLIPD model = First Order Lag plus Integral Plus time Delay model
F1 (s), F2 (s) = Transfer functions of components of some PID controller
structures
G c (s) = PID controller transfer function
G CL (s) = Closed loop transfer function
544
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Appendix 1: Glossary of Symbols and Abbreviations
545
G CL ( jω) = Desired closed loop frequency response
G m (s) = Process model transfer function
G p (s) = Process transfer function
G p ( jω) = Process transfer function at frequency ω
G p ( jω) = Magnitude of G p ( jω)
G p ( jωφ ) = Magnitude of G p ( jω) , at frequency ω , corresponding to phase lag
of φ
∠G p ( jω) = Phase of G p ( jω)
± h = Relay amplitude (relay autotuning)
IE = Integral of Error =
∞
∫ e( t )dt
0
∫
IAE = Integral of Absolute Error =
∞
0
e( t ) dt
IMC = Internal Model Controller
IPD model = Integral Plus time Delay model
I 2 PD model = Integral Squared Plus time Delay model
ISE = Integral of Squared Error =
∫
∞
0
e 2 ( t )dt
ISTES = Integral of Squared Time multiplied by Error, all to be Squared =
∫
∞
0
[ t 2 e( t )]2 dt
ISTSE = Integral of Squared Time multiplied by Squared Error
=
∫
∞
0
t 2 e 2 ( t )dt
ITSE = Integral of Time multiplied by Squared Error =
∫
ITAE = Integral of Time multiplied by Absolute Error =
K c = Proportional gain of the controller
KH =
9
2
2τ m K m
τm2
ξ T τ 
2
− Tm1 − m m1 m  (Hwang, 1995)

9

 18
∞
0
∫
te 2 ( t )dt
∞
0
t e( t ) dt
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K H1 =
 τ m 2 τ m Tm 1.884K c K u τ m 
−
+


18
9ω u
 18

9
2K m τ m
+
2
2
2
2
2
9  τm
49Tm
7T τ
4.71K u K c (τm + 1.6Tm ) 1.775K c K u 


−
+
+ m m−
2
2K m τm  324
324
162
81ωu
81ωu


(Hwang, 1995)
K H2 =
9
2K m τ m
 τm
1.884K u K c τ m 
ξ τ T
9
2
− Tm1 − m m m1 +
x1

+
2
18
9
9
ω
u

 2K m τ m
2
2
with
4
2
2
2
2
2
3
3


τ
49τm ξm Tm1 τm Tm1 7Tm1ξm τm 10τm Tm1 ξ m
4
−
+
+
− x2 
x1 =  Tm1 + m +
324
81
9
81
9


and
x2 =
2
2
2
3
0.471K u K c 10τ m
4T τ (8ξ m τ m + 9Tm1 )  1.775K u K c τ m
+ m1 m
−

2
81
ωu
81ω u

 81
(Hwang, 1995)
K i = Integral gain of the controller
K L = Gain of a load disturbance process model
K m = Gain of the process model
K p = Gain of the process
K u = Ultimate proportional gain
∧
K u = Ultimate proportional gain estimate determined from relay autotuning
K φ = Proportional gain when G c ( jω)G p ( jω) has a phase lag of φ
K 0 = Weighting parameter used in some two degree of freedom PID controller
structures
K x % = Proportional gain required to achieve a decay ratio of 0.01x
K φ0 = Controller gain when G c ( jω)G p ( jω) has a phase lag of φ 0
min = Minimum
max = Maximum
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FA
547
M s = Closed loop sensitivity
M max = Maximum value of closed loop sensitivity, M s
M t = Complementary closed loop sensitivity
n = Order of a process model with a repeated pole
N = Parameter that determines the amount of filtering on the derivative term on
some PID controller structures
OS = Closed loop response overshoot (often a percentage)
PI controller = Proportional Integral controller
PID controller = Proportional Integral Derivative controller
r = Desired variable (time domain)
2
r1 =
0.7071M max − M max 1 − 0.5M max
2
2
M max − 1
(Chen et al.,1999a)
R(s) = Desired variable (Laplace domain)
s = Laplace variable
SOSPD model = Second Order System Plus time Delay model
SOSIPD model = Second Order System plus Integral Plus time Delay model
t = Time
Tar = Average residence time = time taken for the open loop process step
response to reach 63% of its final value
Td = Derivative time of the controller
TCL = Desired closed loop system time constant
TCL 2 = Desired parameter of second order closed loop system response
(1)
( 2)
( 3)
TCL , TCL , TCL = Desired dynamic parameters of the closed loop system
response
Tf = Time constant of the lag in some PID controller structures
Ti = Integral time of the controller
TL = Time constant of a load disturbance FOLPD process model
Tm = Time constant of a FOLPD process model
Tm1, Tm2, Tm3, Tm4 = Time constants of second, third or fourth order process models,
as appropriate
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Tm1i , Tm2i , Tm3i , Tm4i = Time constants of a general process model
Tp = Time constant of a FOLPD process
Tp1, Tp2, Tp3 = Time constants of second or third order process, as appropriate
TR = Closed loop rise time
TS = Closed loop settling time
Tu = Ultimate period
∧
T u = Ultimate period estimate determined from relay autotuning
Tx % = Period of the waveform with a decay ratio of 0.01x , when the closed loop
system is under proportional control
TOSPD model = Third Order System Plus time Delay model
u(t) = Manipulated variable (time domain)
u ∞ = Final value of the manipulated variable (time domain)
U(s) = Manipulated variable (Laplace domain)
V = Closed loop response overshoot (as a fraction of the controlled variable final
value)
y(t) = Controlled variable (time domain)
y ∞ = Final value of the controlled variable (time domain)
Y(s) = Controlled variable (Laplace domain)
α , β , χ , δ , ε , φ , ϕ = Weighting factors in some PID controller structures
2
ε=
6Tm1 + 4ξ m Tm1 τ m + K H K m τ m
ε1 =
6Tm1 + 4Tm1 ξ m τ m + K u K m τ m
2
2τ m Tm1 ω u
2
(Hwang, 1995)
2Tm ωu + K H1 K m τ m ωu − 1.884 K u K c
(Hwang, 1995)
0.471K c K u ω H1 τ m
2
ε2 =
(Hwang, 1995)
2
2Tm1 τ m ω H
2
ε0 =
2
2
6Tm1 ωu + 4Tm1ξ m τ m ω u + K H 2 K m τ m ωu − 1.884 K u K c τ m
2
2
(0.471K u K c τ m + 2 τ m Tm1 ωu )ω H 2
(Hwang, 1995)
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549
± ε 4 = Relay hysteresis width (relay autotuning)
φ = Phase lag
φc = Phase of the plant at the crossover frequency of the compensated system,
with a ‘conservative’ PI controller (Pecharromán and Pagola, 2000)
φ m = Phase margin
φω = Phase lag at an angular frequency of ω
κ = 1 Km Ku
λ = Parameter that determines robustness of compensated system.
τ = τ m (τ m + Tm )
τ CL = Desired closed loop system time delay
τ L = Time delay of a load disturbance process model
τ m = Time delay of the process model
τ am = τ 'm − τ m , τ 'm is obtained using the tangent and point method of Ziegler and
Nichols (1942). Subsequently, τ m is estimated from the process step
response (Schaedel, 1997).
