[Communications and Control Engineering] Tadeusz Kaczorek - Polynomial and rational matrices Applications in dynamical systems theory (2006, Springer)

Communications and Control Engineering
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Tadeusz Kaczorek
Polynomial and
Rational Matrices
Applications in Dynamical Systems Theory
123
Tadeusz Kaczorek, Prof. dr hab. inż.
Institute of Control and Industrial Electronics
Faculty of Electrical Engineering
Warsaw University of Technology
00-662 Warsaw
ul. Koszykowa 75m. 19
Poland
Series Editors
E.D. Sontag · M. Thoma · A. Isidori · J.H. van Schuppen
British Library Cataloguing in Publication Data
Kaczorek, T. (Tadeusz), 1932Polynomial and rational matrices : applications in
dynamical systems theory. - (Communications and control
engineering)
1. Automatic control - Mathematics 2. Electrical
engineering - Mathematics 3. Matrices 4. Linear systems
5. Polynomials
I. Title
629.8’312
ISBN-13: ISBN-13: 9781846286049
ISBN-10: ISBN-10: 1846286042
Library of Congress Control Number: 2006936878
Communications and Control Engineering Series ISSN 0178-5354
ISBN 978-1-84628-604-9
e-ISBN 1-84628-605-0
Printed on acid-free paper
© Springer-Verlag London Limited 2007
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
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Preface
This monograph covers the selected applications of polynomial and rational
matrices to the theory of both continuous-time and discrete-time linear systems. It
is an extended English version of its preceding Polish edition, which was based on
the lectures delivered by the author to the Ph.D. students of the Faculty of
Electrical Engineering at Warsaw University of Technology during the academic
year 2003/2004.
The monograph consists of eight chapters, an appendix and a list of references.
Chapter 1 is devoted to polynomial matrices. It covers the following topics:
basic operations on polynomial matrices, the generalised Bézoute theorem, the
CayleyHamilton theorem, elementary operations on polynomial matrices, the
choosing of a basis for a space of polynomial matrices, equivalent polynomial
matrices, reduced row matrices and reduced column matrices, the Smith canonical
form of polynomial matrices, elementary divisors and zeros of polynomial
matrices, similarity of polynomial matrices, the Frobenius and Jordan canonical
forms, cyclic matrices, pairs of polynomial matrices, the greatest common divisors
and the smallest common multiplicities of matrices, the generalised Bezoute
identity, regular and singular matrix pencil decompositions, and the
WeierstrassKronecker canonical form of a matrix pencil.
Rational functions and matrices are discussed in Chap. 2. With the basic
definitions and operations on rational functions introduced at the beginning, the
following issues are subsequently addressed: decomposition into the sum of
rational functions, operations on rational matrices, the decomposition of a matrix
into the sum of rational matrices, the inverse matrix of a polynomial matrix and its
reducibility, the McMillan canonical form of rational matrices, the first
factorization of rational matrices and the application of rational matrices in the
synthesis of control systems.
Chapter 3 addresses normal matrices and systems. A rational matrix is called
normal if every non-zero minor of size 2 of the polynomial matrix of the
denominator is divisible by the minimal polynomial of this matrix. It has been
proved that a rational matrix is normal if and only if its McMillan polynomial is
equal to the smallest common denominator of all the elements of the rational
matrix. Further, the following issues are discussed: the fractional forms of normal
vi
Preface
matrices, the sum and product of normal matrices, the inverse matrix of a normal
matrix, the decomposition of normal matrices into the sum of normal matrices, the
structural decomposition of normal matrices, the normalisation of matrices via
feedback and electrical circuits as examples of normal systems.
The problem of the realisation of normal matrices is addressed in Chap. 4. The
problem formulation is provided; further the following issues are discussed:
necessary and sufficient conditions for the existence of minimal and cyclic
realisations, methods of computing the realisation with the state matrix in both the
Frobenius and Jordan canonical forms, structural stability and the computation of
the normal transfer function matrix
Chapter 5 is devoted to normal singular systems. In particular it focuses on
discrete singular systems, cyclic pairs of matrices, the normal inverse matrices of
cyclic pairs, normal transfer matrices, reachability and cyclicity of singular systems,
cyclicity of feedback systems, computation of equivalent standard systems for
singular systems. It is shown that electrical circuits consisting of resistances and
inductances or resistances and capacities, together with ideal voltage (current)
sources, constitute examples of singular continuous-time systems. Both the Kalman
decomposition and the structural decomposition of the transfer matrix are generalised
to the case of singular systems.
Polynomial matrix equations, both rational and algebraic, are discussed in
Chap. 6. The chapter begins with unilateral polynomial equations with two
unknown matrices. Subsequently the following issues are addressed: the
computation of minimal degree solutions to matrix equations, bilateral polynomial
equations, the computation of rational solutions to polynomial equations, matrix
equations of the m-th order, the Kronecker product of matrices and its applications,
and the methods for computing solutions to Sylvester and Lapunov matrix
equations.
Chapter 7, the last one, is devoted to the problem of realisation and perfect
observers for linear systems. A new method for computing minimal realisation for
a given improper transfer matrix is provided together with the existence conditions;
subsequently the methods for computing full and reduced order observers, as well
as functional perfect observers, for 1D and 2D systems are given.
In Chap. 8 some new results (published and unpublished) are presented on
positive linear discrete-time and continuous-time systems with delays: asymptotic
and robust stability, reachability, minimum energy control and positive realisation
problem.
The Appendix contains some basic definitions and theorems pertaining to the
controllability and observability of linear systems.
The monograph contains some original results of the author, most of which
have already been published.
It is haped that this monograph will be of value to Ph.D. students and
researchers from the field of control theory and circuit theory. It can be also
recommended for undergraduates in electrical engineering, electronics,
mechatronics and computer engineering.
I would like to express my gratitude to Professors M. Busáowicz and J. Klamka,
the reviewers of the Polish version of the book, for their valuable comments and
Preface
vii
suggestions, which helped to improve this monograph. I also wish to thank my
Ph.D. students, the first readers of the manuscript, for their remarks.
I wish to extend my special thanks to my Ph.D. students Maciej Twardy,
Konrad Markowski and Stefan KrzemiĔski for their valuable help in the
preparation of this English edition.
T. Kaczorek
Contents
Notation .................................................................................................................xv
1 Polynomial Matrices............................................................................................1
1.1 Polynomials ...................................................................................................1
1.2 Basic Notions and Basic Operations on Polynomial Matrices.......................5
1.3 Division of Polynomial Matrices ...................................................................9
1.4 Generalized Bezoute Theorem and the Cayley–Hamilton Theorem ...........16
1.5 Elementary Operations on Polynomial Matrices .........................................20
1.6 Linear Independence, Space Basis and Rank of Polynomial Matrices ........23
1.7. Equivalents of Polynomial Matrices...........................................................27
1.7.1 Left and Right Equivalent Matrices ...................................................27
1.7.2 Row and Column Reduced Matrices..................................................30
1.8 Reduction of Polynomial Matrices to the Smith Canonical Form ...............32
1.9 Elementary Divisors and Zeros of Polynomial Matrices .............................37
1.9.1 Elementary Divisors...........................................................................37
1.9.2 Zeros of Polynomial Matrices ............................................................39
1.10 Similarity and Equivalence of First Degree Polynomial Matrices.............42
1.11 Computation of the Frobenius and Jordan Canonical Forms of Matrices..45
1.11.1 Computation of the Frobenius Canonical Form of a Square
Matrix ..............................................................................................45
1.11.2 Computation of the Jordan Canonical Form of a Square Matrix......47
1.12 Computation of Similarity Transformation Matrices.................................49
1.12.1 Matrix Pair Method ..........................................................................49
1.12.2 Elementary Operations Method........................................................54
1.12.3 Eigenvectors Method........................................................................57
1.13 Matrices of Simple Structure and Diagonalisation of Matrices .................59
1.13.1 Matrices of Simple Structure............................................................59
1.13.2 Diagonalisation of Matrices of Simple Structure .............................61
1.13.3 Diagonalisation of an Arbitrary Square Matrix by the Use of a
Matrix with Variable Elements ........................................................65
1.14 Simple Matrices and Cyclic Matrices ........................................................67
1.14.1 Simple Polynomial Matrices ............................................................67
x
Contents
1.14.2 Cyclic Matrices ................................................................................69
1.15 Pairs of Polynomial Matrices.....................................................................75
1.15.1 Greatest Common Divisors and Lowest Common Multiplicities of
Polynomial Matrices ........................................................................75
1.15.2 Computation of Greatest Common Divisors of a Polynomial
Matrix ..............................................................................................77
1.15.3 Computation of Greatest Common Divisors and Smallest Common
Multiplicities of Polynomial Matrices .............................................78
1.15.4 Relatively Prime Polynomial Matrices and the Generalised Bezoute
Identity .............................................................................................84
1.15.5 Generalised Bezoute Identity ...........................................................86
1.16 Decomposition of Regular Pencils of Matrices .........................................87
1.16.1 Strictly Equivalent Pencils ...............................................................87
1.16.2 Weierstrass Decomposition of Regular Pencils................................92
1.17 Decomposition of Singular Pencils of Matrices ........................................95
1.17.1 Weierstrass–Kronecker Theorem .....................................................95
1.17.2 Kronecker Indices of Singular Pencils and Strict Equivalence of
Singular Pencils .............................................................................102
2 Rational Functions and Matrices ...................................................................107
2.1 Basic Definitions and Operations on Rational Functions ..........................107
2.2 Decomposition of a Rational Function into a Sum of Rational Functions.116
2.3 Basic Definitions and Operations on Rational Matrices ............................124
2.4 Decomposition of Rational Matrices into a Sum of Rational Matrices .....128
2.5 The Inverse Matrix of a Polynomial Matrix and Its Reducibility ..............132
2.6 Fraction Description of Rational Matrices and the McMillan Canonical
Form..........................................................................................................136
2.6.1 Fractional Forms of Rational Matrices.............................................136
2.6.2 Relatively Prime Factorization of Rational Matrices .......................146
2.6.3 Conversion of a Rational Matrix into the McMillan Canonical
Form...............................................................................................152
2.7 Synthesis of Regulators .............................................................................155
2.7.1 System Matrices and the General Problem of Synthesis of
Regulators ......................................................................................155
2.7.2 Set of Regulators Guaranteeing Given Characteristic Polynomials
of a Closed-loop System ...............................................................159
3 Normal Matrices and Systems........................................................................163
3.1 Normal Matrices ........................................................................................163
3.1.1 Definition of the Normal Matrix ......................................................163
3.1.2 Normality of the Matrix [Is – A]-1 for a Cyclic Matrix ....................164
3.1.3 Rational Normal Matrices ................................................................168
3.2 Fraction Description of Normal Matrices ..................................................170
3.3 Sum and Product of Normal Matrices and Normal Inverse Matrices ........175
3.3.1 Sum and Product of Normal Matrices ..............................................175
3.3.2 The Normal Inverse Matrix..............................................................180
3.4 Decomposition of Normal Matrices...........................................................182
Contents
xi
3.4.1 Decomposition of Normal Matrices into the Sum of Normal
Matrices .........................................................................................182
3.4.2 Structural Decomposition of Normal Matrices ................................185
3.5 Normalisation of Matrices Using Feedback...............................................191
3.5.1 State-feedback .................................................................................191
3.5.2 Output-feedback ...............................................................................197
3.6 Electrical Circuits as Examples of Normal Systems..................................200
3.6.1 Circuits of the Second Order ............................................................200
3.6.2 Circuits of the Third Order...............................................................203
3.6.3 Circuits of the Fourth Order and the General Case ..........................210
4 The Problem of Realization ...........................................................................219
4.1 Basic Notions and Problem Formulation...................................................219
4.2 Existence of Minimal and Cyclic Realisations ..........................................220
4.2.1 Existence of Minimal Realisations...................................................220
4.2.2 Existence of Cyclic Realisations ......................................................224
4.3 Computation of Cyclic Realisations ..........................................................226
4.3.1 Computation of a Realisation with the Matrix A in the Frobenius
Canonical Form..............................................................................226
4.3.2 Computation of a Cyclic Realisation with Matrix A in the Jordan
Canonical Form..............................................................................232
4.4 Structural Stability and Computation of the Normal Transfer Matrix .......244
4.4.1 Structural Controllability of Cyclic Matrices ...................................244
4.4.2 Structural Stability of Cyclic Realisation .........................................245
4.4.3 Impact of the Coefficients of the Transfer Function on the System
Description.....................................................................................247
4.4.4 Computation of the Normal Transfer Matrix on the Basis of Its
Approximation ...............................................................................249
5 Singular and Cyclic Normal Systems.............................................................255
5.1 Singular Discrete Systems and Cyclic Pairs ..............................................255
5.1.1 Normal Inverse Matrix of a Cyclic Pair ...........................................257
5.1.2 Normal Transfer Matrix ...................................................................260
5.2 Reachability and Cyclicity.........................................................................264
5.2.1 Reachability of Singular Systems.....................................................264
5.2.2 Cyclicity of Feedback Systems ........................................................267
5.3 Computation of Equivalent Standard Systems for Linear Singular
Systems .....................................................................................................272
5.3.1 Discrete-time Systems and Basic Notions ......................................272
5.3.2 Computation of Fundamental Matrices ............................................276
5.3.3 Equivalent Standard Systems ...........................................................279
5.3.4 Continuous-time Systems ...............................................................282
5.4 Electrical Circuits as Examples of Singular Systems ................................285
5.4.1 RL Circuits .......................................................................................285
5.4.2 RC Circuits.......................................................................................288
5.5 Kalman Decomposition .............................................................................291
5.5.1 Basic Theorems and a Procedure for System Decomposition..........291
xii
Contents
5.5.2 Conclusions and Theorems Following from System
Decomposition ................................................................................295
5.6 Decomposition of Singular Systems..........................................................298
5.6.1 Weierstrass–Kronecker Decomposition ...........................................298
5.6.2 Basic Theorems ................................................................................299
5.7 Structural Decomposition of a Transfer Matrix of a Singular System.......305
5.7.1 Irreducible Transfer Matrices...........................................................305
5.7.2 Fundamental Theorem and Decomposition Procedure.....................306
6 Matrix Polynomial Equations, and Rational and Algebraic Matrix
Equations.......................................................................................................313
6.1 Unilateral Polynomial Equations with Two Variables...............................313
6.1.1 Computation of Particular Solutions to Polynomial Equations ........313
6.1.2 Computation of General Solutions to Polynomial Equations...........319
6.1.3 Computation of Minimal Degree Solutions to Polynomial Matrix
Equations .......................................................................................322
6.2 Bilateral Polynomial Matrix Equations with Two Unknowns ...................325
6.2.1 Existence of Solutions......................................................................325
6.2.2 Computation of Solutions.................................................................328
6.3 Rational Solutions to Polynomial Matrix Equations..................................332
6.3.1 Computation of Rational Solutions ..................................................332
6.3.2 Existence of Rational Solutions of Polynomial Matrix Equations ...333
6.3.3 Computation of Rational Solutions to Polynomial Matrix
E
qua tions.........................................................................................334
6.4 Polynomial Matrix Equations ....................................................................336
6.4.1 Existence of Solutions......................................................................336
6.4.2 Computation of Solutions.................................................................337
6.5 The Kronecker Product and Its Applications.............................................340
6.5.1 The Kronecker Product of Matrices and Its Properties ....................340
6.5.2 Applications of the Kronecker Product to the Formulation of Matrix
Equations .......................................................................................343
6.5.3 Eigenvalues of Matrix Polynomials .................................................345
6.6 The Sylvester Equation and Its Generalization..........................................347
6.6.1 Existence of Solutions......................................................................347
6.6.2 Methods of Solving the Sylvester Equation .....................................349
6.6.3 Generalization of the Sylvester Equation .........................................357
6.7 Algebraic Matrix Equations with Two Unknowns ....................................358
6.7.1 Existence of Solutions......................................................................358
6.7.2 Computation of Solutions.................................................................360
6.8 Lyapunov Equations ..................................................................................361
6.8.1 Solutions to Lyapunov Equations.....................................................361
6.8.2 Lyapunov Equations with a Positive Semidefinite Matrix ...............363
7 The Realisation Problem and Perfect Observers of Singular Systems ......367
7.1 Computation of Minimal Realisations for Singular Linear Systems .........367
7.1.1 Problem Formulation........................................................................367
7.1.2 Problem Solution..............................................................................369
7.2 Full- and Reduced-order Perfect Observers...............................................376
Contents xiii
7.2.1 Reduced-order Observers ................................................................378
7.2.2 Perfect Observers for Standard Systems ..........................................384
7.3 Functional Observers .................................................................................392
7.4 Perfect Observers for 2D Systems .............................................................396
7.5 Perfect Observers for Systems with Unknown Inputs ...............................400
7.5.1 Problem Formulation .....................................................................400
7.5.2 Problem Solution ...........................................................................402
7.6 Reduced-order Perfect Observers for 2D Systems with Unknown Inputs 409
7.6.1 Problem Formulation........................................................................408
7.6.2 Problem Solution..............................................................................411
8 Positive Linear Systems with Delays..............................................................421
8.1 Positive Discrete-time and Continuous-time Systems ..............................421
8.1.1 Discrete-time Systems .....................................................................421
8.1.2 Continuous-time Systems ................................................................424
8.2 Stability of Positive Linear Discrete-time Systems with Delays ...............425
8.2.1 Asymptotic Stability.........................................................................425
8.2.2 Stability of Systems with Pure Delays .............................................432
8.2.3 Robust Stability of Interval Systems ................................................434
8.3 Reachability and Minimum Energy Control..............................................437
8.3.2 Minimum Energy Control ................................................................442
8.4 Realisation Problem for Positive Discrete-time Systems ..........................446
8.4.1 Problem Formulation........................................................................446
8.4.2 Problem Solution..............................................................................447
8.5 Realisation Problem for Positive Continuous-time Systems with Delays 456
8.5.1 Problem Formulation........................................................................456
8.5.2 Problem Solution..............................................................................457
8.6 Positive Realisations for Singular Multi-variable Discrete-time Systems
with Delays ...............................................................................................463
8.6.1 Problem Formulation........................................................................463
8.6.1 Problem Solution..............................................................................466
A Selected Problems of Controllability and Observability of Linear
Systems .........................................................................................................473
A.1 Reachability .............................................................................................473
A.2. Controllability .........................................................................................477
A.3 Observability............................................................................................480
A.4 Reconstructability ....................................................................................483
A.5 Dual System.............................................................................................485
6 Stabilizability and Detectability...................................................................485
References ..........................................................................................................487
Index ....................................................................................................................501
Notation
A
AT
A*
A-1
Adj A
A(s)
AJ
AS(s)
det A
Dn-1(O)
L[iuc]
L[i, j]
L[i+jub(s)]
mun
P[iuc]
P[i, j]
P[i+jub(s)]
P[i+jub(s)]
tr A
rank A
Im A
Ker A
W(A)
M(O)
<(O)
O
matrix
transpose of A
conjugate of A
inverse of A
adjoint (adjugate) of A
polynomial matrix
Jordan canonical form of A
Smith canonical form of A(s)
determinant of A
greatest common divisor of all the elements of Adj [OIn-A]
multiplication of the i-th row by the number cz0
interchange of the i-th and j-th rows
addition of the j-th row multiplied by the polynomial b(s)
to the i-th row
dimension of a matrix with m rows and n columns
multiplication of the i-th column by the number c z 0
interchange of the i-th and j-th columns
addition of the j-th column multiplied by the polynomial
b(s) to the i-th column
addition of the j-th column multiplied by the polynomial
b(s) to the i-th column
trace of A
rank of A
image of A
kernel of A
characteristic polynomial of A
characteristic polynomial of a matrix
minimal polynomial of a matrix
eigenvalue
xvi
Notion
In
0n
Mij
|| ||
mu n
mu n
,
mu n
[s]
(s)
[s], [s]
(s)
p(s)
mu n
s(s)
[s-1]
mun
p
(s)
mun
(s)
s
mun
[s-1]
+
(s)
[s]
identity matrix of size n
zero matrix of size n
minor of a matrix
Kronecker product
norm
set of mun matrices with entries from the field of complex
numbers , real numbers
set of mun polynomial matrices
set of mun rational matrices
set of polynomials with coefficients from the field ,
field of complex functions of the variable s
set of rational causal functions with coefficients from the
field
set of stable rational functions with coefficients form the
field
set of finite rational functions with coefficients from the
field
set of rational causal mun matrices with coefficients from
the field
set of rational stable mun matrices with coefficients from
the field
set of rational finite mun matrices with the coefficients
from the field
set of nonnegative real numbers
set of rational numbers
set of rational functions
set of polynomials of the variable s
1
Polynomial Matrices
1.1 Polynomials
Letting be a field, e.g., of the real numbers , the complex numbers , the
rational numbers , the rational functions W(s) of a complex variable s, etc.,
w( s )
n
¦a s
i
i
a0 a1s ... an s n
(1.1.1)
i 0
is called a polynomial w(s) in the variable s over the field , where ai for
i = 0,1,...,n are called the coefficients of this polynomial.
The set of polynomials (1.1.1) over the field will be denoted by [s].
If an z 0, then the nonnegative integral n is called the degree of a polynomial
and is denoted deg w(s), i.e., n = deg w(s). The polynomial (1.1.1) is called monic,
if an = 1 and zero polynomial, if ai = 0 for i = 0,1,…,n. The sum of two
polynomials
w1 ( s )
w2 ( s)
a0 a1s ... an s n ,
m
b0 b1s ... bm s ,
(1.1.2a)
(1.1.2b)
is defined in the following way
w1 ( s ) w2 ( s )
n
­m
½
i
(
a
b
)
s
ai s i , n ! m°
¦
°¦ i i
i m 1
°i 0
°
n
°
°
i
(ai bi ) s , n m
®
¾.
¦
i 0
°
°
m
° n
°
i
i
° ¦ (ai bi ) s ¦ bi s , m ! n°
i n 1
¯i 0
¿
(1.1.3)
2
Polynomial and Rational Matrices
If n > m, then the sum is a polynomial of degree n, if m > n then the sum is a
polynomial of degree m. If n = m and an+bn z 0, then this sum is a polynomial of
degree n and a polynomial of degree less than n, if an+bn = 0. Thus we have
deg > w1 ( s ) w2 ( s) @ d max ª¬deg > w1 ( s ) @ , deg > w2 ( s )@º¼ .
(1.1.4)
In the same vein we define the difference of two polynomials.
A polynomial whose coefficients are the products of the coefficients ai and the
scalar O, i.e.,
O w( s)
n
¦O a s
i
,
i
(1.1.5)
i 0
is called the product of the polynomial (1.1.1) and the scalar O (a scalar can be
regarded as a polynomial of zero degree).
A polynomial of the form
w1 ( s ) w2 ( s )
n m
¦c s
i
(1.1.6a)
i
i 0
is called the product of the polynomials (1.1.2), where
i
ci
¦a b
k i k
, i
0, 1, ! , n m
k 0
( ak
0 for k ! n, bk
(1.1.6b)
0 for k ! m).
From (1.1.6a) it follows that
deg > w1 ( s ) w2 ( s ) @
nm,
(1.1.7)
since anbm z 0 for an z 0, bm z 0.
Let w2(s) in (1.1.2) be a nonzero polynomial and n > m, then there exist exactly
two polynomials q(s) and r(s) such that
w1 ( s )
w2 ( s )q ( s ) r ( s ) ,
(1.1.8)
where
deg > r ( s ) @ deg > w2 ( s ) @
m.
(1.1.9)
The polynomial q(s) is called the integer part when r(s) z 0 and the quotient
when r(s) = 0, and r(s) is called the remainder.
Polynomial Matrices
3
If r(s) = 0, then w1(s) = w2(s)q(s); we say then that polynomial w1(s) is divisible
without remainder by the polynomial w2(s), or equivalently, that polynomial w2(s)
divides without remainder a polynomial w1(s), which is denoted by w1(s) | w2(s).
We also say that the polynomial w2(s) is a divisor of the polynomial w1(s).
Let us consider the polynomials in (1.1.2). We say that a polynomial d(s) is a
common divisor of the polynomials w1(s) and w2(s) if there exist polynomials
w 1(s) and w 2(s) such that
w1 ( s )
d ( s ) w1 ( s ), w2 ( s )
d ( s ) w2 ( s ) .
(1.1.10)
Polynomial dm(s) is called a greatest common divisor (GCD) of the
polynomials w1(s) and w2(s), if every common divisor of these polynomials is a
divisor of the polynomial dm(s). A GCD dm(s) of polynomials w1(s) and w2(s) is
determined uniquely up to multiplication by a constant factor and satisfies the
equality
d m ( s)
w1 ( s )m1 ( s ) w2 ( s )m2 ( s ) ,
(1.1.11)
where m1(s) and m2(s) are polynomials, which we can determine using Euclid’s
algorithm or the elementary operations method.
The essence of Euclid’s algorithm is as follows. Using division of polynomials
we determine the sequences of polynomials q1,q2,…,qk and r1,r2,…,rk satisfying the
following properties
­ w1 w2 q1 r1 ½
°w r q r °
° 2 1 2 2 °
°°r1 r2 q3 r3
°°
®
¾.
°"""""" °
°rk 2 rk 1qk rk °
°
°
¯°rk 1 rk qk 1
¿°
(1.1.12)
We stop computations when the last nonzero remainder rk is computed and rk-1
is found to be divisible without remainder by rk. With r1,r2,…,rk-1 eliminated from
(1.1.12) we obtain (1.1.11) for dm(s) = rk. Thus the last nonzero remainder rk is a
GCD of the polynomials w1(s) and w2(s).
Example 1.1.1.
Let
w1
w1 ( s )
s 3 3s 2 3s 1, w2
w2 ( s )
s2 s 1 .
(1.1.13)
4
Polynomial and Rational Matrices
Using Euclid’s algorithm we compute
w1
w2 q1 r1 , q1
s 4, r1
6s 3,
w2
r1q2 r2 , q2
1
1
s , r2
6
12
(1.1.14)
3
.
4
Here we stop because r1 is divisible without remainder by r2.
Thus r2 is a GCD of the polynomials in (1.1.13). Elimination of r1 from (1.1.14)
yields
w1 (q2 ) w2 (1 q1q2 )
r2 ,
that is,
1·
§1
s 3 3s 2 3s 1 ¨ s ¸ s 2 s 1
6
12
©
¹
2·
§1 2 7
¨ s s ¸
6
12
3
©
¹
3
.
4
The polynomials in (1.1.2) are called relatively prime (or coprime) if and only
if their monic GCD is equal to 1. From (1.1.11) for dm(s) = 1 it follows that
polynomials w1(s) and w2(s) are coprime if and only if there exist polynomials
m1(s) and m2(s) such that
w1 ( s )m1 ( s ) w2 ( s )m2 ( s ) 1 .
(1.1.15)
Dividing both sides of (1.1.11) by dm(s), we obtain
1 w1 ( s )m1 ( s ) w2 ( s )m2 ( s ) ,
(1.1.16)
where
wk ( s )
wk ( s )
for k
d m (s)
1, 2,! .
Thus if dm(s) is a GCD of the polynomials w1(s) and w2(s), then polynomials
w 1(s) and w 2(s) are coprime.
Let s1,s2,…,sp be different roots of multiplicities m1,m2,…,mp
(m1+m2+…+mp = n), respectively, of the equation w(s) = 0. The numbers
s1,s2,…,sp are called the zeros of polynomial (1.1.1). This polynomial can be
uniquely written in the form
w( s )
an ( s s1 ) m1 ( s s2 ) m2 ...( s s p )
mp
.
(1.1.17)
Polynomial Matrices
5
1.2 Basic Notions and Basic Operations on Polynomial Matrices
A matrix whose elements are polynomials over a field
matrix over the field (briefly polynomial matrix)
A( s)
ª¬ aij ( s ) º¼ i 1,...,m
j 1,..., n
is called a polynomial
ª a11 ( s ) ! a1n ( s ) º
« #
%
# »» , aij ( s )  ( s ) .
«
«¬ am1 ( s ) ! amn ( s ) »¼
(1.2.1)
An ordered pair of the number of rows m and columns n, respectively, is called
the dimension of matrix (1.2.1) and is denoted by mun. A set of polynomial
matrices of dimension mun over a field will be denoted by mun[s].
The following matrix is an example of a 2u2 polynomial matrix over the field
of real numbers
A 0 (s)
ª s 2 2s 1
s2 º
2u2
« 2
»  [ s] .
2
2
3
3
3
s
s
s
s
¬
¼
(1.2.2)
Every polynomial matrix can be written in the form of a matrix polynomial. For
example, the matrix (1.2.2) can be written in the form of the matrix polynomial
A0 (s)
ª1 0 º 2 ª 2 1º
ª1 2 º
« 2 3» s « 1 1» s «3 3»
¬
¼
¬
¼
¬
¼
A 2 s 2 A1s A 0 .
(1.2.3)
Let a matrix of the form (1.2.1) be expressed as the matrix polynomial
A( s)
A q s q ... A1s A 0 , A k  mun , k
0, 1, ..., q .
(1.2.4)
If Aq is not a zero matrix, then number q is called its degree and is denoted by
q = deg A(s). For example, the matrix (1.2.2) (and also (1.2.3)) has the degree two
q = 2.
If n = m and det Aq z 0, then matrix (1.2.4) is called regular.
The sum of two polynomial matrices
q
A( s)
B( s )
ª¬ aij ( s ) º¼ i 1,...,m
j 1,...,n
ª¬bij ( s ) º¼ i 1,...,m
j 1,..., n
¦A s
k
k
and
k 0
t
¦ Bk s k
k 0
of the same dimension mun is defined in the following way
(1.2.5)
6
Polynomial and Rational Matrices
A ( s ) B( s )
q
­ t
k
k
°¦ ( A k B k ) s ¦ A k s
k
k
t
0
1
°
°° q
k
®¦ ( A k B k ) s
°k 0
t
° q
k
k
°¦ ( A k B k ) s ¦ B k s
k q 1
¯° k 0
m
¬ª aij ( s ) bij ( s ) ¼º ij 1,...,
1,...,n
½
q ! t°
°
°°
q t ¾.
°
°
q t°
¿°
(1.2.6)
If q = t and Aq + Bq z 0, then the sum in (1.2.6) is a polynomial matrix of
degree q, and if Aq + Bq = 0, then this sum is a polynomial matrix of a degree not
greater than q. Thus we have
deg > A( s ) B( s )@ d max > deg [ A( s )], deg [B( s )]@ .
(1.2.7)
In the same vein, we define the difference of two polynomial matrices.
A polynomial matrix where every entry is the product of an entry of the matrix
(1.2.1) and the scalar O is called the product of the polynomial matrix (1.2.1) and
the scalar O
O A( s) ª¬O aij ( s ) º¼ i 1,...,m .
j 1,..., n
From this definition for O z 0, we have deg [OA(s)] = deg [A(s)].
Multiplication of two polynomial matrices can be carried out if and only if the
number of columns of the first matrix (1.2.1) is equal to the number of rows of the
second matrix
t
B( s )
ª¬bij ( s ) º¼ i 1,...,n
j 1,..., p
¦B s
k
k
.
(1.2.8)
k 0
A polynomial matrix of the form
C( s )
ª¬ cij ( s ) º¼ i 1,...,m
j 1,..., p
A ( s )B ( s )
q t
¦C s
k
k
(1.2.9)
k 0
is called the product of these polynomial matrices, where
Ck
k
¦A B
l
k l
k
0,1,..., q t
l 0
(Al
0, l ! q, Bl
0, l ! t ) .
(1.2.10)
Polynomial Matrices
7
From (1.2.10) it follows that Cq+t = AqBt and this matrix is a nonzero one if at
least one of the matrices Aq and Bt is nonsingular, in other words one of the
matrices A(s) and B(s) is a regular one. Thus we have the relationship
deg > A(s)B(s) @ = deg > A(s)@ + deg > B(s)@ if at least one of these
matrices is regular,
(1.2.11)
deg > A(s)B(s)@ d deg > A(s) @ + deg > B(s) @ otherwise.
For example, the product of the polynomial matrices
A( s)
B( s )
ª s2 s
2 s 2 s 1º ª 1 2 º 2 ª 1 1 º
ª 0 1º
s «
s«
« 2
» «
»
»
»,
2
2s 2 ¼ ¬ 1 2 ¼
¬2 0¼
¬ 1 2 ¼
¬ s 2s 1
ª 2s 2 s 3 º ª 2 1 º
ª 2 3º
« s 1 1 s 1» «1 1 » s « 1 1»
¬
¼
¬
¼ ¬
2
2¼
is the following polynomial matrix
A ( s )B ( s )
ª7 s 2 2s 1
«
¬ 4s 4
5
2
s 2 92 s 1º
»
s 2 6s 1 ¼
ª7 52 º 2 ª 2 92 º
ª 1 1 º
« 0 1 » s « 4 6 » s « 4 1» ,
¬
¼
¬
¼
¬
¼
whose degree is smaller than the sum deg [A(s)] + deg [B(s)], since
A 2 B1
ª 1 2 º ª 2 1 º
« 1 2 » « 1 1 »
¬
¼¬
2¼
ª0 0 º
«0 0 » .
¬
¼
The matrix (1.2.4) can be written in the form
A( s)
s q A q ... sA1 A 0 ,
(1.2.12)
since multiplication of the matrix Ai (i = 1,2,…,q) by the scalar s is commutative.
Substituting the matrix S in place of the scalar s into (1.2.4) and (1.2.12), we obtain
the following, usually different, matrices
A p (S)
A q S q ... A1S A 0 ,
A l (S)
S q A q ... SA1 A 0 .
The matrix Ap(S) (Al(S)) is called the right-sided (left-sided) value of the matrix
A(s) for s = S.
Let
8
Polynomial and Rational Matrices
C( s )
A ( s ) B( s ) .
It is easy to verify that
C p (S)
A p (S) B p (S)
Cl (S)
A l (S) B l (S) .
and
Consider the polynomial matrices in (1.2.5).
Theorem 1.2.1. If the matrix S commutes with the matrices Ai for i = 1,2,…,q and
Bj for j = 1,2,…,t, then the right-sided and the left-sided value of the product of the
matrices in (1.2.5) for s = S is equal to the product of the right-sided and left-sided
values respectively, of these matrices for s = S.
Proof. Taking into account the polynomial matrices in (1.2.5) we can write
D( s )
A ( s )B ( s )
t
§ q
i ·§
j ·
¨ ¦ Ai s ¸ ¨ ¦ B j s ¸
©i 0
¹© j 0
¹
¦¦ A B s
q
§ q i ·§ t j ·
¨ ¦ s Ai ¸ ¨ ¦ s B j ¸
©i 0
¹© j 0
¹
¦¦ s
t
i
i j
j
i 0 j 0
and
D( s )
A ( s )B ( s )
q
t
i j
Ai B j .
i 0 j 0
Substituting the matrix S in place of the scalar s, we obtain
q
D p (S)
t
¦¦ A B S
i
i j
j
i 0 j 0
t
§ q
i ·§
j ·
¨ ¦ AiS ¸ ¨ ¦ B j S ¸
©i 0
¹© j 0
¹
A p (S)B p (S) ,
since BjS = SBj for j = 1,2,…,t and
p
Dl (S)
q
¦¦ S
i j
Ai B j
i 0 j 0
since SAi=AiS for i = 1,2,…,q. § p i ·§ q j ·
¨ ¦ S Ai ¸ ¨ ¦ S B j ¸
©i 0
¹© j 0
¹
A l (S)Bl (S ) ,
Polynomial Matrices
9
1.3 Division of Polynomial Matrices
Consider the polynomial matrices A(s) and B(s) where det A(s) z 0 and
deg A(s) < deg B(s). The matrix A(s) may be not regular, i.e., the matrix of
coefficients of the highest power of variable s may be singular.
Theorem 1.3.1. If det A(s) z 0, then for the pair of polynomial matrices A(s) and
B(s), deg B(s) > deg A(s) there exists a pair of matrices Qp(s), Rp(s) such that the
following equality is satisfied
B( s )
Q p ( s) A( s ) R p ( s ), deg A( s ) ! deg R p ( s ) ,
(1.3.1a)
and there exists a pair of matrices Ql(s), Rl(s) such that the following equality
holds
B( s )
A( s )Ql ( s ) R l ( s ), deg A ( s ) ! deg R l ( s ) .
(1.3.1b)
Proof. Dividing the elements of matrix B(s) Adj A(s) by a polynomial det A(s), we
obtain a pair of matrices Qp(s), R1(s) such that
B( s )Adj A( s )
Q p ( s ) det A( s ) R1 ( s ), deg > det A ( s )@ ! deg R1 s .
(1.3.2)
Post-multiplication of (1.3.2) by A(s)/det A(s) yields
B( s )
Q p ( s) A( s) R p ( s ) ,
(1.3.3)
since Adj A(s) A(s) = In det A(s), where
R p (s)
R1 ( s ) A ( s )
.
det A( s )
(1.3.4)
From (1.3.4) we have
deg R p ( s)
deg R1 ( s ) deg A( s ) deg > det A( s ) @ deg A ( s ) ,
since deg [det A(s)] > deg R1(s).
The proof of equality (1.3.1b) is similar. Remark 1.3.1. The pairs of matrices Qp(s), Rp(s) and Ql(s), Rl(s) satisfying the
equality (1.3.1) are not uniquely determined (are not unique), since
B( s ) [Q p ( s ) C( s )]A( s ) R p ( s ) A( s )C( s )
(1.3.5a)
10
Polynomial and Rational Matrices
and
B( s )
A( s )[Ql ( s ) C( s )] R l ( s ) A ( s )C( s )
(1.3.5b)
are satisfied for an arbitrary matrix C(s) satisfying
deg >C(s) A ( s ) @ deg A ( s ), deg > A ( s )C(s) @ deg A ( s ) .
Example 1.3.1.
For the matrices
ª s 1º
« 1 1» , B( s )
¬
¼
A( s)
s º
ªs
« 1 s 2 1»
¬
¼
determine the matrices Qp(s), Rp(s) satisfying the equality (1.3.1a).
In this case, det A1 = 0 and det A(s) = s+1.
We compute
Adj A( s )
ª1 1º
«1 s » , B( s )Adj A( s)
¬
¼
ª0
« 2
¬s
s 2 s º
»,
s 3 s 1¼
and with (1.3.2) taken into account we have
ª0
« 2
¬s
s 2 s º
»
s 3 s 1¼
s º
ª 0
ª0 0 º
« s 1 s 2 s 2 » ( s 1) «1 1» ,
¬
¼
¬
¼
i.e.,
Q p (s)
s º
ª 0
« s 1 s 2 s 2 » , R1 ( s )
¬
¼
ª0 0 º
«1 1» .
¬
¼
According to (1.3.4) we obtain
R p (s)
R1 ( s ) A ( s )
det A( s )
ª0 0º
«1 0 » .
¬
¼
Consider two polynomial matrices
A( s)
B( s )
A n s n A n1s n1 ... A1s A 0 ,
m
B m s B m1s
m 1
... B1s B 0 .
(1.3.6a)
(1.3.6b)
Polynomial Matrices
11
Theorem 1.3.2. If A(s) and B(s) are square polynomial matrices of the same
dimensions, and A(s) is regular (det An z 0), then there exist exactly one pair of
polynomial matrices Qp(s), Rp(s) satisfying the equality
B( s )
Q p ( s) A( s) R p ( s ) ,
(1.3.7a)
and exactly one pair of polynomial matrices Ql(s), Rl(s) satisfying the equality
B( s )
(1.3.7b)
A( s )Ql ( s ) R l ( s )
where
deg A( s) ! deg R p ( s ), deg A( s ) ! deg R l ( s ) .
Proof. If n > m, then Qp(s) = 0 and Rp(s) = B(s). Assume that m t n. By the
assumption det An z 0 there exists the inverse matrix An-1. Note that the matrix
BmAn-1sm-nA(s) has a term in the highest power of s, equal to Bmsm. Hence
B( s )
B m A n1s mn A( s ) B (1) ( s ) ,
where B(1)(s) is a polynomial matrix of degree m1 d m-1 of the form
B (1) ( s )
m1
m1 1
B (1)
B (1)
... B1(1) s B 0(1) .
m1 s
m1 1 s
If m1 t n, then we repeat this procedure, taking the matrix B (m1) instead of the
1
matrix Bm, and obtain
B (1) ( s )
1 m1 n
B (1)
A( s ) B (2) ( s ) ,
m1 A n s
B (2) ( s )
m2
m2 1
B (2)
B (2)
... B1(2) s B (2)
(m2 m1 ) .
0
m2 s
m2 1 s
where
Continuing this procedure, we obtain the sequence of polynomial matrices B(s),
B(1)(s), B(2)(s),…, of decreasing degrees m, m1, m2,…, respectively. In step r, we
obtain the matrix B(r)(s) of degree mr < n and
B( s )
1 m1 n
B m A n1s mn B (1)
... B (mrr1)1 A n1s mr 1 n A ( s ) B ( r ) ( s ) ,
m1 A n s
that is the equality (1.3.7a) for
12
Polynomial and Rational Matrices
Q p (s)
1 m1 n
B m A n1s mn B (1)
... B (mrr1)1 A n1s mr 1 n ,
m1 A n s
R p (s)
B ( r ) ( s ).
(1.3.8)
Now we will show that there exists only one pair Qp(s), Rp(s) satisfying
(1.3.7a). Assume that there exist two different pairs Qp(1)(s), Rp(1)(s) and Qp(2)(s),
Rp(2)(s) such that
B( s)
(1)
Q (1)
p ( s) A( s) R p ( s)
(1.3.9a)
B( s )
(2)
Q (2)
p ( s) A(s) R p ( s) ,
(1.3.9b)
and
where deg A(s) > deg Rp(1)(s) and deg A(s) > deg Rp(2)(s). From (1.3.9) we have
(2)
ª¬Q (1)
º
p ( s) Q p ( s) ¼ A( s)
(1)
R (2)
p (s) R p (s) .
(1.3.10)
For Qp(1)(s) z Qp(2)(s) the matrix [Qp(1)(s) - Qp(2)(s)]A(s) is a polynomial matrix
of a degree greater than n, and [Rp(2)(s) - Rp(1)(s)] is a polynomial matrix of a
degree less than n. Hence from (1.3.10) it follows that Qp(1)(s) = Qp(2)(s) and
Rp(1)(s) = Rp(2)(s). Similarly one can prove that
Ql ( s )
m1 n
A n1B m s mn A n1B (1)
... A n1B (mrr1)1 s mr 1 n ,
m1 s
R l ( s)
B ( r ) ( s ).
(1.3.11)
The matrices Qp(s), Rp(s) (Ql(s), Rl(s)) are called, respectively: the right (left)
quotient and the remainder from division of the matrix B(s) by the matrix A(s).
From the proof of Theorem 1.3.2 the following algorithm for determining
matrices Qp(s) and Rp(s) (Ql(s) and Rl(s)) ensues.
Procedure 1.3.1.
Step 1: Given matrix An compute An-1.
Step 2: Compute
B m A n1s mn A( s )
A( s ) A n1B m s mn
and
B (1) ( s )
B( s ) B m A n1s mn A( s )
m1
B (1)
... B1(1) s B (1)
0
m1 s
Polynomial Matrices
B (1) ( s )
13
m1
B (1)
... B1(1) s B (1)
.
0
m1 s
Ǻ( s ) A( s ) A n1B m s mn
Step 3: If m1 t n, then compute
m1 n
1 m1 n
B (1)
A( s ) A( s ) A n1B (1)
m1 A n s
m1 s
and
1 m1 n
B (1) ( s ) B (1)
A(s)
m1 A n s
B (2) ( s )
m2
B (2)
... B1(2) s B (2)
m2 s
0
m1 n
B (1) ( s ) A( s ) A n1B (1)
m1 s
B (2) ( s )
m2
B (2)
... B1(2) s B (2)
.
m2 s
0
Step 4: If m2 t n, then substituting in the above equalities m1 and B(1)(s) by m2 and
B(2)(s), respectively, compute B(3)(s). Repeat this procedure r times until
mr < n.
Step 5: Compute the matrices Qp(s), Rp(s) (Ql(s), Rl(s)).
Example 1.3.2.
Given the matrices
ª s2 1 s º
«
» and B( s )
s2 s ¼
¬ s
A( s)
ª s 4 s 2 1 s 3 s 2 2s º
«
»,
2
s3 s 2 ¼
¬ 2s s
determine matrices Qp(s), Rp(s) and Ql(s), Rl(s) satisfying (1.3.7).
Matrix A(s) is regular, since
A2
ª1
«0
¬
0º
and B 4
1»¼
ª1
«0
¬
0º
.
0 »¼
Using Procedure 1.3.1 we compute the following.
Steps 1–3: In this case,
B 4 A 21s 2 A( s )
ª1
«
¬0
0 º ª1 0 º 2 ª s 2 1 s º ª s 4 s s 3 º
s «
»
» «
»
0 ¼
s2 s ¼ ¬ 0
0¼ «¬0 1 »¼ ¬ s
and
B (1) ( s )
B( s ) B 4 A 21s 2 A( s )
ª s 4 s 2 1 s 3 s 2 2s º ª s 4 s 2
«
»«
2
s3 s 2 ¼ ¬ 0
¬ 2s s
s3 º
»
0 ¼
ª 1
2s 3 s 2 2s º
« 2
».
s3 s 2 ¼
¬ 2s s
14
Polynomial and Rational Matrices
Since m1 = 3, n = 2, and
B3(1)
ª0 2 º
«0 1 » ,
¬
¼
we have
B3(1) A 21s A( s )
ª 0 2 º ª1 0 º ª s 2 1
«0 1 » «0 1 » s «
¬
¼¬
¼ ¬ s
s º
»
2
s s¼
ª 2s 2
« 2
¬s
2s 3 2s 2 º
»
s3 s 2 ¼
and
B (2) ( s )
B (1) ( s ) B3(1) A 21s A( s )
ª 1
2s 3 s 2 2s º ª 2s 2
« 2
»«
s3 s 2 ¼ ¬ s 2
¬2s s
2s3 2s 2 º
»
s3 s 2 ¼
ª 2 s 2 1 3s 2 2 s º
« 2
».
2
¬ s s s s 2¼
Step 4: We repeat the procedure, since m2 = 2 = n. Taking into account that
B (2)
2
ª 2 3º
« 1 1» ,
¬
¼
we compute
1
B(2)
2 A 2 A( s)
ª 2 3º ª1 0 º ª s 2 1 s º
»
« 1 1» «0 1 » «
s2 s¼
¬
¼¬
¼¬ s
ª 2s 2 3s 2 3s 2 s º
« 2
»
s 2 2s ¼
¬ s s 1
and
B (3) ( s )
1
B (2) ( s ) B (2)
2 A 2 A( s )
ª 2s 2 3s 2 3s 2 s º
« 2
»
s 2 2s ¼
¬ s s 1
ª 2 s 2 1 3s 2 2s º
« 2
»
2
¬ s s s s 2¼
3s º
ª 3s 3
« 2 s 1 3s 2 » .
¬
¼
Step 5: The degree of this matrix is less than the degree of the matrix A(s). Hence,
according to (1.3.8), we obtain
Polynomial Matrices
15
1
B 4 A 21s 2 B3(1) A 21s B (2)
2 A2
Q p (s)
ª1 0 º 2 ª 0 2 º
ª 2 3º
«0 0 » s «0 1 » s « 1 1»
¬
¼
¬
¼
¬
¼
ª s 2 2 2 s 3º
«
»
s 1 ¼
¬ 1
and
B (3) ( s )
R p (s)
3s º
ª3s 3
« 2 s 1 3s 2 » .
¬
¼
We compute Ql(s) and Rl(s) using Procedure 1.3.1.
Steps 1–3: We compute
ª s 2 1 s º ª1 0 º ª1 0 º 2
«
»«
»«
»s
s 2 s ¼ ¬0 1 ¼ ¬0 0¼
¬ s
A( s ) A 21B 4 s 2
ªs4 s2
« 3
¬ s
0º
»
0¼
and
B (1) ( s )
B( s ) A( s ) A 21B 4 s 2
ªs4 s2
« 3
¬ s
0º
»
0¼
ª s 4 s 2 1 s3 s 2 2s º
«
»
2
s3 s 2 ¼
¬ 2s s
ª
1
s3 s 2 2s º
« 3
».
2
3
¬ s 2s s s s 2 ¼
Taking into account that m1 = 3 > n = 2 and
B3(1)
ª 0 1º
« 1 1» ,
¬
¼
we compute
A( s ) A 21B3(1) s
ª s 2 1 s º ª1 0 º ª 0 1º
«
»«
»«
»s
s 2 s ¼ ¬0 1 ¼ ¬ 1 1¼
¬ s
ª s2
« 3 2
¬s s
s3 s 2 s º
»
s 3 2s 2 ¼
and
B (2) ( s )
B (1) ( s ) A( s ) A21B 3(1) s
ª s2
« 3 2
¬s s
s3 s 2 s º
»
s 3 2s 2 ¼
ª
1
s 3 s 2 2s º
« 3
»
2
3
¬ s 2s s s s 2 ¼
ªs 2 1
º
s
« 2
».
2
3
s
s
2
s
s
2
¬
¼
16
Polynomial and Rational Matrices
Step 4: We repeat the procedure, since m2 = 2 = n. Taking into account that
B (2)
2
ª 1 0 º
« 3 2 » ,
¬
¼
we have
A ( s ) A 21B (2)
2
ª s 2 1 s º ª1 0 º ª 1 0 º
«
»«
»«
»
s 2 s ¼ ¬0 1 ¼ ¬ 3 2 ¼
¬ s
ª s 2 3s 1
º
2s
«
»
2
2
3
2
2
2
s
s
s
¬
¼
and
B (3) ( s )
ªs 2 1
º
s
« 2
»
2
s
s
s
s
3
2
2
¬
¼
s º
ª 3s 2
.
« s
3s 2 »¼
¬
B (2) ( s ) A( s ) A 21B (2)
2
ª s 2 3s 1
º
2s
«
»
2
2
2 s 2 s ¼
¬ 3s 2 s
Step 5: The degree of this matrix is less than the degree of matrix A(s). Hence
according to (1.3.11), we have
Ql ( s )
A 21B 4 s 2 A 21B3(1) s A 21B (2)
2
ª1 0 º 2 ª 0 1º
ª 1 0 º
«0 0 » s « 1 1» s « 3 2 »
¬
¼
¬
¼
¬
¼
s º
ª 3s 2
.
R l ( s ) B (3) ( s ) «
s
3
s
2 »¼
¬
ª s2 1
s º
«
»,
¬s 3 s 2¼
1.4 Generalized Bezoute Theorem and the Cayley–Hamilton
Theorem
Let us consider the division of a square polynomial matrix
F( s)
Fn s n Fn1s n1 " F1s F0  mum [ s ]
(1.4.1)
by a polynomial matrix of the first degree [Ims A], where Fk mum, k = 0,1,…,n
and A mum. The right (left) Rp (Rl) remainder from division of F(s) by [Ims A]
is a polynomial matrix of zero degree, i.e., it does not depend on s.
Theorem 1.4.1. (Generalised Bezoute theorem). The right (left) remainder Rp (Rl)
from division of the matrix F(s) by [Ims - A] is equal to Fp(A) (Fl(A)), i.e.,
Polynomial Matrices
17
Rp
Fp ( A )
Fn A n Fn1A n1 " F1A F0  mum
(1.4.2a)
Rl
Fl ( A )
A n Fn A n1Fn1 " AF1 F0  mum .
(1.4.2b)
Proof. Post-dividing the matrix F(s) by [Ims - A], we obtain
F(s)
Q p (s) I m s A R p ,
and pre-dividing by the same matrix, we obtain
F(s)
> I m s A @ Ql ( s) R l .
Substituting the matrix A in place of the scalar s in the above relationships, we
obtain
Fp ( A )
Q p ( A )( A A ) R p
Fl ( A)
( A A )Q l ( A ) R l
Rp
and
Rl .
The following important corollary ensues from Theorem 1.4.1.
Corollary 1.4.1. A polynomial matrix F(s) is post-divisible (pre-divisible) without
remainder by [Ims A] if and only if Fp(A) = 0 (Fl(A) = 0).
Let M(s) be the characteristic polynomial of a square matrix A of degree n, i.e.,
M ( s ) det > I n s A @ s n an1s n1 " a1s a0 .
From the definition of the inverse matrix we have
>I n s A @ Adj>I n s A @
I nM ( s )
(1.4.3a)
and
Adj > I n s A @> I n s A @ I nM ( s ) .
(1.4.3b)
It follows from (1.4.3) that a polynomial matrix InM(s) is post-divisible and predivisible by [Ins - A]. According to Corollary 1.4.1 this is possible if and only if
InM(A) = M(A) = 0. Thus the following theorem has been proved.
18
Polynomial and Rational Matrices
Theorem 1.4.2. (CayleyHamilton). Every square matrix A satisfies its own
characteristic equation
M ( A)
A n an1A n1 " a1A a0 I n
0.
(1.4.4)
Example 1.4.1.
The characteristic polynomial of the matrix
A
ª1 2 º
«3 4 »
¬
¼
(1.4.5)
is
ªs 1
M ( s ) det > I n s A @ «
¬ 3
2 º
s 4 »¼
s 2 5s 2 .
It is easy to verify that
2
M ( A)
A 2 5A 2I 2
ª1 2 º
ª1 2 º
ª1 0 º
«3 4 » 5 «3 4 » 2 « 0 1 »
¬
¼
¬
¼
¬
¼
ª0 0º
«0 0» .
¬
¼
Theorem 1.4.3. Let a polynomial w(s) [s] be of degree N, and A
N t n. There exists a polynomial r(s) of a degree less than n, such that
w( A )
r ( A) .
nun
, where
(1.4.6)
Proof. Dividing the polynomial w(s) by the characteristic polynomial M(s) of the
matrix A, we obtain
w( s )
q ( s )M ( s ) r ( s ) ,
where q(s) and r(s) are the quotient and remainder on division of the polynomial
w(s) by M(s), respectively, and deg M(s) = n > deg r(s). With the matrix A
substituted in place of the scalar s and with (1.4.4) taken into account, we obtain
w( A )
q ( A )M ( A ) r ( A )
r ( A) .
Example 1.4.2.
The following polynomial is given
w( s )
s 6 5 s 5 3 s 4 5 s 3 2 s 2 3s 2 .
Polynomial Matrices
19
Using (1.4.6) one has to compute w(A) for the matrix (1.4.5). The characteristic
polynomial of the matrix is M(s) = s2 - 5s - 2. Dividing the polynomial w(s) by M(s),
we obtain
s4 s2
w( s )
s 2 5s 2 3s 2 ,
that is
r ( s)
3s 2 .
w( A)
r ( A)
Hence
3A 2I 2
ª1 2 º
ª1 0 º
2«
3«
»
»
¬3 4 ¼
¬0 1 ¼
ª5 6 º
«9 14» .
¬
¼
The above considerations can be generalized to the case of square polynomial
matrices.
Theorem 1.4.4. Let W(s) nun[s] be a polynomial square matrix of degree N, and
A nun, where N t n. There exists, a polynomial matrix R(s) of a degree less than
n such that
Wp ( A )
R p ( A ) and Wl ( A )
R l ( A) ,
(1.4.7)
where Wp(A) and Wl(A) are the right-side and left-side values, respectively, of the
matrix W(s) with A substituted in place of s.
Proof. Dividing the entries of the matrix W(s) by the characteristic polynomial
M(s) of A, we obtain
W( s)
Q( s )M ( s ) R ( s ) ,
where Q(s) and R(s) are the quotient and remainder, respectively, of the division of
W(s) by M(s), and deg M(s) = n > deg R(s). With A substituted in place of the
scalar s and with (1.4.4) taken into account, we obtain
Wp ( A )
Q p ( A)M ( A) R p ( A )
Wl ( A )
Ql ( A )M ( A ) R l ( A )
R p (A)
and
R l (A) .
20
Polynomial and Rational Matrices
Example 1.4.3.
Given the polynomial matrix
W( s)
ª s 6 5s 5 2 s 4 s 2 3s 1
s 5 5s 4 2 s 3 s 1 º
« 4
»,
3
2
2s 6 10 s 5 4s 4 s 2 ¼
¬ s 5 s 3s 5 s 3
one has to compute Wp(A) and Wl(A) for the matrix (1.4.5) using (1.4.7).
Dividing every entry of W(A) by the characteristic polynomial M(s) of matrix
A, we obtain
W( s)
ª s4 1 s3 º 2
ª2s 3 s 1 º
s 5s 2 «
,
« 2
4»
s 2 »¼
¬ 1
¬ s 1 2 s ¼
R( s)
ª2s 3 s 1 º
.
« 1
s 2 »¼
¬
i.e.,
Hence
Wp ( A )
R p ( A)
ª 2 1º
ª3 1º
« 0 1» A «1 2 »
¬
¼
¬
¼
ª 2 1º ª1 2 º ª3 1º
« 0 1» «3 4 » «1 2 »
¬
¼¬
¼ ¬
¼
ª 2 1º
« 2 2»
¬
¼
ª1 2 º ª 2 1º ª3 1º
«3 4 » « 0 1» «1 2 »
¬
¼¬
¼ ¬
¼
ª 5 4º
«7 5 » .
¬
¼
and
Wl ( A)
R l ( A)
ª 2 1º ª3 1º
A«
»«
»
¬ 0 1¼ ¬1 2 ¼
1.5 Elementary Operations on Polynomial Matrices
Definition 1.5.1. The following operations are called elementary operations on a
polynomial matrix A(s) mun[s]:
1. Multiplication of any i-th row (column) by the number c z 0.
2. Addition to any i-th row (column) of the j-th row (column) multiplied by
any polynomial w(s).
3. The interchange of any two rows (columns), e.g., of the i-th and the j-th
rows (columns).
Polynomial Matrices
21
From now on we will use the following notation:
L[iuc]
multiplication of the i-th row by the number c z 0,
P[iuc]
multiplication of the i-th column by the number c z 0,
L[i+juw(s)] addition to the i-th row of the j-th row multiplied by the polynomial
w(s),
P[i+juw(s)] addition to the i-th column of the j-th column multiplied by the
polynomial w(s),
L[i, j]
the interchange of the i-th and the j-th row,
P[i, j]
the interchange of the i-th and the j-th column.
It is easy to verify that the above elementary operations when carried out on rows
are equivalent to pre-multiplication of the matrix A(s) by the following matrices:
i -th column
L m (i, c)
ª1
«
«0
«#
«
«0
«
«#
«
¬0
0 ! 0 ! 0º
»
1 ! 0 ! 0»
# % # % #»
»
 mum ,
0 ! c ! 0 » i -th row
»
# % # % #»
»
0 ! 0 ! 1¼
i
L d (i, j , w( s ))
ª1
«
«0
«#
«
«0
«
«#
«
¬0
0 ! 0 !
1 ! 0 !
# % # %
L z i, j
0
0
#
! 0º
»
! 0»
% #»
»  mum > s @ ,
0 ! 1 ! w( s) ! 0»
»
# % # %
#
% #»
»
0 ! 0 !
0 ! 1¼
i
ª1
«0
«
«#
«
«0
«#
«
«0
«#
«
«¬0
j
j
0
1
#
0
!
!
%
!
0
0
#
0
!
!
%
!
0
0
#
1
!
!
%
!
#
0
#
0
%
!
%
!
#
1
#
0
%
!
%
!
#
0
#
0
!
!
%
!
0º
0 »»
#»
»
0»
.
#»
»
0»
#»
»
1 »¼
(1.5.1)
22
Polynomial and Rational Matrices
The same operations carried out on columns are equivalent to postmultiplication of the matrix A(s) by the following matrices:
i -th column
Pm (i, c)
ª1
«0
«
«#
«
«0
«#
«
«¬ 0
0 ! 0 0 ! 0º
1 ! 0 0 ! 0 »»
# % # # % #»
 nun ,
»
0 ! c 0 ! 0 » i -th row
# % # # % #»
»
0 ! 0 0 ! 1 »¼
i
Pd (i, j , w( s ))
ª1
«0
«
«#
«
«0
«0
«
«#
«0
¬
0 !
1 !
# %
0 !
0 !
# %
0 !
i
Pz (i, j )
ª1
«0
«
«#
«
«0
«#
«
«0
«#
«
¬«0
0 ! 0
1 ! 0
# % #
0 ! 0
# % #
0 ! 1
# % #
0 ! 0
j
0
! 0 ! 0º
0 ! 0 ! 0 »»
#
% # % #»
»
1
! 0 ! 0 »  nun ,
w( s ) ! 1 ! 0 »
»
#
% # % #»
0 ! 0 ! 0 »¼
(1.5.2)
j
!
!
%
!
%
!
%
!
0 ! 0º
0 ! 0 »»
# % #»
»
1 ! 0»
 nun .
# % #»
»
0 ! 0»
# % #»
»
0 " 1 ¼»
It is easy to verify that the determinants of the polynomial matrices (1.5.1) and
(1.5.2) are nonzero and do not depend on the variable s. Such matrices are called
unimodular matrices.
Polynomial Matrices
23
1.6 Linear Independence, Space Basis and Rank of Polynomial
Matrices
Let ai = ai(s), i = 1,…,n be the i-th column of a polynomial matrix A(s) mun[s].
We will consider these columns as m-dimensional polynomial vectors, ai m[s],
i = 1,…,n.
Definition 1.6.1. Vectors ai m[s] are called linearly independent over the field of
rational functions (s) if and only if there exist rational functions wi=wi(s) (s) not
all equal to zero such that
w1a1 w2 a2 ... wn an
0 (zero order) .
(1.6.1)
In other words, these vectors are called linearly independent over the field of
rational functions, if the equality (1.6.1) implies wi = 0 for i = 1,…,n.
For example, the polynomial vectors
a1
ª1º
« s » , a2
¬ ¼
ª s º
«1 s 2 »
¬
¼
(1.6.2)
are linearly independent over the field of rational functions, since the equation
w1a1 w2 a2
ª1 º
ª s º
« s » w1 «1 s 2 » w2
¬ ¼
¬
¼
s º ª w1 º
ª1
« s s 2 1» « w »
¬
¼¬ 2¼
ª0º
«0»
¬ ¼
has only the zero solution
ª w1 º
«w »
¬ 2¼
1
s º ª0º
ª1
« s s 2 1» « 0 »
¬
¼ ¬ ¼
ª0º
«0» .
¬ ¼
We will show that the rational functions wi, i = 1,…,n in (1.6.1) can be replaced
by polynomials pi = pi(s), i = 1,…,n. To accomplish this, we multiply both sides of
(1.6.1) by the smallest common denominator of rational functions wi, i = 1,…,n.
We then obtain
p1a1 p2 a2 ... pn an
0,
(1.6.3)
where pi = pi(s) are polynomials.
For example, the polynomial vectors
a1
ª1 º
« s » , a2
¬ ¼
ª s 1 º
«s2 s»
¬
¼
(1.6.4)
24
Polynomial and Rational Matrices
are linearly dependent over the field of rational functions, since for
w1
1 and w2
1
,
s 1
we obtain
w1a1 w2 a2
ª1º
1 ª s 1 º
« » « 2
»
¬s¼ s 1 ¬s s¼
ª0 º
« ».
¬0 ¼
(1.6.5)
Multiplying both sides of (1.6.5) by the smallest common denominator of rational
functions w1 and w2, which is equal to s + 1, we obtain
ª1º ª s 1 º
( s 1) « » « 2
»
¬s¼ ¬s s¼
ª0º
«0» .
¬ ¼
If the number of polynomial vectors of the space n[s] is larger than n, then
these vectors are linearly dependent. For example, adding to two linearly
independent vectors (1.6.2) an arbitrary vector
a
ª a11 º
2
«a »  [s] ,
¬ 21 ¼
we obtain linearly dependent vectors, i.e.,
0,
p1a1 p2 a2 p3 a
(1.6.6)
for p1, p2, p3 [s] not simultaneously equal to zero.
Assuming, for example, p3 = -1, from (1.6.6) and (1.6.2), we obtain
s º ª p1 º
ª1
« s s 2 1» « p »
¬
¼¬ 2¼
ª a11 º
«a »
¬ 21 ¼
and
ª p1 º
«p »
¬ 2¼
1
s º ª a11 º
ª1
« s s 2 1» « a »
¬
¼ ¬ 21 ¼
ª s 2 1 s º ª a11 º
«
»« »
1 ¼ ¬ a21 ¼
¬ s
ª s 2 1 a11 sa21 º
«
».
«¬ sa11 a21 »¼
Thus vectors a1, a2, a are linearly dependent for any vector a.
Definition 1.6.2. Polynomial vectors bi = bi(s) n[s], i = 1,…,n are called a basis
of space n[s] if they are linearly independent over the field of rational function
Polynomial Matrices
25
and an arbitrary vector a n[s] from this space can be represented as a linear
combination of these vectors, i.e.,
a
p1b1 p2b2 ... pn bn ,
(1.6.7)
where pi [s], i = 1,…,n.
There exist many different bases for the same space. For example, for the space
[s] we can adopt the vectors (1.6.2) as a basis. Solving system of equations for
an arbitrary vector
2
ª a11 º
2
« a »  [ s] ,
¬ 21 ¼
s ºªp º
ª p º ª1
> a1 a2 @ « p1 » « 2 » « p1 »
¬ 2 ¼ ¬ s s 1¼ ¬ 2 ¼
ª a11 º
«a » ,
¬ 21 ¼
we obtain
ª p1 º
«p »
¬ 2¼
1
s º ª a11 º
ª1
« s s 2 1» « a »
¬
¼ ¬ 21 ¼
ª s 2 1 a11 sa21 º
«
».
¬« sa11 a21 ¼»
As a basis for this space we can also adopt
e1
ª1 º
« 0 » , e2
¬ ¼
ª0º
«1 » .
¬ ¼
In this case, p1 = a11 and p2 = a21.
Definition 1.6.3. The number of linearly independent rows (columns) of a
polynomial matrix A(s) num[s] is called its normal rank (briefly rank).
The rank of a polynomial matrix A(s) can be also equivalently defined as the
highest order of a minor, which is a nonzero polynomial, of this matrix.
The rank of matrix A(s) num[s] is not greater than the number of its rows n or
columns m, i.e.,
rank A( s ) d min (n, m) .
(1.6.8)
If a square matrix A(s) nun[s] is of full rank, i.e., rank A(s) = n, then its
determinant is a nonzero polynomial w(s), i.e.,
det A( s )
w( s ) z 0 .
(1.6.9)
26
Polynomial and Rational Matrices
Such a matrix is called nonsingular or invertible. It is called singular when
det A(s) = 0 (the zero polynomial). For example, the square matrix built from
linearly independent vectors (1.6.2) is nonsingular, since
s º
ª1
det «
1
s
1
s 2 »¼
¬
and the matrix built from linearly dependent vectors (1.6.4) is singular, since
ª1 s 1 º
det «
»
2
¬s s s¼
0.
Theorem 1.6.1. Elementary operations carried out on a polynomial matrix do not
change its rank.
Proof. Let
A( s)
L( s ) A( s )P( s )  num [ s ] ,
(1.6.10)
where L(s) nun[s] and P(s) mum[s] are unimodular matrices of elementary
operations on rows and columns, respectively.
From (1.6.10) we immediately have
rank A( s )
rank > L( s ) A( s )P ( s ) @
rank A( s ) ,
since L(s) and P(s) are unimodular matrices. For example, carrying out the operation Ld(2+1u(-s)) on rows of the matrix
built from the columns (1.6.2), we obtain
s º
ª 1 0 º ª1
« s 1 » « s s 2 1»
¬
¼¬
¼
ª1 s º
« 0 1» .
¬
¼
Both polynomial matrices
s º
ª1
ª1 s º
« s s 2 1» and «0 1»
¬
¼
¬
¼
are full rank matrices.
Polynomial Matrices
27
1.7. Equivalents of Polynomial Matrices
1.7.1 Left and Right Equivalent Matrices
Definition 1.7.1. Two polynomial matrices A(s), B(s) mun[s] are called left
(right) or row (column) equivalent if and only if one of them can be obtained from
the other as a result of a finite number of elementary operations carried out on its
rows (columns)
B( s )
L( s ) A( s )
or B( s )
A( s)P( s) ,
(1.7.1)
where L(s) (P(s)) is the product of unimodular matrices of elementary operations
on rows (columns).
Definition 1.7.2. Two polynomial matrices A(s), B(s) mun[s] are called
equivalent if and only if one of them can be obtained from the other as a result of a
finite number of elementary operations carried out on its rows and columns, i.e.,
B( s )
L( s ) A( s )P ( s ) ,
(1.7.2)
where L(s) and P(s) are the products of unimodular matrices of elementary
operations on rows and columns, respectively.
Theorem 1.7.1. A full rank polynomial matrix A(s)
upper triangular matrix of the form
A( s)
L( s ) A( s )
­ ª a11 ( s )
°«
°« 0
°« #
°«
°« 0
°« 0
°° «
®« #
°« 0
°¬
° ª a11 ( s )
°«
°« 0
°« #
°«
¯° ¬ 0
mul
[s] is left equivalent to an
a12 ( s ) ! a1l ( s ) º
a22 ( s ) ! a2l ( s ) »»
#
%
# »
»
! a1l ( s) »
0
!
0
0 »
»
#
%
# »
0
0 »¼
!
for
a12 ( s ) ! a1m ( s ) º
a22 ( s ) ! a2 m ( s ) »»
#
%
# »
»
0
! amm ( s ) ¼
for
m!l
(1.7.3)
m
l
28
Polynomial and Rational Matrices
­ ª a11 ( s ) a12 ( s )
°«
a22 ( s )
°« 0
®
#
°« #
° «¬ 0
0
¯
! a1m ( s ) !
a1l ( s ) º
! a2 m ( s ) ! a2l ( s ) »»
for m l
A ( s) L( s) A ( s )
%
#
%
# »
»
! amm ( s ) ! aml ( s ) ¼
where the elements a 1i(s), a 2i(s),…, a i-1,i(s) are polynomials of a degree less than
a ii(s) for i = 1,2,…,m, and L(s) is the product of the matrices of elementary
operations carried out on rows.
Proof. Among nonzero entries of the first columns of the matrix A(s) we choose
the entry that is a polynomial of the lowest degree and carrying out L[i, j], we
move this entry to the position (1,1). Denote this entry by a 11(s). Then we divide
all remaining entries of the first column by a 11(s). We then obtain
ai1 ( s )
a11 ( s )qi1 ( s ) ri1 ( s ) for i
2, 3, ..., m ,
where qi1(s) is the quotient and ri1(s) the remainder of division of the polynomial
a i1(s) by a 11(s). Carrying out L[i+1u(-qi1(s))], we replace the entry a i1(s) with
the remainder ri1(s). If not all remainders are equal to zero, then we choose this
one, that is the polynomial of the lowest degree, and carrying out operations L[i, j],
we move it to position (1,1). Denoting this remainder by r i1(s), we repeat the
above procedure taking the remainder r 11(s) instead of a 11(s). The degree r 11(s)
is lower than the degree of a 11(s). After a finite number of steps, we obtain the
matrix A (s) of the form
( s)
A
ª a11 ( s ) a12 ( s )
« 0
a22 ( s )
«
« #
#
«
am 2 ( s )
¬ 0
! a1l ( s ) º
! a2l ( s ) »»
.
%
# »
»
! aml ( s ) ¼
We repeat the above procedure for the first column of the submatrix obtained from
the matrix A(s) by deleting the first row and the first column. We then obtain a
matrix of the form
ˆ ( s)
A
ª a11 ( s ) a12 ( s ) a13 ( s )
« 0
a22 ( s ) aˆ23 ( s )
«
« 0
0
aˆ33 ( s )
«
#
#
« #
«¬ 0
0
aˆm 3 ( s )
! a1l ( s ) º
! aˆ2l ( s ) »»
! aˆ3l ( s ) » .
»
%
# »
! aˆml ( s ) »¼
If a 12(s) is not a polynomial of lower degree than the one of a 22(s), then we
divide a 12(s) by a 22(s) and carrying out L[1+2u(-q12(s))], we replace the entry
Polynomial Matrices
29
a 12(s) with the entry a 12(s) = r12(s), where q12(s) and r12(s) are the quotient and
the remainder on the division of a 12(s) by a 22(s) respectively.
Next, we consider the submatrix obtained from the matrix A (s) by removing
the first two rows and the first two columns. Continuing this procedure, we obtain
the matrix (1.7.3). An algorithm of determining the left equivalent matrix of the form (1.7.3)
follows immediately from the above proof.
Example 1.7.1.
The given matrix
A( s)
s
ª 1
« s 1 s 2
«
«¬ s 2 s 3 1
2 º
»
»
»¼
1
2s 2
is to be transformed to the left equivalent form (1.7.3).
To accomplish this, we carry out the following elementary operations:
ª1
o ««0
«¬0
L>1 2us @
ª1
L ª¬3 2u( ( s 2 2)) º¼

o ««0
«¬0
L ¬ª 21u ( s 1) ¼º
L ª31u( s 2 ) º
¬
¼
s
2 º
s
2 º
ª1
»
«
L[2,3]
o «0
1
0 »» 
o
s 2 2s 1» 
«¬0 s 2 2 2 s 1»¼
1
0 »¼
2
0
2
1
0
0 2 s 1
º
».
»
»¼
Theorem 1.7.2. A full rank polynomial matrix A(s)
lower triangular matrix of the form
A( s)
mul
[s] is right equivalent to a
A( s)P( s)
­ ª a11 ( s )
0
°«
° « a21 ( s ) a22 ( s )
°« °«
°¬ am1 ( s ) am 2 ( s )
®
0
° ª a11 ( s )
° « a (s) a ( s)
22
° « 21
«
°
° « a (s) a (s)
m2
¯ ¬ m1
0
0
0 0º
0 0 »»
for n ! m,
»
»
am 3 ( s ) amm ( s ) 0 0 ¼
(1.7.4)
0
0
0 º
0 »»
for n
»
»
amm ( s ) ¼
m,
30
Polynomial and Rational Matrices
­ ª a11 ( s )
0
°«
° « a21 ( s ) a22 ( s )
°° « #
#
®«
° « al1 ( s ) al 2 ( s )
°« #
#
°«
¯° «¬ am1 ( s ) am 2 ( s )
!
!
%
!
%
!
0 º
0 »»
# »
» for n m ,
al 2 ( s ) »
# »
»
aml ( s ) »¼
(1.7.4)
where the elements a i1(s), a i2(s),…, a i-1,i(s) are polynomials of lower degree
than that of a ii(s) for i = 1,2,…,n, and P(s) is the product of unimodular matrices
of elementary operations carried out on columns.
1.7.2 Row and Column Reduced Matrices
The degree of the i-th column (row) of a polynomial matrix is the highest degree of
a polynomial that is an entry of this column (row).
The degree of the i-th column (row) of the matrix A(s) will be denotedn by
deg ci[A(s)] (deg ri[A(s)]) or shortly deg ci (deg ri).
Let Lc (Lr) be the matrix built from the coefficients at the highest powers of
variable s in the columns (rows) of the matrix A(s). For example, for the
polynomial matrix
ªs2 1 s
3s º
«
»
s
2
s
2 »,
«
« s2
s 1 2 s 1»¼
¬
A( s)
(1.7.5)
we have deg A(s) = 2
deg c1
2, deg c2
deg c3
1,
deg r1
deg r3
2, deg r2
and
Lk
ª1 1 3º
« 0 1 0 » , L
w
«
»
¬«1 1 2 »¼
ª1 0 0 º
«1 1 0 » .
«
»
«¬1 0 0 »¼
The matrix (1.7.5) can be written, using the above matrices, as follows
A( s)
2
ª1 1 3º ª s
«
«
»
«0 1 0 » « 0
¬«1 1 2 ¼» ¬« 0
0 0 º ª 1
0 0º
» «
s 0 » « s 2 0 2 »»
0 s ¼» ¬« 0
1 1¼»
1
Polynomial Matrices
31
or
ªs2
«
«0
«0
¬
A( s)
0
0 º ª1 0 0 º ª 1 s
3s º
»«
»
«
s 0 » «1 1 0 » « 2
0
2 »» .
0 s 2 ¼» ¬«1 0 0 ¼» ¬« 0 s 1 2s 1¼»
In the general case for a matrix A(s)
mun
[s], we have
A( s)
L c diag ª¬ s deg c1 , s deg c2 ,..., s deg cl º¼ A( s )
(1.7.6)
A(s)
(s) ,
diag ª¬ s deg r1 , s deg r2 ,..., s deg rm º¼ L r A
(1.7.7)
and
(s) are polynomial matrices satisfying the conditions
where A (s), A
( s ) deg A ( s ) .
deg A ( s ) deg A ( s ), deg A
If m = n and det Lc z 0, then the determinant of the matrix (1.7.6) is a polynomial
of the degree
nk
l
¦ deg
ci ,
i 1
since
det A ( s )
det L k det diag ª¬ s deg c1 , s deg c2 ,..., s deg cl º¼ ...
s nk det L c ...
Similarly, if det Lr z 0, then the determinant of the matrix (1.7.7) is a polynomial
of the degree
nr
m
¦ deg
rj .
j 1
Definition 1.7.3. A polynomial matrix A(s) is said to be column (row) reduced if
and only if Lc (Lr) of this matrix is a full rank matrix.
Thus, a square matrix A(s) is column (row) reduced if and only if det Lc z 0
(det Lr z 0).
For example, the matrix (1.7.5) is column reduced but not row reduced, since
32
Polynomial and Rational Matrices
1
det L c
1
3
0 1
0
1
2
1
1
5,
det L r
0
0
1 1 0
1
0
0.
0
From the above considerations and Theorems 1.7.1 and 1.7.1c the following
important corollary immediately follows.
Corollary 1.7.1. Carrying out only elementary operations on rows or columns it is
possible to transform a nonsingular polynomial matrix to one of column reduced
form and row reduced form, respectively.
1.8 Reduction of Polynomial Matrices to the Smith Canonical
Form
mun
Consider a polynomial matrix A(s)
[s] of rank r.
Definition 1.8.1. A polynomial matrix of the form
A S (s)
0
ªi1 ( s)
« 0 i (s)
2
«
« #
#
«
0
« 0
« 0
0
«
#
« #
« 0
0
¬
!
!
%
!
!
%
!
0
0 ! 0º
0
0 ! 0 »»
#
# % #»
»
ir ( s ) 0 ! 0 »  mun [ s ] .
0
0 ! 0»
»
#
# % #»
0
0 ! 0 »¼
(1.8.1)
r d min(n,m) is called the Smith canonical form of the matrix A(s) mun[s], where
i1(s), i2(s),…,ir(s) are nonzero polynomials that are called invariant, with
coefficients by the highest powers of the variable s equal to one, such that the
polynomial ik+1(s) is divisible without remainder by the polynomial ik(s), i.e.,
ik+1 | ik for k = 1,…,r-1.
Theorem 1.8.1. For an arbitrary polynomial matrix A(s) mun[s] of rank r
(r d min(n,m)) there exists its equivalent Smith canonical form (1.8.1).
Proof. Among the entries of the matrix A(s) we find a nonzero one, which is a
polynomial of the lowest degree in respect to s, and interchanging rows and
columns we move it to position (1,1). Denote this entry by a 11(s). Assume at the
beginning that all entries of the matrix A(s) are divisible without remainder by the
entry a 11(s). Dividing the entries a i1(s) of the first column and the first row
a 1j(s) by a 11(s), we obtain
Polynomial Matrices
ai1 ( s )
a11 ( s )qi1 ( s )
(i
2, 3,..., m),
a1 j ( s )
a11 ( s )q1 j ( s )
(j
2, 3,..., n),
33
where qi1(s) and q1j(s) are the quotients from division of a i1(s) and a 1j(s) by
a 11(s), respectively.
Subtracting from the i-th row (i = 2,3,…,m) the first row multiplied by qi1(s)
and, respectively from the j-th column (j = 2,3,…,m) the first column multiplied by
q1j(s), we obtain a matrix of the form
0
ª a11 ( s )
« 0
a
22 ( s )
«
« #
#
«
am 2 ( s )
¬ 0
0 º
! a2 n ( s ) »»
.
%
# »
»
! amn ( s ) ¼
!
(1.8.2)
If the coefficient by the highest power of s of polynomial a 11(s) is not equal to
1, then to accomplish this we multiply the first row (or column) by the reciprocal
of this coefficient.
Assume next that not all entries of the matrix A(s) are divisible without
remainder by a 11(s) and that such entries are placed in the first row and the first
column. Dividing the entries of the first row and the first column by a 11(s), we
obtain
a1i ( s )
a11 ( s )q1i ( s ) r1i ( s )
(i
a j1 ( s )
a11 ( s )q j1 ( s ) rj1 ( s )
(j
2, 3,..., n),
2, 3,..., m),
where q1i(s), qj1(s) are the quotients and r1i(s), rj1(s) are the remainders of division
of a 1i(s) and a j1(s) by a 11(s), respectively. Subtracting from the j-th row (i-th
column) the first row (column) multiplied by qj1(s) (by q 1i(s)), we replace the
entry a j1(s) ( a 1i(s)) by the remainder rj1(s) (r1i(s)). Next, among these remainders
we find a polynomial of the lowest degree with respect to s and interchanging rows
and columns, we move it to the position (1,1). We denote this polynomial by
r 11(s). If not all entries of the first row and the first column are divisible without
remainder by r 11(s), then we repeat this procedure taking the polynomial r 11(s)
instead of the polynomial a 11(s). The degree of the polynomial r 11(s) is lower
than the degree of a 11(s). After a finite number of steps, we obtain in the position
(1,1) a polynomial that divides without remainder all the entries of the first row
and the first column. If the entry a ik(s) is not divisible by a 11(s), then by adding
the i-th row (or k-th column) to the first row (the first column), we reduce this case
to the previous one.
Repeating this procedure, we finally obtain in the position (1,1) a polynomial
that divides without remainder all the entries of the matrix. Further we proceed in
the same way as in the first case, when all the entries of the matrix are divisible
without remainder by a 11(s).
34
Polynomial and Rational Matrices
If not all entries a ij(s) (i = 2,3,…,m; j = 2,3,…,n) of the matrix (1.8.2) are
equal to zero, then we find a nonzero entry among them, which is a polynomial of
the lowest degree with respect to s, and interchanging rows and columns, we move
it to the position (2,2). Proceeding further as above, we obtain a matrix of the form
ª a11 (s)
« 0
«
« 0
«
« #
« 0
¬
0
a22 (s)
0
0
0
a33 ( s)
"
"
"
#
0
#
am 3 (s)
%
"
0 º
0 »»
a3n (s) » ,
»
# »
amn (s) »¼
where a 22(s) is divisible without remainder by a 11(s), and all elements a ij(s)
(i = 3,4,…,m; j = 3,4,…,n) are divisible without remainder by a 22(s). Continuing
this procedure, we obtain a matrix of the Smith canonical form (1.8.1).
From this proof the following algorithm for determining of the Smith canonical
form follows immediately as, illustrated by the following example.
Example 1.8.1.
To transform the polynomial matrix
A( s)
ª ( s 2) 2
( s 2)( s 3) s 2 º
«
»
( s 2) 2
s 3¼
¬( s 2)( s 3)
(1.8.3)
to the Smith canonical form, we carry out the following elementary operations.
Step 1: We carry out the operation P[1, 3]
A1 ( s )
ª s 2 ( s 2)( s 3)
( s 2) 2 º
«
».
( s 2) 2
( s 2)( s 3) ¼
¬s 3
All entries of this matrix are divisible without remainder by s + 2 with exception of
the entry s + 3.
Step 2: Taking into account the equality
s3
s2
1
1
,
s2
we carry out the operation L[2+1u(-1)]
A 2 ( s)
ª s 2 ( s 2)( s 3) ( s 2) 2 º
«
» .
( s 2)
s2 ¼
¬ 1
Polynomial Matrices
35
Step 3: We carry out the operation L[1, 2]
A3 ( s)
s2
( s 2)
ª 1
« s 2 ( s 2)( s 3) ( s 2) 2
¬
º
».
¼
Step 4: We carry out the operations P[2+1u(s+2)] and P[3+1u(-s-2)]
A 4 ( s)
0
0º
ª 1
« s 2 ( s 2)(2s 5) 0 » .
¬
¼
Step 5: We carry out the operation L[2+1u(-s-2)] and P[2u1 / 2]
A s (s)
0
0º
ª1
«
».
5
«0 ( s 2) ¨§ s ¸· 0 »
2 ¹ »¼
©
¬«
This matrix is of the desired Smith canonical form of (1.8.3).
From divisibility of the invariant polynomials ik+1 | ik, k = 1, ..., r – 1, it follows
that there exist polynomials d1,d2,…,dr, such that
i1
d1 , i2
d1d 2 , ..., ir
d1d 2 ... d r .
Hence the matrix (1.8.1) can be written in the form
A S (s)
ª d1
«0
«
«#
«
«0
«0
«
«#
«0
¬
0
!
0
d1d 2 !
0
#
%
0
! d1d 2 ... d r
#
0
!
0
#
%
#
0
!
0
0 0 ! 0º
0 0 ! 0 »»
# # % #»
»
0 0 ! 0» .
0 0 ! 0»
»
# # % #»
0 0 ! 0 »¼
(1.8.1a)
Theorem 1.8.2. The invariant polynomials i1(s),i2(s),…,ir(s) of the matrix (1.8.1)
are uniquely determined by the relationship
ik ( s )
Dk ( s )
Dk 1 ( s )
for k
1, 2, ..., r ,
(1.8.4)
where Dk(s) is the greatest common divisor of all minors of degree k of matrix A(s)
(D0(s) = 1).
36
Polynomial and Rational Matrices
Proof. We will show that elementary operations do not change Dk(s). Note that
elementary operations 1) consisting of multiplying of an i-th row (column) by a
number c z 0 causes multiplication of minors containing this row (column) by this
number c. Thus this operation does not change Dk(s). An elementary operation 2)
consisting of adding to an i-th row (column) j-th row (column) multiplied by the
polynomial w(s) does not change Dk(s), if a minor of the degree k contains either
the i-th row and the j-th row or does not contain of them. If the minor of the degree
k contains the i-th row, and does not contain the j-th row, then we can represent it
as a linear combination of two minors of the degree k of the matrix A(s). Hence the
greatest common divisor of the minors of the degree k does not change. Finally, an
operation 3), consisting on the interchange of i-th and j-th rows (columns), does
not change Dk(s) either, since as a result of this operation a minor of the degree k
either does not change (the both rows (columns) do not belong to this minor), or
changes only the sign (both rows belong to the same minor), or it will be replaced
by another minor of the degree k of the matrix A(s) (only one of these rows
belongs to this minor).
Thus equivalent matrices A(s) and AS(s) have the same divisors D1(s), D2(s), ...,
Dr(s). From the Smith canonical form (1.8.1) it follows that
D1 ( s )
i1 ( s ),
(1.8.5)
D2 ( s )
i1 ( s ) i2 ( s ),
Dr ( s )
i1 ( s ) i2 ( s )...ir ( s ).
From (5) we immediately obtain the formula (4). Using the polynomials d1,d2,…,dr we can write the relationship (1.8.5) in the
form
D1 ( s )
d1 ,
D2 ( s )
d12 d 2 ,
Dr ( s )
r
1
d d
r 1
2
.
(1.8.6)
...d r
From definition (1.8.1) and Theorems 1.8.1 and 1.8.2, the following important
corollary can be derived.
Corollary 1.8.1. Two matrices A(s), B(s)
they have the same invariant polynomials.
mu n
[s] are equivalent if and only if
Polynomial Matrices
37
1.9 Elementary Divisors and Zeros of Polynomial Matrices
1.9.1 Elementary Divisors
Consider a polynomial matrix A(s) mun[s] of the rank r, whose Smith canonical
form AS(s) is given by the formula (1.8.1).
Let the k-th invariant polynomial of this matrix be of the form
ik ( s)
m
( s s1 ) k1 ( s s2 )
mk2
...( s sq )
mkq
.
(1.9.1)
From divisibility of the polynomial ik+1(s) by the polynomial ik(s) it follows that
mr ,1 t mr 1,1 t ... t m1,1 t 0
.
mr ,q t mr 1,q t ... t m1,q t 0
(1.9.2)
If, for example, i1(s) = 1, then m11 = m12 = … =m 1q = 0.
Definition 1.9.1. Everyone of the expressions (different from 1)
( s s1 ) m11 , ( s s2 ) m12 , ..., ( s sq )
mrq
appearing in the invariant polynomials (1.9.1) is called elementary divisor of the
matrix A(s).
For example, the elementary divisors of the polynomial matrix (1.8.3) are (s+2)
and (s+2, 5).
The elementary divisors of a polynomial matrix are uniquely determined. This
follows immediately from the uniqueness of the invariant polynomial of
polynomial matrices. Equivalent polynomial matrices possess the same elementary
divisors. For a polynomial matrix of known dimensions its rank together with its
elementary divisors uniquely determine its Smith canonical form.
For example, knowing the elementary divisors s 1, (s 1)(s 2), (s 2)2 ,
(s 3), of a polynomial matrix, its rank r = 4 and dimension 4u4, we can write its
Smith canonical form of this polynomial matrix
A s (s)
0
0
0
ª1
º
«0 s 1
»
0
0
«
».
«0
»
0
( s 1)( s 2)
0
«
»
2
0
0
( s 1)( s 2) ( s 3) ¼
¬0
Consider a polynomial, block-diagonal matrix of the form
(1.9.3)
38
Polynomial and Rational Matrices
A( s)
diag [ A1 ( s ), A 2 ( s )]
0 º
ª A1 ( s )
.
« 0
A 2 ( s ) »¼
¬
(1.9.4)
Let AkS(s) be the Smith canonical form of the matrix Ak(s), k = 1,2, and
k
( s sk1 )m11 ,..., ( s skq )
k
mrk
, qk
its elementary divisors.
Taking into account that equivalent polynomial matrices have the same
elementary divisors, we establish that a set of elementary divisors of the matrix
(1.9.4) is the sum of the sets of elementary divisors of Ak(s), k = 1,2.
Example 1.9.1.
Determine elementary divisors of the block-diagonal matrix (1.9.4) for
A1 ( s )
0 º
ªs 1 1
« 0
s 1 1 »» , A 2 ( s )
«
«¬ 0
0
s 1»¼
0 º
ªs 1 1
« 0
s 1
0 »» .
«
«¬ 0
0
s 2 »¼
(1.9.5)
It is easy to check that the Smith canonical forms of the matrices (1.9.5) are
A1S ( s )
0 º
ª1 0
«0 1
» , A ( s)
0
2S
«
»
«¬ 0 0 ( s 1)3 »¼
0
ª1 0
º
«0 1
».
0
«
»
«¬ 0 0 ( s 1) 2 ( s 2) »¼
(1.9.6)
The elementary divisors of the matrices (1.9.5) are thus equal (s 1)3, (s 1)2, and
(s 2), respectively. It is easy to show that the Smith canonical form of the matrix
(1.9.4) with the blocks (1.9.5) is equal to
A S (s)
diag ª¬1 1 1 1 ( s 1) 2
( s 1)3
( s 2) º¼
(1.9.7)
and its elementary divisors are (s - 1)2, (s - 1)3, (s - 2).
Consider a matrix A nun and its corresponding polynomial matrix [Ins - A].
Let
[I n s A]S
diag >i1 ( s), i2 ( s ), ..., in ( s ) @ ,
(1.9.8)
where
ik ( s )
m
( s s1 ) k1 ( s s2 )
mk2
...( s sq )
mkq
, k
1, ..., n ,
(1.9.9)
Polynomial Matrices
39
and s1,s2,…,sq, q d n are the eigenvalues of the matrix A.
Definition 1.9.2. Everyone of the expressions (different from 1)
( s s1 ) m11 , ( s s2 ) m12 , ..., ( s sq )
mnq
appearing in the invariant polynomials (1.9.9) is called the elementary divisor of
the matrix A.
The elementary divisors of the matrix A are uniquely determined and they
determine its essential structural properties.
1.9.2 Zeros of Polynomial Matrices
Consider a polynomial matrix A(s) mun[s] of rank r, whose Smith canonical form
is equal to (1.8.1). From (1.8.5) it follows that
Dr ( s )
i1 ( s )i2 ( s )...ir ( s ) .
(1.9.10)
Definition 1.9.3. Zeros of the polynomial (1.9.10) are called zeros of the
polynomial matrix A(s).
The zeros of the polynomial matrix A(s) can be equivalently defined as those
values of the variable s, for which this matrix loses its full (normal) rank. For
example, for the polynomial matrix (1.8.3) we have
Dr ( s )
( s 2)( s 2.5) .
Thus the zeros of the matrix are s10 = -2, s20 = -2.5.
It is easy to verify that for these values of the variable s, the matrix (1.8.3)
(whose normal rank is equal to 2) has a rank equal to 1.
If the polynomial matrix A(s) is square and of the full rank r = n, then
det A ( s )
cDr ( s )
c is a constant coefficient independent of s
(1.9.11)
and the zeros of this matrix coincide with the roots of its characteristic equation
det A(s) = 0.
For example, for the first among the matrices (1.9.5) we have
det A r ( s )
s 1
1
0
0
s 1
1
0
0
s 1
( s 1)3 .
40
Polynomial and Rational Matrices
Thus this matrix has the zero s = 1 of multiplicity 3. The same result will be
obtained from (1.9.10), since Dr(s) = (s - 1)3 for A1S(s).
Theorem 1.9.1. Let a polynomial matrix A(s)
to r d min(m,n). Then
rank A s
­r
®
¯r di
mun
[s] have a rank (normal) equal
s V A
s
½
¾,
si  V A ¿
(1.9.12)
where VA is a set of the zeros of the matrix A(s) and di is a number of distinct
elementary divisors containing si.
Proof. By definition of zero, it follows that the matrix A(s) does not lose its full
rank if we substitute in place of the variable s a number that does not belong to the
set VA, i.e., rank A(s) = r for sVA. Elementary operations do not change the rank
of a polynomial matrix. In view of this rank A(s) = rank AS(s) = r, where r is the
number of the invariant polynomials (including those equal to 1). If an invariant
polynomial contains si, then this polynomial is equal to zero for s = si. Thus we
have rank A(si) = r - di, siVA, since the number of polynomials containing si is
equal to the number of distinct elementary divisors containing si.
„
For instance, the polynomial matrix (1.9.3) of the full column rank has one
elementary divisor containing s10 = 3, two elementary divisors containing s20 = 2
and three elementary divisors containing s30 = 1. In view of this, according
to (1.9.12) we have
rank A S (3)
3, rank A S (2)
2, rank A S (1) 1 .
Remark 1.9.1.
A unimodular matrix U(s) nun[s] does not have any zeros since det U(s) = c,
where c is certain constant independent of the variable s.
Theorem 1.9.2. An arbitrary rectangular, polynomial matrix A(s)
rank that does not have any zeros can be written in the form
A( s)
where P(s)
­> I m 0@ P ( s ), m n ½
°
°
ªI n º
®
¾,
L
!
s
m
n
(
)
,
«0»
°
°
¬ ¼
¯
¿
nu n
[s] and L(s)
mu m
mun
[s] of full
(1.9.13)
[s] are unimodular matrices.
Proof. If m < n and the matrix does not have any zeros, then applying elementary
operations on columns we can bring this matrix to the form [Im 0]. Similarly, if
Polynomial Matrices
41
m > n and the matrix does not have any zeros, then applying elementary operations
ªI º
on rows we can bring this matrix to the form « n » .
¬0¼
„
Remark 1.9.2.
From the relationship (1.9.13) it follows that a polynomial matrix built from an
arbitrary number of rows or columns of a matrix that does not have any zeros,
never has any zeros.
Theorem 1.9.3. An arbitrary polynomial matrix A(s) mun[s] of rank
r d min (m, n) having zeros can be presented in the form of the product of matrices
A( s)
B( s )C( s ) ,
(1.9.14)
where the matrix B(s) = L-1(s) diag [i1(s),…,ir(s),0,…,0]
containing all the zeros of the matrix A(s), and
C( s )
­
1
°> I m 0@ P ( s ),
°
°°
P 1 ( s ),
®
°
° ªI º
° « n » P 1 ( s ),
°¯ ¬ 0 ¼
n!m
n
m
nm
mu m
is a matrix
½
°
°
°°
.
¾
°
°
°
°¿
(1.9.15)
Proof. Let L(s) mum[s] and P(s) nun[s] be unimodular matrices of elementary
operations on rows and on columns, respectively, reducing the matrix A(s) to the
Smith canonical form AS(s), i.e.,
A S (s)
L( s ) A( s )P ( s ) .
(1.9.16)
Pre-multiplying (1.9.16) by L-1(s) and post-multiplying by P-1(s), we obtain
A( s)
L1 ( s ) A S ( s )P 1 ( s )
A S (s)
­
½
°diag [i1 ( s ), ..., ir ( s), 0, ..., 0] > I m 0@ , n ! m °
°
°
°°
°°
n m¾ .
®diag [i1 ( s ), ..., ir ( s), 0, ..., 0],
°
°
°
°
ªI n º
°diag [i1 ( s ), ..., ir ( s), 0, ..., 0] « » ,
n m°
¬0¼
¯°
¿°
B( s )C( s ) ,
since
42
Polynomial and Rational Matrices
From (1.9.15) it follows that the matrix C(s)
since the matrix P-1(s) is a unimodular matrix.
mun
[s] does not have any zeros,
„
1.10 Similarity and Equivalence of First Degree Polynomial
Matrices
Definition 1.10.1. Two square matrices A and B of the same dimension are said to
be similar matrices if and only if there exists a nonsingular matrix P such that
B
P 1AP
(1.10.1)
and the matrix P is called a similarity transformation matrix.
Theorem 1.10.1. Similar matrices have the same characteristic polynomials, i.e.,
det [ sI B] det [ sI A] .
(1.10.2)
Proof. Taking into account (1.10.1), we can write
det [ sI B] det ª¬ sP 1P P 1AP º¼ det ª¬ P 1 ( sI A)P º¼
det P 1 det [ sI A ]det P det [ sI A ] ,
since det P-1 = (det P)-1. „
Theorem 1.10.2. Polynomial matrices [sI - A] and [sI - B] are equivalent if and
only if the matrices A and B are similar.