ω = Angular frequency
ω bw = − 3 dB closed loop system bandwidth (Shi and Lee, 2002)
ω−6dB = Frequency where closed loop system magnitude is –6dB (Shi and Lee,
2004)
ωc = Maximum cut-off frequency
ωd = Bandwidth of stochastic disturbance signal (Van der Grinten, 1963)
ω CL = 2π TCL
ωg = Specified gain crossover frequency (Shi and Lee, 2002)
1+ KHKm
ωH =
Tm1
ω H1 =
2
2T τ ξ
K K τ
+ m1 m m + H m m
3
6
2
(Hwang, 1995)
1 + K H1 K m
2
0.942 K c K u τ m
τ m Tm K H1 K m τ m
+
−
3
6
3ωu
(Hwang, 1995)
b720_Appendix-01
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Handbook of PI and PID Controller Tuning Rules
550
1 + K H2 K m
ωH 2 =
2
Tm1 +
2
0.942 K c K u τ m
2ξ m τ m Tm1 K H 2 K m τ m
+
−
3
6
3ωu
(Hwang, 1995)
ω M max = Frequency where the sensitivity function is maximised (Rotach, 1994)
ωn = Undamped natural frequency of the compensated system
ωp =
A m φ m + 0.5πA m (A m − 1)
(A
2
m
)
− 1 τm
(see, for example, Hang et al. 1993a, 1993b)
ω pc = Phase crossover frequency
ωr = Resonant frequency (Rotach, 1994)
ωu = Ultimate frequency
^
ω u = Ultimate frequency estimate obtained from relay autotuning
ω φ = Angular frequency when G c ( jω)G p ( jω) has a phase lag of φ
^
ω φ = Estimate of angular frequency when G c ( jω)G p ( jω) has a phase lag of φ
ξ = Damping factor of the compensated system
ξ m = Damping factor of an underdamped process model
ξ des
m = Desired damping factor of an underdamped process model
ξ m ni , ξ mdi = Damping factors of a general underdamped process model
b720_Appendix-02
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Appendix 2
Some Further Details on Process Modelling
Processes with time delay may be modelled in a variety of ways. The modelling
strategy used will influence the value of the model parameters, which will in turn
affect the controller values determined from the tuning rules. The modelling
strategy used in association with each tuning rule, as described in the original
papers, is indicated in the tables in Chapters 3 and 4. Some outline details of
these modelling strategies are provided, together with references that describe the
modelling method in detail. The references are given in the bibliography. For all
models, the label ‘Model: Method 1’ indicates that the model method has not
been defined or that the model parameters are assumed known; interestingly, this
is the modelling method assigned to a majority of tuning rules.
A2.1 FOLPD Model
A2.1.1 Parameters estimated from the open loop process step or impulse
response
Method 2:
Parameters estimated using a tangent and point method (Ziegler and
Nichols, 1942).
Method 3: K m , τ m determined from the tangent and point method of Ziegler
and Nichols (1942); Tm determined at 60% of the total process
variable change (Fertik and Sharpe, 1979).
Method 4: K m , τ m assumed known; Tm estimated using a tangent method
(Wolfe, 1951).
551
b720_Appendix-02
552
Method 5:
Method 6:
Method 7:
Method 8:
Method 9:
Method 10:
Method 11:
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Parameters estimated using a second tangent and point method
(Murrill, 1967).
Parameters estimated using a third tangent and point method
(Davydov et al., 1995).
τ m and Tm estimated using the two-point method; K m estimated
from the open loop step response (ABB, 2001).
τ m and Tm estimated from the process step response:
Tm = 1.4( t 67% − t 33% ) , τ m = t 67% − 1.1Tm ; K m assumed known
(Chen and Yang, 2000). 1
τ m and Tm estimated from the process step response:
Tm = 1.245( t 70% − t 33% ) , τ m = 1.498t 33% − 0.498t 70% ; K m assumed
known (Vítečková et al., 2000b). 2
τ m and Tm estimated from the process step response:
Tm = 0.910(t 75% − t 25% ) , τ m = 1.262 t 25% − 0.262 t 75% ; K m is
estimated from the process step response (Arrieta Orozco and Alfaro
Ruiz, 2003; Arrieta Orozco, 2003). 3
K m estimated from the process step response; τ m = time for which
the process variable does not change; Tm determined at 63% of the
total process variable change (Gerry, 1999).
Method 12: K m estimated from the process step response; τ m = time for which
the process variable has to change by 5% of its total value; Tm
determined at 63% of the total process variable change
(Kristiansson, 2003).
Method 13: K m estimated from the process step response; τ m is the time for
which the process variable has to change until it is outside a
predefined noise band; Tm = 0.25(t 98% − τ m ) (McMillan, 2005). 4
1
2
3
4
t 67% , t 33% are the times taken by the process variable to change by 67% and 33%, respectively,
of its total value.
t 70% is the time taken by the process variable to change by 70% of its total value.
t 75% , t 25% are the times taken by the process variable to change by 75% and 25%, respectively,
of its total value.
t 98% is the time taken by the process variable to change by 98% of its total value.
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553
Method 14: Parameters estimated using a least squares method in the time
domain (Cheng and Hung, 1985).
Method 15: The model parameters are estimated by minimising a function of the
error between the process and model open loop step response
(Zhuang, 1992).
Method 16: Parameters estimated from linear regression equations in the time
domain (Bi et al., 1999).
Method 17: Parameters estimated using the method of moments (Åström and
Hägglund, 1995).
Method 18: Model parameters estimated using the area method (Klán, 2000).
Method 19: Parameters estimated from the process step response and its first
time derivative (Tsang and Rad, 1995).
Method 20: Parameters estimated from the process step response using
numerical integration procedures (Nishikawa et al., 1984).
Method 21: Parameters estimated from the process impulse response (Peng and
Wu, 2000).
Method 22: Parameters estimated using a ‘process setter’ block (Saito et al.,
1990).
Method 23: Parameters estimated from a number of process step response data
values (Kraus, 1986).
Method 24: Parameters estimated in the sampled data domain using two process
step tests (Pinnella et al., 1986).
Method 25: A model is obtained using the tangent and point method of Ziegler
and Nichols (1942); label the parameters K 'm , Tm' and τ 'm .
Subsequently, τ m is estimated from the process step response; then,
a parameter labelled τ am = τ 'm − τ m is defined (Schaedel, 1997).
Method 26: A model is obtained using the tangent and point method of Ziegler
and Nichols (1942); label the parameters K 'm , Tm' and τ 'm .
Subsequently, τ m is estimated from the process step response
(Henry and Schaedel, 2005).
Method 27: Parameters estimated from the open loop step response (Clarke,
2006).
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Handbook of PI and PID Controller Tuning Rules
A2.1.2 Parameters estimated from the closed loop step response
Method 28: K m estimated from the open loop step response. T90% and τ m
estimated from the closed loop step response under proportional
control (Åström and Hägglund, 1988).
Method 29: Parameters estimated using a method based on the closed loop
transient response to a step input under proportional control (Sain
and Özgen, 1992).
Method 30: Parameters estimated using a second method based on the closed
loop transient response to a step input under proportional control
(Hwang, 1993).
Method 31: Parameters estimated using a third method based on the closed loop
transient response to a step input under proportional control (Chen,
1989; Taiwo, 1993).
Method 32: Parameters estimated from a step response autotuning experiment –
Honeywell UDC 6000 controller (Åström and Hägglund, 1995).
Method 33: Parameters estimated from the closed loop step response when
process is in series with a PID controller (Morilla et al., 2000).
Method 34: Parameters estimated by modelling the closed loop response as a
second order system (Ettaleb and Roche, 2000).
A2.1.3 Parameters estimated from frequency domain closed loop information
Method 35: Model parameters estimated using a relay autotuning method (Yu,
2006).
Method 36: Parameters estimated from two points, determined on process
frequency response, using a relay and a relay in series with a delay
(Tan et al., 1996).
Method 37: Tm and τ m estimated from the ultimate gain and period, determined
using a relay in series with the process in closed loop; K m assumed
known (Hang and Cao, 1996).
Method 38: Tm and τ m estimated from the ultimate gain and period, determined
using a relay in series with the process in closed loop; K m
estimated from the process step response (Hang et al., 1993b).
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Appendix 2: Some Further Details on Process Modelling
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555
Method 39: Tm and τ m estimated from data obtained when a relay is introduced
in series with the process in closed loop; K m assumed known
(Padhy and Majhi, 2006a).
Method 40: Parameters estimated from data obtained when a relay is introduced
in series with the process in closed loop (Padhy and Majhi, 2006b).
Method 41: Tm and τ m estimated from the ultimate gain and period, determined
using the ultimate cycle method of Ziegler and Nichols (1942); K m
estimated from the process step response (Hang et al., 1993b).
Method 42: Tm and τ m estimated at ω1800 ; K m estimated at ω 0 0 (Tavakoli
et al., 2006).
Method 43: Tm and τ m estimated from relay autotuning method (Lee and Sung,
1993); K m estimated from the closed loop process step response
under proportional control (Chun et al., 1999).
Method 44: Parameters estimated in the frequency domain from the data
determined using a relay in series with the closed loop system in a
master feedback loop (Hwang, 1995).
Method 45: K u and Tu estimated from relay autotuning method; τ m estimated
from the open loop process step response, using a tangent and point
method (Wojsznis et al., 1999).
Method 46: Parameters estimated by including a dynamic compensator outside
or inside an (ideal) relay feedback loop (Huang and Jeng, 2003).
Method 47: Parameters estimated from measurements performed on the
manipulated and controlled variables when a relay with hysteresis is
introduced in place of the controller (Zhang et al., 1996b).
Method 48: Non-gain parameters estimated using a relay, with hysteresis, in
series with the process in closed loop; K m is estimated with the aid
of a small step signal added to the reference (Potočnik et al., 2001).
Method 49: Parameters estimated using a relay in series with the process in
closed loop (Perić et al., 1997).
Method 50: Parameters estimated using an iterative method, based on data from
a relay experiment (Leva and Colombo, 2004).
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Handbook of PI and PID Controller Tuning Rules
Method 51: K m estimated from relay autotuning method; τ m , Tm estimated
using a least squares algorithm based on the data recorded from the
relay autotuning method, with the aid of neural networks (Huang
et al., 2005a).
Method 52: Parameters estimated from data obtained when an asymmetrical
relay is introduced in series with the process in closed loop (Majhi,
2005).
Method 53: Parameters estimated from data obtained when an asymmetrical
relay is introduced in series with the process in closed loop (Prokop
and Korbel, 2006).
A2.1.4 Other methods
Method 54: Parameter estimates back-calculated from a discrete time
identification method (Ferretti et al., 1991).
Method 55: Parameter estimates determined graphically from a known higher
order process (McMillan, 1984).
Method 56: Parameter estimates determined from a known higher order process
(Skogestad, 2003).
Method 57: The parameters of a second order model plus time delay are
estimated using a system identification approach in discrete time;
the parameters of a FOLPD model are subsequently determined
using standard equations (Ou and Chen, 1995).
Method 58: Parameter estimates back-calculated from a discrete time
identification non-linear regression method (Gallier and Otto, 1968).
Method 59: Parameter estimates determined using a recursive least squares
method (Bai and Zhang, 2007).
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Appendix 2: Some Further Details on Process Modelling
557
A2.2 FOLPD Model with a Zero
Method 2:
Method 3:
Non-delay parameters are estimated in the discrete time domain
using a least squares approach; time delay assumed known (Chang
et al., 1997).
Parameters estimated graphically from the open-loop step response
(Slätteke, 2006).
A2.3 SOSPD Model
A2.3.1 Parameters estimated from the open loop process step response
Method 2:
Method 3:
Method 4:
Method 5:
Method 6:
5
K m , Tm1 and τ m are determined from the tangent and point method
of Ziegler and Nichols (1942) (Shinskey, 1988, page 151); Tm 2
assumed known.
FOLPD model parameters estimated using a tangent and point
method (Ziegler and Nichols, 1942); corresponding SOSPD
parameters subsequently deduced (Auslander et al., 1975).
Parameters estimated using a tangent and point method (Murata and
Sagara, 1977).
Tm1 = Tm 2 . τ m , Tm1 estimated from process step response:
Tm1 = 0.794( t 70% − t 33% ) ,
τ m = 1.937 t 33% − 0.937 t 70% .
Km
5
assumed known (Vítečková et al., 2000b).
Tm1 = Tm 2 . K m , τ m and Tm1 estimated from the process step
response; τ m = time for which the process variable does not change;
Tm1 determined at 73% of the total process variable change
(Pomerleau and Poulin, 2004).
t 70% , t 33% are the times taken by the process variable to change by 70% and 33%, respectively,
of its total value.
b720_Appendix-02
558
Method 7:
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Parameters estimated from the underdamped or overdamped
transient response in open loop to a step input (Jahanmiri and
Fallahi, 1997).
A2.3.2 Parameters estimated from the closed loop step response
Method 8:
Method 9:
Parameters estimated from a step response autotuning experiment –
Honeywell UDC 6000 controller (Åström and Hägglund, 1995).
Parameters estimated from the servo or regulator closed loop
transient response, under PI control (Rotach, 1995).
A2.3.3 Parameters estimated from frequency domain closed loop information
Method 10: In this method, Tm1 = Tm 2 = Tm . Tm and τ m estimated from K u ,
Tu determined using a relay autotuning method; K m estimated
from the process step response (Hang et al., 1993a).
Method 11: Parameters estimated using a two-stage identification procedure
involving (a) placing a relay and (b) placing a proportional
controller, in series with the process in closed loop (Sung et al.,
1996).
Method 12: Parameters estimated using a two-stage identification procedure
involving (a) placing a relay in series with the process in closed loop
and (b) introducing an excitation to the relay feedback system
through an external DC input at the output of the relay (Huang et al.,
2003).
Method 13: Parameters estimated from data determined by inserting a relay in
series with the process in closed loop (Lavanya et al., 2006a).
Method 14: Parameters estimated in the frequency domain from the data
determined using a relay in series with the closed loop system in a
master feedback loop (Hwang, 1995).
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559
Method 15: Parameters estimated from values determined from an experiment
using an amplitude dependent gain in series with the process in
closed loop (Pecharromán and Pagola, 1999).
Method 16: Parameters estimated from data obtained when the process phase lag
is − 90 0 and − 180 0 , respectively (Wang et al., 1999).
Method 17: Model parameters estimated by including a dynamic compensator
outside or inside an (ideal) relay feedback loop (Huang and Jeng,
2003).
Method 18: K m estimated from a relay autotuning method; τ m , Tm1 and
ξ m estimated using a least squares algorithm based on the data
recorded from the relay autotuning method, with the aid of neural
networks (Huang et al., 2005a).
Method 19: Model parameters estimated using a two-stage identification
procedure involving (a) placing a relay and (b) placing a relay which
operate on the integral of the error, in series with the process in
closed loop (Naşcu et al., 2006).
Method 20: Model parameters estimated using a relay autotuning method
(Thyagarajan and Yu, 2003).
A2.3.4 Other methods
Method 21: Parameter estimates back-calculated from a discrete time
identification method (Ferretti et al., 1991).
Method 22: Parameter estimates back-calculated from a second discrete time
identification method (Wang and Clements, 1995).
Method 23: Parameter estimates back-calculated from a third discrete time
identification method (Lopez et al., 1969).
Method 24: Model parameters estimated assuming higher order process
parameters are known (Skogestad, 2003).
Method 25: Parameter estimates back-calculated from a discrete time
identification non-linear regression method (Gallier and Otto, 1968).
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A2.4 SOSPD Model with a Zero
Method 2:
Method 3:
Parameters estimated from a closed loop step response test using a
least squares approach (Wang et al., 2001a).
Tm1 = Tm 2 . K m , τ m , Tm1 and Tm 3 (or Tm 4 ) are estimated from the
process step response; the latter three parameters are estimated using
a tabular approach (Pomerleau and Poulin, 2004).
A2.5 TOSPD Model
Method 2:
Method 3:
Parameters estimated using a tangent and point method (Murata and
Sagara, 1977).
Model parameters estimated using least squares regression in the
time domain; the model input is a step signal (Yu, 2006, p. 107).
A2.6 General Model
Method 2:
A FOLPD model is obtained using the tangent and point method of
Ziegler and Nichols (1942); label the parameters K 'm , Tm' and τ 'm .
(
Method 3:
Method 4:
)
Then, n = 10 τ 'm Tm' + 1 . Subsequently, τ m is estimated from the
open loop step response, and Tar is estimated using ‘known
methods’ (Schaedel, 1997).
Parameters estimated using a tangent and point method (Murata and
Sagara, 1977).
Model parameters Taa , τ m and Tar are estimated using the area
method (Gorez and Klàn, 2000).
A2.7 Non-Model Specific
Modelling methods have not been specified in the tables in most cases, as
typically the tuning rules are based on K u and Tu . Modelling methods have
been specified whenever relevant data have been estimated using a relay method.