Proof. Firstly, we show that if the matrices A and B are similar, then the
polynomial matrices [sI - A] and [sI - B] are equivalent. If the matrices A and B
are similar, i.e., they satisfy the relationship (1.10.1), then
[ sI B ]
ª¬ sI P 1AP º¼
P 1[ sI A]P .
This relationship is a special case (for L(s) = P-1 and P(s) = P) of the relationship
(1.7.2). Thus the polynomial matrices [sI - A] and [sI - B] are equivalent. We will
show now, that if the matrices [sI - A] and [sI - B] are equivalent, then the matrices
A and B are similar. Assuming that the matrices [sI - A] and [sI - B] are
equivalent, we have
[ sI B] L( s )[ sI A]P ( s ) ,
(1.10.3)
Polynomial Matrices
43
where L(s) and P(s) are unimodular matrices. The determinant of the matrix L(s) is
different from zero and does not depend on the variable s. In view of this, the
inverse matrix
Q( s )
L1 ( s )
is a polynomial, unimodular matrix as well. Pre-dividing the matrix Q(s) by
[sI - A] and post-dividing P(s) by [sI - B], we obtain
Q( s ) [ sI A]Q1 ( s ) Q 0 ,
P( s)
P1 ( s )[ sI B] P0 ,
(1.10.4)
(1.10.5)
where Q1(s) and P1(s) are polynomial matrices and the matrices Q0 and P0 do not
depend on the variable s. With (1.10.3) pre-multiplied by Q(s) = L-1(s) we obtain
Q( s )[ sI B] [ sI A]P ( s )
(1.10.6)
and after substitution of (1.10.4) and (1.10.5) into (1.10.6)
[ sI A] >Q1 ( s ) P1 ( s ) @[ sI B] [ sI A]P0 Q 0 [ sI B] .
(1.10.7)
Note that the following equality must hold
Q1 ( s )
P1 ( s ) ,
(1.10.8)
since otherwise the left-hand side of (1.10.7) would be a matrix polynomial of a
degree of at least 2, and the right side a matrix polynomial of degree of at most 1.
After taking into account the equality (1.10.8) from (1.10.7) we obtain
Q 0 [ sI B] [ sI A]P0 .
(1.10.9)
Pre-division of the matrix L(s) by [sI - B] yields
L( s ) [ sI B]L1 ( s ) L 0 ,
(1.10.10)
where L1(s) is a polynomial matrix and L0 is a matrix independent of the variable
s.
We will show that the matrices Q0 and L0 are nonsingular matrices satisfying
the condition
Q0L0
I.
Substitution of (1.10.4) and (1.10.10) into the equality
(1.10.11)
44
Polynomial and Rational Matrices
Q ( s )L ( s )
I
yields
I
Q ( s )L ( s )
>( sI A)Q1 ( s) Q0 @>(sI B)L1 ( s) L0 @
(1.10.12)
[ sI A]Q1 ( s )[ sI B]L1 ( s ) Q 0 [ sI B]L1 ( s ) [ sI A]Q1 ( s )L 0 Q 0 L 0 .
Note that this equality can be satisfied if and only if
[ sI A]Q1 ( s)[ sI B]L1 ( s ) Q 0 [ sI B]L1 ( s ) [ sI A]Q1 ( s )L 0
0 . (1.10.13)
Otherwise the left-hand side of (1.10.12) would be a matrix polynomial of zero
degree and the right-hand side would be a matrix polynomial of at least the first
degree. With (1.10.13) taken into account, from (1.10.12) we obtain the equality
(1.10.11).
From this equality the nonsingularity of the matrices Q0 and L0 as well the
equality L0 = Q0-1 follow immediately.
Pre-multiplication of (1.10.9) by Q0-1 yields
[ sI B] L 0 [ sI A ]P0
and
B
L 0 AP0 , L 0 P0
I.
From these relationships it follows that the matrices A and B are similar.
„
Theorem 1.10.3. Matrices A and B are similar if and only if the matrices [sI A]
and [sI B] have the same invariant polynomials.
Proof. According to Corollary 1.8.1 two matrices are equivalent if and only if they
have the same invariant polynomials. From Theorem 1.10.2 it follows immediately
that the polynomial matrices [sI A] and [sI B] have the same invariant
polynomials if and only if the matrices A and B are similar. Thus the matrices A
and B are similar if and only if the matrices [sI A] and [sI B] have the same
invariant polynomials.
„
Polynomial Matrices
45
1.11 Computation of the Frobenius and Jordan Canonical Forms
of Matrices
1.11.1 Computation of the Frobenius Canonical Form of a Square Matrix
Consider nun matrices of the form
F
1
0
ª0
«0
0
1
«
«#
#
#
«
0
0
0
«
«¬ a0 a1 a2
Fˆ
ª an1
« 1
«
« 0
«
« #
«¬ 0
an2
0
1
#
0
! 0
! 0
0
0
º
ª0
»
«1
»
«
» , F «#
% #
#
»
«
! 0
1
»
«0
«¬0
! an2 an1 »¼
! a1 a0 º
ª an1
« a
»
! 0
0 »
« n2
0 » , F « #
! 0
«
»
% #
# »
« a1
«¬ a0
! 1
0 »¼
0 ! 0
0 ! 0
# % #
1 ! 0
0 ! 1
1 0 !
0 1 !
# # %
0 0 !
0 0 !
a0 º
a1 »»
# »,
»
a2 »
an1 »¼ (1.11.1)
0º
0 »»
#».
»
1»
0 »¼
We say that the matrices in (1.11.1) have Frobenius canonical forms (or normal
canonical forms).
Expanding along the row (or the column) containing a0,a1,…,an1, it is easy to
show that
det > I n s F @ det ª¬I n s F º¼
det ª¬ I n s Fˆ º¼
det ª¬I n s F º¼
s n an1s n1 ... a1s a0 .
(1.11.2)
We will show that the polynomial (1.11.2) is the only invariant polynomial of the
matrix (1.11.1) different from 1. Detailed considerations will be given only for the
matrix F. The proof in the other three cases is similar. Deleting the first column
and the n-th row in the matrix
>I n s F @
ªs
«0
«
«#
«
«0
«¬ a0
1
0
...
0
s
1 ...
0
#
0
#
0
#
s
a1
a2 ! an2
%
!
0
º
»
»
# »,
»
1 »
s an1 »¼
0
(1.11.3)
we obtain the minor Mn1 equal to (1)n1. With the above in mind, a greatest
common devisor of all minors of degree n 1 of this matrix is equal to 1, i.e.,
46
Polynomial and Rational Matrices
Dn1(s) = 1. From the relationship (1.8.4) it follows that the polynomial (1.11.2) is
the only polynomial of the matrix F different from 1.
Let A nun and the monic polynomials
i1 ( s ) 1, ..., i p ( s) 1, i p 1 ( s), ..., in ( s)
be the invariant polynomials of the polynomial matrix [Ins A], where
ip+1(s),…,in(s) are the polynomials of at least the first degree such that ik(s) divides
(without remainder) ik+1(s) (k = p+1,…,n1). The matrix [sI A] reduced to the
Smith canonical form is of the form
[ sI A]S
ª1
«#
«
«0
«
«0
«#
«
¬«0
0
#
0
0
"
%
"
!
# %
0 "
0
0
#
#
1
0
0 i p 1 ( s )
#
0
#
0
0 º
"
% ! »»
0 »
!
».
0 »
"
%
# »
»
" in ( s ) ¼»
(1.11.4)
Let Fp+1,…,Fn be the matrices of the form (1.11.1) that correspond to the invariant
polynomials ip+1(s),…,in(s). From considerations of Sect. 1.10 it follows that the
quasi-diagonal matrix
FA
0
ªFp 1
« 0 F
p2
«
« #
#
«
0
¬ 0
" 0º
" 0 »»
% #»
»
" Fn ¼
(1.11.5)
and A have the same invariant polynomials. Thus according to Theorem 1.10.2, the
matrices A and FA are similar. Hence there exists a nonsingular matrix P such that
A
PFAP 1 .
(1.11.6)
The matrix FA given by (1.11.5) is called a Frobenius canonical form or a normal
canonical form of the square matrix A.
Thus the following important theorem has been proved.
Theorem 1.11.1. For every matrix A nun there exists a nonsingular matrix
P nunsuch that the equality (1.11.6) holds.
Example 1.11.1.
The following matrix is given
Polynomial Matrices
A
ª 1 1 0º
« 0 1 0» .
«
»
«¬ 1 0 2 »¼
47
(1.11.7)
Carrying out the elementary operations: P[1+2u(s1)], L[2+1u(s1)],
P[3+1u(s+2)], L[2u(1)], L[2+3u(s1)2], L[1u(1)], L[2, 3], L[1, 2] on the matrix
0 º
ª s 1 1
« 0
0 »» ,
s 1
«
«¬ 1
0
s 2 »¼
[ sI 3 A ]
we transform this matrix to its Smith canonical form
[ sI 3 A]S
0
ª1 0
º
«0 1
».
0
«
»
«¬0 0 ( s 1) 2 ( s 2) »¼
Thus the matrix A has the only invariant polynomial different from one
( s 1) 2 ( s 2)
i3 ( s )
s 3 4 s 2 5s 2 .
In view of this, the Frobenius canonical form of the matrix (1.11.7) is the following
FA
ª0 1 0º
«0 0 1 » .
«
»
«¬ 2 5 4 »¼
(1.11.8)
1.11.2 Computation of the Jordan Canonical Form of a Square Matrix
Consider an elementary divisor of the form
s s0
m
.
(1.11.9)
We will show that the polynomial (1.11.9) is the only elementary divisor of a
square matrix of the form
48
Polynomial and Rational Matrices
J
J ( s10 , m)
ª s0
«0
«
«#
«
«0
«¬ 0
1
s0
#
0
0
0
1
#
0
0
" 0º
" 0 »»
% # »  mum ,
»
" 1»
" s0 »¼
(1.11.10a)
or
Jc
J '( s10 , m)
ª s0
«1
«
«#
«
«0
«¬ 0
0
s0
#
0
0
" 0
" 0
% #
" s0
" 1
0º
0 »»
# »  mum .
»
0»
s0 »¼
(1.11.10b)
The determinant of the polynomial matrix
[ sI m J ]
ª s s0
« 0
«
« #
«
« 0
«¬ 0
1
s s0
#
0
0
0
1
#
0
0
...
0
...
0
%
#
... s s0
...
0
0 º
0 »»
# »
»
1 »
s s0 »¼
(1.11.11)
is equal to the polynomial (1.11.9).
The minor Mn1 obtained from the matrix (1.11.11) by removing the first and the
m-th columns is equal to (1)m1. Thus a greatest common divisor of all minors of
degree m-1 of the matrix (1.11.11) is equal to 1, Dm-1(s) = 1. From (1.8.4) it follows
that the polynomial (1.11.9) is the only invariant polynomial of the matrix
(1.11.11) different from 1. The proof for the matrix Jc is similar.
The matrices J and Jc are called Jordan blocks of the first and the second type,
respectively.
If q elementary divisors correspond to one eigenvalue, then q Jordan blocks
correspond to this eigenvalue.
Let J1, J2,…,Jp be Jordan blocks of the form (1.11.10a) (or (1.11.10b)),
corresponding to the elementary divisors of the matrix A, where p is the number of
elementary divisors of this matrix. Note that all these elementary divisors of the
matrix A are also the elementary divisors of a quasi-diagonal matrix of the form
Polynomial Matrices
JA
ª J1
«0
«
«#
«
¬« 0
0
J2
#
0
" 0º
" 0 »»
 nun .
% # »
»
" J p »¼
49
(1.11.12)
Matrices having the same elementary divisors also have the same invariant
polynomials. In view of this, according to Theorem 1.10.2, the matrices A and JA,
being matrices having the same invariant polynomials, are similar. Thus there
exists a nonsingular matrix T such that
A
TJ AT1 .
(1.11.13)
The matrix (1.11.12) is called the Jordan canonical form of the matrix A, or shortly
the Jordan matrix. Thus the following important theorem has been proved.
Theorem 1.11.2. For every matrix A nun there exists a nonsingular matrix
T nun such that the equality (1.11.13) holds.
If all elementary divisors of the matrix A are of the first degree (in the
relationship (1.11.9) m = 1), then a Jordan matrix is a diagonal one. Thus we have
the following important corollary.
Corollary 1.11.1. A matrix A is similar to the diagonal matrix consisting of its
eigenvalues if and only if all its elementary divisors are divisors of the first degree.
Example 1.11.2.
The matrix (1.11.7) has only one invariant polynomial different from one and equal
to i(s) = (s 1)2(s 2). Thus this matrix has two elementary divisors (s 1)2 and
(s 2). Hence the Smith canonical form of the matrix (1.11.7) is equal to
JA
ª1 1 0 º
«0 1 0 » .
«
»
«¬0 0 2 »¼
1.12 Computation of Similarity Transformation Matrices
1.12.1 Matrix Pair Method
A cyclic matrix A nun and its Frobenius form FA are given. Compute a nonsingular matrix P nun such that
50
Polynomial and Rational Matrices
PAP 1
ª 0
« 0
«
« #
«
« 0
«¬ a0
FA
1
0
#
0
a1
0
1
#
0
a2
"
0 º
"
0 »»
%
# ».
»
1 »
"
" an1 »¼
For the given matrix A we choose a row matrix c
1un
ª c º
« cA »
»z0.
det «
« # »
« n1 »
¬cA ¼
(1.12.1)
such that
(1.12.2)
Almost every matrix c chosen by a “triall and error” method will satisfy the
condition (1.12.2), since in the space of parameters the elements of the matrix c lie
on a plane.
We choose the matrix P in such a way that the condition (1.12.1) holds and
cP 1
>1
0 " 0@  1un .
(1.12.3)
Letting pi (i = 1,2,…,n) be the i-th row of the matrix P. Using (1.12.1) and (1.12.3),
we can write
ª 0
« 0
«
« #
«
« 0
«¬ a0
ª p1 º
«p »
« 2»A
«# »
« »
¬ pn ¼
and c
1
0
#
0
1
#
0
a1
0
a2
"
"
%
0
0
#
º
» ª p1 º
»«p »
»« 2»
»« # »
1 »« »
"
p
" an1 »¼ ¬ n ¼
(1.12.4)
ª p1 º
«p »
>1 0 " 0@ «« #2 »» .
« »
¬ pn ¼
Carrying out the multiplication and comparing appropriate rows from (1.12.4), we
obtain
p1
c , p2
p1A, p3
p2 A, ! , pn
pn1A .
(1.12.5)
Using (1.12.5) we can compute the unknown rows p1,p2,…,pn of the matrix P.
Thus we have the following procedure for computation of the matrix P.
Polynomial Matrices
51
Procedure 1.12.1.
Step 1: Compute the coefficients a0,a1,…,an-1 of the polynomial
s n an1s n1 " a1s a0 .
det[I n s A ]
(1.12.6)
Step 2: Knowing a0,a1,…,an-1 compute the matrix FA.
Step 3: Choose c 1un such that the condition (1.12.2) holds.
Step 4: Using (1.12.5) compute the rows p1,p2,…,pn of the matrix P.
Example 1.12.1.
The following cyclic matrix is given
A
ª 1 1 0º
« 0 1 0» .
«
»
«¬ 1 0 2 »¼
(1.12.7)
One has to compute a matrix P transforming this matrix by similarity to the
Frobenius canonical form FA.
Using Procedure 1.12.1, we obtain the following:
Step 1: The characteristic polynomial of the matrix (1.12.7) has the form:
det [I n s A]
s 1 1
0
0
s 1
0
1
0
s2
( s 2)( s 1) 2
s 3 4s 2 5s 2. (1.12.8)
Step 2: Thus the matrix FA has the form
FA
ª0 1 0º
«0 0 1 » .
«
»
«¬ 2 5 4 »¼
(1.12.9)
Step 3: We choose c = [1 0 1] satisfying the condition (1.12.2), since
ª c º
det «« cA »»
2
¬«cA ¼»
1
0 1
0 1 2
2 1 4
4.
Step 4: Using (1.12.5), we obtain
p1
c
>1
0 1@ ,
p2
p1 A
>0
1 2@ ,
p3
p2 A
> 2
1 4@ .
52
Polynomial and Rational Matrices
Thus the matrix P has the form
P
ª 1 0 1º
« 0 1 2» .
«
»
«¬ 2 1 4 »¼
ª p1 º
«p »
« 2»
«¬ p3 »¼
(1.12.10)
If we search for a matrix P that satisfies the condition
P 1 AP
FA
ª0
«1
«
«0
«
«#
«¬ 0
0 " 0
a0 º
0 " 0 a1 »»
1 " 0 a2 » ,
»
# % #
# »
0 " 1 an1 »¼
(1.12.11)
then it is convenient to choose a column matrix b
n
det [b, Ab, " , A n1b] z 0 .
in such a way that
(1.12.12)
Let p i (i = 1,…,n) be the i-th column of the matrix P . Using (1.12.11) and
P -1b=[1 0 ... 0]T n, we can write
A > p1
b
p2 "
> p1
p2 "
pn @
> p1
p2 "
ª0
«1
«
pn @ «0
«
«#
«¬0
0 " 0 a0 º
0 " 0 a1 »»
1 " 0 a2 » ,
»
# % #
# »
0 " 1 an1 »¼ (1.12.13)
ª1 º
«0»
pn @ « » .
«# »
« »
¬0¼
Multiplying and comparing appropriate columns from (1.12.13), we obtain
p1
b , p2
Ap1 , p3
Ap2 , " , pn
Apn1 .
(1.12.14)
Using (1.12.14), we can successively compute the columns p 1, p 2,…, p n of the
matrix P . Thus we have the following procedure for computation of the matrix P .
Polynomial Matrices
53
Procedure 1.12.2.
Step 1: Is the same as in Procedure 1.12.1.
Step 2: Knowing the coefficients a1,a2,…,an of the polynomial (1.12.6) compute the
matrix F A.
Step 3: Choose b n such that the condition (1.12.12) is satisfied.
Step 4: Using (1.12.14) compute the columns p 1, p 2,…, p n of the matrix P .
Example 1.12.2.
Find a matrix P transforming the matrix (1.12.7) by similarity into its canonical
form F A.
Using Procedure 1.12.2 we obtain the following:
Step 1: The characteristic polynomial of the matrix (1.12.7) has the form (1.12.8).
Step 2: Thus the matrix F A has the form
FA
ª0 0 2 º
«1 0 5» .
«
»
¬«0 1 4 »¼
(1.12.15)
Step 3: We choose b = [0 1 -1]T, which satisfies the condition (1.12.12), since
2
det ª¬b, Ab, A b º¼
0 1 2
1 1 1
1 2 5
2.
Step 4: Using (1.12.14), we obtain
p1
ª0º
« 1 »,
« »
«¬ 1»¼
p2
Ap1
ª1º
« 1 »,
« »
«¬ 2 »¼
p3
Ap2
ª2º
« 1 ».
« »
«¬ 5»¼
Thus the desired matrix has the form
P
> p1
p2
p3 @
ª0 1 2º
«1 1 1».
«
»
«¬ 1 2 5»¼
The above considerations can be generalised for the remaining canonical
Frobenius forms F̂ A and F A of the matrix A.
54
Polynomial and Rational Matrices
1.12.2 Elementary Operations Method
Substituting into (1.10.5) and (1.10.10) the matrix B instead of the variable s, we
obtain
P (B )
L0 .
P0 , L(B)
(1.12.16)
Thus from the relationship B = L0AP0 it follows that if the matrices A and B are
similar, i.e., B = P-1AP, then the transformation matrix P is given by the following
formula
P
P (B)
1
> L(B) @
,
(1.12.17)
where P(s) and L(s) are unimodular matrices in the equality
[ sI B] L( s )[ sI A]P ( s ) .
(1.12.18)
To compute P(s), using elementary operations, we reduce the matrices [sI A],
[sI B] to the Smith canonical form
[ sI A]S
[ sI B]S
L1 ( s )[ sI A ]P1 ( s ) ,
L 2 ( s )[ sI B]P2 ( s )
(1.12.19)
(1.12.20)
where
P1 ( s )
P11 ( s ) P12 ( s )...P1k1 ( s ) ,
(1.12.21)
P2 ( s )
P21 ( s )P22 ( s )...P2 k2 ( s )
(1.12.22)
where P11(s),P12(s),…,P 1k2 (s) and P21(s),P22(s),…,P 2k2 (s) are matrices of
elementary operations carried out on columns of matrices [sI A] and [sI B],
respectively. The matrices L1(s) and L2(s) are defined similarly. Similarity of the
matrices A and B implies that
> s I A @S > s I B @S .
Taking into account (1.12.19) and (1.12.20) we obtain
L 2 ( s )[ sI B]P2 ( s )
L1 ( s )[ sI A]P1 ( s )
i.e.,
[ sI B] L21 ( s )L1 ( s )[ sI A ]P1 ( s )P21 ( s ) .
(1.12.23)
Polynomial Matrices
55
From a comparison of (1.12.18) and (1.12.21) to (1.12.23) and (1.12.22),
respectively, we obtain:
P(s)
P1 ( s )P21 ( s )
P11 ( s )P12 ( s )...P1k1 ( s )P2k12 ( s )...P221 ( s ) P211 ( s ).
(1.12.24)
Thus we compute the matrix P(s) carrying out elementary operations given by the
matrices on the identity matrix
P11 ( s ), P12 ( s), ..., P1k1 ( s), P2k12 ( s ), ..., P221 ( s ), P211 ( s ) .
When computing the inverse matrices to the matrices of elementary operations we
use the following relationships:
P 1[i u c]
1
P [i, j ]
ª 1º
P « i u » , P 1 > i j u b ( s ) @
¬ c¼
P[ j , i ] P[i, j ].
P >i j u w( s )@ ,
(1.12.25)
From the above considerations, the following algorithm for computation of the
matrix P can be inferred.
Algorithm 1.12.1.
Step 1: Transforming the matrices [sI – B], [sI – A] to the Smith canonical forms,
determine the sequence of elementary operations given by the matrices
P11 ( s ), P12 ( s ), ..., P1k1 ( s ), P21 ( s ), P22 ( s ), ..., P2 k2 ( s ) .
Step 2: Carrying out elementary operations given by the matrices
P11(s),P12(s),…,P 1k2 (s),P 2k2 -1(s) P22-1(s), P21-1(s) on the identity matrix,
compute the matrix P(s).
Step 3: Substituting in the matrix P(s) in place of s the matrix B, compute the
matrix P = P(B).
Example 1.12.3.
Compute a matrix P that transforms the matrix (1.11.7) to the Frobenius canonical
form (1.11.8).
In this case, the matrix FA is the matrix B.
Step 1: To reduce the matrix
sI FA
0 º
ª s 1
«0 s
1 »»
«
«¬ 2 5 s 4 »¼
to its Smith canonical form
56
Polynomial and Rational Matrices
[ sI FA ]S
0
ª1 0
º
«0 1
»,
0
«
»
«¬0 0 ( s 1) 2 ( s 2) »¼
the following elementary operations need to be carried out
L ª¬3 2 u s 4 º¼ , P ª¬ 2 3 u s º¼ , P ª¬1 2 u s º¼ ,
L ª¬3 1 u s 2 4 s 5 º¼ , L ª¬1 u 1 º¼ , L ª¬ 2 u 1 º¼ ,
P > 2, 3@ , P >1, 2@ .
Step 2: In Example 1.11.1 to reduce the matrix [Ins A] to the Smith canonical
form, the following elementary operations are applied
P[1 2 u ( s 1)], L[2 1 u ( s 1)], P[3 1 u (2 s)], L[2 u (1)],
L[2 3 u ( s 1) 2 ], L[1 u (1)], L[2, 3], L[1, 2].
To compute the matrix P(s) the following elementary operations have to be carried
out on the columns of the identity matrix of the third degree
P ª¬1 2 u s 1 º¼ , P ª¬3 1 u 2 s º¼ , P > 2, 3@ , P >1, 2@ ,
P ª¬1 2 u s º¼ , P ª¬ 2 3 u 1 º¼ .
Then we obtain
P(s)
ª 2(1 s )
« 2( s 1) 2
«
1
¬«
1
0º
1 1 »»
0 0 ¼»
ª 0 0 0º
ª 2 0 0 º
ª 2 1 0º
« 2 0 0 » s 2 « 4 0 0 » s « 2 1 1 » .
«
»
«
»
«
»
«¬ 0 0 0 »¼
«¬ 0 0 0 »¼
«¬ 1 0 0»¼
Step 3: We substitute into this matrix the matrix
FA
ª0 1 0º
«0 0 1 »
«
»
«¬ 2 5 4 »¼
in place of the variable s. We obtain
Polynomial Matrices
P
57
ª 2 1 0 º
« 2 3 1» .
«
»
«¬ 1 0 0 »¼
P FA
It is easy to check that this matrix transforms the matrix (1.11.7) to the form FA.
1.12.3 Eigenvectors Method
Let a matrix A nun and its Jordan canonical form (1.11.12), containing p blocks
of the form (1.11.10a), be given. Let the i-th block, corresponding to the
eigenvalue si, have the dimensions miumi (i = 1,…,p). The following matrix
T
ª¬T1 T2 ... Tp º¼ , Ti
(1.12.26)
ª¬ti1 ti 2 ... timi º¼
satisfying (1.11.13) is to be computed.
Post-multiplying (1.11.13) by T we obtain
AT
TJ A
and after taking into account (1.12.26), (1.11.10) and (1.11.12)
ATi
Ti J i for i 1,..., p ,
and
> A Isi @ ti1
0,
> A Isi @ ti 2
ti1 , ..., > A Isi @ timi
ti ,mi 1 ,
i 1,..., p.
(1.12.27)
For the eigenvalue si from the first among the equations in (1.12.27) we compute
the column ti1, knowing ti1 we compute from the second equation the column ti2
and finally from the last equation we compute the column tim .
Repeating these computations successively for i = 1,2,…,p, we obtain the
desired matrix (1.12.26).
i
Example 1.12.4.
Compute the matrix T transforming the matrix
A
ª1
« 1
«
«0
«
¬0
2 1 0º
3 1 0 »»
1 2 0»
»
0 0 1¼
(1.12.28)
58
Polynomial and Rational Matrices
to its Jordan canonical form
JA
ª2
«0
«
«0
«
¬0
1 0 0º
2 1 0 »»
.
0 2 0»
»
0 0 1¼
(1.12.29)
From (1.12.29) it follows that the matrix (1.12.28) has one eigenvalue s1 = 2 of
multiplicity 3 and one eigenvalue s2 = 1 of multiplicity 1.
In this case, the matrix (1.12.26) is of the form
>T1
T
T2 @
>t11
t12 t13 t21 @ .
For i = 1 the equations (1.12.27) take the form
> A Is1 @ t11
> A Is1 @ t12
> A Is1 @ t13
and for i
ª 1
« 1
«
«0
«
¬0
ª 1
« 1
«
«0
«
¬0
ª 1
« 1
«
«0
«
¬0
2 1
1 1
1 0
0 0
2 1
1 1
1 0
0 0
2 1
1 1
1 0
0 0
0º
0 »»
t11
0»
»
1¼
0º
0 »»
t12
0»
»
1¼
0º
0 »»
t13
0»
»
1¼
ª0º
«0»
« »,
«0»
« »
¬0¼
t11 ,
t12 ,
2
> A Is2 @ t21
ª0
« 1
«
«0
«
¬0
2 1 0º
2 1 0 »»
t21
1 1 0»
»
0 0 0¼
ª0º
«0»
« ».
«0»
« »
¬0¼
Solving these equations successively, we obtain
Polynomial Matrices
ª1 º
«0»
« »,
«1 »
« »
¬0¼
t11
ª1 º
«1 »
« »,
«0»
« »
¬0¼
t12
ª0 º
«0 »
« »,
«1 »
« »
¬0 ¼
t13
59
ª0º
«0»
« »,
«0»
« »
¬1 ¼
t21
and the desired matrix has the form
T
>t11
t12 t13 t21 @
ª1
«0
«
«1
«
¬0
1 0 0º
1 0 0 »»
.
0 1 0»
»
0 0 1¼
If blocks have the form (1.11.10b) considerations are similar.
1.13 Matrices of Simple Structure and Diagonalisation of
Matrices
1.13.1 Matrices of Simple Structure
nun
Consider a matrix A
whose characteristic polynomial has the form
\ (O ) det > I n O A @ O n an1O n1 ..... a1O a0 .
(1.13.1)
The roots O1, O2,…,Op (p d n) of the equation \(O) = 0 are called eigenvalues of the
matrix A, and the set of these eigenvalues is called the spectrum of this matrix.
Definition 1.13.1. We say that an eigenvalue Oi has algebraic multiplicity ni, if Oi is
the ni-fold root of the equation \(O) = 0, i.e.,
\ Oi
\ ' Oi
but \
ni
ni 1
Oi
0,
(1.13.2)
Oi z 0, i 1, ..., p,
where \
\ O
.... \
k
d k\ O
, i.e.,
dOk
O
O O1
n1
O O2
n2
.... O O p
np
.
(1.13.3)
We say that an eigenvalue Oi has geometrical multiplicity mi if
rank > I n Oi A @
n mi , i 1, ..., p .
(1.13.4)
60
Polynomial and Rational Matrices
From the Jordan canonical form of the matrix A, it follows that ni > mi for
i = 1,…,p.
Definition 1.13.2. A matrix A nun for which ni = mi for i = 1,…,p, is called a
matrix of simple structure. Otherwise we say that the matrix has a complex
structure.
For example, the matrix
A
ª2 a º
«0 2 »
¬
¼
(1.13.5)
for a = 0 is a matrix of simple structure, since n1 = m1 = 2 and for a z 0 it is a
matrix of complex structure, since n1 = 2, m1 = 1 (rank [I22 A] = 1)).
Theorem 1.13.1. The similar matrices A nun and B = PAP-1, det P z 0, have
eigenvalues of the same algebraic and geometric multiplicities.
Proof. According to Theorem 1.10.1, the similar matrices A and B share the same
characteristic polynomial, i.e.,
det > I n O A @ det > I n O B @ .
(1.13.6)
The equality (1.13.6) implies that the matrices A and B have the same eigenvalues
of the same algebraic multiplicities.
From the relationship
rank > I n Oi B @
for i 1, ..., p,
rank ª¬ P > I n Oi A @ P 1 º¼ rank > I n Oi A @ ,
(1.13.7)
it follows that eigenvalues of the matrices A and B also have the same geometrical
multiplicities.
„
From the Jordan canonical structure and (1.13.4) the following important
corollary ensues.
Corollary 1.13.1. Geometrical multiplicity mi of an eigenvalue Oi, i = 1,…,p of the
matrix A is equal to a number of blocks corresponding to this eigenvalue.
Theorem 1.13.2. A matrix A nun is of simple structure if and only if all its
elementary divisors are of the first degree.
Polynomial Matrices
61
Proof. According to Corollary 1.11.1, the matrix A is similar to the diagonal
matrix consisting of eigenvalues of this matrix if and only if all its elementary
divisors are of the first degree. In this case
rank > I n Oi A @
n ni for i 1,..., p .
(1.13.8)
In view of this, mi = ni for i = 1,…,p, and A is a matrix of simple structure if and
only if all its divisors are of the first degree.
„
Example 1.13.1.
The matrix (1.13.5) is a matrix of simple structure if and only if a = 0, since the
Smith canonical form of the matrix
ª s 2 a º
« 0
s 2 »¼
¬
>I 2 s A @
is equal to
>I 2 s A@ s
>I 2 s A @ s
0 º
ªs 2
, for a 0 ,
« 0
s
2 »¼
¬
0 º
ª1
«
2 » , for a z 0 .
«¬ 0 s 2 »¼
For a = 0, the matrix (1.13.5) has two elementary divisors of the first degree, and
for a z 0, it has one elementary divisor (s 2)2. According to Theorem 1.13.2, the
matrix (1.13.5) is thus of simple structure if and only if a = 0.
1.13.2 Diagonalisation of Matrices of Simple Structure
Theorem 1.13.3. For every matrix A
singular matrix P nun such that
P 1 AP
nun
of simple structure there exists a non-
diag > O1 , O2 , ..., On @
(1.13.9)
where some eigenvalues Oi, i = 1,…,p can be equal.
Proof. From the fact that A is a matrix of simple structure it follows that for every
eigenvalue Oi there are as many corresponding eigenvectors Pi as the multiplicity of
the eigenvalue amounts to
APi
Oi Pi , for i 1, ..., n .
(1.13.10)
62
Polynomial and Rational Matrices
The eigenvectors P1,P2,…,Pn are linearly independent. Hence the matrix
P = [P1,P2,…,Pn] is nonsingular.
From (1.13.10) for i = 1,…,n, we have
AP
P diag > O1 , O2 , ..., On @ .
(1.13.11)
Pre-multiplying (1.13.11) by P-1, we obtain (1.13.9). „
In particular, in the case when A has distinct eigenvalues O1, O2,…,On, the
following important corollary ensues from Theorem 1.13.3.
Corollary 1.13.2. Every matrix A nun with distinct eigenvalues O1, O2,…,On can
be transformed by similarity to the diagonal form diag [O1, O2,…,On].
To compute the eigenvectors P1,P2,…,Pn, we solve the equation
>I n Oi A @ Pi
0, for i 1, ..., n
(1.13.12)
or taking instead of Pi any nonzero column of Adj [InOi - A].
From definition of the inverse matrix
1
>I nO A@
Adj > I n O A @
,
det > I n O A @
we have
>I n O A @ Adj>I n O A @
I n det > I n O A @ .
(1.13.13)
Substituting O = Oi into (1.13.13) and taking into account that det [InOi - A] = 0, we
obtain
>I n Oi A @ Adj>I n Oi A @
0, for i 1, ..., p .
(1.13.14)
From (1.13.14) it follows that every nonzero column of Adj [InOi - A] is the
eigenvector of the eigenvalue Oi of the matrix A.
Example 1.13.2.
Compute a matrix P that transforms the matrix
A
ª 3 1 1 º
1«
1 5 1»»
«
2
«¬ 2 2 4 »¼
(1.13.15)
Polynomial Matrices
63
to the diagonal form.
The characteristic equation of the matrix (1.13.15)
O 32
1
2
12
12
1
O 52
1
2
1
O2
det > I n O A @
O3 6O 2 11O 6 0
has three real roots O –O –O –. To compute the eigenvectors
P1,P2,P3, we compute the adjoint (adjugate) matrix
Adj > I n O A @
ª O 2 92 O 92
« 1
1
« 2O 2
« O2
¬
12 O 32
2
O 72 O 52
O 2
O 32 º
»
12 O 12 » .
O 2 4O 4 »¼
1
2
(1.13.16)
As the eigenvectors P1, P2, P3 of the matrix (1.13.15) we take the third column of
the adjoint matrix successively for O –O –O –. The matrix built
from these vectors (after multiplication of the third column for O – by 2) has the
form
> P1 , P2 , P3 @
P
ª1 1 0 º
«0 1 1 »
«
»
«¬1 0 1 »¼
and its inverse is
P
1
ª 1 1 1 º
1«
1 1 1»» .
2«
«¬ 1 1 1 »¼
Hence
1
P AP
ª 1 1 1 º ª 3 1 1 º ª1 1 0 º
1«
1
1 1 1»» «« 1 5 1»» «« 0 1 1 »»
«
2
2
«¬ 1 1 1 »¼ «¬ 2 2 4 »¼ «¬1 0 1 »¼
Example 1.13.3.
Compute a matrix P that reduces the matrix
ª 1 0 0 º
« 0 2 0 » .
«
»
«¬ 0 0 3»¼
64
Polynomial and Rational Matrices
A
ª2 0 0º
«0 2 0»
«
»
«¬ 1 1 1 »¼
(1.13.17)
to the diagonal form.
The characteristic equation of the matrix (1.13.17)
det > I 3O A @
O2
0
0
0
O 2
0
1
1
O 1
(O 2) 2 (O 1)
0
has one double root O and one root of multiplicity 1, O . The matrix
(1.13.17) is a matrix of simple structure, since
rank > I 3O1 A @
ª 0 0 0º
rank «« 0 0 0 »» 1 .
«¬ 1 1 1 »¼
Thus using similarity transformation the matrix (1.13.17) can be reduced to the
diagonal form.
From the equation
>I 3O1 A @ Pi
ª 0 0 0º
« 0 0 0» P
«
» i
«¬ 1 1 1 »¼
(i 1, 2)
0
it follows that as the eigenvectors P1 and P2 we can adopt
P1
ª1 º
«0» , P
2
« »
«¬1 »¼
ª1 º
«1 » .
« »
«¬ 0 »¼
Solving the equation
>I 3O2 A @ P3
ª 1 0 0 º
« 0 1 0 » P
«
» 3
«¬ 1 1 0 »¼
0,
Polynomial Matrices
we obtain P3
ª0 º
« 0 » . Thus
« »
«¬1 »¼
> P1 ,
P
65
ª1 1 0 º
«0 1 0 » .
«
»
«¬1 0 1 »¼
P2 , P3 @
It is easy to verify that
1
1
P AP
ª1 1 0 º ª 2 0 0 º ª1 1 0 º
«0 1 0 » « 0 2 0» « 0 1 0 »
«
» «
»«
»
«¬1 0 1 »¼ «¬1 1 1 »¼ «¬1 0 1 »¼
ª 2 0 0º
« 0 2 0» .
«
»
«¬ 0 0 1 »¼
1.13.3 Diagonalisation of an Arbitrary Square Matrix by the Use of a Matrix
with Variable Elements
Let a square matrix A and a diagonal matrix / of the same dimension be given. We
will show that an arbitrary matrix A can be transformed to the diagonal form / by
use of a transformation of a matrix with variable elements.
Theorem 1.13.4. For an arbitrary matrix A
/ nun there exists a nonsingular matrix
T
T(t )
nun
e( A ȁ )t
and a given diagonal matrix
(1.13.18)
such that
)T 1
( AT T
ȁ.
(1.13.19)
Proof. From (1.13.18) it follows that this matrix is nonsingular for arbitrary
matrices A and /. Taking into account that
T
( A ȁ )e ( A ȁ ) t
( A ȁ )T ,
we obtain
AT T T1
AT A ȁ T T1
ȁ.
„
66
Polynomial and Rational Matrices
Example 1.13.4.
Compute a matrix T that transforms the matrix
ª2 1º
«0 2»
¬
¼
A
to the diagonal form
ª2 0º
«0 2» .
¬
¼
ȁ
Note that the given matrix A is of the Jordan canonical form and one cannot
transform it to a diagonal form using similarity transformation (with a matrix P
with constant elements) since it does not have a simple structure.
Using (1.13.18) we compute
T
e( Aȁ )t
ª0 1 º
exp «
»t
¬0 0 ¼
ª1 t º
« 0 1» .
¬
¼
Taking into account that
T 1
ª1 t º «0 1 » , T
¬
¼
ª0 1 º
«0 0»
¬
¼
it is easy to check that
ȁ
T 1
AT T
­° ª 2 2t 1º ª0 1 º ½° ª1 t º
®«
¾
2 »¼ «¬0 0 »¼ °¿ «¬ 0 1 »¼
°¯ ¬ 0
ª 2 0º
« 0 2» .
¬
¼
These considerations can be generalised into a matrix A(t) whose elements depend on
time t.
We will show that a square matrix A(t) of dimension nun with elements being
continuous functions of time t can be transformed to the diagonal form
ȁ (t )
diag ª¬ O1 t , O2 t , ..., On t º¼ .
(1.13.20)
Let matrix I(t) be the solution of the matrix differential equation
I t
AI t ,
Satisfying, for example, the initial condition I(0) = In.
(1.13.21)
Polynomial Matrices
67
Let
t
³ ȁ(W )dW
T(t ) I (t )e 0
.
(1.13.22)
It is known that the matrix (1.13.22) is a nonsingular matrix for every t t 0.
We will show that the matrix (1.13.22) satisfies the equation
(t )
T
A(t )T(t ) T(t ) ȁ(t ) .
(1.13.23)
Differentiating the matrix (1.13.22) with respect to t and taking into account
(1.13.21), we obtain
t
T (t ) I(t )e
t
³ ȁ (t )dt
0
I (t )e
³ ȁ (t )dt
0
t
A(t )I (t )e
³ ȁ (t )dt
0
O (t )
t
I (t ) e
³ ȁ (t )dt
0
ȁ (t )
A(t )T(t ) T(t ) ȁ (t ).
From (1.13.23), we obtain
ȁ (t )
T 1 (t ) ª¬ A(t )T(t ) T (t ) º¼
diag > O1 (t ), O2 (t ), ...., On (t ) @ .
Thus the desired matrix is given by the relationship (1.13.22), where the matrix
I(t) is a solution to the equation (1.13.21).
1.14 Simple Matrices and Cyclic Matrices
1.14.1 Simple Polynomial Matrices
Consider a polynomial matrix A(s)
mun
[s] of rank r d min(m,n).
Definition 1.14.1. A polynomial matrix A(s) mun of rank r is called a simple one
if and only if it has only one invariant polynomial distinct from 1.
Taking (1.8.4) into account, we can equivalently define a simple matrix as a
polynomial matrix satisfying the conditions
D1 ( s )
D2 ( s ) ...
Dr 1 ( s ) 1 and Dr ( s )
ir ( s ) ,
(1.14.1)
68
Polynomial and Rational Matrices
where Dk(s), k = 1,…,r is a greatest common divisor of all minors of size k of the
matrix A(s).
Thus the Smith canonical form of the simple matrix A(s) is equal to
A S (s)
­ ª1 0
°«
° «0 1
°« # #
°«
° «0 0
° «0 0
°¬
°diag[1,
°
°° ª1 0
«
® «0 1
°« # #
°«
° «0 0
°«
° «0 0
° «0 0
°«
°« # #
° «0 0
°¬
°¯
0 " 0º
0 " 0 »»
#
#
# % # » for n ! m
»
1
0
0 " 0»
0 ir ( s ) 0 " 0 »¼
for n m
" , 1, ir ( s )]
"
"
%
"
"
"
"
%
"
"
"
%
"
0
0
0
0
#
1
0
0
#
0
0
0
0 º
0 »»
# »
»
0 »
ir ( s ) »
»
0 »
# »
»
0 »¼
Theorem 1.14.1. A polynomial matrix A(s)
only if
rank A si0
r
r
.
for m ! n
mun
(1.14.2)
r
[s] of rank r is simple if and
r 1 for si0  V A ,
(1.14.3)
where VA is the set of zeros of the matrix A(s).
Proof. The normal rank of the matrix A(s) and of its Smith canonical form AS(s) is
the same, i.e., rank A(s) = rank AS(s) = r. From (1.14.2) it follows that the defect of
the rank of the matrix A(s) is equal to 1 if and only if s is a zero of this matrix.
„
From Definition 1.14.1 one obtains the following important corollary.
Corollary 1.14.1. A polynomial matrix A(s) is simple if and only if only one
elementary divisor corresponds to each zero.
Example 1.14.1.
In Example 1.8.1 it was shown that to the polynomial matrix
Polynomial Matrices
A s
ª ( s 2) 2
«
¬( s 2)( s 3)
( s 2)( s 3)
( s 2) 2
s 2º
»
s 3¼
69
(1.14.4)
the Smith canonical form
A S (s)
0
0º
ª1
«0 ( s 2)( s 2.5) 0 »
¬
¼
(1.14.5)
corresponds.
From (1.14.5) it follows that i1(s) = 1, i2(s) = (s+2)(s+2.5) and thus the matrix
(1.14.4) is simple.
It is easy to check that the matrix (1.14.4) loses its full rank equal to 2 for zeros
s1 = 2 and s2 = 2.5, since
A(2)
ª0 0 0 º
«0 0 1 » , A(2.5)
¬
¼
ª 0.25 0.25 0.5º
« 0.25 0.25 0.5 » .
¬
¼
We obtain the same result from the matrix (1.14.5).
1.14.2 Cyclic Matrices
Consider a matrix A
num
.
Definition 1.14.2. A matrix A num is called cyclic if and only if the polynomial
matrix [Ins - A] corresponding to it is simple.
Consider the following matrices
F
Fˆ
ª 0
« 0
«
« #
«
« 0
«¬ a0
ª an1
«
« 1
« 0
«
« #
«¬ 0
1
0
0 º
ª0
»
«1
0
1 "
0
0 »
«
#
# %
#
# » , F «0
»
«
0
0 "
0
1 »
«#
«¬0
a1 a2 " an2 an1 »¼
an2 " a1 a0 º
ª an1
« a
" 0
0
0 »»
« n2
" 0
1
0 » , F « #
«
»
#
% #
# »
« a1
«¬ a0
" 1
0
0 »¼
"
0
0 " 0
0 " 0
1 % 0
# " #
0 " 1
1 0 "
0 1 "
#
# %
0 0 "
0 0 "
a0 º
a1 »»
a2 » ,
»
# »
an1 »¼ (1.14.16)
0º
0 »»
#».
»
1»
0 »¼
We say that the matrices (1.14.6) have Frobenius canonical form.
Expanding the determinant along the row (or column) containing a0,a1,…,an-1 it
is easy to show that the following equality holds
70
Polynomial and Rational Matrices
det > I n s F @ det ¬ª I n s F ¼º
det ª¬I n s Fˆ º¼
det ª¬I n s F º¼
s n an1s n1 " a1s a0 .
(1.14.7)
Theorem 1.14.2. The matrices (1.14.6) are cyclic for arbitrary values of the
coefficients a0,a1,…,an-1.
Proof. We prove the theorem in detail only for the matrix F, since in other cases
the proof is similar.
After deleting the first column and the n-th row from the matrix
[I n s F ]
ªs
«0
«
«#
«
«0
«¬ a0
1 0
s 1
#
#
0 0
a1 a2
" 0
% 0
% #
% s
" an2
0 º
0 »»
# »,
»
1 »
an1 »¼
(1.14.8)
we obtain the minor Mn1 equal to (1)n1. Thus the greatest common divisor Dn1(s)
of the all minors of degree n1 of the matrix (1.14.8) is equal to 1, i.e., Dn1(s) = 1.
The condition (1.14.1) is thus satisfied and the matrix F is cyclic.
„
Theorem 1.14.3. A matrix A = [aij]
satisfied
­ 0
aij ®
¯z 0
­ 0
aij ®
¯z 0
for j ! i 1
is cyclic if the following conditions are
i, j 1, ! , n ,
(1.14.9a)
for i ! j 1
, i, j 1, ! , n .
for i j 1
(1.14.9b)
for j
i 1
,
nun
Proof. If the conditions (1.14.9a) are satisfied then after deleting the first column
and the n-th row from the matrix
[I n s A ]
ª s a11
« a
« 21
« #
«
« an1,1
« an1
¬
a12
s a22
0
a23
#
an1,2
an 2
#
an2,3
an 3
0
"
0
!
%
#
! s an1,n1
an ,n1
"
0
0
º
»
»
# » , (1.14.10)
»
an1,n »
s ann »¼
we obtain the minor Mn1 equal to Mn1=(1)n1a12a23…an-1,n z 0. Thus Dn1(s) = 1
and the condition (1.14.1) is satisfied. In the case of (1.14.9b) the proof is similar.
„
Polynomial Matrices
71
Example 1.14.2.
Determine the conditions under which the matrix
A2
ª a11
«a
¬ 21
a12 º
a22 »¼
(1.14.11)
is or is not a cyclic matrix.
If a21 z 0, then carrying out the elementary operations: L[1+2u1/a21(s a11)],
L[2u(a21)], L[1,2] and L[2ua21] on the matrix
[I 2 s A 2 ]
ª s a11
« a
¬ 21
a12 º
,
s a22 »¼
we obtain its Smith canonical form, which is equal to
>I 2 s A 2 @
M s
ª1 0 º
«1 M ( s ) » ,
¬
¼
(1.14.12)
2
det > I 2 s A @
s a11 a22 s a11a22 a12 a21 .
From (1.14.12) it follows that for a21 z 0, the matrix (1.14.11) is cyclic for any
values of other elements.
We obtain a similar result for a12 z 0.
It is easy to check that for a12 = a21 = 0 the diagonal matrix
A2
ª a11
«0
¬
0º
,
a22 »¼
is cyclic if and only if a11 z a22.
Theorem 1.14.4. A matrix A nun is cyclic if and only if only one Jordan block
corresponds to it every distinct eigenvalue, i.e.,
JA
where
ª J ( s1 ,n1 )
«
« 0
« #
«
« 0
¬
0
J ( s2 ,n2 )
#
0
...
0
...
0
º
»
»
 nun
%
# »
»
... J ( s p ,n p ) »¼
(1.14.13a)
72
Polynomial and Rational Matrices
J ( sk , nk )
ª sk
«0
«
«#
«
«0
«¬ 0
1
sk
0 ...
1 ...
0
0
#
# %
#
0
0 ...
sk
0
0 ...
0
ª sk
«1
«
«#
«
«0
«¬ 0
0
sk
... 0
... 0
0
0
0º
0 »»
# »  nk unk
»
1»
sk »¼
(1.14.13b)
or
J ( sk , nk )
# % # #
0 ... 1 sk
0 ... 0 1
0º
0 »»
# »  nk unk , k
»
0»
sk »¼
1, ..., p .
Proof. The polynomial matrix
Ins J A
diag[I n1 s J ( s1 , n1 ), ..., I n p s J ( s p , n p )]
(1.14.14)
is simple since
rank > I n s J A @
s sk
n 1 for k
1, ..., p .
(1.14.15)
By virtue of Theorem 1.14.1 and Definition 1.14.2 the matrices (1.14.3) and A are
cyclic. If at least two blocks correspond to one eigenvalue sk then defect of the rank
of the matrix (1.14.14) is greater than 1 and the matrix A is not cyclic.
„
Example 1.14.3.
From Theorem 1.14.4 it follows that the matrix
A
ª1 1 0 º
«0 1 0 »
«
»
«¬0 0 a »¼
(1.14.16)
is a cyclic one for a z 1. However, it is not cyclic for a = 1, since two Jordan
blocks correspond to its eigenvalue that is equal to 1
J (1, 2)
ª1 1º
« 0 1» and J (1,1)
¬
¼
>1@ .
From Theorem 1.14.4 for J(sk,nk) = ak, nk = 1, k = 1,…,n one obtains the
following important corollary.
Polynomial Matrices
73
Corollary 1.14.1. The diagonal matrix
A
diag > a1 , a2 , ..., an @  nun
(1.14.17)
is cyclic if and only if ai z aj for i z j.
Theorem 1.14.5. Let O1,O2,…,Op be the eigenvalues of multiplicities n1,n2,…,np,
respectively, of the matrix A nun. This matrix is cyclic if and only if
n
n 1
rank > I n Oi A @ i
rank > I n Oi A @ i
for i 1, ..., p.
(1.14.18)
Proof. It is known that similarity transformation does not change the rank of the
matrix
n
n
rank > I n Oi A @ i
rank > I n Oi J A @ i
for i 1, ..., p,
(1.14.19)
where JA is a Jordan canonical form of the matrix A.
Taking into account (1.11.10) it is easy to verify that
ni
>I n1Oi J (Oi , ni )@
0 for i 1, ..., p .
(1.14.20)
From the Jordan canonical form JA of the matrix A and (1.14.20) it follows that
only one block corresponds to every eigenvalue Oi if and only if the condition
(1.14.18) is satisfied. Thus by virtue of Theorem 1.14.4, the matrix A is cyclic if
and only if the condition (11.14.8) is satisfied.
„
Example 1.14.4.
The matrix (1.14.16) for a z 1 has only one eigenvalue O1 = 1 of multiplicity n1 = 2
and one eigenvalue O2 = a of multiplicity 1.
It is easy to check that
2
>I 3O1 A @
3
>I 3O1 A @
ª0
«0
«
«¬ 0
ª0
«0
«
«¬0
2
rank > I 3O1 A @
1
0
0 º
0 »»
1 a »¼
0
0
0
2
0 º
ª0 0
«0 0
0 »» ,
«
«¬ 0 0 (1 a ) 2 »¼
º
0
0 »» ,
0 (1 a)3 »¼
3
rank > I 3O1 A @
­1 for a z 1
®
¯0 for a 1
74
Polynomial and Rational Matrices
and
>I3O2 A @
ª a 1
«
2
>I3O2 A @ « 0
« 0
¬
­2 for a z 1
2
.
rank > I 3O2 A @ ®
¯0 for a 1
ª a 1 1 0 º
«
a 1 0 »» ,
« 0
«¬ 0
0
0 »¼
rank > I 3O2 A @
2
2(a 1) 0 º
»
(a 1) 2 0 » ,
0
0»
¼
Thus the condition (1.14.18) is satisfied and the matrix is cyclic if and only if
a z 1.
Theorem 1.14.6. A matrix A nun can be transformed by similarity to the
Frobenius canonical form (1.14.6) or to the Jordan canonical form (1.14.13) if and
only if the matrix A is a cyclic one.
Proof. It is known that there exist nonsingular matrices P1 and P2 of similarity
transformation such that
AF
P1AP11 such that J A
P2 AP21
(1.14.21)
if and only if the polynomial matrices [Ins - A], [Ins - AF] and [Ins - JA] are
equivalent, i.e., they have the same invariant polynomials. This takes place if and
only if the matrix A is cyclic. The sufficiency follows immediately by virtue of
Theorems 1.11.1 and 1.11.2.
„
Example 1.14.5.
Consider the matrix (1.14.16). This matrix for a z 1 is cyclic and can be
transformed by similarity into the Frobenius canonical form AF that is equal to
AF
1
0 º
ª0
«0
0
1 »» ,
«
«¬ a 2a 1 2 a »¼
since
0 º
ª s 1 1
« 0
»
s
1
0
«
»
«¬ 0
0
s a »¼
( s 1) 2 ( s a ) s 3 (2 a ) s 2 (2a 1) s a.
det > I 3 s A @
(1.14.22)
Polynomial Matrices
75
For a = 1, the matrix (1.14.16) has the Jordan canonical form with two
blocks corresponding to an eigenvalue equal to 1 and is not cyclic. The
matrix (1.14.16) for a = 1 cannot be transformed by similarity into the
Frobenius canonical form.
1.15 Pairs of Polynomial Matrices
1.15.1 Greatest Common Divisors and Lowest Common Multiplicities of
Polynomial Matrices
Let mun[s] be the set of mun polynomial matrices with complex coefficients in the
variable s.
Definition 1.15.1. A matrix B(s) muq[s] is called a left divisor (LD) of the matrix
A(s) mul[s] if and only if there exists a matrix C(s) qul[s] such that
A( s)
B( s )C( s ) .
(1.15.1)
A matrix C(s) mul[s] is called a right divisor (RD) of A(s)
if there exists a matrix B(s) muq[s] such that (1.15.1) holds.
mul
[s] if and only
Definition 1.15.2. A matrix A(s) qul[s] is called a right multiplicity (RM) of a
matrix B(s) muq[s] if and only if there exists a matrix C(s) qun[s] such that
(1.15.1) holds.
A matrix A(s) mul[s] is called a left multiplicity (LM) of a matrix
C(s) qul[s] if and only if there exists a matrix B(s) mul[s] such that (1.15.1)
holds.
Consider the two polynomial matrices A(s) mul[s] and B(s) mup[s].
Definition 1.15.3. A matrix L(s) muq[s] is called a left common divisor (LCD) of
matrices A(s) mul[s] and B(s) mup[s] if and only if there exist matrices
A1(s) qul[s] and B1(s) qup[s] such that
A( s)
L( s ) A1 ( s ) and B( s )
L( s )B1 ( s ) .
(1.15.2)
A matrix P(s) qul[s] is called a right common divisor (RCD) of matrices
A(s) mul[s] and B(s) pul[s] if and only if there exist matrices A2(s) muq[s] and
B2(s) puq[s] such that
A( s)
A 2 ( s )P ( s ) and B( s )
B 2 ( s )P ( s ) .
(1.15.3)
76
Polynomial and Rational Matrices
Definition 1.15.4. A matrix D(s) pul[s] is called a common left multiplicity
(CLM) of matrices A(s) mul[s] and B(s) qul[s] if and only if there exist matrices
D1(s) mum[s] and D2(s) puq[s] such that
D( s )
D1 ( s ) A( s ) and D( s )
D 2 ( s )B ( s ) .
(1.15.4)
A matrix F(s) mup[s] is called a common right multiplicity (CRM) of matrices
A(s) mul[s] and B(s) muq[s] if and only if there exist matrices F1(s) lup[s] and
F2(s) qup[s] such that
F(s)
A( s )F1 ( s ) and F ( s )
B( s )F2 ( s ) .
(1.15.5)
Definition 1.15.5. A matrix L(s) muq[s] is called a greatest common left divisor
(GCLD) of matrices A(s) mul[s] and B(s) mup[s] if and only if
x the matrix L(s) is a common left divisor of the matrices A(s) and B(s);
x the matrix L(s) is a common right multiplicity of every common left
divisor of the matrices A(s) and B(s), i.e., if A(s) = L1(s)A3(s) and
B(s) = L1(s)B3(s), then L(s) = L1(s)T(s), where L1(s), A3(s), B3(s) and T(s)
are polynomial matrices of appropriate dimensions.
A matrix P(s) qul[s] is called a greatest common right divisor (GCRD) of
matrices A(s) mul[s] and B(s) pul[s] if and only if
1. the matrix P(s) is a common right divisor of the matrices A(s) and B(s);
2. the matrix P(s) is a common left multiplicity of every common right
divisor of the matrices A(s) and B(s), i.e., if A(s)=A4(s)P1(s) and
B(s) = B4(s)P1(s), then P(s) = T(s)P1(s), where A4(s), P1(s), B4(s) and T(s)
are polynomial matrices of appropriate dimensions.
Definition 1.15.6. A matrix D(s) pul[s] is called a smallest common left
multiplicity (SCLM) of matrices A(s) mul[s] and B(s) qul[s] if and only if
1. the matrix D(s) is a common left multiplicity of the matrices A(s) and B(s);
2. the matrix D(s) is a right devisor of every common multiplicity of the
matrices A(s) and B(s), i.e., if D (s) = D3(s)A(s) and D (s)=D4(s)B(s), then
D (s)=T(s)D(s), where D (s), D3(s), D4(s) and T(s) are polynomial
matrices of appropriate dimensions.
A matrix F(s) mup[s] is called a smallest common right multiplicity (SCRM) of
matrices A(s) mul[s] and B(s) muq[s] if and only if
1. the matrix F(s) is a common right multiplicity of the matrices A(s) and
B(s);
2. the matrix F(s) is a left divisor of every common multiplicity of the
matrices A(s) and B(s), i.e., if F (s) = A(s)F3(s) and F (s) = B(s)F4(s), then
F (s) = F(s)T(s), where F (s), F3(s), F4(s) and T(s) are polynomial
matrices of appropriate dimensions.
Polynomial Matrices
77
1.15.2 Computation of Greatest Common Divisors of a Polynomial Matrix
Problem 1.15.1. Given C
that
C
lum
[s], L
lul
[s] a matrix C1 is to be computed such
LC1 ,
(1.15.6)
where L is a lower triangular matrix and rank L t rank C.
Solution. Assume that the matrix L of rank r has the form
L
ª g11
«g
« 21
« #
«
« g r1
« #
«
«¬ gl1
0
g 22
#
gr 2
#
gl 2
! 0
! 0
% #
! g rr
% #
! glr
0 ! 0º
0 ! 0 »»
# % #»
»
0 ! 0»
# % #»
»
0 ! 0 »¼
(1.15.7)
and the matrix C1
C1
ª x11 ! x1m º
« # % # ».
«
»
«¬ xl1 ! xlm »¼
(1.15.8)
The equality (1.15.6) can be written in the form
ª c11 ! c1m º
«# % # »
«
»
¬« cl1 ! clm ¼»
ª g11
«g
« 21
« #
«
« g r1
« #
«
¬« gl1
!
0
g 22 !
0
0
# % #
g r 2 ! g rr
# % #
gl 2 ! glr
0 ! 0º
0 ! 0 »»
ª x ! x1m º
# % # » « 11
»
» # % # » . (1.15.9)
0 ! 0» «
« x ! xlm ¼»
# % # » ¬ l1
»
0 ! 0 ¼»
Carrying out the multiplication and comparing appropriate elements from the
equality (1.15.9), we obtain
c1 j
and
g11 x1 j i.e., x1 j
c1 j
g11
, j 1,! , m
78
Polynomial and Rational Matrices
c2 j
g 21 x1 j g 22 x2 j , x2 j
1
c2 j g 21 x1 j .
g 22
Thus in the general case for i d r we obtain
xij
1
gii
i 1
§
·
¨ cij ¦ gik xkj ¸ .
k 1
©
¹
(1.15.10)
Entries of rows of the matrix C1 with indices (i, j), i = r+1,…,l, j = 1,…,m can be
chosen arbitrarily.
Example 1.15.1.
Given the matrices
C
ª1 s 1 s 1 s 2 º
«
»
1 »,
«1 s 1 s
« 2
0
2 s »¼
¬
L
ª1 s 0 0 º
« 1
s 0 »» ,
«
«¬ 1
1 0 »¼
one has to compute a matrix C1 that satisfies (1.15.6). In this case, rank L = 2.
According to (1.15.10), to compute x1j, we divide the first row of the matrix C by
g11 = 1+s, and then we subtract the first row of the matrix C1 from the second row
of the matrix C and we divide the result by s. We thus obtain:
C1
ª1 1 1 s º
« 1 1
1 ».
«
»
¬« 2 0 2 s »¼
Example 1.15.2.
Given C lum[s], P
C
mum
[s], one has to compute a matrix C2 such that
C2 P ,
(1.15.11)
where P is an upper triangular matrix and rank P t rank C.
Using the transposition, the solution of the dual problem can be reduced to the
solution of 1.15.1
1.15.3 Computation of Greatest Common Divisors and Smallest Common
Multiplicities of Polynomial Matrices
Theorem 1.15.1. A matrix L(s) mum[s] is a GCLD of matrices A(s)
B(s) muq[s] (m d l + q) if and only if
mul
[s] and
Polynomial Matrices
> A( s)
B( s ) @ and
> L( s )
0@
79
(1.15.12)
are right equivalent matrices.
Proof. If the matrices (1.15.12) are right equivalent then there exists a unimodular
matrix
U( s )
ª U11 ( s ) U12 ( s ) º
« U ( s) U (s) » ,
¬ 21
¼
22
> A(s)
ª U ( s ) U12 ( s ) º
B( s ) @ « 11
»
¬ U 21 ( s ) U 22 ( s ) ¼
> A( s)
B( s ) @
such that
> L( s ) 0@
(1.15.13)
V12 ( s) º
,
V22 ( s ) »¼
(1.15.14)
and
ª V (s)
>L(s) 0@ «V11 ( s)
¬
21
where
ª V11 ( s ) V12 ( s ) º
«V (s) V (s)»
¬ 21
¼
22
U 1 ( s ) .
From (1.15.14) we have
A( s)
L( s )V11 ( s ) and B( s )
L( s )V12 ( s ) .
Thus the matrix L(s) is a CLD of the matrices A(s) and B(s). To show that the
matrix L(s) is a GCLD of the matrices A(s) and B(s), we take into account the
relationship
A( s )U11 ( s ) B( s )U 21 ( s )
L( s) ,
(1.15.15)
which ensues from (1.15.13). Hence it follows that every CLD of the matrices A(s)
and B(s) is also an LD of the matrix L(s). Thus the matrix L(s) is a RM of every
CLD of the matrices A(s) and B(s), i.e., a GCLD of these matrices.
Now we will show that if a matrix L(s) is a GCLD of the matrices A(s) and
B(s), then the matrices in (1.15.12) are right equivalent. By assumption we have
A( s)
L( s ) A1 ( s ) , B( s )
L( s )B1 ( s ) ,
(1.15.16)
80
Polynomial and Rational Matrices
where a GCLD of the matrices A1(s) and B1(s) is the identity matrix Im.
From (1.15.16) we have
> A(s)
B( s ) @
> L( s )
ª A ( s ) B1 ( s ) º
0@ « 1
»,
¬ N( s) M ( s) ¼
(1.15.17)
where N(s) and M(s) are arbitrary polynomial matrices.
We will show that there exist matrices N(s) and M(s) such that the matrix
ª A1 ( s ) B1 ( s ) º
« N( s ) M ( s ) »
¬
¼
(1.15.18)
is a unimodular matrix. A GCLD of the matrices A1(s) and B1(s) is the identity
matrix Im. In view of this, there exists a unimodular matrix U1(s) such that
> A1 ( s)
B1 ( s ) @ U1 ( s )
>I m
0@ .
The matrix U1-1(s) is also a unimodular matrix. Thus from the last relationship we
have
> A1 ( s)
B1 ( s ) @
>I m
0@ U11 ( s )
>I m
ª A ( s ) B1 ( s ) º
0@ « 1
».
¬ N( s ) M ( s ) ¼
Thus the matrix (1.15.18) is unimodular and from (1.15.17) it follows that the
matrices (1.15.12) are right equivalent.
„
Corollary 1.15.1. If a matrix L(s) is a GCLD of the matrices A(s) and B(s), then
there exist polynomial matrices U11, (s) U21(s) such that (1.15.15) holds.
The matrix L(s) can be a lower triangular matrix.
Corollary 1.15.2. If the GCLD of the matrices A1(s) and B1(s), is equal to L(s) = I,
then there exist polynomial matrices N(s) and M(s) such that the square matrix
(1.15.18) is a unimodular one.
From (1.15.13) it follows that
A( s )U12 ( s )
B( s )U 22 ( s )
F(s) .
(1.15.19)
Theorem 1.15.2. The matrix F(s) given by the equality (1.15.19) is a SCRM of the
matrices A(s) and B(s).
Polynomial Matrices
81
Proof. From Definition 1.15.4 and (1.15.19) it follows that the matrix F(s) is a
CRM of the matrices A(s) and B(s). One has still to show that the matrix F(s) is a
left divisor of every CRM of the matrices A(s) and B(s). To show this, it suffices to
note that the GCRD of the matrices U12(s), U22(s) is an identity matrix Im-1-q.
„
To compute a GCLD and SCRM of matrices A(s)
one can apply the following algorithm.
mul
[s] and B(s)
muq
[s],
Algorithm 1.15.1.
Step 1: Write the matrices A(s), B(s) and the identity matrices Il, Iq as
ª A( s ) B( s ) º
«
»
0 ».
« Il
« 0
I q »¼
¬
Step 2: Carrying out appropriate elementary operations on the columns of the
matrix [A(s) B(s)] reduce it to the form [L(s) 0]. Carry out the same
elementary operations on the columns of the matrix Il+q. Partition the
resulting matrix U(s) into the submatrices U11(s), U12(s), U21(s), U22(s) of
dimensions corresponding to those of the matrices A(s) and B(s), i.e.,
ª A( s) B( s ) º
0 º
ª L( s )
«
» R «
»
I

o
U
U
s
0
(
)
12 ( s ) » .
« l
»
« 11
« 0
«¬ U 21 ( s ) U 22 ( s ) »¼
I q »¼
¬
(1.15.20)
Step 3: The GCLD and SCRM we seek are equal to L(s) in (1.15.20) and F(s) in
(1.15.19), respectively.
Example 1.15.1.
Compute a GCLD and a GCRD of the matrices
A( s)
ª s 2 2s s º
«
»,
¬ s 2 1¼
B( s )
ª s 2º
« 1 ».
¬
¼
In this case, m = l = 2, q = 1. In order to compute L(s) and U(s), we write the
matrices A(s), B(s) and I2, I1 as follows
82
Polynomial and Rational Matrices
ª A( s) B( s ) º
«
0 »»
« I2
«¬ 0
1 »¼
ª s 2 2s
«
« s2
« 1
«
« 0
« 0
¬
s
1
0
1
0
s 2s º
»
1 »
0 »
»
0 »
1 »¼
and we perform the following elementary operations
ª 0
« 0
P>1 2u(2 s )@
«
P>3 2u( 1) @

o« 1
«
«2 s
«¬ 0
s 2 º
0 º
ª s 2
»
«
1 0 » P[1,2] «1 0
0 »»
P[2,3]
o «0 0
0 0 » 
1 ».
»
«
»
1 1»
«1 1 2 s »
«¬0 1
0 1 »¼
0 »¼
Thus we have
L( s )
ª s 2 º
«1 0 » , U ( s )
¬
¼
ª U11 ( s ) U12 ( s ) º
« U (s) U ( s) »
¬ 21
¼
22
ª0 0
1 º
«
»
«1 1 2 s » .
«0 1
0 »¼
¬
We compute the SCRM of the matrices A(s) and B(s) using (1.15.19)
F(s)
A( s )U12 ( s )
ª s 2 2s s º ª 1 º
«
»«
»
¬ s 2 1¼ ¬ 2 s ¼
ª0º
«0» .
¬ ¼
Theorem 1.15.3. A matrix P(s)Cqul[s] is the GCRD of matrices A(s)Cmul[s] and
B(s)Cpul[s] (m+p t l) if and only if the matrices
ª A( s) º
ª P(s) º
« B( s ) » and « 0 »
¬
¼
¬
¼
(1.15.21)
are left equivalent.
The proof of this theorem is similar to that of Theorem 1.15.1. Carrying out
elementary operations on the rows, we make the following transformation
ª A( s) I m
« B( s ) 0
¬
0 º L ª P ( s ) U11
c ( s ) U12
c (s) º
.

o«
c
c
I p »¼
U 21 ( s ) U 22 ( s ) »¼
¬ 0
(1.15.22)
Polynomial Matrices
83
Carrying out elementary operations on the rows of the matrix Im+p transforming the
matrix ª A( s ) º to the form ª P ( s ) º , we compute the unimodular matrix
¬« B( s ) ¼»
¬« 0 ¼»
Uc( s )
c ( s ) U12
c (s)º
ª U11
« Uc ( s ) U c ( s ) » .
¬ 21
¼
22
Corollary 1.15.3. If the matrix P(s) is a GCRD of the matrices A(s) and B(s), then
there exist polynomial matrices U11c(s) and U12c(s) such that the equality
c ( s ) A( s ) U12
c ( s )B ( s )
U11
P(s)
(1.15.23)
holds.
Corollary 1.15.4. If a GCRD of the matrices A1(s) and B1(s) is equal to P(s) = I,
then there exist polynomial matrices Nc(s) and Mc (s) such that the square matrix
ª A1 ( s) N '( s) º
« B ( s ) M '( s) »
¬ 1
¼
(1.15.24)
is a unimodular one.
Theorem 1.15.4. The matrix D(s) given by
D( s )
Uc21 ( s ) A( s )
Uc22 ( s )B( s )
(1.15.25)
is an SCLM of the matrices A(s) and B(s).
Proof of this theorem is similar to that of Theorem 1.15.2.
An algorithm for computing a GCRD and a SCLM of matrices A(s) and B(s) is
different from Algorithm 1.15.1 only in that instead of the transformation
(1.15.20), we carry out the transformation (1.15.22) and instead of elementary
operations on columns, we carry out elementary operations on rows. The GCRD
we seek is equal to the matrix P(s), and the SCLM that is equal to the matrix D(s)
is computed from (1.15.25).
Remark 1.15.1.
Greatest common divisors and smallest common multiplicities are computed
uniquely up to multiplication by a unimodular matrix. In this sense, they are not
unique, therefore we usually put the indefinite article a before these notions.
84
Polynomial and Rational Matrices
1.15.4 Relatively Prime Polynomial Matrices and the Generalised Bezoute
Identity
Definition 1.15.7. Matrices A(s) mul[s] and B(s) muq[s] are called relatively
left prime (RLP) if and only if only unimodular matrices are their left common
divisors.
Matrices A(s) mul[s] and B(s) pul[s] are called relatively right prime (RRP)
if and only if only unimodular matrices are their right common divisors.
Theorem 1.15.5. Matrices A(s)
matrices
> A( s)
B( s ) @ and
are right equivalent.
Matrices A(s)
>I m
mul
[s], B(s)
mul
[s], B(s)
0@
muq
[s] are RLP if and only if the
(1.15.26)
pul
[s] are RRP if and only if the matrices
ª A( s) º
ªI l º
« B( s ) » and « 0 »
¬
¼
¬ ¼
(1.15.27)
are left equivalent.
Proof. If the matrices (1.15.26) are right equivalent then according to Theorem
1.15.1, the GCLD of the matrices A(s) and B(s) is Im, i.e., these matrices are RLP.
If the matrices A(s) and B(s) are RLP, then the GLCD is a unimodular matrix,
which by use of elementary operations on the columns can by reduced to the form
[Im 0], i.e., the matrices (1.15.26) are right equivalent. The proof of the second part
of the theorem is similar.
„
From Corollary 1.15.1 for L(s) = Im and from Corollary 1.15.3 for P(s) = I1 we
obtain the following.
Corollary 1.15.5. If the matrices A(s) and B(s) are RLP, then there exist
unimodular matrices U11(s) and U21(s) such that
A( s )U11 ( s ) B( s )U 21 ( s )
Im .
(1.15.28)
If the matrices A(s) and B(s) are RRP, then there exist polynomial matrices U11c(s)
and U12c(s) such that
c ( s ) A( s ) U12
c ( s )B ( s )
U11
Il .
(1.15.29)
Polynomial Matrices
85
The matrices U11(s), U21(s) and U11c(s), U12c(s) can be computed using
Algorithm 1.15.1
Example 1.15.2.
Show that the matrices
A( s)
ª s2 sº
«
» , B( s )
¬ s 1 1¼
ª s 2 2º
«
»
¬ s ¼
are RLP and compute polynomial matrices U11(s), and U21(s) for them such that
(28) holds.
We will show that the given matrices A(s) and B(s) have a GCLD equal to I2.
To accomplish this, we write down these matrices and matrix I3 in the form
ª A ( s ) B( s ) º
«
»
0 »
« Il
« 0
I q »¼
¬
ª s2
«
«s 1
« 1
«
« 0
« 0
¬
s
1
0
1
0
s 2 2º
»
s »
0 »
»
0 »
1 »¼
and we carry out the following elementary operations
ª0
«1
P>1 2u( s ) @
«
P>3 2u( s ) @
«1
o
«
«s
«¬ 0
ª
«
«
o «
«
«
«¬
P ª¬3u 12 º¼
P{2,3]
P[1,2]
s
0
0
1
0
1
0
1
s
1
1 s 12 s 2
0
12 s
2º
ª
«
0 »» P[21u( 1)]
«
P ª 23u 12 s º
¬
¼
o«
0 » 
«
»
1 s »
«
«¬
0 1 »¼
1
0
0
0
0
1
0
1
2
s
1
s
1
1 s 12 s 2
1
2
0
12 s
2º
0 »»
0»o
»
s »
1 »¼
º
»
»
».
»
»
»¼
Thus the given matrices A(s) and B(s) have a GCLD equal to I2. Thus these
matrices are RLP.
From the matrix
86
Polynomial and Rational Matrices
U( s )
ª
º
1
1
« 0
»
«
»
« 1 s s 1 s 1 s 2 » ,
2
« 2
»
«
»
1
« 1
s »
0
«¬ 2
»¼
2
we obtain
U11 ( s )
ª 0
« 1
¬ 2
1º
,
s »¼
U 21 ( s )
> 12
0@ .
It is easy to verify that the matrices A(s), B(s), U11(s), U21(s) satisfy (1.15.28).
1.15.5 Generalised Bezoute Identity
Consider the polynomial RLP matrices A(s)
mun
[s], B(s)
mup
[s], (n + p t m).
Theorem 1.15.6. If polynomial matrices A(s) mun[s] and B(s) mup[s] are RLP,
then there exist polynomial matrices C(s), D(s), M1(s), M2(s), M3(s) and M4(s) of
appropriate dimensions such that
ª A( s)
«C( s )
¬
B( s ) º ª M1 ( s )
D( s ) »¼ «¬ M 3 ( s )
M 2 (s) º
M 4 ( s ) »¼
ªI m
«0
¬
0 º
I n p m »¼
(1.15.30)
B( s ) º
D( s ) »¼
ªI m
«0
¬
0 º
.
I n p m »¼
(1.15.31)
and
ª M1 ( s )
«M ( s)
¬ 3
M 2 ( s ) º ª A( s)
M 4 ( s ) »¼ «¬C( s )
Proof. By the assumption that the matrices A(s) and B(s) are RLP it follows that
there exists a unimodular matrix of elementary operations on columns
ª U1 ( s ) U 2 ( s ) º
( n p )u( n p )
[ s]
« U (s) U ( s) »  4
¬ 3
¼
such that > A( s ) B( s ) @ U( s ) > I m 0@ .
U( s)
Post-multiplying the latter equality by the matrix
Polynomial Matrices
U 1 ( s )
ª V1 ( s )
« V ( s)
¬ 3
V2 ( s ) º
,
V4 ( s ) »¼
87
(1.15.32)
we obtain
ª V1 ( s ) V2 ( s ) º
0@ «
»
¬ V3 ( s ) V4 ( s ) ¼
> A( s)
B( s ) @
>I m
> A( s)
B( s ) @
> V1 ( s)
and
V2 ( s ) @ .
The matrix (1.15.32) is unimodular and the following equality holds
B 1 ( s )U( s )
ª A( s ) B( s ) º ª U1 ( s ) U 2 ( s ) º
« V (s) V (s) » « U ( s) U ( s) »
4
4
¬ 3
¼¬ 3
¼
ªI m
«0
¬
0 º
.
I n p m »¼
Thus [C(s) D(s)] = [V3(s) V4(s)] and Mk(s) = Uk(s), for k = 1,2,3,4. The identity
(1.15.31) follows from the equality U(s)U(s)1 = U(s)1U(s) = In+p.
„
The following dual theorem can be proved in a similar way.
Theorem 1.15.7. If polynomial matrices Ac (s) mun[s] and Bc(s) pun[s] are
RRP, then there exist polynomial matrices Cc(s), Dc(s), N1(s), N2(s), N3(s) and
N4(s) of appropriate dimensions, such that
0 º
ª Ac( s ) Cc( s ) º ª N1 ( s ) N 2 ( s ) º ª I n
»,
« Bc( s ) Dc( s ) » « N ( s ) N ( s ) » « 0 I
m p n ¼
¬
¼¬ 3
4
¼ ¬
0 º
ª N1 ( s ) N 2 ( s ) º ª Ac( s ) Cc( s ) º ª I n
».
« N ( s ) N ( s ) » «Bc( s ) Dc( s ) » « 0 I
m p n ¼
¼ ¬
4
¬ 3
¼¬
(1.15.33)
(1.15.34)
1.16 Decomposition of Regular Pencils of Matrices
1.16.1 Strictly Equivalent Pencils
Definition 1.16.1. A pencil [Es A] (or a pair of matrices (E, A)) is called regular
if the matrices E and A are square and
88
Polynomial and Rational Matrices
(1.16.1)
det [Es A] z 0 for some s  Definition 1.16.2. Let Ek, Ak mun for k = 1,2. The pencils [E1s – A1] and
[E2s – A2] (or the pairs of the matrices (E1, A1) and (E2, A2)) are called strictly
equivalent if there exist nonsingular matrices P mum, Q nun (with elements
independent of the variable s) such that
P > E1s A1 @ Q
E2 s A 2 .
(1.16.2)
Let Dk(s, t) (k = 1, ..., n) be the greatest common divisor of the all minors of
degree k of the matrix [Es – At]. According to (1.8.4) the invariant polynomials of
the matrix [Es – At] are uniquely determined by
ik ( s, t )
Dnk 1 ( s, t )
for k
Dnk ( s, t )
1, 2, ..., r .
(1.16.3)
Factoring the polynomials (1.16.3) into appropriate polynomials that cannot be
factored in a given field, we obtain elementary divisors ei(s, t) (i = 1, ..., p) of the
matrix [Es – At]. Substituting t = 1 into ei(s, t), we obtain appropriate elementary
divisors ei(s) = ei(s,1) of the [Es – A]. Knowing ei(s) of the matrix [Es – A], we can
also compute elementary divisors ei(s,t) of the [Es – At] using the relationship
ei(s,t) = tqei(s/t), where q is the degree of the polynomial ei(s).
In this way, we can find all finite elementary divisors of the matrix [Es – At]
with exception of elementary divisors of the form tq. Elementary divisors of the
form tq are called infinite elementary divisors of the matrix [Es – A]. Infinite
elementary divisors appear if and only if det E = 0.
For instance, bringing the pencil
[ Es A ]
ª1 1º
ª1 1 º
«1 1» s «1 2 »
¬
¼
¬
¼
into the Smith canonical form
[Es A]S
0 º
ª1
«0 s 1» ,
¬
¼
we assess that this pencil possesses the finite elementary divisor s + 1 and the
infinitive elementary divisor t, since e(s) = s+1, q = 1 and te(s/t) = s+t.
Consider two square pencils of the same size
>E1s A1 @ and >E2 s A 2 @ such
that det E1 z 0 and det E 2 z 0. (1.16.4)
Polynomial Matrices
89
Theorem 1.16.1. If the condition (1.16.4) is satisfied, then the pencils [E1s – A1]
and [E2s – A2] are equivalent if and only if they are strictly equivalent, i.e.,
unimodular matrices L(s) and P(s) in the equation
E1s A1
L( s) > E2 s A 2 @ P ( s )
(1.16.5)
can be replaced with matrices L and P, which are both independent of the variable
s,
E1s A1
L >E2 s A 2 @ P .
(1.16.6)
Proof. The inverse matrix M(s) = L-1(s) of a unimodular matrix L(s) is also a
unimodular matrix. Pre-multiplying (1.16.5) by M(s), we obtain
M ( s ) > E1 s A1 @
>E2 s A 2 @ P( s) .
(1.16.7)
Pre-dividing the matrix M(s) by [E2s – A2] and post-dividing the matrix P(s) by
[E2s – A2], we obtain
M( s)
> E 2 s A 2 @ Q( s ) M,
P(s )
T( s ) > E1s A1 @ P,
(1.16.8)
where M and P are matrices independent of the variable s.
Substituting (1.16.8) into (1.16.7), we obtain
>E2 s A 2 @>T(s) Q( s)@>E1s A1 @
M > E1s A1 @ > E2 s A 2 @ P . (1.16.9)
This equality holds only for T(s) = Q(s); otherwise the left-hand side of this
equation would be a polynomial matrix of at least second degree, and the righthand side would be a polynomial matrix of at most first degree.
Taking into account T(s) = Q(s) in (1.16.9), we obtain
M > E1s A1 @
>E2 s A 2 @ P .
(1.16.10)
We will show that det M z 0. Pre-dividing the matrix L(s) by E1s - A1, we obtain
L( s )
>E1s A1 @ R (s) L ,
where L is independent of the variable s.
Using (1.16.11), (1.16.7) and (1.16.8) successively, we obtain
(1.16.11)
90
Polynomial and Rational Matrices
I
M ( s )L ( s )
M ( s ) > E1s A1 @ R ( s ) L
M ( s) > E1s A1 @ R ( s ) M ( s )L
>E2 s A 2 @ P(s)R( s) > E2 s A 2 @ Q(s)L ML
>E2 s A 2 @> P( s)R( s) Q(s)L@ ML.
(1.16.12)
The right-hand side of (1.16.12) is a matrix of zero degree (equal to an identity
matrix) if and only if
P ( s ) R ( s ) Q ( s )L
0.
(1.16.13)
With the above taken into account, from (1.16.12) we have
ML
I.
Thus the matrix M is nonsingular and L = M-1. Pre-multiplying (1.16.10) by
L = M-1, we obtain (1.16.6).
From Theorem 1.16.1 we have the following important corollary.
Corollary 1.16.1. If the condition (1.16.4) is satisfied, then notions of equivalence
and strict equivalence of pencils [E1s – A1] and [E2s – A2] are the same.
From the fact that two polynomial matrices are equivalent if and only if they
have the same elementary divisors and from Corollary 1.16.1, the following
theorem ensues immediately.
Theorem 1.16.2. If the condition (1.16.4) is satisfied, then pencils [E1s – A1] and
[E2s – A2] are strictly equivalent if and only if they have the same finite elementary
divisors.
If the condition (1.16.4) is not satisfied, then the pencils [E1s – A1] and
[E2s – A2] might not be equivalent in spite of the fact that they have the same
elementary devisors.
>E1s A1 @
>E2 s A 2 @
ª1 1 2 º
ª2
«1 1 2 » s « 3
«
»
«
«¬1 1 3 »¼
«¬ 3
ª1 1 1º
ª2
«1 1 1» s « 1
«
»
«
¬«1 1 1¼»
¬« 1
1 3º
2 5 »» ,
2 6 »¼
1 1º
2 1»» ,
1 1¼»
(1.16.14)
Polynomial Matrices
91
are not strictly equivalent (since rank E1 = 2, rank E2 = 1), although they have the
same elementary divisor s + 1, because they have different infinite elementary
divisors. Performing elementary operations on the pencil [E1s A1t], we obtain
assertion of this.
Theorem 1.16.3. Two regular pencils [E1s – A1] and [E2s – A2] are strictly
equivalent if and only if they have the same finite and infinite elementary divisors.
Proof. The strict equivalence of the pencils [E1s – A1] and [E2s – A2] implies strict
equivalence of the pencils [E1s – A1t] and [E2s – A2t]. In view of this, the pencils
[E1s – A1] and [E2s – A2] should have the same finite and infinite elementary
divisors. Conversely, let two regular pencils [E1s – A1] and [E2s – A2], which have
the same finite and infinite elementary divisors, be given. Let
s
aO b P , t
cO d P (ad bc z 0) .
(1.16.15)
Substituting (1.16.15) into [E1s – A1t] and [E2s – A2t] yields
>E1s A1t @
> E2 s A 2t @
ª¬E1 aO bP A1 cO d P º¼
ª¬E1O A1P º¼ ,
ª¬E2 aO bP A 2 cO d P º¼ ª¬E 2 O A 2 P º¼ ,
(1.16.16)
where
E1
aE1 cA1 , A1
dA1 bE1 , E 2
aE 2 cA 2 , A 2
dA 2 bE 2 . (1.16.17)
By assumption of regularity of the pencils [E1s – A1t] and [E2s – A2t], one can
choose numbers a and c such that
det E1 z 0 and det E2 z 0 .
(1.16.18)
If the condition (1.16.18) is satisfied, then the pencils
ª¬E1O A1P º¼ and ª¬ E2 O A 2 P º¼
are strictly equivalent and this fact implies that the pencils [E1s – A1t] and
[E2s – A2t], as well as the starting-point pencils [E1s – A1], [E2s – A2], are strictly
equivalent.
„
92
Polynomial and Rational Matrices
1.16.2 Weierstrass Decomposition of Regular Pencils
Assume at the beginning that rectangular matrices E, A
rank [Es A]
qun
are such that
q for some s  .
(1.16.19)
Theorem 1.16.4. If the condition (1.16.19) is satisfied, then there exist full-rank
matrices P qun and Q nun such that
ª I n s A1
[ Es A ] P « 1
¬« 0
0
º
»Q ,
Ns I n2 ¼»
(1.16.20)
where n1 is the greatest degree of the polynomial of the variable s, which is a minor
of degree q of the matrix [Es – A], n1+n2 = n, and N is a nilpotent matrix of index v
(Nv = 0).
Proof. If the condition (1.16.19) is satisfieds then there exists a number c such
that the matrix F = [Ec - A] has full row rank. In this case, there exists the inverse
of this matrix
Fp
1
FT ª¬ FFT º¼  nuq ,
(1.16.21)
which satisfies the condition FFp = Iq.
Note that
> Es A @ > E ( s c ) E c A @ > E ( s c ) F @
T
F ª¬Fp E( s c ) I n º¼ . (1.16.22)
According to the considerations in Sect. 4.2.2, there exists a nonsingular matrix
nun
such that
Fp E
T ¬ª diag J1 , J 0 ¼º T1 ,
n un
(1.16.23)
n un
where J1 = 1 1 is a nonsingular matrix and J0 = 2 2 is a nilpotent matrix with
index v. The matrix T can be chosen in such a way that diag (J1, J0) has the Jordan
canonical form.
Substitution of (1.16.23) into (1.16.22) yields
[Es A ] FT ª¬diag (J1 ( s c) I n1 , J 0 ( s c ) I n2 ) º¼ T 1
FTdiag J1 , J 0 c I n2 ª¬diag I n1 s J11 (I n1 J1c),
J 0 c I n2
1
J 0 s I n2 T 1
P ªdiag I n 1 s A1 , Ns I n2 º Q,
¬
¼
(1.16.24)
Polynomial Matrices
93
where
P
N
FT diag J1 , J 0 c I n2 , A1
J 0 c I n2
1
J0 , Q
J11 J1c I n1 ,
T1.
(1.16.25)
Note that Nv = 0, since J0v = 0 and Nv = (J0c - I n2 )-vJ0v = 0.
Remark 1.16.3.
Transforming A1 and N to the Jordan canonical form, we obtain
diag ª¬ H m1 s I m1 , ..., H mt s I mt , I n1 s J º¼ ,
where
H mi
ª0
«0
«
«#
«
«0
«¬ 0
1 0 ! 0 0º
0 1 ! 0 0 »»
# # % # # »  mi umi (i 1, ..., t )
»
0 0 ! 0 1»
0 0 ! 0 0 »¼
and J is the Jordan canonical form of the matrix A1 and m1+m2+…+mt+n1=n.
Theorem 1.16.4 generalises the classical Weierstrass theorem for the case of a
rectangular pencil, which satisfies the condition (1.16.19).
If q = n, then the matrix P is square and nonsingular
P 1 > Es A @ Q 1
ª I n1 s A1
«
«¬ 0
0
º
»,
Ns I n2 »¼
(1.16.26)
and n1 is equal to the degree of the polynomial det [Es - A].
Theorem 1.16.5. If [Es – A] is a regular pencil, then there exist two nonsingular
matrices P, Q nun such that (1.16.26) holds.
The transformation matrices P and Q appearing in (1.16.26) can be computed
by use of (1.16.25). Another method of computing these matrices is provided
below.
Let si be the i-th root of the equation
det > Es A @ 0
(1.16.27)
94
Polynomial and Rational Matrices
and
mi
dim Ker > Esi A @ .
(1.16.28)
Compute finite eigenvectors vij1 using the equation
>Esi A @ vij1
0, for j 1, ..., mi ,
(1.16.29)
and then (finite) eigenvectors vijk+1 from the equation
>Esi A @ vijk 1
Evijk , for k t 1 .
(1.16.30)
Let
mf
dim Ker E
n rank E .
(1.16.31)
We compute infinite eigenvectors vfj1 from the equations
Ev1fj
(1.16.32)
0, for j 1, ..., mf ,
and then eigenvectors vfjk+1 from the equation
Evfk j 1
Avfk j , for k t 1 .
(1.16.33)
The computed vectors are columns of the desired matrices
P
ª¬ Evijk # Avfk j º¼ ,
Q 1
ª¬ vijk # vfk j º¼ .
(1.16.34)
Using (1.16.29)( 1.16.33) one can easily verify that
>Es A @ ª¬vijk
ª¬Evijk
#
#
vfk j º¼
ªI n1 s A1
Avfk j º¼ «
«¬ 0
0
º
» .
Ns I n2 »¼
(1.16.35)
Pre-multiplying (1.16.35) by [Evijk # Avfjk]-1, we obtain (26) for P and Q given
by (1.16.34).
Example 1.16.1.
Compute the matrices P and Q for a regular pencil whose matrices E and A have
the form
Polynomial Matrices
E
ª1 0 0 º
«0 1 0 » ,
«
»
«¬0 0 0 »¼
A
95
ª1 0 1º
«0 1 0».
«
»
«¬ 1 0 1»¼
In this case,
det [Es A]
s 1
0
1
0
s 1
0
1
0
1
s ( s 1)
and n1 = 2, n2 = 1, s1 = 1, s2 = 0, m1 = dim Ker [Es1 A] = 1.
Using (1.16.29), (1.16.30), (1.16.32) and (1.16.33), we compute successively
>Es1 A @ v1
>Es2 A 2 @ v2
Ev3
ª0 0 1º
«0 0 0 » v
«
» 1
«¬1 0 1 »¼
ª0º
«0» ,
« »
«¬ 0 »¼
ª 1 0 1º
« 0 1 0 » v
«
» 2
¬« 1 0 1 »¼
ª1 0 0 º
«0 1 0 » v
«
» 3
«¬0 0 0 »¼
ª0º
«0» ,
« »
«¬ 0 »¼
v3
ª0 º
«1 » ,
« »
«¬0 »¼
v1
ª0º
«0» ,
« »
¬« 0 ¼»
ª0 º
«0 » ,
« »
«¬1 »¼
Ev1
v2
ª1 º
«0 » ,
« »
¬« 1¼»
Av3
ª1 º
«0 » .
« »
«¬ 1»¼
ª0º
«1 » ,
« »
«¬ 0»¼
Ev2
ª1 º
«0» ,
« »
¬« 0 ¼»
Thus from (1.16.33) we have
P
>Ev1 , Ev2 , Av3 @
ª0 1 1 º
« 1 0 0 » , Q 1
«
»
«¬ 0 0 1»¼
>v1 , v2 , v3 @
ª0 1 0 º
«1 0 0 » .
«
»
«¬ 0 1 1 »¼
1.17 Decomposition of Singular Pencils of Matrices
1.17.1 Weierstrass–Kronecker Theorem
Definition 1.17.1. A pencil [Es – A] (E, A
det [Es – A] for all s when m = n.
mun
) is said to be singular if m z n or
96
Polynomial and Rational Matrices
Let rank [Es – A] = r d min (m, n) for almost every s .
Assume that r < n. In this case, the columns of the matrices [Es – A] are
linearly dependent and the equation
> Es A @ x
(1.17.1)
0
has a nonzero solution x = x(s).
Among the polynomial solutions to (1.17.1) we seek solutions of the minimal
degree p with respect to s having the form
x( s )
x0 x1s x2 s 2 " x p s p .
(1.17.2)
Substituting (1.17.2) into (1.17.1) and comparing coefficients by the same
powers of the variable s, we obtain the equations
Ax0
0, Exi 1 Axi
0, for i 1, ..., p and Ex p
0,
which can be written in the form
0
ªA 0
« E A 0
«
« 0
E A
«
#
#
#
«
« 0
0
0
«
0
0
¬« 0
0 º ª x0 º
«
»
0 »» « x1 »
0 » « x2 »
»
»«
% #
# »« # »
! E A » « x p 1 »
»
»«
! 0 E ¼» ¬« x p ¼»
! 0
! 0
! 0
ª0º
«0»
« »
«0»
« ».
« #»
«0»
« »
¬« 0 ¼»
(1.17.3)
Note that (1.17.3) has a solution if and only if the matrix
Gp
0
ªA 0
« E A 0
«
« 0
E A
«
#
#
#
«
« 0
0
0
«
0
0
«¬ 0
! 0
! 0
0 º
0 »»
! 0 0 »
( p 2) mu( p 1) n
»
% #
# »
! E A »
»
! 0 E »¼
does not have full column rank. By assumption p is minimal, thus we have
rank Gi = (i+1)n, for i = 0,1,…,p1 and rank Gp < (p + 1)n.
Lemma 1.17.1. If (1.17.1) has the solution (1.17.2) of the minimal degree p > 0,
then the pencil [Es – A] is strictly equivalent to the pencil
Polynomial Matrices
ªL p
«0
¬
0 º
,
Es A »¼
97
(1.17.4)
where
Lp
ªs 1
«
«0 s
«0 0
«
«# #
«
«0 0
«
¬0 0
0 " 0 0º
»
1 " 0 0»
s " 0 0»
#
%
#
0 " 1
0 " s
»  pu( p1)
#»
»
0»
»
1¼
and the equation
ª¬Es A º¼ x
(1.17.5)
0
does not have polynomial solutions of degree smaller than p.
Proof. Consider a linear operator [Es – A] mapping n into m. We will show that
one can choose bases in n and m in such a way that the corresponding pencil
[Es – A] has the form
ªL p
«0
¬
Bs c º
.
Es A »¼
(1.17.6)
The linear operator equation corresponding to (1.17.1) is
> Es A @ x
0,
(1.17.7)
where
x
x(s)
x0 x1 s x2 s 2 " x p s p .
Similarly as for (1.17.1) we obtain
Ax0
0,
Exi 1
Axi and i 1, ..., p,
Ex p
0.
(1.17.8)
We will show that the vectors
Ax1 , Ax2 , ..., Ax p
(1.17.9)
98
Polynomial and Rational Matrices
are linearly independent.
Suppose that vector Axk linearly depends on vectors Ax1,…,Axk-1 (k d p), that is
Axk
a1Ax1 ... ak 1Axk 1 for certain ai  .
Using (1.17.8), we obtain
Axk
Exk 1
Exˆk 1
0,
a1Ex0 a2 Ex1 ... ak 1Exk 2
and
where
xˆk 1
xk 1 a1 x0 a2 x1 ... ak 1 xk 2 .
Note that
Axk 1
Axk 1 a1Ax0 a2 Ax1 ... ak 1Axk 2
Exˆk 2
E xk 2 a2 x0 a3 x1 ... ak 1 xk 3
where
xˆk 2
xk 2 a2 x0 a3 x1 ... ak 1 xk 3 .
Similarly,
Axˆk 2
Axk 2 a2 Ax0 a3 Ax1 ... ak 1Axk 3
E xk 3 a3 x1 ... ak 1 xk 4
Exˆk 3 ,
where
xˆk 3
xk 3 a3 x1 ... ak 1 xk 4 .
Continuing this procedure, we obtain
Axˆk 3
Exˆk 4 , ..., Axˆ1
Exˆ0 , Axˆ0
0,
where
xˆk 4
xk 4 a4 x1 ... ak 1 xk 5 , ..., xˆ1
x1 ak 1 x0 , xˆ0
x0 .
Polynomial Matrices
99
Taking into account the above relationships one can easily verify that the vector
x
xˆ ( s)
xˆ0 xˆ1s xˆ2 s 2 ... xˆk 1s k 1 and k d p
is a solution to (1.17.7) of degree smaller than p. This contradiction proves that the
vectors (1.17.9) are linearly independent.