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Appendix 2: Some Further Details on Process Modelling
Method 2:
Method 3:
Method 4:
561
K u and Tu estimated from an experiment using a relay in series
with the process in closed loop (Jones et al., 1997).
K u and Tu estimated with the assistance of a relay (Lloyd, 1994).
ω900 and A p estimated using a relay in series with an integrator
(Tang et al., 2002).
Method 5:
Method 6:
∧
∧
K u and Tu estimated from values determined from an experiment
using an amplitude dependent gain in series with the process in
closed loop (Pecharromán and Pagola, 1999).
Parameters estimated from an experiment using a relay in series
with the process in closed loop (Chen and Yang, 1993).
A2.8 IPD Model
A2.8.1 Parameters estimated from the open loop process step response
Method 2:
Method 3:
Method 4:
Method 5:
τ m assumed known; K m estimated from the slope at start of the
process step response (Ziegler and Nichols, 1942).
K m , τ m estimated from the process step response (Hay, 1998).
K m , τm estimated from the process step response (Clarke, 2006).
K m , τ m estimated from the process step response using a tangent
and point method (Bunzemeier, 1998).
A2.8.2 Parameters estimated from the closed loop step response
Method 6:
Parameters estimated from the servo or regulator closed loop
transient response, under PI control (Rotach, 1995).
b720_Appendix-02
562
Method 7:
Method 8:
Method 9:
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Parameters estimated from the servo closed loop transient response
under proportional control (Srividya and Chidambaram, 1997).
Parameters estimated from the closed loop response, under the
control of an on-off controller (Zou et al., 1997).
Parameters estimated from the closed loop response, under the
control of an on-off controller (Zou and Brigham, 1998).
A2.8.3 Parameters estimated from frequency domain closed loop information
Method 10: Parameters estimated from K u and Tu values, determined from an
experiment using a relay in series with the process in closed loop
(Tyreus and Luyben, 1992).
Method 11: Parameters estimated from values determined from an experiment
using an amplitude dependent gain in series with the process in
closed loop (Pecharromán and Pagola, 1999).
Method 12: Parameters estimated from data determined when a relay is placed in
series, in closed loop, with the closed loop compensated process,
under PI or PID control (NI Labview, 2001).
A2.9 FOLIPD Model
Method 2:
Method 3:
Method 4:
Method 5:
Parameters estimated from the open loop process step response and
its first and second time derivatives (Tsang and Rad, 1995).
Parameters estimated using the method of moments (Åström and
Hägglund, 1995).
Parameters estimated from values determined from an experiment
using an amplitude dependent gain in series with the process in
closed loop (Pecharromán and Pagola, 1999).
Parameters estimated from the open loop response of the process to
a pulse signal (Tachibana, 1984).
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Appendix 2: Some Further Details on Process Modelling
Method 6:
Method 7:
Method 8:
FA
563
Parameters estimated from the waveform obtained by introducing a
single symmetrical relay in series with the process in closed loop
(Majhi and Mahanta, 2001).
K m estimated from the response of the process to a square wave
pulse; other process parameters estimated from the waveform
obtained by introducing a relay in series with the process in closed
loop (Wang et al., 2001a).
Parameters estimated using a relay in series with the process in
closed loop (Perić et al., 1997).
A2.10 SOSIPD model
Method 2:
Parameters estimated from values determined from an experiment
using an amplitude dependent gain in series with the process in
closed loop (Pecharromán and Pagola, 1999).
A2.11 Unstable FOLPD Model
Method 2:
Method 3:
Method 4:
Method 5:
Method 6:
Parameters estimated by least squares fitting from the open loop
frequency response of the unstable process; this is done by
determining the closed loop magnitude and phase values of the
(stable) closed loop system and using the Nichols chart to determine
the open loop response (Huang and Lin, 1995; Deshpande, 1980).
Parameters estimated using a relay feedback approach (Majhi and
Atherton, 2000).
Parameters estimated using a biased relay feedback test (Huang and
Chen, 1999).
Parameters estimated using a single symmetric relay feedback
feedback test (Vivek and Chidambaram, 2005).
Tm and τ m estimated from data obtained when a relay is introduced
in series with the process in closed loop; K m assumed known
(Padhy and Majhi, 2006a).
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A2.12 Unstable SOSPD Model (one unstable pole)
Method 2:
Method 3:
Method 4:
Parameters estimated by least squares fitting from the open loop
frequency response of the unstable process; this is done by
determining the closed loop magnitude and phase values of the
(stable) closed loop system and using the Nichols chart to determine
the open loop response (Huang and Lin, 1995; Deshpande, 1980).
Parameters estimated by minimising the difference in the frequency
responses between the high order process and the model, up to the
ultimate frequency. The gain and delay are estimated from analytical
equations, with the other parameters estimated using a least squares
method (Kwak et al., 2000).
Parameters estimated using a biased relay feedback test (Huang and
Chen, 1999).
b720_Bibliography
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FA
b720_Index
17-Mar-2009
FA
Index
Åström and Hägglund (2004), 20, 98, 174,
360, 381
Åström and Hägglund (2006), 21, 22, 72,
74, 76, 109, 110–111, 145, 177, 227, 276,
321, 326, 356, 362, 390, 424
Atherton and Boz (1998), 175, 274, 477,
503–504
Atherton and Majhi (1998), 476–477
Atkinson (1968), 341
Atkinson and Davey (1968), 324
Auslander et al. (1975), 206
ABB (1996), 325
ABB (2001), 9, 32, 52, 152, 157, 319
Abbas (1997), 57, 100
Aikman (1950), 321
Alenany et al. (2005), 360, 378
Alfa-Laval, 7, 322
Alfaro Ruiz (2005a), 80, 122, 124, 207, 326,
359
Allen Bradley, 5, 8
Alvarez-Ramirez et al. (1998), 357
Andersson (2000), 74, 357
Anil and Sree (2005), 422, 423
Araki (1985), 156
Arbogast and Cooper (2007), 365, 367, 370
Arbogast et al. (2006), 74
Argelaguet et al. (1997), 156
Argelaguet et al. (2000), 156
Arrieta and Vilanova (2007), 125–127
Arrieta Orozco (2003), 39, 43, 47, 49, 122,
123, 124
Arrieta Orozco and Alfaro Ruiz (2003), 39,
43, 47, 49, 122, 123, 124
Arrieta Orozco (2006), 43, 49, 86, 87–88,
91, 94
Arvanitis et al. (2000a), 221, 232, 261, 275
Arvanitis et al. (2000b), 473, 477
Arvanitis et al. (2003a), 373, 376, 377
Arvanitis et al. (2003b), 400–409, 413
Åström (1982), 322, 325
Åström and Hägglund (1984), 325
Åström and Hägglund (1988), 79, 325
Åström and Hägglund (1995), 21, 56, 108,
164, 248, 319, 322, 325, 331, 347, 351,
359, 389, 413
Åström and Hägglund (2000), 19
Bai and Zhang (2007), 70
Bailey, 5, 6, 7, 8, 10
Bain and Martin (1983), 46, 53
Barberà (2006), 49, 94
Bateson (2002), 322, 327, 349
Bekker et al. (1991), 55
Belanger and Luyben (1997), 370
Benjanarasuth et al. (2005), 353, 379
Bequette (2003), 21, 24, 120
Bialkowski (1996), 74, 283, 287, 357, 436
Bi et al. (1999), 57
Bi et al. (2000), 57, 221
Blickley (1990), 324
Boe and Chang (1988), 77
Bohl and McAvoy (1976b), 210, 230, 241,
250
Borresen and Grindal (1990), 32, 79
Boudreau and McMillan (2006), 223, 357
Brambilla et al. (1990), 74, 112, 203, 227
Branica et al. (2002), 109
Bryant et al. (1973), 20, 52, 197–198
Buckley (1964), 21
599
b720_Index
600
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
Buckley et al. (1985), 354
Bunzemeier (1998), 351, 368
Calcev and Gorez (1995), 319, 325
Callender (1934), 359
Callender et al. (1935/6), 18, 23, 30, 78
Camacho et al. (1997), 100
Carr (1986), 324
Chandrashekar et al. (2002), 441, 455–456,
466–468
Chang et al. (1997), 182
Chao et al. (1989), 189, 190, 194, 239, 240,
241–242, 242–243, 244, 245, 246–247
Chau (2002), 326
Chen and Seborg (2002), 58, 103, 221–222,
357, 361, 389
Chen and Yang (1993), 318, 325, 330
Chen and Yang (2000), 68
Chen et al. (1997), 75
Chen et al. (1999a), 64, 354, 362, 389
Chen et al. (1999b), 64, 226
Chen et al. (2006), 283
Cheng and Hung (1985), 86, 93
Cheng and Yu (2000), 354
Chesmond (1982), 322, 327
Chidambaram (1994), 354
Chidambaram (1995a), 107–108
Chidambaram (1995b), 443, 452
Chidambaram (1997), 440, 448, 474
Chidambaram (1998), 443, 452
Chidambaram (2000a), 161, 176
Chidambaram (2000b), 376
Chidambaram (2000c), 441, 465
Chidambaram (2002), 28, 33, 58, 80, 103,
162, 181, 222, 280, 383, 420, 441–442,
449, 466
Chidambaram and Kalyan (1996), 492
Chidambaram and Sree (2003), 353, 361,
362, 376
Chidambaram et al. (2005), 59
Chien (1988), 74, 130, 146, 285, 287, 357,
366, 369, 394, 396
Chien and Fruehauf (1990), 130, 146, 285,
287, 357, 366, 369, 394, 396
Chien et al. (1952), 31, 78
Chien et al. (1999), 160–161, 376
Chien et al. (2003), 277, 282, 284, 286, 290,
292
Chiu et al. (1973), 52, 220
Chun et al. (1999), 75
Clarke (2006), 72, 356
Cluett and Wang (1997), 66–67, 101, 353,
361
Cohen and Coon (1953), 31, 78
Concept, 6
Connell (1996), 79
Controller structures
Classical controller, 6–7, 25, 134–148,
238–250, 286–287, 341–342, 368–370,
395–396, 423, 425, 436, 458–461, 491–
496
Controller with filtered derivative, 6, 122–
133, 236–237, 284–285, 298, 305–307,
316, 336–340, 366–367, 394
Generalised classical controller, 8, 26,
149–151, 251–252, 288, 343–345, 371,
397–398, 462, 483, 508, 516–517
Ideal controller in series with a first order
lag, 6, 24, 118–121, 182, 232–235, 282–
283, 297, 315, 332–335, 364–365, 392–
393, 422, 455–457, 481–482, 490, 513–
515
Ideal PI controller, 5, 18–22, 28–29, 30–
77, 180–181, 183–205, 277–278, 293–
295, 310–311, 318–323, 350–358, 383–
384, 385–387, 430, 438, 440–446, 480,
486–487, 511–512
Ideal PID controller, 4, 5, 23, 78–117,
206–231, 279–281, 296, 303–304, 312–
314, 324–331, 359–363, 388–391, 420–
421, 424, 447–454, 488–489, 506–507
Two degree of freedom controller 1, 8–9,
27, 152–167, 253–263, 289–291, 299–
301, 308–309, 317, 346–348, 372–377,
399–415, 426, 431–435, 437, 439, 463–
472, 484–485, 497–502, 518–519
Two degree of freedom controller 2, 9,
168–169, 378–380, 416–417, 427–428,
473–474, 509–510
Two degree of freedom controller 3, 10–
11, 170–179, 264–276, 292, 302, 349,
381–382, 418–419, 429, 475–479, 503–
505, 520
ControlSoft Inc. (2005), 80
Coon (1956), 350, 385
Coon (1964), 350, 385
Cooper (2006a), 165, 179
Cooper (2006b), 120, 179
Corripio (1990), 324, 341
Cox et al. (1994), 323
b720_Index
17-Mar-2009
Index
Cox et al. (1997), 67
Cuesta et al. (2006), 60
Davydov et al. (1995), 57, 128, 311, 316
De Oliveira et al. (1995), 74
De Paor (1993), 325
De Paor and O’Malley (1989), 442, 452
Desbiens (2007), 181, 203, 278
Devanathan (1991), 45, 90
Dutton et al. (1997), 330
ECOSSE Team (1996a), 45
ECOSSE Team (1996b), 322, 327
Edgar et al. (1997), 38, 137–138, 153, 319,
341, 346
Eriksson and Johansson (2007), 174, 178
Ettaleb and Roche (2000), 57
Faanes and Skogestad (2004), 33, 58, 74,
136, 147
Farrington (1950), 324
Fanuc, 5
Fertik (1975), 19, 42, 49, 136, 140
Fertik and Sharpe (1979), 32
Fischer and Porter, 6, 7, 10
Fisher–Rosemount, 10
Foley et al. (2005), 59
Ford (1953), 359
Foxboro, 6, 7, 10, 134
Frank and Lenz (1969), 33, 40, 41, 43
Friman and Waller (1997), 64, 109, 319
Fruehauf et al. (1993), 56, 99, 352
Fukura and Tamura (1983), 50, 197
Gaikward and Chidambaram (2000), 443,
452
Gain margin, analytical calculation, 522–
528
Gain margin, figures, 530, 532–533
Gain and phase margins, constant, 534–543
PI controller, FOLPD model, 534–536
PI controller, IPD model, 536–539
PD controller, FOLIPD model, 542–543
PID controller, FOLPD model,
classical controller, 539–541
PID controller, SOSPD model,
series controller, 541–542
PID controller, SOSPD model with
a negative zero, classical controller,
542
FA
601
Gallier and Otto (1968), 45, 90, 191, 214
García and Castelo (2000), 328
Genesis, 9
Geng and Geary (1993), 77, 116
Gerry (1998), 19, 89
Gerry (1999), 74, 113
Gerry (2003), 45
Gerry and Hansen (1987), 19
Gong et al. (1996), 130
Gong et al. (1998), 23, 131
Gong (2000), 113
Gonzalez (1994), 56, 100
Goodwin et al. (2001), 129
Gorecki et al. (1989), 55, 62, 99, 105, 352,
360, 362
Gorez (2003), 58, 104, 109
Gorez and Klàn (2000), 220, 312, 313
Gu et al. (2003), 117
Haalman (1965), 19, 40, 239, 352, 388
Haeri (2002), 163
Haeri (2005), 59
Hägglund and Åström (1991), 322