We will show by contradiction that the vectors x0,x1,…,xp are also linearly
independent.
Suppose that these vectors are linearly dependent, that is
0 for some bi  .
b0 x0 b1 x1 ... bp x p
(1.17.10)
In this case, we obtain
b1Ax1 b2 Ax2 ... bp Ax p
0,
since Ax0 = 0.
The vectors (1.17.9) are linearly independent. In view of this,
b1 = b2 =…= bp = 0 and from (1.17.10) we obtain b0x0 = 0. Note that x0 z 0, since
otherwise s-1x(s) would also be a solution of the equation. Hence b0x0 = 0 implies
b0 = 0 and the vectors x0,x1,…,xp are linearly independent. We choose the vectors
(1.17.9) to be the first basis vectors of the space n and the vectors x0,x1,…,xp to be
the first basis vectors of m, respectively. Using (1.17.8), it is easy to verify that in
this case, the pencil [Es A] has the form (1.17.6). Note that (1.17.4) can be
obtained from (1.17.6) by adding to [Bs + C] an appropriate linear combination
with coefficients independent of s from columns Lp and rows [ E s A ].
We will show that (1.17.5) has no solutions of degree smaller than p.
Taking into account (1.17.4), we can write down
ªL p
«0
¬
0 º ªz º
Es A »¼ «¬ y »¼
0,
(1.17.11)
which is equivalent to
Lpz
0,
ª¬Es A º¼ y
0.
(1.17.12)
From the special form of Lp it follows that the equation (Lpz = 0) has a solution of
degree p of the form
zi
(1)i1 s i1z1 (i 1, ..., p 1)
for arbitrary z1, where z1 is the i-th component of vector z. Thus the matrix Gp-1 in
(1.17.3) has full column rank equal to pn.
100
Polynomial and Rational Matrices
The equation [ E s - A ]y = 0 has solution of the minimal degree p if and only if
the matrix Gp-1 in the equation
G p 1
ªA 0
0
«
« E A 0
« 0
E A
«
#
#
« #
« 0
0
0
«
0
0
«¬ 0
!
!
!
%
!
!
0 0 º
»
0 0 »
0 0 »
( p 1)( n p )u p ( n p 1)
»
#
# »
E A »
»
0 E »¼
has full column rank, equal to p(n – p 1). From (1.17.4) it follows that the matrix
Gp-1 in (1.17.3), after the appropriate interchange of rows and columns, can be
written in the form
G p 1
ˆ
ªG
0 º
p 1
«
»,
G p 1 ¼»
¬« 0
where Ĝ p-1 has dimensions p(p + 1)up(p + 1) and corresponds to the equation
Lpz = 0. Note that the condition rank Gp-1 = np implies that rank Ĝ p-1 = p(p+1) and
rank G p-1 = p(np1). Hence the equation Lpz = 0 has no solution of degree
smaller than p. „
In the general case we assume that
1. rank [Es – A]= r < min(m,n);
2. columns and rows of [Es – A] are linearly dependent over
exist x n and v m (independent of s) such that
> Es A @ x
, i.e., there
(1.17.13)
0
and
T
> Es A @
v
0.
(1.17.14)
Let (1.17.13) have p0 linearly independent solutions x1,x2,…,xp . Choosing
these solutions as the first p0 basis vectors of the space n, we obtain a strictly
equivalent pencil that has the form
0
ª¬0np0
Es A º¼ ,
where 0np is a zero-matrix of dimension nup0.
0
(1.17.15)
Polynomial Matrices
101
Similarly, let (1.17.14) have q0 linearly independent solutions
v1,v2,…,vq 0 .Choosing these solutions as the first q0 basis vectors of the space m,
we obtain a strictly equivalent pencil that has the form
0q0 ,n p0 º
»,
sA
E
»¼
ª
«
«¬ 0n, p0
(1.17.16)
] has rows and columns linearly independent over .
where [ E s A
] be linearly dependent over the field of rational
Let the columns of [ E s A
functions C(s) and let the equation
ºx
ªE s A
¬
¼
0
Have a polynomial solution of the minimal degree p1. Applying Lemma 1.17.1 to
], we obtain a strictly equivalent pencil that has the form
the pencil [ E s A
0q0 ,n p0
ª
«
« 0n, p0
«
¬
L p1
0
º
»
0 »,
E1s A1 »¼
(1.17.17)
and the equation
>E1s A1 @ x
(17.18)
0
that has no polynomial solutions of degree smaller than p1.
If (1.17.18) has a polynomial solution of the minimal degree p2, then continuing
this procedure, we obtain a strictly equivalent pencil that has the form
ª
«
«¬ 0np0
0q0 ,n p0
º
»,
diag ¬ª L p1 , L p2 , ..., L pw , E w s A w ¼º »¼
(1.17.19)
where p1 d p2 d…d pw, and the equation [Ews – Aw]x = 0 has no nonzero
polynomial solutions.
If the pencil [Ews – Aw] has linearly dependent rows over the field (s) and the
equation
T
>Ew s A w @
v
0
has polynomial solution of the minimal degree q1, then applying Lemma 1.17.1 to
[Ews – Aw]T, we obtain a strictly equivalent pencil that has the form
102
Polynomial and Rational Matrices
0q0 ,n p0
ª
«
«0
¬ np0
º
»,
diag ª L p1 , L p2 , ..., L pw , LTq1 , E1c s A1c º »
q
¬
¼¼
(1.17.20)
where the equation
T
>E1cs A1c @
v
0
(1.17.21)
has no polynomial solutions of degree smaller than q1.
If (1.17.21) has polynomial solution of the minimal degree q2, then continuing
this procedure, we obtain a strictly equivalent pencil that has the form
0q0 ,n p0
ª
«
«¬ 0np0
º
»,
diag ¬ª L p1 , , ..., L pw , LTq1 , ..., LTqs , E0 s A 0 º¼ »¼
(1.17.22)
where [E0s – A0] is a regular pencil.
Applying Theorem 1.16.4 to the pencil [E0s – A0], we obtain the Weierstrass–
Kronecker canonical form of a singular pencil, that is
ª
«
«¬0np0
0q0 ,n p0
º
» , (1.17.23)
diag ¬ª L p1 , ,..., L pw , L ,..., L , H n1 s I n1 ,..., H nt s I nt , I r s J º¼ »¼
T
q1
T
qs
where the pencil
diag ª¬ H n1 s I n1 ,..., H nt s I nt , I r s J º¼
corresponds to the regular pencil E0s – A0.
Thus we have proven the following Weierstrass–Kronecker theorem about
decomposition of a singular pencil.
Theorem 1.17.1. An arbitrary singular pencil [Es – A] is strictly equivalent to the
pencil (1.17.23).
1.17.2 Kronecker Indices of Singular Pencils and Strict Equivalence of
Singular Pencils
Let us consider a pencil [Es – A] for E, A mun. Let x1(s) be a nonzero polynomial
solution of minimal degree p1 of the equation
> Es A @ x
0.
(1.17.24)
Polynomial Matrices
103
Among polynomial solutions of the equation, linearly which are independent of
x1(s) over (s), we choose a solution x2(s) of minimal degree p2 (p2 t p1). Then
among polynomial solutions of (1.17.24) which are linearly independent of x1(s)
and x2(s) over (s), we choose solutions x3(s) of minimal degree p3 (p3 t p2).
Continuing this procedure we obtain a sequence of linearly independent
polynomial solutions of (1.17.24) of the form
(1.17.25)
x1 ( s ), x2 ( s ),..., xw ( s ) ( w d n)
with degrees
p1 d p2 d " d pw .
(1.17.26)
In the general case, for a given pencil [Es – A] there exist many sequences of the
polynomial solutions (1.17.25) to (1.17.24). We will show that all these sequences
of polynomial solutions have the same sequence of degrees (1.17.26).
Suppose that x 1(s), x 2(s),..., x w(s) with degrees p 1d p 2d…d p w is another
sequence of polynomial solutions to (1.17.24). Let
p1
...
pn1 pn1 1
...
pn2 pn2 1
...
p1
...
pn1 pn1 1
...
pn2 pn2 1
...
and
From this choice of x1(s) and x 1(s) it follows that p1 = p 1. Note that x 1(s) for
i = 1,…, n 1 is a linear combination x1(s),…,x n (s), since otherwise x n 1 (s) in
1
1
(1.17.25) could be replaced with a polynomial vector of degree smaller than p n 1 .
1
Similarly, xi(s) for i = 1,…,n1 is a linear combination x1 ( s ), ..., xn ( s ) . In view of
1
this,
n1= n
p n 1 = p
2
n2 1
1
and
p n 1 = p
1
.
n 1
1
Similarly
it
is
easy
to
show
that
p n 2 1 .
p n 2 1
Definition 1.17.2. Nonnegative integers p1,p2,…,pw are called minimal column
(Kronecker) indices of the pencil [Es – A].
Let v1(s) be the nonzero polynomial solution of the minimal degree q1 of the
equation
T
> Es A @
v
0.
(1.17.27)
Among the polynomial solutions of this equation, which are linearly independent
over (s) of v1(s), we choose a solution v2(s) of minimal degree q2 (q2 t q1).
104
Polynomial and Rational Matrices
Continuing this procedure, we obtain a sequence of polynomial solutions to
(1.17.27) of the form
v1 ( s ), v2 ( s ) , ..., vs ( s ) ( s d n)
(1.17.28)
with degrees
q1 d q2 d d qs .
(1.17.29)
Similarly to (1.17.25) and (1.17.26) one can show that all sequences of polynomial
solutions (1.17.28) to (1.17.27) have the same sequences of minimal degrees
(1.17.29).
Definition 1.17.3. Nonnegative integers q1,q2,…,qs are called minimal row
(Kronecker) indices of the pencil [Es – A].
Lemma 1.17.2. Strictly equivalent pencils have the same minimal column and row
Kronecker indices.
Proof. Take strictly equivalent pencils [E1s – A1] and [E2s – A2], i.e., related by
the relationship [E2s – A2] = P[E1s – A1]Q.
Pre-multiplying the equation
>E1s A1 @ x
(1.17.30)
0
by a nonsingular matrix P and defining a new vector z = Q-1x (Q is a nonsingular
matrix), we obtain
P > E1 s A1 @ QQ 1 x
>E2 s A 2 @ z
0.
(1.17.31)
Thus these pencils have the same minimal column indices, since the degree of x
in (1.17.30) is equal to the degree of z in (1.17.31). Similarly we can prove that
these pencils have the same minimal row indices.
„
Lemma 1.17.3. The Weierstrass–Kronecker canonical form (1.17.23) of the pencil
[Es – A] is completely determined by p0 minimal column indices, which are equal
to zero, nonzero minimal column indices p1,p2,…,pw,q0, minimal row indices equal
to zero, nonzero minimal row indices q1,q2,…,qs and by its finite and infinite
elementary divisors.
Proof. The matrix L pi (i = 1,…,w) has only one minimal column index pi, since
the equation L p z = 0 has only one polynomial solution of degree pi and the rows
i
of the matrix L p are linearly independent. Similarly, the matrix LT q (j = 1,…,s)
i
j
Polynomial Matrices
105
has only one minimal zero index qj, since the equation LT q v = 0 has only one
j
polynomial solution of degree qj and columns of the matrix LT q are linearly
j
independent. It is easy to check that the matrix L p (or LT q ) does not have any
i
j
elementary divisors, since one of its minors, of the greatest degree pi (respectively
qj), is equal to 1 and the other one is equal to s p ( s q ) .
The first p0 columns of the matrix (1.17.23) correspond to polynomial solutions
of (1.17.13). In view of this, the first p0 minimal column indices of
[Es – A] are equal to 0. Dually, the first q0 minimal row indices of [Es – A] are
equal to zero.
Note that the pencil [E0s – A0] in (1.17.22) is regular, hence it is completely
determined by its finite and infinite elementary divisors. From the block-diagonal
form (1.17.23) it follows that the canonical form of the pencil [Es – A] is
completely determined by minimal column and row indices, and finite and infinite
elementary divisors of every diagonal block.
„
i
j
From Lemmas 1.17.2 and 1.17.3 and from the fact that two singular pencils
having the same canonical forms are strictly equivalent, the following Kronecker
theorem can be inferred.
Theorem 1.17.2. (Kronecker) Two singular pencils [E1s – A1], [E2s – A2] for
Ek, Ak mun (k = 1,2) are strictly equivalent if and only if they have the same
minimal column and row indices, as well the same finite and infinite elementary
divisors.
2
Rational Functions and Matrices
2.1 Basic Definitions and Operations on Rational Functions
A quotient of two polynomials l(s) and m(s) in variable s, where m(s) is a nonzero
polynomial,
w( s )
l ( s)
m( s )
(2.1.1)
is called a rational function of the variable s.
The set of rational functions with coefficients from a field will be denoted by
(s). A field can be the field of real numbers , of the complex numbers , of
the rational numbers , or a field of rational functions of another variable z, etc.
We say that rational functions
w1 ( s )
l1 ( s )
, w2 ( s )
m1 ( s )
l2 ( s )
m2 ( s )
(2.1.2)
belong to the same equivalence class if and only if
l1 ( s )m2 ( s )
l2 ( s )m 1 ( s ) .
(2.1.3)
Let l1(s) = a(s) l 1(s) and m1(s) = a(s) m 1(s), where a(s) is a greatest common
divisor of l1(s) and m1(s). Then
w1 ( s )
a ( s )l1 ( s )
a ( s )m1 ( s )
l1 ( s )
,
m1 ( s )
108
Polynomial and Rational Matrices
where l 1(s) and m 1(s) are relatively prime.
Thus the rational function (2.1.1) represents the whole equivalence class. We
say that the rational function (2.1.1) is of standard form if and only if the
polynomials l(s) and m(s) are relatively prime and the polynomial m(s) is monic
(i.e., a polynomial in which the coefficient at the highest power of the variable s is
1).
Zeros of the numerator polynomial l(s) are called finite zeros (shortly zeros),
and zeros of the denominator polynomial m(s) are called finite poles (shortly poles)
of the rational function (2.1.1).
Definition 2.1.1. An order r of the rational function (2.1.1) is a difference of
degrees of denominator m(s) and numerator l(s).
r
deg m( s ) deg l ( s ) ,
(2.1.5)
where deg denotes the degree of a polynomial.
Let
l (s)
lm s m lm1s m1 ... l1s l0 , m( s )
s n an1s n1 ... a1s a0 .(2.1.6)
If the polynomials (2.1.6) are a numerator and denominator, respectively, of the
function (2.1.1), then the order of this function is equal to r = n -m. This function
has m finite zeros and n finite poles. If r = n - m < 0, then this function has a pole
of multiplicity r (s = f) at infinity and if r = n - m > 0, then this function has a zero
of multiplicity r (s = f) at infinity.
Definition 2.1.2. The rational function (2.1.1) is called proper (or causal) if and
only if its order is nonnegative (r = deg m(s) – deg l(s) t 0), and strictly proper (or
strictly causal) if and only if its order is positive (r = deg m(s) – deg l(s) > 0).
Dividing the numerator l(s) by the denominator m(s), the rational function
(2.1.1) can be presented in the form
w( s )
wr s r wr 1s ( r 1) ... ,
(2.1.7)
where r is the order of the function given by (2.1.5), and wr,wr+1,… are coefficients
dependent on the coefficients of the polynomials l(s) and m(s).
For example, division of the polynomial l(s) = 2s + 1 by m(s) = s2 + 2s + 3
yields
w1 ( s )
2s 1
s 2s 3
2
2s 1 3s 2 9 s 4 ...
In this case, m = 1, n = 2, r = n - m = 1, w1 = 2, w2 = -3, w3 = 0, w4 = 9,…
(2.1.8)
Rational Functions and Matrices
109
The coefficients wr,wr+1,… can also be computed in the following way.
From the equality
lm s m lm1 s m1 ... l1 s l0
s n mn1s n1 ... m1s m0
w( s )
wr s r wr 1s ( r 1) ...
(2.1.9)
we obtain
lm s m lm1 s m1 ... l1 s l0
( s n mn1s n1 ... m1s m0 )( wr s r wr 1s ( r 1) ...).
(2.1.10)
Comparing coefficients of the same powers of the variable s, we obtain
wr
lm , wr 1
lm1 mn1 wr , wr 2
lm2 mn1 wr 1 mn2 wr , ...
(2.1.11)
Using (2.1.11) for the function (2.1.8) we obtain the same result as the one yielded
by the polynomial division method.
From Definition 2.1.2 it follows that the function (2.1.1) is proper if and only if
it can be presented in the form
w( s )
w0 w1s 1 w 2 s 2 ...
(2.1.12)
and w0 z 0. It is a strictly proper function if and only if w0 = 0.
Proper and strictly proper functions have no poles at infinity.
The rational functions (2.1.2) are equal if and only if they satisfy the condition
(2.1.3). Rational functions of the form (2.1.7) are equal if and only if their
appropriate coefficients wk, k = r,r+1,… are equal.
A rational function of the form
w1 ( s ) w2 ( s )
l1 ( s )
l (s)
2
m1 ( s ) m2 ( s )
l1 ( s )m2 ( s ) l2 ( s )m1 ( s )
m1 ( s )m2 ( s )
(2.1.13)
is called the sum of two rational functions (2.1.2).
The sum of two rational functions of the form (2.1.7) is by definition equal to
the rational function whose coefficients wk, k = r,r+1,… are the sums of the
appropriate coefficients of these rational functions.
For example, the sum of the rational function (2.1.8) and of
w2 ( s )
2s 2 1
s2
is the rational function
2 s 4 9 s 1 18s 2 36 s 3 72s 4 ...
(2.1.14)
110
Polynomial and Rational Matrices
2 s 4 11s 1 21s 2 36s 3 61s 4 ...
w1 ( s ) w2 ( s )
(2.1.15)
The rational function
w1 ( s ) w2 ( s )
l1 ( s )l2 ( s )
m1 ( s )m2 ( s )
(2.1.16)
is called the product of two rational functions of the form (2.1.2).
The product of two rational proper functions of the form (2.1.12) and
w( s )
w0 w1 s 1 w2 s 2 ...
is by definition equal to
w( s ) w( s )
w0 w0 ( w0 w1 w1w0 ) s 1 ( w0 w2 w1w1 w2 w0 ) s 2 ...
f
k
¦¦ w w
i
k i
sk .
(2.1.17)
k 0 i 0
The product of two arbitrary functions of the form (2.1.7) is similarly defined. For
example, the product of the rational functions (2.1.8) and (2.1.14) is equal to the
following rational function
w1 ( s) w2 ( s)
(2s 1 3s 2 9s 4 ...)(2s 4 9s 1 18s 2 36s 3 ...)
4 14s 1 30s 2 ... .
It is easy to check that with thus defined operations addition and multiplication, the
set of rational function satisfies the conditions of the definition of a field. The
rational function 0/1 is the zero of this field, and the rational function 1/1 is the 1 of
this field (0 denotes the zero polynomial, i.e., a polynomial with zero coefficients
and 1 denotes a polynomial whose coefficients, with the exception of the one by s0
(which is 1), are equal to 0). As regards the set of causal rational functions with
coefficients of polynomials from a field , it constitutes a ring, which will be
denoted by p(s).
The 1 of this ring is constituted by proper rational functions of order r = 0.
A special case of proper rational functions are stable rational functions.
Definition 2.1.3. A proper rational function of the form (2.1.12) is called stable if
and only if the sequence of its coefficients w0,w1,w2,… converges to zero.
A proper rational function of the form (2.1.1) is stable if and only if its all poles
have moduli less than 1.
Stable rational functions with coefficients from the field constitute a ring,
which will be denoted by S(s). A unit of this ring consists of stable rational
functions that have zeros only inside of unit disk (|s|<1).
A special case of rational proper functions are finite causal functions.
Rational Functions and Matrices
111
Definition 2.1.4. A proper rational function of the form (2.1.12) is called finite if
and only if wk = 0 for k > n.
Thus a finite rational function has the form of a polynomial of the variable s-1
wn s n wn1s n1 ... w1s 1 w0 .
A proper rational function of the form (2.1.1) is finite if m(s) = sp.
The set of finite functions with coefficients from a field constitutes a ring,
which will be denoted by [s-1].
Let l(s) be a nonzero polynomial. Then a function of the form
m( s )
l ( s)
w 1 ( s )
(2.1.18)
is called the inverse function of the rational function (2.1.1).
The inverse function w-1(s) of the function (2.1.7) is of the form
w 1 ( s )
wˆ r s r ... wˆ 0 wˆ1s 1 ... wˆ r s r wˆ r 1s ( r 1) ... .
To compute its unknown coefficients
wˆ , ..., wˆ , wˆ , ..., wˆ , wˆ
, ...
r
0 1
r r 1
(2.1.19)
we use the
condition
w( s ) w1 ( s ) 1
(2.1.20)
and the principle (2.1.17) of rational function multiplication.
Substitution of (2.1.7) and (2.1.19) into (2.1.20) yields
f
§ f
i · §
j ·
¨ ¦ wi s ¸ ¨ ¦ wˆ j s ¸
©i r
¹ © j r
¹
f
f
¦ ¦ w wˆ s
i
j
(i j )
1.
(2.1.21)
i r j r
Equality of coefficients at the same powers of the variable s in (2.1.21) implies
r k
¦ w wˆ
q
q r
k q
­1 for k 0
®
¯0 for k 1, 2, ...
(2.1.22)
Solving the above equation system we compute desired coefficients of the function
(2.1.19).
Example 2.1.1.
For the rational function
w( s )
2s 1 3s 2 9s 4 ... ,
112
Polynomial and Rational Matrices
2s 1 3s 2 9s 4 ... ,
w( s )
the inverse function of the form
w 1 ( s )
wˆ 1s wˆ 0 wˆ 1s 1 wˆ 2 s 2 ...
is to be computed.
Equation (2.1.21) in this case takes the form
(2 s 1 3s 2 9s 4 ...)( wˆ 1s wˆ 0 wˆ1s 1 wˆ 2 s 2 ...) 1 .
Equality of coefficients of the same powers of the variable s implies
2wˆ 1
1, 2wˆ 0 3wˆ 1
0, 2 wˆ1 3wˆ 0
0, 2 wˆ 2 3wˆ1 9 wˆ 1
0
3
, wˆ 1
4
9
, wˆ 2
8
and
wˆ 1
1
, wˆ 0
2
3
wˆ 1
2
3
wˆ 0
2
3
9
wˆ1 wˆ 1
2
2
9
.
16
Thus the desired function is of the form
1
3 9
9
s s 1 s 2 ... .
2
4 8
16
w 1 ( s )
An arbitrary rational function in the form of the series (2.1.7) is given. With the
coefficients wk, k = r,r+1,… of this series known, compute the rational function in
the form of the product of the two polynomials (2.1.1), which corresponds to the
given function.
The solution of this problem follows by virtue of the following lemma.
Lemma 2.1.1. Let
w( s )
l ( s)
m( s )
ln1 s n1 ... l1 s l0
s n mn1s n1 ... m1s m0
w1s 1 w2 s 2 w3 s 3 ...
and
Tk
ª w1
«w
« 2
«#
«
¬ wk
w2
w3
#
wk 1
wk º
! wk 1 »»
, k
%
# »
»
" w2 k 1 ¼
!
1, 2, ...
(2.1.23)
Rational Functions and Matrices
113
If the polynomials l(s) and m(s) are relatively prime, then
­z 0 for k
®
¯ 0 for k
det Tk
1, 2, ..., n
n 1
.
(2.1.24)
Proof. To simplify computations we will consider in detail the case when n = 2.
For n > 2 considerations proceed similarly.
Division of l1s + l0 by s2 + m1s + m0 yields
l1s l0
s m1s m0
w1s 1 w2 s 2 w3 s 3 ...
2
(2.1.25)
where
w1
l1 z 0, w2
w4
m1w3 m0 w2 , w5
det T2
l0 m1w1 , w3
m1w2 m0 w1 ,
(2.1.26)
m1w4 m0 w3 ,...,
w1
w2
w1
w2
w2
w3
w2
m1w2 m0 w1
m0 w12 l0 w2
m0l12 l02 m1l0l1 z 0 ,
since by assumption a zero s0 = -l0/l1 of rational function is not equal to its pole,
2
l02 m1l0l1 m0l12
z 0.
l12
§ l0 ·
§ l0 ·
¨ ¸ m1 ¨ ¸ m0
© l1 ¹
© l1 ¹
We will show now that det T3 = 0. Indeed, using (2.1.26) and carrying out
appropriate elementary row operations, we obtain
det T3
w1
w2
w3
w1
w2
w3
w2
w3
w4
w2
w3
w4
w3
w4
w5
w3
m0 w2 m1w3
m0 w3 m1w4
w1
w2
w3
w2
w3
w4
m0 w1 m1w2 w3
0
0
0,
since m0w1+m1w2+w3 = 0. „
At the beginning we consider a case of strictly proper function of the form
w( s )
w1s 1 w2 s 2 ... .
(2.1.27)
114
Polynomial and Rational Matrices
Knowing the coefficients w1,w2,… of the function (2.1.27), we check the rank of
the symmetric matrix (2.1.23) successively for k = 1, 2, ... According to Lemma
2.1.1, if det Tk z 0 for k = 1, ..., n and det Tn+1 = 0, then the desired degree of the
denominator is n, i.e.,
s n mn1s n1 ... m1s m0
m( s )
(2.1.28)
and the numerator l(s) is a polynomial of a degree of at most n–1
ln1 s n1 ... l1s l0 ,
l ( s)
(2.1.29)
since the function (2.1.27) is strictly proper.
Division of (2.1.29) by (2.1.28) yields
w1s 1 w2 s 2 ... ,
w( s )
(2.1.30)
where
w1
ln1 , w2
ln2 ln1mn1 , w3
ln3 mn1 (ln2 ln1mn1 ), ...
(2.1.31)
Solving the following system of 2n equations
w1
ln1
w1 , w2
ln2 ln1mn1 w2 ,
w3
ln3 ln1mn2 mn1 (ln2 ln1mn1 ) w3 ,..., w2 n
w2 n
(2.1.32)
with respect to lk and mk for k = 0, 1, 2, ..., n–1, we compute the desired
polynomials (2.1.28) and (2.1.29).
Example 2.1.2.
The following strictly proper rational function is given
w( s )
2s 1 3s 2 9s 4 18s 5 ... .
Compute the corresponding function of the form (2.1.1).
In this case, the determinants of the matrices (2.1.23) are successively as
follows
det T1
det T3
w1
w2
2
3
w2
w3
3
0
w3
2
3
0
w3
w4
3
0
9
w4
w5
0
9
18
w1
2, det T2
w1
w2
w2
w3
0.
9,
Rational Functions and Matrices
115
Hence n = 2 and the desired polynomials are of the form
m( s )
s 2 m1s m0 , l ( s )
l1s l0 .
Using (2.1.32), we obtain the equations
l1
w1
2, l0 2m1
3, 2m0 (l0 2m1 )m1
0, (l0 2m1 )m0
9
whose solutions are l0 = 1, l1 = 2, m0 = 3, m1 = 2.
Thus the desired function is
w( s )
l ( s)
m( s )
2s 1
.
s 2s 3
2
This result is consistent with (2.1.8).
Now take into account an improper rational function of the form
w r s r ... w1s w0 w1s 1 w2 s 2 ... .
(2.1.33)
We decompose this function into the polynomial
q( s)
w r s r ... w1s w0
(2.1.34)
and the strictly proper function
w( s )
w1s 1 w2 s 2 ... .
(2.1.35)
Using the method presented above we compute the function in the form of the
quotient of the two polynomials, which corresponds to the strictly proper function
(2.1.35), and then we add to it the polynomial (2.1.34), i.e.,
w( s )
where
l ( s )
l ( s)
w r s r ... w1s w0
m( s )
r
m ( s )( w s ... w s w ) l ( s )
r
1
0
l ( s )
,
m( s )
.
Example 2.1.3.
With the given improper rational function
w( s )
2s 4 9s 1 18s 2 36 s 3 72s 4 ...
(2.1.36)
116
Polynomial and Rational Matrices
compute the corresponding function in the form of the quotient of two
polynomials.
In this case, q(s) = 2s – 4 and w( s ) =9s-1 – 18s-2 + 36s-3 - 72s-4
To compute a function of the form (2.1.1) corresponding to the strictly proper
function w ( s ) , we compute the determinants of the matrices (2.1.23) successively
for k = 1, 2, ...
We obtain
det T1
9, det T2
9
18
18
36
0.
Hence n = 1 and the desired polynomials are m(s) = s + m0, l(s) = l0.
Using (2.1.32), we obtain
l0
w1
9, m0l0
w2
18, m0
2.
The desired function, according to (2.1.36), is
w( s )
l (s)
q( s)
m( s )
9
2s 4
s2
2s 2 1
.
s2
The result is consistent with (2.1.14).
2.2 Decomposition of a Rational Function into a Sum of Rational
Functions
An arbitrary rational function (2.1.1) can be uniquely decomposed into a sum of
the strictly proper rational function r(s)/m(s) and the polynomial function q(s), i.e.,
w( s)
r ( s)
q( s) ,
m( s )
(2.2.1)
where deg r(s) < deg m(s).
To decompose the rational function (2.1.1) into a sum (2.2.1), we divide the
polynomial l(s) by m(s) and obtain
l s
q s m s r s ,
(2.2.2)
where q(s) and r(s) are the integer part and remainder, respectively, on division of
l(s) by m(s).
Substitution of (2.2.2) into (2.1.1) yields (2.2.1). If deg l(s) < deg m(s), then
q(s) is a zero polynomial and l(s) = r(s).
Rational Functions and Matrices
117
For example the rational function (2.1.14) can be decomposed into the strictly
proper rational function 9/(s + 2) and the polynomial 2s – 4, since
2s 2 1
s2
9
2s 4 .
s2
Consider a strictly proper rational function of the form
w( s )
l (s)
,
m1 ( s )m2 ( s )...m p ( s )
(2.2.3)
where the polynomials m1(s),m2(s),…,mp(s) are pair-wise relatively prime.
We will show that the rational function (2.2.3) can be uniquely decomposed
into a sum of strictly proper rational functions
lk ( s )
, k
mk ( s )
1, ..., p ,
i.e.,
w( s )
l ( s)
l1 ( s )
l ( s)
,
2
... p
m1 ( s ) m2 ( s )
m p (s)
(2.2.4)
where deg lk(s) < deg mk(s) for k = 1, ..., p.
To simplify the considerations assume that p = 2. Then from (2.2.3) and
(2.2.4), we obtain
l ( s)
m1 ( s )m2 ( s )
l1 ( s )
l ( s)
2
m1 ( s ) m2 ( s )
l1 ( s )m2 ( s ) l2 ( s )m1 ( s )
.
m1 ( s )m2 ( s )
(2.2.5)
Let
l (s)
m2 ( s )
ln1s n1 ln2 s n2 ... l1s l0 , m1 ( s )
s n1 an1 1s n1 1 ... a1s a0 ,
s n2 bn2 1s n2 1 ... b1s b0 ,
l1 ( s )
cn1 1s n1 1 cn1 2 s n1 2 ... c1s c0 ,
l2 ( s )
d n2 1s n2 1 d n2 2 s n2 2 ... d1s d 0 , n
(2.2.6)
n1 n2 .
The equality (2.2.5) yields
l (s)
l1 ( s )m2 ( s ) l2 ( s )m1 ( s )
(2.2.7)
118
Polynomial and Rational Matrices
and substitution of (2.2.6) into (2.2.7) produces
ln1s n1 ln2 s n2 ... l1s l0
(cn1 1s n1 1 cn1 2 s n1 2 ... c1s c0 )
u( s n2 bn2 1s n2 1 ... b1s b0 ) (d n2 1s n2 1 d n2 2 s n2 2 ... d1s d 0 ) (2.2.8)
u( s n1 an1 1s n1 1 ... a1s a0 ).
Comparing the coefficients at the same powers of the variable s in (2.2.8), we
obtain a system of n linear equations of the form
cn1 1 d n2 1
ln1 ,
cn1 2 cn1 1bn2 1 d n2 2 d n2 1an1 1
ln 2 ,
cn1 3 cn1 1bn2 1 cn1 1bn2 2 d n2 3 d n2 2 an1 1 d n2 1an1 2 ,
c1b0 c0b1 d1a0 d 0 a1
c0b0 d 0 a0
(2.2.9)
l1 ,
l0 .
It is easy to check that if m1(s), m2(s) are pair-wise relatively prime, then the matrix
of the coefficients of the system (2.2.9) is nonsingular.
Hence the system has exactly one solution with respect to the desired
coefficients ck, k = 0,2,…,n11 of the polynomial l1(s) and the coefficients dk,
k = 0,1,…,n21 of the polynomial l2(s) for the given coefficients ai, i = 0,1,…,n11,
bj, j = 0,1,…,n21 and lk, k = 0,1,…,n1 of the polynomials m1(s), m2(s), l(s).
Example 2.2.1.
Decompose the rational function
w( s )
l3 s 3 l2 s 2 l1s l0
( s a1s a2 )( s 2 b1s b2 )
2
(2.2.10)
(l0 , l1 , l2 , l3 , a1 , a2 , b1 , b2 given)
into a sum of strictly proper rational functions. In this case,
l3 s 3 l2 s 2 l1s l0
( s 2 a1s a2 )( s 2 b1s b2 )
x s x4
x1s x2
2 3
.
s a1s a2 s b1s b2
2
(2.2.11)
From (2.2.11) we have
l3 s 3 l2 s 2 l1s l0
( x1 s x2 ) s 2 b1s b2 ( x3 s x4 ) s 2 a1 s a2 .
Comparing the coefficients of the same powers of the variable s, we obtain
(2.2.12)
Rational Functions and Matrices
l3
x1 x3 , l2
x1b1 x2 x3 a1 x4 , l1
l0
x2b2 x4 a2 .
x1b2 x2b1 x3 a2 x4 a1 ,
119
(2.2.13)
Equation (2.2.13) can be written in the form
ª1
«b
« 1
«b2
«
¬0
0
1
b1
b2
1
a1
a2
0
0 º ª x1 º
1 »» «« x2 »»
a1 » « x3 »
»« »
a2 ¼ ¬ x4 ¼
ªl3 º
«l »
« 2» .
« l1 »
« »
¬l0 ¼
(2.2.14)
The matrix of coefficients
A
ª1
«b
« 1
«b2
«
¬0
0
1
b1
b2
1
a1
a2
0
0º
1 »»
a1 »
»
a2 ¼
is nonsingular if a1 z b1, a2 z b2 (the polynomials s2+a1s+a2, s2+b1s+b2 are
relatively prime). In this case,
det A
(a2 b2 ) 2 b2 (a1 b1 ) 2 b1 (a1 b1 )(a2 b2 ) .
Solving (2.2.14), we obtain
1
ª x1 º ª 1 0 1 0 º ªl3 º
« x » « b 1 a 1 » «l »
1
1
« 2» « 1
» « 2»
« x3 » «b2 b1 a2 a1 » « l1 » a22 b1a1a2 b2 a12 2b2 a2 b12 a2 b1b2 a1 b22
« » «
» « »
¬ x4 ¼ ¬ 0 b2 0 a2 ¼ ¬l0 ¼
ª a22 b1a1a2 b2 a12 b2 a2
º ª l3 º
b1a2 b2 a1
a2 b2
a1 b1
«
»« »
2
a
b
a
b
)
(
)
(
)
(
)
(
a
b
a
b
a
a
a
b
a
a
b
a
2
1
1
2
2
1
2
2
1
2
2
2
2
1
1
1
» « l2 » .
u«
« (b12 a2 b1b2 a1 b2 a2 b22 ) (b1a2 b2 a1 )
» « l1 »
(a1 b1 )
a2 b2
«
»« »
b2 (a2 b2 )
b2 (b1a2 b2 a1 )
b2 (a1 b1 )
a2 b1a1 b12 b2 ¼» ¬l0 ¼
«¬
Thus the desired decomposition is of the form
w( s )
x3 s x4
x1s x2
.
s 2 a1s a2 s 2 b1s b2
Example 2.2.2.
Decompose the strictly proper rational function
120
Polynomial and Rational Matrices
w( s )
s 2 3s 2
( s 2 3s 2)( s 3)
(2.2.15)
into a sum of two strictly proper rational functions. In this case,
s 2 3s 2, m1 ( s )
l (s)
s 2 3s 2, m2 ( s )
s 3.
According to (2.2.5) and (2.2.6) we seek
l1 ( s )
d.
c1s c0 , l2 ( s )
(2.2.16)
Equation (2.2.7) in this case takes the form
s 2 3s 2
(c1s c0 )( s 3) d ( s 2 3s 2) .
(2.2.17)
Comparing the coefficients at the same powers of the variable s, we obtain the
equations
c1 d
1, c0 3c1 3d
3, 3c0 2d
2,
whose solution is c0 = 6, c1 = 9, d = 10.
Thus the desired decomposition of the function (2.2.15) has the form
s 2 3s 2
( s 3s 2)( s 3)
2
9s 6
10
.
s 2 3s 2 s 3
Consider a strictly proper rational function of the form
w( s )
l ( s)
,
m( s )
m( s )
( s s1 ) n1 ( s s2 ) n2 ...( s s p ) p ,
(2.2.18)
where
n
p
¦n
i
n
deg m( s ) ! deg l ( s ),
(2.2.19)
i 1
and s1,s2,…,sp are distinct poles of the function (2.2.18) with multiplicities
n1,n2,…,np, respectively.
The function (2.2.18) is a special case of the function (2.2.3) for
mk ( s )
s sk
nk
, k
1,..., p .
Rational Functions and Matrices
121
The strictly proper rational function
lk ( s)
, k
( s sk ) nk
1, ..., p ,
can be further decomposed and represented uniquely in the form
lk ( s)
( s sk ) nk
nk
lki
i 1
k
¦ (s s )
nk i 1
.
(2.2.20)
Using the decomposition (2.2.4) of the functions (2.2.18) and (2.2.20), we obtain
w( s )
l ( s)
m( s )
p
p
lk ( s)
¦ (s s )
k 1
k
nk
lki
¦¦ (s s )
nk
k 1 i 1
k
nk i 1
,
(2.2.21)
where the coefficients lki are given by the formula
lki
1 w i 1 l ( s )( s sk ) nk
(i 1)! ws i 1
m( s )
s sk
.
(2.2.22)
This formula can be derived in the following way.
Multiplication of (2.2.21) by ( s sk ) nk yields
l ( s )( s sk ) nk
m( s )
lknk ( s sk )
nk 1
l11
( s sk ) nk
( s sk ) nk
... lm1
lk1 lk 2 ( s sk ) ...
n1
( s s1 )
( s s1 )
l p1
( s sk ) nk
(s s p )
np
... l pn p
( s sk ) nk
.
(s s p )
From (2.2.23) for s = sk we successively obtain
lk 1
l ( s)( s sk ) nk
m( s )
lk 3
1 w 2 § l ( s)( s sk ) nk ·
¨
¸
2 ws 2 ©
m( s )
¹
s sk
, lk 2
w § l ( s )( s sk ) nk ·
¨
¸
m( s )
ws ©
¹
s sk
, ... ,
that is the formula (2.2.22).
Example 2.2.3.
Decompose the strictly proper rational function
s sk
,
(2.2.23)
122
Polynomial and Rational Matrices
w( s )
s 2 3s 2
( s 1) 2 ( s 2)
(2.2.24)
into the sum (2.2.21). In this case, l(s) = s2 + 3s + 2, m(s) = (s - 1)2(s - 2) and
s 2 3s 2
( s 1) 2 ( s 2)
l11
l
l
12 21 .
( s 1) 2 s 1 s 2
Using (2.2.22), we obtain
l11
l ( s )( s 1) 2
( s 1) 2 ( s 2)
l12
w § l (s) ·
¨
¸
ws © s 2 ¹
l21
l ( s )( s 2)
( s 1) 2 ( s 2)
s 1
l ( s)
s2
s 2 3s 2
s2
s 1
s 1
(2s 3)( s 2) ( s 2 3s 2)
( s 2) 2
s 1
s 2
l ( s)
( s 1) 2
s 2
6,
s 1
11,
12.
Thus the desired decomposition of the function (2.2.24) has the form
s 2 3s 2
( s 1) 2 ( s 2)
6
11
12
.
( s 1) 2 s 1 s 2
Now consider an improper rational function of the form
w( s )
l ( s)
, deg l ( s ) ! deg m1 ( s ) deg m2 ( s ) ,
m1 ( s )m2 ( s )
(2.2.25)
where m1(s), m2(s) are relatively prime.
Separating from the function (2.2.25) the polynomial part q(s) (according to the
decomposition (2.2.1)), we obtain
w( s )
l (s)
q( s) ,
m1 ( s )m2 ( s )
(2.2.26)
where deg m1(s)+deg m2(s)> deg l (s).
Using (2.2.5), one can write the function (2.2.26) in the form
w( s )
l1 ( s )
l ( s)
2
q( s) ,
m1 ( s ) m2 ( s )
(2.2.27)
Rational Functions and Matrices
123
where deg m1(s) < deg l 1(s), deg m2(s) > deg l 2(s).
Let p(s) be an arbitrary polynomial. The function (2.2.27) can be then written in
the form
w( s )
§ l1 ( s )
· § l (s)
·
p( s) ¸ ¨ 2
q(s) p( s) ¸
¨
© m1 ( s )
¹ © m2 ( s )
¹
l1 ( s )
l 1 ( s ) m1 ( s ) p ( s ), l2 ( s )
l1 ( s )
l (s)
, (2.2.28)
2
m1 ( s ) m2 ( s )
where
l2 ( s ) m2 ( s )(q ( s ) p( s )) .
The decomposition (2.2.25) is thus not unique, since the polynomial p(s) is an
arbitrary one.
If we separate from functions l1(s)/m1(s), l2(s)/m2(s) the polynomial parts q1(s)
and q2(s), respectively, we obtain
w( s )
l1 ( s )
l ( s )
q1 ( s ) 2
q2 ( s ) ,
m1 ( s )
m2 ( s )
(2.2.29)
where deg m1(s) > deg l 1(s), deg m2(s) > deg l 2(s).
From comparison of (2.2.27) and (2.2.29) it follows that uniqueness of the
decomposition holds for l1 ( s ) l1 ( s ), l2 ( s ) l2 ( s ) and q(s) = q1(s)+q2(s).
Taking p(s) = 0 in (2.2.28) one can represent the function (2.2.25) as a sum
w( s )
w1 ( s ) w2 ( s ), w1 ( s )
l1 ( s)
, w2 ( s )
m1 ( s )
l2 ( s )
q(s) ,
m2 ( s )
w( s )
w1 ( s ) w2 ( s ), w1 ( s )
l1 ( s )
q ( s ), w2 ( s )
m1 ( s )
(2.2.30)
or
l2 ( s )
.
m2 ( s )
(2.2.31)
The decomposition (2.2.30) is called the minimal decomposition of the function
(2.2.25) with respect to m1(s), and the decomposition (2.2.31) is called the minimal
decomposition of the function (2.2.25) with respect to m2(s).
Using the decomposition (2.2.4), one can generalise these considerations to the
case of a rational function of the form (2.2.3).
124
Polynomial and Rational Matrices
2.3 Basic Definitions and Operations on Rational Matrices
A matrix W(s) with m rows and n columns whose entries are rational functions
wij(s) of a variable s with coefficients from a field
W( s)
ª w11 ( s ) w12 ( s )
« w (s) w (s)
22
« 21
« #
#
«
w
(
s
)
w
m 2 (s)
¬ m1
! w1n ( s ) º
! w2 n ( s ) »»
%
# »
»
! wmn ( s ) ¼
(2.3.1)
is called a rational matrix.
The set of rational matrices of dimensions mun of a variable s and with
coefficients from a field will be denoted by mun(s). A field can be the field of
real numbers , of complex numbers , of rational numbers
or of a field of
rational functions of another variable z, etc.
With all the entries wij(s) of the matrix (2.3.1) brought to the common
denominator m(s) with the coefficient at the highest power of s equal to 1, the
matrix can be expressed in the form
W( s)
L( s )
,
m( s )
(2.3.2)
where L(s) mun[s] is a polynomial matrix with coefficients from the field , and
m(s) is a polynomial.
Let
m( s )
n
( s s1 ) n1 ( s s2 ) n2 " ( s s p ) p ,
p
¦n
i
n.
(2.3.3)
i 1
Definition 2.3.1. The matrix (2.3.2) is called irreducible if and only if
L( sk ) z 0mn , k
1, ..., p ,
(2.3.4)
where 0mn is a zero matrix of size mun.
If L(sk) = 0mn, then all entries of the matrix L(s) are divisible by (s-sk) and the
matrix (2.3.2) is reducible by (s-sk).
An irreducible matrix of the form (2.3.2) is called a matrix of standard form.
With the polynomial matrix L(s) expressed as the matrix polynomial
L( s )
L q s q L q 1s q1 ... L1s L 0 ,
we can write the matrix (2.3.2) in the form
(2.3.5)
Rational Functions and Matrices
W( s)
L q s q L q1s q 1 ... L1s L 0
m( s )
.
125
(2.3.6)
For example, for the following rational matrix
W( s)
ª s
« s 1
«
« 2
«¬ s 2
1
s2
s2
s 1
º
s»
»
»
2s »
¼
(2.3.7)
the least common denominator of its entries is the polynomial m(s) = (s+1)(s+2),
whose roots are s1 = 1, s2 = 2.
The rational matrix (2.3.7) of the form (2.3.2) is equal to
W( s)
s 1
ª s( s 2)
1
«
( s 1)( s 2) ¬ 2( s 1) ( s 2) 2
s ( s 1)( s 2) º
2s ( s 1)( s 2) »¼
L( s )
. (2.3.8)
m( s )
This matrix is irreducible, since
L( s1 )
ª 1 0 0 º
« 0 1 0 » , L ( s2 )
¬
¼
ª 0 1 0 º
« 2 0 0 » .
¬
¼
The form (2.3.8) is thus the standard form of the matrix (2.3.7).
The matrix L(s) expressed as a matrix polynomial is equal to
L( s )
ª 0 0 1 º 3 ª1 0 3 º 2 ª 2 1 2 º
ª 0 1 0º
«0 0 2 » s «0 1 6 » s « 2 4 4 » s « 2 4 0 » .
¬
¼
¬
¼
¬
¼
¬
¼
In view of this, the matrix (2.3.7) in the form (2.3.6) is equal to
W( s)
1
( s 1)( s 2)
ª 0 1 0 º °½
°­ ª 0 0 1 º 3 ª1 0 3º 2 ª 2 1 2 º
s «
s «
s«
u ®«
»
»
»
»¾.
°¯ ¬ 0 0 2 ¼
¬0 1 6 ¼
¬ 2 4 4¼
¬ 2 4 0 ¼ °¿
(2.3.9)
Definition 2.3.2. The rational matrix (2.3.2) is called proper (or causal) if and only
if deg m(s) t deg L(s) and strictly proper (or strictly causal) if and only if deg
m(s) > deg L(s).
The matrix (2.3.7) is not a proper one, since as it follows from (2.3.9),
deg L(s) = 3 and deg m(s) = 2.
126
Polynomial and Rational Matrices
Dividing every entry of the matrix L(s) by m(s), one can express the rational
matrix (2.3.2) in the form
W( s)
Wr s r Wr 1s ( r 1) ... ,
(2.3.10)
where r = deg m(s) – deg L(s) is a matrix rank, and Wr,Wr+1,… are matrices of
coefficients and depend on the coefficients of the polynomial m(s) and the
polynomial matrix L(s).
For example, taking into account
s
1
1 s 1 s 2 s 3 ...,
s 1
s2
s2
1
2
3
1 s s s ...,
s 1
the matrix (2.3.7) can be written in the form
ª s
« s 1
«
« 2
«¬ s 2
ª0 0
«0 0
¬
s 1 2 s 2 4s 3 ...,
1
º
s»
s2
»
s2
»
2s »
s 1
¼
1º
ª1 0 0 º ª 1 1 0 º 1 ª 1 2 0º 2
s«
»«
» s « 4 1 0» s ... .
2 »¼
¬0 1 0 ¼ ¬ 2 1 0 ¼
¬
¼
(2.3.11)
In this case, r = –1.
The sum (difference) and the product of rational matrices are defined
analogously to the sum (difference) and the product, respectively, of two rational
functions.
Using (2.3.10), it is easy to show that the sum, the difference and the product of
two strictly proper matrices are themselves strictly proper matrices.
The set of proper (causal) rational matrices of the variable s, with coefficients
from a field and of dimensions mun will be denoted by pmun(s). The entries of
these matrices belong to the ring p(s). Thus we can define a p(s)-unimodular
matrix as a nonsingular matrix whose determinant is a unit of the ring p(s).
Definition 2.3.3. The following operations are called p(s)-elementary operations
on the rows and on the columns of a matrix, respectively
1. Multiplication of the i-th row (column) by a unit of the ring p(s), w(s).
This operation will be denoted by L[iuw(s)] (P[iuw(s)]).
2. Addition to the i-th row (column) of the j-th row (column) multiplied by
an arbitrary proper (causal) rational function w ( s ) . This operation will be
denoted by L[i+iu w ( s ) ] (P[i+iu w ( s ) ]).
3. The interchange of two arbitrary rows (columns) i, j. This operation will
be denoted by L[i, j] (P[i, j]).
Rational Functions and Matrices
Analogously to polynomial matrices, we can define the
proper rational matrices.
127
p(s)-equivalence
of
Definition 2.3.4. Two proper rational matrices W1(s) and W2(s) of the same
dimensions are called p(s)-equivalent if and only if there exist p(s)-unimodular
matrices Lp(s) and Pp(s) such that
W1 ( s )
L p ( s ) W2 ( s )Pp ( s ) .
(2.3.12)
Matrices Lp(s) and Pp(s) are the products of matrices of p(s)-elementary
operations on rows and columns, respectively.
With p(s)-equivalence, every proper rational matrix W(s) pmun(s) can be
converted to the Smith canonical form
WS ( s )
diag ª¬ s d1 , s d2 ,..., s dr , 0,..., 0 º¼  pmun ( s ) ,
(2.3.13)
where d1dd2d…ddJ are nonnegative integers, uniquely determined by the matrix
W(s) and r = rank W(s).
For example, the proper rational matrix
W( s)
ª s
«s 1
«
« 1
«¬ s 1
can be converted by
WS ( s )
s º
s2 »
»
1 »
s ( s 2) »¼
p(s)-equivalence
ª1 0 º
«0 s 1 » (d1
¬
¼
into
0, d 2
1)
by the following p(s)-elementary operations on rows: L[2+1u(-1/s)], L[1u(s+1)/s]
and on columns: P[2+1u-(s+1)/(s+2)], P[2u(s+2)/(1-s)]. The matrices of p(s)
elementary operations are:
L p ( s)
ªs 1 º
0»
« s
«
» , Pp ( s )
« 1 1»
«¬ s
»¼
s 1º
ª
«1 1 s »
«
».
«0 s 2 »
«¬
1 s »¼
Definition 2.3.4. A rational matrix whose entries are stable proper (causal)
functions is called a stable matrix.
128
Polynomial and Rational Matrices
The set of stable matrices with coefficients from a field and of dimensions
mun will be denoted by Smun(s). This set is a subset of the set of proper rational
matrices pmun(s).
A matrix whose elements are finite proper functions is called a finite rational
matrix. The set of finite rational matrices with coefficients from a field and of
dimensions mun will be denoted by mun[s-1].
Consider the rational function
w( s )
l ( s)
,
m( s )
(2.3.14)
such that l(s) and m(s) are relatively prime elements of one of the rings p(s), S(s),
[s-1].
Analogously, one can define the set of rational matrices whose entries are
rational functions of the form (2.3.14).
2.4 Decomposition of Rational Matrices into a Sum of Rational
Matrices
An arbitrary rational matrix of the form (2.3.1) can be decomposed into a sum of a
strictly proper rational matrix R(s)/m(s) and of a polynomial matrix Q(s), i.e.,
W( s)
R ( s)
Q( s ) ,
m( s )
(2.4.1)
where deg m(s) > deg R(s).
In order to decompose the rational matrix (2.3.2) into the sum (2.4.1), we
divide every entry lij(s) of the matrix L(s) by m(s)
lij ( s )
q ij ( s)m( s ) rij ( s ), i 1, ..., m; j 1, ..., n ,
(2.4.2)
where qij(s) and rij(s) are the integer part and the remainder of division,
respectively.
Substituting (2.4.2) into (2.3.2) and defining Q(s) = [qij(s)], R(s) = [rij(s)], we
obtain (2.4.1).
If deg L(s) < deg m(s), then Q(s) is a zero matrix and R(s) = L(s).
A strictly proper rational matrix of the form
W( s)
L( s )
m1 ( s )m2 ( s )...m p ( s )
(2.4.3)
where the polynomials m1(s),m2(s),…,mp(s) are pair-wise relatively prime, is taken
into account.
Rational Functions and Matrices
129
The matrix (2.4.3) can be uniquely decomposed into the sum of p strictly
proper rational matrices
L k ( s)
, k
mk ( s )
1, ..., p ,
i.e.,
W( s)
L ( s)
L1 ( s ) L 2 ( s )
,
... p
m1 ( s ) m2 ( s )
m p (s)
(2.4.4)
where deg mk(s) > deg Lk(s), k = 1,…,p.
In order to carry out the decomposition (2.4.4), one has to apply to every
element lij(s) of the matrix L(s) the procedure introduced in point 2.
Consider the strictly proper matrix (2.3.2) for m(s) of the form (2.2.19).
This matrix is a special case of the matrix (2.4.3) for
mk ( s )
s sk
nk
, k
1,..., p .
The strictly proper rational matrix
Lk (s)
, k
( s sk ) nk
1, ..., p
may be further uniquely decomposed into the form
Lk (s)
( s sk ) nk
nk
L ki
i 1
k
¦ (s s )
nk i 1
, k
1, ..., p .
(2.4.5)
The decomposition (2.4.5) applied to every term of the sum (2.4.4) yields
W( s)
L( s )
m( s )
p
nk
L ki
¦¦ ( s s )
k 1 i 1
k
nk i 1
,
(2.4.6)
where the matrices Lki of the coefficients are given by the formula
L ki
1 w i 1 L( s )( s sk ) nk
(i 1)! ws i 1
m( s )
s sk
, k
1, ..., p; i 1, ..., nk .
(2.4.7)
This formula follows from application of (2.2.22) to every entry of the matrix L(s).
130
Polynomial and Rational Matrices
Example 2.4.1.
Decompose the rational matrix
W( s)
ª s
« ( s 1) 2
«
« 2
«¬ s 2
1 º
s 2»
».
4 »
s 1 »¼
(2.4.8)
We write the matrix in the form (2.4.3)
W( s)
L( s )
,
m1 ( s )m2 ( s )
m1 ( s )
( s 1) 2 , m2 ( s )
(2.4.9)
where
s 2, L( s )
ª s ( s 2)
«
2
¬ 2( s 1)
( s 1) 2
º
».
4( s 1)( s 2) ¼
We want to decompose the matrix (2.4.8) into the form
W( s)
L11
L
L
12 21 .
( s 1) 2 s 1 s 2
(2.4.10)
Using (2.4.7), we obtain
L11
L12
ª 1 0 º
« 0 0» ,
¬
¼
1
dL ( s )
d ( s 2)
( s 2) L( s )
w L( s )
ds
ds
( s 2) 2
ws s 2 s 1
L( s )
s2s
ª1 0 º
«0 4 » ,
¬
¼
(2.4.11)
s 1
L 21
L( s )
( s 1) 2
s 2
ª0 1º
«2 0» .
¬
¼
Substitution of (2.4.11) into (2.4.10) yields the desired decomposition of the matrix
(2.4.8)
W( s)
1 ª 1 0 º
1 ª1 0 º
1 ª0 1º
.
»
«
»
2 «
( s 1) ¬ 0 0 ¼ s 1 ¬ 0 4 ¼ s 2 «¬ 2 0»¼
(2.4.12)
Rational Functions and Matrices
131
Now an improper rational matrix of the form
W( s)
L( s )
m1 ( s )m2 ( s )
(2.4.13)
is taken into account where deg m1(s)+deg m2(s) < deg L(s), and polynomials
m1(s), m2(s) are relatively prime.
Separating from the matrix (2.4.13) the polynomial part Q(s) mun[s]
(according to the decomposition (2.4.1)), we obtain
W( s)
L( s )
Q( s ), deg L( s ) deg m1 ( s ) deg m2 ( s ) . (2.4.14)
m1 ( s )m2 ( s )
With the decomposition (2.4.4) applied, the matrix (2.4.14) can be written in the
form
W( s)
L1 ( s ) L 2 ( s )
Q( s ) ,
m1 ( s ) m2 ( s )
(2.4.15)
where deg m ( s ) ! deg L1( s ) and deg m ( s) ! deg L2 ( s ) .
1
2
Addition to and subtraction from the right-hand side of (2.4.15) of an arbitrary
polynomial matrix P(s) mun[s] yields
W( s)
§ L1 ( s )
· § L 2 (s)
·
P(s) ¸ ¨
Q( s ) P ( s ) ¸
¨
© m1 ( s )
¹ © m2 ( s )
¹
L1 ( s )
L1 ( s ) m1 ( s )P ( s ), L 2 ( s )
L1 ( s ) L 2 ( s )
, (2.4.16)
m1 ( s ) m2 ( s )
where
L 2 ( s ) m2 ( s ) >Q( s ) P( s ) @ .
Thus the decomposition (2.4.16) of the matrix (2.4.13) is not unique.
If we separate from the improper matrices
L k ( s)
, k
mk ( s )
1, ..., p
the polynomial parts Q1(s) and Q2(s), respectively, we obtain
W( s)
L 1 ( s)
L ( s )
Q1 ( s) 2
Q2 (s) ,
m1 ( s )
m2 ( s )
(2.4.17)
132
Polynomial and Rational Matrices
where
deg m1 ( s ) ! deg L 1 (s ) and deg m2 (s ) ! deg L 2 (s ) .
A comparison of (2.4.17) to (2.4.15) implies that the uniqueness of the
decomposition holds for
L 1 ( s )
L1 ( s ), L 2 ( s )
L 2 ( s ) and Q( s )
Q1 ( s ) Q 2 ( s ) .
Taking P(s) = 0 in (2.4.16), one can express the matrix (2.4.14) as the sum
L 2 (s)
Q( s ) , (2.4.18)
m2 ( s )
W( s)
W1 ( s ) W2 ( s ); W1 ( s )
L1 ( s )
, W2 ( s )
m1 ( s )
W( s)
W1 ( s ) W2 ( s ); W1 ( s )
L1 ( s )
Q( s ), W2 ( s )
m1 ( s )
or
L2 (s)
. (2.4.19)
m2 ( s )
The decomposition (2.4.18) is called the minimal decomposition of the matrix
(2.4.13) with respect to m1(s) and the decomposition (2.4.19) is called the minimal
decomposition of the matrix (2.4.13) with respect to m2(s).
Using the decomposition (2.4.4) one can generalise the above considerations to
the case of, rational matrix of the form (2.4.3).
2.5 The Inverse Matrix of a Polynomial Matrix and Its
Reducibility
Consider an invertible (nonsingular) polynomial matrix A(s) nun[s]. Its inverse
matrix is the rational matrix A-1(s) nun(s).
Let U(s), V(s) nun[s] be unimodular matrices of elementary operations on
rows and columns, respectively, that convert this polynomial matrix into the Smith
canonical form AS(s), i.e.,
A S (s)
U( s) A( s )V ( s)
diag >i1 ( s ), i2 ( s ), ..., in ( s ) @ ,
(2.5.1)
where ik(s), k = 1,…,n are the monic invariant polynomials, satisfying the
divisibility condition ik(s) | ik+1(s) for k = 0, 1, ..., n1.
From (2.5.1), we have
A( s )
U 1 ( s ) A S ( s )V 1 ( s )
U 1 ( s )diag >i1 ( s ), i2 ( s), ..., in ( s ) @ V 1 ( s) , (2.5.2)
Rational Functions and Matrices
133
where the inverse matrices U-1(s), V-1(s) are also unimodular ones.
Thus with the following relationships applied, the inverse of the matrix (2.5.2)
can be computed as
A 1 ( s)
ª¬ U 1 ( s ) A S ( s)V 1 ( s ) º¼
1
V ( s ) A S 1 ( s ) U ( s )
1
V ( s )diag >i1 ( s ), i2 ( s),..., in ( s) @ U( s)
V(s)
(2.5.3)
Adj > diag (i1 ( s ), i2 ( s),..., in ( s)) @
U( s),
i1 ( s)i2 ( s)...in ( s)
where the adjoint matrix is of the form
Adj > diag[i1 ( s ), i2 ( s ), ..., in ( s )]@
diag >i2 ( s )i3 ( s )...in ( s ), i1 ( s )i3 ( s )...in ( s ), ..., i1 ( s )i2 ( s )...in1 ( s ) @ .
(2.5.4)
Note that in the general case, reductions will take place in the inverse matrix A-1(s),
since for certain roots of the invariant polynomials, the adjoint matrix (2.5.4) is
equal to a zero matrix. On the other hand, if
i1 ( s )
i2 ( s ) ... in1 ( s ) 1 ,
(2.5.5)
then the matrix (2.5.4) takes the form
Adj > diag [i1 ( s ), i2 ( s ),..., in ( s )]@ diag >in ( s ), in ( s ),..., in ( s ),1@
(2.5.6)
and for all roots of the invariant polynomial in(s) it is a nonzero matrix. In this case,
there are no reductions in the inverse matrix A-1(s). The condition (2.5.5) is also a
necessary one for occurrence of reductions in the matrix A-1(s). If this condition is
not satisfied, the invariant polynomials i1(s),i2(s),…,in-1(s) have at least one
common root. For this root, the adjoint matrix (2.5.4) is equal to zero and the
reduction of this root occurs in the matrix A-1(s). In this way, the following
theorem has been proved.
Theorem 2.5.1. There are no reductions in the inverse matrix A-1(s) if and only if
the polynomial matrix A(s) is a simple matrix. i.e., the condition (2.5.5) is satisfied,
or equivalently, the characteristic polynomial is identical with the minimal
polynomial of this matrix.
Thus the inverse A-1(s) of the simple matrix A(s) is of the form
A 1 ( s)
V ( s )diag ª¬1, 1, ..., 1, in1 ( s) º¼ U( s) .
(2.5.7)
134
Polynomial and Rational Matrices
Example 2.5.1.
Compute the inverse of the following polynomial matrix
A( s)
ª s 1 ( s 1) 2 ( s 2) 2
«
2
¬ 2( s 1)( s 2)
( s 1) 2 ( s 2) º
»
2( s 1)( s 2) ¼
(2.5.8)
and check whether any reductions occur in the inverse matrix.
In this case,
A( s)
0
0º
ª1 s 1º ª s 1
ºª 1
.
«0
»
«
»
«
2 ¼¬ 0
( s 1)( s 2) ¼ ¬ s 2 1 »¼
¬
Hence
A S (s)
0
ªs 1
º
1
« 0
» , U (s)
(
s
1)(
s
2)
¬
¼
ª1 s 1º
1
«0
» , V (s)
2
¬
¼
0º
ª 1
« s 2 1» .
¬
¼
The matrix (2.5.8) is not simple, since i1(s) = s+1 z 1 and the reductions by (s+1)
will take place in the inverse matrix A-1(s).
Using (2.5.3) and (2.5.4), we obtain
A 1 ( s )
V ( s)
diag > ( s 1)( s 2), s 1@
U( s)
( s 1) 2 ( s 2)
0º
ª 1
ª 1
« ( s 2) 1 » diag « s 1
¬
¬
¼
ª 1
« s 1
«
« s2
« s 1
¬
1
ª
º
1 ( s 1) »
º«
1
2
«
»
1
( s 1)( s 2) »¼ «
»
0
2
¬«
¼»
1
º
»
2
».
2
( s 1)( s 2) 1 »
2( s 1)( s 2) »¼
Example 2.5.2.
Show that the matrix [Ins – A]-1 is not reducible for any coefficients a0,a1,…,an-1 of
the matrix
A
ª 0
« 0
«
« #
«
« 0
«¬ a0
1
0
0
1
#
0
a1
#
0
a2
0 º
0 »»
!
# ».
»
1 »
%
! an1 »¼
!
!
(2.5.9)
Rational Functions and Matrices
135
Taking into account that for the matrix (2.5.9)
s
0
1 0 !
s 1 !
0
0
0
0
#
0
#
0
#
0
#
s
#
1
a0
a1
a2 ! an2
det > I n s A @
n
s an1s
n 1
%
!
s an1
... a1s a0 ,
we obtain
1
>I n s A @
Adj > I n s A @
det > I n s A @
ª*
«*
1
«
s n an1s n1 ... a1s a0 « #
«
¬*
* ! * 1º
* ! * *»»
,
# % # #»
»
* ! * *¼
(2.5.10)
where * stands for an entry that does not matter in the considerations.
It follows from (2.5.10) that the matrix [Ins – A]-1 is irreducible, since the entry
(1, n) of the adjoint matrix is equal to 1.
Example 2.5.3.
Show that the matrix [Is – A]-1 is irreducible if and only if the entry a of the matrix
A
ª1 1 0 º
«0 1 0 »
«
»
«¬ 0 0 a »¼
(2.5.11)
is different from 1.
Computation of the inverse [Is – A]-1 of the matrix (2.5.11) yields
1
> Is A @
0 º
ª s 1 1
« 0
s 1
0 »»
«
«¬ 0
s a »¼
0
1
sa
0 º
ª( s 1)( s a )
1
«
0
( s 1)( s a )
0 »» .
( s 1) 2 ( s a) «
«¬
0
0
( s 1) 2 »¼
(2.5.12)
136
Polynomial and Rational Matrices
From (2.5.12) it follows that the matrix [Is – A]-1 for the matrix (2.5.11) is
irreducible if and only if a z 1.
Using elementary operations it is easy to show that for a = 1
> Is A @ S
0 º
ª s 1 1
« 0
s 1 0 »»
«
«¬ 0
0
s 1»¼ S
0
0 º
ª1
«0 s 1
0 »»
«
«¬ 0
0
( s 1) 2 »¼
0 º
ª s 1 1
« 0
s
1
0 »»
«
«¬ 0
s a »¼ S
0
ª
«1 0
«
«
«0 1
«
«
«0 0
¬«
and for a z 1
> Is A @ S
º
»
»
»
0
».
»
( s 1) 2 ( s a ) »
»
(a 1) 2 ¼»
0
2.6 Fraction Description of Rational Matrices and the McMillan
Canonical Form
2.6.1 Fractional Forms of Rational Matrices
We will show that an arbitrary rational matrix of the form (2.3.1) can be written in
the form
W( s)
W(s)
Dl1 ( s )N l ( s ) ,
1
p
N p ( s )D ( s) ,
(2.6.1a)
(2.6.1b)
where
Dl ( s )  mum [ s ] and D p ( s )  nun [ s ]
are nonsingular matrices and Nl(s) mun[s], Np(s) mun[s].
According to the considerations in point 3 an arbitrary matrix of the form
(2.3.1) can be expressed in the standard form (2.3.2).
Taking Dl(s) = Imm(s) and Nl(s) = L(s), we obtain from (2.3.2) the matrix W(s)
of the form (2.6.1a). Taking Dp(s) = Inm(s) and Np(s) = L(s), we obtain the matrix
W(s) of the form (2.6.1b).
Rational Functions and Matrices
137
If Dl(s) = Imm(s), then deg det Dl(s) = m deg m(s), and if Dp(s) = Inm(s), then
deg det Dp(s)=n deg m(s).
Note that premultiplication of the matrices Dl(s) and Nl(s) by an arbitrary nonsingular matrix K(s) mum[s] does not change the matrix (2.6.1a), since
1
> K ( s ) Dl ( s ) @
K ( s )Nl ( s )
Dl1 ( s)K 1 ( s)K ( s)Nl ( s )
Dl1 ( s )N l ( s )
W( s ).
Analogously, post-multiplication of the matrices Dp(s) and Np(s) by an arbitrary
nonsingular matrix K(s) mum[s] does not change the matrix (2.6.1b), since
N p ( s )K ( s ) ª¬ D p ( s )K ( s ) º¼
1
N p ( s )K ( s )K 1 ( s )Dp1 ( s )
N p ( s )Dp1 ( s )
W ( s ).
Thus for a given rational matrix W(s) there are many pairs of matrices (Dl(s),
Nl(s)) and (Dp(s), Np(s)), which give the same matrix W(s). Thus these pairs are
not unique.
If
W( s)
L( s )
m( s )
Dl1 ( s )N l ( s )
N p ( s )D p1 ( s ) ,
(2.6.2)
then
deg m( s ) deg L( s ) d deg Dl ( s ) deg N l ( s ) ,
deg m( s ) deg L( s) d deg D p ( s) deg N p ( s ) .
(2.6.3a)
(2.6.3b)
From (2.6.2), we have
Dl ( s )L( s )
m( s ) N l ( s )
and
deg > Dl ( s )L( s ) @ deg > m( s )N l ( s ) @ .
Taking into account that
deg > Dl ( s )L( s ) @ d deg Dl ( s ) deg L( s )
and
deg > m( s ) N l ( s ) @ deg m( s ) deg N l ( s ) ,
from (2.6.4) we obtain (2.6.3a). The proof of (2.6.3b) is analogous.
(2.6.4)
138
Polynomial and Rational Matrices
If deg m(s) > deg L(s), then from (2.6.3) it follows that deg Dl(s) > deg Nl(s)
and deg Dp(s) > deg Np(s). If, on the other hand deg m(s) t deg L(s), then from
(2.6.3) we have deg Dl(s) t deg Nl(s) and deg Dp(s) t deg Np(s).
Example 2.6.1.
The rational matrix
W( s)
ª 1
«s 3
«
« 1
«s 2
¬
1
s2
1
s3
1
º
( s 2)( s 3) »
»
1
»
( s 2) 2 »¼
(2.6.5)
is to be converted to the forms (2.6.1a) and (2.6.1b).
We write this matrix in the standard form (2.3.2)
W( s)
ª ( s 2) 2
( s 2)( s 3) s 2 º
1
«
»
2
( s 2) ( s 3) ¬ ( s 2)( s 3)
s 3¼
( s 2) 2
L( s )
.
m( s )
(2.6.6)
Taking
Dl ( s )
I 2 ( s 2) 2 ( s 3),
Nl (s)
L( s)
ª ( s 2) 2
( s 2)( s 3) s 2 º
«
»,
(
s
2)(
s
3)
( s 2) 2
s 3¼
¬
we obtain
W( s )
ª ( s 2) 2 ( s 3)
º
0
«
»
2
0
( s 2) ( s 3) ¼
¬
1
ª ( s 2) 2
( s 2)( s 3) s 2 º
u«
».
s
s
s 3¼
(
2)(
3)
( s 2) 2
¬
On the other hand, taking
D p ( s)
we obtain
I 3 ( s 2) 2 ( s 3), N p ( s )
L( s) ,
(2.6.7)
Rational Functions and Matrices
W(s)
139
ª ( s 2) 2
( s 2)( s 3) s 2 º
«
»
( s 2) 2
s 3¼
¬( s 2)( s 3)
1
ª( s 2) 2 ( s 3)
º
0
0
«
»
2
u«
0
( s 2) ( s 3)
0
» .
2
«
0
0
( s 2) ( s 3) »¼
¬
(2.6.8)
In this case,
deg det Dl ( s )
m deg m( s )
2 ˜ 3 6 and
deg det D p ( s )
n deg m( s )
3˜3 9 .
(2.6.9)
From (2.6.4) and (2.6.5), it follows that there are reductions between entries of
Dl-1(s) and Nl(s), as well as Dp-1(s) and Np(s).
Thus the question arises as to under which conditions the reductions do not
occur, i.e., the pairs (Dl(s), Nl(s)) and (Dp(s), Np(s)) are irreducible and the degrees
of the determinants of the matrices Dl(s) and Dp(s) are minimal. The following
theorem gives the answer to this question.
Theorem 2.6.1. The pair (Dl(s), Nl(s)) is irreducible and the degree of the
determinant of Dl(s) is minimal if and only if
rank > Dl ( s ), N l ( s )@
m for all s  .
(2.6.10a)
The pair (Dp(s), Np(s)) is irreducible and the degree of the determinant of Dp(s) is
minimal if and only if
ª D p (s) º
rank «
»
¬ N p ( s) ¼
n for all s  .
(2.6.10b)
Proof. If the condition (2.6.10a) is satisfied, then a unimodular matrix U(s), with
deg det U(s) = 0, can be a greatest common divisor of the matrices Dl(s) and Nl(s).
In this case, Dl(s) and Nl(s) are irreducible and the degree of det Dl(s) is minimal.
The condition (2.6.10a) is also a necessary one. Let Ll(s), which is a
unimodular matrix, be a common left divisor of the matrices Dl(s) and Nl(s), i.e.,
Dl(s) = Ll(s) D l(s), Nl(s) = Ll(s) N l(s). Then for those values of the variable s for
which det Ll(s) = 0, the condition (2.6.10a) is not satisfied, and
W( s)
Dl1 ( s )N l ( s )
1
¬ªL l ( s )Dl ( s ) ¼º Ll ( s )N l ( s )
Dl ( s ) N l ( s ) .
In this case, (Dl(s), Nl(s)) is irreducible and the degree of the determinant of Dl(s) is
not minimal.
140
Polynomial and Rational Matrices
The proof of the second part of the theorem for the pair (Dp(s), Np(s)) is
analogous (dual).
„
Definition 2.6.1. An irreducible pair (Dl(s), Nl(s)) (Dp(s), Np(s)) yielding (2.6.1a)
((2.6.1b)) is called a left (right) minimal fraction form of the rational matrix W(s).
From the proof it follows that minimal fraction forms of a rational matrix are
determined uniquely up to the multiplication by unimodular matrices and that for
minimal Dl(s) and Dp(s) the following equality holds
deg det Dl ( s )
deg det D p ( s ) .
(2.6.11)
To compute a minimal pair ( D l(s), N l(s)), having given nonminimal
(irreducible) pair (Dl(s), Nl(s)) a greatest common left divisor Ll(s) of these
matrices is to be determined. To accomplish this, we apply elementary operations
on the columns of [Dl(s), Nl(s)] and perform the reduction
ª Dl ( s ) N l ( s ) º
ªLl ( s) 0 º
« I
» 
« U
R
o
0
U 2 »» ,
« m
»
« 1
«¬ 0
«¬ U 3
I n »¼
U 4 »¼
mum
U1 U1 ( s)  [ s ], U 4 U 4 ( s )  nun [ s ]
(2.6.12)
(R denotes an elementary operation on columns),
where
ª U1
«U
¬ 3
U2 º
U 4 »¼
is unimodular and partitioned into blocks of dimensions corresponding to those of
Dl(s) and Nl(s).
From (2.6.12), we have
> Dl (s),
ªU
Nl ( s)@ « 1
¬ U3
> Dl (s),
ªU º
Nl ( s)@ « 1 »
¬ U3 ¼
U2 º
U 4 »¼
>Ll ( s), 0@
and
Ll ( s ),
> Dl (s),
ªU º
Nl ( s)@ « 2 »
¬U 4 ¼
0.
(2.6.13)
The matrix U4 is nonsingular. From the second equation of (2.6.13), we obtain
Rational Functions and Matrices
Dl1 ( s )N l ( s )
ª U2 º
« »
¬ U4 ¼
U 2 U 41 .
141
(2.6.14)
is a full rank matrix for all s , since it is a part of a unimodular matrix.
Thus (U4, U2) is a minimal (irreducible) pair. Using (2.6.14) we can compute a
minimal pair ( D l(s), N l(s)) for an arbitrary given pair (Dl(s), Nl(s)).
Knowing a greatest common left divisor Ll(s), we can compute a minimal pair
from the relationship
Dl ( s)
Ll 1 ( s)Dl ( s ), N l ( s )
Ll 1 ( s )N l ( s ) .
(2.6.15)
Analogously, to compute a minimal pair ( D p(s), N p(s)) having given a
nonminimal (reducible) pair (Dp(s), Np(s)) one has to compute a greatest common
right divisor Pp(s) of these matrices. Carrying out elementary operations on rows of
ªD p (s)º
«
»
¬« N p ( s ) ¼»
we make the reduction
ª D p ( s ) I n 0 º L ª Pp ( s ) V1 V2 º
o«
,
« N ( s ) 0 I » 
V3 V4 »¼
m¼
¬ 0
¬ p
V1 V1 ( s )  nun [ s ], V4 V4 ( s )  mum [ s ],
(2.6.16)
where
ª V1
«V
¬ 3
V2 º
V4 »¼
is a unimodular matrix partitioned into blocks of dimensions corresponding to
those of Dp(s) and Np(s).
From (16) we have
ª V1
«V
¬ 3
V2 º ª D p ( s ) º
«
»
V4 »¼ ¬ N p ( s ) ¼
ª Pp ( s ) º
« 0 »
¬
¼
and
V1D p ( s ) V2 N p ( s )
Pp ( s ), V3 D p ( s ) V4 N p ( s )
0.
(2.6.17)
[V3, V4] is a full rank matrix for all s , since it is a part of a unimodular matrix,
and V4 is nonsingular. From the second relationship in (2.6.17), we obtain
N p ( s )Dp1 ( s )
V41V3 .
(2.6.18)
142
Polynomial and Rational Matrices
Using (2.6.18), we can compute a minimal pair ( D p(s), N p(s)) for an arbitrary pair
(Dp(s), Np(s)).
Knowing a greatest common right divisor Pp(s), we can compute a minimal pair
from the equations
D p (s)
D p ( s )Pp1 ( s ), N p ( s )
N p ( s )Pp1 ( s ) .
(2.6.19)
Example 2.6.2.
Using the solution of Example 2.6.1 compute the left and the right minimal
fraction form of the matrix (2.6.5).
It is easy to check that the fraction forms (2.6.7) and (2.6.8) of this matrix are
not minimal ones.
Using the reduction (2.6.12), we compute a greatest common left divisor Ll(s)
of the matrices Dl(s) and Nl(s). With this aim we carry out the following
elementary operations
1º
ª
P > 4 5 u ( s 3)@ , P >3 5 u ( s 2) @ , P «5 4 u » ,
2¼
¬
ª
§ s 5 ·º
P >1 5 u ( s 2)( s 3) @ , P > 2 1 u 2( s 2) @ , P «1 4 u ¨ ¸ » ,
© 4 8 ¹¼
¬
P >5 4 u (4)@ , P > 4 1 u (6 s 40)@ , P >1 u 8@ , P > 2, 5@ ,
ª Dl ( s ) N l ( s ) º
« I
0 »»
« m
«¬ 0
I n »¼
ª( s 2) 2 ( s 3)
0
( s 2) 2
( s 2)( s 3) s 2 º
«
»
2
0
( s 2) ( s 3) ( s 2)( s 3)
( s 2) 2
s 3»
«
«
1
0
0
`0
0 »
«
» R
o
0
1
0
0
0 » 
«
«
»
0
0
1
0
0
«
»
0
0
0
1
0 »
«
«
0
0
0
0
1 »¼
¬
Rational Functions and Matrices
143
0
0
s2
ª
«
0
1
0
«
«
8
0
4
«
0
0
0
«
«
0
0
1
«
( s 3)(2s 5)
0
« 4( s 2)( s 3) (2s 5)
« ( s 3)[4( s 1)( s 2) 2s 5] ( s 2)[2( s 1)( s 3) s 4] ( s 2)
¬
0
0
º
»
0
0
»
»
8(2 s 5)
2( s 2)
»
0
1
».
»
0
0
»
2
(2 s 5)[(4( s 2)( s 3) 2s 5] 1
( s 2) ( s 3) »
( s 3)[4( s 1)( s 2)(2s 5) (2s 5) 2 1] ( s 1)( s 2) 2 ( s 3) »¼
Using (2.6.15) and (2.6.14), we obtain
1
Dl ( s )
º
0
ª s 2 0 º ª( s 2) 2 ( s 3)
«
»
« 0
»
2
1¼ ¬
0
( s 2) ( s 3) ¼
¬
0
ª( s 2)( s 3)
º
«
»,
2
0
(
s
2)
(
s
3)
¬
¼
Ll 1 ( s )Dl ( s )
1
Nl (s)
W(s)
(2.6.20)
( s 2)( s 3) ( s 2) º
ª s 2 0 º ª ( s 2) 2
L ( s)N l ( s) «
»
» «
0
1
( s 2) 2
( s 3) ¼
¬
¼ ¬ ( s 2)( s 3)
s2
s3
1 º
ª
« ( s 2)( s 3) ( s 2) 2 s 3» ,
¬
¼
1
1
1
ª
º
« s 3 s 2 ( s 2)( s 3) »
»,
Dl1 ( s )N l ( s ) «
1
1
« 1
»
(2.6.21)
2
«s 2 s 3
»
s
(
2)
¬
¼
det Dl ( s )
1
l
( s 2)3 ( s 3) 2 ,
144
Polynomial and Rational Matrices
W( s )
U 2 U 41
ª0 8(2 s 5) 2( s 2) º
«0
0
1 »¼
¬
0
(2s 5)[(4( s 2)( s 3) 2s 5] 1
ª 1
(2.6.22)
u «« 0
«¬ ( s 2) ( s 3)[4( s 1)( s 2)(2s 5) (2s 5)2 1]
1
1
1
ª 1
º
0
º
« s 3 s 2 ( s 2)( s 3) »
«
».
( s 2) 2 ( s 3) »»
1
1
« 1
»
2
( s 1)( s 2) ( s 3) »¼
2
«s 2 s 3
»
(
2)
s
¬
¼
Using the reduction (2.6.16), we compute a greatest common right divisor Pp(s).
With this aim we carry out the following elementary operations:
L >5, 4@ , L > 4 5 u ( s 2) @ , L >1 5 u ( s 2)( s 3) @ , L >1, 2@ ,
5 ·º
ª
§ 1
L >3 1 u ( s 2) @ , L « 2 4 u ¨ s ¸ » , L > 2 u (4) @ , L >5 2@ ,
4 ¹¼
© 2
¬
L > 4 2 u (2s 5) @ , L >1 u (1) @ , L >3, 5@ ,
ª D p (s) I n 0 º
« N ( s) 0 I »
m¼
¬ p
2
ª( s 2) ( s 3)
0
0
«
2
0
(
s
2)
(
s
3)
0
«
«
0
0
( s 2) 2 ( s 3)
«
2
( s 2)( s 3)
s2
« ( s 2)
2
« ( s 2)( s 3)
(
s
2)
s3
¬
0
( s 2)( s 3) 1
1
ª 0
« 0
( s 2)
0
0
4
«
«s 2
0
1
0
4
«
0
0
0
4(2 s 5)
« 0
«¬ 0
0
0
s 2 ( s 2)
( s 2)( s 3)
(2 s 5)( s 3)
( s 2)( s 3)
( s 2)(2s 5)
º
»
»
(2 s 5)( s 3) 1
1 ( s 2)(2s 5) » .
»
( s 3)[1 (2s 5) 2 ] ( s 2)[1 (2s 5) 2 ]»
( s 2) 2 ( s 3)
( s 2) 2 ( s 3) »¼
Using (2.6.19) and (2.6.18), we obtain
1 0 0 0 0º
»
0 1 0 0 0»
L
o
0 0 1 0 0 » 
»
0 0 0 1 0»
0 0 0 0 1 »¼
0
0
0
0
1
Rational Functions and Matrices
D p ( s)
145
D p ( s )Pp1 ( s )
ª ( s 2) 2 ( s 3)
º
0
0
«
»
2
( s 2) ( s 3)
0
0
«
»
«
0
0
( s 2) 2 ( s 3) ¼»
¬
0
( s 2)( s 3) º
ª 0
»
u «« 0
s2
0
»
0
1
¬« s 2
¼»
1
0
( s 2)( s 3) º
ª 1
« 0
»,
(
s
2)(
s
3)
0
«
»
0
0
¬« s 2
¼»
N p ( s)
N p ( s )Pp1 ( s )
0
( s 2)( s 3) º
ª 0
ª ( s 2) 2
( s 2)( s 3) s 2 º «
»
s2
0
«
» 0
»
( s 2) 2
s 3¼ «
¬ ( s 2)( s 3)
«¬ s 2
»¼
0
1
s3
s2 º
ª 1
« 2
s 3 »» ,
«
s2
( s 2) 2 ¼»
¬« s 2
N p ( s )D p1 ( s )
W(s)
det D p ( s )
ª 1
«s 3
«
« 1
«s 2
¬
1
s2
1
s3
1
1
º
( s 2)( s 3) »
»,
1
»
( s 2) 2 »¼
(2.6.23)
(2.6.24)
( s 2)3 ( s 3) 2
and
W( s)
V41V3
1
4(2 s 5) 0 º
ª( s 3)[1 (2s 5) 2 ] ( s 2)[1 (2s 5) 2 ]º ª 0
«
» «
2
2
2
( s 2) 1 »¼
s
(
2)
(
3)
(
2)
(
3)
s
s
s
s
¬
¼ ¬
ª 1
«s 3
«
« 1
«s 2
¬
1
s2
1
s3
1
º
( s 2)( s 3) »
».
1
»
( s 2) 2 »¼
(2.6.25)
Comparing (2.6.22) to (2.6.24) and (2.6.21) to (2.6.25), we find that the appropriate
results are the same as well, that deg det D l(s) = deg det D p(s) = 5 and is greater than
the degree of the polynomial m(s), which is 3.
146
Polynomial and Rational Matrices
2.6.2 Relatively Prime Factorization of Rational Matrices
Consider a strictly proper rational matrix G(s)=
lim G ( s ) 0 .
mul
[s], i.e., satisfying the condition
sof
Problem 2.6.1. A strictly proper rational matrix G(s) mup[s] is given. Compute
polynomial relatively left prime matrices A1(s) mum[s] and B1(s) mup[s] such
that
G (s)
A11 ( s )B1 ( s ) .
(2.6.26)
Problem 2.6.2. A strictly proper rational matrix G(s) mup[s] is given. Compute
polynomial relatively right prime matrices A2(s) pup[s] and B2(s) mup[s] such
that
G (s)
B 2 ( s) A 21 ( s) .
(2.6.27)
Expressing a rational matrix G(s) in the form (2.6.26) or (2.6.27) is called
relatively prime factorization. Below we first give the procedure of such a
factorization, at first, to the form (2.6.26).
Procedure 2.6.1.
Step 1: Find the least common denominators mi(s) (i = 1,2,…,p) for the columns
and write the matrix G(s) in the form
G (s)
B( s ) A 1 ( s ) ,
(2.6.28)
where
A( s)
0
ª m1 ( s )
« 0
m
2 (s)
«
« #
#
«
0
«¬ 0
0 º
0 »»
.
%
# »
»
! m p ( s ) »¼
!
!
(2.6.29)
Step 2: Applying appropriate elementary operations on the rows, carry out the
reduction
ª A( s) I p
« B( s ) 0
¬
0 º L ª P( s) U1 ( s ) U 2 ( s ) º

o«
I m »¼
U 3 ( s) U 4 ( s) »¼
¬ 0
and compute the matrices U4(s), U3(s).
Step 3: The desired factorisation (2.6.26) is readily obtained from the relationship
Rational Functions and Matrices
G (s)
U 41 ( s )U 3 ( s ) .
147
(2.6.30)
This procedure can be derived in the following way. From the equality
ª U1 ( s ) U 2 ( s ) º ª A( s ) º
« U ( s ) U ( s ) » «B( s ) »
¼
4
¬ 3
¼¬
ª P( s) º
« 0 »,
¬
¼
we have
U 3 ( s ) A ( s ) U 4 ( s )B ( s )
0,
i.e.,
G ( s)
B( s) A 1 ( s)
U 41 ( s )U 3 ( s ) ,
with the assumption det U4(s) z 0. We will show that indeed det U4(s) z 0.
Let
U 1 ( s )
ª V1 ( s )
« V ( s)
¬ 3
V2 ( s ) º
.
V4 ( s ) »¼
ª A( s) º
« B( s ) »
¬
¼
ª V1 ( s )
«V (s)
¬ 3
V2 ( s ) º ª P( s ) º
V4 ( s ) »¼ «¬ 0 »¼
(2.6.31)
From
it follows that A(s) = V1(s)P(s). Nonsingularity of A(s) implies det V1(s) z 0. On
the other hand, from the relationship
V2 ( s ) º ªI p V11 ( s ) V2 ( s ) º
ª V1 ( s )
»
« V ( s) V ( s) » «
Im ¼
0
4
¬ 3
¼¬
0
ª V1 ( s )
º
« V ( s ) V ( s ) V ( s )V 1 ( s )V ( s ) »
4
3
1
2
¬ 3
¼
(2.6.32)
it follows that
det ª¬ V4 ( s ) V3 ( s )V11 ( s )V3 ( s ) º¼ z 0 .
Premultiplying (2.6.32) by U(s) and taking into account (2.6.31), we obtain
(2.6.33)
148
Polynomial and Rational Matrices
V11 ( s )
ªI p
«
¬«0
Im
V2 ( s ) º
»
¼»
0
ª U1 ( s ) U 2 ( s ) º ª V1 ( s )
º
« U ( s ) U ( s ) » « V ( s ) V ( s ) V ( s ) V 1 ( s ) V ( s ) »
4
3
1
3
¼
4
¬ 3
¼¬ 3
and
Im
U 4 ( s ) ª¬ V4 ( s ) V3 ( s )V11 ( s )V3 ( s ) º¼ .
Hence after considering (2.6.32), we obtain det U4(s) z 0.
The matrices U3(s) and U4(s) are relatively left prime, since [U3(s), U4(s)] is a
part of the unimodular matrix U(s).
The procedure of factorization (2.6.27) is as follows.
Procedure 2.6.2.
Step 1: Find the least common denominators mic (s) (i = 1,2,…,m) for rows and
write the matrix G(s) in the form
G (s)
Ac1 ( s )Bc( s ) ,
(2.6.34)
where
Ac( s )
0
ª m1c( s )
« 0
c2 ( s )
m
«
« #
#
«
0
¬ 0
0 º
0 »»
.
%
# »
»
! m2c ( s ) ¼
!
!
(2.6.35)
Step 2: Applying elementary operations on columns carry out the reduction
ªL( s )
ª Ac( s ) Bc( s ) º
« I
»
R
0 » 
o «« U1 ( s )
« m
«¬ U 3 ( s )
«¬ 0
I l »¼
0º
U 2 ( s ) »»
U 4 ( s ) »¼
and compute the matrices U4(s), U2(s).
Step 3: The desired factorization (2.6.27) is derived from the relationship
G ( s)
U 2 ( s )U 41 ( s ) .
(2.6.36)
Example 2.6.3.
Using Procedures 2.6.1 and 2.6.2, compute the factorizations (2.6.26) and (2.6.27)
for the matrix
Rational Functions and Matrices
G ( s)
ª 1
« s 1
«
« 1
¬« s
1
s
2
s2
2 º
s 1»
».
1 »
s 2 ¼»
149
(2.6.37)
Applying Procedure 2.6.1, we compute the following.
Step 1: We compute the least common denominators of all entries of the respective
columns of this matrix and write them in the form
G (s)
0
ª s ( s 1)
ª s s 2 2( s 2) º «
0
s ( s 2)
« s 1 2s
s 1 »¼ «
¬
«¬ 0
0
1
0
º
» .
0
»
( s 1)( s 2) »¼
Step 2: Carrying out appropriate elementary operations on the rows of the matrix
G (s)
ª s ( s 1)
« 0
«
« 0
«
« s
«¬ s 1
0
0
1 0 0 0 0º
0 1 0 0 0 »»
( s 1)( s 2) 0 0 1 0 0 » ,
»
2( s 2)
0 0 0 1 0»
s 1
0 0 0 0 1 »¼
s ( s 2)
0
0
s2
2s
we obtain
ª1
«0
«
«0
«
«0
«¬0
and
0
4
2( s 2)
( s 1)
0 ( s 1)( s 2)
0
0
0
0
32
3
2
0
s
0
3
0
1 s
3s 2 (3s 2)
3
2
9
2
1
2s
9s
s
2( s 1)
0
s ( s 1)
4 s ( s 1)
1 12 s º
»
1
2 s
»
0 »
»
0 »
s ( s 2) »¼
150
Polynomial and Rational Matrices
P(s)
U 2 (s)
U 4 (s)
( s 2) º
ª1 0
«0 4
( s 1) »» , U1 ( s )
«
¬« 0 0 ( s 1)( s 2) ¼»
1 12 s º
ª s
« 2( s 1)
»
1
2 s » , U3 (s)
«
«¬ 0
0 »¼
0
ª s ( s 1)
º
« 4s ( s 1)
».
s
s
(
2)
¬
¼
ª 32
« 3
« 2
«¬ 0
0
3
2
º
3 »» ,
0
1 »¼
9
2
1 s
2s º
ª s
«3s 2 (3s 2) 9 s » ,
¬
¼
Step 3: Thus
G (s)
1
4
U ( s )U3 ( s )
ª s ( s 1)
« 4s ( s 1)
¬
1
0 º ª s
( s 1) 2 s º
.
s ( s 2) »¼ «¬ 2 3s 3s 2 9s »¼
Now applying Procedure 2.6.2:
Step 1: We compute the least common denominators of all entries of the respective
rows of this matrix and write them in the form
1
G (s)
0 º ª s s 1 2s º
ª s ( s 1)
.
« 0
(
2) »¼ «¬ s 2 2 s
s
s
s »¼
¬
Step 2: Carrying out appropriate elementary operations on the columns of the
matrix
G ( s)
we obtain
s s 1 2s º
0
ª s ( s 1)
« 0
s ( s 2) s 2 2 s
s »»
«
« 1
0
0
0
0»
«
»
1
0
0
0 »,
« 0
« 0
0
1
0
0»
«
»
0
0
1
0»
« 0
« 0
0
0
0
1 »¼
¬
Rational Functions and Matrices
0
0
ª 1
« 0
1
0
«
« 9
9
1
«
4
8
«
0
4
« 0
«
1
« 0
4s
2
«
«
3
3
s
s
« 1 s
4
8
«
«
3
1 3
« 2 s s 2s
2
2 4
¬
0
151
0
º
»
0
0
»
»
4 s
15 »
»
4( s 1)
0
»
»
4( s 1)
4 s »
»
»
0
s »
»
»
0
2(4 3s ) »
¼
and
L( s )
U3 (s)
ª9 9 º
« 4 8 » , U ( s)
2
«
»
¬« 0 0 »¼
1
º
»
2
»
3
s » , U 4 ( s)
»
8
»
1 3 »
s
2 4 ¼»
ª1 0 º
« 0 1 » , U1 ( s )
¬
¼
ª
« 0
«
« 1 3 s
«
4
«
« 2 3 s
2
¬«
ª1
« 4
¬
4 s
4( s 1)
15º
,
0 »¼
4s º
ª 4 s 4( s 1)
«s
s »» .
0
«
«¬ 2 s
0
2(4 3s ) »¼
Step 3: In view of this,
G (s)
U 2 ( s )U 41 ( s )
1
4 s º
ª 4s 4( s 1)
4s
15º «
ª 1
s »» .
0
« 4 4s ( s 1) 0 » « s
¬
¼ « 2s
0
2(4 3s ) »¼
¬
Problem 2.6.3. A rational matrix G(s) mup[s] is given in the form of the left
factorisation (2.6.26). Compute the right relatively prime factorisation (2.6.27) of
this matrix.
Solution. Applying elementary operations on columns we carry out the
transformation
152
Polynomial and Rational Matrices
ª A1 ( s )
«
« Im
« 0
¬
B1 ( s ) º
0 º
ªL1 ( s )
» R «
o «V1 ( s ) V2 ( s ) »» .
0 » 
«¬ V3 ( s ) V4 ( s ) »¼
I p »¼
(2.6.38)
The desired factorisation matrices are given by
B 2 ( s)
V2 ( s ), A 2 ( s )
V4 ( s ) or B 2 ( s )
V2 ( s ), A 2 ( s )
V4 ( s ) . (2.6.39)
The relations in (2.6.39) can be derived in the following way.
From (2.6.38) we have
ª V2 ( s ) º
»
¬ V4 ( s ) ¼
> A1 ( s) , B1 (s)@ «
0,
(2.6.40)
i.e., A1(s)V2(s) = -B1(s)V4(s).
Nonsingularity of A1(s) implies nonsingularity of V4(s). Hence
A11 ( s )B1 ( s )
V2 ( s )V41 ( s ) .
The dual to Problem 2.6.3 can be formulated as follows.
Problem 2.6.3c. A rational matrix G(s) mup[s] is given in the form of the right
factorisation (2.6.27). Compute the left relatively prime factorisation (2.6.26) of
this matrix.
To solve this problem we use Step 2 from Procedure 2.6.1.
The desired factorisation matrices are given by
A1 ( s )
U 4 ( s ), B1 ( s )
or A1 ( s )
U 4 ( s ), B1 ( s )
U 3 ( s ),
U 3 ( s ).
(2.6.41)
We proceed further in the same way as in Problem 2.6.3.
2.6.3 Conversion of a Rational Matrix into the McMillan Canonical Form
Let the following rational matrix be given
W( s)
ª W11 ( s ) ! W1n ( s ) º
« #
%
# »»  mun ( s )
«
«¬ Wm1 ( s ) ! Wmn ( s ) »¼
(2.6.42)
Rational Functions and Matrices
153
whose rank is r d min(n, m). With the monic least common denominator of all
entries of Wij(s) found, we can express the above matrix in the form
W( s)
L( s )
,
m( s )
(2.6.43)
where L(s) mun[s]. Applying elementary operations, we can transform L(s) into
the Smith canonical form
L s ( s)
U ( s )L ( s ) V ( s )
ªi1 ( s )
« 0
«
« #
«
« 0
« 0
«
« #
« 0
¬
0
!
i2 ( s ) !
#
%
0
!
0
!
#
%
0
!