Hägglund and Åström (2002), 68, 165, 376,
471
Hägglund and Åström (2004), 21, 70–71,
202–203, 205, 321, 355–356
Hang and Åström (1988a), 167
Hang and Åström (1988b), 325
Hang and Cao (1996), 167
Hang et al. (1991), 55, 167
Hang et al. (1993a), 63, 223, 233
Hang et al. (1993b), 63, 134, 318, 341
Hang et al. (1994), 236
Hansen (1998), 275
Hansen (2000), 21, 376, 426
Harriott (1964), 327
Harriott (1988), 39, 47, 184
Harris and Tyreus (1987), 146, 178
Harrold (1999), 135, 342
Hartmann and Braun, 6
Hartree et al. (1937), 25
Hassan (1993), 189, 210–211
Hay (1998), 32–33, 79–80, 322, 327, 351,
352, 359
Hazebroek and Van der Waerden (1950),
30–31, 39–40, 352
Heck (1969), 32
Henry and Schaedel (2005), 65
Hill and Adams (1988), 77
b720_Index
602
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
Hill and Venable (1989), 36–37, 83–85
Hiroi and Terauchi (1986), 152, 170
Ho and Xu (1998), 443, 492
Ho et al. (1994), 223
Ho et al. (1995a), 223–224
Ho et al. (1997), 224
Ho et al. (1998), 87, 92
Ho et al. (1999), 63, 87, 92
Ho et al. (2001), 178
Honeywell, 5, 7, 9, 11, 56, 248
Horn et al. (1996), 120, 151
Hougen (1979), 61, 146, 199, 247, 294, 354,
369, 387, 396
Hougen (1988), 199, 248, 369
Huang and Chao (1982), 136, 138, 139, 140,
141, 142, 143, 238, 240, 242, 243, 244,
245, 246
Huang and Chen (1997), 458, 492
Huang and Chen (1999), 458, 492, 502
Huang and Jeng (2002), 19, 46
Huang and Jeng (2003), 97–98, 218–219
Huang and Jeng (2005), 71, 146, 232, 249
Huang and Lin (1995), 463, 464, 497–502
Huang et al. (1996), 38, 46, 185–187, 191–
193, 264–268, 270–273
Huang et al. (1998), 74, 228
Huang et al. (2000), 165, 176, 261, 276,
290, 292
Huang et al. (2002), 121
Huang et al. (2003), 146, 251
Huang et al. (2005a), 71, 77, 146, 148, 232,
235
Huang et al. (2005b), 379, 416
Huba (2005), 429
Huba and Žáková (2003), 353, 387
Hwang (1995), 44, 48, 89, 96–97, 190, 195–
196, 213, 215–217
Hwang and Chang (1987), 318
Hwang and Fang (1995), 44, 51, 89, 97
Idzerda et al. (1955), 318
Intellution, 5
Isaksson and Graebe (1999), 75
Jacob and Chidambaram (1996), 440, 473
Jahanmiri and Fallahi (1997), 251
Jhunjhunwala and Chidambaram (2001),
448, 466
Johnson (1956), 318
Jones and Tham (2006), 296, 302
Jones et al. (1997), 323, 329
Juang and Wang (1995), 100
Jyothi et al. (2001), 29, 182, 291, 383–384,
420–421
Kamimura et al. (1994), 21, 56
Kang (1989), 208
Karaboga and Kalinli (1996), 326
Kasahara et al. (1997), 160
Kasahara et al. (1999), 131–132
Kasahara et al. (2001), 161
Kaya (2003), 381
Kaya and Atherton (1999), 381
Kaya and Scheib (1988), 136, 138, 139, 140,
142, 171, 172
Keane et al. (2000), 218
Keane et al. (2005), 331
Keviczky and Csáki (1973), 19, 20, 53, 194
Khan and Lehman (1996), 48, 55, 57
Khodabakhshian and Golbon (2004), 278
Kim et al. (2004), 163
King (2006), 73, 112
Kinney (1983), 341
Klán (2000), 58, 312
Klán (2001), 376, 390, 413
Klán and Gorez (2000), 319
Klán and Gorez (2005a), 59
Klán and Gorez (2005b), 59
Klein et al. (1992), 32, 61
Kookos et al. (1999), 354
Kosinsani (1985), 34, 82
Kotaki et al. (2005a), 153–154, 164
Kotaki et al. (2005b), 153–155
Kraus (1986), 134
Kristiansson (2003), 69, 149–150, 204, 228,
234, 251, 320, 332, 344
Kristiansson and Lennartson (1998a), 337–
338
Kristiansson and Lennartson (1998b), 338
Kristiansson and Lennartson (2000), 320,
338–339
Kristiansson and Lennartson (2002a), 320,
340, 366, 394
Kristiansson and Lennartson (2002b), 69,
119, 334
Kristiansson and Lennartson (2003), 320,
321
Kristiansson and Lennartson (2006), 69–70,
119, 120, 321, 335, 392
Kristiansson et al. (2000), 343
b720_Index
17-Mar-2009
FA
Index
Kuhn (1995), 57, 128
Kukal (2006), 60, 104
Kuwata (1987), 20, 27, 53, 77,
159, 160, 166–167, 198, 204,
262–263, 294, 295, 296, 300,
317, 352, 375, 386, 413, 415,
437, 438, 439
Kwak et al. (2000), 504–505
116,
231,
301,
430,
158,
259,
311,
435,
Landau and Voda (1992), 231
Larionescu (2002), 314, 316
Latzel (1988), 61, 129–130
Lavanya et al. (2006a), 237
Lavanya et al. (2006b), 73
Lee and Edgar (2002), 121, 168–169, 178–
179, 233, 379, 382, 416, 419, 474, 479
Lee and Shi (2002), 147, 151
Lee and Teng (2003), 313
Lee et al. (1992), 63, 106, 164, 177
Lee et al. (1998), 75, 113, 228
Lee et al. (2000), 454, 474, 489, 505, 507,
509
Lee et al. (2005), 323
Lee et al. (2006), 113, 227, 281, 362, 390,
421, 457, 505
Leeds and Northup, 5, 79
Lennartson and Kristiansson (1997), 337
Leonard (1994), 360
Leva (1993), 322, 336
Leva (2001), 65, 74, 112, 132
Leva (2005), 165
Leva (2007), 74
Leva and Colombo (2000), 132
Leva and Colombo (2004), 165
Leva et al. (1994), 70, 224–225
Leva et al. (2003), 68–69, 132, 202, 226
Leva et al. (2006), 132
Li et al. (1994), 105
Li et al. (2004), 283
Lim et al. (1985), 118, 232
Lipták (1970), 327
Lipták (2001), 33, 80
Litrico and Fromion (2006), 356, 363
Litrico et al. (2006), 356
Litrico et al. (2007), 358
Liu et al. (2003), 417, 427–428
Lloyd (1994), 328
Lopez (1968), 81, 86, 87, 183, 188, 207,
208–209
Lopez et al. (1969), 188, 210
603
Loron (1997), 438
Luo et al. (1996), 326
Luyben (1993), 77
Luyben (1996), 370
Luyben (1998), 443
Luyben (2000), 278
Luyben (2001), 74, 120
Luyben and Luyben (1997), 326
MacLellan (1950), 321, 327
Madhuranthakam et al. (2008), 155, 156,
256, 257, 269, 289–290
Maffezzoni and Rocco (1997), 130
Majhi (1999), 323, 329, 346, 347
Majhi (2005), 50
Majhi and Atherton (1999), 175, 274, 477,
503–504
Majhi and Atherton (2000), 440, 475–476
Majhi and Mahanta (2001), 419
Majhi and Litz (2003), 250
Mann et al. (2001), 54, 102
Mantz and Tacconi (1989), 348
Manum (2005), 21, 222, 442
Marchetti and Scali (2000), 181, 203, 228,
278, 280, 295, 296
Marchetti et al. (2001), 453, 488
Marlin (1995), 38, 46, 81, 91
Mataušek and Kvaščev (2003), 69
Matsuba et al. (1998), 77, 116
Maximum sensitivity, analytical calculation,
529
Maximum sensitivity, figures, 530
Maximum sensitivity, constant, 538
McAnany (1993), 56
McAvoy and Johnson (1967), 187, 240, 324
McMillan (1984), 76, 116, 387, 391, 487,
489
McMillan (1994), 32, 318, 322, 325, 327
McMillan (2005), 22, 59, 322, 325, 335
Mesa et al. (2006), 60
Modicon, 6
Modcomp, 8
Mollenkamp et al. (1973), 52, 220
Moore, 10, 11
Morari and Zafiriou (1989), 130
Morilla et al. (2000), 102
Moros (1999), 31, 80
Murata and Sagara (1977), 34, 46, 183, 191,
293, 310