0
0 ! 0º
0
0 ! 0 »»
#
# % #»
»
ir ( s) 0 ! 0 » ,
0
0 ! 0»
»
#
# % #»
0
0 ! 0 »¼
(2.6.44)
where U(s) mum[s], V(s) nun[s] are unimodular matrices, i1(s),i2(s),…,ir(s) are
the invariant polynomials such that ii+1(s) is divisible (without remainder) by ii(s).
From (2.6.43) and (2.6.44), after reduction of all common factors occurring
simultaneously in m(s) and ik(s), k = 1,…,r, we obtain
WM ( s )
U(s) W( s)V ( s)
ª l1 ( s)
«\ ( s )
« 1
«
« 0
«
« #
«
« 0
«
«
« 0
« #
«
¬« 0
0
L s ( s)
m( s )
!
0
!
0
#
%
#
0
!
0
!
0
#
0
%
!
#
0
l2 ( s )
\ 2 ( s)
lr ( s )
\ r ( s)
where
lk ( s )
\ k (s)
ik ( s )
m( s )
(k
1, 2, ..., r )
º
0 ! 0»
»
»
0 ! 0»
»,
# % #»
»
0 ! 0»
»
»
0 ! 0»
# % #»
»
0 ! 0 ¼»
(2.6.45)
154
Polynomial and Rational Matrices
and li+1(s) is divisible (without remainder) by li(s), and \k-1(s) is divisible (without
remainder) by \k(s).
Definition 2.6.2. A matrix WM(s) given by (2.6.45) is called the McMillan
canonical form of the matrix W(s).
Using the contradiction method, we will show that \1(s) = m(s) and l1(s) = i1(s).
Assume that \1(s) z m(s). In this case, every entry of the matrix L(s) is divisible by
the appropriate factor of the polynomial m(s). Thus the polynomial m(s) cannot be
the least common denominator of all entries of W(s), which contradicts the
assumption. Hence \1(s) = m(s) and this implies immediately that l1(s) = i1(s).
From the above considerations, the following procedure for computation of the
McMillan canonical form (2.6.45) of the matrix W(s) can be derived.
Procedure 2.6.3.
Step 1: Compute the monic least common denominator m(s) of all entries of the
matrix W(s).
Step 2: Writing the matrix W(s) in the form (2.6.43), compute the polynomial
matrix L(s).
Step 3: Applying elementary operations, convert L(s) into the Smith canonical
form LS(s).
Step 4: Reduce common factors occurring in the polynomials m(s) and ik(s), and then
compute the McMillan canonical form (2.6.45).
Example 2.6.4.
Compute the McMillan canonical form of the matrix (2.6.5).
We proceed according to Procedure 2.6.3.
Step 1: The least common denominator of all entries of the given matrix is
m( s )
( s 2) 2 ( s 3) .
Steps 2 and 3: With the least common denominator found, the matrix W(s) takes
the form
W( s)
L( s )
m( s )
ª ( s 2) 2
1
«
2
( s 2) ( s 3) ¬« ( s 2)( s 3)
Applying elementary operations, we convert the matrix
L( s )
ª ( s 2) 2
«
¬« ( s 2)( s 3)
into the Smith canonical form
( s 2)( s 3)
( s 2)
2
s 2º
»
s 3 ¼»
( s 2)( s 3)
( s 2)
2
s 2º
».
s 3 ¼»
Rational Functions and Matrices
ª1
«0
¬
L s ( s)
0
( s 2)( s 2, 5)
155
0º
.
0 »¼
Step 4: The desired McMillan canonical form of the matrix (2.6.5) is
1
ª
« ( s 2) 2 ( s 3)
«
«
0
«
¬
L s ( s)
m( s )
WM ( s )
º
0 »
».
s 2, 5
»
0»
( s 2)( s 3)
¼
0
2.7 Synthesis of Regulators
2.7.1 System Matrices and the General Problem of Synthesis of Regulators
Consider a discrete feedback system (Fig. 2.7.1) consisting of a plant with the
matrix transfer function
T0 z
Dl
Dp
Dl1N l
N p Dp1 ,
Dl z  pu p > z @ , N l
Dp z  mum
> z@,
Np
N l z  pum > z @ ,
Np z 
pum
(2.7.1)
>z@
and a regulator with the matrix transfer function
Tr z
Xl
Xp
Xl1Yl
Yp X p1 ,
Xl z  mum > z @ , Yl
Xp z  pu p
> z@,
Yp
Yl z  mu p > z @ ,
Yp z  mu p
(2.7.2)
> z @.
We assume that the matrices
T0 z  pum z , Tr z  mu p z
are proper
lim T0 z
z of
D0  pum , lim Tr z
z of
or strictly proper D0 = 0 and Dr = 0.
Dr  mu p ,
(2.7.3)
156
Polynomial and Rational Matrices
Fig. 2.1. Discrete-time system with feedback
From the scheme in Fig. 2.1 we can write the equations
yi
T0 z ui
T0 z vi zi , vi
Tr z yi , i  ' ^0, 1, ...` ,(2.7.4)
where yi, ui, vi and zi are vector sequences of plant output, control, regulator output,
and disturbances.
Substituting (2.7.1) and (2.7.2) into (2.7.4), we obtain
Dl yi N l vi
N l zi ,
Xl vi Yl yi
0, i  ' ,
(2.7.5)
which we write in the form
ª Dl
«Y
¬ l
N l º ª yi º
Xl »¼ «¬ vi »¼
ª Nl º
« 0 » zi .
¬ ¼
(2.7.6)
Definition 7.1.1. The polynomial matrix
Sl
ª Dl
«Y
¬ l
N l º

Xl »¼
pm u pm
> z@
(2.7.7a)
will be called the left system matrix of the closed-loop system, and the polynomial
matrix
Sp
ª Xp
« Y
¬ p
Np º

D p »¼
p m u p m
> z@
(2.7.7b)
will be called the right system matrix of the closed-loop system.
If the matrices Dl and Nl are relatively left prime (T0(z)=Dl-1Nl is irreducible),
then according to the considerations in Sect. 1.15.3 there exist a unimodular matrix
of elementary operations on columns
U
ª U11
«U
¬ 21
U12 º
, U11
U 22 »¼
U11 z  pu p > z @ , U 22
U 22 z  mum > z @ , (2.7.8)
Rational Functions and Matrices
157
such that
ªU
N l @ « 11
¬ U 21
> Dl
U12 º
U 22 »¼
ª¬I p
0 º¼ .
(2.7.9)
Postmultiplying (2.7.9) by the unimodular matrix
ª V11 V12 º
«V
»,
¬ 21 V22 ¼
V11 z  pu p > z @ , V22
U 1
V11
(2.7.10)
V22 z  mum
> z@,
we obtain
> Dl
N l @
ª¬I p
ªV
0 º¼ « 11
¬ V21
Dl
V11 ,
Nl
V12 .
V12 º
,
V22 »¼
(2.7.11)
where
(2.7.12)
From (2.7.9) we have
Dl U12
N l U 22
and
Dl1N l
1
U12 U 22
,
(2.7.13)
since det U22 z 0.
Comparison of (2.7.1) to (2.7.13) implies that
Np
U12 , D p
U 22 .
(2.7.14)
According to the considerations in Sect. 1.15.3, provided that Dl and Nl are
relatively prime, there exist polynomial matrices Xp and Yp such that
Dl X p N l Yp
Ip .
(2.7.15)
From (2.7.9) it follows that
Dl U11 N l U 21
Ip .
(2.7.16)
Comparison of (2.7.15) to (2.7.16) yields
Xp
U11 , Yp
U 21 .
(2.7.17)
158
Polynomial and Rational Matrices
The matrices (2.7.14) are relatively right prime. Thus there exist polynomial
matrices Xl and Yl such that
Im .
Yl N p Xl D p
(2.7.18)
From
ª V11
«V
¬ 21
V12 º ª U11
V22 »¼ «¬ U 21
U12 º
U 22 »¼
ªI p
«0
¬
0º
I m »¼
(2.7.19)
it follows that
V21U12 V22 U 22
Im
and if we take into account (2.7.14), we obtain
Im .
V21N p V22 D p
(2.7.20)
Comparison of (2.7.18) to (2.7.20) yields
Xl
V22 , Yl
V21 .
(2.7.21)
From (2.7.19) and (2.7.21), as well as (2.7.14) and (2.7.17), it follows that
Sl
ª Dl
«Y
¬ l
N l º
Xl »¼
ª V11
«V
¬ 21
V12 º
, Sp
V22 »¼
ª Xp
«Y
¬ p
Np º
D p »¼
ª U11
«U
¬ 21
U12 º
.(2.7.22)
U 22 »¼
From (2.7.19) and (2.7.22), we have
Sl S p
S p Sl
I pm .
(2.7.23)
This way the following theorem has been proved.
Theorem 7.1.1. If the transfer matrix T0(z) of the plant is irreducible, then there
exists the transfer matrix TJ(z) of the regulator such that the system matrices (2.7.7)
of the closed-loop system satisfy (2.7.23) and are unimodular.
The general problem of synthesis of regulator for a given plant can be
formulated as follows. With the plant transfer matrix T0(z) given, one has to
compute the transfer matrix Tr(z) of the regulator in such a way that the system
matrix (2.7.7) of the closed-loop system has the desired dynamical properties; for
instance, that it is a unimodular matrix or its determinant is equal to a given
polynomial.
Rational Functions and Matrices
159
In order to compute the system matrices Sp and Sl, one has to proceed in the
following way.
1. Applying elementary operations on columns and carrying out the reduction
ª Dl
«
«I p
«¬ 0
2.
Nl º
ª Ip
»
R
o «« U11
0 » 
«¬ U 21
I m »¼
0 º
U12 »»
U 22 »¼
(2.7.24)
compute the unimodular matrix
Compute the inverse of Sp, which is equal to Sl (since Sl = Sp-1).
2.7.2 Set of Regulators Guaranteeing Given Characteristic Polynomials of a
Closed-loop System
For a feedback system (Fig. 2.1), the transfer matrix (2.1.1) of the plant is given;
the transfer matrix (2.1.2) of the regulator is to be computed in such a way that
det S l
ªD
det « l
¬ Yl
N l º
Xl »¼
cw z ,
(2.7.25)
where w(z) is a given characteristic polynomial of the closed-loop system, and c is
a constant independent of z.
Theorem 7.2.1. Let Dl and Nl be relatively left prime matrices and Xl0 and Yl0 be
matrices of regulator chosen in such a way that the system matrix
Sl0
ª Dl
«Y0
¬ l
N l º
Xl0 »¼
(2.7.26)
is unimodular. The set of transfer matrices satisfying the condition (2.7.25) is given
by the relationships
Xl
PXl0 QN l , Yl
PYl0 QDl ,
where the polynomial matrix P = P(s)
det P
and Q = Q(s)
mum
[z] satisfies the condition
w z
mup
(2.7.27)
[z] is an arbitrary polynomial matrix.
(2.7.28)
160
Polynomial and Rational Matrices
Proof. From (2.1.23) it follows that
S 0p
ª¬Sl0 º¼
1
ª X0p
« 0
¬« Yp
Np º
».
D p ¼»
(2.7.29)
Using (2.1.15) and (2.1.13), we obtain
N l º ª X0p
«
Xl »¼ «¬ Yp0
ª Dl
«Y
¬ l
Sl S 0p
Np º
»
D p »¼
ªI p
«Q
¬
0º
,
P »¼
(2.7.30)
since
Dl X0p N l Yp0
Nl D p ,
(2.7.31)
Yl N p Xl D p .
(2.7.32)
I p , Dl N p
and
Q
Yl X0p Xl Yp0 , P
It is easy to show that det P = det [YlNp + XlDp] is the characteristic polynomial of
the closed-loop system and the condition (2.7.28) is satisfied.
From (2.7.30), we have
det S l S 0p
det Sl det S 0p
ªI
det « p
¬Q
0º
P »¼
det P
w z ,
i.e.,
det Sl
cw z
where c
1
.
det S p0
(2.7.33)
„
Lemma 7.2.1. The matrix pair (2.7.32) is right equivalent to the pair (Yl, Xl) and is
relatively left prime if and only if the pair (Yl, Xl) is relatively left prime.
Proof. From (2.7.32), we have
>Q
P@
> Yl
ª X0
Xl @ « p0
«¬ Yp
Np º
»
D p »¼
> Yl
Xl @ S 0p .
(2.7.34)
Rational Functions and Matrices
161
Thus the pair (2.7.32) is right equivalent to the pair (Yl, Xl), since the matrix Sp0 is
unimodular. According to Definition 1.15.7, the pair (2.7.32) is relatively left
prime if and only if the pair (Yl, Xl) is relatively left prime.
„
So far we have assumed that the matrices Dl and Nl are relatively left prime,
which means that the transfer matrix T0(z) of the system is irreducible.
Now assume that L = L(z) is the greatest common left divisor (GCLD) of the
matrices Dl and Nl, i.e.,
Dl
LDl ,
Nl
LN l ,
L
L z  pu p > z @ .
(2.7.35)
Theorem 7.2.2. Let L be the GCLD of the matrices Dl and Nl; let X l0, Y l0 be the
matrices of regulator chosen in such a way that the system matrix
Sl0
ª Dl
« 0
¬ Yl
Nl º
»
Xl0 ¼
(2.7.36)
is unimodular. The set of transfer functions of the regulator satisfying the condition
(2.7.25) exists if and only if
w z
det L
w z
(2.7.37)
and is determined by the relationships
Xl
PXl0 QN l , Yl
PYl0 QDl ,
(2.7.38)
where
det P
(2.7.39)
w z
and Q = Q (z)
mup
[z] is an arbitrary matrix.
Proof. Taking into account that
S 0p
we can write
ª¬Sl0 º¼
1
ª X0p
« 0
¬« Yp
Np º
»,
D p ¼»
(2.7.40)
162
Polynomial and Rational Matrices
ª Dl
«Y
¬ l
Sl S 0p
N l º ª X0p
«
Xl »¼ «¬ Yp0
Np º
»
D p »¼
ªL 0 º
«Q P » ,
¬
¼
(2.7.41)
since
Dl X0p N l Yp0
Q
L,
Yl X0p Xl Yp0 ,
Dl N p N l D p
P
L Dl N p N l D p
Yl N p Xl D p .
0,
(2.7.42)
From (2.7.41), we have
det S l det S 0p
det L det P ,
(2.7.43)
and taking into account (2.7.37) and (2.7.39), we obtain
det S l
det L det P
det S 0p
cw z
where c
1
.
det S 0p
(2.7.44)
Note that equality in (2.7.44) holds if and only if the condition (2.7.37) is satisfied.
„
3
Normal Matrices and Systems
3.1 Normal Matrices
3.1.1 Definition of the Normal Matrix
Consider a rational matrix in the standard form
W s
L s
,
m s
(3.1.1)
where L(s) mun[s] is a polynomial matrix and m(s) (a monic polynomial) is the
least common denominator of the entries of the matrix W(s). We assume that the
number of rows m and columns n of the matrix (3.1.1) is greater than or equal to
two (that is, m, n t 2).
Definition 3.1.1. A rational matrix of the form (3.1.1) is called normal if and only
if every nonzero second-order minor of the polynomial matrix L(s) is divisible
(without remainder) by the polynomial m(s).
For example, the matrix
W s
ª 1
« s 1
«
« 0
«¬
º
0 »
»
1 »
s 2 »¼
0 º
ªs 2
1
s 1»¼
s 1 s 2 «¬ 0
is normal, since the determinant of the matrix
L s
m s
(3.1.2)
164
Polynomial and Rational Matrices
L s
0 º
ªs 2
,
« 0
s 1»¼
¬
(3.1.3)
which is a second-order minor of this matrix, is divisible (without remainder) by
the polynomial m(s) = (s + 1)(s + 2).
On the other hand, the matrix
ª s2
2
«
« s 1
«
« 0
¬
W s
º
0 »
»
1 »
»
s 1¼
0 º
ªs 2
« 0
s 1»¼
s 1 ¬
1
2
L s
m s
(3.1.4)
is not normal, since the determinant of L(s) is not divisible (without remainder) by
the polynomial m(s) = (s + 1)2.
3.1.2 Normality of the Matrix [Is – A]-1 for a Cyclic Matrix
The inverse matrix [Is – A]-1 for any matrix A
be written in the standard form
1
> Is A @
nun
is a rational matrix, which can
LA s
,
m s
(3.1.5)
where LA(s) nun[s] and m(s) is a least common denominator.
Applying elementary operations on rows and columns, we can reduce [Is – A]
to its Smith canonical form
> Is A @ S
U s > Is A @ V s
diag ª¬i1 s , i2 s , ..., ir s , 0, ..., 0 º¼  nun > s @ ,
(3.1.6)
where U(s) and V(s) are unimodular matrices of elementary operations on rows
and columns; i1(s), i2(s), … ,ir(s) are the monic invariant polynomials satisfying the
divisibility condition ik+1(s) | ik(s) (the polynomial ik+1(s) is divisible without
remainder by the polynomial ik(s), k = 1,…,r-1, and r = rank LA(s)).
The invariant polynomials are given by the formula
ik s
Dk s
, for k
Dk 1 s
1, ..., r ,
D0 s
1 ,
(3.1.7)
where Dk(s) is a greatest common divisor of all k-th order minors of [Is – A].
The minimal polynomial <(s) of a matrix A nun is related to its characteristic
polynomial M(s) = det [Ins – A] in the following way
Normal Matrices and Systems
< s
M s
Dn1 s
.
165
(3.1.8)
From (3.1.7) and (3.1.8) it follows that <(s) = M(s) if and only if
D1 s
D2 s
...
Dn1 s
1.
(3.1.9)
According to Definition 1.14.2, a matrix A nun satisfying condition (3.1.9) (or
equivalently <(s) = M(s)) is called a cyclic matrix.
Theorem 3.1.1. Let A nun and n t 2. Then every nonzero second-order minor of
LA(s) is divisible without remainder by the polynomial m(s) if and only if
<(s) = M(s).
Proof. Sufficiency. If <(s) = M(s), then from (3.1.9) and (3.1.7) it follows that
i1(s) = i2(s) = … = ,in-1(s) = 1, in(s) = <(s) = m(s) and
> Is A @ S
diag ª¬1, 1, ...1, m s º¼ .
(3.1.10)
The adjoint of the matrix (3.1.10) thus has the form
Adj > Is A @S
LA s
diag > m( s ), m( s ), ..., m( s ), 1@ .
(3.1.11)
Hence every nonzero second-order minor of (3.1.11) is divisible without remainder
by the polynomial m(s). According to Binet–Cauchy theorem, every second-order
minor of LA(s) = Adj [U-1(s)[Is - A]SV-1(s)] = V(s) Adj [Is – A]SU(s) is the sum of
the products of the second-order minors of (3.1.11) and of the unimodular matrices
U(s) and V(s). Thus every nonzero second-order minor of LA(s) is divisible without
remainder by the polynomial m(s).
Necessity. It follows from the definition of the standard form that LA(s)/m(s) is an
irreducible fraction and the polynomial m(s) is monic. If <(s) z M(s), then from (8)
it follows that Dn-1(s) z 1 and every nonzero element of Adj [Is – A] is divisible by
Dn-1(s). Expanding the determinant det [Is – A] along any row or column, we
obtain det [Is – A] = Dn-1(s) m (s). But in this case LA(s)/m(s) is not an irreducible
fraction.
„
Example 3.1.1.
Show that every nonzero second-order minor of LA(s) in the inverse matrix
[Is – A]-1 = LA(s)/m(s) is divisible without remainder by the polynomial
m(s) = det [Is – A] for any values of a0, a1, a2 in the Frobenius matrix
166
Polynomial and Rational Matrices
A
ª 0
« 0
«
«¬ a0
1
0
a1
0 º
1 »» .
a2 »¼
(3.1.12)
In this case, we have
1
0
1
s
a1 s a2
s
det > Is A @
0
a0
s 3 a2 s 2 a1s a0
and
1
> Is A @
ªs
«0
«
«¬ a0
1
0 º
s
1 »»
a1 s a2 »¼
1
LA s
,
m s
where m(s) = det [Is – A]=s3 + a2s2 + a1s + a0
LA s
ª s 2 a2 s a1
«
a0
«
«
a0 s
¬
1º
»
s ».
s 2 »¼
s a2
2
s a2 s
a1s a0
(3.1.13)
The second-order minors of the matrix (3.1.13) are
M 11
s 2 a2 s
a1s a0
M 13
a0 s 2 a2 s
a0 s a1s a0
M 22
s 2 a2 s a1
a0 s
1
s2
M 23
s 2 a2 s a1
a0 s
s a2
a1s a0
M 32
s 2 a2 s a1 1
a0
s
M 33
s 2 a2 s a1
a0
s
s2
sm s , M 12
a0 s
a0 s s 2
a0 m s , M 21
0,
s a2
a1s a0
1
s2
m s ,
sm s ,
m s , M 31
m s ,
s a2
s 2 a2 s
s a2 m s .
s a2 1
s 2 a2 s s
(3.1.14)
0,
Normal Matrices and Systems
167
The nonzero minors (3.1.14) are divisible without remainder by the polynomial
m(s).
Note that, since a Frobenius matrix is cyclic, the above considerations are true
for a Forbenius matrix of arbitrary dimensions.
Example 3.1.2.
We will show that [Is – A]-1 is a normal matrix for
A
ª1 1 0 º
«0 1 0 »
«
»
¬« 0 0 a »¼
(3.1.15)
when a z 1 and is not normal for a = 1.
To prove this, we compute
1
> Is A @
0 º
ª s 1 1
« 0
0 »»
s 1
«
«¬ 0
0
s a »¼
1
LA s
,
m s
(3.1.16)
where m(s) = det [Is – A]=(s – 1)2(s – a)
ª s 1 s a
«
0
«
«
0
«¬
LA s
º
»
».
2»
s 1 »¼
sa
s 1 s a
0
0
0
(3.1.17)
The second-order minors of the matrix (3.1.17) are
M 11
M 13
M 32
M 23
M 33
s 1 s a
0
s 1
0
0
s 1 s a
0
0
s 1 s a
0
s 1 s a
0
s 1 m s , M 12
2
0, M 22
0
0, M 21
0
sa
0
s 1 s a
sa
0
s 1 s a
0
0
s 1
s 1 s a
0
0
s 1
sa
0, M 31
0
0
0
s 1
2
sa
0
0
sa m s .
0,
s 1 m s ,
2
(3.1.18)
m s ,
s 1 s a
2
0,
168
Polynomial and Rational Matrices
If a z 1, then LA(s)/m(s) is an irreducible fraction and every nonzero minor (3.1.18)
is divisible by the polynomial m(s). On the other hand, if a = 1, then
0 º
ªs 1 1
«
s 1 « 0
s 1 0 »» , m s
«¬ 0
0
s 1»¼
LA s
s 1
3
and
0 º
ªs 1 1
« 0
s
1
0 »» .
2 «
s 1
«¬ 0
0
s 1»¼
1
1
> Is A @
In this case, the minor
M 21
1
0
0 s 1
s 1
is not divisible by the polynomial (s – 1)2. Thus for a = 1, the matrix (3.1.16) is not
normal.
3.1.3 Rational Normal Matrices
Consider a rational matrix of dimensions mun in the standard form (3.1.1).
Let
WM s
ª l1 s
«
« <1 s
«
« 0
«
«
« #
«
« 0
«¬
U s W s V s
0
!
0
l2 s
<2 s
!
0
#
%
#
0
!
lm s
<m s
º
0 ! 0»
»
»
0 ! 0»
mun
s
»
»
# % #»
»
0 ! 0»
»¼
(3.1.19)
be the McMillan canonical form of the matrix (3.1.1), and
wM s
<1 s < 2 s ! < m s
(3.1.20)
Normal Matrices and Systems
169
the McMillan polynomial of this matrix, where U(s) and V(s) are unimodular
matrices (see Sect. 2.6).
Theorem 3.1.2 Let a rational matrix of the form (3.1.1) satisfy the condition
min (m, n) t 2. In this case, every nonzero second-order minor of the polynomial
matrix L(s) mun[s] is divisible without remainder by the polynomial m(s) if and
only if the McMillan polynomial wM(s) of the matrix (3.1.1) coincides with the
polynomial m(s), i.e.,
wM s
m s .
(3.1.21)
Proof. Sufficiency. If the condition (3.1.21) is satisfied, then <2(s) = <3(s) =…
=<m(s) = 1, since <1(s) = m(s). In this case, the matrix (3.1.19) has the form
WM s
LM s
,
m s
(3.1.22)
where
LM s
ªl1 s
«
« 0
« #
«
¬« 0
0
!
0
l2 s m s
!
0
#
0
%
#
! lm s m s
0 ! 0º
»
0 ! 0»
 mun > s @ . (3.1.23)
# % #»
»
0 ! 0 ¼»
From (3.1.23) it follows that every nonzero second-order minor of the matrix is
divisible without remainder by the polynomial m(s).
According to the Binet–Cauchy theorem, every second-order minor of
L(s) = U-1(s)LM(s)V-1(s) is the sum of the products of second-order minors of the
matrix (3.1.23) and of unimodular matrices U-1(s) and V-1(s). Therefore, every
nonzero second-order minor of the polynomial matrix L(s) is divisible without
remainder by the polynomial m(s).
Necessity. If every nonzero second-order minor of L(s) is divisible without
remainder by the polynomial m(s), then it follows from the Binet–Cauchy theorem
that every nonzero second-order minor of LM(s) is also divisible without remainder
by m(s). The polynomial matrix LM(s) thus has the form (3.1.23), and from (3.1.22)
it follows that <2(s) = <3(s) = … =<m(s) = 1. Hence from (3.1.20), we obtain the
condition (3.1.21).
„
Example 3.1.3.
For the rational matrix (3.1.2) the Smith canonical from of the polynomial matrix
(3.1.3) is
170
Polynomial and Rational Matrices
LS s
ª1
«0
¬
0
º
s 1 s 2 »¼
(3.1.24)
and the McMillan canonical form of the rational matrix (3.1.2)
WM s
1
ª
« s 1 s 2
«
0
¬«
LS s
m s
º
0»
».
1 ¼»
(3.1.25)
The McMillan polynomial of this matrix
wM s
(3.1.26)
s 1 s 2
satisfies the condition (3.1.21). Hence the rational matrix (3.1.2) is normal. This
result is consistent with the considerations in Sect. 3.1.1.
The McMillan canonical form of the rational matrix (3.1.4) is
WM s
LS s
m s
ª1
2 «
s 1 ¬0
1
0
º
s 1 s 2 ¼»
ª 1
«
« s 1
«
« 0
¬
2
º
0 »
» (3.1.27)
s 2»
»
s 1¼
and its McMillan polynomial is
wM s
3
s 1 .
(3.1.28)
The polynomial (3.1.28) does not satisfy the condition (3.1.21), since
m(s) = (s + 1)2 z wM(s) = (s + 1)3.
Thus the rational matrix (3.1.4) is not normal. This result is consistent with the
considerations in Sect. 3.1.1.
3.2 Fraction Description of Normal Matrices
According to Definition 1.14.1, a polynomial matrix A(s) mun[s] is called simple
if and only if it has only one invariant polynomial different from 1. For example,
the Smith canonical form AS(s) of a simple square full rank matrix A(s) mum[s] is
AS s
diag ¬ª1, ... 1, im s ¼º ,
where im(s) is an invariant polynomial different from 1.
(3.2.1)
Normal Matrices and Systems
171
Consider a rational matrix in the canonical form
L s
 mun s ,
m s
W s
where L(s)
matrix W(s).
(3.2.2)
mun
[s] and m(s) is a least common denominator of entries of the
Theorem 3.2.1. Let
W s
Dl1 s N l s
N p s D p1 s  mun s , min m, n t 2 . (3.2.3)
Then the matrix (3.2.3) is normal if and only if the matrices Dl(s)
Dp(s) mun[s] are simple.
mum
[s] and
Proof. Sufficiency. We will consider in detail the case when n > m. The
considerations for m t n are analogous.
Let
LS s
U s L s V s
ªi1 s
«
« 0
« #
«
¬« 0
0
i2 s
#
0
!
!
%
0
0
#
! im s
0
0
#
0
! 0º
»
! 0»
, n!m
% #»
»
! 0 ¼»
(3.2.4)
be the Smith canonical form of L(s), where U(s) and V(s) are unimodular matrices
of elementary operations on rows and columns. It follows from the definition of a
normal matrix that every nonzero second-order minor of L(s) is divisible without
remainder by the polynomial m(s). From the Binet–Cauchy theorem it follows that
also every nonzero second-order minor of LS(s) is divisible without remainder by
the same polynomial. Hence, if the matrix W(s) is normal, the matrix LS(s) has the
form
LS s
ªi1 s
«
« 0
« #
«
¬« 0
0
i2 s m s
#
0
0
0
!
%
#
! im s m s
!
0
0
#
0
! 0º
»
! 0»
,
% #»
»
! 0 ¼»
(3.2.5)
where the fraction i1(s)/m(s) is irreducible (otherwise m(s) would not be a least
common denominator of the entries of the matrix W(s)). In this case, we have
172
Polynomial and Rational Matrices
W s
L s
m s
ª i1 s
0
«
«m s
U s « 0
i 2 s
«
#
« #
« 0
0
¬
1
Dl s N l s ,
!
0
!
0
%
#
º
0 ! 0»
»
0 ! 0» V s
»
# % #»
0 ! 0 »¼
! im s
(3.2.6)
where
Nl s
Dl s
ªi1 s
«
« 0
« #
«
¬« 0
0
i
^diag ª¬m
!
0
#
!
%
0
#
0
! im s
2
s
0 ! 0º
»
0 ! 0»
V s  mun > s @ ,
# % #»
»
0 ! 0 ¼»
(3.2.7)
`
s , 1, ..., 1º¼ U 1 s  mum > s @ ,
Note that deg Dl(s) = deg m(s), since deg det U-1(s) = 0.
Necessity. Let Dl(s) be a simple matrix, i.e.,
ª1
«0
«
Dl s U s « #
«
«0
«0
¬
U s diag ª¬1,
0 ! 0
1 ! 0
0 º
0 »»
# % #
# »V s
»
0 ! 1
0 »
0 ! 0 m s »¼
(3.2.8)
1, ! 1, m s º¼ V s ,
where U (s) and V (s) are unimodular matrices of elementary operations on rows
and columns.
The inverse of the matrix of (3.2.8) has the form
Dl1 s
V 1 s
diag ª¬ m s , m s , ... m s , 1º¼ U 1 s .
m s
(3.2.9)
Every nonzero second degree minor of the diagonal matrix
diag ª¬ m s , m s , ... m s , 1º¼
(3.2.10)
Normal Matrices and Systems
173
is divisible without remainder by the polynomial m(s). Thus it follows from the
Binet–Cauchy theorem that every nonzero second-order minor of
V 1 s diag ª¬ m s , m s , ... m s , 1º¼ U 1 s
is also divisible without remainder by the polynomial m(s).
The rational matrix W(s) = Dl-1(s)Nl(s) is thus normal. The proof for
W(s) = Np(s)Dp-1(s) is analogous.
„
Corollary 3.2.1. Every nonzero k-th order minor (k > 2) of the polynomial matrix
L(s) in the normal rational matrix (3.2.2) is divisible without remainder by the
polynomial mk-1(s).
The corollary follows immediately from the form of the matrix (3.2.5).
Corollary 3.2.2 Every column matrix (3.2.2) for L(s) = [l1(s), l2(s), …, lm(s)]T in
the fraction form Dl-1(s)Nl(s) has the simple matrix Dl(s). Every row rational matrix
(3.2.2) for L(s) = [l1(s), l2(s), …, ln(s)] in the fraction form Np(s)Dp-1(s) has the
simple matrix Dp(s).
Proof. The proof is carried out only for the first case because it is analogous the
second one.
Applying elementary operations on the rows of the matrix
L(s) = [l1(s), l2(s), …, lm(s)]T, we transform it to the form
U s L s
T
0 ... 0 º¼ ,
ª¬l s
(3.2.11)
where U(s) is an unimodular matrix of elementary operations on rows.
Taking into account (3.2.11), we obtain
ªl
«
U s «
m s «
«
¬
W s
L s
m s
Dl s
^diag ª¬m
Nl s
ªl
«
«
«
«
¬
1
s º
»
0 »
# »
»
0 ¼
Dl1 s N l s ,
(3.2.12)
where
s
0
#
`
s , 1, ..., 1º¼ U s  mum > s @ ,
º
»
»  mu1 > s @ .
»
»
0 ¼
(3.2.13)
174
Polynomial and Rational Matrices
Thus the matrix Dl(s) is simple.
In the second case, the matrices Dp(s) and Np(s) have the forms
Dp s
Np s
^
`
V s diag ª¬ m s , 1, ..., 1º¼  nun > s @ ,
1un
ª¬l s , 1, ..., 1º¼  >s@
(3.2.14)
where V(s) is an unimodular matrix of elementary operations on columns.
„
Theorem 3.2.2. Let a least common denominator of the rational matrix (3.2.2)
have the form
m s
s s1
n1
s s2
n2
... s s p
np
.
(3.2.15)
The matrix (3.2.2) is normal if and only if its every second-order minor has a pole
of multiplicity not greater than ni, i = 1,2,…,p at the point s = si, i = 1,2,…,p.
Proof. Sufficiency. Let Wklij be a second-order minor consisting of rows i, j and
columns k, l of the matrix (3.2.2), i.e.,
Wklij
lik s
m s
lil s
m s
lik s l jl s lil s l jk s
l jk s
l jl s
m2 s
m s
m s
,
(3.2.16)
where lij(s) is the (i, j) entry of the matrix L(s).
If the matrix (3.2.2) is normal, then the polynomial lik(s)ljl(s) – lil(s)ljk(s) is
divisible without remainder by the polynomial m(s) and we obtain
Wklij
wklij s
.
m s
(3.2.17)
From (3.2.17) it follows that the minor Wklij has a pole of a multiplicity not greater
than ni, i = 1,2,…,p at the point s = si, i = 1,2,…,p.
Necessity. If on the other hand, the minor Wklij has a pole of multiplicity not greater
than ni, i = 1,2,…,p at the point s = si, i = 1,2,…,p, then from (3.2.16) it follows
that the polynomial lik(s)ljl(s) – lil(s)ljk(s) is divisible without remainder by the
polynomial m(s) and the matrix (3.2.2) is normal.
„
Normal Matrices and Systems
175
Corollary 3.2.3. Let m(s) of the standard rational matrix (3.2.2) have the form
(3.2.15). If the matrix (3.2.2) is normal, then
rank L si
1, for i 1, 2, ..., p .
(3.2.18)
Proof. If the matrix (3.2.2) is normal then from (3.2.9) it follows that (3.2.18)
holds.
„
Corollary 3.2.4. Let the polynomial (3.2.15) have only roots of multiplicity 1
(ni = 1, i = 1,2,…,p). The matrix (3.2.2) is normal if and only if the condition
(3.2.18) is satisfied.
Proof. Sufficiency follows from Corollary 3.2.3. If the condition (3.2.18) is
satisfied, then every nonzero second-order minor of the matrix L(s) is divisible by
m(s), thus the matrix (3.2.2) is normal.
„
3.3 Sum and Product of Normal Matrices and Normal Inverse
Matrices
3.3.1 Sum and Product of Normal Matrices
Consider two normal matrices of the standard form
W1 s
L1 s
m1 s
ª lij1 s º
«
» , W2 s
¬« m1 s »¼
L2 s
m2 s
ª lij2 s º
«
».
¬« m2 s »¼
(3.3.1)
We will state conditions under which the sum and product of normal matrices
of the form (3.3.1) are themselves normal matrices.
Theorem 3.3.1. The sum of normal matrices of the form (3.3.1) and of compatible
dimensions is itself a normal matrix if the polynomials m1(s) and m2(s) are
relatively prime (they do not have any common roots).
Proof. The sum of matrices of the form (3.3.1) is
W s
W1 s W2 s
L1 s
L s
2
m1 s
m2 s
m2 s L1 s m1 s L 2 s
.
m1 s m2 s
(3.3.2)
176
Polynomial and Rational Matrices
If m1(s) and m2(s) do not have common roots and the matrices are of the standard
form (3.3.1) then the right-hand side of (3.3.2) is an irreducible fraction.
The minor Wklij of the matrix (3.3.2) consisting of its rows i, j and columns k, l
has the form
Wklij
lik1 s
l2 s
ik
m1 s
m2 s
lil1 s
l2 s
il
m1 s
m2 s
l1jk s
l 2jk s
l1jl s
m2 s
m1 s
m1 s
l 2jl s
.
(3.3.3)
m2 s
Applying the known rule of addition of rational functions as well as computing the
second-order minors and performing appropriate reductions, we obtain an
irreducible rational function of the form
Wklij
wklij s
,
m1 s m s
(3.3.4)
where wklij(s) is a polynomial.
As is well known, the sum of rational matrices of the form (3.3.1) is itself a
normal matrix if the polynomials m1(s) and m2(s) are relatively prime.
„
From Theorem 3.3.1 the following important corollary can be immediately
derived.
Corollary 3.3.1. If W(s) mun(s) is a normal matrix and P(s)
polynomial matrix, then their sum
W s P s
mun
(s) is a
(3.3.5)
is normal.
Theorem 3.3.2. The product of the normal matrices (3.3.1) (of compatible
dimensions) is a normal matrix if the matrix
L1 s L 2 s
m1 s m2 s
is irreducible.
Proof. If the condition (3.3.6) is satisfied, then
(3.3.6)
Normal Matrices and Systems
L1 s L 2 s
m1 s m2 s
W1 s W2 s
L s
.
m1 s m2 s
177
(3.3.7)
It follows from the Binet–Cauchy theorem that every nonzero second-order minor
of L(s) is the sum of products of the second-order minors of L1(s) and L2(s). Thus
every nonzero second-order minor of W(s) is divisible by the polynomial
m1(s)m2(s), hence (3.3.7) is a normal matrix.
„
Example 3.3.1.
Given the two normal matrices
W1 s
ª 1
« s 1
«
« 0
¬«
º
0 »
»
1 »
s 2 ¼»
0 º
ªs 2
,
«
s 1»¼
s 1 s 2 ¬ 0
W2 s
ª 1
«s 3
«
« 0
¬«
º
0 »
»
1 »
s 4 ¼»
0 º
ªs 4
,
«
s 3»¼
s3 s4 ¬ 0
1
(3.3.8)
1
compute the sum and the product of these matrices and check whether they are
normal.
The sum of the matrices (3.3.8) is
W s
W1 s W2 s
0 º
0 º
ªs 2
ªs 4
1
«
»
«
s 1¼ s 3 s 4 ¬ 0
s 3»¼
s 1 s 2 ¬ 0
1
ª 2s 5 s 2 s 4
«
0
s 1 s 2 s 3 s 4 ¬
1
0
º
»
2s 6 s 1 s 3 ¼
and is itself a normal matrix.
The product of the matrices (3.3.8) is
W s
W1 s W2 s
0 º
0 º
ªs 2
ªs 4
1
«
»
«
s 1¼ s 3 s 4 ¬ 0
s 3»¼
s 1 s 2 ¬ 0
1
ª s2 s4
«
0
s 1 s 2 s 3 s 4 ¬
1
0
º
»
s 1 s 3 ¼
178
Polynomial and Rational Matrices
and is itself a normal matrix as well. This result is consistent with Theorems 3.3.1
and 3.3.2, since the matrices (3.3.8) satisfy the assumptions of these two theorems.
Example 3.3.2.
Compute the sum and the product of the normal matrices
W1 s
W2 s
ª 1
º
0 »
« s 1
«
»
1 »
« 0
«¬
s 2 »¼
ª 1
º
0 »
«s 2
«
»
1 »
« 0
«¬
s 3 »¼
0 º
ªs 2
1
,
s 1»¼
s 1 s 2 «¬ 0
(3.3.9)
0 º
ªs 3
.
«
s 2 »¼
s2 s3 ¬ 0
1
The sum of the matrices (3.3.9) is
W1 s W2 s
2s 3
ª
0
« s 1 s 2
«
«
2s 5
0
«
2 s3
s
¬
0
ª 2s 3 s 3
º
1
«
»
0
2s 5 s 1 ¼
s 1 s 2 s 3 ¬
ª 1
« s 1
«
« 0
¬«
º ª 1
0 » «
s2
»«
1 » «
0
s 2 ¼» ¬«
º
0 »
»
1 »
s 3 ¼»
º
»
»
»
»
¼
and this not a normal matrix. The matrices (3.3.9) do not satisfy the assumption of
Theorem 3.3.1, since the polynomials m1(s) = (s + 1)(s + 2) and m2(s) = (s + 1)
u(s + 3) are not relatively prime.
The product of the matrices (3.3.9) is
1
ª
0
« s 1 s 2
«
W1 s W2 s
«
1
0
«
2
s
s3
¬
ªs 3 0 º
1
s 1»¼
s 1 s 2 s 3 «¬ 0
º
»
»
»
»
¼
and it is not a normal matrix. The matrix (3.3.9) does not satisfy the assumption
(3.3.6), since the matrix
Normal Matrices and Systems
L1 s L 2 s
m1 s m2 s
1
s 1 s 2
2
s3
ª s2 s3
«
0
¬
179
0
º
»
s 1 s 2 ¼
is reducible by s + 2.
Now the following problem will be solved.
Problem 3.3.1. There are two normal matrices given in the fraction forms
W1 s
D11 s N1 s , W2 s
D2 1 s N 2 s ,
(3.3.10)
where the matrices D1(s) and D2(s) are simple and the condition of irreducibility
(3.3.6) is satisfied. Compute an irreducible pair (D(s), N(s)) such that
W1 s W2 s
D1 s N s .
(3.3.11)
Taking into account (3.3.10), we obtain
W1 s W2 s
D11 s N1 s D21 s N 2 s .
(3.3.12)
From the assumption (3.3.6) it follows that the matrix N1(s)D2-1(s) is irreducible.
Applying the procedure introduced in Sect. 3.2.6, we can compute for the pair
N1(s), D2(s) an irreducible pair ( D 2(s), N 1(s)) such that
N1 s D21 s
D21 s N1 s .
(3.3.13)
Substituting (3.3.13) into (3.3.12), we obtain
W1 s W2 s
D11 s D21 s N1 s N 2 s
D 1 s N s ,
(3.3.14)
where
D s
D2 s D1 s , N s
N1 s N 2 s .
(3.3.15)
The dual problem to Problem 3.3.1 can be formulated as follows.
Problem 3.3.1c. Two normal matrices are given in the fraction form
W1 s
N1 s D11 s , W2 s
N 2 s D21 s ,
(3.3.16)
where D1(s) and D2(s) are simple and the irreducibility condition (3.3.6) is
satisfied. Compute an irreducible pair (D(s), N(s)) such that
180
Polynomial and Rational Matrices
W1 s W2 s
N s D1 s .
(3.3.17)
Taking into account (3.3.16), we obtain
W1 s W2 s
N1 s D11 s N 2 s D21 s .
(3.3.18)
From the irreducibility condition (3.3.6) it follows that D1-1(s)N2(s) is an
irreducible matrix. Applying the procedure introduced in Sect. 3.2.6 we can
compute for the pair (D1(s), N2(s)) an irreducible pair ( N 2(s), D 1(s)) such that
D11 s N 2 s
N 2 s D11 s .
(3.3.19)
Substituting (3.3.19) into (3.3.18), we obtain
W1 s W2 s
N1 s N 2 s D11 s D21 s
N s D1 s ,
(3.3.20)
where
N s
N1 s N 2 s , D s
D2 s D1 s .
(3.3.21)
3.3.2 The Normal Inverse Matrix
Consider a rational nonsingular matrix W(s) nun(s). We will provide the
conditions under which the inverse W-1(s) is a normal matrix.
Let
W s
Dl1 s N l s
N p s Dp1 s  nun s
nt2 .
(3.3.22)
The assumption det W(s) z 0 implies nonsingularity of the matrices Nl(s)
and Np(s) nun[s]. From (3.3.22), we have
W 1 s
N l1 s Dl s
D p s N p1 s .
nun
[s]
(3.3.23)
From (3.3.23) and Theorem 3.2.1 the following theorem ensues immediately.
Theorem 3.3.3. The inverse matrix (3.3.23) is normal if and only if the matrices
Nl(s) and Np(s) are simple.
The following example shows that the inverse matrix (3.3.23) may be a normal
matrix even when W(s) is not.
Normal Matrices and Systems
181
Example 3.3.3.
The matrix
ª s2
2
«
« s 1
«
« 0
¬
W s
º
0 »
»
1 »
»
s 1¼
0 º
ªs 2
« 0
s 1»¼
s 1 ¬
1
(3.3.24)
2
is not normal, since
0 º
ªs 2
det «
s 1»¼
¬ 0
s 1 s 2
is not divisible by (s + 1)2.
The inverse of the matrix (3.3.24) has the form
W
1
ª s 1 2
«
« s2
«¬ 0
s
º
0 »
»
s 1»¼
1 ª s 1
«
s 2 «¬ 0
2
º
»
s 1 s 2 »¼
0
(3.3.25)
and is normal, since
ª s 1
det «
«¬ 0
2
º
»
s 1 s 2 »¼
0
s 1
3
s2
is divisible by (s + 2).
According to Theorem 3.2.1 the matrix (3.3.22) is normal if and only if Dl(s)
and Dp(s) are simple matrices. Thus the inverse (3.3.23) of the normal matrix
(3.3.22) is normal if and only if the matrices Nl(s) and Np(s) are simple. Thus the
following theorem has been established.
Theorem 3.3.4. The inverse (3.3.23) of the normal matrix (3.3.22) is a normal
matrix if and only if Dl(s), Dp(s), Nl(s) and Np(s) are simple matrices.
Example 3.3.4.
The matrix
W s
ª 1
«s 1
«
« 0
¬«
º
0 »
»
s »
s 2 ¼»
0 º
ªs 2
« 0
s s 1 »¼
s 1 s 2 ¬
1
(3.3.26)
182
Polynomial and Rational Matrices
is normal. This matrix can be expressed in the form (3.3.22) as
1
W s
0 º ª1 0 º
ªs 1
« 0
s 2 »¼ «¬0 s »¼
¬
Dl s
Dp s
1
0 º
ª1 0 º ª s 1
,
«0 s » « 0
s 2 »¼
¬
¼¬
where
0 º
ªs 1
, Nl s
« 0
s 2 »¼
¬
Np s
ª1 0 º
«0 s » .
¬
¼
(3.3.27)
It is easy to check that the matrices (3.3.27) are simple.
The inverse W-1(s) of (3.3.26) is
W 1 s
0 º
ªs 1
«
s
2 »»
« 0
s ¼
¬
1 ªs s 1
«
s¬ 0
0 º
».
s 2¼
It is a normal matrix, since
ªs s 1
det «
¬ 0
0 º
»
s 2¼
s s 1 s 2
is divisible by s.
3.4 Decomposition of Normal Matrices
3.4.1 Decomposition of Normal Matrices into the Sum of Normal Matrices
Consider a normal matrix in the standard form (3.1.1), where the polynomial
m(s) = m1(s)m2(s) with m1(s) and m2(s) relatively prime, i.e.
W s
L s
.
m1 s m2 s
(3.4.1)
We will show that a matrix of the form (3.4.1) may be decomposed into the
sum of two normal matrices
L1 s
,
m1 s
L2 s
,
m2 s
Normal Matrices and Systems
183
i.e.,
L s
m1 s m2 s
L1 s
L s
2
.
m1 s
m2 s
(3.4.2)
Note that L1(s)/m1(s) and L2(s)/m2(s) are irreducible, otherwise a matrix of the
form (3.4.1) for L(s) = m2(s)L1(s) + m1(s)L2(s) would be also a reducible matrix.
L(s)/m1(s) is a normal matrix, since every nonzero second-order minor of L(s) is
divisible by m1(s). In view of this we have
L s
m1 s
D11 s N1 s ,
(3.4.3)
where D1(s) is a simple matrix such that det D1(s) = m1(s).
Pre-multiplying (3.4.2) by D1(s) and taking into account (3.4.3), we obtain
N1 s
m2 s
D1 s
L1 s
L s
D1 s 2
m1 s
m2 s
and
N1 s
L s
D1 s 2
m2 s
m2 s
D1 s
L1 s
.
m1 s
(3.4.4)
We will show that L1(s)/m1(s) is a normal matrix.
Note that the left-hand side of (3.4.4) is analytic at the roots of the polynomial
m1(s), and the right-hand side is analytic at the roots of the polynomial m2(s). Thus
the matrix
N1 ( s )
D1 ( s )
L1 ( s )
m1 ( s )
is polynomial and
L1 s
m1 s
D11 s N1 s ,
(3.4.5)
where D1-1(s) N 1(s) is irreducible and D1(s) is simple. Thus L1(s)/m1(s) is a normal
matrix.
The proof that L2(s)/m2(s) is a normal matrix follows in the same vein.
Thus the following theorem has been established.
184
Polynomial and Rational Matrices
Theorem 3.4.1. If the polynomials m1(s) and m2(s) are relatively prime, then a
normal matrix of the form (3.4.1) may be decomposed into the sum of the two
normal matrices
L1 s
m1 s
and
L2 s
.
m2 s
Matrices L1(s) and L2(s) can be computed by the method introduced in Sect.
3.2.4. If a matrix of the form (3.4.1) is strictly proper, then the decomposition
(3.4.2) is unique.
Example 3.4.1.
Decompose the following normal matrix
W s
2s 4
ª
« s 1 s 3
«
«
0
«
¬
º
0 »
»
s 3»
»
s 2¼
ª 2 s 2 8s 8
º
0
1
«
»
3
2
s 1 s 2 s 3 ¬
0
s 7 s 15s 9 ¼
(3.4.6)
into the sum of two normal matrices L1(s)/m1(s) and L2(s)/m2(s), taking
m1 s
s 2 3s 2 and m2 s
s 1 s 2
s 3.
We shall look for the matrices L1(s) and L2(s) of the forms
L1 s
ª a1s a0
« 0
¬
0 º
, L2 s
b1s b0 »¼
ªc1s c0
« 0
¬
0 º
.
d1s d 0 »¼
(3.4.7)
Taking into account (3.4.6) and (3.4.7), we obtain
ª 2s 2
1
«
s 1 s 2 s 3 ¬
L1 s
L s
m2 s
2
m1 s
m2 s
1
2
s 3s 2 s 3
0
º
8s 8
0
»
3
2
0
s 7 s 15s 9 ¼
L1 s m1 s L2 s
m1 s m2 s
ª s 3 a1s a0 s 2 3s 2 c1s c0
«
0
«¬
º
».
s 3 b1s b0 s 3s 2 d1s d 0 »¼
2
(3.4.8)
Normal Matrices and Systems
185
Comparison of the coefficients of the same powers of the variable s in (3.4.8)
yields c1 = 0, d1 = 1 and
a1 c0 3c1 2, 3a1 a0 2c1 3c0
b1 d 0 3d1 7, 3b1 b0 2d1 3d 0
8, 3a0 2c0
15, 3b0 2d 0
8,
9.
(3.4.9)
Solving (3.4.9), we obtain
a1 1, a0
2, c1 1, c0
1, b1 1, b0
1, d 0
3, d1 1 .
The desired matrices L1(s) and L2(s) are
0 º
ªs 2
, L2 s
« 0
s 1»¼
¬
L1 s
0 º
ª1
« 0 s 3» .
¬
¼
It is easy to check that the matrices
0 º
ªs 2
1
,
s 1»¼
s 1 s 2 «¬ 0
0 º
1 ª1
s 3 ¬«0 s 3¼»
L1 s
m1 s
L2 s
m2 s
are normal.
3.4.2 Structural Decomposition of Normal Matrices
Consider a rational matrix of the standard form
L s
,
m s
W s
where L(s)
(3.4.10)
mun
[s] and m(s) is a monic polynomial.
Theorem 3.4.2. The matrix (3.4.10) is normal if and only if
P s Q s m s G s ,
L s
where P(s)
m
[s], Q(s)
1un
[s], G(s)
(3.4.11)
mun
[s] and
deg P s deg m s , deg Q s deg m s .
(3.4.12)
186
Polynomial and Rational Matrices
Proof. Sufficiency. If (3.4.11) holds, then the computation of the second-order
minor built from the rows i, j and the columns k, l of the matrix L(s) yields
pi s qk s m s gik s
p j s qk s m s g jk s
Li,k,lj s
ij
kl
m s l
pi s ql s m s g il s
p j s ql s m s g jl s
(3.4.13)
s ,
where pi(s), qk(s) and gik(s) are entries of P(s), Q(s) and G(s); lklij(s) is a
polynomial. From (3.4.13) it follows that the minor Lklij(s) is divisible by m(s).
Thus the matrix (3.4.10) is normal.
Necessity. If the matrix (3.4.10) is normal, then every nonzero second-order minor
of the matrix (3.4.11) is divisible by m(s). In this case the matrix (3.4.11) has the
form
L s
ªi1 s
«
0
U s «
« #
«
¬« 0
0
i2 s m s
#
0
!
!
0
0
%
#
! im s m s
0 ! 0º
»
0 ! 0»
V s , (3.4.14)
# % #»
»
0 ! 0 ¼»
n!m
where i1(s), i2(s), … ,im(s) are the invariant polynomials; U(s) and V(s) are
unimodular matrices.
The matrix (3.4.14) can be decomposed into the sum of two matrices
(L(s) = L1(s) + m(s)L2(s))
L1 s
ª1
«0
i1 s U s «
«#
«
¬0
L2 s
0
ª0
«0 i s
2
U s «
«#
#
«
0
0
¬«
0
0
#
0
! 0º
! 0 »»
V s i1 s U1 s V1 s ,
% #»
»
! 0¼
!
0
0 ! 0º
!
0
0 ! 0 »»
V s ,
%
#
# % #»
»
! im s 0 ! 0 ¼»
(3.4.15)
where U1(s) is the first column of the matrix U(s), and V1(s) is the first row of the
matrix V(s).
Taking
P s
i1 s U1 s , Q s
V1 s , G s
L2 s ,
(3.4.16)
Normal Matrices and Systems
187
we obtain the desired decomposition (3.4.11).
If the condition (3.4.12) is not satisfied, then the division of every entry pi(s)
(qk(s)) of the vector P(s) (or Q(s)) by m(s) yields
P s
m s K1 s P s , Q s
m s K2 s Q s ,
(3.4.17)
where deg P (s) < deg m(s), deg Q (s) < deg m(s); K1(s) and K2(s) are a
polynomial column and row vector, respectively.
Substituting (3.4.17) into (3.4.11), we obtain
L s
P s Q s m s G s ,
(3.4.18)
G s
G s m s K1 s K 2 s P s K 2 s K1 s Q s .
where
„
Example 3.4.2.
Compute the structural decomposition of the following rational matrix
W s
L s
m s
0 º
ªs 2
.
«
s 1»¼
s 1 s 2 ¬ 0
1
(3.4.19)
Applying the elementary operations L[2 + 1], P[1 + 2u(-1)], L[2 + 1u(s + 1)],
P[2 + 1u(-s-1)] to the matrix
L s
0 º
ªs 2
,
« 0
s 1»¼
¬
(3.4.20)
we obtain its Smith canonical form
LS s
U s L s V s
ª1
«0
¬
0
º
,
s 1 s 2 »¼
as well as the unimodular matrices
U s
Therefore,
1 º
ª 1
« s 1 s 2» , V s
¬
¼
ª 1 s 1
«
s2
¬ 1
º
».
¼
188
Polynomial and Rational Matrices
L s
U 1 s L S s V 1 s
ª s2
« s 1
¬
1º ª1
1 »¼ «¬ 0
0
º ª s 2 s 1º
.
s 1 s 2 »¼ «¬ 1
1 »¼
In this case, according to (3.4.16) we obtain
P s
i1 s U1 s
ª s2
« s 1
¬
º
», Q s
¼
V1 s
>s 2
s 1@
and
0 º 1
ª0
U 1 s «
»V s
¬0 im s ¼
0
1º ª0
ª s2
º ª s 2 s 1º
« s 1 1 » «0 s 1 s 2 » «
1 »¼
¬
¼¬
¼¬ 1
G s
ª 1 1º
s 1 s 2 «
»,
¬1 1¼
where U1(s) and V1(s) are the first column of U-1(s) and the first row of V-1(s),
respectively.
The desired decomposition of L(s) has the form
0 º
ªs 2
« 0
s 1¼»
¬
ª s2 º
ª 1 1º
« s 1 » > s 2 s 1@ s 2 s 1 «
».
¬1 1¼
¬
¼
We will show that one does not necessarily have to compute the Smith
canonical form of L(s) in order to obtain the decomposition (3.4.11).
Applying elementary operations on rows and columns, we can write the
polynomial matrix L(s) in the form:
U s L s V s
ª 1
i s «
¬k s
w s º
»,
L s ¼
(3.4.21)
where U(s) and V(s) are unimodular matrices of elementary operations
w(s) 1u(n-1)[s], k(s) m-1[s], L (s) (m-1)u(n-1)[s] and i(s) [s].
This follows immediately from the possibility of reduction of L(s) to its Smith
canonical form LS(s).
Let
P s
ª 1 º
U 1 s i s «
», Q s
¬k s ¼
ª¬1 w s º¼ V 1 s .
(3.4.22)
Normal Matrices and Systems
189
Since the second-order minors of L(s) are divisible by m(s), the entries of the
matrix i(s)[ L (s) – k(s)w(s)] are divisible by m(s), that is,
m s Lˆ s ,
i s ª¬L s k s w s º¼
where L̂ (s)
(m-1)u(n-1)
(3.4.23)
[s].
Defining
G s
01,n1 º 1
ª 0
U 1 s «
»V s ,
ˆ
¬«0m1 L s ¼»
(3.4.24)
we obtain from (3.4.21)–( 3.4.24)
L s
ª 1
w s º 1
U 1 s i s «
»V s
¬k s L s ¼
­°
ª 0
ª 1 º
ª1 w s ¼º «
U 1 s ®i s «
»
¬
«¬ 0 m 1
¬k s ¼
°¯
P s Q s m s G s ,
01,n 1 º ½° 1
»¾ V s
m s Lˆ s »¼ °¿
which is the desired decomposition (3.4.11).
Example 3.4.3.
Compute the decomposition (3.4.11) of the polynomial matrix (3.4.20).
Carrying out the elementary operations L[1 + 2], P[1 + 2u(-1)] on the matrix
(3.4.20), we obtain
U s L s V s
s 1º
ª 1
« s 1 s 1» ,
¬
¼
where
U s
ª1 1º
«0 1» , V s
¬
¼
ª 1 0º
« 1 1 » , i s
¬
¼
1.
In this case, using (3.4.22)(3.4.24) we obtain
P s
ª 1 º
U 1 s i s «
»
¬k s ¼
ª1 1º ª 1 º
«0 1 » « s 1»
¬
¼¬
¼
ª s2
« s 1
¬
º
»,
¼
190
Polynomial and Rational Matrices
Q s
G s
ª¬1 w s º¼ V 1 s
>1
ª1 0 º
s 1@ «
»
¬1 1 ¼
0 º 1
ª0
U 1 s «
»V s
ˆ
¬« 0 L s »¼
ª 1 1º
s 1 s 2 «
».
¬1 1¼
>s 2
ª1 1º ª 0
«0 1 » «0
¬
¼¬
s 1@ ,
0
º ª1 0 º
»
s 1 s 2 ¼ «¬1 1 »¼
Thus the desired decomposition (3.4.20) has the form
0 º
ªs 2
« 0
s 1»¼
¬
ª s2 º
« s 1 » > s 2 s 1@ s 1 s 2
¬
¼
ª 1 1º
«1 1».
¬
¼
The result is consistent with that obtained in Example 3.4.2.
Corollary 3.4.1. Let s1,s2,…,sp be the poles (not necessarily distinct) of the rational
matrix (3.4.10). Then
rank L sk
1, for k
1, 2, ..., p .
(3.4.25)
The condition (3.4.10) follows from (3.4.11), since rank P(sk) = rank Q(sk) = 1.
Corollaries 3.3.4 and 3.4.1 give the following criterion of normality of the
matrix (3.4.10).
Criterion 3.4.1. If the poles s1,s2,…,sp of a matrix are distinct, then this matrix is
normal if and only if the condition (3.4.25) is satisfied. If the poles are multiple,
then the matrix (3.4.10) is not normal when
rank L sk ! 1 for certain k  ^1, 2, ..., p` .
(3.4.26)
Example 3.4.3.
The rational matrix
W s
m s
ª 1
« s 1
«
« 1
«
¬s 2
1
s2
1
º
s 1 s 2 »
»,
»
1
1
»
s 1
s2
¼
1 º
ªs 2 s 1
s 1 s 2 , L s «
s
s
s
1»¼
1
2
¬
L s
m s
(3.4.27)
Normal Matrices and Systems
191
has only the distinct poles s1 = 1, s2 = 2.
This matrix is not normal, since
rank L s1
ª1 0 1 º
rank «
»
¬0 1 0¼
2 ! 1.
We will obtain the same result by checking divisibility of the second-order minors
of L(s) by m(s).
The minor
s 1
1
2
s 1 s 2
s 2 s 1
s2 s 1
is not divisible by m(s) = (s + 1)(s + 2).
Note that in the case of poles of multiplicities greater than 1, the condition
(3.4.25) is not a sufficient condition for normality of the matrix (3.4.10).
For example, the matrix with the double pole at the point s1 = 1
W s
ª s2
2
«
« s 1
«
« 0
¬
L s
m s
º
0 »
»
1 »
»
s 1¼
0 º
ªs 2
« 0
s 1»¼
s 1 ¬
1
2
is not normal although it satisfies the condition (3.4.10), since
rank L s1
ª1 0 º
rank «
» 1.
¬0 0¼
3.5 Normalisation of Matrices Using Feedback
3.5.1 State-feedback
Consider the system
x
y
Ax Bu ,
Cx ,
with the state-feedback in the form
(3.5.1a)
(3.5.1b)
192
Polynomial and Rational Matrices
u
v Kx ,
(3.5.2)
where v m is a new input, and K
into (3.5.1a) yields
x
mun
is a gain matrix. Substitution of (3.5.2)
A BK x Bv .
(3.5.3)
The transfer matrix of the closed-loop system has the form
1
Tz (s) C > I n s (A BK ) @ B .
(3.5.4)
The problem of normalisation of a transfer matrix using state-feedback can be
formulated in the following way.
Problem 3.5.1. Given a system in the form (3.5.1), with the matrix A non-cyclic
and the pair (A, C) unobservable, compute the matrix K in such a way that the
closed-loop transfer matrix of the system (3.5.4) is normal.
The solution to this problem is based on the following lemma.
Lemma 3.5.1. If the matrix A is in its Frobenius canonical form
A
ª0 #
« k
«¬
I n1 º
nun
» , k
»¼
> k1
k 2 ... kn @ ,
(3.5.5)
then for any nonzero matrix C the observability of the pair (A, C) can be always
assured by an appropriate choice of k.
Proof. The pair (A, C) is observable if and only if
ªI s A º
rank « n
»
¬ C ¼
n for all s  .
Applying elementary operations on rows and columns, we can transform the matrix
ª s 1 0
«0
s 1
«
« #
#
#
«
0
0
«0
« k1 k2 k3
«
« c11 c12 c13
« #
#
#
«
¬« c p1 c p 2 c p 3
!
!
%
0
0
#
!
s
! kn1
! c1,n1
%
#
! c p ,n1
0
0
#
º
»
»
»
»
1 »
s kn »
»
c1n »
# »
»
c pn ¼»
Normal Matrices and Systems
193
to the following form
ª 0
« 0
«
« #
«
« 0
« p0 s
«
« p1 s
« #
«
«¬ p p s
1 0 !
0 1 !
p0 s
s n kn s n1 ... k2 s k1 ,
pi s
cin s n1 ... ci 2 s ci1 , i 1, 2, ..., p .
#
0
0
#
0
0
%
!
!
0
#
0
0
#
0
!
%
!
0º
0 »»
#»
»
1» ,
0»
»
0»
#»
»
0 »¼
(3.5.6)
where
Carrying out appropriate elementary operations on the rows of the matrix (3.5.6)
and appropriately choosing k1,…,kn, we obtain
ª 0 1 0
« 0 0 1
«
«# #
#
«
0
0
0
«
«a 0 0
«
«0 0 0
«# #
#
«
«¬ 0 0 0
!
!
%
!
!
!
%
!
0º
0 »»
#»
»
1»
and a z 0 .
0»
»
0»
#»
»
0 »¼
(3.5.7)
The matrix (3.5.7) for a z 0 is a full column rank matrix and thus the pair (A, C) is
observable.
„
Theorem 3.5.2. Let the matrix A of the system (3.5.1) by noncyclic and the pair
(A, C) unobservable. Then there exists a matrix K such that the transfer matrix
(3.5.4) is normal if and only if the pair (A, B) is controllable.
Proof. Necessity. As is well-known, the pair (A+ BK, B) is controllable if and
only if the pair (A, B) is controllable. If the pair (A, B) is not controllable, then the
transfer matrix (3.5.4) is not normal. Thus if the pair (A, B) is not controllable,
then there exists no K such that the transfer matrix (3.5.4) is normal.
194
Polynomial and Rational Matrices
Sufficiency. If the pair (A, B) is controllable, then there exists a nonsingular matrix
T nun(s) such that
A
TAT
A ij  1
di ud j
ª A11 ! A1m º
« # %
# »» , B
«
¬« A m1 ! A mm ¼»
TB
ªB1 º
« # »,
« »
«¬B m »¼
(3.5.8a)
, Bi  di um ,
where
A ij
­ ª0 I di 1 º
°«
» for i
°¬ ai ¼
®
°ª 0 º
° « a » for i z j
¯ ¬« ij ¼»
aij
[a0ij a1ij ... adijj 1 ], bi
j
ª0º
«b » ,
¬« i ¼»
, Bi
(3.5.8b)
where
[0" 0 1 bi,i 1 ... bim ] and d1 , ..., d m
are controllability indices satisfying the condition
m
¦d
i
n.
i 1
Let
1
Bˆ
ªb1 º
« »
«b2 »
«#»
« »
¬bm ¼
K
ª K1 º
( m1)un
, k
« k » , K1  ¬ ¼
ª1 b12
«0 1
«
«# #
«
¬0 0
" b1m º
" b2 m »»
% # »
»
" 1 ¼
1
(3.5.9)
and
> k1
k2 ... kn @  1un .
(3.5.10)
Using (3.5.8a) and (3.5.9), one can easily check that
B
BBˆ
diag [b1 , ..., bm ], bi
[0
"
0
1]T  di ,
(3.5.11)
Normal Matrices and Systems
195
where T denotes the transpose.
Let
K
Bˆ 1KT1
ni
¦d
ª an1 en1 1 º
«
»
#
«
»
« a e
»,
nm 1
nm 1 1
«
»
«¬ anm k »¼
(3.5.12)
where
i
k
,
k 1
and a ni is the ni-th row of the matrix A i, ei is the i-th row of the identity matrix In
and k is given by (3.5.10).
Using (3.5.9), (3.5.11) and (3.5.12), one can easily check that
Az
T(A BK )T1
ª0 1
«0 0
«
«# #
«
«0 0
«¬ k1 k2
A BKT1
0 !
1 !
# %
0º
0 »»
# ».
»
0 ! 1»
k3 ! kn »¼
ˆ ˆ 1KT1
A BBB
A BK
(3.5.13)
The matrix (3.5.13) is cyclic and k will be chosen in such a way that the pair
(Az, C) is observable. According to Lemma 3.5.1, if Az has the Frobenius canonical
form (3.5.13), then it is always possible to choose the elements k1,…,kn in such a
way that the pair (Az, C) is observable. If Az is cyclic, the pair (A, B) is
controllable and the pair (Az, C) is observable, then the transfer matrix (3.5.4) is
normal.
„
In a general case there exist many gain matrices K normalising the transfer
matrix.
If the pair (A, B) is controllable, then we can compute the matrix k using the
following procedure.
Procedure 3.5.1.
Step 1: Compute a nonsingular matrix T that transforms the pair (A, B) to the
ˆ B .
canonical form (3.5.8), and A, B, B,
Step 2: Using (3.5.12) compute K and
196
Polynomial and Rational Matrices
ˆ
K = BKT
(3.5.14)
for the unknown matrix k.
Step 3: Choose k in such a way that the pair (Az, C) is observable.
Step 4: Compute the desired matrix K substituting k (computed in Step 3) into
(3.5.14).
Example 3.5.1.
Consider the system (3.5.1) with matrices
A
ª0 1
«
«0 2
«0 0
«
¬0 0
0 0º
0 1»»
,B
0 1»
»
0 2 ¼
ª0
«1
«
«0
«
¬0
0º
2 »»
, C
0»
»
1¼
ª0 1 0 0 º
«0 0 0 1 » , D
¬
¼
0.
(3.5.15)
It is easy to check that A is not cyclic, the pair (A, B) is controllable and the pair
(A, C) is not observable.
We seek a matrix
K
ª k11 k12
«k k
2
¬ 1
k13
k3
k14 º
k4 »¼
such that the closed loop transfer matrix (3.5.4) is normal.
Applying the above procedure, we obtain the following.
Step 1: The matrices (3.5.15) already have the canonical form (3.5.8) and
A
Bˆ
ª A11
«A
¬ 21
ªb1 º
«b »
¬ 2¼
1
A12 º
A 22 »¼
ª0 1
«
«0 2
«0 0
«
¬0 0
ª1 2 º
«0 1 »
¬
¼
1
0
0º
0 1»»
,
0 1»
»
0 2 ¼
ª1 2 º
«0 1 » ,
¬
¼
B
B
BBˆ
Step 2: Using (3.5.12) and (3.5.16), we compute
K
and
ª a2 e3 º
«
»
¬ a4 k ¼
ª 0
« k
¬ 1
2
k2
1
k3
ªB1 º
«B »
¬ 2¼
1 º
2 k4 »¼
ª0
«1
«
«0
«
¬0
ª0
«
«1
«0
«
¬0
0º
0 »»
.
0»
»
1¼
0º
2 »»
,
0»
»
1¼
(3.5.16)
Normal Matrices and Systems
K
2
1
1 º
ª1 2 º ª 0
«0 1 » « k k k 2 k »
¬
¼¬ 1
2
3
4¼
2 2 k 2 1 2 k3 2 k 4 3 º
.
2 k4 »¼
k2
k3
ˆ
BKT
ª 2k1
« k
¬ 1
197
(3.5.17)
Step 3: The pair ( A z , C) for
Az
ª0
«0
«
«0
«
¬ k1
1
0
0
1
0
k2
0
k3
0º
0 »»
1»
»
k4 ¼
is observable for k1 z 0 and arbitrary k2, k3, k4, since
ª C º
rank «
»
¬CA c ¼
ª0
«0
«
«0
«
¬ k1
1
0
0
0
0
k2
1
k3
0º
1 »»
0»
»
k4 ¼
4
for k1 z 0 and arbitrary k2, k3, k4.
Step 4: The desired gain matrix has the form (3.5.17) for k1 z 0 and arbitrary
k2, k3, k4.
3.5.2 Output-feedback
Consider the system (3.5.1) with an output-feedback in the form
u
v Fy ,
(3.5.18)
where F mup is a gain matrix.
From (3.5.1a) and (3.5.18), we have
x
A BFC x Bv .
(3.5.19)
The closed-loop transfer matrix has the form
1
Tc ( s) C > I n s ( A BFC) @ B .
(3.5.20)
The problem of normalisation of the transfer matrix using an output-feedback
can be formulated in the following way. Given the system (3.5.1) with the
198
Polynomial and Rational Matrices
noncyclic matrix A, the controllable pair (A, B) and the observable pair (A, C),
compute the matrix F in such a way that the closed-loop transfer matrix (3.5.20) is
normal. If the pair (A, C) is not observable, then the pair (A + BFC , C) is not
observable, and the closed-loop transfer matrix (3.5.20) is not normal, regardless of
the matrix F. Thus the problem of normalisation of the transfer function using the
output-feedback has a solution only if the pair (A, C) is observable. If additionally
the pair (A, B) is controllable, then the problem of normalisation reduces to
computation of the matrix F in such a way that the closed-loop system matrix
 z = A + BFC is cyclic. Let K = FC. In this case, using the approach provided in
the proof of Theorem 3.5.2, one can compute K, which is given by (3.5.14), in
such a way that  z = A + BK is a cyclic matrix. From the Kronecker–Capelli
theorem it follows that the equation K = FC has a solution for the given C and K if
and only if
rank C
ªC º
rank « » .
¬K ¼
(3.5.21)
Thus the following theorem has been proved.
Theorem 3.5.2. Let the pair (A, B) be controllable, the pair (A, C) observable, and
A be a cyclic matrix. Then there exists a matrix F such that the transfer matrix
(3.5.20) is normal if and only if the condition (3.5.21) is satisfied.
If the condition (3.5.21) is satisfied, then applying elementary operations on the
columns of the matrix K = FC, we obtain
[K1 0] F[C1 0],
K1  mu p , C1  pu p
(3.5.22)
and det C1 z 0, since by assumption C is a full row rank matrix.
From (3.5.22), we obtain
F
K1C11 .
(3.5.23)
Example 3.5.2.
Consider the system (3.5.1) with the matrices
A
ª0 1
«0 2
«
«0 0
«
¬0 0
0 0º
0 1»»
,B
0 1»
»
0 2 ¼
ª0
«1
«
«0
«
¬0
0º
2 »»
,C
0»
»
1¼
ª1 1 0 0.5º
«0 2 1 1 » .
¬
¼
(3.5.24)
It is easy to check that the pair (A, B) is controllable, the pair (A, C)
observable, and A is not a cyclic matrix.
We seek a matrix
Normal Matrices and Systems
F
ª f11
«f
¬ 21
199
f12 º
f 22 »¼
such that the closed-loop transfer matrix is a normal matrix.
In the same way as in Example 3.5.1 we compute the matrix K and from
(3.5.17) for k1 = 1, k2 = 0, k3 = -1/2, k4 = 2, we obtain
ª 2 2 0 1º
« 1 0 0.5 0 » .
¬
¼
K
(3.5.25)
In this case, the condition (3.5.21) is satisfied, since
ª1 1 0 0.5º
ªCº
rank «
rank « »
»
¬1 2 1 0 ¼
¬K ¼
ª 1 1 0 0.5º
«0 2 1
1 »»
rank «
2.
«2 2 0
1 »
«
»
¬ 1 0 0.5 0 ¼
rank C
Applying elementary operations on the columns of the matrix
ªC º
«K »
¬ ¼
ª1
«0
«
«2
«
¬ 1
1
2
0
1
0.5º
1 »»
,
2 0
1 »
»
0 0.5 0 ¼
we obtain
ªC1 0 º
«K 0»
¬ 1 ¼
0
ª1
«0
1
«
«2
0
«
1
0.5
¬
0
0
0
0
0º
0 »»
.
0»
»
0¼
Using (3.5.23), we obtain the desired matrix
F
K1C11
ª2 0 º
« 1 0.5» .
¬
¼
(3.5.26)
200
Polynomial and Rational Matrices
3.6 Electrical Circuits as Examples of Normal Systems
3.6.1 Circuits of the Second Order
Consider an electrical circuit with its scheme given in Fig. 3.6.1, with known
resistances R1, R2, inductance L, capacity C, and source voltages e1 and e2.
Fig. 3.1 A circuit of the second order
Taking as state variables the current i in the coil and the voltage uC on the
capacitor, we can write the equations
e1
e2
di
uC ,
dt
§ duC
·
¨ C dt i ¸ R2 uC .
©
¹
R1i L
We write these equations in the form of the state equation
d ªi º
dt «¬uC »¼
ª R1
« L
«
« 1
«¬ C
1 º
ª1
»
i
L ª º «L
»
«
1 » «¬uC »¼ «
0
«¬
CR 2 »¼
º
0 »
ªe º
»« 1».
1 » ¬e2 ¼
CR 2 »¼
Denoting
x
ªi º
«u » , A
¬ C¼
ª R1
« L
«
« 1
«¬ C
1 º
L »
», B
1 »
CR 2 »¼
ª1
«L
«
«0
«¬
º
0 »
», u
1 »
CR 2 »¼
ª e1 º
«e » ,
¬ 2¼
(3.6.1)
we obtain
x
Ax Bu .
(3.6.2a)
Normal Matrices and Systems
201
Take as an output the voltage on the coil y1=L di/dt and the current i2 of the
voltage source e2, y2 = i2. In this case, the output equation takes the form
y
ª y1 º
«y »
¬ 2¼
ªe1 R1i uC º
«
»
« e2 uC »
R2
«¬
»¼
C
ª R1
«
« 0
«¬
1 º
1 »» , D
R 2 »¼
Cx Du ,
(3.6.2b)
0 º
1 »» .
R 2 »¼
(3.6.3)
where
ª1
«
«0
«¬
The matrix A is cyclic and its characteristic polynomial
R1
L
1
C
1
L
s
m s
det > Is A @
s
1
CR 2
(3.6.4)
§R
R1 R 2
1 ·
s2 ¨ 1 ¸s L
CR
LCR 2
©
2 ¹
is the same as the minimal one.
The inverse matrix
1
> Is A @
1
LCR2 s R1CR2 L s R1 R2
ª L sCR2 1
u«
LR2
¬
2
CR2
º
»
sLR2C R1 R2C ¼
(3.6.5)
is a normal matrix.
B and C are square nonsingular matrices. Hence the pair (A, B) is controllable
and the pair (A, C) is observable. The transfer matrix of this circuit is irreducible
and has the form
202
Polynomial and Rational Matrices
1
C > Is A @ B D
T s
1
R
1 º
ª
1 º « s 1
L
L »
»
1 »» «
1 »
« 1
s
R 2 »¼ «
CR 2 »¼
¬ C
1
LCR 2 s 2 R1CR 2 L s R1 R 2
ª R1
«
« 0
«¬
ª sR1CR 2 R1 R 2
«
u«
1
¬«
ª1
«L
«
«0
«¬
º
0 » ª1
»«
1 » «0
«
CR 2 »¼ ¬
sL
º ª1
» «
sL R1 » «
0
»¼ «¬
R2
0 º
1 »»
R 2 »¼
(3.6.6)
0 º
1 »» .
R 2 »¼
It is easy to check that (3.6.6) is a normal matrix.
Now we will perform structural decomposition of the matrix (3.6.5).
Postmultiplying the polynomial matrix
CR 2
ª sLCR 2 L
º
« LR
sLCR 2 R1CR 2 »¼
2
¬
L s
(3.6.7)
by the matrix
V
ª 0 1º
« 1 0 » ,
¬
¼
we obtain
L s V
CR 2
ª
« sLCR R CR
2
1
2
¬
s C LR 2 L º
.
LR 2 »¼
Using the notation adopted in Sect. 3.4.2 we obtain in this case
U s
ª1 0 º
«0 1 » , V s
¬
¼
k s
sL R1 , L s
P s
ª 1 º
U 1 s i s «
»
¬k s ¼
V, i s
CR2 , w s
sL L
C
and
CR2
ª
«
«¬ sLCR2 R1CR2
º
»,
»¼
L
,
CR2
Normal Matrices and Systems
ª
L
« sL CR
¬
2
Q s
ª¬1 w s º¼ V 1 s
G s
0 º 1
ª0
U 1 s «
V s
ˆ »
¬«0 L s »¼
203
º
1» ,
¼
ª 0
« R CL2
¬ 2
0º
.
0 »¼
It is easy to check that the following equality holds
1
>Is A @
1 ª L sCR2 1
«
m s ¬
LR2
P s Q s
G s .
m s
CR2
º
»
sLR2C R1 R2C ¼
(3.6.8)
Note that the structural decomposition of the matrix (3.6.5) yields the structural
decomposition of the transfer matrix (3.6.6), since
T s
1
C > Is A @ B D
CP s Q s B
CG s B D
m s
P s Q s
G s ,
m s
(3.6.9)
where
P s
CP s , Q s
Q s B, G s
CG s B D .
3.6.2 Circuits of the Third Order
Consider the electrical circuit with its scheme given by Fig. 3.2, with known
resistances R1, R2, inductance L, capacities C1, C2 and source voltages e1 and e2. As
the state variables we take the current in the coil i and the voltages u1 and u2 on the
capacitors, as the outputs we take the voltages on the resistances R1, y1 = R1i and
R2, y2 = R2i2. Using Kirchoff’s laws we can write the following equations for this
circuit
di
u1 ,
dt
du
e2 u2 R2C2 2 u1 ,
dt
du1
du2
C1
i C2
,
dt
dt
e1
R1i L
204
Polynomial and Rational Matrices
Fig. 3.2 A circuit of the third order
which can be written in the form of the state equation
ªiº
d « »
u1
dt « »
«¬u2 »¼
ª R
« 1
« L
« 1
«
« C1
«
« 0
¬
1
L
1
R 2 C1
1
R 2 C2
º
ª1
»
«
»ª i º «L
1 »« » «
» u1 « 0
R 2 C1 » « » «
«¬u2 »¼
«
1 »
»
«0
R 2C2 ¼
¬
0
º
0 »
»
1 » ª e1 º
.
»
R 2 C1 » «¬e2 »¼
1 »
»
R 2C2 ¼
Denoting
x
ªiº
«u » , A
« 1»
¬«u2 ¼»
ª R
« 1
« L
« 1
«
« C1
«
« 0
¬
1
L
1
R2C1
1
R2C2
º
»
»
1 »
»,B
R2C1 »
1 »
»
R2 C2 ¼
0
ª1
«
«L
«
«0
«
«
«0
¬
º
0 »
»
1 »
» ,u
R2C1 »
1 »
»
R2C2 ¼
ª e1 º
«e » ,(3.6.10)
¬ 2¼
we obtain
x
Ax Bu .
(3.6.11a)
Taking into account that
y1
R1i,
y2
R 2C2
du2
dt
e2 u1 u2 ,
we obtain the output equation of the form
Normal Matrices and Systems
ª y1 º
«y »
¬ 2¼
y
R1i
ª
º
«e u u »
¬ 2 1 2¼
ª R1
«0
¬
ªiº
0 º « » ª 0 0 º ª e1 º
u1 1 1»¼ « » «¬ 0 1 »¼ «¬ e2 »¼
«¬u2 »¼
205
0
(3.6.11b)
Cx Du
where
C
ª R1
«0
¬
0 0º
, D
1 1»¼
ª0 0º
«0 1 » .
¬
¼
(3.6.12)
A is a cyclic matrix since the minor obtained after elimination of the first row and
the third column of the matrix [Is – A] is equal to -1/(R2C1C2), therefore the
greatest common divisor of Adj [Is – A] is 1. The characteristic polynomial
(minimal) of A is
R1
L
1
C1
1
L
s
m s
det > Is A @
0
s
1
R 2 C1
1
R 2 C2
0
1
R 2 C1
s
1
R 2 C2
§R
R1
R1
1
1 · 2 § 1
s3 ¨ 1 ¸s ¨
© L R 2 C1 R 2 C2 ¹
© LC1 LR 2 C1 LR 2 C2
2R1
2 ·
1
2
.
¸s LR 2 C1C2 LR 22 C1C2
R 2 C1C2 ¹
(3.6.13)
The inverse
1
> Is A @
where
ª
R
«s 1
L
«
« 1
« « C1
«
« 0
¬
1
L
s
1
R 2 C1
1
R 2C2
º
»
0
»
1 »
»
R 2 C1 »
1 »
s
»
R 2 C2 ¼
1
L s
,
m s
(3.6.14)
206
Polynomial and Rational Matrices
L s
ª 2 § 1
1 ·
2
«s ¨
¸s 2
R
C
R
C
R
C
©
¹
2
1
2
2
2
1C2
«
«
1
1
«
s
«
C1
R 2 C1C 2
«
1
«
«
R 2C1C2
¬
º
»
»
»
R1
1
»
s
LR 2 C1
R 2 C1
»
»
§ 1
R1 ·
R1
1 »
2
s ¨
¸s LR 2 C1 LC1 »¼
© R 2 C1 L ¹
1
1
s
L
LR 2 C 2
§ 1
R ·
R1
1 ¸s s2 ¨
LR 2 C 2
© R 2C2 L ¹
R1
1
s
R 2 C2
LR 2 C 2
1
LR 2 C1
(3.6.15)
is a normal matrix, since all nonzero second order minors of the matrix (3.6.15) are
divisible without remainder by the polynomial (3.6.13).
The pair (A, B) of this circuit is controllable, since the matrix built from the
first three columns of [B AB] is nonsingular
ª1
«
«L
«
det « 0
«
«
«0
¬
0
1
R 2 C1
1
R 2C2
R1 º
»
L2 »
1 »
»
LC1 »
»
0 »
¼
1
.
L2 R 2 C1C2
(3.6.16)
If R1 z 0, then the pair (A, C) is observable too, since the matrix built from the first
three rows of
ªC º
« AC »
¬
¼
is nonsingular
Normal Matrices and Systems
ª
« R1
«
«
det « 0
«
2
« R1
« L
¬
º
0»
»
»
1»
»
»
0»
¼
0
1
R1
L
R12
.
L
207
(3.6.17)
The transfer matrix of this circuit has the form
T s
1
C > Is A @ B D
ª
R
«s 1
L
«
ª R1 0 0 º « 1
« 0 1 1» « C
¬
¼«
1
«
« 0
¬
ˆ
0
0
L
s
ª
º
«
» m s ,
0
1
¬
¼
1
L
s
1
R 2 C1
1
R 2 C2
º
0
»
»
1 »
»
R 2 C1 »
1 »
s
»
R 2 C2 ¼
1
ª1
«
«L
«
«0
«
«
«0
¬
º
0 »
»
1 »
» (3.6.18)
R 2 C1 »
1 »
»
R 2C2 ¼
where
Lˆ s
ª R1 2 § R1
R1 ·
2 R1
« s ¨
¸s 2
L
L
L
L
R
C
R
C
R
©
2 1
2 2 ¹
2 C1C 2
«
«
1
s
«
C1
L
¬«
R1
2 R1 º
s
»
LR 2 C1
LR 22 C1C2 » (3.6.19)
»
R
1
s3 1 s 2 s »
L
LC1
¼»
is an irreducible matrix, since det L̂ (s) is divisible without remainder by the
polynomial (3.6.13).
We will perform the structural decomposition of the matrix (3.6.14). To
accomplish this, we write the matrix (3.6.15) in the form
208
Polynomial and Rational Matrices
1
LR2C1
L s
ª
§
LC1 ·
C
2L
2
R2C1 s 1
« LR2C1s ¨ L ¸s C
R
C
C
2 ¹
2 2
2
©
«
(3.6.20)
«
§
·
LC
R1C1
L
2
1
«
u
LR2 s LR2C1s ¨ R1 R2C1 ¸ s «
C2
C2
© C2
¹
«
LC1
RC
L
«
s 1 1
«
C2
C2
C2
¬
1
º
»
Ls R1
»
2
LR2C1s L R1 R2C1 s R1 R2 »¼
and then we postmultiply it by the matrix
V
ª0 0 1 º
«0 1 0» .
«
»
«¬1 0 0 »¼
(3.6.21)
Then we obtain
L s V
1
LR2C1
ª
1
«
«
«
u«
Ls R1
«
«
« LR2C1 s 2 L R1 R2C1 s R1 R2
«¬
§
LC ·
2L º
LR2 C1 s 2 ¨ L 1 ¸ s »
C
R
2 ¹
2 C2 »
©
»
L
LR2 s ».
C2
»
»
L
»
C2
»¼
In this case,
R2 C1 s C1
C2
§ LC
·
RC
LR2C1s 2 ¨ 1 R1 R2C1 ¸ s 1 1
C2
© C2
¹
LC1
R1C1
s
C2
C2
(3.6.22)
Normal Matrices and Systems
209
1
,
LR2C1
U s
I3 , i s
w s
ª
C1
« R2C1s C2
¬
k s
Ls R1
ª
º
« LR C s 2 L R R C s R R » ,
2 1
1 2 1
1
2¼
¬
L s
ª
§ LC1
·
RC
2
R1 R2C1 ¸ s 1 1
« LR2C1s ¨ C
C2
©
¹
2
«
«
LC1
RC
s 1 1
«
C2
C2
¬«
§
LC ·
2L º
LR2C1s 2 ¨ L 1 ¸ s »,
C2 ¹
R 2 C2 ¼
©
(3.6.23)
Lº
»
C2 »
.
»
L
»
C2
¼»
LR2 s Using (3.4.22)–(3.4.24) and (3.6.23), we obtain
P s
ª 1 º
U 1 s i s «
»
¬k s ¼
1
ª
º
1 «
»
Ls R1
»
LR 2 C1 «
«¬ LR 2 C1s 2 L R1R 2 C1 s R1 R 2 »¼
ª
º
1
«
»
LR 2 C1
«
»
«
»
R1
1
«
»,
s
R 2 C1
LR 2 C1
«
»
«
»
« s 2 § 1 R 1 · s R1 1 »
¨
¸
«
LR 2 C1 LC1 »¼
© R 2 C2 L ¹
¬
Q s
ª¬1 w s º¼ V 1
ª
C1
«1 R 2 C1s C
2
¬
ª0 0 1 º
§
LC1 ·
2L º «
»
LR 2 C1s ¨ L » «0 1 0»
¸s C
R
C
©
2 ¹
2 2 ¼
¬«1 0 0 ¼»
2
ª
§
LC1 ·
2L
2
« LR 2 C1s ¨ L ¸s C2 ¹
R 2 C2
©
¬
G (s)
R 2 C1s 0
ª0
º 1
U 1 ( s ) «
»V
ª
º
L
0
i
(
s
)
(
s
)
k
(
s
)
w
(
s
)
¬
¼¼
¬
C1
C2
º
1» ,
¼
ª 0 0 0º
« x 0 0» ,
«
»
«¬ y z 0 »¼
(3.6.24)
210
Polynomial and Rational Matrices
where
§
R
L
L · 2 § R1
1
2L ·
Ls 3 ¨ R1 1 2
¸s ¨
¸s
R
C
R
C
R
C
R
C
C
R
2 1
2 2 ¹
2 2
1
2 C1C2 ¹
©
© 2 1
2 R1
1
,
C1C2 R2 R22C1C2
x
§
·
§
LC1
RC
3L · 2
L
LR 2 C1s 4 ¨ 2 L R1R 2 C1 ¸ s 3 ¨ R 2 1 1 2R 1 ¸s
C
C
R
C
R
©
¹
©
2
2
2 1
2 C2 ¹
§ 1
3R1
R1
R 2R 1
1
2L ·
¨
,
s 2
2 ¸
C
R
C
C
R
C
C
C
R
C1C1R 22
© 2
2 2
1
2 1
2 1 2 ¹
y
z
§ RRC
§ R
C ·
RC R
2 ·
R2C1s 3 ¨ 1 2 1 1 1 ¸ s 2 ¨ 2 1 1 1 ¸s
L
C
L
C
L
L
R
©
©
2 ¹
2
2 C2 ¹
§ 2 R1
1 ·
¨
¸.
© R2C2 L C2 L ¹
The structural decomposition of the matrix (3.6.14) yields the structural
decomposition of the transfer function (3.6.18).
3.6.3 Circuits of the Fourth Order and the General Case
Consider an electrical circuit with its scheme given in Fig. 3.3, with known
resistances R1, R2, R3, inductances L1, L2, capacities C1, C2, as well the source
voltages e1, e2 and e3.
As the state variables we take the currents i1 and i2 in the coils and the voltages
u1, u2 on the capacitors, as the outputs y1 and y2 we take the voltage on the coil L1
and the current in the capacitor C2, respectively.
Fig. 3.3 A circuit of the fourth order
Normal Matrices and Systems
211
Using Kirchoff’s laws we can write the following equations for this circuit
di1
u1 R3 i1 i2 ,
dt
du
e2 u2 R2C2 2 u1 ,
dt
di
e3 e2 R3 i1 i2 L2 2 ,
dt
du1
du2
,
C1
i1 C2
dt
dt
e1
R1i1 L1
which can be written in the form of the state equation
ª i1 º
« »
d « i2 »
dt « u1 »
« »
¬u2 ¼
ª R1 R3
« L
1
«
«
R3
« L
2
«
«
1
«
« C1
«
0
«
¬
R3
L1
R3
L2
1
L1
º
ª1
»
«L
»
« 1
» ª i1 º «
0 »« » « 0
» « i2 » «
1 » « u1 » «
»« » « 0
R2C1 » ¬u2 ¼ «
«
1 »
»
«0
R2C2 ¼
¬
0
0
0
1
R2C1
0
1
R2C2
0
1
L2
1
R2C1
1
R2C2
º
0»
(3.6.25)
»
1»
ªe º
L2 » « 1 »
» e .
»« 2»
0 » «¬ e3 »¼
»
»
0»
¼
Denoting
x
ª i1 º
«i »
« 2 », u
« u1 »
« »
¬u 2 ¼
A
ª R1 R3
« L
1
«
«
R3
« L
2
«
«
1
«
« C1
«
0
«
¬
ª e1 º
« »
«e2 » ,
«¬ e3 »¼
R3
L1
R3
L2
1
L1
0
0
1
R2C1
0
1
R2C2
º
»
»
»
0 »
», B
1 »
»
R2C1 »
1 »
»
R2C2 ¼
0
ª1
«L
« 1
«
«0
«
«
«0
«
«
«0
¬
0
1
L2
1
R2C1
1
R2C2
º
0»
»
1»
L2 »
»,
»
0»
»
»
0»
¼
212
Polynomial and Rational Matrices
we obtain
Ax Bu .
x
(3.6.26a)
Taking into account
y1
di1
R1 R3 i1 R3i2 u1 e1 ,
dt
du
1
1
1
C2 2 u1 u2 e2 ,
dt
R2
R2
R2
L1
y2
we obtain the output equation
y
ª y1 º
«y »
¬ 2¼
ª R1 R 3
«
«
0
«¬
R 3
0
1
1
R2
ª i1 º
0 º « » ª1
» i2
«
1 « »
» « u1 » «0
R 2 »¼ « » «¬
¬u 2 ¼
0
1
R2
0 º ª e1 º
» «e »
0» « 2 »
»¼ «¬ e3 »¼
(3.6.26b)
Cx Dy,
where
C
ª R1 R3
«
«
0
¬«
R3
0
1
1
R2
0 º
»
1 , D
»
R2 ¼»
ª1
«
«0
¬«
0
1
R2
0º
».
0»
¼»
(3.6.27)
To show that A is a cyclic matrix, we transform the matrix [Is – A] by
similarity (which does not change the characteristic polynomial) to the form
P > Is A @ P T
We then obtain
Is PAPT for P
ª0
«1
«
«0
«
¬0
1 0 0º
0 0 0 »»
0 1 0»
»
0 0 1¼
PT
P 1
P .
Normal Matrices and Systems
213
PAPT
ª0
«1
«
«0
«
¬0
1 0
0 0
0 1
0 0
ª R3
« L
« 2
« R3
« L
« 1
«
« 0
«
«
« 0
¬
ª R1 R3
« L
1
«
0º «
R3
L2
0 »» ««
0» «
1
»«
1 ¼ « C1
«
0
«
¬
R
0
3
L2
R1 R3
L1
1
L1
1
C1
1
R2C1
0
1
R2C2
R3
L1
R3
L2
1
L1
0
0
1
R2C1
0
1
R2C2
º
»
»
» ª0
0 »«
» «1
1 » «0
»«
R2C1 » ¬ 0
1 »
»
R2C2 ¼
0
1 0 0º
0 0 0 »»
0 1 0»
»
0 0 1¼
º
»
»
»
0 »
».
1 »
»
R2C1 »
1 »
»
R2C2 ¼
0
(3.6.28)
Note that the minor obtained from [Is – PAPT] by elimination of the first row
and the fourth column is equal to R3/(L1R2C1C2). The greatest common divisor of
the entries of Adj [Is – PAPT] is 1, thus A is a cyclic matrix.
The characteristic polynomial (minimal) of the matrix A is
s
det > Is A @
R1 R 3
L1
R3
L2
R3
L1
s
R3
L2
1
C1
0
0
0
s
1
L1
0
0
0
1
R 2 C1
1
R 2C2
1
R 2 C1
s
1
R 2 C2
§ L L C L1 L2C1 R3 R2C1C2 L2 R1 R2C1C2 L2 R3 R2C1C2 L1 · 3
s4 ¨ 1 2 2
¸s
R2 L1 L2C1C2
©
¹
§ R C L R3 L2C2 R3 L2C1 L1 R3C2 R1 L2C1 R1 L2C2 L1 R3C1
¨ 2 2 2
R2 L1 L2C1C2
©
R3
R1 R2 R3C1C2 · 2 § L2 R1 R3C2 R2 R3C2 R1 R3C1 ·
.
¸s ¨
¸s R2 L1 L2C1C2 ¹
R
L
L
C
C
R
L
L
2 1 2 1 2
2 1 2 C1C2
©
¹
The inverse matrix
(3.6.29)
214
Polynomial and Rational Matrices
1
> Is A @
R1 R 3
ª
«s L
1
«
«
R3
«
L2
«
«
1
« C1
«
«
0
«
¬
R3
L1
s
1
L1
R3
L2
0
0
1
s
R 2 C1
0
1
R 2C2
º
»
»
»
0
»
»
»
1
»
R 2 C1 »
1 »
s
»
R 2 C2 ¼
1
0
L s , (3.6.30)
m s
where
L s
ª s 3 R2C1C2 L2 s 2 C1C2 R2 R3 C1L2 C2 L2 sR3 C1 C2
«
C1C2 R2 L2
«
2
«
s R2 R3C1C2 sR3 C1 C2
«
«
C1C2 R2 L2
«
2
s L 2 R2C2 s L2 R2 R3C2 R3
«
«
C1C2 R2 L2
«
«
sL2 R3
«
C1C2 R2 L2
¬
s 2 R2 R3C1C2 sR3 C1 C2
C1C2 R2 L1
3
2
ª s L1 R2C1C2 s L1C2 L1C1 C1R1R2C2 R3 R2C1C2 º
«
»
«¬ s R1C2 R1C1 R3C2 R3C 1 R2C2 1
»¼
C1C2 R2 L1
sR2 R3C2 R3
C1C2 R2 L1
R3
C1C2 R2 L1
(3.6.31)
Normal Matrices and Systems
215
s 2 L 2 R2C2 s L2 R2 R3C2 R3
L1 L2 R2C2
sR2 R3C2 R3
L1 L2 R2C2
ª s 3 L1 L2 R2C2 s 2 R2 R3 L1C2 R1 R2C2 L2 R2 R3C2 L2 L1 L2 º
«
»
¬« s R1 R2 R3C2 R3 R1 L2 R3 L2 R1 R3
¼»
L1 L2 R2C2
s 2 L1 L2 s L1 R3 R1 L2 R3 L2 R1 R3
L1 L2 R2 L2
sL2 R3
L1 L2C1 R2
º
»
»
»
R3
»
L1 L2C1 R2
»
»
s 2 L1 L2 s L1 R3 R1 L2 R3 L2 R1 R3
»
»
L1 L2C1 R2
»
»
ª s 3 L1 L2C1 R2 s 2 L1 L2 L1 R2 R3C1 R1 R2C1 L2 R2 R3C1 L2 º »
«
» »
¬« s L1 R3 R1 L2 R1 R2 R3C1 R3 L2 R2 L2 R1 R3 R2 R3 ¼» »
»¼
L1 L2C1 R2
is a normal matrix, since all nonzero second-order minors of the matrix (3.6.31) are
divisible without remainder by the polynomial (3.6.29).
The pair (A, B) of this circuit is controllable, since the matrix built from the
first four columns of the matrix [B AB] is nonsingular
ª1
«
« L1
«
«0
det «
«
«0
«
«
«0
¬«
0
0
1
L2
1
L2
1
R2C1
0
1
R2C2
0
R1 R3 º
»
L12
»
»
R3
»
L1 L2 »
»
1
»
L1C1 »
»
»
0
¼»
The pair (A, C) is observable, since
1
.
L12 L2 R2C1C2
(3.6.32)
216
Polynomial and Rational Matrices
ªC º
«CA »
¬
¼
is nonsingular
ªC º
det «
»
¬CA ¼
R1 R3
ª
«
0
«
« ª L C R 2 2L C R R º
2 1 1 3
«« 2 1 1
»
det « «¬ L2C1 R32 R32 L1C1 L1 L2 »¼
«
L1 L2C1
«
«
1
«
R2C1
¬
1
1
R2
R3
0
R1 R3 L2 R32 L1 L2
L1 L2
0
0
1
R2
R1 R2C1 R2 R3C1 L1
L1 R2C1
C1 C2
R22C1C2
º
»
»
»
»
1
» z 0.
R2C1 »
»
C1 C2 »
R22C1C2 »¼
(3.6.33)
The transfer matrix of this circuit has the form
T s
1
C > Is A @ B D
ª R1 R3
«
«
0
«¬
R1 R3
R3
1
ª
«s L
L1
L1
1
«
«
R3
R3
0
s
«
L2
L2
u«
«
1
1
0
s
« C
R
1
2 C1
«
«
1
0
0
«
R2C2
¬
ª1 0 0 º ˆ
» L s ,
«
«0 1 0» m s
R2
¬«
¼»
R3
0
º
»
»
»
0
»
»
»
1
»
R2C1 »
1 »
s
»
R2C2 ¼
0
1
1
1
R2
ª1
«L
« 1
«
«0
«
«
«0
«
«
«0
¬
0 º
»
1
»
R2 »¼
0
1
L2
1
R2C1
1
R2C2
º
0»
»
1»
L2 »»
»
0»
»
»
0»
¼
(3.6.34)
Normal Matrices and Systems
217
where
Lˆ s
ª§ L1 L2 R2C1C2 s 4 L1 L2C1 R3 L1 R2C1C2 L1 L2C2 s 3 ·
Ǭ
¸
2
¸
«¨© L1 R3C2 L1 R3C1 s
¹
«
2
C
L
s
R
C
s
2 2
3 2
«¬
§
LL
LC L ·
L1 R2C1C2 R3 L1 L2C2 s 3 ¨ L1C1 R3 1 2 1 2 2 ¸ s 2 R2
R2C1 ¹
©
§LRC
LR ·
¨ 1 3 2 1 3 ¸s
R2 ¹
© R2C1
L1 L2C1C2 s 4 L1 R3C1C2 R1 L2C1C2 R3 L2C1C2 R3 L1C1
L1 L2C1 L1 L2C2 · 3 §
¸ s ¨ R1 R3C1C2 L2C2 R2
R2 ¹
R2
©
R1 L2C2 R3 L2C1 R3 L1C2 R1 L2C1 R3 L2C2 · 2
¸s R2
R2
R2
R2
R2 ¹
§ R1 R3C1 R1 R3C2 L2 L2C2 ·
R3 C2 R3
¨
¸s R
R
R
R
C
R
R2C1
2
2
2
2 1 ¹
2
©
L1 R3 R2C1C2 s 3 R3 L1 C2 C1 s 2 º
»,
R3C2 s
»¼
4
3
m s s R2C1C2 L1 L2 s R1 R2C1C2 L2 R2 R3C1C2 L2
(3.6.35)
R2 R3C1C2 L1 C1 L1 L2 C2 L1 L2 s 2 R3C2 L1 R1C1 L2
R1C2 L2 R1 R2 R3C1C2 R3C1 L2 R3C2 L2 R2C2 L2 R3C1 L1
s R1 R3C1 R1 R3C2 R2 R3C2 L2 R3 .
This is an irreducible and normal matrix, since all nonzero second degree
minors of the matrix (3.6.35) are divisible without remainder by the polynomial
(3.6.29).
Analogously to the two previous cases, we can perform the structural
decomposition of the inverse matrix (3.6.30) and transfer matrix (3.6.34).
The above considerations can be generalised into electrical circuits of an
arbitrary order.
From the above considerations we can derive two important corollaries
pertaining to electrical circuits of the n-th order (n is not less than 2), with at least
two inputs m t 2 and at least two outputs p t 2, that is, min (n, m, p) t 2.
218
Polynomial and Rational Matrices
Corollary 3.6.1. Every matrix A of an electrical circuit of the second order (n = 2)
is cyclic, and the inverse [Is – A]-1, as well as transfer matrix T(s) =
=C[Is – A]-1B + D are normal.
Corollary 3.6.2. The matrices A of typical electrical circuits consisting of
resistances, inductances, capacities and source voltages (currents) are cyclic
matrices and the inverses [Is – A]-1 are normal matrices. In particular cases, the
values of R, L, C can be chosen in such a way that the pair (A, B) are not
controllable or/and the pair (A, C) are not observable. In these cases, the transfer
matrix
T s
Lˆ s
m s
may be reducible and then is not a normal matrix, i.e., not all nonzero second-order
minors of the polynomial matrix L̂ (s) are divisible without remainder by the
polynomial m(s).
Remark 3.6.1.
If (A, B) is not a controllable pair, then some pole-zero cancellations occur in
Adj > Is A @ B
.
det > Is A @
Analogously, if (A, C) is not an observable pair, then some pole-zero cancellations
occur in
CAdj > Is A @
.
det > Is A @
4
The Problem of Realization
4.1 Basic Notions and Problem Formulation
Consider a continuous system given by the equations
x
y
Ax Bu ,
Cx Du ,
(4.1.1a)
(4.1.1b)
where x n, u m, y p are the state, the input and the output vectors,
respectively, and A nun, B num, C pun and D pum.
The transfer matrix of the system (4.1.1) is given by
T s
1
C > Is A @ B D .
(4.1.2)
For the given matrices A, B, C and D there exists only one transfer matrix (4.1.2).
On the other hand, for a given proper transfer matrix T(s) there are many matrices
A, B, C and D satisfying (4.1.2).
Definition 4.1.1. The quadruplet of the matrices: A nun, B num, C pun and
D pum satisfying (4.1.2), is called a realisation of the given transfer matrix
T(s) pum(s). It will be denoted Rn,m,p(T) or briefly Rn,m,p.
Definition 4.1.2. A realisation Rn,m,p is called minimal if the matrix A has the
minimal (least) dimension among all realisations of T(s). A minimal realisation
will be denoted by R n,m,p.
Definition 4.1.3. A minimal realisation R n,m,p is called cyclic (or simple) if the
matrix A is cyclic. A cyclic realisation will be denoted by R̂ n,m,p.
220
Polynomial and Rational Matrices
The matrix D for a given proper transfer matrix T(s) can be computed using the
formula
D
lim T s ,
(4.1.3)
s of
which results from (4.1.2), since
1
lim > Is A @
s of
0.
From (4.1.2) and (4.1.3), we have
Tsp s
T s D
1
C > Is A @ B .
(4.1.4)
Having the proper matrix T(s) and using (4.1.4) we can compute the strictly proper
matrix Tsp(s).
The realisation problem can be formulated in the following way.
With a proper rational matrix T(s) pum(s) given, compute the realisation Rn,m,p
of this matrix.
The minimal realisation problem can be formulated in the following way.
With a proper rational matrix T(s) pum(s) given, compute a minimal
realisation R n,m,p of this matrix.
The problem of cyclic realisation is formulated as follows. With a proper
rational matrix T(s) pum(s) given, compute a cyclic realisation R̂ n,m,p of this
matrix.
In the case of a strictly proper transfer matrix Tsp(s) pum(s), the realisation
problem reduces to the computation of only three matrices A, B, C satisfying
(4.1.4).
4.2 Existence of Minimal and Cyclic Realisations
4.2.1 Existence of Minimal Realisations
The theorem stated below provides us with necessary and sufficient conditions for
the existence of a minimal realisation R n,m,p for a given rational proper transfer
matrix T(s) pum(s).
Theorem 4.2.1. A realisation (A, B, C, D) of a matrix T(s) is minimal if and only
if (A, B) is a controllable pair and (A, C) is an observable pair.
Proof. We will show by contradiction that if (A, B) is a controllable pair and
(A, C) is an observable pair, then the realisation is minimal.
The Problem of Realization
221
Let (A, B, C), A nun and ( A,B,C ), A  nun be two different realizations for
n ! n of the matrix T(s). From (4.1.4), we have
1
C > Is A @ B
1
(4.2.1)
C ª¬Is A º¼ B
and
CA i B
CA i B, i
0, 1, ... .
(4.2.2)
From the assumption that (A, B) and ( A, B ) are controllable pairs and that
(A, C) and ( A,C ) are observable pairs, it follows that
rank S
rank S
rank H
rank H
n,
n,
(4.2.3a)
(4.2.3b)
where
ª C º
« CA »
»,
S [B AB ! A n1B], H «
« # »
«
n 1 »
¬CA ¼
ª C º
«
»
CA »
S = ª¬B AB … A n-1B º¼ , H = «
.
« # »
«
n-1 »
¬«CA ¼»
(4.2.3c)
(4.2.3d)
From (4.2.2) we have
HS
ª C º
« CA »
«
» ªB AB … A n-1B º
¼
« # »¬
« n-1 »
¬CA ¼
ª CB
CAB
«
2
« CAB CA B
« #
#
«
n 1
«¬CA B CA n B
and
CA n1B º
»
! CA n B »
»
%
#
»
2 n 1
! CA
B »¼
!
ª CB
CAB
«
2
CAB
CA
B
«
« #
#
«
n1
«¬CA B CA n B
CA n1B º
»
! CA n B »
»
%
#
»
2 n 1
! CA
B »¼
!
ª C º
«
»
« CA » ªB AB ! A n1B º
¼
« # »¬
«
n 1 »
«¬CA »¼
HS
222
Polynomial and Rational Matrices
rank HS .
rank HS
(4.2.4)
The relationships rank HS = n, rank HS n and (4.2.4) lead to a contradiction
since by assumption n ! n .
Now we will show that if (A, B) is not a controllable pair or/and (A, C) is not
an observable pair, then (A, B, C) is not a minimal realisation.
If (A,B) is not a controllable pair, then there exists a nonsingular matrix P such
that
A
PAP 1
ª A1
«0
¬
A2 º
, B
A 3 »¼
PB
ªB1 º
« 0 », C
¬ ¼
CP 1
>C1
C2 @ ,
(4.2.5)
A1  n1un1 , B1  n1um , A 3  nn1 u nn1 , C1  pun1
where (A1, B1) is a controllable pair and
1
C > Is A @ B
1
C1 > Is A1 @ B1 .
(4.2.6)
From (4.2.6) it follows that (A1, B1, C1) is a realisation whose matrix A1 is of
smaller size than that of A. Thus (A, B, C) is not a minimal realisation if (A,B) is
not a controllable matrix. The proof that if (A, C) is not observable, then (A,B,C)
is not a minimal realisation, is analogous.
„
Theorem 4.2.2. If the triplet of matrices (A, B, C) is a minimal realisation R n,m,p
of a strictly proper transfer matrix T(s) pum(s), then the triplet (PAP-1, BP, CP-1)
is also a minimal realisation of the transfer matrix T(s) for an arbitrary nonsingular
matrix P nun.
Proof. We will show that the matrices PAP-1, BP, CP-1 satisfy the condition
(4.1.4). Substituting these matrices into (4.1.4), we obtain
1
CP 1 ª¬Is PAP 1 º¼ PB
1
1
CP 1 ª P > Is A @ P 1 º PB
¬
¼
1
CP 1P > Is A @ P 1PB
C > Is A @ B ,
since PP-1 = I.
„
If (A, B, C) and ( A,B,C ) are two minimal realisations of the transfer matrix
T(s), then there exists only one nonsingular matrix P such that
A
PAP 1 , B
BP, C
CP 1 .
(4.2.7)
The Problem of Realization
223
Example 4.2.1.
Given two minimal realisations (A, B, C) and ( A,B,C ) of a transfer matrix T(s),
compute a nonsingular matrix P satisfying (4.2.7).
From the assumption that (A, B, C) and ( A,B,C ) are two minimal realisations
of the transfer matrix T(s), it follows that they satisfy the equality (4.2.2) and
HS ,
HS
(4.2.8)
where n = n , and the matrices H, S, H and S are given by (4.2.3c) and (4.2.3d).
The condition (4.2.3) implies that det [SST] = det [ S S T] z 0 and det [HTH] =
= det [ H T H ] z 0. Post-multiplying (4.2.8) by S T, and computing H from the
resulting relationship, we obtain
1
H
HSST ª¬SST º¼
P
SST ª¬SST º¼ .
HP ,
(4.2.9)
where
1
(4.2.10)
On the other hand, pre-multiplying (4.2.8) by H T and computing S from the
resulting relationship, we obtain
S
1
ª¬ H T H º¼ HT HS
P 1S ,
(4.2.11)
where
1
P 1
ª¬ H T H º¼ HT H .
(4.2.12)
Equality of the first m columns of (4.2.11) and the first p rows of (4.2.9) yields
B
P 1B, C
CP .
(4.2.13)
One can easily verify that
HAS
H AS .
(4.2.14)
Pre-multiplying (4.2.14) by H T, post-multiplying it by S T and then computing A
from the resulting relationship, we obtain
224
Polynomial and Rational Matrices
A
1
ª¬ HT H º¼ HT H A SST ª¬SST º¼
1
P 1AP .
(4.2.15)
To show that P is the only feasible matrix, suppose that a matrix P also satisfies
(4.2.7). In this case, the equality HP = H P yields H(P - P ) = 0, which implies
that P = P , since H is a full column rank matrix.
4.2.2 Existence of Cyclic Realisations
We will provide the necessary and sufficient conditions for the existence of a
cyclic realisation R̂ n,m,p(A,B,C) for a given rational proper transfer matrix
T(s) pum(s).
Theorem 4.2.3. If A is a cyclic matrix and (A, B) is a controllable pair, then
W s
Adj> Is A @ B
det > Is A @
(4.2.16)
is an irreducible and normal matrix.
If A is a cyclic matrix and (A,C) is an observable pair then
W s
CAdj > Is A @
det > Is A @
(4.2.17)
is an irreducible and normal matrix.
Proof. According to Theorem 2.5.1
Adj> Is A @
det > Is A @
is an irreducible matrix if A is a cyclic matrix, i.e., [Is – A] is a simple matrix, and
at the same time, according to Theorem 3.1.1, it is a normal matrix as well.
If (A, B) is a controllable pair, then there exist two polynomial matrices M(s)
and N(s) such that
> Is A @ M
s BN s
I.
(4.2.18)
Pre-multiplying (4.2.18) by [Is – A]-1, we obtain
M s Adj> Is A @ B
N s
det > Is A @
1
>Is A @
.
(4.2.19)
The Problem of Realization
225
From (4.2.19) it follows immediately that the matrix (4.2.16) is irreducible.
Normality of the matrix (4.2.16) follows from normality of the matrix [Is – A] and
the Binet–Cauchy theorem.
The proof for (A, C) being an observable pair is analogous (dual).
„
Theorem 4.2.4. The rational matrix
CAdj > Is A @ B
det > Is A @
W s
(4.2.20)
is irreducible if and only if the matrices A, B, C constitute a cyclic realization
(A, B, C ) R̂ n,m,p of the matrix W(s) pum(s).
Proof. Necessity. If the matrices A, B, C do not constitute a cyclic realisation, then
A is not a cyclic matrix or (A, B) is not a controllable pair or (A, C) is not an
observable pair. If A is not a cyclic matrix, then
1
> Is A @
Adj > Is A @
det > Is A @
is a reducible matrix. If (A, B) is not a controllable pair, then
Adj > Is A @ B
det > Is A @
is a reducible matrix and if (A, C) is an unobservable pair then
CAdj > Is A @
det > Is A @
is reducible as well.
Sufficiency. According to Theorem 4.2.3, if A is a cyclic matrix and (A, B) is a
controllable pair, then the matrix (4.2.16) is irreducible, and if (A, C) is an
observable pair, then the matrix (4.2.17) is irreducible. Thus if the matrices A, B, C
constitute a cyclic realisation, then the matrix (4.2.20) is irreducible.
„
Theorem 4.2.5. There exists a cyclic realisation for a rational proper (transfer)
matrix T(s) pum(s) if and only if T(s) is a normal matrix.
226
Polynomial and Rational Matrices
Proof. Necessity. If there exists a cyclic realisation (A, B, C, D) of the matrix T(s),
then [Is – A]-1 is a normal matrix and according to the Binet–Cauchy theorem
[Is – A]-1B is a normal matrix. Normality of the matrix C[Is – A]-1 follows by
virtue of Theorem 4.2.3.
Sufficiency. If
T( s)
L( s )
m( s )
is a normal matrix, then using (4.2.3) we can compute the matrix D and the strictly
proper matrix (4.2.4), and in turn compute the cyclic matrix A with the dimensions
nun, n = deg m(s), the controllable pair (A, B), and the observable pair (A, C).
„
4.3 Computation of Cyclic Realisations
4.3.1 Computation of a Realisation with the Matrix A in the Frobenius
Canonical Form
The problem of computing a cyclic realisation (AF, B, C, D) for a rational matrix
T(s), with the matrix AF in the Frobenius canonical form, can be formulated in the
following way.
Given a rational proper matrix T(s) pum(s), compute a minimal realisation
(AF, B, C, D) R̂ n,m,p with the matrix AF in the Frobenius canonical form
AF
ª 0
« 0
«
« #
«
« 0
«¬ a0
1
0
!
0
1
!
#
0
#
0
%
a1
!
a2 !
0 º
0 »»
# ».
»
1 »
an1 »¼
(4.3.1)
Given T(s) and using (4.1.3) we can compute the matrix D, and in turn the
strictly proper rational matrix
Tsp s
T s D
1
C > Is A F @ B
L s
.
m s
(4.3.2)
Thus the problem is reduced to computing a minimal realization (AF, B, C) R̂ n,m,p
of the strictly proper matrix Tsp(s) pum(s).
The Problem of Realization
227
The characteristic polynomial m(s) of the matrix (4.3.1), which is equal to the
minimal polynomial <(s), has the form
m s
< s
det > Is A @
s n an1s n1 ! a1s a0 .
(4.3.3)
One can easily show that Adj [Is - AF] of the matrix (4.3.1) has the form
Adj > Is A F @
ªw s
«
¬M s
ªs
«0
«
Adj « #
«
«0
«¬ a0
1 0 !
s 1 !
#
0
a1
0
0
# % #
0 ! s
a2 ! an 2
0
0
º
»
»
# »
»
1 »
s an1 »¼
(4.3.4)
1 º
nun
»  >s@ ,
k s ¼
where
w s
k s
mn1
ª¬ mn1 s
mn2 s
! m1 s º¼ ,
T
ª¬ s s 2 ! s n1 º¼ ,
s s n1 an1s n2 ! a2 s a1
mn2 s
m1 s
(4.3.5)
s n2 an1s n3 ! a3 s a2
s an1
and M(s) (n-1)u(n-1)[s] is a polynomial matrix depending on the coefficients
a0,a1,…,an-1.
In order to perform the structural decomposition of the inverse [Is - AF]-1, we
reduce the matrix (4.3.4) to the form (3.4.14). To this end, we pre-multiply the
matrix (4.3.4) by
U s
ª 1
« k s
¬
01,n1 º
I n1 »¼
(4.3.6a)
and post-multiply it by the unimodular matrix
V s
Now we obtain
ª 0n11,
« 1
¬
I n1 º
.
w s »¼
(4.3.6b)
228
Polynomial and Rational Matrices
01,n1
ª 1
º
«0
»,
¬ n11, M s k s w s ¼
U s Adj> Is A F @ V s
(4.3.7)
where [Is - AF]-1 is a normal matrix. Every nonzero second-order minor is divisible
without remainder by m(s). Thus every entry of M (s) = M(s) – k(s)w(s) is
divisible without remainder by m(s). Therefore, we have
M s
ˆ s , M
ˆ ( s )  n1 u n1 > s @ .
m s M
(4.3.8)
Taking into account that
U 1 s
ª 1
«k s
¬
01,n1 º
, V 1 s
I n1 »¼
ªw s
«
¬ I n1
1 º
»
0n11, ¼
(4.3.9)
as well as (4.3.8) and (4.3.7), we obtain
Adj > Is A F @
ª 1
U 1 s «
¬« 0n11,
01,n1 º 1
V s
ˆ s »»
m s M
¼
(4.3.10)
PF s Q F s m s G F s ,
where
PF s
QF s
GF s
ª 1 º
U 1 s «
»
¬0n11, ¼
ª¬1 01,n1 º¼ V 1
ª 0
U 1 s «
«¬0n11,
ª 1 º
«
» ,
¬k s ¼
s
ª¬ w s
(4.3.11)
1º¼ ,
01,n1 º 1
V s
ˆ s »»
M
¼
ª 01,n1
«ˆ
«¬ M s
0 º
».
0n11, »¼
From (4.3.2) and (4.3.10), we have
L s
CAdj> Is A F @ B
P s Q s m s G s ,
where
CPF s Q F s B m s CG F s B
(4.3.12)
The Problem of Realization
ª 1
C«
¬k s
P s
CPF s
Q s
QF s B
G s
CG F s B.
ª¬ w s
229
º
»,
¼
(4.3.13)
1º¼ B,
Let Ci be the i-th column of the matrix C, and Bi the i-th row of the matrix B,
i = 1,2,…,n.
Taking into account (4.3.13) and (4.3.5) we obtain
ª 1 º
« s »
P s >C1 C2 ! Cn @ « »
« # »
« n1 »
¬s ¼
C1 C2 s ! Cn s n1 P1 P2 s P3 s 2 ! Pn s n ,
ª B1 º
«B »
Q s ª¬ mn1 s mn2 s ! m1 s 1º¼ « 2 »
« # »
« »
¬B n ¼
B1mn1 s B 2 mn2 s ! B n1m1 s B n
(4.3.14)
B1s n1 an1B1 B 2 s n2 an2 B1 an1B 2 B 3 s n3
! a1B1 a2 B 2 ! B n
Q1 Q 2 s Q3 s 2 ! Q n s n1
where
Pi
Ci , for i 1, 2, ! , n ,
Qn
B1 , Q n1
an1B1 B 2 , Q n2
Q1
a1B1 a2 B 2 ! an1B n1 B n .
(4.3.15a)
an2 B1 an1B 2 B3 , ! ,
(4.3.15b)
With Qn, Qn-1, ..., Q1 known we can recursively compute from (4.3.15b) the rows
Bi, i = 1,2,…,n of the matrix B
B1
Qn , B2
Q n1 an1B1 , B3
Bn
Q1 a1B1 a2 B 2 ! an1B n1.
Q n2 an2 B1 an1B 2 , ! ,
(4.3.17)
From the above considerations we can derive the following procedure for
computing the desired cyclic realisation (AF, B, C, D) of a given transfer matrix
T(s) pun(s).
230
Polynomial and Rational Matrices
Procedure 4.3.1.
Step 1: Using (4.1.3), compute the matrix D pum and the strictly proper matrix
(4.3.2).
Step 2: With the coefficients a0,a1,…,an-1 of the polynomial m(s) known, compute
the matrix AF given by (4.3.1).
Step 3: Performing the decomposition of the polynomial matrix L(s), compute the
matrices P(s) and Q(s).
Step 4: Using (4.3.15a) and (4.3.17), compute the matrices C and B.
Example 4.3.1.
Using Procedure 4.3.1, compute the cyclic realisation of the rational matrix
ª s3 s 1
s3 s 2 2s 2 º
1
«
».
s 3 s 2 2 s 1 ¬ s 3 s 2 2 s 2 s 3 2 s 2 5s 2 ¼
T s
(4.3.18)
It is easy to check that the matrix (4.3.18) is normal. Thus its cyclic realization
exists. Using Procedure 4.3.1, we compute
Step1: Using (4.1.3) and (4.3.2), we obtain
D
lim T s
s of
ª 1 1 º
« 1 2»
¬
¼
(4.3.19)
and
Tsp s
T s D
ª s 2 s 2 1º
1
«
».
s s 2s 1 ¬ 1
s¼
3
2
(4.3.20)
Step 2: In this case, a0 = 1, a1 = 2, a2 = 1 and
AF
ª0 1 0º
«0 0 1 ».
«
»
«¬ 1 2 1»¼
Step 3: In order to perform the structural decomposition of the matrix
L s
ª s 2 s 2 1º
«
»
s¼
¬ 1
it suffices to interchange its columns, i.e., to post-multiply it by
(4.3.21)
The Problem of Realization
231
ª0 1 º
«1 0 »
¬
¼
V s
and compute P(s) and Q(s)
ª s 2 s 2 1 º ª 0 1 º ª1 s 2 s 2 º
«
»«
»
» «
s ¼ ¬1 0 ¼ ¬ s
1 ¼
¬ 1
0
ª1 º
ª0
º
2
« s » ª¬1 s s 2 º¼ « 0 s 3 s 2 2s 1» ,
¬ ¼
¬
¼
L s V s
that is
ª1 º
«s» , Q s
¬ ¼
P s
ª0 1 º
ª¬1 s 2 s 2 º¼ «
»
¬1 0 ¼
ª¬ s 2 s 2 1º¼ .
Step 4: Taking into account that
ª1 º ª 0 º
« 0 » «1 » s
¬ ¼ ¬ ¼
P s
P1 P2 s
Q s
Q1 Q 2 s Q3 s 2
and
> 2 1@ >1
0@ s >1 0@ s 2 ,
from (4.3.15a) and (4.3.17), we obtain
B2
ª1 º
ª0º
ª0 º
« 0 » , C2 P2 «1 » , C3 P3 «0 » , B1
¬ ¼
¬ ¼
¬ ¼
Q 2 a2 B1 >1 0@ 1>1 0@ > 0 0@ ,
B3
Q1 a1B1 a2 B 2
C1
P1
> 2 1@ 2 >1
0@
>0 1@
Q3
>1
0@ ,
.
Hence the desired matrices B and C are
B
ª B1 º
« »
«B 2 »
«¬ B3 »¼
ª1 0 º
«
»
«0 0 » , C
«¬0 1 »¼
>C1
C2
C3 @
ª1 0 0 º
«0 1 0 » .
¬
¼
(4.3.22)
232
Polynomial and Rational Matrices
It is easy to check that (AF, B) (determined by (4.3.21) and (4.3.22)) is a
controllable pair and (AF, C) is an observable pair. Thus the obtained realisation is
cyclic.
4.3.2 Computation of a Cyclic Realisation with Matrix A in the Jordan
Canonical Form
The problem of computing the cyclic realisation (AJ, B, C, D) R̂ n,m,p of a given
transfer matrix T(s) with the matrix AJ in the Jordan canonical form can be
formulated as follows.
Given a normal rational matrix T(s) pum(s), compute the minimal realisation
(AJ, B, C, D) R n,m,p with the matrix AJ in the Jordan canonical form
0
ª J1
«0
«
«#
«
¬« 0
AJ
0º
0 »»
% # »
»
! J p ¼»
!
!
J2
#
0
diag ª¬ J1
J 2 ! J p º¼ ,
(4.3.23a)
with
J ci
J ci
ª si
«0
«
«#
«
«0
«¬ 0
1
si
ª si
«1
«
«0
«
«#
«¬ 0
0
#
0
0
si
1
#
0
0 ! 0
1 ! 0
# % #
0 ! si
0 ! 0
0 ! 0
0 ! 0
si ! 0
#
% #
0 ! 1
0º
0 »»
# »  mi umi ,
»
1»
si »¼
0º
0 »»
0 »  mi umi ,
»
#»
si »¼
(4.3.23b)
where i = 1,2,…,p, and s1,s2,…,sp are different poles with multiplicities m1.m2,…
…,mp, respectively,
p
¦m
i
n
i 1
of the matrix T(s).
With the matrix T(s) given, and using (4.1.3) we compute the matrix D, and
then the strictly proper rational matrix (4.1.4).
The Problem of Realization
233
The problem has been reduced to the computation of the minimal realization
(AJ, B, C) R n,m,p of the strictly proper matrix Tsp(s) pum(s).
Firstly consider the case of poles of multiplicity 1 (m1 = m2 = … = mp = 1) of
the matrix
Tsp ( s )
L( s )
,
m( s )
m s
s s1 s s2 ! s sn , si z s j , for i z j , i, j 1, ! , n, (4.3.24)
where
and s1,s2,…,sn are real numbers.
In this case, Tsw (s) can be expressed in the following form
n
Ti
¦ss
Tsp s
i 1
,
(4.3.25)
i
where
Ti
L si
lim s si Tsp s
, i 1, ! , n .
n
s o si
–
(4.3.26)
si s j
j 1
j zi
From (4.3.26) and (3.4.11) it follows that
rank Ti
1, i 1, ! , n .
(4.3.27)
We decompose the matrix Ti into the product of the two matrices Bi and Ci of rank
equal to 1
Ti
Ci B i , rank Ci
rank Bi
1, i 1, ! , n .
(4.3.28)
We will show that the matrices
AJ
diag > s1
s 2 ! sn @ , B
ª B1 º
«B »
« 1», C
« # »
« »
¬B n ¼
are a minimal realisation of the matrix Tsw (s).
>C1
C1 ! Cn @ (4.3.29)
234
Polynomial and Rational Matrices
To this end, we compute
1
C > Is A J @ B
>C1
n
ª 1
C1 ! Cn @ diag «
¬ s s1
Ci B i
n
Ti
¦ss ¦ss
i 1
i
i 1
1
s s2
ª B1 º
« »
1 º « B1 »
!
»
s sn ¼ « # »
« »
¬B n ¼
Tsp s .
i
Thus the matrices (4.3.29) are a realisation of the matrix Tsp(s).
It is easy to check that
rank > Is A J B @
ª s s1
« 0
rank «
« #
«
¬ 0
0
!
0
s s2 !
0
#
0
B1 º
B2 »»
# »
»
Bn ¼
%
#
! s sn
n
for all s , since rank Bi = 1 for i = 1,…,n.
Analogously to the above
ª Is A J º
rank «
»
¬ C ¼
ª s - s1
« 0
«
rank « #
«
« 0
«¬ C1
0
!
s - s2 !
#
0
%
!
C2
!
0 º
0 »»
# »
»
s - sn »
Cn »¼
n
for all s , since rank Ci = 1 for i = 1,…,n.
Thus (AJ, B) is a controllable pair and (AJ, C) is an observable pair. Hence the
realisation (4.3.29) is minimal.
The desired cyclic realisation (4.3.29) can be computed using the following
procedure.
Procedure 4.3.2.
Step 1: Using (4.3.26) compute the matrices Ti for i = 1,…,n.
Step 2: Decompose the matrices Ti into the product (4.3.28) of the matrices Bi and
Ci, i = 1,…,n.
Step 3: Compute the desired cyclic realisation (4.3.29).
Example 4.3.2.
Given the normal strictly proper matrix
The Problem of Realization
1
ª
«
s 1
«
1
«
« s 1 s 2
¬
Tsw s
1 º
s 1»
»
1 »
s 1 »¼
ª s 2 s 2º
1
,
s 2 »¼
s 1 s 2 «¬ 1
235
(4.3.30)
compute its cyclic realisation (AJ, B, C).
In this case, m(s) = (s + 1)(s + 2) and the matrix (4.3.30) has the real poles
s1 = -1 and s2 = -2. Using Procedure 4.3.2 we obtain the following.
Step 1: Using (4.3.26), we obtain
T1
T2
lim s s1 Tsp s
s o s1
lim s s2 Tsp s
s o s2
1º
ª 1
ª1 1º
« 1
»
«1 1» ,
«
»
1
¬
¼
¬ s 2 ¼ s 1
ªs 2 s 2º
« s 1 s 1»
ª 0 0º
«
»
« 1 0 » .
s 2»
¬
¼
« 1
¬« s 1 s 1 »¼ s 2
(4.3.31)
Step 2: We decompose the matrices (4.3.31) into the products (4.3.28)
T1
C2
ª1 1º
«1 1» C1B1 , C1
¬
¼
ª0º
« 1» , B 2 >1 0@
¬ ¼
ª1º
«1» , B1
¬¼
>1 1@ ,
T2
ª 0 0º
« 1 0 »
¬
¼
C2 B 2 ,
.
Step 3: Thus the desired cyclic realisation of the matrix (4.3.30) is
AJ
C
ª s1
«
¬0
>C1
0º
s2 »¼
C2 @
ª 1 0 º
« 0 2 » , B
¬
¼
1
0
ª
º
«1 1» .
¬
¼
ª B1 º
«B »
¬ 2¼
ª1 1 º
«1 0 » ,
¬
¼
(4.3.32)
If the matrix Tsp(s) has complex conjugated poles, then using Procedure 4.3.2,
we obtain the cyclic realisation (4.3.29) with complex entries. In order to obtain a
realisation with real entries, we additionally transform the complex realisation
(4.3.29) by the similarity transformation.
Let the equation m(s) = 0 have r distinct real roots s1,s2,…,sr and q distinct pairs
of complex conjugated roots a1 + jb1, a1 – jb1,…,aq + jbq, aq jbq, r + q = n.
Let the complex realisation (4.3.29) have the form
236
Polynomial and Rational Matrices
AJ
B
C
s2 … sr
diag[ s1
ª B1 º
«
# »»
«
« Br »
«
»
« c1 jd1 »
« c1 jd1 » ,
«
»
# »
«
« c jd »
q»
« q
«¬ c q jd q »¼
ª¬C 1 C2 … Cr
a1 jb1
a1 jb1 … aq jbq
aq jbq ],
(4.3.33)
g1 jh1
g 1 jh1 … g q jhq
gq jhq º¼ .
In this case, the similarity transformation matrix P has the form
P
diag >1 … 1 D1 … D1 @  C nu n , D1
1 ª1 j º
.
2 «¬1 j »¼
(4.3.34)
Using (4.3.33) and (4.3.34), we obtain
AJ
P 1A J P
B
P 1B
C
CP
diag[ s1 … s r
A1 … A q ],
ª B1 º
« # »
«
»
« Br »
«
»
« 2c1 » ,
« 2d »
« 1»
« # »
« 2c »
« q»
«¬ 2dq »¼
ª¬C1 … C r
(4.3.35)
g1 h1 … g q
hq º¼ ,
since
0 º
ªa jbk
D1 1 « k
D
ak jbk »¼ 1
¬ 0
ª ck jd k º ª 2ck º
D1 1 «
» «
» , > gk jhk
¬ ck jd k ¼ ¬2d k ¼
Ak
ªak
«b
¬ k
bk º
,
ak »¼
gk jhk @ D1
Thus the realisation (4.3.35) has only real entries.
(4.3.36)
> gk
hk @ .
The Problem of Realization
237
Example 4.3.3.
Given the normal matrix
Tsp s
s 3 º
ª 1
1
,
«
2
s 3 s 4 s 2 ¬ s 4 s 2»¼
3
(4.3.37)
2
compute its real cyclic realisation (AJ, B, C).
The matrix (4.3.37) has one real root s1 = 1 and the pair of the complex
conjugated roots s2 = 1 + j, s3 = 1 j since
s s1
s s2 s s3
s 1
s 3 3s2 4 s 2 .
s 1 j s 1 j
Applying Procedure 4.3.2, we obtain the following.
Step 1: Using (4.3.26) we obtain
T1
lim s s1 Tsp s
T2
lim s s2 Tsp s
so s1
s3º
4 s 2 »¼ s
1
ª1 2º
« 1 2 » ,
¬
¼
so s2
1
ª 1
«
s 1 s 1 j ¬ s2
T3
1
ª 1
s 2 s 2 «¬ s 2
2
s 3 º
»
4s 2¼ s
1 j
1º
ª 1
« 2 1 j 2 » ,
«
»
¬ j 1 j2 ¼
(4.3.38)
lim s s3 Tsp s
s os3
1
ª 1
s 1 s 1 j «¬ s 2
s 3 º
4 s 2 »¼ s
1 j
ª 1
« 2
«
¬ j
1º
1 j »
2 .
»
1 j2 ¼
Step 2: Decomposing the matrices (4.3.38) into the products (4.3.28), we obtain
T1
T2
T3
ª1 2º
ª1 º
« 1 2» C1B1 , C1 « 1» ,
¬
¼
¬ ¼
1
1
ª
º
« 2 1 j 2 » C B , C
2 2
2
«
»
1
2
j
j
¬
¼
1
1
ª
º
« 2 1 j 2 » C B , C
3 3
3
«
»
1 j2 ¼
¬ j
B1
>1
2@ ,
ª 1º
« » , B2
¬ 2 j¼
ª 1 º
«
» , B3
¬ 2 j ¼
1º
ª 1
«¬ 2 1 j 2 »¼ ,
1º
ª 1
« 2 1 j 2 » .
¬
¼
Step 3: The desired cyclic realisation (4.3.29) with complex entries is
238
Polynomial and Rational Matrices
AJ
B
ª s1
«0
«
«¬ 0
ª B1 º
«B »
« 2»
«¬ B 3 »¼
0
s2
0
0º
0 »»
s3 »¼
ª
« 1
«
« 1
« 2
« 1
«
¬ 2
0
0
ª 1
« 0 1 j
0
«
«¬ 0
1 0
º
2 »
»
1
1 j » , C >C1
2»
1»
1 j »
2¼
º
»,
»
j »¼
(4.3.39)
1 º
ª1 1
C2 C 3 @ «
».
¬ 1 2 j 2 j ¼
In order to compute a real realization, we perform the similarity transformation
(4.3.34) on the realisation (4.3.39)
P
diag >1 D1 @
ª
«1
«
«0
«
«
«0
¬«
0
1
2
1
2
º
0 »
»
1»
.
j
2»
»
1
j »
2 ¼»
Using (4.3.35), we obtain
AJ
B
P 1 AJ P
ª
«1
«
«0
«
«
«0
¬«
º
0 »
»
1
1»
j
2
2»
»
1
1
j »
2
2 ¼»
0
1
ª
«1
0
0 º«
ª 1
«
»
0 » «0
« 0 1 j
«
0
1 j¼» «
¬« 0
«0
¬«
ª 1 0 0 º
« 0 1 1» ,
«
»
«¬ 0 1 1»¼
ª
ºª
º
0 »«1
2 »
«1 0
«
»«
»
1
1
1
1
P 1B « 0
1 j »
j » «
«
2
2 »« 2
2»
«
»«
»
1
1
1
1
«0
j » «
1 j »
«¬
2
2 »¼ ¬« 2
2 »¼
ª1 2º
«
»
« 1 2» ,
¬« 0 1»¼
0
1
2
1
2
º
0 »
»
1 »
j
2»
»
1
j »
2 ¼»
The Problem of Realization
C
CP
ª1 1
« 1 2 j
¬
ª
«1
«
1 º«
0
2 j»¼ «
«
«0
¬«
0
1
2
1
2
º
0 »
»
1
j »
2 »
1»
j »
2 »¼
239
ª1 1 0 º
« 1 0 2» .
¬
¼
Let in a general case
s s1
m s
m1
s s2
m2
… s sp
mp
p
,
¦m
i
n,
i 1
where s1 ,s2 ,…, sp are real or complex conjugated poles.
In this case, the matrix Tsw (s) can be expressed as
Tsp s
p
mi
¦¦
i 1 j 1
Tij
s si
mi j 1
,
(4.3.40)
where
Tij
1
d j 1 ª
s si
j 1 ! d s j 1 ¬
mi
Tsp s º .
¼| s si
(4.3.41)
Let only one Jordan block J i of the form (4.3.23b) correspond to the i-th pole si
with multiplicity mi , and the matrices B and C have the form
B
ª B1 º
« »
« B2 » , C
« # »
« »
«¬ B p »¼
¬ªC 1 C2 … C p ¼º
(4.3.42a)
where
Bi
ª B i1 º
«B »
« i2 » , C
i
« # »
«
»
«¬Bimi »¼
Taking into account that
ªCi 1 Ci 2 … Cim º , i 1,2, … , p .
i ¼
¬
(4.3.42b)
240
Polynomial and Rational Matrices
ª 1
«s s
i
«
«
« 0
«
« #
«
«
« 0
¬
1
> Is J i @
1
…
2
s si
1
s si
…
#
%
0
…
1
º
»
»
»
1
mi 1 »
s si
» , i 1,2, … , p , (4.3.43)
»
#
»
»
1
»
s si ¼
s si
mi
we can write
1
Ci > Is Ji @ Bi
1
s si
mi
¦C
B ik ik
k 1
mi 1
1
s si
2
¦C
ik
B ik 1 … k 1
1
s si
mi
Ci1B imi .
(4.3.44)
A comparison of (4.3.40) to (4.3.44) yields
Tij
j
¦C
ik
Bi,mi j k , for i 1, …, p, j 1, … , mi .
(4.3.45)
k 1
From (4.3.45) for j = 1, we obtain
Ti1
Ci1Bimi .
(4.3.46)
With the matrix Ti1 given, we decompose it into the column matrix Ci1 and the row
matrix B imi . Now for (4.3.45), with j = 2, we obtain
Ti 2
Ci1 Bi,mi 1 Ci 2 Bi,m i .
(4.3.47)
With Ti2 and Ci1 , B i,mi known, we take as the vector Ci2 this column of the matrix
Ti2 that corresponds to the first nonzero entry of the matrix B i,mi and we multiply it
by the reciprocal of this entry. Then we compute
Ti(21 )
Ti 2 Ci 2 Bi,mi
Ci1Bi,mi 1
(4.3.48)
and Bi,m i 1 for the known vector Ci1 .
From (4.3.45), for j = 3, we have
Ti3
Ci1 Bi,mi 2 Ci2 Bi,mi 1 Ci 3Bi,mi .
(4.3.49)
The Problem of Realization
241
With Ti3 and Ci2, Bi,mi 1 known, we can compute
Ti 3
Ti 3 Ci 2 B i,mi 1
(4.3.50)
Ci1Bi,mi 2 Ci 3B i,mi
and then, in the same way as Ci2, we can choose Ci3 and compute Bi,mi 2 . Pursuing
the procedure further, we can compute Ci1 Ci2,…, Ci,mi and Bi1 Bi2,…, Bi,mi .
If the structural decomposition of the matrix L(s) of the following form is given
L s
P s Q s m s G s ,
(4.3.51)
then
s si
mi
L s
mi s
Tsw s
P s Q i s s si
mi
G s , i 1, ! , p, (4.3.52)
where
mi s
m s
s si
mi
, Qi s
Q s
.
mi s
(4.3.53)
Taking into account (4.3.53), we can write (4.3.41) in the following form
Tij
1
d j 1
ª P s Qi s º¼
j 1 ! ds j 1 ¬
s si
for i 1, ! , p, j 1, ! , mi , (4.3.54)
since
d j 1 ª
s si
ds j 1 ¬
mi
G s º
¼
s si
0 for j 1, ! , mi , i 1, ! , p .
From (4.3.54) it follows that the matrices Tij depend only on the matrices P(s) and
Q(s) and do not depend on the matrix G(s).
Knowing P(s) and Q(s) and using (4.3.54), we can compute the matrices Tij for
i = 1,…,p and j = 1,…,mi.
It is easy to check that for the matrices (AJ, B, C) determined by (4.3.23) and
(4.3.42), (AJ, B) is a controllable pair and (AJ, C) is an observable pair. Thus these
matrices constitute a cyclic realisation. If the poles s1,s2,…,sp are complex
conjugated, then, according to (4.3.34), in order to obtain a real cyclic realisation
one has to transform them by the similarity transformation.
From the above considerations, one can derive the following important
procedure for computing the cyclic realisation (AJ, B, C) for a given normal,
strictly proper matrix Tsw (s) with multiple poles.
242
Polynomial and Rational Matrices
Procedure 4.3.3.
Step 1: Compute the poles s1,s2,…,sp of the matrix Tsp(s) and their multiplicities
m1,m2,…,mp.
Step 2: Using (4.3.41) or (4.3.54) compute the matrices Tij for i = 1,…,p and
j = 1,…,mi.
Step 3: Using the procedure established above, compute the columns Ci1
Ci2,…, Ci,mi of the matrix Ci and the rows Bi1 Bi2,…, Bi,mi of the matrix Bi
for i = 1,…,p.
Step 4: Using (4.3.23) and (4.3.42) compute the desired realisation (AJ, B, C).
Example 4.3.3.
Given the normal matrix
1
Tsp s
s 1
2
s2
2
2
ª
s 1
«
¬« s 1 s 2
2
s 1 º
»,
s 2 ¼»
(4.3.55)
compute its cyclic realisation (AJ, B, C).
Applying Procedure 4.3.3, we obtain the following.
Step 1: The matrix (4.3.55) has the two double real poles: s1 = 1, m1 = 2, s2 = 2,
m2 = 2.
Step 2: Using (4.3.41), we obtain
T11
2
s 1 Tsp s
1
s2
2
s s1
2
ª
s 1
«
¬« s 1 s 2
d ª
2
T12
s 1 Tsp s º s s1
¬
¼
ds
2
ª
s 1
d ­° 1
«
®
ds ¯° s 2 2 «¬ s 1 s 2
T21
2
s 1 º
»
s 2 ¼» s
ª0 0 º
«0 1 » ,
¬
¼
1
2
s 1 º ½°
»¾
s 2 »¼ ¿°
s
1
ª0 0 º
«1 1» ,
¬
¼
2
s 2 Tsp s
s s2
2
ª
s 1
2 «
s 1 ¬« s 1 s 2
1
2
s 1 º
»
s 2 »¼ s
d ª
2
T22
s 2 Tsp s º s s2
¬
¼
ds
2
s 1
s 1
d ­° 1 ª
«
®
2
ds ¯° s 1 «¬ s 1 s 2
s2
2
ª1 1º
«0 0 » ,
¬
¼
2
º ½°
»¾
»¼ ¿° s
2
ª 0 0º
« 1 1 » .
¬
¼
The Problem of Realization
243
Step 3: Using (4.3.46) and (4.3.47), we obtain
ª0 0 º
ª0º
«0 1 » C11B12 , C11 «1 » , B12
¬
¼
¬ ¼
ª0 0 º
T12 «
» C11B11 C12 B12 .
¬1 1¼
We choose
T11
C12
B11
T21
T22
ª0º
« 1» thus C11B11
¬ ¼
>1 0@ ,
ª1
«0
¬
ª0
« 1
¬
1º
0 »¼
0º
1 »¼
T12 C12 B12
C21B 22 , C21
ª1 º
«0» ,B 22
¬ ¼
>0 1@ ,
ª0 0 º ª 0 º
«1 1» « 1» > 0 1@
¬
¼ ¬ ¼
>1
ª0 0º
«1 0 » ,
¬
¼
1@ ,
C21B 21 C22 B 22 .
We choose
C22
B 21
ª0º
« 1» thus C21B 21
¬ ¼
> 0 0@ .
T22 C22 B 22
ª 0 0º ª 0 º
« 1 1 » « 1» >1 1@
¬
¼ ¬ ¼
ª0 0º
«0 0» ,
¬
¼
Step 4: Using (4.3.23) and (4.3.42), we obtain the desired realisation
AJ
C
ª 1 1 0 0 º
« 0 1 0 0 »
«
», B
« 0 0 2 1 »
«
»
¬ 0 0 0 1¼
>C11
C12
C21 C 22 @
ª B11 º ª1 0 º
« B » «0 1 »
« 12 » «
»,
« B 21 » « 0 0 »
« » «
»
¬ B 22 ¼ ¬1 1¼
ª0 0 1 0 º
«1 1 0 1» .
¬
¼
A question arises: Is it possible, using the similarity transformation, to obtain a
cyclic realisation from a noncyclic realisation and vice versa? The following
theorem provides us with the answer.
Theorem 4.3.1. A realisation (PAP-1, PB, CP-1, D)Rn,m,p for an arbitrary
nonsingular matrix P is a cyclic realisation if and only if (A, B, C, D) Rn,m,p is a
cyclic realisation.
Proof. According to Theorem 4.2.2 (PAP-1, PB, CP-1, D) is a minimal realisation
if and only if (A, B, C) is a minimal realisation. We will show that the similarity
transformation does not change the invariant polynomials of A. Let U and V be the
244
Polynomial and Rational Matrices
unimodular matrices of elementary operations on the rows and columns of [Is – A]
transforming this matrix to its Smith canonical form, i.e.,
> Is A @S
U s > Is A @ V s .
(4.3.56)
Let U (s) = U(s)P-1 and V (s) = PV(s). U (s) and V (s) are also unimodular
matrices for any nonsingular matrix P, since det U (s) = det U(s) det P-1 and
det V (s) = det P det V(s), with det P and det P-1 independent of the variable s. We
will show that the matrices U (s) and V (s) reduce the matrix [Is – PAP-1] to its
Smith canonical form [Is – A]S.
Using the definition of U (s) and V (s), and (4.3.56), we obtain
U s [Is PAP 1 ] V s
U s > Is A @ V s
U s P 1P > Is A @ P 1PV s
>Is A @S .
Thus the matrices [Is – PAP-1], [Is – A] have the same invariant polynomials.
Hence (PAP-1, PB, CP-1, D) is a cyclic realisation if and only if (A, B, C, D) is a
cyclic realisation.
„
4.4 Structural Stability and Computation of the Normal Transfer
Matrix
4.4.1 Structural Controllability of Cyclic Matrices
A matrix A nun is called a cyclic matrix if its minimal polynomial <(s) coincides
with its characteristic polynomial, <(s) = det [Is – A].
Definition 4.4.1. A nun is called a structurally stable matrix if and only if there
exist such a positive number H0 that for any matrix B nun and any H satisfying the
condition |H| < H0 all the matrices A + BH are stable.
Theorem 4.4.1. A cyclic matrix A
nun
is structurally stable.
The proof of this theorem can be found in [189]; it is based on the following
two facts:
1.
If A nun is a nonsingular matrix then all the matrices A + B are also
nonsingular whenever
B D
(4.4.1)
The Problem of Realization
2.
for some D > 0.
If A nun has rank A = r, then rank [A + B] t r for the matrix B
satisfying the condition (4.4.1).
245
nun
Noncyclic matrices are not structurally stable but for a noncyclic matrix
A nun one can always choose a matrix B nun and a sufficiently small number H
(|H| > 0) so that the sum A + BH is a cyclic matrix.
Only for a particular choice of the matrix B and H is the sum A + BH a
noncyclic matrix. As it is known, a matrix in the Frobenius canonical form
A
ª 0
« 0
«
« #
«
« 0
«¬ a0
1
0
0
1
!
#
0
#
0
%
!
a1
!
a2 !
0 º
0 »»
# »
»
1 »
an1 »¼
(4.4.2)
is a cyclic matrix regardless of the values of the coefficients a0,a1,a2,..,an1.
For example, the matrix
A
ª1 1 0 º
«0 1 0 »
«
»
«¬ 0 0 a »¼
(4.4.3)
is a cyclic matrix for all the values of the coefficient a z 1, and it is a noncyclic
matrix only for a = 1.
Let 'A nun be regarded as a disturbance (uncertainty) to the nominal matrix
A nun, and take HB = 'A. Then, according to Theorem 4.4.1, since A is cyclic,
the matrix A + 'A is also cyclic.
4.4.2 Structural Stability of Cyclic Realisation
A minimal realisation (A, B, C, D) R̂ n,m,p with the cyclic matrix A is called a
cyclic realisation.
Theorem 4.4.2. Let (A1, B1, C1, D1)Rn,m,p be a cyclic realisation and
(A2, B2, C2, D2)Rn,m,p another realisation of the same dimensions. Then there exist
such a number H0 > 0 that all the realisations
A1 H A 2 , B1 H B 2 ,C 1 H C2 , D1 H D2  Rn,m,p for H H 0 ,
are cyclic realisations.
246
Polynomial and Rational Matrices
Proof. According to Theorem 4.4.1, if A1 is a cyclic matrix, then all the matrices
A1 + HA2 are cyclic for |H| < H0. If (A1, B1) is a controllable pair then
(A1 + HA2, B1 + HB2) is also controllable for all |H| < H1. Analogously, if (A1, C1) is
an observable pair, then (A1 + HA2, C1 + HC2) is also observable for all |H| < H2.
Thus the realisation (A1 + HA2, B1 + HB2, C1 + HC2) is a minimal one for
|H| < min(H1, H2) = H0, and with (A1 + HA2) being a cyclic matrix it is a cyclic
realisation as well.
„
Example 4.4.1.
A cyclic realisation (A1, B1, C1)R3,3,1 is given with
A1
ª0
«0
«
«¬ a10
1
0º
1 »» , B1
a12 »¼
0
a11
ª0º
«0» , C
1
« »
«¬1 »¼
>1
0 0@ ,
(4.4.4)
where a10, a11, a12 are arbitrary parameters.
The question, arises for which values of the parameters a20, a21, a22, b and c in
the matrices
A2
ª0
«0
«
¬« a20
1
0
a21
0º
1 »» , B 2
a22 ¼»
ª0 º
«0 » , C
2
« »
¬«b ¼»
>0
c 0@
(4.4.5)
is the realisation (A1 + A2, B1 + B2, C1 + C2)R3,3,1 a cyclic one?
We denote
A
A1 A 2
ª0
«0
«
«¬ a0
>1 c
2
0
a1
0º
2 »» , B
a2 »¼
B1 B 2
ª 0 º
« 0 »,
«
»
«¬1 b »¼
0@ ,
C
C1 C2
ak
a1k a2 k , for k
where
0,1, 2,
(4.4.6)
A is a cyclic matrix for all the values of the parameters a20, a21, and a22. (A, B) is a
controllable pair for those values of the parameters a20, a21, a22 and b, for which
det [B, AB, A2B] z 0, that is
The Problem of Realization
det
0
0
4 1 b
0
2 1 b
2 a2 1 b
1 b
a2 1 b
2a1 a
2
2
8 1 b
3
247
z 0 for b z 1. (4.4.7)
1 b
(A, C) is an observable pair for those values of the parameters a20, a21, a22 and c,
for which
ª C º
det «« CA »» z 0 ,
«¬CA 2 »¼
that is
1
c
ª C º
«
»
det « CA »
0
2
«¬CA 2 »¼ 2ca0 2ca1
for a1c 2 z 2 a2 c a0 c3 ,
0
2c
4 2ca2
4 ª¬ 2 a2 c a0 c3 a1c 2 º¼ z 0,
(4.4.8)
and taking (4.4.6) into account, we obtain
a20 c3 a21c 2 a22 c z a11c 2 a10 c3 a12 c 2 .
(4.4.9)
Thus (A, B, C) is a cyclic realisation for the parameters a20, a21, a22, b and c in the
matrices (4.4.5) satisfying the condition (4.4.9) and b z 1.
4.4.3 Impact of the Coefficients of the Transfer Function on the System
Description
Consider the transfer matrix
T s
0 º
ªs 2
.
«
s 1 a »¼
s 1 s 2 ¬ 0
1
This matrix is normal if and only if a = 0, since the polynomial
s2
0
0
s 1 a
s 1 a s 2
is divisible without remainder by (s + 1)(s + 2) if and only if a = 0.
(4.4.10)
248
Polynomial and Rational Matrices
For a = 0 there exists a cyclic realisation (A, B, C) R̂ 2,2,2 of the matrix
(4.4.10) with
A
ª 1 0 º
« 0 2 » , B
¬
¼
ª1 0 º
«0 1 » , C
¬
¼
ª1 0 º
«0 1» ,
¬
¼
(4.4.11)
which can be computed using Procedure 4.3.2.
Applying Procedure 4.3.2 for a z 0, we obtain
lim s s1 T s
T1
s o s1
ª1
«0
¬
0º
a »¼
C 1B 1 ,
1 ªs 2
s 2 «¬ 0
ª1
«0
¬
C1
s o s2
ª0
«0
¬
0 º
1 a »¼
C 2B 2 ,
0º
, B1
1 »¼
1 ªs 2
s 1 «¬ 0
lim s s 2 T s
T2
0
º
s 1 a »¼
C2
ª1
«0
¬
0º
,
a »¼
0
º
s 1 a »¼
ª0º
«1 » , B 2
¬ ¼
>0
s 1
s 2
1 a @.
Thus the desired minimal realisation is
A
C
ªI 2 s1
« 0
¬
>C1
0º
s2 »¼
C2 @
ª 1
«0
«
«¬ 0
ª1 0
«0 1
¬
0 0º
1 0 »» , B
0 2 »¼
0º
.
1 »¼
ª B1 º
«B »
¬ 2¼
0 º
ª1
«0
»,
a
«
»
«¬ 0 1 a »¼
(4.4.12)
To the cyclic realisation (4.4.11) corresponds a system described by the
following state equations
x1
x2
x1 u1 ,
y1
x1 ,
y2
x2 .
2 x2 u2 ,
(4.4.13)
To the minimal realisation (4.4.12) corresponds a system described by the
following state equations
Problem of Realisation
x1
x2
x1 u1 ,
x3
2 x3 1 a u2 ,
y1
x1 ,
y2
x2 x3 .
249
x2 au2 ,
(4.4.14)
Note that for a = 0 in (4.4.14) we do not obtain (4.4.13), and the pair (A, B) of
the system (4.4.14) becomes not controllable.
The above considerations can be generalised to the case of linear systems of
any order.
4.4.4 Computation of the Normal Transfer Matrix on the Basis of Its
Approximation
Consider a transfer matrix
Tp s
L s
 pum s ,
m s
(4.4.15)
whose coefficients differ from the coefficients of a normal transfer matrix
T s
L s
 pum s .
m s
(4.4.16)
The problem of computing the normal transfer function on the basis of its
approximation can be formulated in the following way.
With the transfer matrix (4.4.15) given, one has to compute the normal transfer
matrix (4.4.16), which is a good approximation of the matrix (4.4.15).
Below we provide a method for solving the problem. The method is based on
the structural decomposition of the matrix (4.4.15) [168].
Applying elementary operations, we transform the polynomial matrix
L (s) pum[s] into the form
U s L s V s
ª 1
i s «
¬k s
w s º
».
M s ¼
(4.4.17)
where U(s) and V(s) are polynomial matrices of elementary operations on rows and
columns, respectively; i(s) is a polynomial and
w s  1u m1 > s @ , k s  p 1 > s @ , M s  p 1 u m 1
>s@ .
(4.4.18)
250
Polynomial and Rational Matrices
Pre-multiplication of the matrix
L1 s
ª 1
«
¬k s
w s º
»
M s ¼
(4.4.19)
by the unimodular matrix
U1 s
ª 1
«k s
¬
01,p 1 º
I p 1 »¼
and post-multiplication by the unimodular matrix
V1 s
ª 1
«
¬ 0m1,1
w s º
»
I m1 ¼
yields
U1 s L1 s V1 s
01,m1
ª 1
º
«0
».
M
s
k
s
w
s
¬ p11,
¼
(4.4.20)
m s M1 s R s ,
(4.4.21)
In this method we take
M s k s w s
where
M1 s  p 1 u m 1
>s@ , R
s 
p 1 u m 1
>s@ ,
deg R s deg m s .
In the further considerations we omit the polynomial matrix R(s).
From (4.4.17) and (4.4.20), we have
L s
U 1 s i s L1 s V 1 s
01,m1
ª 1
º 1
1
U 1 s i s U11 s «
» V1 s V s
M
s
k
s
w
s
0
¬ p 11,
¼
01,p 1 º ª 1
01,m1
ª 1
º
U 1 s i s «
» «0
»
I
M
k
s
s
k
s
w
s
p 1 ¼ ¬ p 11
,
¬
¼
ª 1
u«
¬0m11,
w s º 1
»V s .
I m1 ¼
(4.4.22)
Problem of Realisation
251
Using (4.4.21) and (4.4.22) and omitting R(s), we obtain
ª 1
U 1 s i s «
¬k s
ª 1
w s º 1
u«
»V s
0
¬ m11, I m1 ¼
L s
01,p 1 º ª 1
I p 1 »¼ «¬ 0 p 11,
01,m1
º
m s M1 s »¼
(4.4.23)
and
T s
L s
m s
P s Q s
G s ,
m s
(4.4.24)
where
ª 1 º
1
i s U 1 s «
» , Q s ª¬1 w s º¼ V s ,
k
s
¬
¼
(4.4.25)
01,m1 º 1
ª 0
1
G s i s U s «
»V s .
¬ 0 p 1,1 M1 s ¼
The above considerations yield the following procedure for solving our
problem.
P s
Procedure 4.4.1.
Step 1: Applying elementary operations, transform the matrix L (s) into the form
(4.4.17) and compute the polynomial i(s) as well the unimodular matrices
U(s) and V(s).
Step 2: Choose M1(s) and R(s).
Step 3: Using (4.4.25), compute the matrices P(s), Q(s) and G(s).
Step 4: Using (4.4.24), compute the desired normal transfer matrix T(s).
Example 4.4.2.
Provided that the parameter a is small enough (close to zero), compute the normal
transfer matrix for the matrix (4.4.10).
Using Procedure 4.4.1, we obtain the following.
Step 1: In this case, m (s) = m(s) = (s + 1)(s + 2) and
L s
0 º
ªs 2
.
« 0
s 1 a »¼
¬
Applying the elementary operations L[1 + 2] and P[1 + 2u(1)], we obtain
(4.4.26)
252
Polynomial and Rational Matrices
0 º ª 1 0º
ª1 1º ª s 2
«0 1» « 0
s 1 a »¼ «¬ 1 1 »¼
¬
¼¬
s 1 a º
ª
1
«
s 1 aº
1 a »
».
1 a «
»
s 1 a¼
s 1 a »
« s 1 a
«¬ 1 a
1 a »¼
U s L s V s
ª 1 a
« s 1 a
¬
Thus
ª1 1º
« 0 1» , V s
¬
¼
1 a , U s
i s
V 1 s
ª1 0 º
«1 1 » , w s
¬
¼
ª 1 0º
1
« 1 1 » , U s
¬
¼
s 1 a
, k s
1 a
s 1 a
,M s
1 a
Step 2: In this case,
M s k s w s
s 1 a
s 1 a
2
1 a
1 a
s 1 s 2 a s 2
m s M1 s R s .
2
1 a
2
We take
M1 s
1
1 a
2
and R s
a
s2
1 a
2
.
Step 3: Using (4.4.25), we obtain
ª 1 º
i s U 1 s «
»
¬k s ¼
1
ª
º
ª1 1º «
»
1 a «
s
a
1
»
»
¬0 1 ¼ « ¬ 1 a ¼
Q s ª¬1 w s º¼ V 1 s
P s
ª s 1 a º ª1 0 º
«¬1
1 a »¼ «¬1 1 »¼
ªs 2
«¬ 1 a
ª s2 º
« s 1 a » ,
¬
¼
s 1 a º
,
1 a »¼
ª1 1º
«0 1 » ,
¬
¼
s 1 a
.
1 a
The Problem of Realization
253
0 º 1
ª0
i s U 1 s «
»V s
¬ 0 M1 s ¼
0 º
ª0
ª1 1º «
ª1 0 º
1 ª 1 1º
1 »» «
1 a «
»
»
«
».
«
¬ 0 1 ¼ « 0 1 a 2 » ¬1 1 ¼ 1 a ¬ 1 1 ¼
¬
¼
G s
Step 4: Thus the desired matrix is
T s
P s Q s
G s
m s
ªs 2
«1 a
1
«
s 1 s 2 « a
«¬ 1 a
a
º
»
1 a
».
2
s 1 2a 1 2a a »
»¼
1 a
(4.4.27)
Note that for a = 0 in (4.4.27), we obtain the normal transfer matrix
T s
P s Q s
G s
m s
ªs 2
s 1 s 2 «¬ 0
1
which can be also obtained from (4.4.10) for a = 0.
0 º
,
s 1»¼
5
Singular and Cyclic Normal Systems
5.1 Singular Discrete Systems and Cyclic Pairs
Consider the following discrete system
Exi 1
yi
Axi Bui , i  ' {0, 1, ...} ,
Cxi Dui
(5.1.1a)
(5.1.1b)
where xi n, ui m, yi p are the state, input and output vectors, respectively, at
the discrete instant i, and E, A nun, B num, C pun, D pum.
The system (5.1.1) is called singular if det E = 0 and standard if det E z 0.
We assume that det E = 0 and
det[Ez A ] z 0, for some z  (the complex numbers field).
(5.1.2)
A system of the form (5.1.1) satisfying the condition (5.1.2) is called a regular
system. The transfer matrix of the system (5.1.1) is given by
T(z )
1
C > Ez A @ B D .
(5.1.3)
This matrix can be written in the standard form
T(z )
P (z )
,
d (z )
(5.1.4)
where P(z) pum[z] ( pum[z] is the set of polynomial matrices of dimensions pum),
d(z) is the minimal monic common denominator of all the elements of the matrix
T(z).
256
Polynomial and Rational Matrices
Applying elementary operations on rows and columns, we can reduce the
matrix P(z) pum[z] to its Smith canonical form
PS ( z )
diag >i1 ( z ), i2 ( z ), ..., ir ( z ), 0, ..., 0@  pum [ z ] ,
where i1(z),…,ir(z) are the monic invariant polynomials satisfying the divisibility
condition ik+1(z)|ik(z), k = 1,…,r-1 (the polynomial ik(z) divides without remainder
the polynomial ik+1(z), k = 1,…,r-1), and r = rank P(z).
The invariant polynomials are given by
Dk ( z )
Dk 1 ( z )
ik ( z )
D0 ( z ) 1 , k
1, ..., r ,
(5.1.6)
where Dk(z) is a greatest common divisor of all the k-th order minors of the matrix
P(s).
The characteristic polynomial M(z) = det [Ez – A] of the pair (E, A) and the
minimal polynomial <(s) are related in the following way
<( z)
M ( z)
.
Dn1 ( z )
(5.1.7)
Definition 5.1.1. (E, A) is called a cyclic pair if and only if <(z) = M(z).
It follows from (5.1.6) that (E, A) is a cyclic pair if and only if
Dn1 ( z ) 1 or equivalently
i1 ( z )
i2 ( z ) " ir 1 ( z ) 1, ir ( z )
<( z)
(5.1.8)
d ( z ).
Theorem 5.1.1. (E, A) is a cyclic pair if the matrices E = [eij]
A = [aij] nun satisfy one of the following conditions
­ 0 for
j ! i and aij ®
¯z 0 for
j ! i 1
and
(5.1.9a)
eij
0, for
eij
­ 0 for i ! j 1
i, j 1, ..., n . (5.1.9b)
0, for i ! j and aij ®
¯z 0 for i j 1
j
i 1
i, j 1, ..., n
nun
or
Proof. If the condition (5.1.9a) is satisfied, then the minor Mn1 (obtained by
deletion of the n-th row and first column) of the matrix [Ez – A] equals
Mn1 = a12a23…an1,n z 0. Thus Dn1(z) = 1 and from (5.1.7), we have M(z) = <(z).
Singular and Cyclic Normal Systems
257
The proof of the condition (5.1.9b) is analogous (dual).
„
It follows immediately from Theorem 5.1.1 that (E, A) is a cyclic pair if
E
ªI n1 0 º
« 0 0» , A
¬
¼
ª 0 I n1 º
«
» (the Frobenius canonical form) . (5.1.10)
¬ a ¼
5.1.1 Normal Inverse Matrix of a Cyclic Pair
For any pair (E, A) satisfying the condition (5.1.2) the inverse [Ez – A]-1 can be
written as
1
> Ez A @
P( z )
,
d ( z)
(5.1.11)
where P (z) = PE,A(z) nun[z] and d (z) is the minimal monic denominator. In this
case, rank [Ez – A] = rank P (z) = n.
Definition 5.1.2. The matrix (5.1.11) is called normal if and only if every nonzero
second-order minor of the polynomial matrix P (z) is divisible without remainder
by the polynomial d (z).
Theorem 1.5.2. Let E, A nun, n t 2 and the assumption (5.1.2) be satisfied. The
matrix (5.1.11) is normal if and only if (E, A) is a cyclic pair.
Proof. Sufficiency. If (E, A) is a cyclic pair and the conditions (5.1.8) hold, then
the Smith canonical form of [Ez – A] is
[Ez A]S
U( z )[Ez A] V ( z )
diag [1, ..., 1, d ( z )]  nun [ z ] ,
(5.1.12)
where U(z) nun[z] and V(z) nun[z] are unimodular matrices of elementary
operations on rows and columns, respectively.
The adjoint of the matrix (5.1.12) has the form
Adj [Ez A]S
diag [d ( z ), ..., d ( z ), 1]  pum [ z ] .
(5.1.13)
Thus every nonzero second-order minor of the matrix (5.1.13) is divisible without
remainder by the polynomial d (z).
Taking into account that
258
Polynomial and Rational Matrices
P( z )
(c
Adj [Ez A] cV ( z ) Adj [Ez A]S U( z )
det U( z )V ( z ))
(5.1.14)
and using the Binet–Cauchy theorem, we find that every nonzero second-order
minor of the matrix (5.1.14) is divisible without remainder by d (z). Thus (5.1.11)
is a normal matrix.
Necessity. Let
[Ez A]S
diag [ p1 ( z ), p1 ( z ) p2 ( z ), ..., p1 ( z ) p2 ( z ) " pn ( z )]  nun [ z ], (5.1.15)
where some of the polynomials p1(z),p2(z),…,pn(z) may be equal to 1.
We will show that if every nonzero second-order minor of the matrix P (z) is
divisible without remainder by d (z), then p1(z) = p2(z) = … = pn-1(z) = 1 and the
condition (5.1.8) holds.
Note that the inverse of the matrix (5.1.15) has the form
[Ez A]S1
Pˆ ( z )
,
dˆ ( z )
(5.1.16)
where
Pˆ ( z )
dˆ ( z )
diag [ p2 ( z ) p3 ( z )... pn ( z ), p3 ( z ) p4 ( z ) " pn ( z ), ..., pn ( z ),1],
p1 ( z ) p2 ( z )... pn ( z ).
(5.1.17)
From (5.1.16)( 5.1.17) it follows that every nonzero second-order minor of the
matrix P̂ (z) is divisible without remainder by d̂ (z) if and only if p1(z) = p2(z) = …
= pn-1(z) = 1.
Moreover, note that a nonzero second-order minor of the unimodular matrices
U(z) and V(z) is not divisible by d̂ (z). From the relationships
[Ez – A]1
-1
= V(z)[Ez – A]S U(z), (11) and the Binet–Cauchy theorem it follows that if every
nonzero second-order minor of the matrix P (z) is divisible without remainder by
d (z). Then the conditions (5.1.8) hold true and (E, A) is a cyclic pair.
„
Example 5.1.1.
The matrix pair
E
ª1 0 0 º
«0 1 0 » , A
«
»
«¬0 0 0 »¼
is cyclic, since
ª 0 1 0º
« 0 0 1»
«
»
«¬ 1 2 0 »¼
(5.1.18)
Singular and Cyclic Normal Systems
M ( z ) det[Ez A]
z 1 0
0 z 1
1 2 0
259
2z 1
and
[Ez A]S
0 º
ª1 0
«0 1
0 »» .
«
«¬ 0 0 2 z 1»¼
Therefore, <(z) = M(z).
In this case, the inverse (5.1.11) has the form
[Ez A]1
ª z 1 0 º
«0 z 1»
«
»
«¬1 2 0 »¼
1
P( z )
,
d ( z)
(5.1.19)
where
P( z )
0
1º
ª2
« 1
0
z »» , d ( z )
«
«¬ z 2 z 1 z 2 »¼
2z 1 .
The nonzero second-order minors of the matrix P (z)
M 32
M 22
M 13
2
1
2 z 1, M 23
1 z
2
z
1
1
z
2
z (2 z 1), M 21
2
0
z 2 z 1
0
z 2 z 1
2 z 1, M 11
0
2(2 z 1),
1
2 z 1,
2 z 1 z 2
0
z
2 z 1 z 2
(5.1.20)
z (2 z 1)
are divisible without remainder by the polynomial d (z) = 2z + 1. The inverse
(5.1.19) is thus a normal matrix.
Example 5.1.2.
The matrix pair
260
Polynomial and Rational Matrices
E
ª1 0 0 º
«0 1 0 » , A
«
»
«¬0 0 0 »¼
ª0 0 0 º
«0 0 1 »
«
»
«¬ 0 0 1»¼
(5.1.20)
is not cyclic, since
z 0
M ( z ) det[Ez A]
0
0 z 1
0 0
2
z , [Ez A]S
1
ª1 0 0 º
«0 z 0»
«
»
«¬ 0 0 z »¼
and
Dn1 ( z )
z, < ( z )
M ( z)
Dn ( z )
z.
Therefore, <(z) z M(z).
In this case the inverse matrix (5.1.11) has the form
1
[ Ez A ]
ªz 0 0 º
«0 z 1»
«
»
«¬ 0 0 1 »¼
1
ª1 0 0 º
1«
0 1 1 »»
z«
«¬ 0 0 z »¼
P z
,
d z
(5.1.21)
where
P z
ª1 0 0 º
«0 1 1 » , d z
«
»
«¬ 0 0 z »¼
z.
In this case, the minor
M 33
1 0
0 1
of P (z) is not divisible by d (z) = z. Thus the matrix (5.1.21) is not normal.
5.1.2 Normal Transfer Matrix
The transfer matrix (5.1.3) of the system (5.1.1) can be always written in the
standard form (5.1.4). If m > p and rank C = p, rank B = m, then r = rank P = p
and the Smith canonical form of the matrix P(z) is
Singular and Cyclic Normal Systems
PS ( z )
261
U( z )P( z )V ( z )
0
ªi1 ( z )
« 0
i2 ( z )
«
« #
#
«
0
0
¬«
!
!
0
0
0 !
0 !
%
#
# %
! ip ( z) 0 !
0º
0 »»
 pum [ z ],
»
»
0 ¼»
(5.1.22)
where U(z) pup[z] and V(z) mum[z] are unimodular matrices of elementary
operations on rows and columns, respectively.
From (5.1.22) and (5.1.4) we obtain the McMillan canonical form of the matrix
T(z)
PS ( z )
d ( z)
TM ( z )
ª n1 ( z )
« q ( z)
« 1
«
« 0
«
« #
«
« 0
«
¬
0
U( z )P( z )V ( z )
d ( z)
!
0
n2 ( z )
!
q2 ( z )
0
#
%
0
!
#
np ( z)
q p ( z)
º
0 ! 0»
»
»
0 ! 0»
pum
»  ( z ),
# % #»
»
0 ! 0»
»
¼
(5.1.23)
where
ik ( z )
d ( z)
nk ( z )
, for k
qk ( z )
1, ..., p n1 ( z )
i1 ( z ), q1 ( z )
d ( z) ,
nk(z) and qk(z) are relatively prime and nk+1(z)|nk(z) and qk+1(z)|qk(z), k = 1,…,p-1,
where pum(z) is the set of rational matrices of dimensions pum.
The polynomial
q( z )
(5.1.24)
q1 ( z )q2 ( z )...q p ( z )
is called the McMillan polynomial of the matrix T(z).
From (5.1.22)–( 5.1.24) it follows that deg q(z) t deg d(z) and
q( z )
d ( z ) if and only if qk ( z ) 1, for k
2, ..., p and q1 ( z )
d ( z ). (5.1.25)
Theorem 5.1.3. Let T(z) have the form (5.1.4) and min (m, p) t 2. T(z) is a normal
matrix if and only if q(z) = d(z).
262
Polynomial and Rational Matrices
Proof. Sufficiency. If q(z) = d(z) then according to (5.1.25) qk = 1 for k = 2,…,p,
and the relationship (23) takes the form
TM ( z )
PS ( z )
,
d ( z)
(5.1.26)
and
T( z )
U 1 ( z )TM ( z )V 1 ( z )
P( z )
U 1 ( z )PS ( z )V 1 ( z )
P( z )
,
d ( z)
where
(5.1.27a)
and
PS ( z )
0
ªi1 ( z )
« 0
i1 ( z )t2 ( z )d ( z )
«
« #
#
«
0
0
¬«
0 " 0º
(5.1.27b)
0 " 0 »»
 pum [ z ]
# % #»
»
! i1 ( z )t p ( z )d ( z ) 0 ! 0 ¼»
"
"
%
0
0
#
U-1(z) and V-1(z) are unimodular matrices and some of the polynomials tk(z),
k=2,…,p may be equal to 1.
From (5.1.27) it follows that every nonzero second-order minor of PS(z) is
divisible without remainder by d(z). Applying the Binet–Cauchy theorem to
(5.1.27a) we find that every nonzero second-order minor of P(z) is divisible
without remainder by d(z). Thus T(z) is a normal matrix.
Necessity. If T(z) is a normal matrix then every nonzero second-order minor of
P(z) (but also of PS(z)) is divisible without remainder by d(z). This implies that
PS(z) has the form (5.1.27b), and from (5.1.26) it follows that qk(z) = 1 for
k = 2,…,p. In this case from (5.1.25) we have q(z) = d(z).
„
Example 5.1.3.
Consider the transfer matrix
T( z )
0
3 º
1 ª2
.
«
2
2 z 1 ¬ z 2 z 1 z z »¼
In this case, d(z) = 2z + 1
(5.1.28)
Singular and Cyclic Normal Systems
P( z )
0
3 º
ª2
« z 2 z 1 z 2 z » .
¬
¼
263
(5.1.29)
The Smith canonical form of the matrix (5.1.29) is
PS ( z )
0
0º
ª1
«0 2 z 1 0 »
¬
¼
and the McMillan canonical from of the matrix (5.1.28)
ª 1
º
« 2 z 1 0 0» .
«
»
1 0¼
¬ 0
TM ( z )
Therefore, q(z) = d(z) = 2z + 1.
The nonzero second-order minors of the matrix (5.1.29)
M3
M1
2
0
2(2 z 1), M 2
z 2z 1
0
3
2
2z 1 z z
2
z
3
2
z z
2 z 2 z,
3(2 z 1)
are divisible without remainder by d(z). The matrix (5.1.28) is normal.
Example 5.1.4.
Writing the transfer matrix
T( z )
ª 1
« z 1
«
« 0
«
«
« 0
«¬
º
»
»
1 »
( z 1) 2 »
»
1 »
z 1 ¼»
0
(5.1.30)
in the standard form (5.1.4), we obtain d(z) = (z + 1)2 and
P( z )
ªz 1 0 º
« 0
1 »» .
«
z 1¼»
¬« 0
The Smith canonical from of the matrix (5.1.31) is
(5.1.31)
264
Polynomial and Rational Matrices
PS ( z )
0 º
ª1
«0 z 1»
«
»
«¬0
0 »¼
and the McMillan canonical form of the matrix (5.1.30) is
TM ( z )
ª 1
« ( z 1) 2
«
«
« 0
« 0
¬«
º
0 »
»
1 ».
z 1»
»
0 »¼
Therefore, q(z) = (z + 1)3 z d(z) = (z + 1)2.
The minor
M3
z 1 0
0
1
of the matrix (5.1.31) is not divisible by d(z).
Thus the matrix (5.1.30) is not normal.
5.2 Reachability and Cyclicity
5.2.1 Reachability of Singular Systems
Consider the singular system (5.1.1). If the relationship (5.1.2) holds, then
[Ez A]1
f
¦P ĭ z
i
( i 1)
,
(5.2.1)
i where
μ
rank E deg det [Ez A] 1
is the nilpotent index and )i are the fundamental matrices satisfying the
relationship
Eĭi Aĭi 1
ĭi E ĭi 1A
­I for i 0
®
¯0 for i z 0
(5.2.2)
Singular and Cyclic Normal Systems
265
and )i = 0 for i < -P.
The solution xi to (5.1.1a) with the initial condition x0 is
xi
ĭi Ex0 i P 1
¦ĭ
Bu j , i  ' .
i j 1
(5.2.3)
j 0
Substituting (5.2.3) into the right-hand side of (5.1.1a), and taking into account
(5.2.2), we obtain
Axi Bui
Aĭi Ex0 i P 1
¦ Aĭ
B u j B ui
i j 1
j 0
iP
ª
º
E «ĭi 1Ex0 ¦ ĭi j Bu j »
j 0
¬
¼
Exi 1.
Therefore, (5.2.3) satisfies (5.1.1a) and is its solution.
Definition 5.2.1. The system (5.1.1) is said to be reachable in k steps if for every
xf n there exists an input sequence ui m, i = 0,1,…,k + P 1, which steers the
state of this system from x0 = 0 to xf.
The system (5.1.1) is called reachable if there exists k such that the system is
reachable in k steps.
Theorem 5.2.1. The system (5.1.1) is reachable in n P steps if and only if
rank [ĭ μ B ,...,ĭ 1B ,ĭ 0 B ,...,ĭ nP 1B] n .
(5.2.4)
Proof. From (5.2.3), for x0 = 0, i = n - P, xn - P = xf, we have
xf
n 1
¦ĭ
j 0
Bu j
n P j 1
ªun1 º
« # »
«
»
«u n P »
».
¬ªĭ P B ,..., ĭ 1B , ĭ0 B ,...,ĭ n P 1B ¼º «u
« nP 1 »
«
»
« # »
«¬ u0 »¼
(5.2.5)
From (5.2.5) it follows that there exists an input sequence u0,…,un-1 for every xf if
and only if the condition (5.2.4) is satisfied.
„
266
Polynomial and Rational Matrices
Theorem 5.2.2. The system (5.1.1) with single input (m = 1) is reachable in n P
steps if and only if the characteristic polynomial M(z) = det [Ez – A] of the pair
(E, A) coincides with the minimal polynomial <(z) of this pair, that is, M(z) = <(z).
Proof. According to Theorem 5.2.1, for m = 1 (B = b), the system (5.1.1) is
reachable in n P steps if and only if
rank ª¬ĭ P b,..., ĭ 1b ,ĭ0b,..., ĭ nP 1b º¼
n.
(5.2.6)
From (5.1.7) and the equality
det [Ez A ] [Ez A ]1
Adj[Ez A ] ,
we obtain
< ( z )[Ez A]1
Adj[Ez A]
Dn1 ( z )
H q z q " H1 z H 0 .
(5.2.7)
Let
<( z)
z n1 an1 1 z n1 1 " a1 z a0 .
(5.2.8)
Substitution of (5.2.8) and (5.2.1) into (5.2.7) yields
( z n1 an1 1 z n1 1 " a1 z a0 )
u(ĭ P z P 1 " ĭ 2 z ĭ 1 ĭ0 z 1 ĭ1 z 2 ")
H q z q " H1 z H 0 .
(5.2.9)
Comparing the coefficients by z1 in the above equality, we obtain
ĭ n1
an1 1ĭ n1 1 " a1ĭ1 a0ĭ 0 .
(5.2.10)
If <(z) z M(z), then deg M(z) > n1 and it follows from (5.2.10) and the equality
P = rank E – deg M + 1 that the condition (5.2.6) is not satisfied, since the column
)n-P-1b is linearly dependent on )0, b,)1, b….
„
Example 5.2.1.
Consider the single input system with the matrices E and A as in (5.1.20), and
Singular and Cyclic Normal Systems
b
267
ª0 º
«0 » .
« »
«¬1 »¼
In Example 5.1.2 we have proved that M ( z )
[Ez A]1
ªz 0 0 º
«0 z 1»
«
»
«¬0 0 1 »¼
z 2 , <( z)
z and
1
ĭ 1 ĭ 0 z 1 ,
(5.2.11)
where
ª0 0 0º
«0 0 0» , ĭ
0
«
»
¬«0 0 1 »¼
ĭ 1
ª1 0 0 º
«0 1 1 » .
«
»
¬« 0 0 0 »¼
In this case, (5.2.10) takes the form )k = 0 for k = 1,2,….
Using (5.2.6), we obtain
rank >ĭ 1b ,ĭ 0b , ĭ1b @
ª0 0 0º
rank ««0 1 0 »»
«¬1 0 0 »¼
2n
3.
(5.2.12)
Thus the considered system is not reachable. The same result is obtained by the use
of Theorem 5.2.2.
5.2.2 Cyclicity of Feedback Systems
Consider the system (5.1.1) with a state feedback of the form
ui
vi Kxi ,
(5.2.13)
where vi m is the new input vector and K mun is a feedback matrix.
Substituting (5.2.13) into (5.1.1a), we obtain
Exi 1
A z xi Bvi ,
(5.2.14)
where
Az
A BK .
(5.2.15)
268
Polynomial and Rational Matrices
First, consider a single input system (m = 1) with the matrices E, A, b in the
canonical form
E
ªI n1 0 º
nun
«
» , A
0
¬
¼
ª0 I n1 º
nun
«
» , b
a
¬
¼
ª0º
« #»
« »,
«0»
« »
¬1 ¼
(5.2.16)
a [a0 a1 ... ar 1 1 0 ... 0]  1un .
Theorem 5.2.3. Let the matrices E, A, b be of the form (5.2.16). The closed-loop
system pair (E, Az), Az = A + bk is cyclic if and only if (E, A) is a cyclic pair.
Proof. Necessity. If the matrices E, A, b have the forms as in (5.2.16), then the
system is reachable for an arbitrary matrix k and according to Theorem 5.2.2, the
reachability of the closed-loop system implies the cyclicity of the pair (E, Az).
Sufficiency. If the matrices E, A, b have the forms as in (5.2.16), then
<( z) M ( z)
det[Ez A]
z r ar 1 z r 1 ! a1 z a0 .
(5.2.17)
Using (5.2.16) and k = [k1 k2 … kn], we obtain
Az
A bk
ª0º
« »
ª 0 I n1 º « # »
k
«
»
¬ a ¼ «0»
« »
¬1 ¼
ª 0 I n1 º
«
»,
¬ a ¼
(5.2.18)
where
a
ak
[ k1 a0 , k2 a1 , ..., kr ar 1 , k r 1 1, kr 2 , ..., kn ] ,
(E, Az) is a cyclic pair, since <z(z) = det [Ez – Az].
„
The following important corollary can be derived from Theorem 5.2.3.
Corollary 5.2.1. If the matrices E, A, b have the canonical forms as in (5.2.16),
then the cyclicity of (E, A) is invariant with respect to the state feedback.
If the system (5.1.1) is not reachable and the pair (E, A) is not cyclic, then in
the example below we will show that it is possible to choose the matrix k in such a
way that (E, Az) is a cyclic pair.
Singular and Cyclic Normal Systems
269
Example 5.2.2.
Let the matrices E, A of the system (5.1.1) have the forms as in (5.1.20), and
ª1 º
«0 » .
« »
«¬1 »¼
b
Using (5.2.11), we obtain
ª0 1 0º
rank «« 0 1 0 »»
«¬1 0 0 »¼
rank >ĭ 1b ,ĭ 0b , ĭ1b @
2n
3.
Thus the system is not reachable.
For k = [0 1 0], we have
Az
A bk
ª 0 0 0 º ª1 º
«0 0 1 » «0 » 0 1 0
@
«
» « »>
«¬ 0 0 1»¼ «¬1 »¼
ª0 1 0 º
«0 0 1 » ,
«
»
«¬0 1 1»¼
and
M ( z ) det[Ez A z ]
z 1 0
z 1
0 1 1
0
z2 z .
It is easy to check that <(z) = M(z). Thus (E, Az) is a cyclic pair although (E, A) is
not a cyclic pair.
Consider the system (5.1.1) with m-inputs (m > 1), with its matrices E, A, B
having the following canonical forms
E
A
ai
diag [E1 ,..., Em ], Ei
diag [ A1 ,..., A m ], A i
ªI qi º
0 »  ( qi 1)u( qi 1) , n
«
0
»¼
¬«
ª 0 I qi º
( q 1)u( qi 1)
,
«
» i
a
¬ i ¼
[ a0i ,..., arii 1 , 1, 0,...., 0], B
m
m ¦ qi ,
diag [B1 ,..., B m ], Bi
i 1
(5.2.19)
ª0 º
« #»
« »  qi 1.
«0 »
« »
¬1 ¼
270
Polynomial and Rational Matrices
Theorem 5.2.4. Let the matrices E, A, B have the forms as in (5.2.19) and let the
(E, A) be a noncyclic pair. Then there exists a feedback matrix K such that (E, Az),
Az = A + BK is a cyclic pair.
Proof. If the matrices E, A, B have the forms as in (5.2.19), then the system is
reachable. (E, A) is not a cyclic pair if at least two pairs (E1, A1),…, (Em, Am) have
at least one common eigenvalue. The feedback matrix K = diag [K1,…,Km] is
chosen in such a way that all the pairs (Ei, Azi), Azi = Ai + BiKi, i=1,…,m have
distinct eigenvalues.
Let
z qi 1 d qi i z qi " d1i z d 0i , i 1,..., m
M zi ( z )
(5.2.20)
be the desired characteristic polynomial of the pair (Ei, Azi), that is,
det > Ei z A zi @ M zi z .
(5.2.21)
Choosing matrices of the form
Ki
ª¬ a0i d 0i , " , arii 1 d rii 1 , 1 d rii , d rii 1 ,..., d qi i º¼ , i 1, ..., m, (5.2.22)
and using (5.2.19), we obtain
A zi
Ai Bi K i
ª 0 I qi º
«
» , i 1, ..., m ,
¬ di ¼
(5.2.23)
where
di
[d 0i , d1i , ..., d qi i ] .
(5.2.24)
The matrix (5.2.23) satisfies the condition (5.2.21) and (E, Az) is a cyclic pair. „
Example 5.2.3.
Consider the system (5.1.1) with the matrices
E
ª1
«0
«
«0
«
¬0
0 0 0º
0 0 0 »»
, A
0 1 0»
»
0 0 0¼
ª0 1 0 0º
« 1 2 0 0 »
«
», B
«0 0 0 1»
«
»
¬ 0 0 2 4 ¼
ª0
«1
«
«0
«
¬0
0º
0 »»
.
0»
»
1¼
(5.2.25)
The matrices (5.2.25) have the canonical forms (5.2.19); (E, A) is not a cyclic
pair, since the characteristic polynomials of the pairs
Singular and Cyclic Normal Systems
z 1
det > E1 z A1 @
E1
1
ª1 0 º
«0 0 » , A1
¬
¼
2
2 z 1, det > E2 z A 2 @
ª0 1º
« 1 2 » , E 2
¬
¼
ª1 0 º
«0 0» , A 2
¬
¼
z 1
2
4
271
4 z 2,
ª0 1º
« 2 4»
¬
¼
(5.2.26)
have the common eigenvalue z = 0.5.
Let the desired characteristic polynomials of the pairs (E1, Az1) and (E2, Az2) be
Mz1 = z + 1 and Mz2 = z + 2. Using (5.2.22), (5.2.23), (5.2.25) and choosing
appropriately the entries of the vector (5.2.24), we obtain
K1
ª¬ a10 d 01 , a11 d11 º¼ [0 , 1], K 2
A z1
A1 B1K1
ª¬ a02 d 02 , a12 d12 º¼ [0 , 3] ,
and
ª0 1º
« 1 1» , ǹ z 2
¬
¼
A2 B2K 2
ª0 1º
« 2 1» .
¬
¼
Therefore,
K
diag [K1 K 2 ]
ª0 1 0 0 º
«0 0 0 3» ,
¬
¼
and
Az
A BK
ª0 1 0 0º
« 1 1 0 0 »
«
».
«0 0 0 1»
«
»
¬ 0 0 2 1¼
It is easy to check that M(z) = <(z) = (z + 1)(z + 2). Thus (E, Az) is a cyclic pair.
Consider an unreachable system of the form (5.1.1), which satisfies the
following conditions
rank [R , B]
rank [R , AR ]
rank[R , ER ]
rank R ,
rank R ,
rank R ,
where R is the reachability matrix given by
(5.2.27)
(5.2.28)
(5.2.29)
272
Polynomial and Rational Matrices
R
ª¬ĭ P B ,...,ĭ 1B ,ĭ0 B ,...,ĭ n P 1B º¼ .
(5.2.30)
Using the method presented in [156], we can compute a nonsingular matrix
T nun such that
A
B
TAT1
TB
ª A1
«0
¬
A2 º , E
A 3 »¼
TET1
ª E1 E 2 º
« 0 E »,
3¼
¬
ªB1 º
r ur
rum
(n r)u(n r)
«0 » , A1 , E1  , B1  , A 3 , E3  ¬ ¼
(5.2.31)
where the subsystem (E1, A1, B1) is reachable.
Theorem 5.2.5. Let a system of the form (5.1.1) be unreachable and (E, A) be a
noncyclic pair. There exists a feedback matrix K such that (E, Az), Az = A + BK is
a cyclic pair if and only if (E3, A3) (pertaining to the decomposition (5.2.31)) is a
cyclic pair.
Proof. Sufficiency. If (E3, A3) is a cyclic pair and (E, A) is not, then the minimal
polynomials <1(z) and <3(z) of the pairs (E1, A1) and (E3, A3) have at least one
common divisor different from a constant. By assumption, the subsystem
(E1, A1, B1) is reachable. Therefore, we can choose the matrix K1 in such a way
that the pair (E1, A1 + B1K1) has the minimal polynomial with no common term
with the polynomial <3(z). Thus in this case, (E, Az) is a cyclic pair.
Necessity. The necessity follows immediately from the fact that (E, Az) is a cyclic
pair if and only if (E3, A3) is a cyclic pair.
„
5.3 Computation of Equivalent Standard Systems for Linear
Singular Systems
5.3.1 Discrete-time Systems and Basic Notions
Consider the singular system (5.1.1) satisfying the assumptions (5.1.2). Below we
present a method of computing equivalent standard systems (E = I) for singular
systems, based on elementary operations.
In the forthcoming considerations we will use the following two types of
elementary row operations:
1. Typical elementary operations consisting of [11]:
x Multiplication of the row i by a nonzero scalar a; we will denote this
operation by L(i u a).
Singular and Cyclic Normal Systems
273
Addition of the row j multiplied by a nonzero scalar b to the row i; we
will denote this operation by L(i + ju b).
x The interchange of the rows i and j; we will denote this operation by
L(i, j).
2. Multiplication of every one of the last k rows of a matrix by a variable z;
we will denote this operation by Lk(z).
Note that elementary operations of the first type are equivalent to the premultiplication of a matrix by a nonsingular matrix L , and elementary operations
of the second type to post-multiplication by a matrix of the form [In-k, Ikz], which is
obtained from the identity matrix of size n by multiplication of its last k rows by
the variable z. Using elementary operations, we will prove the following theorem,
on which the proposed method of computing equivalent standard systems for
singular systems is based.
x
Theorem 5.3.1. There exists a nonsingular polynomial matrix
L(z )
L 0 L1 z " L P z P
(5.3.1)
with its determinant det L(z) = lzq such that
L( z )[Ez A] [Iz A]
(5.3.2)
if and only if the condition (5.1.2) is satisfied, where A  nun and P is the
nilpotent index of the pair (E, A); the numbers l and q will be determined in the
proof.
Proof. [Iz - A ] is a nonsingular matrix for every matrix A . From (5.3.2) it follows
immediately that the condition (5.1.2) must be met. In order to prove the
sufficiency of the condition (5.3.2), we transform the matrix E to the form
ªE1 º
«0 »
¬ ¼
using elementary operations, where E1 has the full row rank equal to r1, that is,
L1E
ª E1 º
« 0 » , L1A
¬ ¼
ª A1 º
r un
r un
« ˆ » , E1  1 , A1  1
«¬ A1 »¼
(5.3.3)
and L 1 is a matrix of elementary operations on the rows of the matrix E.
Pre-multiplying (5.1.1a) by L 1 for B = 0, and taking into account (5.3.3), we
obtain
E1 xi 1
A1 xi ,
(5.3.4a)
274
Polynomial and Rational Matrices
0
ˆ x.
A
1 i
(5.3.4b)
Incrementing by 1 the index i in (5.3.4b), we obtain
0
ˆ x .
A
1 i 1
(5.3.5)
Equations (5.3.4a) and (5.3.5) can be combined to a one equation, that is
ª E1 º
« ˆ » xi 1
«¬ A1 »¼
ª A1 º
« 0 » xi .
¬ ¼
(5.3.6)
Note that incrementing the index i in (5.3.4b) is equivalent to pre-multiplying the
equation
ª E1 z º
« 0 » X (z )
¬ ¼
ª A1 º
« ˆ » X (z )
¬« A1 ¼»
(5.3.7)
by the matrix
diag ª¬I r1 , I nr1 z º¼ ,
where X(z) is a Z-transform of the vector xi. Transition from (5.1.1a), for B = 0, to
(5.3.6) is thus equivalent to pre-multiplication of (5.1.1a), for B = 0, by the matrix
diag ª¬ I r1 , I nr1 z º¼ L1 .