Murrill (1967), 32, 33, 40, 41, 79, 81, 86, 87
b720_Index
604
17-Mar-2009
FA
Handbook of PI and PID Controller Tuning Rules
Naşcu et al. (2006), 235
NI Labview (2001), 323, 329–330, 350, 351,
359
Normey-Rico et al. (2000), 168, 378
Normey-Rico et al. (2001), 378
Nomura et al. (1993), 20, 23, 27, 56, 99–
100, 160
O’Connor and Denn (1972), 90
O’Dwyer (2001a), 63, 146, 249, 287, 355,
390
O’Dwyer (2001b), 135–136, 342
Ogawa (1995), 75, 357
Ogawa and Katayama (2001), 162
Okada et al. (2006), 54
OMEGA Books (2005), 152, 348
Omron, 8
Oppelt (1951), 31
Ou and Chen (1995), 139, 142
Ou et al. (2005a), 364, 446
Ou et al. (2005b), 365, 393
Ou et al. (2005c), 132
Oubrahim and Leonard (1998), 263, 415
Ozawa et al. (2003), 163
Padhy and Majhi (2006a), 177, 478
Padhy and Majhi (2006b), 73
Pagola and Pecharromán (2002), 323
Panda et al. (2004), 204, 228, 232, 234, 249
Panda et al. (2006), 357
Paraskevopoulos et al. (2004), 464–465, 469
Paraskevopoulos et al. (2006), 444–445,
459–461, 492–496
Park et al. (1997), 212, 214
Park et al. (1998), 478
Parr (1989), 32, 79, 318, 322, 323, 324, 327,
328
Paz Ramos et al. (2005), 360
Pecharromán (2000a), 257, 258, 374, 410,
431
Pecharromán (2000b), 258–259, 374–375,
411, 432–433
Pecharromán and Pagola (2000), 257–258,
374, 410, 432
Pemberton (1972a), 34, 220
Pemberton (1972b), 219–220
Peng and Wu (2000), 81
Penner (1988), 358
Perić et al. (1997), 77, 116, 387, 391
Pessen (1953), 341
Pessen (1954), 341
Pessen (1994), 86, 148, 318
Pettit and Carr (1987), 324
Phase margin, analytical calculation, 522–
528
Phase margin, figures, 530, 532–533
Pinnella et al. (1986), 54
PMA (2006), 33, 80
Polonyi (1989), 296
Pomerleau and Poulin (2004), 199, 277, 282
Potočnik et al. (2001), 66
Poulin and Pomerleau (1996), 352, 386,
395–396, 486, 491
Poulin and Pomerleau (1997), 395–396, 491
Poulin and Pomerleau (1999), 353, 387
Poulin et al. (1996), 201, 249, 286, 396, 487
Pramod and Chidambaram (2000), 449
Prashanti and Chidambaram (2000), 470–
471
Process models
Delay model, 12, 18–27
Delay model with a zero, 12, 28–29
Fifth order system plus delay model, 13,
303–309
FOLIPD model, 14, 385–419
FOLIPD model with a zero, 14, 420–423
FOLPD model, 13, 30–179
FOLPD model with a zero, 13, 180–182
General model, 14, 310–317
General model with integrator, 15, 438–
439
IPD model, 14, 350–382
IPD model with a zero, 14, 383–384
I 2 PD model, 14, 424–429
Non-model specific, 14, 318–349
SOSIPD model, 14, 430–435
SOSIPD model with a zero, 14, 436
SOSPD model, 13, 183–276
SOSPD model with a zero, 13, 277–292
TOSIPD model, 14, 437
TOSPD model, 13, 293–302
Unstable FOLPD model, 15, 440–479
Unstable FOLPD model with a zero, 15,
480–485
Unstable SOSPD model (one unstable
pole), 15, 486–505
Unstable SOSPD model (two unstable
poles), 15, 506–510
Unstable SOSPD model with a zero, 15,
511–520
b720_Index
17-Mar-2009
Index
Prokop et al. (2000), 57
Prokop et al. (2005), 319
Pulkkinen et al. (1993), 75
Ramasamy and Sundaramoorthy (2007),
330–331, 333, 344–345
Rao and Chidambaram (2006a), 516
Rao and Chidambaram (2006b), 516, 520
Ream (1954), 23, 52, 197
Reswick (1956), 31
Rice (2004), 362, 370
Rice (2006), 377
Rice and Cooper (2002), 364, 369
Rivera et al. (1986), 74, 112
Rivera and Jun (2000), 120, 228, 233, 364,
390, 393
Robbins (2002), 319, 326
Rotach (1994), 327–328
Rotach (1995), 199, 220, 352, 361
Rotstein and Lewin (1991), 446, 453, 488
Rovira et al. (1969), 45, 48, 90, 93
Rutherford (1950), 321, 327
Sadeghi and Tych (2003), 91, 92, 93
Sain and Özgen (1992), 79
Saito et al. (1990), 99
Sakai et al. (1989), 32
SATT Instruments, 11
Schaedel (1997), 65, 118, 128, 144,
248, 297, 298, 311, 315
Schlegel (1998), 21
Schneider (1988), 55
Seki et al. (2000), 325
Setiawan et al. (2000), 158
Shamsuzzoha and Lee (2006a), 151,
288, 363
Shamsuzzoha and Lee (2006b), 454
Shamsuzzoha and Lee (2007a), 115,
229–230, 262, 363, 377, 391, 415,
489
Shamsuzzoha and Lee (2007b), 150
Shamsuzzoha and Lee (2007c), 169
Shamsuzzoha and Lee (2007d), 508
Shamsuzzoha et al. (2004), 151
Shamsuzzoha et al. (2005), 454, 489
Shamsuzzoha et al. (2006a), 398
Shamsuzzoha et al. (2006b), 510
Shamsuzzoha et al. (2006c), 114, 229,
390, 422
Shamsuzzoha et al. (2007), 371, 462
201,
252,
166,
454,
363,
FA
605
Shen (1999), 254
Shen (2000), 255–256
Shen (2002), 158, 347
Shi and Lee (2002), 147
Shi and Lee (2004), 252
Shigemasa et al. (1987), 159
Shin et al. (1997), 329
Shinskey (1988), 19, 35, 39, 137, 138, 153,
156, 208, 238, 351, 368, 372, 458
Shinskey (1994), 19, 39, 138, 157, 184, 239,
254, 351, 368, 372, 385, 395, 399
Shinskey (1996), 19, 38, 39, 137, 152, 184,
187, 239, 253, 254, 368–369, 372
Shinskey (2000), 33, 135
Shinskey (2001), 33
Shinskey (2003), 39, 138
Siemens, 6
Sklaroff (1992), 56, 248
Skoczowski (2004), 59, 249
Skoczowski and Tarasiejski (1996), 314
Skogestad (2001), 20, 352
Skogestad (2003), 20, 21, 22, 58, 248, 352,
353, 396, 425
Skogestad (2004a), 358
Skogestad (2004b), 221, 353, 418, 429
Skogestad (2004c), 222
Skogestad (2006a), 61
Skogestad (2006b), 61, 249
Slätteke (2006), 180, 181
Smith (1998), 67
Smith (2002), 50, 56, 74, 357, 358
Smith (2003), 93, 228, 326
Smith and Corripio (1997), 34, 46, 53, 138,
140, 144
Smith et al. (1975), 53, 220, 247
Somani et al. (1992), 62, 106, 200–201
Square D, 9
Sree and Chidambaram (2003a), 181, 182
Sree and Chidambaram (2003b), 481, 484–
485
Sree and Chidambaram (2003c), 480, 485
Sree and Chidambaram (2004), 511, 513,
518–519
Sree and Chidambaram (2005a), 467
Sree and Chidambaram (2005b), 361
Sree and Chidambaram (2006), 353, 362,
376, 389, 442, 444, 450–452, 456, 465,
466–467, 468, 474, 477, 480, 481–482,
483, 485, 502, 511, 514–515, 517, 519
Sree et al. (2004), 59, 104, 442, 449–450
b720_Index
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Handbook of PI and PID Controller Tuning Rules
Srinivas and Chidambaram (2001), 162, 465
Srividya and Chidambaram (1997), 354
St. Clair (1997), 32, 135, 341
Streeter et al. (2003), 331
Sung et al. (1996), 212, 214
Suyama (1992), 53, 99, 220
Suyama (1993), 159
Syrcos and Kookos (2005), 90
Tachibana (1984), 388
Taguchi and Araki (2000), 157, 260, 299–
300, 375, 411–412, 433
Taguchi and Araki (2002), 157, 412
Taguchi et al. (1987), 156, 257, 410
Taguchi et al. (2002), 375, 412, 433–434
Tan et al. (1996), 70, 108, 201
Tan et al. (1998a), 393