What we have described above is the essence of a step of the shuffle algorithm,
which is carried out on the pair (E, A). If
ª E1 º
« ˆ »
¬« A1 ¼»
is a nonsingular matrix, then pre-multiplying (5.3.6) by the inverse
1
ª E1 º
« ˆ » ,
¬« A1 ¼»
we obtain
Singular and Cyclic Normal Systems
275
1
xi 1
Axi , where A
ª E1 º ª A1 º
« ˆ » « ».
«¬ A
¼ ¬0 ¼
1»
(5.3.8)
On the other hand, if
ª E1 º
« ˆ »
¬« A1 ¼»
is a singular matrix, then we repeat the procedure described above for (5.3.6),
choosing the matrix of elementary operations L 2 in such a way that
ª E1 º
L2 «
ˆ »
¬« A1 ¼»
ªE2 º
ª A1 º
« 0 » , L 2 «0 »
¬ ¼
¬ ¼
ªA2 º
r un
« ˆ » , E2 , A 2  2
¬« A 2 ¼»
(5.3.9)
and the matrix E2 has the full row rank r2 t r1.
With P steps completed, we obtain a nonsingular matrix
ª EP º
«
»,
ˆ »
«¬ A
P¼
since by assumption the regularity condition (5.1.2) is met, and elementary
operations on the rows of [Ez – A] do not change its rank. Note that the desired
matrix L(z) is of the form
L( z )
P
L P 1 – diag ª¬ I ri , I nri z º¼ L i ,
(5.3.10)
i 1
where
1
L P 1
ª Eμ º
«
» .
ˆ »
«¬ A
μ¼
The matrix (5.3.10), with the multiplications carried out and the ordering with
respect to the successive powers of the variable z accomplished, takes the form
(5.3.1).
Computing the determinant of the matrix (5.3.10), we obtain
276
Polynomial and Rational Matrices
P
ª
º
det «L P 1 – diag ª¬ I ri , I nri z º¼ Li »
i 1
¬
¼
det L( z )
P 1
P
– det L – det diag ª¬I
i
i 1
i 1
ri
, I nri z º¼
lz q ,
since
z nri , where l
det diag ª¬I ri , I nri z º¼
P 1
– det L
i 1
P
i
and q d – (n ri ) .
i 1
Thus the theorem has been proved.
„
Definition 5.3.1. The matrix[Iz - A ] is called the standard form of the regular
pencil (E, A).
5.3.2 Computation of Fundamental Matrices
From (5.3.2), we have [Iz - A]-1L-1(z) = [Iz - A ]-1 and
[Ez A]1
[Iz A]1 L( z ) .
(5.3.11)
Substituting
ª¬Iz A º¼
1
f
i
¦A z
( i 1)
,
(5.3.12)
i 0
as well as (5.2.1) and (5.3.1) into (5.3.11), we obtain
ĭ P z P 1 " ĭ 2 z ĭ 1 ĭ0 z 1 ĭ1 z 2 "
Iz 1 Az 2 " L 0 L1 z " L μ z P .
(5.3.13)
Comparing the coefficients at the same powers of the variable z, we obtain
ĭ P
ĭ0
L P , ĭ1P
L1P AL P ," ,ĭ 1
L 0 AL1 " A P L P , ĭ1
L1 AL 2 " A P 1L P ,
AL 0 A 2 L1 " A P 1L P , "
(5.3.14)
With the matrices L0,L1,…,LP and A known, and with (5.3.14) taken into account,
we can compute successively )-P,)1-P,…,)-1,)0,)1,….
We compute the matrices Li, i = 0,1,…,P and A following the procedure
provided in the proof of Theorem 5.3.1. The procedure for computing both these
Singular and Cyclic Normal Systems
277
and fundamental matrices )j, j = P,1P,…,1,0,1,… will be illustrated by the
following example.
Example 5.3.1.
Let
E
ª1 0 0 º
«0 1 0 » , A
«
»
«¬0 0 0 »¼
ª1 0 0 º
«0 0 1 » .
«
»
«¬ 0 1 0 »¼
(5.3.15)
It is easy to check that the matrices (5.3.15) satisfy the condition (5.1.2), since
z 1 0 0
z 1 1 z .
0
1 0
0
det [Ez A]
Note that in this case, E has the form
ªE1 º
«0 » , E1
¬ ¼
ª1 0 0 º
«0 1 0 » .
¬
¼
Thus r1 = 2, L 1 = I3. Using the shuffle algorithm, we compute
>E, A @
ª1 0 0 1 0 0 º
ª1 0 0 1 0 0 º
«0 1 0 0 0 1 » 
L1 ( z )
o ««0 1 0 0 0 1 »»
«
»
«¬0 0 0 0 1 0 »¼
«¬0 1 0 0 0 0 »¼
ª1 0 0 º
L2 «« 0 1 0 »»
¬« 0 1 1 »¼
ª1 0 0 1 0 0 º
L1 ( z )

o «« 0 1 0 0 0 1 »» 
o
«¬ 0 0 0 0 0 1 »¼
ª1 0
0º
ª1 0 0 1 0 0 º L3 ««0 1 0 »» ª1 0 0 1 0 0 º
«¬ 0 0 1»¼
«
»
o ««0 1 0 0 0 1 »» .
« 0 1 0 0 0 1 » 
«¬0 0 1 0 0 0 »¼
«¬ 0 0 1 0 0 0 »¼
Thus
A
ª1 0 0 º
«0 0 1 »
«
»
«¬ 0 0 0 »¼
278
Polynomial and Rational Matrices
and
L(z )
L3L1 (z )L 2 L1 (z )L1
ª1 0 0 º ª1 0 0 º ª1 0 0 º ª1 0 0 º ª 1 0 0 º
«0 1 0 » «0 1 0 » «0 1 0 » « 0 1 0» « 0 1 0»
«
»«
»«
»«
»«
»
«¬0 0 1»¼ «¬0 0 z »¼ «¬0 1 1 »¼ «¬ 0 0 z »¼ «¬ 0 0 1 »¼
0 º
ª1 0
«0 1
0 »» L 0 L1 z L 2 z 2 ,
«
«¬0 z z 2 »¼
where
L0
ª1 0 0 º
«
»
«0 1 0 » , L1
«¬0 0 0 »¼
ª0 0 0º
«
»
«0 0 0» , L 2
«¬ 0 1 0 »¼
ª0 0 0 º
«
»
«0 0 0 » .
«¬0 0 1»¼
It is easy to check that in this case, the relationship (5.3.2) is satisfied.
Using (5.3.14), we obtain
ĭ 2
ĭ 1
ĭ0
ª0 0 0 º
L 2 ««0 0 0 »» ,
«¬0 0 1»¼
ª0 0 0 º ª1 0 0 º ª 0 0 0 º
L1 AL 2 ««0 0 0 »» «« 0 0 1 »» «« 0 0 0 »»
«¬0 1 0 »¼ «¬ 0 0 0 »¼ «¬ 0 0 1»¼
ª0 0 0 º
« 0 0 1» ,
«
»
«¬ 0 1 0 »¼
L0 AL1 A 2 L 2
ª1 0 0 º ª 1 0 0 º ª 0 0 0 º ª 1 0 0 º ª 0 0 0 º
«0 1 0 » «0 0 1 » « 0 0 0 » «0 0 0» « 0 0 0 »
«
» «
»«
» «
»«
»
«¬0 0 0 »¼ «¬ 0 0 0 »¼ «¬ 0 1 0 »¼ «¬0 0 0»¼ «¬ 0 0 1»¼
ª1 0 0 º ª 1
ĭ1 AL 0 A L1 A L 2 ««0 0 1 »» ««0
«¬0 0 0»¼ «¬0
ª1 0 0 º ª 0 0 0 º ª1 0 0 º ª 0
««0 0 0 »» ««0 0 0»» ««0 0 0 »» ««0
«¬0 0 0 »¼ «¬0 1 0»¼ «¬0 0 0 »¼ «¬0
2
ª1 0 0º
« 0 0 0» ,
«
»
«¬ 0 0 0»¼
0 0º
1 0 »»
0 0 »¼
3
0
0º
0 0 »»
0 1»¼
ª1 0 0 º
«0 0 0» .
«
»
«¬ 0 0 0»¼
Singular and Cyclic Normal Systems
279
With the fundamental matrices )i (i = -P, 1-P,…) and x0, as well as ui (i +)
known, and with (5.2.3) taken into account, we can compute a solution to (5.1.1a).
5.3.3 Equivalent Standard Systems
Consider (5.1.1a), which can be written in its operator form as
[ Ez A ] X ( z )
BU( z ) Ex0 z ,
(5.3.16)
where X(z) and U(z) are the Z transforms of the vectors xi and ui, respectively.
Pre-multiplying (5.3.16) by the matrix (5.3.1), and taking into account (5.3.2),
we obtain
ª¬ Iz A º¼ X ( z )
P
B j z jU ( z ) L j Ex0 z j 1 ,
(5.3.17)
0, 1, ..., P .
(5.3.18)
¦
j 0
where
Bj
L j B, for j
Applying the Z inverse transform to (5.3.17), we obtain
xi 1
Axi B 0ui B1ui 1 ! B P ui P .
(5.3.19)
Pre-multiplying (5.1.1a) by a nonsingular matrix of elementary operations on
the rows of L 1, and taking into account (5.3.3), we obtain
E1 xi 1 A1 xi B1ui ,
x B u ,
0 A
1 i
1 i
(5.3.20a)
(5.3.20b)
where
L1B
ªB1 º
r1um
« » , B1  .
¬B1 ¼
The set of vectors x0 satisfying (5.3.20b) for i = 0 is called the set of admissible
initial conditions, and is denoted 0.
Theorem 5.3.2. Let the condition (5.1.2) be satisfied. Then (5.1.1a) and (5.3.19)
have the same solution for x0 0,
280
Polynomial and Rational Matrices
i 1
A i x0 ¦ A i j 1 B 0u j B1u j 1 " B P u j P .
xi
(5.3.21)
j 0
Proof. Substituting (5.3.21) into (5.3.19), we obtain
Axi B 0 ui B1ui 1 " B P ui P
i 1
ª
º
A « A i x0 ¦ A i j 1 B 0u j B1u j 1 " B P u j P »
j 0
¬
¼
B 0ui B1ui 1 " B P ui P
i
A i 1 x0 ¦ A i j B 0u j B1u j 1 " B P u j P
xi 1.
j 0
Thus (5.3.21) is a solution to (5.3.19).
Substituting (5.3.1) into (5.3.2) and comparing the coefficients by the same
powers of the variable z, we obtain
L0 A
A , L 0 E L1A
L2E
L3 A ,! , L P 1E
I , L1E
L2 A,
LP A, LP E
(5.3.22)
0.
Substituting (5.3.21) into (5.1.1a), using (5.3.22) and taking into account the
constraints imposed on the set 0, we obtain
Axi Bui
i 1
ª
º
A « A i x0 ¦ A i j 1 B 0u j B1u j 1 " B P u j P » Bui
j 0
¬
¼
i
ª
º
E « A i 1 x0 ¦ A i j B 0u j B1u j 1 " B P u j P »
j 0
¬
¼
Thus (5.3.21) is also a solution to (5.1.1a) for x0
Exi1.
0.
„
By virtue of this theorem, all the known results pertaining to standard systems
(for example, reachability or controllability criteria) can be easily applied to
singular systems.
Example 5.3.2.
Given an equation of the form (5.1.1a), the matrices E and A, being the same as in
(5.3.15), and
Singular and Cyclic Normal Systems
ª1 º
«2» ,
« »
«¬1 »¼
B
281
(5.3.23)
find an equivalent equation of the form (5.3.19).
To this end, we use (5.3.18) and the results from Example 5.3.1. We compute
B0
B2
L0B
L 2B
ª1
«0
«
«¬ 0
ª0
«0
«
¬« 0
0 0 º ª1 º
1 0 »» «« 2 »»
0 0 »¼ «¬1 »¼
0 0 º ª1 º
0 0 »» «« 2 »»
0 1¼» ¬«1 ¼»
ª1 º
«2» , B
1
« »
«¬ 0 »¼
ª 0 0 0º ª1 º
« 0 0 0» « 2»
«
»« »
«¬ 0 1 0»¼ «¬1 »¼
L1B
ª0 º
«0 » .
« »
¬« 1¼»
ª0 º
«0 » ,
« »
«¬ 2»¼
.
Thus the desired equation of the form (5.3.19) is
xi 1
Axi B0ui B1ui 1 B 2ui 2
ª1 0 0 º
ª1 º
ª0 º
ª0 º
«0 0 1 » x « 2 » u «0 » u «0 » u .
«
» i « » i « » i 1 « » i 2
«¬ 0 0 0 »¼
«¬ 0 »¼
«¬ 2»¼
«¬ 1»¼
(5.3.24)
The set of admissible initial conditions 0 is in this case given by the
relationship x02 + u0 = 0, where x02 is the value of the second entry of the vector
xiT = [xi1 xi2 xi3] (T denotes the transpose) at the initial point i = 0.
According to (5.3.21), a solution to (5.3.24) for x0 0 is
xi
­ ª x10 u0 º
°« 3
»
° « x0 2u0 »
°«
»
°¬ 2u1 u2 ¼
° 1
° ª x0 º ª1 0
°« » «
® « 0 » «0 0
°« 0 » «0 0
°¬ ¼ ¬
° ªu
º
° « i 1
»
° « 2ui 1
»
° «
»
°¯ ¬ 2ui ui 1 ¼
for
0º
1 »»
0 »¼
i 1
i 1
ª u0
º ª1 0 0 º
«
» «
»
« 2u0
» «0 0 1 »
«¬ 2u1 u2 »¼ «¬0 0 0 »¼
for i ! 1
i 2
ªu1
º
« 2u
» "
« 1
»
«¬ 2u2 u3 »¼
(5.3.25)
282
Polynomial and Rational Matrices
It is easy to verify that the solution (5.3.25) for x0 0 satisfies also (5.1.1a)
with the matrices (5.3.15) and (5.3.23). One can also obtain the solution (5.3.25)
using (5.3.23) and the matrices )i, i = 2,1,0,1,… computed in Example 5.3.1.
5.3.4 Continuous-time Systems
Consider a linear continuous-time system described by the following equations
Ex Ax Bu , x(0)
y Cx Du ,
x0 ,
(5.3.26a)
(5.3.26b)
where x = x(t) n is the state vector, u = u(t) m is the input vector, y = y(t)
the output vector and E, A nun B num, C pun, D pum.
The system (5.3.26) is called regular if E = I, and singular if det E = 0. If
det [Es A ] z 0 for some s  ,
p
is
(5.3.27)
then (5.3.26) is called a singular system with a regular pencil.
Let
L(s )
L 0 L1s " L P s P
(5.3.28)
be a polynomial matrix such that
L ( s ) > Es A @
ª¬Is A º¼ ,
(5.3.29)
where A  nun, and P is the nilpotent index of the pair (E, A). We compute the
matrix (5.3.28) in the same way as for the discrete-time system (5.3.1), applying
the procedure of elementary operations introduced in the proof of Theorem 5.3.1.
Pre-multiplying (5.3.26a) in its operator form
> Es A @ X ( s )
BU ( s ) Ex0
by the matrix (5.3.28), and taking into account (5.3.29), we obtain
ª¬Is A º¼ X ( s )
P
¦
B j s jU (s ) L j Ex0 s j ,
(5.3.30)
j 0
where X(s) and X(s) are the Laplace transforms of the vectors x(t) and u(t),
respectively, and Bj, (j = 0,1,…,P is given by (5.3.18).
Taking the inverse Laplace transform of (5.3.30), we obtain
Singular and Cyclic Normal Systems
x
j
P
§
·
Ax ¦ ¨ B j u (j ) B j ¦ u (k 1)G (j k ) L j Ex0G (j ) ¸ x0G ,
j 0©
k 1
¹
283
(5.3.31)
where
d ju
,
dt j
u( j)
G(j) denotes the j-th distributive derivative of Dirac’s pseudo-function (impulse
function) G. When
j
P
¦B ¦u
( k 1)
j
j 0
G ( j k ) L j Ex0G ( j ) x0G
0,
k 1
then (5.3.31) reduces to
x
P
Ax ¦ B j u (j ) .
(5.3.32)
j 0
Theorem 5.3.3. Let the condition (5.3.27) be satisfied. Then the equations
(5.3.26a) and (5.3.31) have the same solution for x0 0
x(t )
e At x0 j
P t
(5.3.33)
ª
º
§
·
¦ ³ «e A (t W ) B j ¨ u (j ) (W ) ¦ u (k 1) (W )G (j k ) (W ) ¸ L j Ex0G (j ) (W ) » dW .
j 0 0 ¬
k 1
©
¹
¼
The proof of this theorem follows analogously to that of Theorem 5.3.2.
A solution to (5.3.32) is
x(t )
P
t
e At x0 ¦ ³ e A ( t W ) B j u ( j ) (W )dW .
(5.3.34)
j 0 0
With (5.3.14) taken into account, and with Li, i = 0,1,…,P and A known, one can
compute the fundamental matrices )i for i = P,1P,….
Example 5.3.3.
Consider the following equation
ª0 1 0º
«1 0 0 » x
«
»
«¬ 0 1 0 »¼
ª1 0 0 º
ª 2º
«0 0 0 » x « 1 » e t ,
«
»
« »
«¬0 0 1 »¼
«¬ 1»¼
(5.3.35)
284
Polynomial and Rational Matrices
whose matrices E and A are singular, but the pencil Es A is regular, since
det [Es A]
1 s
0
s
0
0
0
s 1
s2 .
Let the initial conditions for t = 0 be
x10
1, x20
1, x30
(5.3.36)
2
and
ª1
>E, A @
0 0º
ª 0 1 0 1 0 0 º L1 «« 0 1 0»» ª0 1 0 1 0 0 º
«1 0 0 0 0 0 » 
¬« 1 0 1 ¼»
o ««1 0 0 0 0 0 »»
«
»
«¬ 0 1 0 0 0 1 »¼
¬«0 0 0 1 0 1 ¼»
ª0 1
0º
ª0 1 0 1 0 0 º L2 ««1 0 0 »» ª1 0 0 0 0 0 º
«¬ 0 1 1»¼
L1 ( s )

o ««1 0 0 0 0 0 »» 
o «« 0 1 0 1 0 0 »» .
«¬1 0 1 0 0 0 »¼
«¬ 0 0 1 0 0 0 »¼
In this case,
L(s )
L 2 L1 (s )L1
ª 0 1 0 º ª1 0 0 º ª 1 0 0 º
«1 0 0 » « 0 1 0 » « 0 1 0 »
«
»«
»«
»
«¬ 0 1 1»¼ «¬0 0 s »¼ «¬ 1 0 1 »¼
ª0 1 0 º
«1 0 0 »
«
»
«¬ s 1 s »¼
where
L0
ª0 1 0 º
«1 0 0 » , L
1
«
»
«¬0 1 0 »¼
ª0 0 0 º
«0 0 0 » , P
«
»
«¬1 0 1»¼
1.
From the relationship
L1 > Ex
we have
Ax Bu @
ªE1 º
« » x
¬0 ¼
ª A1 º
ª B1 º
« ˆ » x « ˆ »u ,
¬« B1 ¼»
¬« A1 ¼»
L 0 L1s,
Singular and Cyclic Normal Systems
ˆ x Bˆ u
A
1
1
> 1
0 1@ x [3]u
0,
thus for t = 0, we obtain x30 – x10 – 3 = 0.
The last relationship determines the set of admissible initial conditions
that the initial conditions (5.3.36) belong to the set 0.
Using (5.3.18), we compute
B0
L0B
ª0 1 0 º ª 2 º
«1 0 0 » « 1 »
«
»« »
«¬ 0 1 0 »¼ «¬ 1»¼
285
ª1 º
« 2» , B
« » 1
«¬1 »¼
ª0 0 0 º ª 2 º
«0 0 0 » « 1 »
«
»« »
«¬1 0 1»¼ «¬ 1»¼
L1B
0.
Note
ª0º
« 0» .
« »
«¬3 »¼
Thus the desired equivalent equation of the form (5.3.31) is
x
Ax B 0u B1u
ª0 0 0 º
ª1º
«1 0 0 » x « 2 » e t ,
«
»
« »
«¬ 0 0 0 »¼
«¬ 2 »¼
and its solution for feasible initial conditions (5.3.36) is
t
x(t )
At
e x0 ³ e
0
A ( t W )
B 0u (W ) B1u (W ) dW
ª 1º
« 1» e t .
« »
«¬ 2 »¼
It is easy to verify that the above is also a solution to (5.3.35).
5.4 Electrical Circuits as Examples of Singular Systems
5.4.1 RL Circuits
We will show that electrical circuits built from resistances and inductances (R, L),
or resistances and capacities (R, C), and ideal voltage sources are examples of the
singular continuous-time systems.
Let the following be given for the circuit in Fig. 5.1: resistances Rk, k = 1,…,8,
inductances of coils L1, L2 and source voltages e1 and e2. Denote mesh currents by
i1,i2,i3,i4. Using the mesh method, we can write the equations
286
Polynomial and Rational Matrices
di1
R1 R3 R5 i1 R3i3 R5i4 ,
dt
di
L2 2 R4 R6 R7 i2 R4i3 R7 i4 ,
dt
0 R3i1 R4i2 R2 R3 R4 i3 e1 ,
L1
0
(5.4.1)
R5i1 R7 i2 R5 R7 R8 i4 e2 .
Fig. 5.1. The scheme of an RL circuit
With the mesh currents x1 = i1, x2 = i2, x3 = i3, x4 = i4 chosen as state variables,
we can write (5.4.1) in the form
Ex
E
Ax Bu ,
ª1
«0
«
«0
«
¬0
0 0 0º
1 0 0 »»
, x
0 0 0»
»
0 0 0¼
(5.4.2)
ª i1 º
«i »
« 2», A
« i3 »
« »
¬i4 ¼
ª R11
« L
« 1
«
« 0
«
« R31
«
¬ R41
R11
ª0 0º
«0 0»
«
» , u ª e1 º ,
«e »
«1 0 »
¬ 2¼
«
»
0
1
¬
¼
R1 R3 R5 , R13
R22
R4 R6 R7 , R23
R32
R33
R2 R3 R4 , R44
R5 R7 R8 .
B
R31
R3 , R14
R4 , R24
0
R13
L1
R22
L2
R23
L2
R32
R33
R42
0
R1
R42
R41
R7 ,
R5 ,
R14 º
L1 »
»
R23 »
,
L2 »»
0 »
»
R44 ¼
(5.4.3)
Singular and Cyclic Normal Systems
287
Note that all the entries of the matrix A off the main diagonal are nonnegative.
Thus A is a Metzler matrix.
Let the outputs of the considered system be voltages on the coil L1,
y1 = L1 di1/dt and on the resistor R6, y2 = R6i2. Thus in this case, the output equation
is
y
ª y1 º
«y »
¬ 2¼
ª R3i3 R5i4 R4i1 º
«
»
R6i2
¬
¼
Cx Du ,
(5.4.4)
where
ª y1 º ª R3i3 R5i4 R4i1 º
»,
«y » «
R6i2
¬ 2¼ ¬
¼
0 R3 R5 º
ª R
ª0 0º
C « 4
, D «
»
».
R6 0 0 ¼
¬0 0¼
¬ 0
y
Thus the considered circuit is an example of a singular continuous-time system.
In a general case, consider an n-mesh circuit with the given resistances R1,R2,…
and inductances L1,L2,…,LJ and m source voltages e1,e2,…,en. Let i1,i2,…,in be the
mesh currents of this circuit. Applying the mesh method as in the case of the circuit
in Fig. 5.1, we obtain the equation of the form (5.4.2), where
x
>i1
T
i2 ! in @ , u
>e1
ª A1
ª I r 0º
num
« 0 0»  , A « A
¬
¼
¬ 3
I r – the identity matrix of the size r.
E
A1
A3
R ij
ª R11
« L
« 1
« R21
«
« L2
« #
«
« Rr1
«¬ Lr
ª Rr 1,1
«R
« r 2,1
« #
«
¬« Rr 1,n
R12
L1
R22
L2
#
Rr 2
Lr
T
e2 ! em @ ,
A2 º
 num , B  num ,
A4 »¼
ª R1,r 1
!
« L
« 1
« R2,r 1
!
«
« L2
« #
%
«
« R3,r 1
!
« L
¬ r
ª Rr 1,r 1 Rr 1,r 2
«R
« r 2,r 1 Rr 2,r 2
« #
#
«
Rn ,r 2
«¬ Rn ,r 1
R1r º
L1 »
»
R2 r »
!
L2 »» , A 2
%
# »
»
R
! rr »
Lr »¼
!
! Rr 1,1r º
! Rr 2,r »»
, A4
%
# »
»
! Rnr »¼
­! 0 for i j
R ji ®
i, j 1,..., n.
¯t 0 for i z j
R1n º
L1 »»
R2 n »
»
L2 » ,
# »
»
Rm »
Lr »¼
! Rr 1,n º
! Rr 2,n »»
, (5.4.5)
%
# »
»
! Rnn »¼
288
Polynomial and Rational Matrices
4.5.2 RC Circuits
Consider the circuit shown in Fig. 5.2, with given resistances Rk, k = 1,…,5, the
capacity C and the source voltage e.
Applying Kirchhoff’s first law for this circuit, we can write the equations
duc
G4 v1 uC G2 v1 uC ,
dt
di
G5 2
e v1 G1 v1 v2 G4 v1 uC ,
dt
G1 v1 v2
G2 v2 uC G3v2 ,
C
(5.4.6)
where v1 and v2 are the node’s potentials, uC the voltage on the capacity and
Gk = 1/Rk, for k = 1,…,5.
Choosing as state variables x1 = uC, x2 = v1, x3 = v2, we can write the equations
(5.4.6) in the form (5.4.2), where
ª1 0 0 º
ªuC º
«0 0 0 » , x « v » ,
«
»
« 1»
«¬0 0 0 »¼
«¬ v2 »¼
G13 º
ª G11 G12
« C
C
C »
«
»
A « G21 G22 G23 » , B
« G
G32 G33 »
« 31
»
¬
¼
E
ª0º
« »
«G 5 » , u
¬« 0 ¼»
(5.4.7)
e.
Let the outputs of this circuit be the current i3 in the resistor R3, y1 = i3 and the
voltage on the capacitor uC. Also in this case, the output equation has the form
(5.4.4), where
Fig. 5.2. Scheme of an RC circuit
Singular and Cyclic Normal Systems
C
289
ª0 0 G3 º
«1 0 0 »
¬
¼
and D is the zero matrix.
Note that also in this case, A is a Metzler matrix and det E = 0. Thus the
considered circuit is an example of a singular continuous-time system.
In a general case, the considerations for RC circuits are dual to the above for RL
circuits.
Theorem 5.4.1. The inverse Rn1 to resistance matrix
Rn
ª R11
«R
« 21
« #
«
¬ Rn1
R12
R22
#
Rn 2
! R1n º
! R2 n »»
%
# »
»
! Rnn ¼
(5.4.8a)
in the mesh method is a matrix with nonnegative elements, where
R ij
­! 0 for i j
R ji ®
¯t 0 for i z j
n
and R ii t ¦ R ij , i, j 1, ..., n . (5.4.8b)
j 0
j zi
Proof. The proof follows by induction with respect to n. For n = 1 the thesis is true,
since R1 = R11 > 0 and R1-1 = R11-1 > 0. With the assumption of the validity of the
thesis for k, we will prove its validity for k + 1.
Let
R k 1
ª Rk
«v
¬ k
uk
º
,
Rk 1,k 1 »¼
(5.4.9)
where
vk
ª¬ Rk 1,1
Rk 1,2 ! Rk 1,k º¼ ,
uk
» R1,k 1
R2,k 1 ! Rk ,k 1 ¼º .
T
The inverse Rk+11 is
R k11
ª 1 Rk1uk vk Rk1
« Rk Rk
«
«
v R 1
k k
«
Rk
¬«
Rk1uk º
»
Rk »
,
1 »
»
Rk ¼»
290
Polynomial and Rational Matrices
where
Rk
R k 1,k 1 vk R k 1uk .
By assumption Rk-1 +kuk. Therefore, Rk+1-1 +(k+1)u(k+1) when R k > 0. It is known
that Rk+1 is positive definite and Rk+1 > 0. From (5.4.9) we have
det R k 1
ªR k
det «
¬ vk
uk º
R k 1,k 1 »¼
det R k det R k 1,k 1 vk R k 1uk
ªR k
det «
¬ vk
uk
º
R k 1,k 1 vk R k 1uk »¼
R k det R k .
From the last relationship it follows that det Rk+1 > 0 and det Rk > 0 imply
R k > 0.
„
Theorem 5.4.2. The matrix A given by (5.4.5) has its inverse (A)-1 with
nonnegative entries (A)1 +nun
Proof. Note that A can be written in the form
LR n ,
A
(5.4.10)
where
L
diag > L1
L 2 ! L n 1 ! 1@  nun .
(5.4.11)
Taking into account that (A)1 = Rn1L1, we find that (A)1 is the product of
two matrices with nonnegative entries, since
L1
ª1
diag «
¬ L1
Hence (A)1
+
1
L2
!
1
Lr
º
1 ! 1»  nun .
¼
nun
.
„
Electrical circuits built from resistances, inductances, capacities and voltage
sources are examples of singular continuous-time systems only in the case of
appropriately chosen values of these parameters.
Singular and Cyclic Normal Systems
291
5.5 Kalman Decomposition
5.5.1 Basic Theorems and a Procedure for System Decomposition
Consider a linear continuous-time or discrete-time system that is neither
controllable nor observable. Such a system can be decomposed into the following
four disjoint parts:
1. reachable and unobservable,
2. reachable and observable,
3. unreachable and unobservable,
4. unreachable and observable.
Theorem 5.5.1. Given a system that is neither controllable nor observable, there
exists a nonsingular matrix P, such that
A
B
PAP 1
PB
ª A11
« 0
«
« 0
«
¬ 0
ª B1 º
«B »
« 2», C
«0»
« »
¬0¼
A12
A 22
A13
0
0
0
A 33
0
A14 º
A 24 »»
,
A 34 »
»
A 44 ¼
>0
C2
0 C4 @ ,
(5.5.1)
where (A11, B1, 0) stands for the reachable and unobservable part, (A22, B2, C2)
stands for the reachable and observable part, (A33, 0, 0) stands for the unreachable
and unobservable part, and (A44, 0, C4) stands for the unreachable and observable
part.
Fig. 5.3. Kalman decomposition of a system
292
Polynomial and Rational Matrices
The proof of the above theorem is based on the possibility of decomposition of
an uncontrollable system into controllable and uncontrollable parts, and of an
unobservable one into observable and unobservable parts. The presented proof is
based on a geometrical approach and provides us with a practical procedure for the
system decomposition.
Procedure 5.5.1.
Step 1: Compute the reachability and observability matrices
R
ª¬B AB } A n-1B º¼ , O
ª C º
« CA »
«
».
« # »
«
n-1 »
¬CA ¼
(5.5.2)
Step 2: Compute
x
the reachable subspace
XS
x
Ker RT ,
the observable subspace
XO
x
(5.5.3)
the unreachable subspace
XS
x
Im R ,
Im OT ,
the unobservable subspace
XO
Ker O .
Step 3: Compute the subspaces (as the products and sums of the subspaces (5.5.3))
X1
XS ˆ XO ,
X2
X S ˆ X S XO ,
X3
XO ˆ XS XO ,
X4
X S ˆ XO.
Step 4: From the basis vectors of the subspace (5.5.4) build the matrix P1.
Step 5: Using (5.5.1), compute the matrices A , B , C .
(5.5.4)
Singular and Cyclic Normal Systems
293
Example 5.5.1.
Decompose a linear system with the following matrices
ª 4 2 1 3º
« 1 1 0 1 »
», B
A «
« 1 0 2 1 »
«
»
¬ 3 2 1 2 ¼
C >1 2 0 1@ , D > 0
ª1
« 1
«
«0
«
¬1
0º
1 »»
,
0»
»
0¼
(5.5.5)
0@ .
Using Procedure 5.5.1, we compute:
Step 1: The reachability matrix R and the observability matrix O
R
ª¬B AB A 2 B A 3B º¼
O
ª C º
« CA »
«
»
«CA 2 »
«
3»
¬CA ¼
ª1 2
«3 2
«
«1 2
«
¬3 2
0
0
0
0
1º
3»»
.
1»
»
3¼
Step 2: We have
x the reachable subspace
XS
x
1º
0 »»
,
0»
»
1¼
the observable subspace
XO
x
Im R
ª0
«1
Im «
«0
«
¬0
Im OT
ª0 1 º
«1 0 »
»,
Im «
«0 0 »
«
»
¬0 1¼
the unreachable subspace
ª1
« 1
«
«0
«
¬1
0 1 2 1
1 1 1 1
0 0 0 0
0 1 2 1
0 1 2 º
1 1 1»»
,
0 0 0»
»
0 1 2 ¼
294
Polynomial and Rational Matrices
XS
x
ª0 1 º
«0 0 »
»,
Ker «
«1 0 »
«
»
¬0 1¼
Ker RT
the unobservable subspace
XO
ª0
«0
Ker «
«1
«
¬0
Ker O
1º
0 »»
.
0»
»
1¼
Step 3: Using (5.5.4), we obtain
X1
X3
X S ˆ XO
ª1 º
«0»
« »,
«0»
« »
¬1 ¼
XO ˆ X S XO
X2
ª0º
«0»
« »,
«1 »
« »
¬0¼
X S ˆ XS XO
X4
X S ˆ XO
ª0º
«1 »
« »,
«0»
« »
¬0¼
ª1º
«0»
« ».
«0»
« »
¬ 1¼
Step 4: Using the basis vectors of the subspace, we compute
P 1
ª1
«0
«
«0
«
¬1
0
1
0
0
0 1º
0 0 »»
, and P
1 0»
»
0 1¼
ª 12
«0
«
«0
«1
¬2
0
1
0
0
0 12 º
0 0 »»
.
1 0»
»
0 12 ¼
Step 5: Using (5.5.1), we compute the desired matrices in the canonical form
ª1 2 1 6 º
« 0 1 0 2 »
», B
A PAP 1 «
«0 0 2 2 »
«
»
¬0 0 0 1 ¼
C CP 1 > 0 2 0 2@ .
PB
ª1
« 1
«
«0
«
¬0
0º
1 »»
,
0»
»
0¼
Singular and Cyclic Normal Systems
295
5.5.2 Conclusions and Theorems Following from System Decomposition
The following conclusions immediately follow from Fig. 5.3
1.
The input u affects directly only two parts of the system, i.e., the
reachable and unobservable part and the reachable and observable part.
Thus if the remaining two parts at the initial time instant have the zero
initial conditions (x3(0) = 0, x4(0) = 0), then their state variables are zero
for all time instances t > 0.
2.
The output of the system is connected to its input only through the
reachable and observable part.
3.
The output is indirectly affected by the dynamics of the unreachable and
unobservable part.
4.
The output is also affected by the dynamics of the unreachable and
observable part.
5.
The dynamics of the reachable and unobservable part is affected by the
dynamics of the remaining three parts, but the dynamics of the first part
does not affect the dynamics of the remaining three parts of the system.
6.
With the input u and the output y known, we are not able to determine the
initial conditions of the reachable and unobservable part nor of the
unreachable and unobservable part since the output y does not depend
directly on the dynamics of these parts of the system.,
7.
It follows from the triangular form of the matrix A that its characteristic
polynomial (and also that of A) is the product of the characteristic
polynomials of the matrices A11, A22, A33, A44, i.e.,
^
`
det > I n O A @ det ª¬I n O P 1AP º¼ det P 1 ¬ªI n O A ¼º P det ¬ªI n O A ¼º
det > I n1O A11 @ det > I n 2 O A 22 @ det > I n3O A 33 @ det > I n 4 O A 44 @ .
Remark 5.5.1.
For a continuous-time system O = s and for a discrete-time system O = z.
Theorem 5.5.2. The transfer matrix of a linear system with the matrices A, B, C
and D is equal to the transfer matrix of the reachable and observable part of the
system, i.e.,
T O
1
1
C > I n O A @ B D C2 > I n 2 O A 22 @ B 2 D .
Proof. Using (5.5.1) we can write
T O
1
1
C > I n O A @ B D CP > I n O A @ P 1B D
1
CP ª¬ P 1 I n O A P º¼ P 1B D
1
C ª¬I n O A º¼ B D
(5.5.6)
296
Polynomial and Rational Matrices
>0
C2
ª B1 º
«B »
u« 2» D
«0»
« »
¬0¼
A12
ªI n1O A11
«
I n 2 O A 22
0
0 C4 @ «
«
0
0
«
0
0
¬
>0
ª> I n1O A11 @1
«
«
0
u«
0
«
«
0
¬«
C2
A13
A14
0
I n 3O A 33
0
º
A 24 »»
A 34 »
»
I n 4 O A 44 ¼
1
0 C4 @
1
>I n 2 O A 22 @
1
>I n3O A33 @
0
0
0
º
» ª B1 º
» «B 2 »
»« »D
»« 0 »
« »
1 »
>I n 4O A 44 @ ¼» ¬ 0 ¼
1
C2 > I n 2 O A 22 @ B 2 D,
where * denotes the matrices that are insignificant for these considerations.
„
Theorem 5.5.3. Using the state-feedback
u
K1 x1 K 2 x2 K 3 x3 K 4 x4 ,
(5.5.7)
one can arbitrarily assign the eigenvalues of the matrices A11 and A22 only, and
using the output feedback
u
Fy ,
(5.5.8)
one can arbitrarily assign the eigenvalues of the matrix A22 only.
Proof. Substituting (5.5.7) into the equation x
x
Px
Ax Bu , we obtain
Az x ,
(5.5.9)
where
Az
A B > K1 K 2
ª A11 B1K1
« BK
2 1
«
«
0
«
0
¬
K3
K4 @
A12 B1K 2
A13 B1K 3
A 22 B 2 K 2
Ǻ2K 3
0
0
A 33
0
A14 B1K 4 º
A 24 Ǻ 2 Ȁ 4 »»
.
»
A 34
»
A 44
¼
(5.5.10)
Singular and Cyclic Normal Systems
297
From (5.5.10) it follows that by an appropriate choice of the matrices K1 and K2,
one can arbitrarily assign the eigenvalues of the matrices A11 and A22 only. Now
substituting
y
Cx
C2 x2 C4 x4
into (5.5.8), and the consequent result into the equation
x
Ax B u ,
we obtain
x
ˆ x,
A
z
where
ˆ
A
z
A BFC
ª A11
« 0
«
« 0
«
¬ 0
A12 B1FC2
A13
A 22 B 2 FC2
0
A 33
0
0
0
A14 B1FC4 º
A 24 B 2 FC4 »»
.
»
A 34
»
A 44
¼
(5.5.11)
It follows from (5.5.11) that by an appropriate choice of the matrix F, one can
arbitrarily assign only the eigenvalues of the matrix A22 of the reachable and
observable part.
„
As it is known, a linear system is externally stable (BIBO) if the steady state
component of its response is bounded for every bounded input.
Theorem 5.5.4. A linear system is externally stable if and only if its reachable and
observable part is asymptotically stable.
Proof. The proof will be accomplished only for the continuous-time system, since
for the discrete-time system it is analogous. It is well-known that the steady state
component of the output y under the input u is given by
t
y t
³g
t W u W dW .
(5.5.12)
0
The above formula implies that this component is bounded for every bounded input
if and only if
298
Polynomial and Rational Matrices
t
h t
³g
W dW
0
is bounded for every t, since
t
y t d
³g
W dW u t
h t u t .
0
The step characteristic h(t) is bounded if and only if the impulse characteristic
g(t) tends to 0 as tof. This occurs if and only if the transfer matrix of the
reachable and observable part has all its poles on the left half-plain. These poles
coincide with the eigenvalues of A22, since there are no pole-zero cancellations
(this part is reachable and observable).
„
Theorem 5.5.5. A linear system is stabilizable (detectable) if and only if its
unreachable and unobservable, as well as its unreachable and observable parts (its
reachable and unobservable, as well as its unreachable and unobservable parts) are
asymptotically stable.
Proof. By virtue of Theorem 5.5.3, it follows that with the state-feedback one can
assign eigenvalues only of the reachable and unobservable part, as well as the
reachable and observable part. Both these parts are reachable, thus with an
appropriate choice of the feedback matrices, the eigenvalues of the matrices A11
and A22 can be arbitrarily assigned. Thus a system is stabilizable if and only if its
two remaining parts are asymptotically stable. The proof of the second part of the
theorem is dual.
„
5.6 Decomposition of Singular Systems
5.6.1 Weierstrass–Kronecker Decomposition
We will show that the Kalman decomposition of standard systems can be
generalised into the case of singular systems.
Consider a singular system of the form
Ex Ax Bu ,
y Cx ,
(5.6.1a)
(5.6.1b)
Singular and Cyclic Normal Systems
where x n is the state vector u
respectively, and E, A  nun, B 
pencil (E, A) is regular, i.e.,
m
and y
,C
num
p
299
are the vectors of input and output,
. We assume that det E = 0 and the
pum
det > Es A @ z 0, for some s  .
(5.6.2)
As it is known, there exist nonsingular matrices P, Q  nun such that a system
of the form (5.6.1) can be decomposed into the following two subsystems:
1. the standard subsystem (slow)
2.
x1
A1 x1 B1u ,
y1
C1 y1 ,
(5.6.3a)
(5.6.3b)
the strictly singular subsystem (fast)
x2 B 2u ,
Nx2
(5.6.4a)
(5.6.4b)
C 2 y2 ,
y2
where
0º
(5.6.5)
»,
N¼
ª A1 0 º
ªB º
, A1  n1un1 , PB « 1 » ,
N  n2 un2 , PAQ «
»
¬B 2 ¼
¬ 0 I n2 ¼
B1  n1um , B 2  n2 um , CQ >C1 C2 @ , C1  pun1 , C2  pun2 ,
ª x1 º
«x »
¬ 2¼
y
Q 1 x, x1  n1 , x2  n2 , PEQ
y1 y2 , n1
deg det > Es A @ , n2
ª I n1
«
¬0
n n1 ,
N is a nilpotent matrix with its nilpotent index P, i.e., NP1 z 0 and NP = 0.
5.6.2 Basic Theorems
Consider a singular system of the form (5.6.1) satisfying the regularity condition
(5.6.2). We decompose the system into two subsystems: the standard subsystem
(5.6.3) and the strictly singular subsystem (5.6.4). According to Theorem 5.5.1, the
standard subsystem (5.6.3) can be decomposed into the four disjoint parts
300
Polynomial and Rational Matrices
ª x11 º
« x »
« 12 »
« x13 »
« »
¬ x14 ¼
y1
ª A11
« 0
«
« 0
«
¬ 0
>0
A12
A13
A 22
0
A 33
0
0
0
A14 º ª x11 º ª B11 º
A24 »» «« x12 »» «« B12 »»
u,
A 34 » « x13 » « 0 »
»« » « »
A 44 ¼ ¬ x14 ¼ ¬ 0 ¼
ª x11 º
«x »
0 C14 @ « 12 » ,
« x13 »
« »
¬ x14 ¼
C12
(5.6.6)
where
ª A11 A12 A13 A14 º
« 0 A
0 A 24 »»
22
P1A1P11 «
, P1B1
« 0
0 A 33 ǹ 34 »
«
»
0
0 A 44 ¼
¬ 0
C1P11 > 0 C12 0 C14 @ ,
x1i  n1i , i 1, 2, 3, 4,
4
¦n
1i
ª B11 º
«B »
« 12 » ,
« 0 »
« »
¬ 0 ¼
(5.6.7)
n1 ,
i 1
P1 is a nonsingular transformation matrix (det P1 z 0). With N, B2, C2 regarded as
the matrices of the regular system (det [In2s – N] z 0), we also decompose the
system according to Theorem 5.5.1 into the four disjoint parts
ª N11 N12 N13
« 0 N
0
22
«
« 0
0 N 33
«
0
0
0
¬
>0 C22 0 C24 @ ,
P2 NP21
C2 P21
N ij  n2 i un2 j
C2 j  pun2 j
N14 º
N 24 »»
, PǺ
N 34 » 2 2
»
N 44 ¼
ª B 21 º
«% »
« 22 » ,
« 0 »
« »
¬ 0 ¼
(5.6.8)
, i, j 1, 2, 3, 4, B 2i  n2 i um , i 1, 2,
, j
2, 4,
4
¦n
2i
n2 ,
i 1
P2 is a nonsingular (det P2 z 0) transformation matrix, and Nii (i = 1,2,3,4) are
nilpotent matrices. Using (5.6.8) we can write the equations of the strictly singular
subsystem (5.6.4) in the form
Singular and Cyclic Normal Systems
ª N11
« 0
«
« 0
«
¬ 0
y2
N14 º ª x21 º
0 N 24 »» «« x22 »»
N 22
0 N 33 N 34 » « x23 »
»« »
0
0 N 44 ¼ ¬ x24 ¼
ª x21 º
«x »
>0 C22 0 C24 @ «« x22 »» .
23
« »
¬ x24 ¼
N12
N13
ª x21 º ª B 21 º
« x » «B »
« 22 » « 22 » u ,
« x23 » « 0 »
« » « »
¬ x24 ¼ ¬ 0 ¼
301
(5.6.9)
Defining
xi
ª x1i º
n1i un2 i
«x »  ¬ 2i ¼
and
Eii
Bi
ªI 0 º
ª A ii 0 º
«0 N » , A ii « 0 I » , i 1, 2, 3, 4,
¬
¼
ii ¼
¬
0
0 º
ª
ª B1i º
ª A ij
« B » , i 1, 2, Eij « 0 N » , A ij « 0
ij ¼
¬
¬ 2i ¼
¬
i, j 1, 2, 3, 4 i z j , C j
»C1 j
C2 j ¼º , j
0º
,
0 »¼
(5.6.10)
2, 4,
we can write (5.6.6) and (5.6.9) in the form
ªE11 E12
«
« 0 E22
« 0
0
«
0
«¬ 0
y
ª¬ 0 C2
E13
0
E33
0
E14 º ª x1 º
»« »
E 24 » « x2 »
E34 » « x3 »
»« »
E 44 »¼ «¬ x4 »¼
ª A11
«
« 0
« 0
«
«¬ 0
A12
A 22
A13
0
0
0
A 33
0
A14 º ª x1 º ª B1 º
»« » « »
A 24 » « x2 » «B 2 »
u,
A 34 » « x3 » « 0 »
»« » « »
A 44 »¼ ¬ x4 ¼ ¬ 0 ¼
(5.6.11)
ª x1 º
«x »
0 C4 º¼ « 2 » .
« x3 »
« »
¬ x4 ¼
Theorem 5.6.1. For an uncontrollable and unobservable singular system of the
form (5.6.1) there exist nonsingular matrices of strong equivalence, which
transform that system into the form (5.6.11) where
1. the singular subsystem ( E11 , A11 , B1 , 0 ) is controllable and unobservable,
2. the singular subsystem ( E22 , A 22 , B 2 , C2 ) is controllable and observable,
302
Polynomial and Rational Matrices
3.
the singular subsystem ( E33 , A 33 , 0, 0 ) is uncontrollable and unobservable,
4.
the singular subsystem ( E44 , A 44 , 0, C4 ) is uncontrollable and observable.
Proof. The matrices of strong equivalence that transform the system (5.6.1) into
the form (5.6.11) are the products of the matrices of strong equivalence P, Q of the
Weierstrass–Kronecker decomposition that decompose this system into the
subsystems (5.6.3) and (5.6.4), the similarity matrix
ª P1
«0
¬
0º
P2 »¼
that transforms these subsystems to the forms (5.6.6) and (5.6.9), respectively, and
the matrices of change of variables that define subvectors x i, i = 1,2,3,4. Using the
controllability conditions for the subsystem ( E11 , A11 , B1 , 0 ), we obtain
ªIs A11
rank «
¬ 0
rank ª¬ E11s A11 , B1 º¼
B11 º
0
N11s I B 21 ¼»
n11 n21 ,
since rank [Is – A11, B11] = n1, and N11 is a nilpotent matrix and [N11s – I] is a
nonsingular matrix for all finite s . Thus the first condition is satisfied. Now we
check the second condition, which is satisfied as well, since
rank > Is N11 , B 21 @ s
0
rank > N11 B 21 @
n21 .
Thus the subsystem ( E11 , A11 , B1 , 0 ) is controllable. The unobservability of this
subsystem follows by the fact that its matrix C 1 = 0. We will show that the
subsystem ( E22 , A 22 , B 2 , C2 ) is controllable and observable. Using the controllability
condition, we obtain
rank ª¬ E22 s A 22 , B 2 º¼
ª Is A 22
rank «
¬ 0
B12 º
0
N 22 s I B 22 »¼
n12 n22 ,
since rank [Is – A22, B22] = n12 and N22 is a nilpotent matrix and [N22s – I] is a
nonsingular matrix for all finite s . In the same vein
rank ª¬E22 , B2 º¼
ªI 0
«0 N
22
¬
B12 º
B 22 »¼
n12 n22 ,
rank > N 22
B 22 @
since
rank > Is N 22 , B 22 @ s
0
n22 .
Singular and Cyclic Normal Systems
303
Thus the conditions are met and the subsystem is controllable. In order to show
that this subsystem is also observable, we use the observability conditions for
(5.6.10). We obtain
ª E s A 22 º
rank « 22
»
C2
¬
¼
ª Is A 22
rank «« 0
«¬ C12
0 º
N 22 s I »»
C22 »¼
n12 n22 ,
since
ªIs A 22 º
rank «
»
¬ C12 ¼
n12 ,
N22 is a nilpotent matrix and [N22s – I] is a nonsingular matrix for all finite s . In
the same vein,
ªE º
rank « 22 »
¬ C2 ¼
ª I
rank «« 0
«¬C12
0 º
N 22 »»
C22 »¼
n12 n22 ,
since
ªIs N 22 º
rank «
»
¬ C22 ¼
ªN º
rank « 22 »
¬ C22 ¼
n22 .
Thus the conditions are met and the subsystem is also observable. The
uncontrollability and unobservability of the subsystem ( E33 , A 33 , 0, 0 ) follows from
the fact that its matrices B 3 and C 3 are the zero ones. The observability of the
subsystem ( E44 , A 44 , 0, C4 ) can be proved in a similar way.
„
Defining
xˆ1
ª x1 º
« x » , xˆ2
¬ 2¼
ª x3 º
«x »
¬ 4¼
and
Eˆ 11
ª E11 E12 º ˆ
«
» , E12
¬ 0 E22 ¼
ª E13
«
¬ 0
E13 º ˆ
» , E 22
E24 ¼
ª E33
«
¬ 0
E34 º
»,
E 44 ¼
304
Polynomial and Rational Matrices
ˆ
A
11
Bˆ 1
ª A11
«
¬ 0
A12 º
ˆ
» , A12
A 22 ¼
ª B1 º
ˆ
« » , C1
¬B 2 ¼
ª A13
«
¬ 0
ˆ
ª¬ 0 C2 º¼ , C
2
A14 º
ˆ
» , A 22
A 24 ¼
ª A 33
«
¬ 0
A 34 º
»,
A 44 ¼
ª¬ 0 C4 º¼ ,
we can write (5.6.11) in the form
ˆ
ˆ º ª xˆ º ª A
ªEˆ 11 E
12
1
11
«
»« » «
ˆ
«¬ 0 E22 »¼ «¬ xˆ2 »¼ «¬ 0
ˆ
ˆ C
ˆ º ª x1 º .
y ª¬C
1
2¼«
»
¬ xˆ2 ¼
ˆ º ª xˆ º ªBˆ º
A
12
1
» « » « 1 » u,
ˆ
ˆ
x
A 22 »¼ ¬ 2 ¼ ¬ 0 ¼
(5.6.12)
In the same way as in the proof of Theorem 5.6.1, one can show that the
subsystem ( Eˆ 11 , Eˆ 11 , Bˆ 1 ) is controllable. Hence the singular system (5.6.12) is said to
be in its controllable canonical form. On the other hand, defining
x1
ª x4 º
« x » , x2
¬ 2¼
ª x3 º
«x » ,
¬ 1¼
we can write (5.6.11) in the form
ªE
0 º ª x1 º ª A
11
11
«
«
»
«
»
E
E
x
A
¬ 21
22 ¼ ¬ 2 ¼
¬ 21
0 º ª x1 º ,
y ª¬C
1
¼ « x »
¬ 2¼
0 º ª x1 º ª B 1 º
u,
» «¬ x2 »¼ « B »
A
¬ 2¼
22 ¼
(5.6.13)
where
E 11
A
11
B 1
ªE44
«
¬E24
0 º » , E21
E 22 ¼
ªE34
«
¬ E14
0 º » , E22
E12 ¼
ªE33
«
¬ Ǽ13
ª A 44
ª A 34
ª ǹ 33
0 º
0 º
«
» , A 21 «
» , A 22 «
¬ A 24 A 22 ¼
¬ A14 A12 ¼
¬ A13
ª0º
« B » , C1 ª¬C2 C4 º¼ , C2 ª¬ 0 C4 º¼ .
¬ 2¼
0 º
»,
E11 ¼
0 º
»,
A11 ¼
In the same vein, it can be shown that the subsystem ( E 11 , E 11 , B 1 ) is
observable. For this reason, we say that it is in the controllable canonical form.
Singular and Cyclic Normal Systems
305
5.7 Structural Decomposition of a Transfer Matrix of a Singular
System
5.7.1 Irreducible Transfer Matrices
We will show that the structural decomposition of a transfer matrix can be
generalised into the case of singular systems.
Consider a discrete-time system described by the equations
Exi 1
yi
Axi Bui , i  ' {0, 1, ...} ,
Cxi Dui ,
(5.7.1a)
(5.7.1b)
where xi , and yi p are the state, input and output vectors, respectively, at the
discrete instant i, and E, A nun, B num, C pun, D pum.
We assume that det E = 0 and
det[Ez A] z 0, for some z  ,
(5.7.2)
where is the field of complex numbers.
The transfer matrix of the system (5.7.1) is given by the formula
T( z )
1
C > Ez A @ B D .
(5.7.3)
This matrix can be written in the following form
T( z )
P( z )
,
d ( z)
(5.7.4)
where P(z) pum[z] ( pum[z] is the set of polynomial matrices of dimensions pum),
and d(z) is a least common denominator of all the entries of T(z).
A transfer matrix is of the standard form if and only if it is irreducible (that is
for all zeros of the polynomial d(z) the matrix P(z) is not the zero matrix) and d(z)
is a monic polynomial.
According to Definition 3.1.1, the standard matrix (5.7.4) is called normal if
and only if every nonzero second-order minor of P(z) is divisible without
remainder by the polynomial d(z).
Theorem 5.7.1. The matrix
CAdj[Ez A]B
det[Ez A]
(5.7.5)
306
Polynomial and Rational Matrices
is irreducible if and only if the following conditions are simultaneously satisfied:
1. (E, A) is a cyclic pair,
2. rank [Ez – A, B] = n for all finite z ,
ª Ez A º
3. rank «
» n for all finite z .
¬ C ¼
Proof. As it is known,
Adj[Ez A]
det[Ez A]
is a normal and irreducible matrix if and only if (E, A) is a cyclic matrix. The
matrices Ez A and B are relatively prime if and only if the condition 2 above is
satisfied. Relative primeness of the matrices Ez A and B is equivalent to the
existence of polynomial matrices M(z) and N(z) such that [152]
[Ez A]M ( z ) BN( z )
In .
(5.7.6)
Pre-multiplying (5.7.6) by [Ez A]-1, we obtain
M( z) Adj[Ez A]B
N( z ) [Ez A]1 .
det[Ez A]
(5.7.7)
Form (5.7.7) it follows immediately that
Adj[Ez A]B
det[Ez A]
is an irreducible matrix. The proof that
CAdj[Ez A]B
det[Ez A]
is an irreducible matrix is analogous (dual).
„
5.7.2 Fundamental Theorem and Decomposition Procedure
Theorem 5.7.1. The matrix (5.7.4) is normal if and only if
T( z )
Q( z ) R ( z )
G( z) ,
d ( z)
(5.7.8)
Singular and Cyclic Normal Systems
307
where
Q( z )  p [ z ], R ( z )  1um [ z ], G ( z )  pum [ z ]
and
deg Q( z ) deg d ( z ), deg R ( z ) deg d ( z ) .
(5.7.9)
Proof. If P(z) = Q(z)R(z) + d(z)G(z), then computing the second-order minor built
from the rows i, j and columns k, l of the matrix P(z), we obtain
Pki,,lj ( z )
qi ( z )rk ( z ) d ( z ) g ik ( z )
qi ( z )rl ( z ) d ( z ) g il ( z )
q j ( z )rk ( z ) d ( z ) g jk ( z ) g j ( z )rl ( z ) d ( z ) g jl ( z )
(5.7.10)
d ( z ) pklij ( z ),
where qi(z), rk(z) and gik(z) are entries of Q(z), R(z) and G(z), respectively, and
pklij(z) is a polynomial.
From (5.7.10) it follows that the minor Pk,li,j(z) is divisible without remainder
by d(z). Thus the matrix (5.7.8) is normal.
Now we will show that if T(z) is a normal matrix, then it can be expressed in
the form (5.7.8).
Applying elementary operations on rows and columns, we transform the matrix
P(z) into the form
U( z )P( z )V ( z )
w( z ) º
ª 1
,
i( z ) «
(
)
P ( z ) »¼
k
z
¬
(5.7.11)
where U(z) and V(z) are unimodular matrices of elementary operations, i(z) [z]
and
w( z )  1u( m1) [ z ], k ( z )  p 1[ z ], P( z )  ( p1)u( m1) [ z ] .
Let
Q( z )
ª 1 º
U 1 ( z )i ( z ) «
» , R( z)
¬k ( z)¼
>1
w( z ) @ V 1 ( z ) .
(5.7.12)
By divisibility of nonzero second-order minors of the matrix P(z) by d(z) and
by (5.7.11) it follows that the entries of
i ( z ) ª¬ P ( z ) k ( z ) w( z ) º¼
308
Polynomial and Rational Matrices
are divisible without remainder by d(z), that is,
i ( z ) ª¬ P( z ) k ( z ) w( z ) º¼
d ( z )Pˆ ( z ), where Pˆ ( z )  ( p 1)u( m1) [ z ] .
(5.7.13)
Defining
G( z)
ª 0 01,m1 º 1
U 1 ( z ) «
» V ( z) ,
«¬ 0 p1 Pˆ ( z ) »¼
(5.7.14)
we obtain from (5.7.11)–( 5.7.14)
P( z )
w( z ) º 1
ª 1
U 1 ( z )i ( z ) «
» V ( z)
¬ k ( z ) P( z ) ¼
ª 0
ª 1 º
°­
U 1 ( z ) ®i ( z ) «
>1 w( z )@ «
»
¬k ( z)¼
¯°
¬«0 p 1
01,m1 º °½ 1
» ¾ V ( z)
d ( z )Pˆ ( z ) »¼ °¿
Q( z )R ( z ) d ( z )G ( z ),
which is the desired decomposition (5.7.8).
If the conditions (5.7.9) are not satisfied, then dividing every entry qi(z) (rk(z))
of the vector P(z) (or R(z)) by d(z), we obtain
Q( z )
d ( z )K1 ( z ) Q( z ), R ( z )
d ( z )K 2 ( z ) R ( z ) ,
(5.7.15)
where
det Q( z ) det d ( z ), det R ( z ) det d ( z )
and K1(z), as well as K2(z) are polynomial column and row vectors, respectively.
Substituting (5.7.15) into (5.7.8), we obtain
T( z )
Q( z ) R ( z )
G( z) ,
d ( z)
(5.7.16)
G( z)
G ( z ) d ( z )K 1 ( z )K 2 ( z ) Q ( z ) K 2 ( z ) K 1 ( z ) R ( z ) .
(5.7.17)
where
The proof of Theorem 5.7.2 provides us with the following procedure for the
structural decomposition (5.7.8) of a given transfer matrix T(z).
Singular and Cyclic Normal Systems
309
Procedure 5.7.1.
Step 1: Given a matrix T(z) bring it to the standard form (5.7.4).
Step 2: Applying elementary operations bring the polynomial matrix P(z) to the
form (5.7.11), compute the unimodular matrices U(z) and V(z) of these
elementary operations, then compute i(z), k(z), w(z) and P (z).
Step 3: Using (5.7.12)( 5.7.14) compute Q(z), R(z) P̂ (z) and G(z).
Step 4: Using (5.7.8) compute the desired structural decomposition.
Example 5.7.1.
Consider a singular discrete-time system of the form (5.7.1) with the matrices
E
C
ª1
«0
«
«¬0
ª1
«0
¬
0 0º
1 0 »» , A
0 0 »¼
0 0º
, D
1 0 »¼
ª 0 1 0º
« 0 0 1» , B
«
»
«¬ 2 1 0 »¼
ª1 0 º
«0 0» ,
«
»
«¬ 0 1 »¼
(5.7.18)
0,
(E, A) is a cyclic pair since its characteristic polynomial
det[Ez A]
z 1
0
0
2
1
0
z
1
z2
(5.7.19)
coincides with its minimal polynomial Dn 1 ( z ) 1 .
In order to obtain the Smith canonical form of [Ez – A], we pre-multiply it by
the unimodular matrix
U( z )
ª 1 0 0 º
« 0 1 0 »
«
»
«¬ 1 0 1 »¼
and post-multiply by the unimodular matrix
V( s)
We obtain
ª0 0 1 º
«1 0 z » .
«
»
«¬ z 1 z 2 »¼
310
Polynomial and Rational Matrices
> E z A @S
U( z )[Ez A]V ( s )
ª 1 0 0 º ª z 1 0 º ª0 0 1 º
« 0 1 0 » « 0 z 1» «1 0 z »
«
»«
»«
»
«¬ 1 0 1 »¼ «¬ 2 1 0 »¼ «¬ z 1 z 2 »¼
0 º
ª1 0
«0 1
0 »» .
«
«¬0 0 z 2 »¼
The inverse [Ez – A]-1 has the form
1
[Ez A ]
Adj[Ez A]
det[Ez A ]
0
1º
ª 1
1 «
0
z »» .
2
z2«
«¬ 2 z z 2 z 2 »¼
(5.7.20)
It is easy to verify that this matrix is normal and irreducible.
Using (5.7.3), we obtain
T( z )
1
C > Ez A @ B D
1
ª z 1 0 º ª1 0 º
ª1 0 0 º «
» «
»
« 0 1 0 » « 0 z 1» « 0 0 »
¬
¼«
¬ 2 1 0 »¼ «¬ 0 1 »¼
(5.7.21)
1 ª 1 1º
.
z 2 «¬ 2 z »¼
This matrix is normal.
In order to compute the structural decomposition of the matrix (5.7.21), we
apply Procedure 5.7.1.
Step 1: The matrix (5.7.21) is already in standard form, with d(z) = z + 2 and
P( z )
ª 1 1º
« 2 z » .
¬
¼
(5.7.22)
Step 2: The polynomial matrix is already in the desired form (5.7.11), with
U( z )
V( z)
ª1 0 º
«0 1 » , i ( z ) 1, k ( z )
¬
¼
2, w( z ) 1, P( z )
z.
Step 3: Using (5.7.12), (5.7.13) and (5.7.14), we obtain
Q( z )
ª 1 º
U 1 ( z )i ( z ) «
»
¬k ( z)¼
i ( z ) ¬ª P ( z ) k ( z ) w( z ) ¼º
that is, P̂ (z) = 1 and
ª1º
1
« 2 » , R ( z ) >1 w( z ) @ V ( z ) [1 1],
¬ ¼
d ( z )Pˆ ( z ) z 2,
Singular and Cyclic Normal Systems
ª 1 º ª 1º
U 1 ( z )i ( z ) «
» « »,
¬ k ( z ) ¼ ¬ 2 ¼
R ( z ) >1 w( z ) @ V 1 ( z ) >1 1@ .
Q( z )
Step 4: Thus the desired structural decomposition of the matrix (5.7.21) is
T( z )
Q( z ) R ( z )
G( z)
d ( z)
ª0 0º
1 ª1º
[1 1] «
«
»
».
z 2 ¬ 2¼
¬0 1 ¼
311
6
Matrix Polynomial Equations, and Rational and
Algebraic Matrix Equations
6.1 Unilateral Polynomial Equations with Two Variables
6.1.1 Computation of Particular Solutions to Polynomial Equations
Consider the following equation
AX BY
C,
(6.1.1)
where A = A(s) lup[s], B = B(s) luq[s], C = C(s) lum[s], X = X(s) pum[s]
and Y = Y(s) qum[s].
Given the matrices A, B and C, compute matrices X and Y satisfying (6.1.1).
The following problem will be called the dual to the above one.
Given the polynomial matrices
A
A( s )  pum [ s ], B
B( s )  qum [ s ], C C( s )  lum [ s ] ,
compute polynomial matrices X = X(s)
equation
XA YB
lup
[s] and Y = Y(s)
C.
luq
[s] satisfying the
(6.1.2)
Using the transpose we can transform (6.1.2) into (6.1.1).
Theorem 6.1.1. Equation (6.1.1) has a solution if and only if one of the following
conditions is met:
1. [A, B, C] and [A, B, 0] are right equivalent matrices,
314
Polynomial and Rational Matrices
2.
a greatest common left divisor (GCLD) of the matrices A and B is a left
divisor of the matrix C.
Proof. Let X0, Y0 be a solution to (6.1.1), that is, AX0 + BY0 = C.
Then
[ A, B, C]
> A,
ª I 0 X0 º
B, AX 0 BY0 @ [ A, B, 0] ««0 I Y0 »» .
«¬ 0 0 I »¼
According to Definition 1.7.1 [A, B, C] and [A, B, 0] are right equivalent matrices,
since
ª I 0 X0 º
«0 I Y »
0»
«
«¬ 0 0 I »¼
(6.1.3)
is a unimodular matrix.
Conversely, if [A, B, C] and [A, B, 0] are right equivalent matrices, then there
exists a unimodular matrix P = P(s) such that
[ A, B, C] [ A, B, 0]P ,
(6.1.4)
where the matrix P is of the form
ª I 0 R1 º
«0 I R » .
2»
«
«¬ 0 0 I »¼
(6.1.5)
From (6.1.4) it follows that AR1 + BR2 = C. Thus the pair R1, R2 constitutes a
solution to (6.1.1).
Now we will show that if (6.1.1) has the solution X0, Y0, then GCLD of the
matrices A and B is a left divisor of the matrix C. Let L be a GCLD of the matrices
A and B, that is,
A
LA1 , B
LB1 ,
(6.1.6)
where A1, B1 are polynomial matrices. Substitution of (6.1.6) into the equation
AX0 BY0
yields
C
(6.1.7)
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
L A1X0 B1Y0
C.
315
(6.1.8)
Thus the matrix L is a left divisor of the matrix C.
Now we will show that if L is a left divisor of C, then (6.1.1) has a solution. By
assumption C = LC1, where C1 is a polynomial matrix. On the other hand, the
assumption that L is a GCLD of A and B implies the existence of polynomial
matrices U11 and U21 such that
AU11 BU 21
L.
(6.1.9)
Post-multiplying (6.1.9) by C1, and taking into account that LC1 = C, we obtain
AU11C1 BU 21C1
LC1
C.
Thus the matrices
X0
U11C1 ,
Y0
U 21C1
(6.1.10)
are a solution to (6.1.1).
Ŷ
Theorem 6.1.2. Equation (6.1.2) has a solution if and only if one of the following
conditions is met:
ªAº
ªAº
«
»
1. « B » and «« B »» are left equivalent matrices,
«¬C »¼
«¬ 0 »¼
2.
a greatest common right divisor (GCRD) of the matrices A and B is a right
divisor of the matrix C.
The proof of this theorem is dual to that of Theorem 6.1.1.
The proof of Theorem 6.1.1 immediately provides us with the following
procedure for computing a particular solution X0, Y0 to (6.1.1).
Procedure 6.1.1.
Step 1: Applying Algorithm 1.15.1 compute a GCLD of A and B, i.e., the matrix L
and the polynomial matrices U11, U21.
Step 2: Compute the matrix C1 satisfying LC1 = C.
Step 3: Using the relationships X0 = U11C1, Y0 = U21C1, compute the desired
particular solution X0, Y0 to (6.1.1).
The procedure for computing the solution to (6.1.2) is analogous (dual).
316
Polynomial and Rational Matrices
Example 6.1.1.
Using Procedure 6.1.1 compute a particular solution to the equation
ª s 2 s º
ª1 s º
X
«
»Y
«0 s »
¬
¼
¬1 s 1¼
ª 0
s 2 s 1º
«
».
s
¬1 s
¼
(6.1.11)
It is easy to verify that the matrices
A
ª1 s º
«0 s » , B
¬
¼
ª s 2 s º
«
», C
¬1 s 1¼
ª 0
s 2 s 1º
«
»
s
¬1 s
¼
satisfy the conditions of Theorem 6.1.1. According to Procedure 6.1.1 we carry out
the following steps
Step 1: In order to compute a GCLD of the matrices A and B we carry out the
elementary operations
P[3 4 u s ], P[2 1 u ( s )], P[4 1 u ( s )], P[2 3 u s ], P[4 3 u (1)], P[2,3]
on the column of the matrix
ª1 s s 2
«
«0 s 1 s
«1 0
0
«
0
«0 1
«0 0
1
«
0
«¬ 0 0
sº
»
1»
0»
»,
0»
0»
»
1 »¼
bringing it to the form
ª1
«0
«
«1
«
«0
«0
«
¬«0
Thus we have
0 0
1 0
0 s
0 1
1
s
s
s2
0
0
s
0
º
»
»
»
».
»
1 »
»
1 s ¼»
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
L
ª1
«0
¬
0º
, U11
1»¼
ª1 0 º
« 0 0 » , U 21
¬
¼
317
ª0 1 º
«0 s » .
¬
¼
Step 2: In this case,
C1
C
ª 0
s 2 s 1º
«
».
s
¬1 s
¼
Step 3: The desired solution to (6.1.11) is of the form
X0
U11C
Y0
U 21C
s 2 s 1º ª0 s 2 s 1º
ª1 0 º ª 0
«
» «
»,
«0 0 »
0
s
¬
¼ ¬1 s
¼ ¬0
¼
2
sº
s s 1º ª 1 s
ª0 1º ª 0
.
» «
«0 s » «
2»
s
¬
¼ ¬1 s
¼ ¬ s (1 s ) s ¼
It follows from the proof of Theorem 6.1.1 that a solution to (6.1.1) can also be
computed using elementary operations on the columns of the block matrix
ªA
«
«I p
«0
¬
B
0
Iq
Cº
»
0»
0 »¼
(6.1.12)
such that they transform it to the form
ªA
«
«I p
«0
¬
B
0
Iq
0 º
»
X» .
Y »¼
(6.1.13)
Indeed, post-multiplication of the first column (block) of the matrix (6.1.12) by
X, and the second one by Y, and addition of the result to the third column of this
matrix yields
ªA
«
«I p
«0
¬
B
0
Iq
C AX BY º
»
X
»
»
Y
¼
and further, with (6.1.1) taken into account, the matrix (6.1.13).
(6.1.14)
318
Polynomial and Rational Matrices
Note that (6.1.1) has many different solutions X and Y, since the transformation
of the matrix (6.1.12) into the form (6.1.13) can be accomplished using different
sequences of elementary operations on the columns of this matrix.
Example 6.1.2.
Consider (6.1.11). We will show that it has also a solution different form that
obtained in Example 6.1.1.
Carrying out the elementary operations:
P[5 1 u ( s 2 )], P[6 1 u ( s 1)], P[5 3 u (1)], P[6 4 u ( s)] ,
we transform the matrix
ªA
«
«I p
«0
¬
B
0
Iq
Cº
»
0»
0 »¼
ª1 s s 2
«
«0 s 1 s
«1 0
0
«
0
«0 1
«0 0
1
«
0
«¬0 0
s
1
0
0
0
1
0
s2
1 s
0
0
0
0
s 1º
»
s
»
»
0
»
0
»
»
0
»
0
»¼
(6.1.15)
to the form
ª1 s s 2
«
«0 s 1 s
«1 0
0
«
0
«0 1
«0 0
1
«
0
«¬0 0
s
1
0
0
0
1
0
0
s2
0
1
0
0 º
»
0 »
( s 1) »
».
0 »
0 »
»
s »¼
A comparison of (6.1.16) to (6.1.13) yields the following result
X1
ªs2
«
¬0
s 1º
» , Y1
0 ¼
ª1 0 º
«0 s » ,
¬
¼
which is a solution to (6.1.11).
In order to obtain the solution
X2
ª0 s 2 s 1º
«
» , Y2
0
¬0
¼
sº
ª 1 s
« s (1 s ) s 2 »
¬
¼
(6.1.16)
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
319
coinciding with that obtained in Example 6.1.1, one has to transform the matrix
(6.1.16) into the form
ª1 s s 2
«
«0 s 1 s
«1 0
0
«
0
1
0
«
«0 0
1
«
0
«¬0 0
s
1
0
0
0
1
º
0
0
»
0
0
»
0
s 2 s 1»
».
0
0
»
»
s
s 1
»
s 2 »¼
s2 s
(6.1.17)
Analogously, a solution to (6.1.2) can be computed by transforming the matrix
ªA I p
«
«B 0
«C 0
¬
0º
»
Iq »
0 »¼
(6.1.18a)
by elementary operations on rows into the form
ªA I p
0 º
«
»
Iq » .
«B 0
« 0 X Y »
¬
¼
(6.1.18b)
6.1.2 Computation of General Solutions to Polynomial Equations
With particular solutions to (6.1.1) and (6.1.2) known, we will seek their general
solutions.
Theorem 6.1.3. If matrices X0, Y0 are a particular solution to (6.1.1), then the
general solution to this equation is of the form
X0 B1T, Y
X
Y0 A1T .
where B1 = B1(s) pu(p+q-n)[s], A1 = A1(s)
polynomial matrices staisfying the equation
AB1
T
(p+q-n)um
(6.1.19)
qu(p+q-n)
[s]
are
right
BA1 ,
[s] is an arbitrary polynomial matrix and rank [A B] = n.
coprime
(6.1.20)
320
Polynomial and Rational Matrices
Proof. By assumption we have (6.1.7). Subtracting sidewise (6.1.7) from (6.1.1),
we obtain
ª X X0 º
A( X X0 ) B(Y Y0 ) [ A B] «
»
¬ Y Y0 ¼
0.
(6.1.21)
In order to compute the general solution to (6.1.1), one has to compute the
general solution to the equation
[ A B]Z
0,
(6.1.22)
where Z (p+q)um[s].
Taking into account the Sylvester inequality
0 d rank [ A B]Z t rank [ A B] rank Z ( p q )
and rank [A B] = n along with (6.1.22), we obtain
rank Z d p q n .
Let
Z
Z1T
ª B1 º
« A »T,
¬ 1¼
(6.1.23)
where Z1 (p+q)u(p+q-n)[s] is a full rank polynomial matrix with its rank equal to
p + q n.
Substituting (6.1.23) into (6.1.22), we obtain
ª B1 º
[ A B] «
»T
¬ A1 ¼
0.
(6.1.24)
The relationship (6.1.24) is equivalent to (6.1.20) for any matrix T.
Thus from (6.1.21)(6.1.23) we have
ª X X0 º
«
»
¬ Y Y0 ¼
ª B1 º
« A »T ,
¬ 1¼
which is the desired general solution (6.1.19).
Ŷ
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
321
Theorem 6.1.4. If matrices X0, Y0 are a particular solution to (6.1.2), then the
general solution to this equation is of the form
X
X0 TB 2 , Y
Y0 TA 2 ,
where B2 = B2(s) (p+q-n)up[s], A2 = A2(s)
matrices satisfying the condition
B2 A
T
lu(p+q-n)
(6.1.25)
(p+q-n)uq
[s] are left coprime polynomial
A 2B ,
(6.1.26)
[s] is an arbitrary polynomial matrix, and
n
ªAº
rank « » .
¬B ¼
The proof of this theorem is analogous (dual) to that of Theorem 6.1.3.
It follows from the considerations in Section 1.15.2 that
A1
U 22 , B1
U12 ,
(6.1.27)
since AU12 = BU22. Substituting (6.1.10) and (6.1.27) into (6.1.19), we obtain the
general solution to (6.1.1)
X
U11C1 U12 T, Y
U 21C1 U 22 T .
(6.1.28)
These relationships yield the following procedure for computing the general
solution to (6.1.1).
Procedure 6.1.2.
Step 1: Applying Algorithm 1.15.1, compute a GCLD of the matrices A and B;
compute the unimodular matrix
ª U11 U12 º
«U
»,
¬ 21 U 22 ¼
(6.1.29)
Step 2: Using the method provided in Section 1.15.2, compute the matrix C1
satisfying the relationship
C
LC1 .
Step 3: Using (6.1.28) compute the desired general solution.
(6.1.30)
322
Polynomial and Rational Matrices
Example 6.1.3.
Using the results from Example 6.1.1, we compute the general solution to (6.1.11).
With L, U11, U21, C1 known (computed in Example 6.1.1) and
U12
ªs s º
« 1 0 » , U 22
¬
¼
1 º
ªs
«s2 1 s» ,
¬
¼
and using (6.1.28), we obtain
X
Y
s 2 s 1º ª s s º ª t11 t12 º
ª1 0 º ª 0
»«
»
«0 0» «
»«
s
¬
¼ ¬1 s
¼ ¬ 1 0 ¼ ¬t21 t22 ¼
ª s (t11 t21 ) s 2 s (1 t12 t22 ) 1º
«
»,
t11
t12
¬
¼
2
1 º ª t11 t12 º
s s 1º ª s
ª 0 1º ª 0
U 21C1 U 22 T «
«
»« 2
»
»
»«
s
¬0 s ¼ ¬1 s
¼ ¬ s 1 s ¼ ¬t21 t22 ¼
U11C1 U12 T
ª s (t11 1) t21 1
« 2
¬ s (t11 1) s (1 t21 ) t21
s 2 s (t12 1) t22 º
»,
s 2 (t12 1) st22 t22 ¼
where t11, t12, t21 and t22 are arbitrary polynomials of the variable s.
6.1.3 Computation of Minimal Degree Solutions to Polynomial Matrix
Equations
Suppose that B1 is a regular polynomial matrix (the matrix of coefficients by the
highest power of the variable s is nonsingular) or a nonsingular one. In this case, if
deg X0 t deg B1, then applying Algorithm 1.15.1 one can compute polynomial
matrices U1 and V1 such that
X0
B1U1 V1 ,
deg V1 deg B1 .
(6.1.31)
Substituting this relationship into the first formula in (6.1.19), we obtain
X
V1 B1 (U1 T) .
Taking T = U1, we obtain a solution to (6.1.1) in the form
X
V1 , Y
Y0 A1U1 .
(6.1.32)
Thus we have the following procedure for computing a minimal degree (with
respect to X) solution to (6.1.1).
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
323
Procedure 6.1.3.
Step 1: Applying Algorithm 1.15.1, compute a GCLD of the matrices A and B
(i.e., the matrix L) and a unimodular matrix of the form (6.1.29).
Step 2: Applying the algorithm given in Sect. 1.15.1, compute the matrix C1
satisfying (6.1.30).
Step 3: Compute X0 = U11C1.
Step 4: If B1 is a regular or nonsingular matrix, then applying Algorithm 1.15.1
compute the matrices U1 and V1 satisfying (6.1.31).
Step 5: Using (6.1.32), compute the desired solution.
Example 6.1.4.
Compute a minimal degree (with respect to X) solution to (6.1.11).
We use the results obtained in Example 6.1.1. In this case,
B1
U12
ª s sº
« 1 0 »
¬
¼
is a nonsingular matrix since det B1 = s, but not a regular one. In order to compute
the matrices U1 and V1 satisfying (6.1.31), we use Algorithm 1.15.1.
We obtain
Adj B1X0
ª0 s º ª0 s 2 s 1º
»
«
»«
0
¬1 s ¼ ¬ 0
¼
0
ª0
º
« 0 s 2 s 1»
¬
¼
and then we divide every its entry by det B1 = s
Adj B1X0
U1 det B1 R
ª0
«0
¬
0 º
ª0
s«
»
s 1¼
¬0
0º
,
1 »¼
that is,
U1
ª0
«0
¬
0 º
, R
s 1»¼
V1
1
B1R
det B1
ª0
«0
¬
0º
.
1»¼
Hence
1ª s
s «¬ 1
s º ª0 0º
0 »¼ «¬0 1 »¼
According to (6.1.32) the desired solution is
ª0 1 º
«0 0» .
¬
¼
324
Polynomial and Rational Matrices
ª0 1º
«0 0 » ,
¬
¼
Y0 A1U1
X
V1
Y
(6.1.33)
1 º ª 0
sº ªs
0 º
ª 1 s
« s (1 s ) s 2 » « s 2 1 s » « 0 s 1»
¬
¼ ¬
¼¬
¼
ª 1 s
« s (1 s )
¬
1º
.
1»¼
Note that a minimal degree solution (with respect to X) to (6.1.1) can be also
computed by transforming it to the form (6.1.13) and carrying out elementary
operations on the columns of the first of these matrices that yield the minimal
degree of X.
Example 6.1.5.
In order to compute the minimal degree (with respect to X) solution (6.1.33) to
(6.1.11), we carry out the following elementary operations on the matrix (6.1.15)
L[6 1 u (1)], L[5 3 u (1)], L[6 3 u 1], L[5 4 u ( s 2 s )], L[6 4 u (1)].
Then matrix (6.1.15) becomes
ª1 s s 2
«
«0 s 1 s
«1 0
0
«
0
1
0
«
«0 0
1
«
0
¬«0 0
s
1
0
0
0
1
0
0º
»
0
0»
0
1»
».
0
0»
s 1 1 »
»
s 2 s 1¼»
(6.1.34)
The comparison of the matrices (6.1.34) and (6.1.13) yields the desired solution
(6.1.33).
If A1 is a regular or nonsingular matrix, then there exist polynomial matrices U2
and V2 such that
Y0
A1U 2 V2 , deg V2 deg A1 .
Substituting this relationship into the second formula in (6.1.19), we obtain
Y
A1 (U 2 T) V2 .
Taking T = U2 we obtain a minimal degree solution (with respect to Y), which
is of the form
X
X0 B1U 2 , Y
V2 .
(6.1.35)
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
325
The procedure for computing a minimal degree solution (with respect to Y)
(6.1.35) to (6.1.1) is analogous to Procedure 6.1.3.
In order to compute a minimal degree solution with respect to
ª Xº
«Y » ,
¬ ¼
we exploit the freedom of choice of the matrix T in the solution (6.1.19). We
choose the matrix in such a way that the degree of [X Y]T is minimal.
Writing (6.1.19) in the form
ªXº
«Y »
¬ ¼
ª X0 º ª B1 º
« »«
»T ,
¬ Y0 ¼ ¬ A1 ¼
and applying elementary operations on columns, we can choose the entries of T in
such a way that the degrees of X and Y are minimal. From (6.1.1) written in the
form
ªXº
[ A, B] « »
¬Y¼
C
it follows that the minimal degree of the matrix [X Y]T cannot be less than the
difference of the degrees of the matrices C and [A, B].
6.2 Bilateral Polynomial Matrix Equations with Two Unknowns
6.2.1 Existence of Solutions
Consider the equation
AX YB
C,
(6.2.1)
where A = A(s) lup[s], B = B(s) qum[s], C = C(s) lum[s], X = X(s) pum[s]
and Y = Y(s) luq[s].
With A, B and C known, one has to compute polynomial matrices X and Y
satisfying (6.2.1).
Theorem 6.2.1. Equation (6.2.1) has a solution if and only if the matrices
ª A 0º ª A Cº
«0 B » , «0 B »
¬
¼ ¬
¼
(6.2.2)
326
Polynomial and Rational Matrices
are equivalent.
Proof. We will show that if (6.2.1) has a solution X0, Y0, then the matrices (6.2.2)
are equivalent. Substituting C = AX0 + Y0B into the second of the matrices of
(6.2.2), we obtain
ª A Cº
«0 B »
¬
¼
ª A AX0 Y0 B º
«
»
B
¬0
¼
ªIl
«0
¬
Y0 º ª A 0 º ªI p
I q »¼ «¬ 0 B »¼ «¬ 0
X0 º
».
Im ¼
(6.2.3)
The matrices (6.2.2) are equivalent since
ªIl
«0
¬
Y0 º ª I p
,
I q »¼ «¬ 0
X0 º
»
Im ¼
(6.2.4)
are unimodular matrices.
Now we will show that if the matrices (6.2.2) are equivalent, then (6.2.1) has a
solution.
Let
AS
U1 A AU 2 A
diag > a1 , a2 , ..., ar , 0, ..., 0@ ,
BS
U1B BU 2 B
diag >b1 , b2 , ..., bs , 0, ..., 0@
(6.2.5)
be the Smith canonical forms of the matrices A and B, where a1, a2, ..., ar;
b1,b2,…,bs are the invariant polynomials of A and B, respectively, and U1A, U2A,
U1B, U2B are unimodular matrices of elementary operations on rows and columns.
Pre-multiplying (6.2.1) by U1A and post-multiplying it by U2B, we obtain
U1 A AU 2 A U 21A XU 2 B U1 A YU1B1 U1B BU 2 B
U1 ACU 2 B ,
and with (6.2.5) taken into account
A S X YB S
C,
(6.2.6)
where
X
U 21A XU 2 B
ª x11
«x
« 21
« #
«
¬« x p1
x12
x22
#
xp2
x1m º
" x2 m »»
,
% # »
»
" x pm ¼»
"
(6.2.7)
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
ª y11
«y
« 21
« #
«
¬« yl1
U1 A YU 21B
Y
C
y12 "
y22
#
yl 2
ª c11 c12
«c
« 21 c22
«#
#
«
¬ cl1 cl 2
U1 ACU 2 B
y1q º
" y2 q »»
,
% # »
»
" ylq ¼»
327
(6.2.8)
" c1m º
" c2 m »»
.
% # »
»
" clm ¼
(6.2.9)
With (6.2.5)( 6.2.9) in mind we can write (6.2.6) in the form
ª a1
«0
«
«#
«
«0
«0
«
«#
«0
¬
0
"
0
a2 "
0
ª y11
«y
21
«
« #
«
¬« yl1
#
0
% #
" ar
0
#
"
%
0
#
0
"
0
y12 "
y22 "
# %
yl 2 "
ª c11 c12
«c
« 21 c22
«#
#
«
¬ cl1 cl 2
0 " 0º
0 " 0 »»
ª x11
# % #» «
» x21
0 " 0» «
« #
0 " 0» «
» « x p1
# % #» ¬
0 " 0 »¼
ªb1 0 "
«0 b "
2
y1q º «
«# # %
»
y2 q » «
0 0 "
# »«
«
» 0 0 "
ylq ¼» «
«# # %
«0 0 "
¬
x12
x22
#
xp2
0
0
#
bs
0
#
0
x1m º
" x2 m »»
% # »
»
" x pm »¼
"
0 " 0º
0 " 0 »»
# % #»
»
0 " 0»
0 " 0»
»
# % #»
0 " 0 »¼
(6.2.10)
" c1m º
" c2 m »»
.
% # »
»
" clm ¼
Carrying out the multiplication and comparing appropriate entries, we obtain
ai xij b j yij
cij for i 1, 2, ..., r; j 1, 2, ..., s,
ai xij
cij for i 1, 2, ..., r ; j
b j yij
cij for i
cij
0 for i
s 1, s 2, ..., m,
r 1, r 2, ..., l ; j 1, 2, ..., s,
r 1, r 2, ..., l ; j
s 1, s 2, ..., m.
(6.2.11)
328
Polynomial and Rational Matrices
Thus (6.2.1) has a solution if the equations in (6.2.11) have solutions.
It is easy to show that these equations have solutions if the matrices (6.2.2) are
equivalent. Note that the matrices
ª U1 A
« 0
¬
ª U1 A
« 0
¬
0 º ª A 0 º ªU2 A
U1B »¼ «¬ 0 B »¼ «¬ 0
0 º ª A Cº ªU 2 A
U1B »¼ «¬ 0 B »¼ «¬ 0
0 º
U 2 B »¼
0 º
U 2 B »¼
ªAS
« 0
¬
ªAS
«
¬ 0
0 º
,
B S »¼
(6.2.12)
Cº
»
BS ¼
(6.2.13)
are equivalent if and only if there exist x ij, y ij satisfying the equations in (6.2.11),
since in this case, carrying out elementary operations on rows and columns one can
transform the matrix (6.2.1) into the form (6.2.12).
Thus we have shown that (6.2.1) has a solution if and only if the matrices
(6.2.2) are equivalent.
6.2.2 Computation of Solutions
First, we introduce a method of computing a particular solution X0, Y0 to (6.2.1),
which is based on elementary operations.
Pre-multiplying (6.2.3) by
ªI l
«0
¬
Y0 º
I q »¼
and post-multiplying it by
ªI p
«0
¬
X0 º
,
I m »¼
we obtain
ªIl
«0
¬
Y0 º ª A C º ªI p
I q »¼ «¬ 0 B »¼ «¬ 0
ªIl
«0
¬
Y0 º ªI l
I q ¼» ¬« 0
X0 º
I m »¼
ªA 0 º
« 0 B» ,
¬
¼
(6.2.14)
since
Y0 º
I q ¼»
ªI l
«0
¬
0º
,
I q ¼»
ªI p
«0
¬
X0 º ªI p
I m »¼ «¬ 0
X0 º
I m »¼
ªI p
«0
¬
0º
.
I m »¼
From (6.2.14) it follows that a particular solution X0, Y to (6.2.1) can be
computed by adding such a combination of the rows of B and such a combination
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
329
of the columns of A to the matrix C, as to replace the matrix C by the zero matrix
in the second of the matrices in (6.2.2). One can accomplish this in many ways.
Thus (6.2.1) has many different particular solutions. We will illustrate this in the
following simple example.
Example 6.2.1.
Compute two different particular solutions to (6.2.1) for the matrices
A
ª1 s 0 º
«0 s s 2 » , B
¬
¼
>1
s@ , C
ª s
«s 2
¬
0 º
.
2s 2 »¼
(6.2.15)
It is easy to verify that the matrices
ªA 0 º
« 0 B»
¬
¼
ª1 s 0
«0 s s 2
«
«¬0 0 0
0
0º
ª A Cº
0 0 »» , «
B 0 »¼
1 s »¼ ¬
ª1 s 0
«0 s s 2
«
«¬ 0 0 0
s
s
1
2
0 º
2 s 2 »»
s »¼
are equivalent. Thus for the matrices (6.2.15) the equation (6.2.1) has a solution.
Particular solutions to (6.2.1) for the matrices (6.2.15) computed in the above way
are as follows:
X0
ª s 2 s 2s 2 º
«
»
2 s » , Y0
« s
« 0
0 »¼
¬
X0
ª s 2 1 3s 2 s º
«
»
3s » , Y0
« s 1
« 0
0 »¼
¬
ª0º
«0»
¬ ¼
(6.2.16a)
and
ª1º
«s» ,
¬ ¼
(6.2.16b)
since according to (6.2.14) the following equations are satisfied
ª1 0 0 º ª1 s 0
«0 1 0» «0 s s 2
«
»«
«¬0 0 1 »¼ «¬ 0 0 0
ª1 s 0
«0 s s 2
«
¬«0 0 0
s
s2
1
s
s 2
1
0 º
2s 2 »»
s ¼»
ª1
«
0 º «0
2»
2s » «0
«
s »¼ «0
«0
¬
0
1
0
0
0
0 s 2 s 2 s 2 º
»
2 s »
s
0
1
0
0 »
»
0
1
0 »
0
0
1 »¼
330
Polynomial and Rational Matrices
and
ª1 0 1º ª1 s 0
«
»«
2
«0 1 s » «0 s s
«¬ 0 0 1 »¼ «¬ 0 0 0
ª1 s 0
«0 s s 2
«
«¬0 0 0
s
s
1
s
s 2
1
ª1
«
0 º «0
2s 2 »» «0
«
s »¼ «0
«0
¬
0 s 2 1 3s 2 s º
»
0 s 1
3s »
1
0
0 »
»
0
1
0 »
0 0
0
1 »¼
0
1
0
0
0 º
2 s 2 »» .
s »¼
2
The proof of Theorem 6.2.1 provides us with the following procedure for
computing the general solution X, Y to (6.2.1).
Procedure 6.2.1.
Step 1: Applying the algorithm introduced in Section 1.7.1, compute the Smith
canonical forms of the matrices A and B. Compute the corresponding
unimodular matrices U1A,U2AU1B, and U2B,.
Step 2: From (6.2.9) compute the matrix C .
Step 3: Write the equations (6.2.11) and compute their solutions.
Step 4: Compute the desired solution
X
U 2 A XU 21B , Y
U1A1 YU1B .
(6.2.17)
Example 6.2.2
Using Procedure 6.2.1 compute a solution of (6.2.1) for the matrices (6.2.15).
Applying Procedure 6.2.1, we compute
Step 1:
AS
U1 A AU 2 A
BS
U1B BU 2 B
ª1 s s 2 º
0 º ª1 s 0 º «
»
0 1 s »
»
«
2»«
1 ¼ ¬0 s s ¼
«0 0 1 »
¬
¼
ª1 s º
[1][1 s ] «
» [1 0]
¬0 1¼
ª1
«0
¬
ª1
«0
¬
0 0º
,
s 0 »¼
and the unimodular matrices
U1 A
ª1 0 º
«0 1» , U 2 A
¬
¼
ª1 s s 2 º
«
»
«0 1 s » , U1B
«0 0 1 »
¬
¼
[1], U 2 B
ª1 s º
« 0 1» .
¬
¼
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
331
Step 2: From (6.2.9) we obtain
C
0º ª s
1 ¼» ¬« s 2
ª1
«0
¬
U1 ACU 2 B
sº
1 ¼»
0 º ª1
2s 2 ¼» ¬«0
ª s
« 2
¬s
s2
º
».
2s s ¼
2
3
Step 3: In this case, (6.2.6) takes the form
ª x11
ª1 0 0 º «
«0 s 0 » « x21
¬
¼ «x
¬ 31
x12 º
ª y11 º
x22 »» « » [1
y
x32 »¼ ¬ 21 ¼
0]
s 2 , sx21 y21
s 2 , sx22
s2 º
»
2s 2 s3 ¼
ª s
« 2
¬s
which yields
x11 y11
s, x12
2s 2 s 3 .
Solving these equations for
y11
y1 ,
sy 2 ,
y21
we obtain
x11
s y1 , x12
s 2 , x21
s y2 ,
ª s y1
«
« s y2
«¬ x31
s2
2s s 2 .
x22
Step 4: Therefore,
X
ª x11
«x
« 21
¬« x31
x12 º
x22 »»
x32 ¼»
º
»
2s s » ,
x32 »¼
2
Y
ª y11 º
«y »
¬ 21 ¼
ª y1 º
« sy » ,
¬ 2¼
where y1, y2, x 31 and x 32 are arbitrary polynomials in the variable s. Thus
according to (6.2.17) the desired solution takes the form
X
U 2 A XU
1
2B
ª1 s s 2 º ª s y1
«
»«
« 0 1 s » « s y2
«0 0 1 » « x31
¬
¼¬
ª s 2 (1 x31 ) s (1 y2 ) y1
«
s(1 x31 ) y2
«
«
x31
¬
1
Y
U1A1 YU1B
s2
º
1
» ª1 s º
2s s » «
0 1»¼
x32 »¼ ¬
2
s 3 x31 s 2 (2 x32 y2 ) sy1 º
»
s 2 x31 s (2 x32 ) y2 » ,
»
sx31 x32
¼
ª1 0 º ª y1 º
« 0 1» « sy » [1]
¬
¼ ¬ 21 ¼
ª y1 º
« sy » .
¬ 2¼
(6.2.18)
332
Polynomial and Rational Matrices
Substituting
y1
y2
x31
x32
0
into (6.2.18), we obtain the particular solution (6.2.16a) and substituting y1 = y2= 1
along with x 31 = x 32 = 0, the particular solution (6.2.16b).
6.3 Rational Solutions to Polynomial Matrix Equations
6.3.1 Computation of Rational Solutions
Consider a polynomial matrix equation of the form (6.1.1). A pair of rational
matrices
X
X( s )  pum ( s ), Y
Y( s )  qum ( s )
satisfying this equation will be called its solution.
Theorem 6.3.1. If
rank [ A, B] l ,
(6.3.1)
then a rational solution to (6.1.1) has the form
1
X
AT ª¬ AAT BBT º¼ C B1T,
Y
BT ª¬ AAT BBT º¼ C A1T,
1
(6.3.2)
where the matrices A1 and B1 satisfy the condition
AB1
(6.3.3)
BA1
and T is an arbitrary rational or polynomial matrix.
Proof. If the condition (6.3.1) is satisfied, then the matrix
ª AT º
[ A, B ] « T »
¬«B ¼»
AAT BBT
is nonsingular. Substituting (6.3.2) into (6.1.1) and taking into account the
condition (6.3.3), we obtain
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
AX BY
1
ª¬ AAT BBT º¼ ª¬ AAT BBT º¼ C > BA1 AB1 @ T
333
C
for an arbitrary matrix T.
Note that if we choose the matrix T in (6.3.2) appropriately, we can obtain in
some cases a polynomial solution to (6.1.1). It is easy to show that if a GCLD of
the matrices A and B is a LD of C, then there exists a matrix T such that X and Y
determined by (6.3.2) are polynomial matrices.
6.3.2 Existence of Rational Solutions of Polynomial Matrix Equations
Consider the polynomial matrix equation
B,
XA
(6.3.4)
where
A
A( s )  muk [ s ], B
B( s )  puk [ s ]
are given and X = X(s) pum(s) is the matrix we seek.
We seek a solution X that is a proper rational matrix, i.e., satisfying the
condition
lim X( s )
s of
K,
(6.3.5)
where K pum is a nonzero matrix.