Tan et al. (1998b), 120, 364, 393
Tan et al. (1999a), 135, 326, 342
Tan et al. (1999b), 446, 453
Tan et al. (2001), 326, 342
Tang et al. (2002), 323, 330
Tang et al. (2007), 121, 133, 147–148
Tavakoli and Fleming (2003), 47
Tavakoli and Tavakoli (2003), 123
Tavakoli et al. (2005), 158, 375
Tavakoli et al. (2006), 47
Taylor, 11
Thomasson (1997), 74, 75, 357
Thyagarajan and Yu (2003), 205
Tinham (1989), 324
Toshiba, 7, 9, 159
Trybus (2005), 60, 222–223, 313–314
Tsang and Rad (1995), 144, 397
Tsang et al. (1993), 144, 149, 397
Tuning rules:
Direct synthesis, frequency domain
criteria, 21, 29, 61–73, 105–112, 118–
120, 129–130, 146, 149–150, 164–165,
177–178, 181, 182, 199–203, 223–227,
232, 236, 249–250, 261–262, 276, 278,
280, 286–287, 290–291, 294, 297, 298,
301, 303–304, 305–307, 308–309, 311,
314, 315, 319–321, 331, 334–335, 336–
340, 347–348, 349, 354–356, 362, 366,
369, 376, 381, 383–384, 387, 389–390,
392, 394, 396, 398, 413–414, 419, 420–
421, 424, 438, 442–445, 452, 459–461,
470–472, 473, 477–478, 487, 492–496,
504–505
Direct synthesis, time domain criteria, 20–
21, 23, 27, 28, 52–61, 98–105, 118,
128–129, 144–145, 149, 158–164, 168,
175–176, 181, 182, 197–199, 219–223,
232, 247–248, 251, 259–260, 274–276,
277, 280, 282–283, 284, 286, 290, 292,
294, 300, 311, 312–314, 316, 317, 330–
331, 333, 344–345, 352–353, 360–362,
375–376, 378–379, 383, 386–387, 388–
389, 396, 397, 413, 416, 418, 420, 423,
425, 426, 429, 430, 435, 438, 439, 440–
442, 448–452, 455–456, 458, 465–469,
473, 476–477, 480, 481–482, 483, 484–
485, 488, 492, 502, 503–504, 506, 511–
512, 513–515, 516, 518–519, 520
Minimum performance index, regulator
tuning, 19, 33–45, 81–90, 122–123,
136–139, 152–155, 171, 180, 183–190,
207–213, 238–243, 253–256, 264–269,
289, 293, 310, 341, 346, 351–352, 360,
368–369, 372–373, 385–386, 388, 395–
396, 399–404, 447, 458, 463–464, 486,
491, 497–499
Minimum performance index, servo
tuning, 19, 45–51, 90–97, 123–124,
140–143, 156, 172–173, 191–197, 213–
217, 243–247, 257, 269–273, 279, 290,
293, 310, 347, 360, 373, 378, 386, 388,
405–409, 440, 448, 464, 475–476, 500–
501
Minimum performance index, servo/
regulator tuning, 125–127, 257–259,
299–300, 448
Minimum performance index, other
tuning, 19–20, 51–52, 97–98, 156–158,
174, 218–219, 296, 332, 343–344, 347,
352, 360, 374–375, 381, 410–412, 431–
434, 464–465
Other tuning rules, 22, 205, 321–323,
327–330, 358
Process reaction curve methods, 18–19,
23, 25, 30–33, 78–80, 134–136, 152,
170, 206–207, 350–351, 359, 368, 385
Robust, 21–22, 23, 24, 74–76, 112–115,
120–121, 130–133, 146–148, 151, 165–
166, 168–169, 178–179, 181, 182, 203–
204, 227–230, 233–235, 236–237, 251–
252, 278, 280–281, 283, 285, 287, 288,
295, 296, 302, 314, 316, 357–358, 362–
363, 364–365, 366–367, 369–370, 371,
b720_Index
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Index
377, 379–380, 382, 387, 390–391, 393,
394, 396, 415, 416–417, 419, 421, 422,
427–428, 436, 446, 453–454, 457, 462,
472, 474, 478–479, 482, 488–489, 490,
505, 507, 508, 509–510, 517
Ultimate cycle, 22, 26, 76–77, 116–117,
148, 151, 166–167, 169, 204–205, 230–
231, 235, 250, 262–263, 278, 283, 295,
296, 301, 318–319, 324–326, 335, 341–
342, 348, 358, 363, 370, 371, 387, 391,
415, 435, 437, 487, 489
Turnbull, 7
Tyreus and Luyben (1992), 326, 352
Urrea et al. (2006), 22
Valentine and Chidambaram (1997a), 57,
102
Valentine and Chidambaram (1997b), 448
Valentine and Chidambaram (1998), 440
Van der Grinten (1963), 20, 98, 219, 388
VanDoren (1998), 152, 348
Velázquez-Figueroa (1997), 386, 387, 388
Venkatashankar and Chidambaram (1994),
443
Vilanova (2006), 121
Vilanova and Balaguer (2006), 133
Visioli (2001), 360, 447
Visioli (2002), 179
Visioli (2005), 74
Visioli (2006), 61
Vítečková (1999), 53, 220–221, 353, 389
Vítečková (2006), 61, 105, 199, 223
Vítečková and Víteček (2002), 54, 103
Vítečková and Víteček (2003), 54, 103
Vítečková et al. (2000a), 53, 220–221, 353,
389
Vítečková et al. (2000b), 53, 221
Vivek and Chidambaram (2005), 450
Voda and Landau (1995), 53, 64
Vrančić (1996), 294, 301, 305, 319, 336–
337, 348
Vrančić and Lumbar (2004), 307, 308–309
Vrančić et al. (1996), 294, 319
Vrančić et al. (1999), 303–304, 306, 337
Vrančić et al. (2004a), 294, 307, 308–309
Vrančić et al. (2004b), 294
Wade (1994), 322, 327
Wang (2000), 226
FA
607
Wang (2003), 23, 75–76, 113–114, 237
Wang and Cai (2001), 414
Wang and Cai (2002), 414, 472
Wang and Clements (1995), 220
Wang and Cluett (1997), 352–353, 361
Wang and Cluett (2000), 66–67, 101
Wang and Jin (2004), 488, 506
Wang and Shao (1999a), 225
Wang and Shao (1999b), 68
Wang and Shao (2000a), 68
Wang and Shao (2000b), 226
Wang et al. (1995a), 91, 93, 279
Wang et al. (1995b), 319, 334
Wang et al. (1999), 225
Wang et al. (2000a), 67
Wang et al. (2000b), 280
Wang et al. (2001a), 280
Wang et al. (2001b), 419
Wang et al. (2002), 144–145
Wang et al. (2005), 365
Wills (1962a), 214, 219, 230
Wills (1962b), 207, 213
Wilton (1999), 51–52, 218
Witt and Waggoner (1990), 134–135, 137,
139, 141, 143
Wojsznis and Blevins (1995), 116
Wojsznis et al. (1999), 116–117, 326
Wolfe (1951), 19, 31, 350
Xing et al. (2001), 92, 95, 96
Xu and Shao (2003a), 414
Xu and Shao (2003b), 472
Xu and Shao (2003c), 414
Xu and Shao (2004a), 414, 418
Xu and Shao (2004b), 71
Xu et al. (2004), 71
Xu et al. (2005), 60
Yang and Clarke (1996), 63, 223, 249
Yi and De Moor (1994), 336
Yokogawa, 5, 8
Young (1955), 322
Yu (1988), 35, 40–41, 42
Yu (1999), 326
Yu (2006), 21, 22, 24, 26, 130, 151, 295,
357, 366, 371
Zhang (1994), 56, 286
Zhang (2006), 457
Zhang and Xu (2002), 472, 490
b720_Index
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FA
Handbook of PI and PID Controller Tuning Rules
Zhang et al. (1996a), 328
Zhang et al. (1996b), 147
Zhang et al. (1999a), 364, 369
Zhang et al. (1999b), 357, 393
Zhang et al. (2000), 457
Zhang et al. (2002a), 379
Zhang et al. (2002b), 121, 132, 147, 233
Zhang et al. (2002c), 169
Zhang et al. (2003), 169
Zhang et al. (2005), 93
Zhang et al. (2006), 380
Zhong and Li (2002), 75, 236, 358
Zhuang (1992), 41, 44, 48, 50, 51, 86, 88,
92, 95, 172–173
Zhuang and Atherton (1993), 41, 44, 48, 50,
51, 86, 88, 92, 95, 107, 172–173
Ziegler and Nichols (1942), 30, 78, 318,
324, 350, 359
Zou and Brigham (1998), 362
Zou et al. (1997), 112, 362