Let rank A = k d m. Applying elementary operations on columns, we transform
the matrix
ªAº
«B »
¬ ¼
so that Ac in
ª Ac º
« Bc »
¬ ¼
is a column-reduced matrix, i.e., the matrix of coefficients by the highest degrees
of its column has full column rank.
Theorem 6.3.2. There exists a rational solution X to (6.3.4) if and only if
334
Polynomial and Rational Matrices
deg ki Ac t deg ki Bc, i 1, ..., k ,
(6.3.6)
where deg kl Ac denotes the degree of the i-th column of the matrix Ac.
Proof. If the condition (6.3.6) is satisfied, then we can choose k rows from Ac in
such a way that the matrix of coefficients by the highest column degrees of this
minor M is nonsingular. Without loss of generality one can assume
M
> I k 0@ Ac .
(6.3.7)
This minor has the same column degrees as the matrix and it is columnreduced.
It is easy to verify that the matrix
X
BcM 1 > I k 0@
(6.3.8)
is a proper rational solution to the equation XAc = Bc, hence also to (6.3.4), since
elementary operations on the columns of
ªAº
«B »
¬ ¼
do not change a solution to (6.3.4).
Let Ai (Bi) be the i-th column of A (B). From (6.3.4) we have
XA i
(6.3.9)
B i , i 1, ..., k .
If X is a rational proper solution, then the condition (6.3.6) is satisfied. Using the transpose, we can transform the polynomial matrix equation
AX
(6.3.10)
B
into the form (6.3.4), where
X
XT , A
AT and B
BT .
6.3.3 Computation of Rational Solutions to Polynomial Matrix Equations
From the proof of Theorem 6.3.2 we obtain the following procedure for computing
a proper rational solution X of polynomial matrix equation (6.3.4) (a solution that
satisfies condition (6.3.6)).
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
335
Procedure 6.3.1.
Step 1: Applying elementary operations on columns, transform the matrix [A B]T
in such a way that the matrix of coefficients by the highest column degrees
of the minor M = [Ik 0]Ac consisting of the k first row of Ac in [Ac Bc]T is
nonsingular.
Step 2: Using (6.3.8) compute the desired solution.
Example 6.3.1.
Compute a proper rational solution X to (6.3.10) for
ªs
«
¬1
A
2s
s
2
s 1º
», B
3s ¼
ª s º
« s 2» .
¬
¼
With the above matrices transposed, we obtain
A
A
T
1º
ª s
«
»
s2 » , Ǻ
«2s
« s 1 3s »
¬
¼
BT
>s
s 2@ ,
A is a column-reduced matrix, since the matrix of coefficients by the highest
column degrees is
ª1 0 º
«2 1 » ,
«
»
«¬1 0 »¼
thus a full column rank matrix.
Hence A = Ac and B = Bc. It is easy to verify that in this case, the condition
(6.3.6) is satisfied and the equation has a proper rational solution. Applying the
Procedure 6.3.1, we compute the following.
Step 1: In this case,
M
> I 2 0@ A
ª s 1º
.
«
2»
¬ 2s s ¼
Step 2: From (6.3.8) we obtain
1
X
ª s 1 º ª1
BM 1 > I 2 0@ [ s s 2] «
«
2»
¬ 2s s ¼ ¬0
1
ª s 3 2 s 2 4s s 2 s 0 º¼ .
s3 2s ¬
0 0º
1 0 »¼
336
Polynomial and Rational Matrices
Thus the desired solution is
ª s 3 2s 2 4s º
»
1 «
2
« s s ».
3
s 2s «
»
0
¬
¼
XT
X
6.4 Polynomial Matrix Equations
6.4.1 Existence of Solutions
Consider the equations
A m X1m A m1X1m1 " A1X1 A 0
m
2
X Am X
m 1
2
A m1 " X 2 A1 A 0
0,
(6.4.1)
0,
(6.4.2)
where Am,Am-1,…,A0, X1 and X2 are square matrices of size n.
With Am,Am-1,…,A0 given, we can compute the matrices X1 and X2 satisfying
(6.4.1) and (6.4.2).
Theorem 6.4.1. Every solution X1 to the matrix equation (6.3.1) satisfies the scalar
equation
0,
w( X1 )
(6.4.3)
and every solution X2 of the matrix equation (6.4.2) satisfies the scalar equation
0,
w( X 2 )
(6.4.4)
where
w(O )
det ª¬ A m O m A m1O m1 " A1O A 0 º¼ .
(6.4.5)
Proof. Using the polynomial matrix
W (O )
A m O m A m1O m1 " A1O A 0 ,
(6.4.6)
we can write (6.4.1) and (6.4.2) in the form
Wp ( X1 )
0,
Wl ( X 2 )
0,
(6.4.7)
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
337
where Wp(X), Wl(X) are left and right value of W(O) respectively (with X
substituted in place of O). According to the generalised Bezoute theorem the
polynomial matrix (6.4.6) is right divisible without remainder by OIn – X1, and left
divisible by OIn – X2, if X1 and X2 are solutions to (6.4.1) and (6.4.2), that is,
Q1 (O ) > O I n X1 @
W (O )
> O I n X 2 @ Q 2 (O ) ,
(6.4.8)
where Q1(O)and Q2(O) are polynomial matrices. Hence
det Q1 (O )M1 (O ) M 2 (O ) det Q 2 (O ) ,
det W (O )
w(O )
(6.4.9)
where
M1 (O ) det > O I n X1 @ , M 2 (O ) det > O I n X 2 @ .
By the Cayley–Hamilton theorem we have
M1 ( X1 ) 0, M 2 ( X 2 ) 0 .
(6.4.10)
From (6.4.10) and (6.4.9) we obtain (6.4.3) and (6.4.4).
Ŷ
6.4.2 Computation of Solutions
Let O1,O2,…,OJ be the roots of the equation w(O) = 0 with multiplicities p1,p2,…,pJ,
respectively, that is
w(O )
O O1
p1
O O2
p2
" O Or
pr
.
(6.4.11)
From (6.4.3) and (6.4.4) it follows that this polynomial is a zeroing polynomial
of X1 and X2, hence it is divisible without remainder by the minimal polynomial of
X1 (or X2).
Thus the minimal polynomial of X1 (X2) is of the form
\ (O )
O O1
m1
O O2
m2
mr
" O Or
,
(6.4.12)
where mi d pi for i = 1,2,…,r, and the elementary divisors of this matrix X1 (X2) are
O Oi
1
where
qi1
, O Oi2
qi2
, " , O Ois
qis
,
(6.4.13)
338
Polynomial and Rational Matrices
i j  {1, 2, ..., r}, qi j d mij , for j 1, 2, ..., and
s
¦q
j 1
ij
n.
According to the considerations in Section 1.10, the matrix X1 has the form
X1
T1X1J T11 ,
(6.4.14)
where T1 is a similarity transformation matrix (and a nonsingular one), and X1J the
Jordan canonical matrix built from the blocks (1.11.10) corresponding to the
elementary divisors (6.4.13). Substituting (6.4.14) into (6.4.1) and taking into
account
X1i
T1X1i J T11 , i 1, 2, ..., m ,
we obtain
A m T1X1mJ T11 A m1T1X1mJ1T11 " A1T1X1J T11 A 0
0,
and post-multiplying this equation by T1, we have
A m T1X1mJ A m T1X1mJ1 " A1T1X1J A 0 T1
0.
(6.4.15)
Substituting
X2
T21X 2 J T2
(6.4.16)
into (6.4.2) and taking into account that
Xi2
T21Xi2 J T2 , i 1, 2, ..., m ,
we obtain
T21X m2 J T2 A m T21X m2 J1T2 A m1 ... T21X 2 J T2 A1 A 0
0,
and pre-multiplying this equation by T2, we have
X 2mJ T2 A m X 2mJ1T2 A m1 ... X 2 J T2 A1 T2 A 0
0.
(6.4.17)
With (6.4.13) known, we can compute X1J (X2J ).
According to (6.4.14) (and (6.4.16)), finding the desired matrix X1 (X2) has
been reduced to the finding of the matrix T1 (T2) by solving (6.4.15) (and (6.4.17)).
The foregoing considerations provide us with the following procedure for
computing the solution X1 (X2) to (6.4.1) (and (6.4.2)).
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
339
Procedure 6.4.1.
Step 1: Using (6.4.5) compute the polynomial w(O).
Step 2: Solving w(O) = 0 compute O1,O2,…,OJ and their multiplicities p1,p2,…,pO.
Step 3: Choose the elementary divisors of X1 (X2) and compute the Jordan
canonical form X1J (X2J).
Step 4: Solving (6.4.15) (or (6.4.17)) compute the matrix T1 (T2).
Step 5: Using (6.4.14) (or (6.4.16)) compute the desired solution X1 (X2).
Example 6.4.1.
Applying Procedure 6.4.1, compute a solution X to the equation
ª1 1 º 2 ª 0
X «
«0
1»¼
¬
¬ 1
1º
7º
ª 1
X«
»
»
1 ¼
¬ 1 1¼
0.
In this case,
A2
ª1 1 º
, A1
«0
1»¼
¬
ª 0
« 1
¬
1º
, A0
1 »¼
7º
ª 1
« 1 1» .
¬
¼
According to Procedure 6.4.1, we compute
Step 1:
w(O )
det ª¬ A2 O 2 A1O A0 º¼
O2 1
O2 O 7
O 1 O 2 O 1
(O 1) O 2 O 2 O 3 .
Step 2: It is easy to verify that the roots of the equation
(O 1)(O 2)(O 2 O 3)
0
are as follows
O1 1, O2
2, O3
1
1 j 11 , O4
2
1
1 j 11 .
2
Step 3: As elementary divisors of X we take (O 1) and (O 2) .
Thus
XJ
0º
ª1
«0 2» .
¬
¼
Step 4: In this case, (6.4.15) is of the form
340
Polynomial and Rational Matrices
ª1 1º ªt11 t12 º ª1 0 º
« 0 1 » «t t » « 0 4 »
¬
¼ ¬ 21 22 ¼ ¬
¼
t12 º ª1
0 º ª 1 1 º ªt11 t12 º
0.
t22 »¼ «¬ 0 2 »¼ «¬ 1 1»¼ «¬t21 t22 »¼
A 2 TX 2J A1TX J A 0 T
ª 0 1º ªt11
«
»«
¬ 1 1¼ ¬t21
With the operations of multiplication and addition carried out and the entries of
the resulting matrix set to zero, we obtain
7t21
0,
3t12 t22
Taking t11 1, t12
T
ª t11 t12 º
«t
»
¬ 21 t22 ¼
0,
t21
0,
t22 3t12
0.
1 , we obtain from the above equations
ª1 1 º
.
«0
3»¼
¬
Step 5: From (6.4.14), we obtain the desired solution
X
TX J T
1
0 º ª1
ª1 1 º ª1
«0
» « 0 2 » «1
3
¬
¼¬
¼¬
1º
3 »¼
1
ª1
«0
¬
1 º
.
2 »¼
6.5 The Kronecker Product and Its Applications
6.5.1 The Kronecker Product of Matrices and Its Properties
Definition 6.5.1. The Kronecker product A B of the matrices A = [aij]
B = [bij] puq is a block matrix of the form
A
B
ª a11B a12 B
«a B a B
22
« 21
« #
#
«
¬ am1B am 2 B
" a1n B º
" a2 n B »»
 mpunq .
%
# »
»
" amn B ¼
For instance, the Kronecker product A B of the matrices
A
ª1 0 1º
,
«2 1
3 »¼
¬
B
ª 2 1º
« 1
2 »¼
¬
mun
and
(6.5.1)
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
341
is
A
ª 2 1 0 0 2 1 º
« 1 2 0 0 1 2 »
«
».
« 4 2 2 1 6 3»
«
»
¬ 2 4 1 2 3 6 ¼
B
Note that usually A B z B A.
Directly from Definition 6.5.1 we have the following properties of the
Kronecker product
1.
O A B A OB O A B ,
2.
AB
C
A
C,
CB
3. A (B C) A B A C ,
4. A (B C) ( A B) C ,
5. ( A B)T AT BT ,
where O is a scalar and A, B, C are matrices.
Theorem 6.5.1. If A=[aij] mum, C = [cij]
then the following equality holds
(A
B)(C
D)
mum
, and B = [bij]
BD .
AC
nun
, D=[dij]
nun
,
(6.5.2)
Proof. Note that the entry ekv of the matrix E = A B is equal to ekv= arubst, where
k
(r 1)n s, v
(u 1)n t ,
r , u 1, 2, ..., m; s, t
1, 2, ..., n .
Analogously, the entry fvl of the matrix F = C D is equal to fvl = cuidtj, where
l = (i – 1)n + j, i= 1,2,…,m; j = 1,2,…,n. Hence the entry gkl of the matrix
G = (A B)(C D) is equal to
mn
g kl
mn
¦e
f
kv vl
v 1
¦a
b c dtj .
ru st ui
v 1
Since v = (u – 1)n + t, we have
m
g kl
¦a
n
c
ru ui
u 1
Now, note that
¦b d
st
t 1
tj
.
(6.5.3)
342
Polynomial and Rational Matrices
m
¦a
c
ru ui
u 1
is the (r, i) entry, i.e., the entry placed in the r-th row and the i-th column of the
matrix AC, and
n
¦b d
st
tj
t 1
is the (s, j) entry of the matrix BD. Hence the expression (6.5.3) is the (k, l) entry
of the matrix AC BD. mum
Theorem 6.5.2. Let A
A
B
A
det > A
B@
In
, B
. Then
B ,
Im
det A
nun
n
(6.5.4)
det B
m
.
(6.5.5)
If A and B are nonsingular matrices, then
A
B
1
A 1
B 1 .
(6.5.6)
Proof. Equation (6.5.4) can be obtained from (6.5.2) for C = Im and B = In. From
(6.5.4) we have
det > A
B@
det > A
I n @ det > I m
B@ ,
In @
ªA 0 " 0 º
«0 A " 0»
»
det «
«# # % #»
«
»
¬ 0 0 " A¼
and with
det > A
det A
n
taken into account, we obtain (6.5.5).
From (6.5.2) for C = A-1 and D = B-1, we obtain
A
B A 1
B 1
Im
In
I mn .
Pre-multiplying the above equality by (A B)-1, we obtain (6.5.6).
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
343
6.5.2 Applications of the Kronecker Product to the Formulation of Matrix
Equations
Consider the equation
C,
AXB
where A 
mun
(6.5.7)
qup
,B
,C
mup
are given and X 
nuq
is the unknown.
Theorem 6.5.3. Equation (6.5.7) is equivalent to
ª¬ A
BT º¼ x
c,
(6.5.8)
where
x
> x1 ,
T
x2 , ..., xn @ , c
>c1 ,
T
c2 , ..., cm @
and xi along with ci are the i-th rows of X and C, respectively.
Proof. From (6.5.7) for the entry cij in , we have
cij
n
¦a
ai Xb j
ik
i 1, ..., m; j 1, ..., p ,
xk b j ,
(6.5.9)
k 1
where ai is the i-th row of A, bj the j-th column of B, and aij is the (i, j) entry of A.
From Definition 6.5.1 and (6.5.8) we have
cij
ai
bTj x
n
¦a
ik
xk b j ,
i 1, ..., m; j 1, ..., p .
(6.5.10)
k 1
From a comparison of (6.5.9) to (6.5.10) it follows that (6.5.7) and (6.5.8) are
equivalent.
Taking in (6.5.7) B = Iq (p = q) and A = In (m = n), we obtain from Theorem
6.5.3 the following two important corollaries.
Corollary 6.5.1. The equation
AX
C,
A  mun ,
is equivalent to the equation
C  muq
(6.5.11)
344
Polynomial and Rational Matrices
A
c.
Iq x
(6.5.12)
Corollary 6.5.2. The equation
XB
C,
B  qu p ,
C  nuq
(6.5.13)
is equivalent to the equation
BT x
In
c.
(6.5.14)
For instance, using (6.5.12) one can write the system of liner equations
ª a11 a12 a13 º ª x11
«
»«
« a21 a22 a23 » « x21
«¬ a31 a32 a33 »¼ «¬ x31
x12 º
x22 »»
x32 »¼
ªb11 b12 º
«b
»
« 21 b22 »
«¬b31 b32 »¼
in the following form
ª a11
«
«0
« a21
«
«0
«a
« 31
«¬ 0
0
a12
0 a13
0 º ª x11 º
»
a11 0 a12 0 a13 » «« x12 »»
0 a22 0 a23 0 » « x21 »
»« »
a21 0 a22 0 a23 » « x22 »
0 a32 0 a33 0 » « x31 »
»« »
a31 0 a32 0 a33 »¼ «¬ x32 »¼
ªb11 º
«b »
« 12 »
«b21 »
« ».
«b22 »
«b »
« 31 »
«¬b32 »¼
Consider the matrix equation
A1XB1 A 2 XB 2 ! A k XB k
C,
(6.5.15)
where Aj, Bj, j = 1,…,k, C and X are square matrices of the same size n.
From the rows x1,x2,…,xn of X and the rows c1,c2,…,cn of C we build the n2dimensional vectors
x
T
> x1 , x2 , ..., xn @
, c
T
>c1 , c2 , ..., cn @
.
With AjXBj, j = 1,…,k written in the equivalent form [Aj BjT]x for j = 1,…,k,
we can write (6.5.15) as
Dx
c,
(6.5.16)
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
345
where
D
B1T A 2
A1
B 2T ! A k
Bk T .
(6.5.17)
Now consider the matrix equation
C,
AX XB
where A =
Let
x
mun
(6.5.18)
mum
, B =
T
> x1 , x2 , ..., xn @
, C =
, c
num
are given and X =
T
>c1 , c2 , ..., cn @
num
is the unknown.
,
where xi and ci are the i-th rows of X and C, respectively. Using (6.5.12) and
(6.5.14) we can subordinate the vector (A Im)x to AX, and the vector (In BT)x to
XB. Thus we can write (6.5.18) in the form
A
BT x
Im In
c.
(6.5.19)
6.5.3 Eigenvalues of Matrix Polynomials
Consider a polynomial of degree p of two independent variables x and y with
complex coefficients cij of the following form
p
w( x, y )
i
¦c x y
ij
j
,
(6.5.20)
i, j 0
Let A and B be square matrices of sizes m and n, respectively, with their entries
being either real or complex.
Consider a square matrix of size mn, given by the formula
p
w( A, B)
¦c A
ij
i
Bj ,
(6.5.21)
i, j 0
where Ai Bj is the Kronecker product of Ai and Bj (see Definition 6.5.1).
Theorem 6.5.4. If O1,O2,…,Om are the eigenvalues of A, and P1,P2,…,Pn are the
eigenvalues of B, then w(Oi, Pj) for i = 1,2,…,m; j = 1,2,…,n are the eigenvalues of
the matrix w(A, B) defined by (6.5.21).
Proof. Let TA and TB be nonsingular matrices transforming A and B to their
respective Jordan canonical forms AJ and BJ, i.e.,
346
Polynomial and Rational Matrices
AJ
TA ATA1 , B J
TB BTB1 .
(6.5.22)
On the main diagonal of AJ are the eigenvalues O1,O2,…,Om, and on the main
diagonal of BJ the eigenvalues P1,P2,…,Pn.
It follows from the definition of the Kronecker product that on the main
diagonal of the matrix AJ BJ are the eigenvalues OiPi, for i = 1,2,…,m;
j = 1,2,…,n. Hence on the main diagonal of w(AJ, BJ) are the eigenvalues w(Oi, Pj),
for i = 1,2,…,m; j = 1,2,…,n. We will show that w(AJ, BJ) and w(A, B) are similar
matrices, thus having the same eigenvalues.
Taking into account (6.5.22) and
A1A 2 A 3
B1B 2 B3
A1
B1 A 2
TA ATA1
TB BTB1
Ǻ2
A3
B3 ,
TB
ǹ
B TA1
we can write
AJ
BJ
TA
TB1 .
With the equality
TA1
TB1
TA
TB
1
taken into consideration, we have
AJ
BJ
TA
TB
A
B TA
TB
1
.
Thus AJ BJ and A B are similar matrices. Hence
w( A J , B J )
TA
TB w A, B TA
TB
1
;
w(AJ, BJ) and w(A, B) as similar matrices have the same eigenvalues. From Theorem 6.5.4 for w(x, y) = x + y and w(x, y) = xy, we have the following
corollaries.
Corollary 6.5.3. If O1,O2,…,Om are the eigenvalues of A, and P1,P2,…,Pn are the
eigenvalues of BT, then Oi + Pj for i = 1,2,…,m; j = 1,2,…,n are the eigenvalues of
the matrix
A
In Im
BT .
(6.5.23)
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
347
Corollary 6.5.4. If O1,O2,…,Om are the eigenvalues of A, and P1,P2,…,Pn the
eigenvalues of B, then OiPj for i = 1,2,…,m; j = 1,2,…,n are the eigenvalues of the
matrix A B.
6.6 The Sylvester Equation and Its Generalization
6.6.1 Existence of Solutions
Consider the following Sylvester equation
AX XB
C,
(6.6.1)
where A, B are square matrices of size m and n, respectively, and C and X are
rectangular matrices of dimension mun. With A, B and C given, one has to
compute the matrix X satisfying (6.6.1).
From the rows x1,x2,…,xm of X and the rows c1,c2,…,cm of C we build the mndimensional vectors
x
T
> x1 , x2 , ..., xm @
, c
T
>c1 , c2 , ..., cm @
.
(6.6.2)
Using (6.5.19) we can write (6.6.1) as
Dx
c,
(6.6.3)
where
D
A
In Im
BT
(6.6.4)
is a square matrix of size mn.
Theorem 6.6.1. Equation (6.6.1) has one solution if and only if the matrices A and
B do not have the same eigenvalues.
Proof. There exists one solution to (6.6.3) if and only if D is a nonsingular matrix.
D is nonsingular if and only if all its eigenvalues are nonzero. According to
Corollary 6.5.3, the numbers Oi - Pj, for i = 1,2,…,m; j = 1,2,…,n, are the
eigenvalues of the matrix (6.6.3). Thus D has nonzero eigenvalues if and only if
the matrices A and B do not have common eigenvalues. In this case, D is a
nonsingular matrix and (6.6.3) (thus also (6.6.1)) has exactly one solution
x
D1c .
(6.6.5)
348
Polynomial and Rational Matrices
Note that the Lyapunov equation
AT P PA
(6.6.6)
Q
is a particular case of the Sylvester equation (6.6.1) for X = P, A = AT, B = A and
C = Q.
In the particular case for C = 0, (6.6.1) takes the form
AX XB
0.
(6.6.7)
If A and B do not have the same eigenvalues, then D is a nonsingular matrix and
the equation Dx = 0 has only the zero solution x = 0. Thus we have the following
corollary.
Corollary 6.6.1. If the matrices A and B do not have the same eigenvalues, then
(6.6.7) has only the zero solution X = 0. If A and B have at least one common
eigenvalue, then (6.6.7) has a nonzero solution.
Theorem 6.6.2. If all the eigenvalues of A and – B have negative real parts, then
the unique solution to (6.6.1) is
f
³ e At Ce Bt dt .
X
(6.6.8)
0
Proof. Substituting (6.6.8) into (6.6.1) we obtain
f
AX XB
³ Ae At Ce Bt e At Ce Bt B dt
0
f
d
³ ª¬ e At Ce Bt º¼ dt
dt
0
e At Ce Bt
f
0
C,
since the matrices A and – B are asymptotically stable and
lim e At Ce Bt
i of
0.
In this case, A and B do not have common eigenvalues and according to
Theorem 6.6.1 there exists, only one solution to (6.6.1).
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
349
6.6.2 Methods of Solving the Sylvester Equation
6.6.2.1 The Kronecker Product Method
When A and B do not have common eigenvalues, we can solve (6.6.1) by the use
of the following procedure.
Procedure 6.6.1.
Step 1: From the rows of X and C build the vectors x and c of the form (6.6.2).
Step 2: Using (6.6.4) compute the matrix D.
Step 3: Using (6.6.5) compute the vector x and then the desired matrix X.
Example 6.6.1.
Solve (6.6.1) with respect to X for the matrices
A
ª 0 1 0 º
« 0 0 1» , C
«
»
«¬1 3 3 »¼
ª 2 1 º
« 0 3» , B
¬
¼
ª1 0 1 º
« 0 1 1» .
¬
¼
Applying Procedure 6.6.1, we obtain
Step 1: Using (6.6.2) we compute the vector
cT
[1
0
1
0
1
1] .
Step 2: According to (6.6.4) the matrix D is
D
A
In Im
B
T
ª 2 1 º
«
»
¬ 0 3¼
I3 I 2
ª0 0 1º
«1 0 3»
«
»
«¬0 1 3»¼
ª 2 0 1 1 0 0 º
« 1 2 3 0 1 0 »
«
»
« 0 1 5 0 0 1 »
«
».
« 0 0 0 3 0 1»
« 0 0 0 1 3 3»
«
»
¬« 0 0 0 0 1 6 ¼»
Step 3: Using (6.6.5) we obtain the desired solution of the form
X
ª x11
«x
¬ 21
x12
x22
x13 º
x23 »¼
1 ª 119 34 59 º
.
288 «¬ 9 126 27 »¼
(6.6.9)
350
Polynomial and Rational Matrices
6.6.2.2 Integration Method
If the matrices A and –B have all their eigenvalues with negative real parts, then
we can solve (6.6.1) using the following procedure.
Procedure 6.6.2.
Step 1: Compute the minimal polynomials \A(O),\B(O) of the matrices A and –B.
Step 2: Compute eAt and e-Bt.
Step 3: From (6.6.8) compute the desired solution X.
Example 6.6.2.
Using Procedure 6.6.2 solve (6.6.1) (with respect to X) for the matrices (6.6.9).
The matrices A and –B have all their eigenvalues with negative real parts.
Using Procedure 6.6.2, we obtain the following.
Step 1: In this case, the characteristic polynomials of the matrices A and –B are the
same as their minimal polynomials
\ A (O ) M A (O ) det > O I m A @
O2
1
O 3
0
O 1
0
O
\ B (O ) M B (O ) det > O I n B @
1
(O 2)(O 3) ,
0
(O 1)3 .
1
3 O 3
Step 2: The matrix A has the two single eigenvalues O1 = 2, O2 = 3, and the
matrix –B has the one eigenvalue O1 = O2 = O3 = P1 1 of multiplicity 3. Using the
Sylvester formula, we obtain
e At
Z1eO1t Z 2 eO2t
1
O1 O2
1º 3t
ª1 1º 2t ª 0
« 0 0 » e « 0 1» e
¬
¼
¬
¼
A O2 I m eO1t ª e 2t
«
¬« 0
1
O2 O1
A O1I m eO2t
e 2t e 3t º
»
e 3t ¼»
and
1
I 3e P1t (B I 3 )te P1t (B I 3 ) 2 t 2 e P1t
2
ª1 0 0 º
ª1 1 0º
ª1 2 1º
«0 1 0 » e t « 0 1 1 » te t 1 « 1 2 1» t 2 e t
«
»
«
»
»
2«
«¬0 0 1 »¼
«¬ 1 3 2 »¼
«¬ 1 2 1 »¼
e Bt
Z11e P1t Z12te P1t Z13t 2 e P1t
1 2
ª1 t 12 t 2
º
t t2
2t
«
»
2
1 2
1 2
1 t t
t 2 t » e t .
« 2t
« t 12 t 2 3t t 2 1 2t 12 t 2 »
¬
¼
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
351
Step 3: From (6.6.8) we have
f
X
³ e At Ce Bt dt
0
f
ª e 2 t
³«
«0
0 ¬
e 2t e 3t º ª1 0
1º
»
3t » «
e ¼» ¬ 0 1 1 ¼
1 2
ª1 t 12 t 2
º
t t2
2t
«
»
2
1 2
1 2
u« 2 t
t 2 t » e t dt
1 t t
« t 12 t 2
3t t 2 1 2t 12 t 2 »¼
¬
ª e 3t 1 t e4t t t 2 e3t 1 2t e 4 t 1 4t 2t 2
³ «
e 4 t t t 2
e 4t 1 4t 2t 2
0 «
¬
e 3t t e4t t 2 3t 1 º
1 ª 119 34 59 º
» dt
.
4 t
2
288 «¬ 9 126 27 »¼
»
e 1 3t t
¼
f
The result is consistent with that obtained in Example 6.6.1.
6.6.2.3 Minimal Polynomial Method
The method is based on the following theorem.
Theorem 6.6.3. Let
s m am1s m1 ! a1s a0
<A s
be the minimal polynomial of A
mum
, and
s n bn1s n1 ... b1s b0
<B s
the minimal polynomial of B nun. Let these polynomials be relatively prime
(without common zeros). A solution to (6.6.1) takes the form
1
>Cn bn1Cn1 ! b1C1 @ ,
(6.6.10)
X
ª¬ < B A º¼
X
>Cm am1Cm1 ! a1C1 @ ª¬ < A B º¼ ,
or
1
(6.6.11)
where
k
Ck
¦A
i 1
i 1
CB k i , C0
0, k
1, 2, ... .
(6.6.12)
352
Polynomial and Rational Matrices
Proof. Using (6.6.1) and (6.6.12) we can write
C0
A 0 X XB 0
C1
AX XB
C2
0
C
2
2
AC CB
3
3
A 2C ACB CB 2
A X XB
C3
A X XB
Ck
A k X XB k
k
¦A
(6.6.13)
i 1
CB k i .
i 1
Taking into account that Ck = AkX – XBk, k = 1, 2, ..., we write the expression
b1C1 + b2C2 + ... + bn–1Cn –1 + Cn in the form
< B A X X< B B
b1C1 b2C2 ! bn1Cn1 Cn .
(6.6.14)
Then invoking that <B(B) = 0 and pre-multiplying (6.6.14) by [<B(A)]-1, we
obtain the desired equation (6.6.10).
Analogously, we write a1C1 + a2C2 + ... + am–1Cm –1 + Cm in the form
< A A X X< A B
a1C1 a2C2 ! am1Cm1 Cm .
(6.6.15)
Then invoking that <A(A) = 0, and post-multiplying (6.6.15) by [<A(B)]-1, we
obtain the desired equation (6.6.11).
If (6.6.1) has exactly one solution, then we can compute it using the following
procedure, which ensues from Theorem 6.6.3.
Procedure 6.6.3.
Step 1: Compute the minimal polynomial (characteristic) <A(s) (<B(s)) of the
matrix A (B).
Step 2: Using (6.6.12) compute the rows C1,C2,…,Cm (Cn).
Step 3: From (6.6.11) (or (6.6.10)) compute the desired solution X.
Example 6.6.3.
Using Procedure 6.6.3 solve (6.6.1) (with respect to X) for the matrices (6.6.9).
In this case, we have the following.
Step 1: The minimal polynomial of A is the same as its characteristic polynomial
<A s
det > Is A @
The matrix <A(B) is
s2
1
0
s3
s 2 5s 6 .
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
<A B
353
B 2 5B 6I
2
ª 0 1 0 º
ª 0 1 0 º ª 6 0 0 º
« 0 0 1» 5 «0 0 1» « 0 6 0 »
«
»
«
» «
»
¬«1 3 3 »¼
¬«1 3 3 ¼» ¬«0 0 6 ¼»
ª 6 5 1 º
« 1 3 8 » .
«
»
¬« 8 23 27 »¼
Step 2: Using (6.6.12) we obtain
ª1 0 1 º
«0 1 1» ,
¬
¼
ª0 1
ª1 0 1 º «
«
» «0 0
¬0 1 1¼ «
¬1 3
C1
C
C2
0º
1»»
3 »¼
AC CB
ª 2 1 º ª 1 0 1 º
« 0 3» «0 1 1» ¬
¼¬
¼
ª 1 3 0 º
« 1 6 1» .
¬
¼
Step 3: From (6.6.11) we have
X
>C2 5C1 @ ª¬ < A B º¼
1
ª 6 5 1 º
ª 4 3 5 º«
»
«
» « 1 3 8 »
¬ 1 1 6 ¼ « 8 23 27 »
¬
¼
1
1 ª 119 34 59 º
.
288 «¬ 9 126 27 »¼
The result is consistent with that obtained in Examples 6.6.1 and 6.6.2.
6.6.2.4 Auxilary Equation Method
The method is based on the following theorem.
Theorem 6.6.4. Let A and B have distinct eigenvalues. The solution X to (6.6.1)
has the form
X
MA ī
where the matrix *
K TA ī īK B
Iq NB ,
mun
(6.6.16)
is a solution to the equation
ª1
«0
«
«#
«
¬0
0 ! 0º
0 ! 0 »»
,
# % #»
»
0 ! 0¼
(6.6.17)
354
Polynomial and Rational Matrices
MA
ª¬C A
AC A ! A m1C A º¼ , N B
ª CB º
« C B »
« B »,
« # »
«
n 1 »
¬C B B ¼
1
ª 0
« 0
0
«
« #
#
«
0
« 0
«¬ b0 b1
1
0 !
0 º
ª 0
« 0
0
1 !
0 »»
«
0
0 !
1 » ,KB
KA « 0
«
»
#
# %
# »
« #
«¬ a0 a1 a2 ! am1 »¼
C C AC B , C A  ruq , C B  qun , rank C A
(6.6.17)
0
1
#
0
b2
rank C B
!
0 º
!
0 »»
%
# »,
»
!
0 »
! bn1 »¼
q,
and ai, i = 0,1,…,m1, bj, j = 0,1,…,n1 are the coefficients of the minimal
polynomial <A(s) and <B(s) of the matrices A and B.
Proof. By virtue of the Cayley–Hamilton theorem, we have
AM A
ª¬C A
ª
« AC A
¬
AC A
M A ª¬ K TA
A 2C A ! A m1C A
ª0
«I
« q
! A m1C A º¼ « 0
«
«#
«0
¬
0
0
Iq
#
0
m 1
º
¦ ai A i C A »
i 1
¼
! 0 a0 I q º
! 0
a1I q »»
! 0 a2 I q »
»
% #
# »
! I q am1I q »¼
(6.6.18a)
I q º¼ .
Analogously
NBB
ª¬K B
ª CB B º
« C B2 »
B
«
»
«
»
«
»
n 1
« CB B
»
n 1
«
»
i
« C B ¦ bi B »
i 0
¬
¼
ª 0
« I
« q
« #
«
« 0
« b0 I q
¬
I q º¼ N B .
Using (6.6.16) and (6.6.18), we obtain
Iq
0
0
Iq
#
#
0
0
b1I q
b2 I q
!
0 º
!
0 »»
%
# »
»
!
Iq »
! an1I q »¼
(6.6.18b)
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
AX XB
M A ª¬ K
AM A ī
T
A
^
^K
^K
I q º¼ ī
M A » K TA
MA
ȂA
I q º¼ ª¬ ī
T
A
ī
T
A
ī īK B
Iq NB M A ī
Iq NB M A ī
I q º¼ ª¬ ī
I q īK B
355
Iq NBB
I q ª¬ K B
I q º¼ ª¬K B
I q º¼ N B
`
I q º¼ N B
(6.6.19)
`
Iq NB
`
Iq NB .
Note that
C
C AC B
­§ ª1 º
·
°¨ « »
¸
° 0
M A ®¨ « » >1 0 ! 0@ ¸
¨
¸
°¨ « # »
¸¸
«
»
¨
° ¬0 ¼
¹
¯©
½
°
°
Iq ¾ NB .
°
°
¿
(6.6.20)
Comparing (6.6.20) to (6.6.19), we have
K TA ī īK B
Iq
§ ª1 º
·
¨« »
¸
¨ «0 » >1 0 ! 0@ ¸
¨ «# »
¸
¨¨ « »
¸¸
© ¬0 ¼
¹
Iq ,
which is exactly (6.6.17).
Thus solving (6.6.1) is reduced to the solving of (6.6.17) with respect to *, with
the matrices KA and KB known.
If (6.6.1) has exactly one solution, then it can be computed using the following
procedure, which ensues from Theorem 6.6.4.
Procedure 6.6.4.
Step 1: Compute the minimal polynomials
<A s
s m am1s m1 ! a1s a0 ,
<B s
s n bn1s n1 ! b1s b0
(6.6.21)
of the matrices A and B.
Step 2: With the coefficients ai, i = 0,1,…,m1 and bj, j = 0,1,…,n1 of the
polynomials (6.6.21) known, solve (6.6.17).
Step 3: Solving (6.6.17) compute the matrix *.
356
Polynomial and Rational Matrices
Step 4: Decompose the matrix C into the product of the matrices CA and CB such
that rank CA = rank CB = rank C.
Step 5: Using (6.6.16) compute the desired solution X.
Example 6.6.4.
Using Procedure 6.6.4 compute the solution X to (6.6.1) for the matrix (6.6.9).
In this case, we have the following.
Step 1: The minimal polynomials of the matrices (6.6.9) are the same as their
characteristic polynomials
<A s
det > Is A @
s2
1
0
s3
s
<B s
det > Is B @
1
s 2 5s 6,
0
0 s
1
1 3 s 3
s 3 3s 2 3s 1.
Step 2: Thus in this case, (6.6.17) takes the form
ª0 1 0 º
T
ª0 1º
«
»
ī
ī
« 6 5»
«0 0 1 »
¬
¼
«¬1 3 3 »¼
ª1 0 0 º
«
».
¬0 0 0¼
This equation is equivalent to the following system of equations (see 6.6.2.1)
ª 0 0 1 6 0
« 1 0 3 0 6
«
« 0 1 3 0 0
«
« 1 0 0 5 0
« 0 1 0 1 5
«
«¬ 0 0 1 0 1
0 º ª x1 º
0 »» «« x2 »»
6 » « x3 »
»« »
1» « x4 »
3» « x5 »
»« »
8 »¼ «¬ x6 »¼
ª1 º
«0»
« »
«0»
« ».
«0»
«0»
« »
«¬ 0 »¼
(6.6.22)
ª 227 53 23 º
« 288 144 288 »
«
»
«
».
« 265 79 37 »
«
»
¬ 1728 864 1728 ¼
(6.6.23)
Step 3: The solution to (6.6.22) is
*
ª x1
«x
¬ 4
x2
x5
x3 º
x6 »¼
Step 4: We decompose C into the product of the matrices
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
CA
ª1 0 º
«0 1 » , C B
¬
¼
ª1 0 1 º
« 0 1 1» .
¬
¼
357
(6.6.24)
Step 5: Using (6.6.16), (6.6.23) and (6.6.24), we obtain
X
>C A
AC A @> ī
ª CB º
I 2 @ «« C B B »»
2
¬«C B B ¼»
1 ª 119 34 59 º .
288 «¬ 9 126 27 »¼
The result is consistent with those obtained in the foregoing three examples.
6.6.3 Generalization of the Sylvester Equation
Consider the following equation
XA FXE
HC ,
(6.6.25)
where A, E nun, F rur, H rup and C pun.
We will show that solving (6.6.25) with respect to X run is equivalent to
solving the Sylvester equation.
Equation (6.6.25) is called the generalised Sylvester equation.
Theorem 6.6.5. Let
det > Es A @ z 0 for some s  (6.6.26)
and the spectra of the pair (E, A) and the matrix F be disjoint. The equation
(6.6.25) has a solution if and only if the following Sylvester equation has a solution
AX XB
C,
(6.6.27)
where
A
C
1
1
>I r s1 F @  rur , B E >Es1 A @
1
1
> Es1 F @ HC > Es1 A @  run .
 nun ,
Proof. If the condition (6.6.26) is satisfied, then there exists a number s1 such
that [Es1 – A] is a nonsingular matrix. With the matrix XEs1 added to and
subtracted from the left-hand side of (6.6.25), we obtain
X > Es1 A @ > I r s1 F @ XE
HC .
(6.6.28)
358
Polynomial and Rational Matrices
Pre-multiplying (6.6.28) by [Irs1 – F]-1 and post-multiplying it by[Es1 – A]-1, we
obtain (6.6.27).
Note that the eigenvalues of A are the reciprocals of the eigenvalues of F, and
det ª¬ I n s B º¼
det ªI n s E Es1 A
¬
1
º.
¼
6.7 Algebraic Matrix Equations with Two Unknowns
6.7.1 Existence of Solutions
Consider the following matrix equation
XA BY XCY
D,
(6.7.1)
where the matrices A nuq, B pum, C num and D puq are known.
One has to compute the matrices X pun and Y muq satisfying the equation
(6.7.1). From now on we will use the MoorePenrose pseudo-inverse of a matrix,
which we will shortly call the pseudo-inverse.
The pseudo-inverse of A mun, denoted A+ nun, is a matrix that satisfies the
following conditions
AA A
A,
(6.7.2a)
A AA A ,
(6.7.2b)
AA ,
(6.7.2c)
A A.
(6.7.2d)
AA A A
T
T
For an arbitrary A mun there exists only one pseudo-inverse A+ mun. It can
be computed using the SVD decomposition of A [158]. If A nun is a nonsingular
matrix, then A+ = A-1.
Theorem 6.7.1. Equation (6.7.1) has a solution if
rank ª¬ D BC A º¼ d max n, m ,
(6.7.3)
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
359
where C+ is the pseudo-inverse of C.
Proof. Taking
Za
A CY ,
(6.7.4a)
we can write (6.7.1) in the following form
XZ a BY
D.
(6.7.5a)
Solving (6.7.4a) with respect to Y, we obtain
Y
C Z a A ,
(6.7.6a)
and substituting (6.7.6a) into (6.7.5a) we have
X BC Z a
D BC A ,
D + BC+A puq can be expressed as the product of the two matrices H
F nuq (obviously not unique)
D BC A
HF ,
(6.7.7a)
pun
and
(6.7.8)
if the condition (6.7.3) is met.
In this case, with (6.7.7a) and (6.7.8) taken into account, we obtain
X + BC+ = H and Za = F. With H and F known, we can compute the desired
matrices from the following relationships
X
H BC , Y
C F A .
(6.7.9a)
On the other hand, taking
Zb
B XC ,
(6.7.4b)
we can write (6.7.1) as
XA Zb Y
D.
(6.7.5b)
Solving (6.7.4b) with respect to X, we obtain
X
Zb B C ,
and substituting (6.7.6b) to (6.7.5b), we have
(6.7.6b)
360
Polynomial and Rational Matrices
Zb C A Y
D BC A .
(6.7.7b)
Thus D + BC+A can be expressed as the product (6.7.8) provided the condition
(6.7.3) is met. In this case, with (6.7.7b) and (6.7.8) taken into account, we have
Zb = H and C+A + Y = F. With H and F known, we can compute
X
H B C , Y
F C A .
(6.7.9b)
Ŷ
Remark 7.7.1.
The decomposition (6.7.8) is not unique. Therefore, (6.7.1) has many different
solutions X, Y.
6.7.2 Computation of Solutions
If the condition (6.7.3) is met, then we can compute a solution X, Y to (6.7.1) using
the following procedure, which ensues from the proof of Theorem 6.7.1.
Procedure 6.7.1.
Step 1: Compute the pseudo-inverse C+ of C and the matrix D + BC+A.
Step 2: Decompose the matrix D + BC+A puq into the product of H
F nuq.
Step 3: Using (6.7.9a), compute the desired solution X, Y.
pun
and
Example 6.7.1.
Using Procedure 6.7.1, compute a solution X, Y to (6.7.1) for the matrices
A
ª1 º
«2» , B
¬ ¼
ª 1 0º
« 1 1 » , C
¬
¼
ª 2 3º
«1 1» , D
¬
¼
ª 10 º
« 10 » .
¬
¼
(6.7.10)
In this case, m = p = n = 2, q = 1, and C is a nonsingular matrix.
Hence
ª 1 3 º
« 1 2 » ,
¬
¼
10
ª
º ª 1 0 º ª 1 3 º ª 1 º
D BC A «
»«
»«
»« »
¬ 10 ¼ ¬ 1 1 ¼ ¬ 1 2 ¼ ¬ 2 ¼
C
C1
The condition (6.7.3) is met, since
rank ª¬ D BC A º¼
ª 5º
rank « » 1 n
¬8¼
2.
ª 5 º
« 8 ».
¬ ¼
(6.7.11)
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
361
Thus (6.7.1) has a solution.
Applying Procedure 6.7.1, we obtain the following.
Step 1: The matrix D + BC+A has the form (6.7.11).
Step 2: The matrix (6.7.11) can be decomposed into the product of two different
matrices. We consider two cases of this decomposition
D BC A
D BC A
ª 5º
«8»
¬ ¼
ª 5 º
«8»
¬ ¼
H1F1 , for H1
H 2 F2 , for H 2
ª1 0 º
ª 5º
(6.7.12a)
« 0 1 » , F1 « 8 » ,
¬
¼
¬ ¼
ª0 5º
ª 7 º
« 1 1» , F2 « 1» . (6.7.12b)
¬
¼
¬ ¼
Step 3: Using (6.7.9a), we obtain for the case (6.7.12a)
X
H1 BC
ª1 0 º ª 1 0 º ª 1 3 º
« 0 1 » « 1 1 » « 1 2 »
¬
¼ ¬
¼¬
¼
Y
C F1 A
ª 1 3 º § ª 5º ª 1 º ·
« 1 2 » ¨ « 8 » « 2» ¸
¬
¼©¬ ¼ ¬ ¼¹
ª 2 3º
«0 2 » ,
¬
¼
ª 24 º
«18 » ,
¬ ¼
(6.7.13a)
for the case (6.7.12b)
X
H 2 BC
Y
C F2 A
ª 0 5 º ª 1 0 º ª 1 3 º
« 1 1» « 1 1 » « 1 2 »
¬
¼ ¬
¼¬
¼
ª 1 3 º § ª 7 º ª 1 º ·
« 1 2 » ¨ « 1» « 2 » ¸
¬
¼©¬ ¼ ¬ ¼¹
ª1 2 º
«0 1 » ,
¬
¼
ª 1º
« 2 ».
¬ ¼
(6.7.13b)
It is easy to check that the matrices (6.7.13a) and (6.7.13b) satisfy (6.7.1) for
the matrices (6.7.10).
6.8 Lyapunov Equations
6.8.1 Solutions to Lyapunov Equations
Definition 6.8.1. The matrix equations
XA AT X
AX XA
T
Q ,
Q
(6.8.1a)
(6.8.1b)
362
Polynomial and Rational Matrices
are called the Lyapunov equations if the matrices A nun and Q nun (positive
definite or positive semidefinite) are given, and X nun (positive definite) is the
matrix we seek.
Theorem 6.8.1. Let A be asymptotically stable and Q be a symmetric, positive
definite (or semidefinite) matrix. Then (6.8.1a) has exactly one solution of the form
f
X
³e
AT t
Qe At dt ,
(6.8.2a)
0
which is a positive definite (semidefinite) matrix, and (6.8.1b) has exactly one
solution of the form
f
X
³e
At
T
Qe A t dt ,
(6.8.2b)
0
which is a positive definite (semidefinite) matrix.
Proof. Substituting (6.8.2a) into (6.8.1a), we obtain
f
XA AT X
³e
0
AT t
f
T
Qe At dtA AT ³ e A t Qe At dt
f
d AT t At
³0 dt e Qe dt
0
f
AT t
e Qe
At
Q
0
since by assumption A is asymptotically stable and eAt o 0 for t o f.
We will show that if Q is positive definite (semidefinite), then the matrix
(6.8.2a) is positive definite (semidefinite), that is, its quadratic form is positive
definite (semidefinite), zTXz > 0 (zTXz t 0) for every z z 0.
Using (6.8.2a) we can write
f
z T Xz
³z
T
T
e A t Qe At zdt .
(6.8.3)
0
The matrices
T
e A t and e At
are nonsingular for every t t 0. Thus if Q is a positive definite (semidefinite)
matrix, then it follows from (6.8.3) that zTXz > 0 for every z z 0 (zTXz t 0 for every
z).
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
363
In order to show that (6.8.1a) has exactly one solution, we assume that it has
two different solutions X1 and X2, that is,
X1A AT X1
Q and X 2 A AT X 2
Q .
(6.8.4)
Subtracting these equations one from another, we obtain
X1 X 2 A AT X1 X 2
0.
(6.8.5)
Pre-multiplying (6.8.5) by e A t and post-multiplying it by e At , we obtain
T
T
e A t ª¬ X1 X 2 A AT X1 X 2 º¼ e At
0
and
d ª AT t
X1 X 2 e At º
e
¼
dt ¬
0.
(6.8.6)
It follows form (6.8.6) that
T
e A t X1 X 2 e At
is a constant matrix for all t. Evaluating it for t = 0 and taking into account (6.8.6),
we obtain X1 – X2 = 0, since eAt | t = 0 = I. We obtain the same result for t = f, since
eAt o f for t o f.
The proof for (6.8.1b) is analogous.
„
6.8.2 Lyapunov Equations with a Positive Semidefinite Matrix
In many cases, the matrix Q in the Lyapunov equation is of the form Q = CCT or
Q = BBT, i.e., it is a positive semidefinite matrix.
Theorem 6.8.2. If A
equation
XA AT X
nun
is an asymptotically stable matrix, then the Lyapunov
CT C
(6.8.7)
has exactly one positive definite solution of the form
f
X
³e
0
AT t
CT Ce At dt
(6.8.8)
364
Polynomial and Rational Matrices
if and only if (A, C) is an observable pair.
Proof. According to Theorem 6.8.1, the solution to (6.8.7) has the form (6.8.8). We
will prove the thesis by contradiction. Suppose the solution is not positively
definite. Then there exists a vector z such that Xz = 0. In this case, we have from
(6.8.8)
f
z T Xz
³ Ce
At
2
(6.8.9)
z dt
0
that is, CeAtz = 0.
We differentiate the above relationship and evaluate its derivatives for t = 0.
We obtain CAkz = 0, for k = 0,1,…,n-1, i.e.,
ª C º
« CA »
«
»z
« # »
«
n 1 »
¬CA ¼
0.
(6.8.10)
By assumption (A, C) is an observable pair. Thus from (6.8.10) we have z = 0,
which contradicts the supposition that the matrix (6.8.8) is not positive definite.
Thus the solution (6.8.8) is a positive definite matrix.
Also by contradiction, we will show now that the asymptotical stability of A
and positive definiteness of the matrix (6.8.8) imply the observability of the pair
(A, C).
Suppose that (A, C) is unobservable, that is,
ª Is A º
rank «
» n, for all s  .
¬ C ¼
In this case there exist an eigenvector x of the matrix A (Ax = sx) such that
Cx = 0, and from (6.8.7) we obtain
x XAx x AT Xx
x CT Cx
Cx
2
x denotes the complex conjugate of x ;
that is,
s s x Xx
Cx
2
0, since Cx
where s denotes the conjugate of s.
0,
(6.8.11)
Matrix Polynomial Equations, and Rational and Algebraic Matrix Equations
365
By assumption A is an asymptotically stable matrix hence s + s < 0. Thus
from (6.8.11) we have x*Xx = 0. This leads to contradiction, since by assumption X
is a positive definite matrix.
„
Theorem 6.8.3. If (A, C) is an observable pair, then A is an asymptotically stable
matrix if and only if there exists, exactly one symmetric positive definite matrix X
satisfying (6.8.7).
Proof. According to Theorem 6.8.2, if A is an asymptotically stable matrix and
(A, C) is an observable pair, then (6.8.7) has exactly one solution, which is positive
definite and has the form (6.8.8). We will show that if (A, C) is an observable pair
and X is positively definite, then A is asymptotically stable. Let x be the
eigenvector of A corresponding to an eigenvalue s. In the same way as in the case
of Theorem 6.8.2, we can show that
s s x Xx
2
Cx .
(6.8.12)
By assumption (A, C) is an observable matrix, hence Cx z 0 and x*Xx > 0,
since X is positive definite. Thus from (6.8.12) we have s + s < 0, thus A is
asymptotically stable. „
Let us sum up made hitherto considerations, in the following important
theorem.
Theorem 6.8.4. Let X be the solution to (6.8.7). In this case, we have
1. If X is a positive definite matrix and (A, C) is an observable pair, then A
is an asymptotically stable matrix.
2. If A is an asymptotically stable matrix and (A, C) is an observable pair,
then X is a positive definite matrix.
3. If A is an asymptotically stable matrix and X is a positive definite matrix,
then (A, C) is an observable pair.
Now consider the following Lyapunov equation
AX XAT
BBT .
(6.8.13)
Taking into account that the controllability of the pair (A, B) is a dual notion to the
observability of the pair (A, C), we immediately have the following theorems.
Theorem 6.8.5. If A is an asymptotically stable matrix, then (6.8.13) has exactly
one solution X, which is symmetric and positive definite, if and only if, (A, B) is a
controllable pair.
366
Polynomial and Rational Matrices
Theorem 6.8.6. Let (A, B) be a controllable pair. Then A is an asymptotically
stable matrix if and only if there exists exactly one solution, which is symmetric
and positive definite, to (6.8.13).
Theorem 6.8.7. Let X be the solution to (6.8.13). In this case, we have the
following.
1.
If X is a positive definite matrix and (A, B) is a controllable pair, then A
is an asymptotically stable.
2.
If A is asymptotically stable matrix and (A, B) is a controllable pair, then
X is a positive definite matrix.
3.
If A is an asymptotically stable matrix and X is a positive definite matrix,
then (A, B) is a controllable pair.
7
The Realisation Problem and Perfect Observers of
Singular Systems
7.1 Computation of Minimal Realisations for Singular Linear
Systems
7.1.1 Problem Formulation
Consider the following continuous-time, singular system
Ex Ax B 0 u B1u ,
y Cx Du ,
(7.1.1a)
(7.1.1b)
where x n, u m and y p are the vectors of the state, input and output,
respectively; E, A nun, B0, B1 num, C pun, D pum.
We assume that det E = 0 and (E,A) is a regular pair, i.e.,
(7.1.2)
det[Es A] z 0, for some s  ,
where is the field of the complex numbers.
The transfer matrix of the system (7.1.1) is
T( s )
1
C > Es A @ (B 0 sB1 ) D .
(7.1.3)
The transfer matrix (7.1.3) is called proper (strictly proper) if and only if
lim T( s )
s of
K  pum and K z 0 (K
0) .
(7.1.4)
368
Polynomial and Rational Matrices
Otherwise, we call it improper.
Equation (7.1.1) can be written as
Ex Ax Bu ,
y Cx Du ,
(7.1.5a)
(7.1.5b)
where
ª E B1 º
ª xº
 n un , x « »  n ,
«0
»
0 ¼
¬
¬u ¼
ªB0 º
ªA 0 º
n um
A «
 n un , Ǻ «
» , n
»
ǿ
0
I
m¼
¬
¬ m¼
C [C 0]  pun , D D.
E
nm ,
(7.1.5c)
Equation (7.1.1) can be also written as
Ex
y
x B u ,
A
Cx ,
(7.1.6a)
(7.1.6b)
where
A
ª A B0 º
n un
«0 I » , B
m¼
¬
ª 0 º
n um
pun
« I »  , C [C D]  .
¬ m¼
(7.1.6c)
Definition 7.1.1. The matrices E, A, B0, B1, C and D are called a realisation of the
transfer matrix T(s) pum(s) (the set of rational matrices of dimensions pum in the
variable s), if they satisfy the relationship (7.1.3). A realisation (E, A, B0, B1, C, D)
is called a minimal realisation if the matrices E and A have minimal dimensions
among all realizations of T(s).
A realisation
, B , C
)
(E, A, B, C, D) or (E, A
is a minimal one if and only if the system (7.1.5), or respectively, the system
(7.1.6), is completely controllable and completely observable.
The system (7.1.5) is completely controllable if and only if
rank ª¬E, B º¼
n and rank ª¬Es A, B º¼
where n is the dimension of the state vector x .
n for all finite s  ,
(7.1.7)
The Realisation Problem and Perfect Observers of Singular Systems
369
The system (7.1.5) is completely observable if and only if
ªE º
rank « »
¬C ¼
ª Es A º
n and rank «
»
¬ C ¼
n for all finite s  .
(7.1.8)
The realisation problem can be formulated in the following way. Given a
rational improper matrix T(s) pum(s), compute a realization (E, A, B0, B1, C, D )
and a minimal realisation
, B , C
) .
(E, A, B, C, D) or (E, A
A solution to the problem by the method presented below was proposed for the
first time in [72].
7.1.2 Problem Solution
An arbitrary rational matrix T(s)
T( s )
pum
(s) can be written as
P( s)
,
d (s)
(7.1.9)
where P(s)is a polynomial matrix of dimension pum and
d (s)
d q s q d q 1s q 1 " d1s d 0
(7.1.10)
is the least common denominator of all the entries of T(s).
Let N = deg P(s) be the degree of the polynomial matrix P(s) and N > q.
The proposed method is based on the following theorem.
Theorem 7.1.1. Let
s
Z 1 O , d O z 0 and N ! q .
The rational matrix
T(Z )
T( s )|s Y 1 O
P (Z )
d (Z )
in the variable Z is a proper matrix, i.e.,
deg d (Z )
N t deg P (Z ) .
(7.1.11)
370
Polynomial and Rational Matrices
Proof. Substituting s = Z1 + O into T(s) we obtain the improper rational matrix in
the variable Z-1
T(Z 1 O )
P(Z 1 O )
,
d (Z 1 O )
(7.1.12)
since the degree of P(Z-1 + O) with respect to Z-1 is N , and the degree d(Z-1 + O) is
q. With both the numerator and denominator of (7.1.12) multiplied by ZN, we
obtain (7.1.11), where
deg d (Z )
N t deg P(Z ) ,
since by assumption d(O) z 0.
„
Note that Theorem 7.1.1 allows us to transform the problem of computing the
realization (E, A, B0, B1, C, D) of T(s) to the problem of computing the realisation
(AZ, BZ, CZ, DZ) of the proper matrix T (Z).
The realisation (AZ, BZ, CZ, DZ) of the matrix T (Z) can be computed using
one of the following well-known methods.
Let
E
AY , A
I n O AY , B 0
O BY , B1
BY , C CY , D
DY .(7.1.13)
Substituting (7.1.13) and s = Z1 + O into (7.1.3) we obtain
T( s )
C[Es A ]1 (B 0 sB1 ) D
CZ [ AZ (Z 1 O ) (I n O AZ )]1 (O BZ BZ s ) DZ
CZ [ AZ Z 1 I n ]1 (O s )BZ DZ
CZ [I nZ AZ ]1 BZ DZ ,
since Z1 =s - O.
Thus the following theorem has been proved.
Theorem 7.1.2. If (AZ, BZ, CZ, DZ) is the realisation of the matrix T (Z) given by
(7.1.11), then the matrices (E, A, B0, B1, C, D) defined by (7.1.13) constitute a
realisation of the matrix T(s).
The foregoing consideration endows us with the following procedure for
computing the realisation (E, A, B0, B1, C, D) of T(s) and the minimal realizations
, B , C
).
(E, A, B, C, D) , (E, A
The Realisation Problem and Perfect Observers of Singular Systems
371
Procedure 7.1.1.
Step 1: Write the matrix T(s) in the form (7.1.9) and choose the scalar O in such a
way that d(O) z 0.
Step 2: Substitute s = Z1 + O into T(s) and multiplying both the numerator and the
denominator of (7.1.12) by ZN and compute T (Z).
Step 3: Using one of the well-known methods provided in [150], compute the
realisation (AZ, BZ, CZ, DZ) of T (Z).
Step 4: Using (7.1.13), compute the desired realisation (E, A, B0, B1, C, D) of T(s)
and the minimal realisation (E, A, B, C, D) or (E, A , B , C ) .
Remark 7.1.1. For two different values of O we obtain in a general case two
different realizations (AZ, BZ, CZ, DZ) and the corresponding two different
realisations (E, A, B0, B1, C, D).
Remark 7.1.2. If d(0) z 0, then it is convenient to assume O = 0. In this case, we
obtain the following from (7.1.13)
E
AY , A
In , B0
0, B1
B Z , C CZ , D
DZ .
(7.1.13)
Applying the above procedure we compute the realisation (E, A, B0, B1, C, D)
of the following transfer function
T( s )
aN s N " a1s a0
for N ! q .
s q bq1s q1 " b1s b0
(7.1.14)
Step 1: In this case,
P(s)
aN s N " a1s a0 , d ( s )
s q bq 1s q 1 " b1s b0 .
(7.1.15)
We choose O in such a way that d(O) z 0.
Step 2: Substituting s = Z1 + O into (7.1.14), we obtain
T(Z 1 O )
aN (Z 1 O ) N " a1 (Z 1 O ) a0
, (7.1.16)
(Z 1 O ) q bq 1 (Z 1 O ) q1 " b1 (Z 1 O ) b0
and with both the numerator and denominator of (7.1.16) multiplied by ZN we
obtain
T(Z )
a0Z N " aN
b0Z N " Z N q
a0
a Z N 1 " a1Z a0
.
N N 1 N 1
b0 Z b0Z " bq 1Z N q
Step 3: A controllable realization of the transfer function (7.1.17) is
(7.1.17)
372
Polynomial and Rational Matrices
AZ
CZ
ª0
«0
«
«#
«
«0
«¬0
1
0
"
0
1
"
#
0
#
0
%
"
0 1 "
0 º
0 »»
# »  N u N , BZ
»
1 »
b0 »¼
[a0 a1...aN 1 ], DZ
ª0 º
« #»
« »  N ,
«0 »
« »
¬1 ¼
ª a0 º
« ».
¬ b0 ¼
Note that if N > q, then det AZ = 0 and E is a singular matrix.
Step 4: Using (7.1.13) and (7.1.18), we obtain the desired realization (E, A, B0, B1,
C, D) of the transfer function (7.1.14).
Example 7.1.1.
Compute two realizations (E, A, B0, B1, C, D) of the following transfer function
T (s)
s 2 2s 3
.
s 1
(7.1.19)
Applying the above procedure, we choose two different values of O. We obtain
the following.
Step 1: In this case,
P( s )
s 2 2s 3 and d ( s)
s 1 .
We choose O = 0 and O = 1, since d(0) = 1 and d(1) = 2.
Step 2: Substituting s = Z1 and s = Z1 + 1 into (7.1.19), we obtain
T (Z 1 )
Z 2 2Z 1 3
,
Z 1 1
(7.1.20a)
and
T (Z 1 1)
Z 2 4Z 1 6
,
Z 1 2
(7.1.20b)
respectively. With both the numerator and the denominator of (7.1.20) multiplied
by Z2, we obtain
The Realisation Problem and Perfect Observers of Singular Systems
T1 (Z )
3Z 2 2Z 1
Z2 Z
3
T2 (Z )
6Z 2 4Z 1
2Z 2 Z
3
Z 1
,
Z2 Z
373
(7.1.21a)
and
1
2
Z 12
,
Z 2 12 Z
(7.1.21b)
respectively.
Step 3: The realisations of T1 (Z ) and T2 (Z ) are
A1Z
ª0 1 º
1
«0 1» , BZ
¬
¼
AZ2
ª0 1 º
2
« 0 1 » , BZ
¬
2¼
ª0º
1
«1 » , CZ
¬ ¼
[1, 1], D1Z
(7.1.22a)
[3]
and
ª0 º
2
«1 » , CZ
¬ ¼
ª1 1º
2
«¬ 2 , 2 »¼ , DZ
[3] ,
(7.1.22b)
respectively.
Step 4: Using (7.1.13) and (7.1.22), we obtain the desired realisations of the
transfer function (7.1.19)
E1
A1Z
C1
C1Z
E2
AZ2
ª0 1 º
ª1 0 º 1
«0 1» , A1 «0 1 » , B0
¬
¼
¬
¼
[1, 1], D1 D1Z [3]
ª0º 1
«0» , B1
¬ ¼
B1Z
ª0 º
« 1» ,
¬ ¼ (7.1.23a)
and
2
1
B
2
BZ
ª0 1 º
, A2 In
«
1»
¬0 2 ¼
ª 0º
2
« 1» , C2 CZ
¬ ¼
O AZ2
ª1 1 º
2
«0 1 » , B 0
¬
2¼
ª1 1º
«¬ 2 , 2 »¼ , D2
2
DZ
O BZ2
ª0 º
«1 » ,
¬ ¼
(7.1.23b)
[3].
respectively.
It is easy to verify that the matrices (7.1.23) are indeed realisations of the
transfer function (7.1.19).
374
Polynomial and Rational Matrices
Theorem 7.1.3. The singular system (7.1.5) is both completely controllable and
completely observable if (AZ, BZ) is a controllable pair and (AZ, CZ) is an
observable pair.
Proof. In order to prove the complete controllability of the system (7.1.5), one has
to show that the conditions (7.1.7) are satisfied for this system.
We carry out the proof in detail for a SISO system (m=1, p=1). Without loss of
generality, we can assume that the matrices AZ, BZ and CZ have the form (7.1.18).
Using (7.1.5c), (7.1.13) and (7.1.7), we obtain
BZ
ª E B1 B 0 º
ªA
rank «
rank « Z
»
0 I m ¼
0
¬0
¬ 0
0 0º
ª0 1 0 " 0
«0 0 1 " 0
0 0 »»
«
«# # # % #
#
#»
rank «
» N n.
0 0»
«0 0 0 " 1
« 0 0 1 " b0 1 O »
«
»
0 1»¼
«¬ 0 0 0 " 0
rank ª¬ E, B º¼
O BZ º
I m »¼
Thus the first of the conditions (7.1.7) is satisfied. The second is met as well,
since
ª Es A B1s B 0 º
rank «
I m I m »¼
¬ 0
ª A s ( I n O A Z ) BZ s O BZ º
rank « Z
I m I m »¼
0
¬
"
0
0
0 0º
ª 1 s O
«0
1 s O "
0
0 0 »»
«
rank « 0
"
#
#»
0
0
sO
«
»
s O " b0 ( s O ) 1 s O »
0
«0
«¬ 0
" "
1 1»¼
0
0
rank ª¬ Es A, B º¼
N
n,
for all finite s
Analogously, in order to prove the complete observability of the system (7.1.5),
one has to show that the conditions (7.1.8) are met for this system. Using (7.1.5c),
(7.1.13) and (7.1.8), we obtain
The Realisation Problem and Perfect Observers of Singular Systems
ªE º
rank « »
¬C ¼
ª0
«0
«
«#
«
rank « 0
«0
«
«0
«a
¬ 0
ª E B1 º
rank «« 0
0 »»
«¬C 0 »¼
ª AZ
rank «« 0
«¬ CZ
BZ º
0 »»
0 »¼
0º
0 »»
0»
»
1»
0»
»
0»
»
¼
n.
1
0
0
1
#
0
0
# %
0 "
1 "
0
a1
0 " 0
a2 " aN 1
"
"
0
0
#
1
b0
N
375
Thus the first condition of (7.1.8) is met. The second one is met as well, since
ªEs A B1s º
ª A Z s (I n O A Z ) BZ s º
«
»
I m » rank ««
I m »»
rank « 0
0
CZ
0 »¼
0 ¼»
¬« C
¬«
"
0
0
0º
ª 1 s O
«0
"
1
0
0 »»
sO
«
«0
"
0
0
sO
0»
rank «
» N n.
s O " b0 ( s O ) 1 s »
0
«0
«0
1»
0
"
"
0
«
»
a1
aN 1
"
"
0 ¼»
¬« a0
ª Es A º
rank «
»
¬ C ¼
„
Remark 7.1.3.
Analogously one can prove that the system (7.1.6) is both completely controllable
and completely observable, if (AZ, BZ) is a controllable pair and (AZ, CZ) is an
observable pair.
The foregoing considerations lead to the following important corollary that the
matrices (7.1.13) are a minimal realisation of the transfer matrix (7.1.9).
With the variable s replaced by z, we can apply the method for computing a
minimal realization of a discrete-time singular system as well. The considerations
can be generalised into the case of singular two-dimensional systems.
376
Polynomial and Rational Matrices
7.2 Full- and Reduced-order Perfect Observers
Consider the following continuous-time singular system
Ex
Ax Bu ,
y
Cx ,
x
dx
, x
dt
x 0
(7.2.1a)
x0
(7.2.1b)
where
x t  n , u
u t  m , y
y t  p
are the vectors of the state, input and output, respectively; E, A nun, B num,
C pun.
We will henceforth assume that det E = 0, rank B = m, rank C = p and
det [Es – A] z 0 for certain s (the field of complex numbers).
Consider also a continuous-time singular system described by the equation
Exˆ
Axˆ Bu K Cxˆ y ,
where x̂ = x̂ (t)
in (7.2.1) and K
n
xˆ 0
xˆ0 ,
(7.2.2)
is the state vector, with u, y and E, A, B, C being the same as
.
nup
Definition 7.2.1. The system (7.2.2) is called a full-order perfect observer for the
system (7.2.1) if and only if x̂ (t) = x(t) for t > 0 and arbitrary initial conditions x0,
x̂ .
Theorem 7.2.1. There exists a perfect observer of the form (7.2.2) for the system
(7.2.1) if it is completely observable, that is,
ª Es A º
rank «
»
¬ C ¼
n,
(7.2.3a)
for all finite s and
ªEº
rank « »
¬C ¼
n.
(7.2.3b)
Proof. Let e(t) = x(t) - x̂ (t), t t 0. From (7.2.1) and (7.2.2) we have
Ee
Ex Exˆ
A KC e .
(7.2.4)
The Realisation Problem and Perfect Observers of Singular Systems
377
If the assumptions (7.2.3) are met, then there exists a matrix K such that
det ª¬Es A KC º¼ D z 0 ,
(7.2.5)
for all s , where D is a scalar and independent of s.
If the condition (7.2.5) is met, then from the expansion
ª¬ Es A KC º¼
1
f
¦P ĭ s
i 1
i
i it follows that )0 = 0 and according to (5.3.34) the solution to (7.2.4) is
e t
e ĭ0
A KC t
ĭ0 Ex0
0 for t ! 0 ,
that is x̂ (t) = x(t), for t > 0.
„
Another proof of this theorem is provided in [115, 116].
If the conditions (7.2.3) are met, then we can obtain an observer of the form
(7.2.2) using the following procedure.
Procedure 7.2.1.
Step 1: Choose the matrix K so that the condition (7.2.5) is met.
Step 2: Using (7.2.2) compute the desired observer.
Example 7.2.1.
Compute an observer of the form (7.2.2) for the system (7.2.1) with
E
ª1 0 0 º
«
»
«0 1 0 » , A
¬«0 0 0 »¼
ª0 1 0º
«
»
«1 2 0 » , B
¬« 0 0 1 »¼
ª1 0 º
«
»
«0 1» , C
¬«1 2 »¼
>1
0 1@ . (7.2.6)
In this case, n =3, m = 2, p = 1. The conditions (7.2.3) are met, since
ª Es A º
rank «
»
¬ C ¼
for all finite s , and
1
0º
ªs
« 1 s 2 0 »
»
rank «
«0
1»
0
«
»
1¼
0
¬1
3,
378
Polynomial and Rational Matrices
ª1
«0
rank «
«0
«
¬1
ªE º
rank « »
¬C ¼
0
1
0º
0 »»
0 0»
»
0 1¼
3.
Thus there exists a perfect observer of the form (7.2.2) for this system.
Step 1: Using (7.2.5) for K = [k1 k2 k3]T, we obtain
det ª¬ Es A KC º¼
s k1
1
k2 1 s 2
k3
0
k1
k2
k3 1
2
k3 1 s 2k3 k1 2 s 2k1 k2 k3 1 .
The condition (7.2.5) is satisfied for k1 = 0, k3 = 1 and k2 z 0 (k2 is arbitrary).
For k2 = 1, one has K = [0 1 1]T.
Step 2: The desired observer has the form
ª1 0 0 º
« 0 1 0 » xˆ
«
»
«¬ 0 0 0 »¼
ª0 1 0 º
ª0 1 º
ª0º
« 2 2 1» xˆ « 0 1» u «1 » y .
«
»
«
»
« »
«¬ 1 0 0 »¼
«¬1 2 »¼
«¬1 »¼
7.2.1 Reduced-order Observers
Without losing generality we can assume
C
>C1
C2 @ , det C1 z 0 ,
where C1 pup, C2
In this case,
Q
ªC11
«
«¬ 0
pu(n-p)
.
C11C2 º
»
I n p »¼
(7.2.7)
is a nonsingular matrix and
C CQ
ª¬I p
0 º¼ .
(7.2.8)
The Realisation Problem and Perfect Observers of Singular Systems
379
Defining the new state vector
x
ª x1 º
p
n p
« x » , x1  , x2  ,
¬ 2¼
Q 1 x
we obtain from (7.2.1) and (7.2.8)
Ex Ax Bu ,
y Cx ,
(7.2.9a)
(7.2.9b)
where
E
ª E11 E12 º
«
»
¬E 21 E22 ¼
EQ,
A
E11 , A11  pu p , E22 , A 22  ª ǹ11
«
¬ A 21
n p u n p
A12 º
»
A 22 ¼
AQ,
(7.2.9c)
.
From (7.2.9) it follows that for a given output y the vector x1 is known. Thus a
reduced-order observer should reconstruct only the vector x2.
Consider the following continuous-time, singular system
Eˆ 2 xˆ2
w
ˆ xˆ Bˆ u D
ˆ yD
ˆ y ,
A
2
0
1
xˆ2 0
xˆ20 ,
(7.2.10a)
ˆuH
ˆ yH
ˆ y ,
Fˆ xˆ2 G
0
1
(7.2.10b)
n-p
, u and y are the same as in (7.2.1), w = w(t) n-p,
ˆ , Bˆ , D
ˆ, H
ˆ , A
ˆ , D
ˆ , Fˆ , G
ˆ , H
ˆ are real matrices of appropriate dimensions and
E
2
0
1
0
1
where x̂ = x̂ (t)
det Ê 2 = 0.
Definition 7.2.2. The system (7.2.10) is called a reduced-order perfect observer for
the system (7.2.1) if and only if w(t) = x2(t) for t > 0 and arbitrary initial conditions
x0, x̂ 20.
If
ªE º
rank « 12 »
¬E22 ¼
ª C1C º
rank E « 1 2 » n p ,
¬« I n p ¼»
then there exists a matrix of elementary operations on rows P
ªE º
P « 12 »
¬ E22 ¼
ª0º
«E » ,
¬ 2¼
(7.2.11)
nun
such that
(7.2.12)
380
Polynomial and Rational Matrices
where E2 (n-p)u(n-p) is a singular matrix. Pre-multiplying (7.2.9a) by P and using
(7.2.12), we obtain
E11 x1 A11 x1 A12 x2 B1u ,
E21 x1 E2 x2 A 21 x1 A 22 x2 B 2u ,
(7.2.13a)
(7.2.13b)
where
ª E11 º
«E »
¬ 21 ¼
ªE º ªB º
P « 11 » , « 1 »
¬E21 ¼ ¬ B 2 ¼
A11  pu p
, B1  pum
ªA
PB, « 11
¬ A 21
, A 22  A12 º
A 22 »¼
n p u n p
PA,
, B2  (7.2.13c)
n p um
.
Substituting x1 = y into (7.2.13a) and (7.2.13b), we obtain
E2 x2
A 22 x2 u ,
(7.2.14a)
(7.2.14b)
y
A12 x2 ,
u
B 2u A 21 y E21 y
where
is the new input vector, and
y
E11 y A11 y B1u
the new output. According to Theorem 7.2.1, there exists a perfect observer for the
system (7.2.14) if
det > E2 s A 22 @ z 0 ,
(7.2.15)
for some s
ªE s A 22 º
rank « 2
»
¬ A12 ¼
n p ,
(7.2.16a)
for all finite s and
ªE º
rank « 2 »
¬ A12 ¼
n.
(7.2.16b)
The Realisation Problem and Perfect Observers of Singular Systems
381
We will show that the condition (7.2.16a) is met if and only if (7.2.3a) is
satisfied.
Using (7.2.9c) and (7.2.13c), we can write
ª Es A º
rank «
»
¬ C ¼
0 º ª Es A º ªQ 0 º °½
°­ ª P
rank ® «
»«
»¾
»«
°¯ ¬ 0 I n p ¼ ¬ C ¼ ¬ 0 I n p ¼ ¿°
ª E11s A11
A12 º
«
»
rank « E21s A 21 E2 s A 22 »
«
»
Ip
0
¬
¼
ª E s A 22 º
p rank « 2
».
¬ A12 ¼
Thus the conditions (7.2.16a) and (7.2.3a) are equivalent.
Theorem 7.2.2. There exists a reduced-order perfect observer of the form (7.2.10)
for the system (7.2.1) if the conditions (7.2.11), (7.2.15), (7.2.3a) and (7.2.16b) are
met.
Proof. As already proved, the conditions (7.2.16a) and (7.2.3a) are equivalent. If
the conditions (7.2.16) and (7.2.15) are met, then there exists a matrix K (n-p)up
such that
det > E2 s A 22 KA12 @ D z 0 ,
(7.2.17)
for all s .
In this case, there exists a reduced-order perfect observer of the form
E2 xˆ2
A 22 xˆ2 u K A12 xˆ2 y ,
xˆ2 t
x2 t ,
(7.2.18)
such that
t !0.
(7.2.19)
If det E2 z 0, then there is no K satisfying the condition (7.2.17).
„
If the conditions (7.2.11), (7.2.15), (7.2.3a) and (7.2.16b) are met, then a
reduced-order perfect observer of the form (7.2.10) for the system (7.2.1) can be
computed using the following procedure.
Procedure 7.2.2.
Step 1:With C = [C1 C2] known, compute the matrix Q (given (7.2.7)) along with
the matrices E , A .
Step 2:Compute P satisfying (7.2.12) along with the matrices E2, A22, A12, A21,
A11.
382
Polynomial and Rational Matrices
Step 3:Compute K satisfying the condition (7.2.17).
Step 4: Using the equality
E2 xˆ2
A 22 xˆ2 u K A12 xˆ2 y ,
(7.2.20)
compute the desired reduced order-perfect observer. An estimate x̂ (t) of
the state vector x(t) is given by
xˆ t
ªC11 y t C11C2 xˆ2 t º
«
».
xˆ2 t
¬
¼
ªy t º
Q«
»
¬ xˆ2 t ¼
(7.2.21)
Example 7.2.2.
Compute a reduced-order perfect observer of the form (7.2.20) for the system
(7.2.1) with
E
ª1 0 1º
«0 1 0 » , A
«
»
«¬ 0 0 0 »¼
ª 0 1 1º
«1 2 0 » , B
«
»
«¬ 0 0 1 »¼
ª 1 0º
« 0 1» , C
«
»
«¬ 1 2 »¼
>1
0 1@ . (7.2.22)
In this case, n =3, m = 2, p = 1, C1 = [1], C2 = [0 -1] and there exists a reduced
order-perfect observer, since
§ ª C1C º ·
rank ¨ E « 1 2 » ¸
¨ « I n p » ¸
¼¹
© ¬
§ ª1 0 1º ª 0 1 º ·
¨
¸
rank ¨ «« 0 1 0 »» ««1 0»» ¸
¨ «0 0 0 » «0 1 » ¸
¼¬
¼¹
©¬
and
ª Es Aº
rank «
»
¬ C ¼
1 1 s º
ªs
« 1 s 2
0 »»
rank «
«0
1 »
0
«
»
1 ¼
0
¬1
for all finite s .
Step 1: Using (7.2.7) and (7.2.22), we obtain
3,
ª0 0 º
rank ««1 0 »» 1
«¬0 0 »¼
The Realisation Problem and Perfect Observers of Singular Systems
Q
A
ª1 0 1 º
«0 1 0 » , E
«
»
«¬0 0 1 »¼
ªC11
«
«¬ 0
C11C2 º
»
I n p »¼
AQ
ª 0 1 1º
«1 2 1 » .
«
»
«¬ 0 0 1 »¼
EQ
ª1 0 0 º
«0 1 0» ,
«
»
«¬ 0 0 0 »¼
Step 2: In this case, P = I3 satisfy (7.2.12) and
ª1 0 º ª A11
«0 0 » , « A
¬
¼ ¬ 21
E2
ª B1 º
«B »
¬ 2¼
PB
B
A12 º
A 22 »¼
PA
ª 0 1 1º
«1 2 1 » ,
«
»
¬« 0 0 1 »¼
A
ª 1 0º
« 0 1» .
«
»
«¬ 1 2 »¼
The conditions (7.2.15) and (7.2.16b) are met since
det > E2 s A 22 @
s 2 1
0
1
2s
and
ªE º
rank « 2 »
¬ A12 ¼
ª1 0 º
rank «« 0 0 »»
«¬1 1»¼
2.
Step 3: Using (7.2.17) for K = [k1 k2]T, we obtain
det ¬ª E2 s A 22 KA12 ¼º
s 2 k1
k1 1
k2
k2 1
k2 1 s k2 1 k1 2 k 2 k1 1 .
The condition (7.2.17) is satisfied for k2 = 1, k1 z 1.
For k1 = 2, we have
K
ª k1 º
«k »
¬ 2¼
ª2º
«1 » and det ¬ª E2 s A 22 KA12 ¼º 1 .
¬ ¼
383
384
Polynomial and Rational Matrices
Step 4: The desired reduced-order perfect observer is
ª1 0 º «0 0 » xˆ2
¬
¼
ª 4 1º
ª 0 1º
ª1 º
ª 2º
«1 0 » xˆ2 « 1 2 » u « 0» y «1 » y >1 0@ u ,
¬
¼
¬
¼
¬ ¼
¬ ¼
and the estimate x̂ (t) is given by
ªC11 y t C11C2 xˆ2 t º
«
»
xˆ2 t
¬
¼
xˆ t
ª1 0 1 º
«0 1 0 » ª y t º .
«
» « xˆ t »
«¬0 0 1 »¼ ¬ 2 ¼
Remark 7.2.1.
If
ªE º
rank « 12 »
¬ E22 ¼
n p ,
(7.2.23)
then there exists a standard reduced-order observer for the singular system (7.2.1).
The procedure for computing such an observer is provided in [152].
7.2.2 Perfect Observers for Standard Systems
Consider the following continuous-time standard system
x
Ax B u , x 0
y
Cx ,
x0 ,
(7.2.24a)
(7.2.24b)
with the feedback
u
v Fy
v FCx ,
(7.2.25)
where F mup and v m is the new input.
Substituting (7.2.25) into (7.2.24a), we obtain
Ex
Ax Bv, x 0
y
Cx ,
E
I n BFC .
x0 ,
(7.2.26a)
(7.2.26b)
where
(7.2.27)
The Realisation Problem and Perfect Observers of Singular Systems
385
The matrix F is chosen in such a way as to assure the matrix (7.2.27) is singular.
Then we build for the singular system (7.2.26) a full-order perfect observer,
according to considerations in Sect. 7.2.
We will show that for the standard system (7.2.24)
ªI s A º
rank « n
»
¬ C ¼
n for all s  ,
(7.2.28)
if and only if for the singular system (7.2.26)
ª Es A º
rank «
»
¬ C ¼
n for all finite s  .
(7.2.29)
Using (7.2.27), we obtain
ª Es A º
ª I A BFCs º
rank «
rank « n
»
»
C
¬ C ¼
¬
¼
§ ªI n BFs º ª I n s A º ·
ªI s A º
=rank ¨ «
rank « n
»,
¨ 0 I p » «¬ C »¼ ¸¸
¬ C ¼
¼
©¬
¹
for all s  .
We will also show that for the singular system (7.2.26)
ªE º
rank « »
¬C ¼
n for an arbitrary F .
(7.2.30)
Using (7.2.27), we can write
ªE º
rank « »
¬C ¼
ª I BFC º
rank « n
»
C
¬
¼
§ ªI n
rank ¨ «
¨
©¬ 0
BF º ª I n º ·
¸
I p »¼ «¬ C »¼ ¹¸
ªI º
rank « n »
¬C¼
n.
As it is known, if the condition (7.2.28) is met, then there exists a nonsingular
matrix P nun such that
A
1
P AP
C CP
ª A11 ! A1 p º
«
»
« # % # », B
« A p1 ! A pp »
¬
¼
ª¬C1 C2 ! C p º¼ ,
P 1B,
(7.2.31a)
386
Polynomial and Rational Matrices
where
ª 0
º
ai »  di udi , A ij
«I
»¼
¬« di 1
A ii
Ci
>0
n
¦d
ci @  pudi , ci
¬ª0 aij ¼º  d i ud j
i z j , i, j 1, ... ,
T
ª¬0 ! 0 1 c1,i 1 ! c1 p º¼ ,
(7.2.31b)
p
i
.
i 1
Let
ˆ
C
diag ª¬cˆ1 , ! , cˆ p º¼ , cˆ i
>0
! 0 1@  1udi .
It is easily verifiable that
ˆ,
C CC
(7.2.32)
where
C
0
ª1
«c
« 21 1
« #
#
«
c
c
¬« p1 p 2
! 0º
! 0 »»
.
% #»
»
! 1 ¼»
(7.2.33)
Note that
CB
CPP 1B
CB ,
(7.2.34)
and
E
I n BFC
P 1 I n BFC P
P 1EP .
(7.2.35)
Using (7.2.35) and (7.2.32), we obtain
E
ˆ ,
I n BFC
(7.2.36)
F
.
FC
(7.2.37)
where
The Realisation Problem and Perfect Observers of Singular Systems
387
Theorem 7.2.2. Let the condition (7.2.28) be met and the matrices A , C have the
form (7.2.31). There exists a matrix F such that
E
ªI t
« 1
«0
«0
«¬
e1
0
e2
0º
»
0 » , t1 t2
I t2 »»
¼
n 1, e1  t1 , e2  t2 ,
(7.2.38)
if and only if
CB z 0 .
(7.2.39)
Proof. Necessity.
Bº
ª I
det « n
»
¬ FC I m ¼
ªI BFC B º
det « n
I m »¼
0
¬
det > I n BFC@ ,
ªI
det « n
¬0
det > I m FCB @ .
but we also have
Bº
ª I
det « n
»
¬ FC I m ¼
B
º
I m FCB »¼
Hence
det E
det > I n BFC@ det > I m FCB @ .
If CB = 0, then det E = 1 for an arbitrary F.
Sufficiency. If CB = CB z 0, then also ĈB z 0, since det C z 0. Hence for at
least one k we have ĉ k b k = b kk z 0, where b k is the k-th row of B and ĉ k is the
k-th column of Ĉ = [ ĉ ij]. With the entries of F chosen in the following way
f ij
­ 1
, for i j
°
® bkk
°0,
otherwise
¯
k
,
(7.2.40)
we obtain
E
ˆ
I n BFC
I n f kk bk cˆk
ªIt
« 1
«0
«0
¬«
e1
0
e2
0º
»
0»,
I t2 »»
¼
388
Polynomial and Rational Matrices
where
e1
T
1
ªbk1 bk 2 ! bk ,k 1 º¼ , e2
bkk ¬
T
1
ªbk ,k 1 bk ,k 2 ! bkn º¼ .
bkk ¬
„
Theorem 7.2.3. There exists a feedback matrix K satisfying
det ª¬Es A KC º¼ D z 0 ,
(7.2.41)
if and only if the conditions (7.2.28) and (7.2.39) are met.
Proof. Sufficiency. If the conditions (7.2.28) and (7.2.39) are satisfied, then using
(7.2.41), (7.2.31), (7.2.32), and (7.2.35), we obtain
det ª¬ P 1 Es A KC
ˆ
det ª¬Es A P 1KC º¼ det ªEs A KC
¬
det ª¬Es A KC º¼
P º¼
(7.2.42)
º,
¼
where
K
.
P 1KC
(7.2.43)
Without loss of generality, in order to simplify the considerations, we assume
E
ªI n1
« 0
¬«
eº
, e
0 »¼»
> e1
T
e2 ! en1 @ .
(7.2.44)
Let
ª A n en 1
1
1
¬
i
§
·
¨ ni ¦ d j ¸ ,
j 1
©
¹
K
A n2 en2 1 ! A n p 1 en p1 1
A np k º ,
¼
(7.2.45)
where A i is the i-th column of A , ei is the i-th column of the identity matrix In,
and k = [k1 k2 … kn]T n. Using (7.2.31) and (7.2.45), it is easy to verify that
ˆ
A KC
ª 0
º
k» .
«I
»¼
¬« n1
(7.2.46)
The Realisation Problem and Perfect Observers of Singular Systems
389
Taking into account (7.2.42), (7.2.44) and (7.2.46), we obtain
ˆ º
det ªEs A KC
¬
¼
e1s k1 º
ªs 0 ! 0
« 1 s ! 0
e2 s k2 »»
«
«# # % #
»
#
«
»
« 0 0 ! s en1s k n1 »
«¬ 0 0 ! 1
»¼
kn
e1s k1 e2 s k2 s ! en1s k n1 s n2 k n s n1
det ª¬Es A KC º¼
(7.2.47)
k1 e1 k2 s ! en2 kn1 s n2 en1 kn s n1.
Comparing both sides of (7.2.41) and (7.2.47), we obtain
k
>D
T
e1 ! en1 @ .
(7.2.48)
The necessity can be proved analogously to that for standard systems.
„
Theorem 7.2.4. There exists a full-rank perfect observer for the system (7.2.24) of
the following form
Ex
Ax Bu K Cx y ,
(7.2.49)
if the conditions (7.2.28) and (7.2.39) are met.
Proof. If the assumption (7.2.39) is met, then a matrix F can be chosen so that the
closed-loop system (7.2.26) is singular. According to Theorem 7.2.3, if the
conditions (7.2.28) and (7.2.39) are met, then there exists a matrix K satisfying
(7.2.41) and there exists a perfect observer of the form (7.2.49).
„
If the conditions (7.2.28) and (7.2.39) are met, then a perfect observer of the
form (7.2.49) can be obtained using the following procedure.
Procedure 7.2.3.
Step 1: Compute a matrix P satisfying (7.2.31).
Step 2: Using (7.2.40) compute the matrix F , then
F
1
FC
(7.2.50)
390
Polynomial and Rational Matrices
and
E
I n BFC .
Step 3: Using (7.2.48) and (7.2.43), compute K and
K
1 .
PKC
(7.2.51)
Step 4:Compute the desired observer
Ex
A KC x Bu Ky .
(7.2.52)
Example 7.2.3.
For the standard system (7.2.24) with
A
ª0
«
«1
«0
«
¬0
1
2
1
3
0 2º
0 1»»
, B
0 0»
»
1 1¼
ª1º
« »
« 0 », C
« 1»
« »
¬1¼
ª0 1 0 0 º
«0 1 0 1 » ,
¬
¼
(7.2.53)
one has to compute the perfect observer (7.2.52) with D = 1.
It is easily verifiable that the considered system satisfies the conditions (7.2.28)
and (7.2.39), since
ªI s A º
rank « 4
»
¬ C ¼
1
0 2 º
ªs
« 1 s 2 0
1 »»
«
«0
1
s
0 »
rank «
»
3 1 s 1»
«0
«0
1
0
0 »
«
»
1
0
1 »¼
«¬ 0
4 for all s  and
CB
ª0 º
«1 » .
¬ ¼
Using Procedure 7.2.3 we obtain the following.
Step 1: The matrices (7.2.53) already have the desired form (7.2.31) A = A,
B = B, C = C and P = I4.
The Realisation Problem and Perfect Observers of Singular Systems
Step 2: Using (7.2.53) and (7.2.40), we obtain
F
>0
1
FC
1@ , F
>0
ª1 0 º
1@ «
»
¬1 1 ¼
1
>1
1@
and
E
ª1
«0
«
«0
«
¬0
I 4 BFC
0 0 1º
0 »»
.
0 1 1»
»
0 0 0¼
1 0
Step 3: Using (7.2.48) and (7.2.45) and taking into account that
e1
1, e2
0, e3
>1
1 and A 2
T
2 1 3@ , A 4
>2
we obtain
k
K
K
>D
e1
e2
ª¬ A 2 e3
1
PKC
T
e3 @
>1
A 4 k º¼
ª 1 3º
« 2 0 » 1
«
Ȼ
« 0 0 » «¬1
«
»
¬ 3 0 ¼
T
1 0 1@ ,
ª 1 3º
« 2 0 »
«
»,
«0 0»
«
»
¬ 3 0 ¼
ª2
1
« 2
0º
«
«0
1 »¼
«
¬ 3
3º
0 »»
.
0»
»
0¼
Step 4: The desired observer has the form
ª1
«0
«
«0
«
¬0
0
1
0
0
0 1º
0 0 »»
x
1 1»
»
0 0¼
ª0
«1
«
«0
«
¬0
0
0
1
0
T
1 0 1@ ,
0 1º
ª1º
ª 2 3º
»
«
»
« 2 0 »
0 1»
0
»y.
x « »u «
« 1»
«0 0»
0 0»
»
« »
«
»
1 1¼
¬1¼
¬ 3 0 ¼
391
392
Polynomial and Rational Matrices
7.3 Functional Observers
Consider the continuous-time singular system (7.2.1). We seek a system of the
form
Ez
w
F z Gu H y , z 0
z 0 ,
Lz ,
(7.3.1a)
(7.3.1b)
which reconstructs the desired linear function of the state vector Kx, where
K mun is known, z n is the state vector, w m is the output vector, and u, y as
well as E are the same as for the system (7.2.1); F nun, G num, H nup,
L mun.
Definition 7.3.1. The system (7.3.1) is called a full-order functional observer for
the system (7.2.1) if and only if
w t
Ȁx t
for t ! 0 ,
(7.3.2)
and arbitrary initial conditions x0, z0.
Let
e
xz.
(7.3.3)
Using (7.3.3), (7.2.1) and (7.3.1), we obtain
Ee
Ex Ez
A HC x Fz B G u .
(7.3.4)
If we choose
F
A HC,
B
G,
(7.3.5)
equation (7.3.4) takes the form
Ee
Fe .
(7.3.6)
From (7.3.1b) for L = K, (7.3.2) and (7.3.3), we have
Kx w
Ke .
(7.3.7)
From Definition 7.3.1 and (7.3.7) it follows that the system (7.3.1) is a
functional observer for the system (7.2.1) if and only if e(t) = 0 for t > 0. This
condition is met if and only if there exists a matrix H such that
The Realisation Problem and Perfect Observers of Singular Systems
det ª¬Es A HC º¼
D,
393
(7.3.8)
where D is a nonzero scalar and independent of s.
Theorem 7.3.1. Let the condition (7.2.3) be satisfied. A full-order perfect observer
for the system (7.2.1) exists if and only if
rank A
a º ,
rank ª¬ A
¼
(7.3.9)
where
A
ª a10
«a
« 11
« #
«
¬ a1r
p s
a20
a21
#
a2 r
! an 0 º
ª a0 D º
»
«
! an1 »
a »
, a « 1 » ,
« # »
% # »
»
«
»
! anr ¼
¬ ar ¼
det > Es A @ ar s r ar 1s r 1 ! a1s a0 , r d rank E n , (7.3.10)
and
pk s
det ª¬h1 s ! h k 1 s cT h k 1 s
akr s r ! ak 1s ak 0 , k 1, ! , n,
ª¬h1 s
! h n s º¼
! h n s º¼
(7.3.11)
T
T
¬ªE s A ¼º
is the determinant of ETs - AT with its k-th column replaced by cT.
Proof. Using the Binet–Cauchy theorem, one can easily show that
det ª¬Es A HC º¼
p s h1 p1 s ! k n pn s ,
(7.3.12)
where
HT
>h1
! hn @ .
From (7.3.8) and (7.3.12), we have
h1 p1 s h 2 p2 s ! k n pn s
Dp s .
(7.3.13)
Comparing the coefficients at the same powers of the variable s, we obtain
from (7.3.13) the following equation
394
Polynomial and Rational Matrices
AH
a .
(7.3.14)
It follows by the Kronecker–Capelli theorem that (7.3.14) has a solution H if
and only if the condition (7.3.9) is met.
„
If the conditions of Theorem 7.3.1 are met, then a perfect observer of the form
(7.3.1) can be obtained using the following procedure.
Procedure 7.3.1.
Step 1:Using (7.3.11), compute the polynomials p1(s),…,pn(s) and check if the
condition (7.3.9) is met. If it is, go to Step 2, otherwise the problem is
unsolvable.
Step 2: For a given value of the scalar D, compute the matrix H satisfying (7.3.14).
H can be also computed by choosing its elements in such a way that
di
di H
0 for i 1, ..., q ,
(7.3.15)
and d0 = d0(H) = D, where
det ª¬ET s AT CT HT º¼
det ª¬Es A HC º¼
d q s q ! d1s d 0 .
Step 3: Using (7.3.5), compute F, G and L = K.
Example 7.3.1.
Compute a functional perfect observer of the form (7.3.1) for the system (7.2.1)
with
E
ª1 0 0 º
«0 1 0 » , A
«
»
«¬0 0 0 »¼
ª 0 1 0º
« 0 0 1» , B
«
»
«¬ 1 0 0 »¼
ª1 0 º
«0 1 » , C
«
»
«¬1 1»¼
>1
0 0@ , (7.3.16)
so that it reconstructs a linear function Kx for
K
ª1 2 3 º
« 2 1 2 » and D
¬
¼
2.
In this case, the condition (7.2.3) is met, since
ª Es A º
rank «
»
¬ C ¼
ª s 1 0 º
«0 s 1»
»,
rank «
«1 0 0 »
«
»
¬1 0 0 ¼
(7.3.17)
The Realisation Problem and Perfect Observers of Singular Systems
395
for all finite s .
Using Procedure 7.3.1, we obtain the following.
Step 1: From (7.3.11) and (7.3.10), we have
T
T
¬ªE s A ¼º
p1 s
ª¬h1 s
det »cT
p2 s
h2 s
h2 s
T
det »h1 s
c
h 3 s º¼
h 3 s º¼
1
0
0
h 3 s ¼º
s 1 1
1 0 0
0 0 0
s
p3 s
p s
det ª¬h1 s
T
h2 s
det > Es A @
c º¼
s 1 0
0 s 1
1
ªs 0
«
« 1 s
«¬ 0 1
0 1
s 0 0
1 0
0
0
1º
0 »» ,
0 »¼
(c
C),
0,
1
1 s 0
0 1 0
1,
1,
0
and
A
ª a10
«a
« 11
«¬ a12
a20
a21
a22
a30 º
a31 »»
a32 »¼
ª0 0 1 º
« 0 0 0 » , a
«
»
«¬ 0 0 0 »¼
ª a 0 D º
« a »
« 1 »
«¬ a2 »¼
ª 1º
«0».
« »
«¬ 0 »¼
(7.3.18)
From (7.3.18), it follows that the condition (7.3.9) is satisfied.
Step 2: The equation (7.3.14) for HT = [h1 h2 h3] has the form
ª0 0 1 º ª h 1 º
«0 0 0» « h »
«
»« 2»
«¬0 0 0 »¼ «¬ h3 »¼
ª 1º
«0»
« »
«¬ 0 »¼
and its solution is HT = [h1 h2 -1], where h1 and h2 are arbitrary. The same result is
obtained with the use of the second method, which relies on the relationship
(7.3.15), since
396
Polynomial and Rational Matrices
det ª¬Es A HC º¼
s h1
h2
1 h3
1 0
s 1 1 h3
0 0
D
2.
Step 3: Using (7.3.5) and (7.3.17), we obtain
F
A HC
ª h1 1 0 º
« h 0 1» , G
« 2
»
«¬ 2 0 0 »¼
B
ª1 0 º
«0 1 » , L
«
»
«¬1 1»¼
Ȁ
ª1 2 3 º
«
».
¬ 2 1 2 ¼
The desired functional perfect observer is
ª1 0 0 º
ª h1 1 0 º
ª1 0 º
ªh1 º
«0 1 0 » z « h 0 1 » z «0 1 » u « h » y,
«
»
« 2
»
«
»
« 2»
«¬0 0 0 »¼
«¬ 2 0 0 »¼
«¬1 1»¼
«¬ 1»¼
ª1 2 3 º
w «
» z.
¬ 2 1 2 ¼
The foregoing considerations can be extended into the case of reduced-order
functional perfect observers [71, 108, 115].
7.4 Perfect Observers for 2D Systems
Let
+ be the set of nonnegative integers.
Consider a two-dimensional (2D) system described by the singular second
Fornasini–Marchesini model
Exi 1, j 1
yij
A1 xi 1, j A 2 xi , j 1 B1ui 1, j B 2ui , j 1 ,
(7.4.1a)
Cxij ,
(7.4.1b)
where xij n, uij m and yij p are vectors of state, input and output, respectively,
and E, Ak nun, Bk num, k = 1,2, C pun.
We assume that det E = 0 and
det > Ez A k @ z 0 for some z  and k
1 or k
2.
(7.4.2)
The boundary conditions for (7.4.1) are
xi 0 for i  ' and x0 j for j  ' .
(7.4.1c)
The Realisation Problem and Perfect Observers of Singular Systems
397
We assume that the boundary conditions are subject to a jump-like change for i = 0
and j = 0.
Consider the following 2D singular system
A1 xi 1, j A 2 xi , j 1 B1ui 1, j B 2ui , j 1 Dyi1, j Fyi , j 1 ,
Exi 1, j 1
wij
Cxij Guij Hyij ,
(7.4.3a)
(7.4.3b)
with the boundary conditions
xi 0 for i  ' and x0 j for j  ' ,
(7.4.3c)
where
xij  n , wij  n , E, A k , C  nun , B k , G  num , k
1, 2, D, F, H  nu p .
Definition 7.4.1. The system (7.4.3) is called a perfect observer of the system
(7.4.1) if and only if
wij
xij , for i, j  ' (7.4.4)
and for arbitrary boundary conditions of the form (7.4.1c) and (7.4.3c).
Consider the following particular case of the system (7.4.3)
Exi 1, j 1
A1 xi 1, j A 2 xi , j 1 B1ui 1, j B 2ui , j 1
(7.4.5a)
K1 Cxi 1, j yi 1, j K 2 Cxi , j 1 yi , j 1 ,
wij
xij , i, j  ' ,
where K1, K2
(7.4.5b)
nup
.
Theorem 7.4.1. The system (7.4.5) is a perfect observer of the system (7.4.1) if
ªCº
rank « » for k 1 or k 2 ,
¬Ak ¼
ªE º
ª Ez A k º
rank « » n and rank «
» n for all finite z  ¬C ¼
¬ C ¼
rank C
and k = 2 or k = 1.
(7.4.6)
(7.4.7)
398
Polynomial and Rational Matrices
Proof. Let
eij
xij xij , i, j  ' .
(7.4.8)
Using (7.4.8), (7.4.1a) and (7.4.5a), we obtain
Eei 1, j 1
Exi 1, j 1 Exi 1, j 1
A1 K1C ei 1, j A 2 K 2C ei , j 1 . (7.4.9)
If the condition (7.4.6) is met for k = 1, then K1 can be chosen in such a way
that A1 = K1C, and from (7.4.9), we obtain
Eei 1, j 1
A 2 K 2 C ei , j 1 .
(7.4.10)
If the condition (7.4.7) is met for k = 2, then there exists a matrix K2 such that
det ª¬Ez A 2 K 2C º¼ D z 0, for some z  ,
(7.4.11)
and according to the considerations in Sect. 7.1 eij = 0 and wij = x ij for i, j +. The
proof for k = 2 is analogous.
„
If the conditions (7.4.6) and (7.4.7) are satisfied, then a perfect observer of the
form (7.4.5) of the system (7.4.1) can be obtained using the following procedure.
Procedure 7.4.1.
Step 1: Compute K1 so that A1 = K1C.
Step 2: With the matrices E, A2, C and scalar D given, compute the matrix K2 so that
det ¬ªEz A 2 K 2C ¼º D z 0 .
(7.4.12)
To this end, we can apply the method of elementary operations, provided in
[122].
Step 3: Using (7.4.5) compute the desired observer.
Remark 7.4.1. In the foregoing considerations one can interchange the role of the
matrices A1 and A2, and K1 and K2, respectively.
Example 7.4.1.
Compute a perfect observer of the form (7.4.5) for the system (7.4.1) with
The Realisation Problem and Perfect Observers of Singular Systems
E
B1
ª1
«0
«
«0
«
¬0
0
1
0
0
0
0
0
0
0º
0 »»
, A1
1»
»
0¼
ª1 0 º
«0 2 »
«
», B
2
«1 1»
«
»
¬1 0 ¼
ª0
«1
«
«0
«
¬0
1
2
0
0
0
0
1
0
ª0 1 º
«1 1»
«
», C
«1 0 »
«
»
¬0 1 ¼
0º
0 »»
, A2
0»
»
1¼
> 1
ª 1
«1
«
« 2
«
¬ 3
0 1
0 1
0 2
0 3
0º
0 »»
,
0»
»
0¼
399
(7.4.13)
0 1 0@ .
The system satisfies the conditions (7.4.6) and (7.4.7), since
ªCº
ªEº
rank « » , rank « »
A
¬C ¼
¬ 2¼
rank C
ª Ez A1 º
4 and rank «
»
¬ C ¼
4,
for all finite z . Thus there exists a perfect observer of the form (7.4.5) for this
system. Taking into account Remark 7.4.1 and applying Procedure 7.4.1, we obtain
the following.
Step 1: In this case, K2 = [1 1 2 3]T, since A2 = K2C.
Step 2: Using (7.4.12) it is easily verified that for K1 = [0 1 1 0]T and D = 1, we
obtain
det ª¬Ez A1 K1C º¼
1
z
0 0
0 z 2 1 0
1
0
0 z
0
0
0 1
1.
Step 3: The desired observer is
ª1
«0
«
«0
«
¬0
ª0
«1
«
«1
«
¬0
0
1
0
0
0
0
0
0
0º
ª0
»
«0
0»
xi 1, j 1 «
«1
1»
»
«
0¼
¬0
1º
ª0º
« 1»
1»»
ui , j 1 « » yi 1, j
«1»
0»
»
« »
1¼
¬0¼
1
2
0
0
0
1
0
0
0º
ª1 0 º
»
«0 2 »
0»
»u
xi 1, j «
«1 1» i1, j
0»
»
«
»
1¼
¬1 0 ¼
ª1º
« 1»
« » yi , j 1.
«2»
« »
¬3¼
400
Polynomial and Rational Matrices
With only slight modifications the foregoing considerations apply to 2D
systems described by the Roesser model
ª xh º
E « iv1, j »
¬« xi , j 1 ¼»
yij
>C1
ª A11
«A
¬ 21
A12 º ª xih, j º ª B11 º
u ,
« »
A 22 »¼ ¬« xiv, j ¼» «¬ B 22 »¼ ij
(7.7.14)
ª xh º
C2 @ « iv, j » , i, j  ' ,
«¬ xi , j »¼
where
xih, j  n1 , xiv, j  n2
are the horizontal state vector and vertical state vector, respectively, ui,j
yi,j p are the vectors of the state, input and output, respectively;
ªA
E, « 11
¬ A 21
A12 º ª B11 º
,
,
A 22 »¼ «¬B 22 »¼
>C1
m
and
C2 @
are real matrices of appropriate dimensions.
If E = diag [E1 E2], (E1 n1un1, E2 n2un2), then the model (7.4.14) can be
written in the form (7.4.1), where
xij
B1
ª xih, j º
0 º
ª 0
ªA
, A 2 « 11
« v » , A1 «
»
«¬ xi , j »¼
¬ 0
¬ A 21 A 22 ¼
ª 0 º
ªB11 º
«B » , B 2 « 0 » , C >C1 C2 @ .
¬ ¼
¬ 22 ¼
A12 º
,
0 »¼
These considerations can be generalised into the case of the singular (2D)
general model [147].
7.5
7.5.1
Perfect Observers for Systems with Unknown Inputs
Problem Formulation
Consider the following linear continuous-time system
x
y
Ax Bu Dv ,
Cx ,
(7.5.1a)
(7.5.1b)
The Realisation Problem and Perfect Observers of Singular Systems
401
where x = dx/dt, x n is the state vector, u q is the input vector, v m is the
vector of unknown disturbances, y p is the output vector; A nun, B nuq,
D num, C pun. We assume that rank C = p < n and rank D = m.
We seek an r-th order perfect observer of the form
E1 z Fz Gu Hy,
xˆ Pz Qy,
(7.5.2)
an observer that for t > 0 exactly reconstructs the state vector x in the presence of
the unknown disturbance v, where z r is the state vector of the observer, x̂ is an
estimate of x, E1, F rur, det E1 = 0, G ruq, H rup, P nur and Q nup.
Let e r be an error of the observer defined as
z Tx ,
e
(7.5.3)
where T run.
Differentiating (7.5.3) with respect to t and using (7.5.1) along with (7.5.2), we
obtain
E1e
E1 z E1Tx
Fz Gu HCx E1TAx E1TBu Ǽ1TDv
Fe FT E1TA HC x G E1TB u E1TDv .
If
E1TB
G,
FT E1TA HC
E1TD
(7.5.4)
0,
(7.5.5)
0,
(7.5.6)
then
E1e
Fe .
(7.5.7)
Note that
xˆ x
Pz QCx x
Pz QCx PTx PTx x
Pe QC PT I n x
if
Pe,
402
Polynomial and Rational Matrices
PT QC
>P
ªT º
Q@ « »
¬C ¼
In .
(7.5.8)
According to the considerations in Sect. 7.1, if
det E1 s F
D z0,
(7.5.9)
where D does not depend on s, then e = 0 for t > 0.
The problem of a reduced-order perfect observer with unknown disturbances
can be formulated in the following way. Given the matrices A,B,C,D compute
E1,F,G,H,T,P,Q in such a way that the relationships (7.5.4), (7.5.5), (7.5.6), (7.5.8)
and (7.5.9) hold true.
7.5.2
Problem Solution
The relationship (7.5.5) can be written as
>F
ªTº
H@ « »
¬C ¼
E1TA .
If rank F = r, then from the Sylvester inequality it follows that
r + n – (r + p) d rank E1TA, and taking into account det E1 = 0, we obtain
r > n p.
Since rank E1TD = 0, we have rank E1T + m – n d 0 and rank E1T < n m.
Hence rank E1 < n m rank. Thus we have p t m.
Lemma 7.5.1. There exist a pair of nonsingular matrices (L, R) that transform the
system matrices into the form
LAR
D2
ª A1 A 2 º
« A A » , CR
4¼
¬ 3
ª0 I m p n º
pum
«0
» ,
0
¬
¼
A
C
ª¬0 I p º¼ , LD
if and only if rank C = p and rank D = m (p d m), where A1
A3 pu(n-p), A4 pup, D1 = [In-p 0] (n-p)um.
D
ª D1 º
« D » , (7.5.10)
¬ 2¼
(n-p)u(n-p)
, A2
(n-p)up
,
Proof. As it is known, if rank C = p, then there exists a nonsingular matrix R1 such
that CR1 = [C1 C2], where C2 pup and rank C2 = p. Thus there exists a
nonsingular matrix R such that
The Realisation Problem and Perfect Observers of Singular Systems
CR
CR1R 2
>C1
0 º
ª I
C2 @ « n1p
1 »
¬ C2 C1 C2 ¼
403
ª¬0 I p º¼ .
Analogously, using the matrix
L2
ˆ 1
ª D
1
« 1
ˆ
ˆ
D
¬« 1 D2
0 º
»
I nm ¼»
and performing an appropriate partition into the blocks D1 and D2 of D, we obtain
(7.5.10).
„
Note that in the course of transformation of the matrices of the system (7.5.1)
into the form (7.5.10), the state vector is also transformed, according to the
relationship x̂ = R1x. It follows from the condition p < n that D2 is not a full rank
matrix.
Let r = 2n – m p. We choose the matrices E1 and F to be of the form
E1
ªI n p
« 0
¬
0 º
, F
0nm »¼
ª 0
«D I
¬ nm
I n p º
.
0 »¼
(7.5.11)
It is easily verifiable that the matrices (7.5.11) satisfy the condition (7.5.9).
Let
T
ª T1
«T
¬ 3
T2 º
,
T4 »¼
where
T1  nm u n p , T2  nm u p , T3  n p u n p , T4  n p u p
and
X
.
FT E1TA
(7.5.12)
Note that the equation HC = -X has a solution if and only if for the given
matrices C and X
rank C
ª Xº
rank « » .
¬C ¼
(7.5.13)
404
Polynomial and Rational Matrices
From (7.5.13) it follows that this condition can be met if and only if the entries
of the first n p columns of X are zero.
Let
[a ], i 1, ..., n, j 1, ..., n .
T [tij ], i 1, ..., r , j 1, ..., n and A
ij
Using (7.5.10) and (7.5.11), we obtain
X
ªt
I n p º « 11
#
0 »¼ «
«tr ,1
¬
ª 0
«D I
¬ n p
ª tnm1,1
« #
«
«t2 nn p ,1
«
« D t11
« #
«
¬« D tnm ,1
!
%
!
!
%
!
ª t11
« #
! t1,n º «
» «t
% # » « n p ,1
0
! tr ,n »¼ «
« #
«
¬« 0
tnm1,n º ª c11
# »» « #
«
t2 nm p ,n » « cn p ,1
»«
D t1,n » « 0
# » « #
» «
D tnm,n ¼» ¬« 0
!
%
!
!
%
!
!
%
!
!
%
!
t1,n º
# »»
ª a11 ! a1,n º
t n p ,n » «
»
» # % # »
0 »«
« an ,1 ! an ,n »
¼
# »¬
»
0 ¼»
(7.5.14)
c1,n º
# »»
cn p ,n »
»,
0 »
# »
»
0 ¼»
where
cij
n
¦t
i ,k
ak , j .
k 1
The condition (7.5.13) and D z 0 imply ti,j = 0, for i = 1,…,nm and
j = 1,…,np, that is, T1 = 0, this in turn implies rank D2 < m.
From (7.5.8) it follows that
ªT º
rank « »
¬C ¼
ª T1
«
rank «T3
«0
¬
T2 º
»
T4 »
I p »¼
n.
If T1 = 0, then rank T3 = np.
Let
T2
ª t1,n p 1 ! t1,n º
«
»
%
# »  n p u p .
« #
«tn p ,n p1 ! tn p ,n »
¬
¼
(7.5.15)
The Realisation Problem and Perfect Observers of Singular Systems
405
The equalities
tn m i , j
ci , j
n
¦t
n
a
i ,l l , j
l 1
¦
ti ,l al , j , for i, j 1, ..., n p
l n p 1
are equivalent to
T3
T2 A3 .
(7.5.16)
The condition (7.5.15) for T1 = 0 implies rank T3 = n p. If p < n p, then this
condition cannot be met. Otherwise if p t n p, T 2 has full rank n p and
rank A3 = np. This explains the choice made earlier that r = 2n – m p. It
guaranties that rank T3 = n p.
Let X1 be the matrix built from the columns n p + 1,…,2n – m p of X.
Taking into account that HC = H[0 Im] = X = [0 X1], we obtain
H X1 .
(7.5.17)
From (7.5.8), we have
R
>P
ªTº
Q@ « » R
¬C ¼
>P
ª TR º
Q@ « » .
¬C¼
(7.5.18)
R is a nonsingular matrix, hence
>P
Q@
ªTR º
R« » ,
¬C¼
where denotes the Moore–Penrose pseudo-inverse.
The following procedure ensues from the foregoing considerations.
Procedure 7.5.1.
Step 1:Compute the nonsingular matrices L and R transforming the matrices of
the system (7.5.1) into the form (7.5.10).
Step 2:Choose the matrices E1 and F of the form (7.5.11)
Step 3:Choose T1 = 0 and T2 with rank n-m.
Step 4: Using the computed in Step 3 ti,j and (7.5.16), compute ti,j,
i = nm+1,…,2nmp, j = 1,…,np.
Step 5:Taking arbitrary values of ti,j (i = nm+1,…,2nmp, j = np+1,…,n) and
using (7.5.4) along with (7.5.17), compute the matrices G and H.
Step 6:Using (7.5.18) compute P and Q.
406
Polynomial and Rational Matrices
From (7.5.10), we have
ª L 0 º ª Is A D º ª R 0 º
«
»«
0 ¼» ¬« 0 I ¼»
¬0 I¼ ¬ C
ªLRs A1
A2
«
Is A 4
« A3
«
Ip
0
¬
D1 º
»
D2 » .
0 »¼
(7.5.19)
Assume at the beginning that rank D = m. Using the matrix D1 and applying
elementary operations we can eliminate from LRs – A1 the entries dependent on s;
with use of Ip, the same can be done for Is – A4. Hence
ª Is A D º
rabk «
0 »¼
¬ C
n m for all s  ,
if and only if rank A3 = np. From (7.5.19) it follows that the condition p t np is
satisfied if p t m, since p + m t n.
Thus the following theorem has been proved.
Theorem 7.5.1. Applying Procedure 7.5.1, one can compute the desired perfect
observer if and only if
1. p t m,
ª Is A D º
n m for all s  .
2. rank «
0 »¼
¬ C
Example 7.5.1.
Compute a perfect observer of the form (7.5.2) for the system (7.5.1) with
A
ª 1 0 0 0 0 º
« 0 2 0 0 0 »
«
»
« 0 0 3 0 0 » , Ǻ
«
»
« 0 0 0 4 0 »
«¬ 0 0 0 0 5»¼
D
ª1
«0
«
«0
«
«0
«¬0
0º
1 »»
0» , C
»
0»
0 »¼
ª0
«0
«
«1
«
«0
«¬0
0º
0 »»
0» ,
»
1»
0 »¼
ª0 0 1 0 0 º
«0 0 0 1 0 » .
«
»
«¬0 0 0 0 1 »¼
Applying Procedure 7.5.1 we obtain, the following.
Step 1: The matrices (7.5.20) already have the desired forms (7.5.10).
(7.5.20)
The Realisation Problem and Perfect Observers of Singular Systems
407
Step 2: In this case, m = 2, p = 3 and we choose
E1
ª1
«0
«
«0
«
«0
«¬ 0
0 0 0 0º
1 0 0 0 »»
0 0 0 0» , F
»
0 0 0 0»
0 0 0 0 »¼
ª 0 0 0 1 0º
« 0 0 0 0 1»
«
»
«D 0 0 0 0 » .
«
»
« 0 D 0 0 0»
«¬ 0 0 D 0 0 »¼
Step 3: In this example,
>T1
ª0 0 1 0 0º
«0 0 0 1 0» .
«
»
¬«0 0 0 0 1 »¼
T2 @
Step 4: Using (7.5.16), we obtain
T
ª0
«0
«
«0
«
«1
«¬0
0
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0º
0 »»
1» .
»
0»
0 »¼
(7.5.21)
Taking into account (7.5.20) and (7.5.12), we obtain
X
ª0
«0
«
«0
«
«0
«¬0
0 3
0 0
0 D
0 0
0 0
0
4
0
D
0
0 º
0 »»
0 ».
»
0 »
D »¼
Step 5: Using (7.5.17) and (7.5.22), we obtain
H
ª 3
« 0
«
« D
«
« 0
«¬ 0
0
4
0
D
D
0º
0 »»
0» ,
»
0»
0 »¼
(7.5.22)
408
Polynomial and Rational Matrices
and from (7.5.4)
G
ª1
«0
«
«0
«
«0
«¬0
0º
1 »»
0» .
»
0»
0 »¼
Step 6: From (7.5.18) and (7.5.21), we have
P
0
0
ª 0
« 0
0
0
«
« 0, 5 0
0
«
« 0 0, 5 0
«¬ 0
0 0, 5
1 0º
0 1 »»
0 0» , Q
»
0 0»
0 0 »¼
0
0 º
ª 0
« 0
0
0 »»
«
« 0, 5 0
0 ».
«
»
« 0 0, 5 0 »
«¬ 0
0 0, 5»¼
Thus the desired observer is
ª1
«0
«
«0
«
«0
«¬ 0
xˆ
0 0 0 0º
1 0 0 0 »»
0 0 0 0 » z
»
0 0 0 0»
0 0 0 0 »¼
0
0
ª 0
« 0
0
0
«
«0, 5 0
0
«
« 0 0, 5 0
«¬ 0
0 0, 5
ª0 0 0
«0 0 0
«
«D 0 0
«
«0 D 0
«¬ 0 0 D
1 0º
ª1
«0
0 1 »»
«
0 0» z «0
»
«
0 0»
«0
»
«¬ 0
0 0¼
0º
ª 3
« 0
1 »»
«
0 » u « D
»
«
0»
« 0
»
«¬ 0
0¼
1 0º
0
0 º
ª 0
»
«
0 1»
0
0 »»
« 0
0 0 » z «0, 5 0
0 »y.
»
«
»
0 0»
« 0 0, 5 0 »
«¬ 0
0 0 »¼
0 0, 5»¼
0
4
0
D
D
0º
0 »»
0» y,
»
0»
0 »¼
The Realisation Problem and Perfect Observers of Singular Systems
409
7.6 Reduced-order Perfect Observers for 2D Systems with
Unknown Inputs
7.6.1 Problem Formulation
Consider the following 2D system
Exi 1, j
yij
A 0 xij A1 xi , j 1 Buij Dvij ,
(7.6.1a)
i, j  ' ,
Cxij ,
(7.6.1b)
where xij n is the state vector, uij m the input vector, vij q the vector of
unknown disturbances, yij p the output vector; E, A0, A1 nun, B num, D nuq,
C pun.
We assume that det E = 0 and
rank C
p.
(7.6.2)
The boundary conditions for (7.6.1a) have the form
x0 j , for j  ' (7.6.3)
Consider the following singular 2D system
E1 zi 1, j
xˆij
F0 zij F1 zi , j 1 Guij H 0 yij H1 yi , j 1 ,
(7.6.4a)
Pzij Qyij ,
(7.6.4b)
with the boundary conditions
(7.6.4c)
z0 j for j  ' where x̂ ij n is an estimate of xij and zij r, E1, F0, F1
H0, H1 rup, det E1 = 0.
rur
, G
rum
,
Definition 7.6.1. The singular system (7.6.4) is called a reduced-order perfect
observer of the system (7.6.1) with unknown disturbances, if
xˆij
xij , for i, j  ' ,
and arbitrary boundary conditions of the form (7.6.3) and (7.6.4c).
Let
(7.6.5)
410
Polynomial and Rational Matrices
eij
(7.6.6)
zij TExij
be the error of the observer, with T
obtain
E1ei 1, j
E1 zi 1, j E1TExi 1, j
run
. Using (7.6.6), (7.6.4) and (7.6.1), we
F0 eij TExij
F1 ei , j 1 TExi , j 1 Guij H 0Cxij H1Cxi , j 1
E1ȉǹ 0 xij E1TA1 xi , j 1 E1TBuij E1TDvij
F0 eij F1ei , j 1 F0 TE H 0C E1TA 0 xij
(7.6.7)
F1TE Ǿ1C E1TA1 xi , j 1
G E1TB uij E1TDvij .
If
F0 TE H 0C E1TA 0
0,
F1TE H1C E1TA1
0,
G
(7.6.8a)
E1TB ,
E1TD
(7.6.8b)
0,
(7.6.8c)
then
E1ei 1, j
F0 eij F1ei , j 1 .
(7.6.9)
Form (7.6.4b), (7.6.6) and (7.6.1b), we have
xˆij xij
P eij TExij QCxij xij
xˆij xij
Peij ,
Peij PTE QC I n xij
and
(7.6.10)
if and only if
PTE QC
>P
ªTE º
Q@ « »
¬C¼
In .
(7.6.11)
Note that a pair of matrices P, Q satisfying (7.6.11) can be found if and only if
The Realisation Problem and Perfect Observers of Singular Systems
ªTE º
rank « »
¬C¼
n.
411
(7.6.12)
From the equality
ªTE º
«C»
¬ ¼
ªT 0 º ª E º
«0 I » « » ,
p ¼ ¬C ¼
¬
it follows that the condition (7.6.12) implies
ªEº
rank « »
¬C ¼
n.
(7.6.13)
Henceforth we will assume that the condition (7.6.13) is met.
The problem of computing a perfect observer can be formulated in the
following manner.
With the matrices E, A0, A1, B, C, D given, one has to compute the matrices of
the observer (7.6.4) E1, F0, F1, G, H0, H1, P, Q so that the conditions (7.6.8) and
(7.6.11) are met.
7.6.2 Problem Solution
Lemma 7.6.1. Let the conditions (7.6.2) and (7.6.13) be met, and p + rank E = n.
Then there exist nonsingular matrices U, V nun such that
E
Ak
ªI r c 0º
« 0 0 » , r c rank E, C CV ª¬0 I p º¼ ,
¬
¼
k
k
ª A11
º
A12
nrc u nrc
k
, k 0,1, A11
,
 r cu r c , A k22  UA k V « k
k »
(7.6.14)
¬ A 21 A 22 ¼
UEV
UB
ª B1 º
n r c um
r cu m
, UD
«B » , B1  , B 2  ¬ 2¼
D1  r cu q , D2  n r c uq
ª D1 º
«D » ,
¬ 2¼
.
Proof. As it is well-known there exist nonsingular matrices U, V1
UEV1
ªI rc
«0
¬
0º
.
0 »¼
Let CV1 = [C1 C2], C1 pu(n-p), C2
(7.6.13) imply det C2 z 0. Hence the matrix
nun
such that
(7.6.15)
pup
. The assumptions (7.6.2) and
412
Polynomial and Rational Matrices
V2
0 º
ª I rc
« C C C1 »
2 ¼
¬ 2 1
(7.6.16)
is nonsingular and
>C1
CV
C2 @ V2
ª¬0 I p º¼ , UEV
ªI rc
«0
¬
0º
0 »¼
E,
(7.6.17)
where V = V1V2.
„
Henceforth we assume that the matrices E and C are of the form (7.6.14).
Lemma 7.6.2. If
det > E1 z1 F0 F1 z2 @ D ,
(7.6.18)
where D is a nonzero scalar independent of z1 and z2, then a solution to (7.6.9)
satisfies the condition
eij
0, for i, j ! 0 .
(7.6.19)
Proof. Let e(z1, z1) be the 2D Z transform of eij, defined as
e z1 , z2
Z ª¬eij º¼
f
f
i j
2
ij 1
¦¦ e z
z
.
(7.6.20)
i 0 j 0
Taking into account that
Z ª¬ei 1, j º¼
z1 ª¬e z1 , z2 e 0, z2 º¼ ,
Z ª¬ei , j 1 º¼
z2 ª¬e z1 , z2 e z1 , 0 º¼ ,
where
e 0, z2
f
j
0j 2
¦e
z , e z1 , 0
j 0
f
i
i0 1
¦e
z ,
i 0
we obtain from (7.6.9)
e z1 , z2
1
>E1 z1 F0 F1 z2 @
¬ªE1 z1e 0, z2 F1 z2 e z1 , 0 º¼ .
(7.6.21)
The Realisation Problem and Perfect Observers of Singular Systems
413
If the condition (7.6.18) is met, then
n1
1
>E1 z1 F0 F1 z2 @
n2
k l
k , l 1 2
¦¦ T
z z ,
(7.6.22)
k 1 l 1
where
F01 , T1,2
T1,1
F01F1T1,1 , T2,1
F01Ǽ1T1,1 , !
and the pair (n1, n2) is the nilpotent index. Note that (7.6.18) implies det F0 z 0.
Substituting (7.6.22) into (7.6.21), we obtain
e z1 , z2
n1
n2
k l
k , l 1 2
¦¦ T
k 1 l 1
z z ª¬E1 z1e 0, z2 F1 z2 e z1 , 0 º¼ .
(7.6.23)
From (7.6.23) and (7.6.20) it follows that eij = 0, for i, jt 0.
„
If r =rc + 1 and
E1
F1
ªI rc 0º
ª 0 I rc º
rur
r ur
« 0 0 »  , F0 « D 0 »  ,
¬
¼
¬
¼
ªF1c 0 º
r ur
r cur c
« 0 0 »  , F1c  ,
¬
¼
(7.6.24)
then the condition (7.6.18) is met, since
ª I z Fc z
det « rc 1 1 2
D
¬
I r c º
» D.
0 ¼
The choice of T is of crucial importance for the problem solution.
Equation (7.6.8a) can be written as
ªH0 º
«H » C
¬ 1¼
ª E1TA 0 F0 TE º
« E TA F TE » .
¬ 1 1 1 ¼
For C = [0 Ip], (7.6.25) has the solution
H
ªH0 º
«H » ,
¬ 1¼
(7.6.25)
414
Polynomial and Rational Matrices
if and only if
ªWº
rank « »
¬C¼
rank C ,
(7.6.26)
where
W
ªE1TA 0 F0 TE º
« E TA F TE »
¬ 1 1 1 ¼
> W1
W2 @ , W1  2 rurc , W2  2 ru p .
Lemma 7.6.3. Let the matrices E1, F0, and F1 have the form (7.6.24). The
considered problem has a solution if T is chosen in such a way that
ªT º
rank « 1 »
¬T3 ¼
D
T11
rc ,
(7.6.27a)
Ker > T1 T2 @ ,
0, T12c
(7.6.27b)
T1A10 T2 A 30 , F1cT1
T1A11 T2 A13 ,
(7.6.27c)
where
T
ª T1
«T
¬ 3
T2 º
, T2  rcu p , T3  1urc , T4  1u p , T1
T4 ¼»
ª T11 º
«T » ,
¬ 12 ¼
ª A1k A 2k º
,
« k
k»
¬ A3 A 4 ¼
A1k  rcurc , A k2  rcu p , A 3k  purc , A k4  pu p , k 0,1.
c
c
T11  1urc , T12  r 1 ur , T12c
>0
ªT º
I rc @ « 1 » , A k
¬ T3 ¼
Proof. If the condition (7.6.27a) is met, then (7.6.12) holds true, since
0º
ªTE º
and rank « »
0 »¼
¬C¼
TE
ª T1
«T
¬ 3
E1T
ªT1 T2 º
«0 0»
¬
¼
With
ª T1
«
rank «T3
«0
¬
0º
»
0»
I p »¼
n.
The Realisation Problem and Perfect Observers of Singular Systems
415
it is easy to show that (7.6.27b) implies the condition (7.6.8c).
The condition (7.6.26) is met if and only if W1 = 0. Taking into account that
W
> W1
W2 @
ªE1TA 0 F0 TE º
« E TA F TE »
¬ 1 1 1 ¼
ª T1A10 T2 A 30 T12c T1A 02 T2 A 04 º
«
»
0
D T1
«
»,
«T1A11 T2 A13 F1c T1 T1A12 T2 A14 »
«
»
0
0
¬«
¼»
we obtain
W1
ª T1A10 T2 A 30 T12c º
«
»
D T1
«
»
«T1A11 T2 A13 F1c T1 »
«
»
0
«¬
»¼
0.
(7.6.28)
If the conditions (7.6.27c) are met, then (7.6.28) holds true.
If (7.6.28) holds, then from (7.6.25) we have H = W2, and from (7.6.8b) we can
compute the matrix G. If the condition (7.6.27a) is met, then from (7.6.11) we can
compute the matrices P,Q. In a general case (7.6.11) has many solutions.
„
From (7.6.27) it follows that r t rc + q.
Lemma 7.6.4. Let the matrices E1, F0, F1 have the form (7.6.24). There exists
T run satisfying the conditions (7.6.27), if and only if
(7.6.29)
ptq
and
ªE z A 0 A1 z2
rank « 1 1
C
¬
where
Dº
0 »¼
n q for all z1 , z2  u ,
(7.6.29b)
is the field of complex numbers.
Proof. Note that there exists a matrix T such that
ªE1T 0 º
rank «
»
¬ 0 Ip ¼
rc p .
(7.6.30)
416
Polynomial and Rational Matrices
Using the Sylvester inequality along with (7.6.29), (37.6.0), (7.6.24), and
(7.6.8c), we obtain
­° ª E1T 0 º ª E1 z1 A 0 A1 z2
rank ® «
»«
C
¯° ¬ 0 I p ¼ ¬
D º ½°
¾
0 »¼ ¿°
ª E TEz1 E1TA 0 E1TA1 z2 E1TD º
rank « 1
C
0 »¼
¬
ª T z T1A10 T2 A 30 T1A11 T2 A13 z2
rank « 1 1
0
«¬
(7.6.31)
T1A 02 T2 A 04 T1A12 T2 A14 z2 º
» t rc q
Ip
»¼
where E1TD = 0.
The condition (7.6.31) is equivalent to the following one
rank [T1 z1 T1A10 T2 A 30 T1A11 T2 A13 z2 ] t r c q p ,
(7.6.32)
which can be met if and only if (7.6.29) holds true.
„
Theorem 7.6.1. Let r t rc + q, rc + p = n and the condition (7.6.13) be satisfied.
The considered problem of the synthesis of a perfect observer has a solution if and
only if the conditions (7.6.29) are met.
Proof. The condition (7.6.13) implies (7.6.27a). There exists T such that
T12c
>0
ªT º
I rc @ « 1 »
¬T3 ¼
>T1
ªA0 º
T2 @ « 10 » ,
¬ A3 ¼
if and only if
rank > T1
T2 @
ªA0 º
rank « 10 »
¬ A3 ¼
rc .
A proper choice of F1c always makes the condition
F1c T1
satisfied.
T1A11 T2 A13
(7.6.33)
The Realisation Problem and Perfect Observers of Singular Systems
417
From (7.6.32) it follows that the condition (7.6.33) is met if and only if (7.6.29)
is met.
„
The foregoing considerations yield the following procedure for computing the
observer (7.6.4).
Procedure 7.6.1.
Step 1:Compute the matrices U, V that transform the matrices E, C, Ak, B, D,
k = 0,1 into the form (7.6.14).
Step 2:Choose the matrices E1, F0, F1 that are of the form (7.6.24)
Step 3:Choose the matrix T that satisfies the condition (7.6.27) for r t rc + q.
Step 4:Compute
H
ªH0 º
«H »
¬ 1¼
ª T1A 02
«
«
« T1A12
«
«¬
T2 A 04 º
»
0
».
T2 A14 »
»
0
»¼
(7.6.34)
Step 5: Using (7.6.8b) and (7.6.11) compute G, P, and Q.
Example 7.6.1.
Compute a perfect observer of the form (7.6.4) for the system (7.6.1) with
E
B
ª1 0
«0 0
«
¬«0 0
ª1º
« 2 »,
« »
«¬ 1»¼
0º
0 »» , A 0
0 ¼»
ª1º
D «« 0 »» ,
«¬ 1»¼
ª 1 2 1 º
« 2 0 3» , A
1
«
»
«¬ 1 1 2 ¼»
C
ª0 1 0 º
«0 0 2 » ,
«
»
¬« 2 1 1¼»
(7.6.35)
ª0 1 0º
«0 0 1 » .
¬
¼
In this case, n = 3, rc = 1, m = q = 1, p = 2, r = 2. The conditions (7.6.29) are
met, since
ª E z A 0 A1 z2
rank « 1 1
C
¬
for all (z1, z2) u .
Dº
0 »¼
ª z1 1
« 2
«
rank «1 2 z2
«
« 0
«¬ 0
2 z2
1
0
3 2 z2
1 z2
2 z 2
1
0
0
1
1º
0 »»
1»
»
0»
0 »¼
4,
418
Polynomial and Rational Matrices
Let
T
ª t11 t12
«t
¬ 21 t22
t13 º
.
t23 »¼
Applying Procedure 7.6.1 we obtain, the following.
Step 1: Matrices (7.6.35) already have the desired forms.
Step 2: We choose
E1
ª1 0 º
« 0 0 » , F0
¬
¼
ª 0
« D
¬
1º
, F1
0 »¼
ªf
«0
¬
Step 3: The conditions (7.6.27) are met if
t11
0, t13
0, t21
2t12 z 0
and t12, t22, t23 are arbitrary.
Step 4: Using (7.6.34), we obtain
H
ªH0 º
«H »
¬ 1¼
ª T1A 02
«
«
« T1A12
«
¬«
T2 A 04 º
»
0
»
T2 A14 »
»
0
¼»
ª 0 3t12 º
«0 0 »
«
».
« 0 2t12 »
«
»
¬0 0 ¼
Step 5: Using (7.6.8b) and (7.6.11), we obtain
G
E1TB
P
ª
« p12
«
« p21
«
« p31
«¬
ª 0 t12
«0 0
¬
ª1º
0º « »
2
0 »¼ « »
«¬ 1»¼
and
1 º
2t12 »
»
0 », Q
»
0 »
»¼
The observer we seek is
ª0 0 º
«
»
«1 0 » .
«¬0 1 »¼
ª 2t12 º
« 0 »,
¬ ¼
0º
.
0 »¼
The Realisation Problem and Perfect Observers of Singular Systems
ª1 0 º
ª 0 1º
ª f 0º
«0 0 » zi 1, j « D 0 » zij « 0 0» zi , j 1
¬
¼
¬
¼
¬
¼
ª 2t12 º
ª 0 3t12 º
ª0 2t12 º
« » uij «
» yij «0 0 » yi , j 1 ,
¬ 0 ¼
¬0 0 ¼
¬
¼
1 º
ª
« p12 2t »
ª0 0 º
12
«
»
xˆij « p21
0 » zij ««1 0 »» yij ,
«
»
«¬0 1 »¼
0 »
« p31
«¬
»¼
where D, f, p12, p21, p31 are arbitrary.
419
8
Positive Linear Systems with Delays
8.1 Positive Discrete-time and Continuous-time Systems
8.1.1 Discrete-time Systems
num
be the set of num matrices with entries from the field of real numbers and
. The set of num matrices with real nonnegative entries will be denoted by
num
and +n = +nu1 The set of nonnegative integers will be denoted by +
+
Consider the discrete-time linear system with delays described by the equations
Let
n
=
nu1
xi 1
q
h
¦A
x
k i k
k 0
yi
¦ B j ui j , i  ' ,
(8.1.1a)
j 0
Cxi Dui ,
(8.1.1b)
where h and q are positive integers, xi n, ui m, yi p are the state, input and
output vectors, respectively, and Ak nun (k = 0,1,…,h), Bj num (j = 0,1,…,q),
C pun, D pum.
The initial conditions for (8.1.1a) are given by
x i  n , (i
0,1,..., h), u j  m ( j 1, 2,..., q ).
Theorem 8.1.1. The solution to (8.1.1a) is given by
(8.1.2)
422
Polynomial and Rational Matrices
xi
ĭ(i ) x0 1 h j 1
¦ ¦ ĭ(i k )A
k 1 j
xj j h k 1
1 q j 1
¦ ¦ ĭ(i k )B
k 1 j
uj
j q k 1
(8.1.3)
q
i 1
¦¦ ĭ(i 1 k j )B k u j ,
j 0 k 0
where
1
h
· °½
°­§
Z 1 ®¨ zI n ¦ A k z k ¸ z ¾
k 0
¹ °¿
°¯©
ĭ(i )
(8.1.4)
is the state-transition matrix and Z1 denotes the inverse z-transform.
The state-transition matrix )(i) satisfies the equation
ĭ(i 1)
A 0ĭ(i ) A1ĭ(i 1) ... A hĭ(i h) ,
(8.1.5)
with the initial conditions
ĭ(0)
I n , ĭ(i )
(8.1.6)
0, for i 0.
Proof. It is easy to verify that (8.1.3) satisfies the initial conditions (8.1.2).
Substituting (8.1.3) into (8.1.1a) and using (8.1.5) and (8.1.6), we obtain
q
h
¦A
x
k i k
k 0
h
¦ B j ui j
¦A
j 0
k 0
1 h j 1
¦
¦ ĭ(i 2k )B
h
ª
«ĭ(i k ) x0 ¦ ĭ(i 2k ) A k j 1 x j
k 0
¬
i k 1 q
º q
u j ¦ ¦ ĭi (i 2k j 1)B k u j » ¦ B j ui j
j 0 k 0
¼ j0
k j 1
j q k 1
ĭ(i 1) x0 k
1 h j 1
¦ ¦ ĭ(i k 1)A
k j 1
xj
j h k 1
1 q j 1
¦
¦ ĭ(i k 1)B
i
q
u j ¦¦ ĭ(i k j )B k u j
k j 1
j q k 1
xi 1.
j 0 k 0
Then (8.1.3) satisfies (8.1.1a).
Definition 8.1.1. The system (8.1.1) is called (internally) positive if xi +n and
yi +p (i +) for every x-i +n, u-j +m, i = 0,1,…,h, j = 1,2,…,q and all inputs
ui +m, i +.
Theorem 8.1.2. The system (8.1.1) is internally positive if and only if
A k  nun , (k
0,1,..., h), B j  num , ( j
0,1,..., q), C  pun , D  pum . (8.1.7)
Positive Linear Systems with Delays
423
Proof. Defining
ª ui º
«u »
« i 1 »
« # »  m ,
«
»
«ui q1 »
« ui q »
¬
¼
xi
ª xi º
« x »
« i 1 »
« # »  n , ui
«
»
« xi h1 »
«¬ xi h »¼
A1 " A h1
A
ª A0
«I
« n
« #
«
«0
«¬ 0
B
ªB0
«0
«
«#
«
«0
«¬ 0
B1 " B q 1 B q º
0 " 0
0 »»
# %
#
# »,
»
0 " 0
0»
0 " 0
0 »¼
C
>C
0
#
0
0
"
%
"
"
0
#
0
In
0 " 0@ , D
Ah º
0 »»
# »,
»
0»
0 »¼
>D
0 " 0@ ,
(8.1.8)
(8.1.9a)
(8.1.9b)
(8.1.9c)
(8.1.1) can be written in the form
xi 1
yi
Axi B ui , i  ' ,
x D
u ,
C
i
i
(8.1.10a)
(8.1.10b)
where n (h 1)n, m (q 1)m and
x0
ª x0 º
« x »
« 1 »
« # »  n , u0
«
»
« x h1 »
«¬ x h »¼
ª u0 º
«u »
« 1 »
« # »  m .
«
»
«u q 1 »
« u q »
¬
¼
In [127] it is shown that system (8.1.10) is positive if and only if
(8.1.11)
424
Polynomial and Rational Matrices
 pun , D
 pum .
A  nun , B  num , C
(8.1.12)
and D
Hence, system (8.1.1) is positive if and only if the matrices A B C
satisfy conditions (8.1.12) that are equivalent to (8.1.7).
8.1.2 Continuous-time Systems
Consider the multivariable continuous-time system with delays
x (t )
q
h
¦ A x(t id ) ¦ B u(t jd ),
i
i 0
y (t )
j
(8.1.13)
j 0
Cx(t ) Du (t ),
where x(t) n, u(t) m, y(t) p are the state, input and output vectors,
respectively and Ai nun, i = 1,…,h, Bj num, j = 0,1,…,q, C pun, D pum and
d > 0 is a delay.
Initial conditions for (8.1.13a) are given by
x0 (t ) for t  [hd , 0] and u0 (t ) for t  [ hq , 0].
(8.1.14)
The solution x(t) of (8.1.13) satisfying (8.1.14) can be found by the use of the
step method [67, pp.49].
Definition 8.1.2. The system (8.1.13) is called (internally) positive if for every
x0(t) +m, t[-hd, 0], u0(t) +m, t[-qh, 0] and all inputs u(t) +, t t 0, we have
x(t) +m and y(t) + for t t 0.
Let Mn be the set of nun Metzler matrices, i.e., the set of nun real matrices with
nonnegative off-diagonal entries.
Theorem 8.1.3. The system (8.1.13) is positive if and only if A0 is a Metzler
matrix and matrices Ai, i = 1,…,q, Bj, j = 0,1,…,q, C, D have nonnegative entries,
i.e.,
A 0  M n , A i  nun , i 1,.., h, B j  num , j
C  pun , D  pum .
0,1,..., q,
(8.1.15)
Proof. To simplify the notation, the essence of proof will be shown for h = q = 1.
Using the step method [67, pp. 49] and defining the vectors
Positive Linear Systems with Delays
x t
ª x t
«
« x td
«
#
«
¬« x t kd
z0 t
ª A1 t d
«
«
«
«
¬
º
»
», u t
»
»
¼»
ª u t
«
«u td
«
#
«
¬«u t kd
º
»
»,
»
»
¼»
425
(8.1.16)
B1u t d º
»
0
»,
»
#
»
0
¼
and the matrices
0 º
ªB 0
«B
0 »»
« 1
0 »,B « 0
A
»
«
#
# % #
# »
«#
»
«¬ 0
0
0 " A1 A 0 ¼
C >C 0 " 0@ , D > D 0 " 0@ ,
ª A0
«A
« 1
«0
«
« #
«¬ 0
0
A0
A1
0
"
0
0
0 "
A0 "
0
0
B0
B1
0
"
0
0 "
B0 "
0
0
#
#
%
#
0
0
" B1
0º
0 »»
0 »,
» (8.1.17)
# »
Ǻ 0 »¼
we may write the equations (8.1.13) in the form
x (t )
Ax (t ) Bu (t ) z0 (t ) t  [0, d ],
y (t )
Cx (t ) Du (t ).
(8.1.18)
It is well-known [127] that the system (8.1.18) is positive if and only if the
matrix A is a Metzler matrix and the matrices B, C and D have nonnegative
entries. From the structure of the matrices (8.1.17), it follows that the system
(8.1.13) is positive if and only if (8.1.15) holds.
„
8.2 Stability of Positive Linear Discrete-time Systems with Delays
8.2.1 Asymptotic Stability
Consider the positive discrete-time linear system with delays described by the
homogeneous equation
426
Polynomial and Rational Matrices
h
A 0 xi ¦ A k xi k , i  ' ,
xi 1
(8.2.1)
k 1
where h is a positive integer and Ak
Defining
xi
ª xi º
«x »
« i 1 »  n , n
« # »
« »
¬ xi h ¼
+
nun
(k = 0,1,…,h).
(h 1)n and A
ª A0
«I
« n
« #
«
¬0
A1 " A h º
0 " 0 »»
 nun , (8.2.2)
# % # »
»
0 In 0 ¼
we may write (8.2.1) in the form
xi 1
(8.2.3)
Axi , i  ' .
The positive system (8.2.3) is called asymptotically stable if its solution
xi
A i x0
satisfies the condition
lim xi
i of
0 for every x0  n .
It is well-known that the positive system (8.2.3) is asymptotically stable if and
only if all eigenvalues z1,z2,…,z n of the matrix A have moduli less than 1, i.e.,
zk 1, for k
1, 2, ! , n .
(8.2.4)
Theorem 8.2.1. [127]. The positive system (8.2.3) is asymptotically stable if and
only if all coefficients a i (i = 0,1,…, n -1) of the characteristic polynomial
det > I n z A I n @
z n an 1 z n 1 ! a1 z a0
(8.2.5)
are positive, i.e., a i > 0, for i = 0,1,…, n 1.
Theorem 8.2.2. [165]. The positive system (8.2.3) is asymptotically stable if and
only if all principal minors of the matrix
A
ª¬ aij º¼
are positive, i.e.,
In A
Positive Linear Systems with Delays
a11
a21
a11 ! 0,
a12
! 0,
a22
a11
a21
a31
a12
a22
a32
a13
a23 ! 0,! , det A ! 0 .
a33
427
(8.2.6)
Using elementary row and column operations (that do not change the value of
the determinant), we obtain
ªI n z A 0
« I
n
«
det « 0
«
#
«
«¬ 0
det > I n z A @
A1
In z
I n
#
0
! A h1
0
0
#
I n
!
!
%
!
2
ª 0
«
« I n
det « 0
«
« #
« 0
¬
I n z A 0 z A1 ! A h1
ª 0
«
« I n
det « 0
«
« #
« 0
¬
0
0
I n
#
0
det ª¬I n z
h 1
0
I n
#
0
!
!
!
%
!
!
!
%
!
0
0
0
#
I n
0
0
#
I n
A h º
0 »»
0 »
»
# »
I n z »¼
A h º
»
0 »
0 » !
»
# »
I n z »¼
(8.2.7)
I n z h1 A 0 z h ! A h1 z A h º
»
0
»
»
0
»
#
»
»
0
¼
A 0 z h ! A h1 A h º¼
z n an 1 z n 1 ! a1 z a0 .
Therefore, we have the following theorem.
Theorem 8.2.3. The positive system with delays (8.2.1) is asymptotically stable if
and only if all roots of the equation
det ª¬ I n z h1 A 0 z h ! A h1 A h º¼
z n an 1 z n 1 ! a1 z a0
have moduli less than 1.
Using elementary row and column operations, we may write
0 (8.2.8)
428
Polynomial and Rational Matrices
det > I n ( z 1) A @
ªI n z 1 A 0
«
I n
det «
«
#
«
0
«¬
A1
In z 1
#
0
%
#
!
I n
!
I n
In
A2
0
#
0
#
0
ª 0
«
I
det « n
« #
«
«¬ 0
º
»
0
»
»
#
»
z 1 »¼
0
2
I n z 1 A 0 z 1 A1
0
#
0
Ah
A 2 ! A h1
0
#
!
%
0
#
0
!
I n
In
0
!
ª 0
«
« I n
det « 0
«
« #
« 0
¬
#
º
»
»
»
»
z 1 »¼
0
#
I n z 1 A 0 z 1 A1
Ah
In
%
Ah
2
ªI n z 1 A 0
«
I n
det «
«
#
«
0
«¬
! A h1
!
0
! A h1
!
0
º
»
» !
»
»
z 1 »¼
0 Mh z º
»
0
0 »
0
0 »,
»
#
# »
I n
0 »¼
0
#
0 !
I n !
#
%
0
!
(8.2.9)
where
Mh z
In z 1
h 1
h
A 0 z 1 A1 z 1
h 1
! A h1 z 1 A h . (8.2.10)
Theorem 8.2.4. The positive system with time-delays (8.2.1) is asymptotically
stable if and only if all coefficients a i (i = 0,1,…, n -1) of the characteristic
polynomial
det M q z
z n a n 1 z n 1 ! a1 z a0
(8.2.11)
are positive, i.e., a I > 0, for i = 0,1,…, n -1.
Proof. From (8.2.9) and (8.2.11) it follows that the characteristic equation
det [In(z + 1) – A] = 0 is equal to det Mh(z) = 0. Applying Theorem 8.2.3 to the
Positive Linear Systems with Delays
429
system (8.2.1) written in the form (8.2.3), we obtain the hypothesis of Theorem
8.2.4.
Applying Theorem 8.2.4 to the system with delays (8.2.1) written in the form
(8.2.3), we obtain the following theorem.
Theorem 8.2.5. The positive system with delays (8.2.1) is asymptotically stable if
and only if all principal minors of the matrix
A
A1 A q 1
ªI n A 0
« I
n
«
« «
¬ 0
In A
In
0
0
I n
Aq º
0 »»
»
»
In ¼
(8.2.12)
are positive.
Example 8.2.1.
Consider the positive system (8.2.1) for n = 2, h = 1 with
A0
ª 0,1 0, 2 º
«0, 2 0,1 » , A1
¬
¼
ª0, 4 0 º
« 0 a» , B
¬
¼
(8.2.13)
0.
Find values of the parameter a t 0 for which the system is asymptotically stable.
In this case, the matrix (8.2.12) has the form
A
ªI n A 0
« I
n
¬
A1 º
I n »¼
ª 0, 9 0, 2 0, 4 0 º
«
a »»
0
« 0, 2 0, 9
.
« 1
0
1
0»
«
»
1
0
1¼
¬ 0
(8.2.14)
Using Theorem 8.2.5 for the system, we obtain
a11
0, 9 ! 0,
a11
a21
a31
a12
a22
a32
a13
a23
a33
a11
a21
a12
a22
0, 9 0, 2
0, 2 0, 9
0, 9 0, 2 0, 4
0, 2 0, 9
0
1
0
1
0, 77 ! 0,
0, 5 0, 2
0, 2 0, 9
0, 41 ! 0,
430
Polynomial and Rational Matrices
0, 9 0, 2 0, 4 0
0, 2 0, 9
0
a
det A
1
0
1
0
0
1
0
1
0, 5
0, 2
0, 2
0, 9
0, 41 0, 5a ! 0.
Hence the system is asymptotically stable for 0 d a d 0,82. The same result can
be obtained by the use of Theorems 8.2.4 or 8.2.3.
It will be shown that the instability of the positive system (without delays)
xi 1
A 0 xi , A 0  nun
(8.2.15)
always implies instability of the positive system with delays (8.2.1).
Theorem 8.2.6. The positive system (with delays) (8.2.1) is unstable if the positive
system (without delays) (8.2.15) is unstable.
Proof. By Theorem 8.2.5, the system (8.2.15) is unstable if at least one of the
principal minors of the matrix
A0
ª¬ aij0 º¼
I n A0
is not positive. The system (8.2.1) is unstable if at least one of the principal minors
of the matrix
In A
ªI n A 0
« I
n
«
« #
«
¬ 0
A1 ! A q 1
In
!
0
#
0
%
!
#
I n
A q º
0 »»
# »
»
In ¼
(8.2.16)
is not positive. From (8.2.16) it follows that if at least one of the principal minors
of the matrix In – A0 is not positive, then at least one of the principal minors of the
matrix (8.2.16) is also not positive. Therefore, the instability of the system (8.2.15)
always implies the instability of the system (8.2.1).
From Theorem 8.2.5, we have the following important corollary.
Corollary 8.2.1. If the positive system (8.2.15) is unstable, then it is not possible to
stabilize the system (8.2.1) by a suitable choice of the matrices Ak, k = 1,…,q.
Theorem 8.2.7. The positive system (8.2.1) is unstable if at least one diagonal
entry of the matrix A0 = [aij0] is greater than 1, i.e.,
Positive Linear Systems with Delays
akk0 ! 1, for some k  1, 2,! , n .
431
(8.2.17)
Proof. It is known [127, Theorem 2.15] that the positive system (8.2.15) is
unstable if for at least one k(1,2,…,n) (8.2.17) holds. In this case, by Theorem
8.2.5 the positive system (8.2.1) is also unstable.
Example 8.2.2.
Consider the positive system (8.2.1) for n = 2, q = 1 with
ª a11
«a
¬ 21
ª0,1 0, 2 º
, A1
«0
2 »¼
¬
A0
a12 º
a22 »¼
aij t 0, i, j 1, 2 .
(8.2.18)
The system (8.2.15) with A0 of the form (8.2.18) is unstable, since one of the
eigenvalues of A0 is equal 2. The same result follows from Theorem 8.2.6, since
a220 = 2 > 1. In this case, the matrix (8.2.16) has the form
In A
ªI n A 0
« I
n
¬
A1 º
I n »¼
ª0, 9 0, 2 a11
« 0
1 a21
«
« 1
0
1
«
1
0
¬ 0
a12 º
a22 »»
.
0 »
»
1 ¼
(8.2.19)
Applying Theorem 8.2.5 to (8.2.19), we obtain
a11
0, 9 ! 0,
a11
a12
a13
a21
a31
a22
a32
a23
a33
a11
a21
a12
a22
0, 9 0, 2
0
1
0, 9 0, 2 a11
0
1
a21
1
0
1
0, 9 0,
0, 9 a11
0, 2
a21
1
(8.2.20)
0, 9 a11 0, 2a21 ,
det A
0, 9 0, 2 a11
a12
0
1
a21
a22
0, 9 a11
0, 2 a12
1
0
1
0
a21
1 a22
0
1
0
1
0, 9 a11 1 a22 a21 0, 2 a12 .
From (8.2.20), it follows that for any entries of the matrix A1, the system
(8.2.1) with (8.2.18) is unstable, since the second-order principal minor is negative.
432
Polynomial and Rational Matrices
8.2.2 Stability of Systems with Pure Delays
The system (8.2.1) is a system with pure delay if Ak { 0 for k = 0,1,…,h1. In
such a case, this system is described by the homogeneous equation
xi 1
(8.2.21)
A h xi h , i  ' .
From (8.1.9a), it follows that the matrix Ap of the equivalent system
xi 1
A p xi
without delays has the form (with n (h 1)n )
Ap
ª0
«I
« n
«#
«
¬0
0 " Ah º
0 " 0 »»
 nun .
# % # »
»
0 In 0 ¼
(8.2.22)
The system (8.2.21) is asymptotically stable if and only if wh(z) z 0 for |z| t 1,
where
wh ( z )
det( z h1I n A h ).
(8.2.23)
From Theorems 8.2.1 and 8.2.2 we have the following theorem.
Theorem 8.2.8. [30, 31]. The positive system (8.2.21) with pure delay is
asymptotically stable if and only if one of the following equivalent conditions
holds:
1.
all coefficients of the polynomial wh(z+1) are positive, where wh(z) has
the form (8.2.23),
2.
all principal minors of the matrix A p I n A p of the form
Ap
ª In
« I
« n
« #
«
¬ 0
0
"
In
"
#
0
%
I n
A h º
0 »»
# »
»
In ¼
(8.2.24)
are positive.
Proof. From the structure of the matrix (8.2.24) it follows that all principal minors
of order from 1 to nh of A p are always positive. Moreover, all principal minors of
Positive Linear Systems with Delays
433
A p of order from nh+1 to (h + 1)n are positive if and only if all principal minors
of the matrix
D
(8.2.25)
In Ah
are positive.
From the above, it follows that if the system (8.2.21) with fixed delay h > 0 (h
is a positive integer) is asymptotically stable, then the system xi+1 = Ahxip, where p
is any positive integer, is also asymptotically stable. Hence, asymptotic stability of
the positive system (8.2.21) with pure delay does not depend on the delay.
„
Positivity of all principal minors of (8.2.24) is necessary and sufficient for
asymptotic stability of the positive system without delay, described by the equation
[127]
xi 1
(8.2.26)
A h xi , i  ' .
It is well-known [127] that the system (8.2.26) is asymptotically stable if and
only if all eigenvalues of the matrix Ah have moduli of less than 1.
From the above and [127], we have the following.
Theorem 8.2.9. [30]. The positive system (8.2.21) with pure delay is
asymptotically stable if and only if one of the following equivalent conditions hold:
1. all principal minors of the matrix (8.2.25) are positive,
2. all coefficients of the polynomial
det[( z 1)I n A h ]
z n an1 z n1 ... a0
(8.2.27)
are positive, i.e., a i > 0 for i = 0,1,…,n1.
Lemma 8.2.1. [30]. The positive system (8.2.21) is not stable if at least one
diagonal entry of the matrix Ah = [ahij] is greater than 1, i.e., ahkk > 1, for some
k(1,2,…,n).
Example 8.2.3.
Consider the positive system (8.2.21) with
Ah
ª a 0.2 0 º
« 0.4 0.1 0.1» .
«
»
«¬ 1 0.3 b »¼
(8.2.28)
434
Polynomial and Rational Matrices
Find values of the parameters a t 0 and b t 0 for which the system is
asymptotically stable.
In this case, matrix (8.2.25) has the form
D
0 º
ª1 a 0.2
« 0.4 0.9 0.1» .
«
»
«¬ 1 0.3 1 b »¼
(8.2.29)
Computing all principal minors of (8.2.29), from condition 1) of Theorem
8.2.9, we obtain
'1 1 a ! 0, ' 2
0.82 0.9a ! 0, ' 3
0.77 0.87 a 0.82b 0.9ab ! 0.
These inequalities can be written in the form
(8.2.30)
a 0.9111, 0.77 0.87 a 0.82b 0.9ab ! 0.
Hence, the system is asymptotically stable for a and b satisfying (8.2.30) and
for any fixed delay (h = 1,2,…).
8.2.3 Robust Stability of Interval Systems
Let us consider a family of positive discrete-time systems with delays
xi 1
h
¦A
x , A k  [ A k , A k ]  nun ,
(8.2.31)
k i k
k 0
where
akij  [akij
, akij
], akij
d akij
, with A k
[akij
], A k
[akij
], for k
0,1,..., h.
The family (8.2.31) is called an interval family or an interval system with
delays.
The interval positive system (8.2.31) is called robustly stable, if the system
(8.2.1) is asymptotically stable for all Ak[Ak-, Ak+] (k = 0,1,..,h).
If Ak[Ak-, Ak+] k = 0,1,..,h, then for the equivalent system (8.2.3), we have
AAI, where A is of the form (8.2.2), AI = [A-, A+] and
A
ª A 0
«
« In
« #
«
¬« 0
A1 " A h º
»
0 " 0 »
, A
# % # »
»
0 I n 0 ¼»
ª A 0
«
« In
« #
«
¬« 0
A1 " A h º
»
0 " 0 »
.
# % # »
»
0 I n 0 ¼»
(8.2.32)
Positive Linear Systems with Delays
435
Theorem 8.2.10. [31]. The interval positive delay system (8.2.31) is robustly
stable if and only if the positive system without delays
xi 1
A xi , i  ' is asymptotically stable or, equivalently, the positive system with delays
xi 1
h
A 0 x0 ¦ A k xi k , i  ' (8.2.33)
k 1
is asymptotically stable.
Proof. The proof follows directly from the fact that all eigenvalues of any
nonnegative matrix A[A-, A+] have moduli less than 1 if and only if all
eigenvalues of A+ have moduli less than 1 [31].
From Theorem 8.2.10 it follows that robust stability of the interval system
(8.2.31) does not depend on the matrices Ak- +nun, k = 0,1,…,h. Therefore, we
may have Ak = 0 for k = 0,1,…,h. Moreover, if the system (8.2.1) is
asymptotically stable for any fixed Ak = Akf +nun k = 0,1,…,h, then this system is
also asymptotically stable for all Af[0, Akf], k = 0,1,…,h.
From the above and Theorems 8.2.1 and 8.2.2 we have the following theorem
and lemma.
Theorem 8.2.11. [31]. The interval positive delay system (8.2.31) is robustly
stable if and only if one of the following equivalent conditions holds:
1. all coefficients of the polynomial w+(z +1) are positive, where
w ( z 1)
h
det[( z 1) h1 I n ¦ A k ( z 1) hk ],
(8.2.34)
k 0
2.
all principal minors of the matrix A A
ªI n A 0
«
« I n
« #
«
«¬ 0
are positive.
A1
"
In
"
#
%
0
I n
A h º
»
0 »
# »
»
I n »¼
I n A of the form
(8.2.35)
436
Polynomial and Rational Matrices
Lemma 8.2.2. [31]. The interval positive delay system (8.2.31) is not robustly
stable if the positive system (without delays) xi+1 = A0+xi is unstable, or at least one
diagonal entry of the matrix A0+ is greater than 1.
Consider a family of positive discrete-time linear systems with delays
xi 1
h
¦a x
k i k
(8.2.36)
, ak  [ak , ak ],
k 0
where
0 d ak and ak d ak , for k
0,1,..., h.
The positive interval system without delays equivalent to (8.2.36) is described
by
xi 1
A s xi , A s  [ A s , A s ]  nun , n
h 1.
(8.2.37)
From Theorem 8.2.10 we have the following theorem.
Theorem 8.2.12. The interval positive system (8.2.36) with delays is robustly
stable if and only if the positive system without delays
xi 1
A s xi , i  ' ,
(8.2.38)
A s
ª a0
«
«1
«#
«
¬« 0
(8.2.39)
where
a1 " ah º
»
0 " 0»
# % # »
»
0 1 0 ¼»
is asymptotically stable or, equivalently, the positive delays system
xi 1
h
¦a
x , i  ' ,
k i k
k 0
is asymptotically stable, that is,
h
' n
1 ¦ ak ! 0.
k 0
(8.2.40)
Positive Linear Systems with Delays
437
Let us consider the interval positive system with pure delay
xi 1
A h xi h , A h  [ A h , A h ]  nun .
(8.2.41)
Theorem 8.2.13. [30, 31]. The interval positive system (8.2.41) with pure delay is
robustly stable if and only if the positive delay system
xi 1
A h xi h , i  ' (8.2.42)
is asymptotically stable or, equivalently, the positive system without delays
xi 1
(8.2.43)
A h xi , i  ' is asymptotically stable.
From Theorem 8.2.13 it follows that robust stability of the interval system
(8.2.41) does not depend on the matrix Ah +nun. Therefore, we may have Ah = 0
From the above and Theorem 8.2.9, we have the following theorem.
Theorem 8.2.14. [30, 31]. The interval positive system (8.2.41) with pure delay is
robustly stable if and only if one of the following equivalent conditions hold:
1. all principal minors of the matrix
D
2.
I n A h
(8.2.44)
are positive,
all coefficients of the polynomial
det[( z 1)I n A h ]
z n aˆn1 z n1 ... aˆ0 ,
(8.2.45)
are positive.
Lemma 8.2.3. [30, 31]. The positive interval system (8.2.41) is not robustly stable
if at least one diagonal entry of the matrix Ah+ = [ahij+] is greater than 1, i.e.,
ahkk+ > 1, for some k(1,2,…,n).
8.3 Reachability and Minimum Energy Control
Consider the positive discrete-time linear system (8.1.1) for h = q with initial the
conditions (8.1.2). The considerations for h z q are similar.
Definition 8.3.1. A state xf +n is called reachable in N steps if there exists a
sequence of inputs ui +m, i = 0,1,…,N1 that transfers the system (8.1.1) from
zero initial conditions (8.1.2) to the state xf.
438
Polynomial and Rational Matrices
Definition 8.3.2. If every state xf
called reachable in N steps.
+
n
is reachable in N steps, then the system is
Definition 8.3.3. If for every state xf +n there exists a natural number N such that
the state xf is reachable in N steps, then the system is called reachable.
Recall that the set  n is called a cone if the following implication holds: if
x , then Dx for every D + The cone is called convex if for any x1, x2
every point of the line segment x = (1-O)x1+Ox2 , for 0 d O d 1. The cone is
called solid if its interior contains the sphere K(x, r) with the centre at the point
x and radius r.
Theorem 8.3.1. The set of reachable states of the positive system (8.1.1) is a
positive convex cone. This cone is solid if and only if there exists an N + such
that the rank of the reachability matrix
RN
(8.3.1)
[Ȍ ( N 1), Ȍ ( N 2), " , Ȍ (1), Ȍ (0)]
is equal to n, where
Ȍ (i )
h
¦ ĭ(i k )B
k
(8.3.2)
,
k 0
and )(i) is the state-transition matrix.
Proof. For
x i
0 (i
0,1,..., h), u j
0 ( j 1, 2,..., q) and i
N !0
solution (8.3.1) or (8.1.1a) has the form
xN
N 1 h
¦¦ ĭ( N 1 k j )B u
k
j
R N u0N ,
(8.3.3)
j 0 k 0
where RN has form (8.3.1) with <(i) defined by (8.3.2) and
u0N
ª u0 º
« u »
« 1 ».
« # »
«
»
¬u N 1 ¼
(8.3.4)
If rank RN = n, then from (8.3.3) it follows that if u0N steers system (8.1.1) from
zero initial conditions to xN, then Du0N steers this system from zero initial
Positive Linear Systems with Delays
439
conditions to DxN for every D . Therefore, the set of states that are reachable in N
steps is a cone.
Let N denote the positive cone of reachable states of the positive system
(8.1.1).
If
xN
R N u0N  % N and x N
R N u0N  % N ,
then
(1 O ) xN O x N
(1 O )R N u0N O R N u0N
R N [(1 O )u0N O u0N ] R N v0N  % N ,
where
v0N
(1 O )u0N O u0N .
Hence, the cone N is convex.
Let K(0,H) be the sphere with a centre x = 0 and radius H. From the assumption
rank RN = n it follows that the system is reachable if the input is unbounded. In this
case, there exists an input 'u0N that steers the state of system (8.1.1) to an arbitrary
point inside the sphere. From the linearity of the system and superposition
principle it follows that the input u0N + 'u0N may steer the system to an arbitrary
point inside the sphere K(x,H), where u0N is the input that steers the system (8.1.1)
to x. The input 'u0N can be chosen so that all entries of u0N + 'u0N are
nonnegative and K(x,H)  N. Hence, the cone N is solid. On the other hand, if N
contains the sphere K(x,H), then there exists an input 'u0N that steers the system
(8.1.1) to an arbitrary point inside the sphere K(0,H) only if rank RN = n.
The cone N of the reachable states of the positive system (8.1.1) usually
increases with N, i.e., % N1  % N2 for N2 > N1.
The following theorem gives the conditions under which this cone in invariant
with respect to N.
Theorem 8.3.2. The cone N of the reachable states of the positive system (8.3.1)
is invariant for N ! n (h 1)n if and only if rank RN = n and the coefficients of
the characteristic polynomial
h
§
·
det ¨ z h1I n ¦ A k z hk ¸
k 0
©
¹
det( zI n A )
are nonpositive, i.e., ak d 0 for k = 0,1,…, n 1.
z n an 1 z n 1 ... a1 z a0 (8.3.5)
440
Polynomial and Rational Matrices
Proof. In the same way as in [127] it can be proved that
ĭ(n j )
an 1ĭ(n j 1) ... a1ĭ( j 1) a0ĭ( j ), j  ' .
(8.3.6)
Hence, )( n + j) for any j + is a linear nonnegative combination of )(j + k)
(k = 0,1,…, n 1) if and only if ak d 0 k = 0,1,…, n 1
From (8.3.2), for i = n + j we have
Ȍ (n j )
h
¦ ĭ(n j k )B
k
, j  ' .
(8.3.7)
k 0
Because Bk +num for k = 0,1,…,h, the matrix <( n + j) for any j + is a linear
nonnegative combination of )( n + j k) k = 0,1,…,h Hence, if rank RN = n, then
X n +1 = X n for all j + if and only if all the coefficients ak (k = 0,1,…, n 1) of
polynomial (4) are nonpositive.
By Definition 8.3.3 the positive system (8.1.1) is reachable if and only if the
reachability cone is equal to +n
Denote by Im+ RN the positive image of the matrix RN +nuNm, i.e.,
Im R N
{y  n : y
R N u , u  Nm }.
(8.3.8)
Theorem 8.3.3. The positive system (8.1.1) is reachable if and only if there exists
an N + such that rank RN = n and
1. Im+ RN = +n, where RN is defined by (8.3.1);
2. n linearly independent columns can be chosen from RN so that the matrix
R N constructed from them is a monomial matrix (every row and every
column has only one positive entry and the remaining entries are equal to
zero);
3. n linearly independent columns can be chosen from RN so that the matrix
R N constructed from them has the inverse R N-1 with nonnegative entries,
i.e., R N-1 +nuN.
Proof. If xN = xf in (8.3.13), then
xf
R N u0N .
(8.3.9a)
From (8.3.9) it follows that for every xf +n there exists u0N +Nm if and only
if the condition 1) is satisfied. If 1) is satisfied, then n linearly independent
columns (being a base of +n) can be chosen from RN if and only if in every row
and every column only one entry is positive and all the remaining entries are zero.
The matrix constructed from these columns is a monomial matrix. The inverse
Positive Linear Systems with Delays
441
matrix of a positive matrix is positive if and only if it is a monomial matrix [127].
Therefore, conditions 2) and 3) are equivalent.
From the above it follows that if the conditions of Theorem 8.3.2 hold, then the
cone of reachable states of the positive system (8.1.1) is invariant for
N t n = (h + 1)n. This means that if this system is not reachable in N = n steps,
then it is not reachable in N t n steps (it is not reachable).
In certain cases the cone of reachable states may be invariant for N < n This
follows from the fact that if m > 1, then condition rank RN = n may be satisfied for
N < n . In such a case, if the conditions of Theorem 8.3.4 hold, then positive
system (8.3.1) is reachable in N < n steps.
Theorem 8.3.4. The positive system (8.1.1) is reachable if there exists an N +
such that the rank of the reachability matrix RN of the form (8.3.1) is equal to n and
R TN [R N R TN ]1  R Nmun .
(8.3.10)
Moreover, if (8.3.10) holds then the sequence of controls ui +m, i = 0,1,…,N1 that transfer the system (8.1.1) from zero initial conditions (8.1.2) to the desired
final state xf +n can be computed from
u0N
R TN [R N R TN ]1 x f
ª u0 º
« u »
« 1 ».
« »
«
»
¬u N 1 ¼
(8.3.11)
Proof. If rank RN = n, then det (RN RNT) z 0 and the matrix RNT[RNRNT]-1 is well
definite. If (8.3.9) holds and xf +n, then u0N +Nm and
xN
R N u0N
R N R TN [R N R TN ]1 x f
xf .
(8.3.12)
Theorem 8.3.5. If matrix
[ A 0 , A1 ,..., A h , B]
does not contain n linearly independent monomial columns, then the positive
system (8.3.1) is not reachable.
Proof. If the positive system (8.1.1) is reachable, then RN has n linearly
independent monomial columns. From (8.3.2) it follows that this is possible only if
[A0,A1,…,Ah,B] has n linearly independent monomial columns.
442
Polynomial and Rational Matrices
Theorem 8.3.6. If the positive system (8.1.1) is reachable, then it is reachable in N
steps, with N t E[n/q], where E[n/q] denotes the minimal positive number greater
than or equal to n/q, and q is number of linearly independent monomial columns of
B.
Proof. Each matrix )(k)B (k = 0,1,…,N1) of the reachability matrix RN may have
maximum q linearly independent monomial columns. Hence, if the positive system
(8.1.1) is reachable, then Nq = n.
8.3.2 Minimum Energy Control
Consider the positive system (8.1.1) with h = q and the performance index
I (u )
N 1
T
i
¦u
Qui ,
(8.3.13)
i 0
where QRmum is a symmetric positive definite weighting matrix such that
Q 1  m um
(8.3.14)
and N is the number of steps in which the system (8.1.1) is transferred to the state
xf.
Control sequence ui +m, i = 0,1,…,N1 that minimizes the performance index
(8.3.13) is called a minimal one. The problem of minimum energy control was first
solved in [182].
The minimum energy control problem for the positive system (8.1.1) with h = q
can be stated as follows. Given are the matrices AkR+nun and BjR+num
(k,j = 0,1,…,h), the number of steps N, the final state xf +n and a weighting
matrix Q such that (8.3.14) holds. Find a control sequence ui +m, i = 0,1,…,N1
that transfers the system (8.1.1) from zero initial conditions to the desired final
state xf +n and minimizes the performance index (8.3.13).
Define the matrix
W
R N Q N R TN  nun ,
(8.3.15)
where RN is the reachability matrix of the form (8.3.1) and
QN
diag[Q 1 ,..., Q 1 ]  Nmu Nm .
(8.3.16)
From (8.3.15) it follows that the matrix W is nonsingular if and only if the
matrix RN has full row rank, i.e., the necessary condition of reachability of the
positive system (8.1.1) holds.
Define the sequence of inputs uˆ0 , uˆ1 , ..., uˆ N 1 by
Positive Linear Systems with Delays
uˆ0N
ª uˆ0 º
« uˆ »
« 1 »
« # »
«
»
¬uˆ N 1 ¼
Q N R TN W 1 x f .
From (8.3.17) it follows that u0N
443
(8.3.17)
+
Nm
for any xf
Q N R TN W 1  Nmun .
+
n
if and only if
(8.3.18)
Theorem 8.3.7. Let the following assumptions hold:
1. positive system (8.1.1) is reachable in N steps,
2. condition (8.3.18) is satisfied,
3. u i +m, i = 0,1,…,N1 is any sequence of inputs that transfer the system
(8.1.1) from zero initial conditions (8.1.2) to the desired final state xf +n.
Then the sequence of inputs û 0, û 1,..., û N1 defined by (8.3.17) also transfer
system (8.1.1) from zero initial conditions to the state xf +n, minimizes
performance index (8.3.13) and
I (uˆ ) d I (u ).
(8.3.19)
Moreover, the minimal value of (8.3.13) is given by
I (uˆ )
x Tf W 1 x f .
(8.3.20)
Proof. If the positive system (8.1.1) is reachable in N steps and (8.3.18) holds, then
û i +m, i = 0,1,…,N1.
From (8.3.3) for u0N = û 0N and (8.3.17) it follows that
xN
R N uˆ0N
R N Q N R TN W 1 x f
xf ,
(8.3.21)
because RN Q NRNTW-1 = In. Hence, the sequence of inputs (8.3.17) provides
xN = xf.
Since both u 0, u 1,..., u N1 and û 0, û 1,..., û N1 transfer the system (8.3.11)
from zero initial conditions to xf +n, xf = RN u 0N = RN û 0N and
R N (uˆ0N u0N )
0.
From (8.3.17) it follows that
R TN W 1 x f
Hence,
Q N1uˆ0N .
(8.3.22)
444
Polynomial and Rational Matrices
(uˆ0N u0N ) T R TN W 1 x f
ˆ uˆ N
(uˆ0N u0N ) T Q
N 0
0,
(8.3.23)
where
ˆ
Q
N
Q N1
diag[Q,..., Q]  Nmu Nm .
(8.3.24)
Using (8.3.23) it is easy to show that
ˆ uN
(u0N )T Q
N 0
ˆ uˆ N (u N uˆ N )T Q
ˆ (u N uˆ N ).
(uˆ0N )T Q
N 0
0
0
N
0
0
(8.3.25)
The last term in (8.3.25) is always nonnegative. Hence, inequality (8.3.19) is true.
Substitution (8.3.17) into (8.3.13) yields
I (uˆ )
N 1
T
i
¦ uˆ
Quˆi
ˆ uˆ N
(uˆ0N )T Q
N 0
ˆ (Q R T W 1 x )
(Q N R TN W 1 x f )T Q
N
N
N
f
i 0
x Tf W 1R N Q N R TN W 1 x f
x Tf W 1 x f ,
since
ˆ Q
Q
N
N
I Nm and W 1R N Q N R TN
In .
The optimal control that minimizes performance index (8.3.13) depends on the
weighting matrix Q. From a comparison of (8.3.11) and (8.3.17) it follows that
control sequence (8.3.11) minimizes performance index (8.3.13) with Q = Im. This
means that u0N computed from (8.3.11) is the minimum energy control with a
performance index
N 1
I (u)
¦u u .
T
i i
i 0
Theorem 8.3.8. Let the weighting matrix have the form Q = aIm, a ! 0. Then
û 0N = u0N, where û 0N and u0N are defined by (8.3.17) and (8.3.11), respectively. In
such a case, the optimal value of the performance index can be computed from the
formula
I (uˆ )
ax Tf [R N R TN ]1 x f .
Proof. If Q = aIm, then from (8.3.16) and (8.3.15) it follows that
(8.3.26)
Positive Linear Systems with Delays
QN
a 1I Nm , W
a 1R N R TN .
445
(8.3.27)
Hence,
uˆ0N
Q N R TN W 1 x f
a 1R TN a(R N R TN )1 x f
R TN (R N R TN )1 x f
u0N . (8.3.28)
Substitution of the second formula of (8.3.27) into (8.3.20) gives (8.3.26).
Example 8.3.1.
Consider the positive system (8.1.1) with h = q = 2 and the matrices
A0
ª0 0 0 º
«0 0 0 » , A
1
«
»
¬« 0 0 0.4 ¼»
ª0.1 0 0 º
« 0 0 0» ,
«
»
¬« 0 0 0 »¼
ª1 0 º
ª0
«0 0» , B
«0
2
«
»
«
«¬ 0 0 »¼
«¬ 0
A2
B0
ª0 0 º
«1 0 » , B
1
«
»
«¬0 0 »¼
0º
0 »» .
1 »¼
0 0º
ª 0
« 0 0.1 0 » ,
«
»
¬« 0.5 0 0 ¼»
(8.3.29a)
(8.3.29b)
Find the optimal control that transfers this system from zero initial conditions to
the final state xf = [1 2 4]T in three steps and minimizes the performance index
(8.3.13) with
Q
ª 1 1º
« 1 2 » .
¬
¼
The necessary condition for reachability in three steps is satisfied because the
reachability matrix
R3
[Ȍ (2), Ȍ (1), Ȍ (0)]
ª0 0 1 0 0 0º
«0 0 0 0 1 0»
«
»
«¬ 0 1 0 0 0 0 »¼
(8.3.30)
has a full row rank equal to 3.
It is easy to check that the conditions of Theorem 8.3.4 are satisfied and the
system is reachable in three steps.
The optimal control sequence computed from (8.3.17) has the form
uˆ0
ª4º
« 4 » , uˆ1
¬ ¼
ª1 º
« 0.5» , uˆ2
¬ ¼
ª 2º
«1 » .
¬ ¼
(8.3.31)
446
Polynomial and Rational Matrices
According to (8.3.20), the minimal value of the performance index (8.3.13) is
I (uˆ ) 18.5.
The control sequence, which also transfers system (8.1.1) with the matrices
(8.3.29) from zero initial conditions to the final state xf = [1 2 4]T, can be
computed from (8.3.11). This control is of the form
u0
ª0º
« 4 » , u1
¬ ¼
ª1 º
« 0 » , u2
¬ ¼
ª 2º
«0» .
¬ ¼
(8.3.32)
The optimal value of (8.3.13) for control sequence (8.3.32) is equal to
I (u ) 37 ! I (uˆ ) 18.5.
8.4 Realisation Problem for Positive Discrete-time Systems
8.4.1 Problem Formulation
Consider the multi-input discrete-time linear system with delays described by the
equations
xi 1
yi
A 0 xi A1 xi 1 B 0ui B1ui 1 ,
Cxi Dui
(8.4.1a)
i  ' ,
(8.4.1b)
where xi n, ui m and yi are the state vector, input vector and scalar output,
respectively, and Ak nun, Bk num, k = 0,1, C 1un, D 1um.
The initial conditions for (8.4.1a) are given by
x i  n , for i
0,1 and u j  , for
j 1.
(8.4.2)
Definition 8.4.1. The system (8.4.1) is called (internally) positive if for every
xk +n, k = 0,1, x1 +m and all inputs ui +, i + we have xi +n and yi + for
i +.
By Theorem 8.1.1 the system (8.4.1) is positive if and only if
A k  nun , B k  num , k
0,1, C  1un , D  1um .
(8.4.3)
The transfer matrix of (8.4.1) is given by
T z
C ª¬ I n z A 0 A1 z 1 º¼
1
B 0 B1 z 1 D .
(8.4.4)
Positive Linear Systems with Delays
447
Definition 8.4.2. Matrices (8.4.3) are called a positive realisation of a given proper
rational function T(z) if they satisfy the condition (8.4.4). A realisation (8.4.3) is
called minimal if the dimension nun of Ak, k = 0,1 is minimal among all
realisations of T(z).
The positive minimal realisation problem can be stated as follows. Given a
proper rational matrix T(z), find a positive minimal realisation (8.4.3) of T(z).
Conditions for solvability of the positive minimal realisation problem will be
established and a procedure for computation of a positive minimal realisation
(8.4.3) of T(z) will be presented below.
8.4.2 Problem Solution
The transfer matrix (8.4.4) can be written in the form
T z
C Adj ª¬ I n z 2 A 0 z A1 º¼ B 0 z B1
D
det ª¬ I n z 2 A 0 z A1 º¼
N z
C Adj ª¬ I n z 2 A 0 z A1 º¼ B 0 z B1
N z
D,
d z
(8.4.5)
where
ª¬ n j ,2 n1 z 2 n1 ... n j ,1 z n j ,0 º¼
j
det ª¬ I n z 2 A 0 z A1 º¼
d z
1,..., m
,
(8.4.6)
z 2 n a2 n1 z 2 n1 ... a1 z a0
and Adj stands for the adjoint matrix.
From (8.4.4), we have
D
lim T z ,
(8.4.7)
z of
since
lim ª¬I n z 2 A 0 z A1 º¼
z of
1
0.
The strictly proper part of T(z) is given by
Tsp z
T z D
N z
.
d z
(8.4.8)
Therefore, the positive minimal realization problem has been reduced to finding
the matrices
448
Polynomial and Rational Matrices
A k  nun , B k  num , k
0,1, C  1un
(8.4.9)
for a given strictly proper rational matrix (8.4.8).
Lemma 8.4.1. If the matrices A0 and A1 have the following forms
A0
ª0
«0
«
«0
«
«#
«0
«
«¬ 0
0 ! 0
0 ! 0
0 ! 0
# % #
0 ! 0
0 ! 0
a1 º
a3 »»
a5 »
» , A1
# »
a2 n 3 »
»
a2 n1 »¼
ª0
«1
«
«0
«
«#
«0
«
«0
¬
0 0 ! 0
0 0 ! 0
1 0 ! 0
a0
a2
a4
#
# # % #
0 0 ! 0 a2 n2
0 0 ! 1
a2 n1
º
»
»
»
» , (8.4.10)
»
»
»
»
¼
then
det ª¬I n z 2 A 0 z A1 º¼
z 2 n a2 n1 z 2 n1 ! a1 z a0 .
(8.4.11)
Proof. Expansion of the determinant with respect to the n-th column yields
det ª¬I n z 2 A 0 z A1 º¼
z2 0 ! 0
1 z 2 ! 0
0 1 ! 0
#
# % #
0 0 ! z2
0
0
a1 z a0
a3 z a2
a5 z a4
#
a2 n3 z a2 n2
! 1 z 2 a2 n1 z a2 n1
z 2 n a2 n1 z 2 n1 ! a1 z a0 .
Remark 8.4.1.
Let P = D P be a generalized permutation matrix where D is a diagonal matrix with
positive diagonal entries and P is a permutation matrix (obtained from the identity
matrix In by permutation of rows and columns). Then for the matrices A0, A1
defined by (8.4.10)
A0
PA 0 P 1  nun , A1
PA1P 1  nun
(8.4.12)
and
det ª¬I n z 2 A 0 z A1 º¼
z 2 n a2 n1 z 2 n1 ! a1 z a0 ,
(8.4.13)
Positive Linear Systems with Delays
449
since
det ª¬I n z 2 A 0 z A1 º¼
det P
and P-1
+
det P 1
det P det ª¬I n z 2 A 0 z A1 º¼ det P 1 ,
1
nun
.
Lemma 8.4.2. If the matrices A0, A1 have the form (8.4.10a), then the n-th row
Rn(z) of the adjoint matrix Adj [Inz2 – A0z – A1] has the form
Rn z
ª1 z 2 ! z 2 n1 º .
¬
¼
(8.4.14)
Proof. Taking into account that
Adj ª¬I n z 2 A 0 z A1 º¼ ª¬I n z 2 A 0 z A1 º¼
I n det ª¬I n z 2 A 0 z A1 º¼ ,(8.4.15)
it is easy to verify that
R n z ª¬I n z 2 A 0 z A1 º¼
det ª¬I n z 2 A 0 z A1 º¼ > 0 0 ! 1@ . (8.4.16)
Let
C
>0
0 ! 1@
b0j
ª b10j º
« 0 »
«b2 j » , b1
j
« # »
« 0»
¬« bnj ¼»
(8.4.17)
and
ª b11j º
« 1 »
«b2 j » ,
« # »
« 1»
¬« bnj ¼»
j 1,..., m ,
be the j-th column of the matrices B0 and B1, respectively.
Then from (8.4.6) and (8.4.14), we have
450
Polynomial and Rational Matrices
C Adj ª¬I n z 2 A 0 z A1 º¼ b 0j z b1j
ª1 z 2 ! z 2 n1
¬
ª b10j z b11j º
« 0
»
b z b21 j »
º« 2j
¼«
»
#
« 0
»
1
¬« bnj z bnj ¼»
n j ,2 n1 z 2 n1 n j ,2 n1 z
2 n 1
(8.4.18)
... n j ,1 z n j 0 , j 1,..., m.
Comparing the coefficients at the same powers of z of the equality (8.4.18), we
obtain
b11j
n j 0 , b10j
n j1 , b21 j
n j 2 , b20 j
n j 3 , ! , bnj1
n j ,2 n1 , bnj0
n j ,2 n1
for j 1,..., m,
and
B0
B1
n21
ª n11
« n
n
23
« 13
« #
#
«
«¬ n1,2 n1 n2,2 n1
n20
ª n10
« n
n22
« 12
« #
#
«
«¬ n1,2 n1 n2,2 n1
nm1 º
nm 3 »»
,
%
# »
»
! nm ,2 n1 »¼
!
nm 0 º
!
nm 2 »»
.
%
# »
»
! nm ,2 n1 »¼
!
!
(8.4.19)
Theorem 8.4.1. There exists a positive minimal realization (8.4.3) of T(z) if the
following conditions are satisfied.
1. T f lim T z  1um .
z of
2.
The conditions
n jk t 0, j 1,..., m, k
ak t 0, k
0,1,..., 2n 1
0,1,..., 2n 1 ,
(8.4.20)
(8.4.21)
hold
Proof. The condition (8.4.1) implies D +1um. If the condition (8.4.21) is satisfied,
then the matrices A0 and A1 of the forms (8.4.10) have nonnegative entries and
their dimension is minimal for the given polynomial d(z). If additionally the
condition (8.4.20) is satisfied, then B0, B1 +num.
Positive Linear Systems with Delays
451
If the conditions of Theorem 8.4.1 are satisfied, then a positive minimal
realization (8.4.3) of T(z) can be found by the use of the following procedure.
Procedure 8.4.1.
Step 1: Using (8.4.7) and (8.4.8) find D and the strictly proper matrix Tsp(z).
Step 2: Knowing the coefficients ak, k = 0,1,…,2n1 of d(z) find the matrices
(8.4.10).
Step 3: Knowing the coefficients njk, j = 1,…,m, k = 0,1,…,2n1 find the matrices
B0, B1.
Example 8.4.1.
Given the transfer matrix
1
N z
d z
T z
6
5
4
1
z6 2z5 z3 2z 2 z 2
3
2
6
5
(8.4.22)
4
3
u ª¬ 2 z 3z 2 z z 4 z z 3 z 2 z z z z 1º¼ ,
find a positive minimal realization (8.4.3).
Using the procedure we obtain the following.
Step 1: From (8.4.7), we have
D
lim T z
z of
> 2 1@
(8.4.23)
and
Tsp z
T z D
N z
d z
1
ª z 5 2 z 4 z 3 z 1 z 4 2 z 2 1º¼ .
6
5
3
z 2z z 2z2 z 2 ¬
(8.4.24)
Step 2: Taking into account that
a0
a2
a5
2, a1
a3
1, a4
0
and using (8.4.10), we obtain
A0
ª 0 0 a1 º
«0 0 a »
3»
«
«¬ 0 0 a5 »¼
ª0 0 1 º
«0 0 1 » , A
1
«
»
«¬ 0 0 2 »¼
ª 0 0 a0 º
«1 0 a »
2»
«
«¬ 0 1 a4 »¼
ª0 0 2º
«1 0 2 » . (8.4.25)
«
»
«¬ 0 1 0 »¼
In this case, the third row R3(z) of the matrix Adj [Inz2 – A0z – A1] has the form
452
Polynomial and Rational Matrices
ª¬1 z 2
R3 z
z 4 º¼ .
(8.4.26)
Step 3: In this case, the quality (8.4.18) has the form
ª b10j z b11j º
«
»
z 4 ¼º «b20 j z b21 j »
« b30 j z b31 j »
¬
¼
2
»1 z
(8.4.27)
n j 5 z 5 n j 4 z 4 n j 3 z 3 n j 2 z 2 n j1 z n j 0 ,
j 1, 2,
where
n15
1, n14
2, n13
1, n12
n25
0, n24
1, n23
0, n22
0, n11 1, n10
2, n21
1,
0, n20
1.
From (8.4.27), we have
b110
1, b210
1
12
1
22
1, b310
b
1. b
B0
ª b110 b120 º
« 0
0 »
«b21 b22 »
«b310 b320 »
¬
¼
1, b111
1
32
1
1, b21
1
31
2, b
1, b
0, b120
0, b220
0, b320
0,
2
and
ª1 0 º
«1 0 » , B
«
» 1
«¬1 0 »¼
ª b111 b121 º
« 1
1 »
«b21 b22 »
1
1 »
«b31
b32
¬
¼
ª1 1 º
«0 2» , C
«
»
«¬ 2 1 »¼
>0
0 1@ . (8.4.28)
The desired positive minimal realisation of (8.4.22) is given by (8.4.23),
(8.4.25) and (8.4.28).
Up to now, the degree of the polynomial d(z) has been even, equal to 2n. Now
let us assume that the degree of the denominator is odd.
Consider the system with one delay in state and two delays in control
xi 1
yi
A 0 xi A1 xi 1 B 0ui B1ui 1 B 2 ui 2 ,
Cxi Dui ,
(8.4.29a)
(8.4.29b)
i  ' ,
where
xi  n , ui  m , yi  , A k  nun , k
C
1un
, D
1um
0,1, B j  num , j
.
The matrix transfer function of (8.4.29) has the form
0,1, 2,
.
Positive Linear Systems with Delays
C ª¬I n z A 0 A1 z 1 º¼
T z
1
453
B 0 B1 z 1 B 2 z 2 D
(8.4.30)
Nc z
D,
dc z
C Adj ª¬I n z 2 A 0 z A1 º¼ B 0 z 2 B1 z B 2
D
z det ª¬ I n z 2 A 0 z A1 º¼
where
C Adj ª¬I n z 2 A 0 z A1 º¼ B 0 z 2 B1 z B 2
Nc z
ª¬ ncj ,2 n z 2 n ncj ,2 n1 z 2 n1 ... ncj ,1 z ncj ,0 º¼
j
1,..., m
,
(8.4.31)
z det ª¬ I n z 2 A 0 z A1 º¼
dc z
z 2 n1 a2 n1 z 2 n ... a1 z 2 a0 z
2 n 1
¦ d cz
i
d 2cn1
i
1, d 0c
0,
i 0
and the coefficients ak, k = 0,1,…,2n1 are of the polynomial d(z) defined by
(8.4.6).
Knowing the coefficients di’ = ai1, i = 1,…,2n of the polynomial d’(z) we may
find the matrices A0 and A1 of the form (8.4.10).
Choosing the matrix C of the form (8.4.17) in similarly to in the previous case,
we obtain
C Adj ª¬I n z 2 A 0 z A1 º¼ b0j z 2 b1j z b 2j
ª1 z 2 ! z 2 n1
¬
ª b10j z 2 b11j z b12j º
« 0 2
»
b z b21 j z b22 j »
º « 2j
¼«
»
#
« 0 2
»
1
2
¬« bnj z bnj z bnj ¼»
ncj ,2 n z 2 n ncj ,2 n1 z 2 n1 ... ncj ,1 z ncj 0 ,
(8.4.32)
j 1,..., m,
where bjk is the j-th column of the matrix Bk, k = 0,1,2.
Comparing the coefficients at the same powers of z of the equality (8.4.32), we
obtain the following 2n+1 equalities:
b12j
ncj 0 , b11j
b21 j
ncj 3 , b20 j b32j
0
n 1 j
b
2
nj
b
bn02, j bn21, j
ncj1 , b10j b22 j
ncj 2 ,
ncj 4 , b31 j
1
n1, j
ncj ,2 n1 , b
ncj ,2 n2 , bnj1
ncj 5 , !
(8.4.33)
ncj ,2 n3 ,
ncj ,2 n1 , bnj0
ncj ,2 n
454
Polynomial and Rational Matrices
with 3n unknown entries bij0,bij1,bij2, i = 1,…,n, j = 1,…,m, of the matrices
B0, B1, B2. Note that we may choose arbitrarily n1 entries of the matrix B0, for
example, bij0 = 0 for i = 1,…,n1 and find the remaining nonnegative entries of the
matrices from (8.4.33).
Therefore, the following theorem has been proved.
Theorem 8.4.2. There exists a positive minimal realization (8.4.3) of the proper
matrix transfer matrix
Nc z
D,
dc z
T z
(8.4.34)
with Nc(z) and dc(z) defined by (8.4.31) if the following conditions are satisfied
T f
lim T z  1um .
z of
The coefficients of Nc(z) and dc(z) satisfy the conditions
ncjk t 0 , j 1,..., m , k
0,1,..., 2 ,
d ic t 0 , i 1,..., 2n, and d 0c
0,
(8.4.35)
(8.4.36)
To find a positive minimal realization (8.4.3) of (8.4.34), the Procedure 8.4.1
with slight modification can be used.
Example 8.4.2.
Given the transfer matrix
1
z 2 z 3z 3 2 z 2 z
u ª¬ z 5 z 4 3z 3 1 2 z 5 4 z 4 5 z 3 3 z 2 2 º¼
T z
5
4
(8.4.37)
find the positive minimal realisation (8.4.3).
Using the procedure, we obtain the following.
Step 1: From (8.4.7), we have
D
and
lim ȉ z
z of
>1
2@
(8.4.38)
Positive Linear Systems with Delays
Tsp z
T z D
Nc z
dc z
1
ª z 4 2 z 2 z 1 z 3 z 2 2 z 2 º¼ .
z 5 2 z 4 3z 3 2 z 2 z ¬
455
(8.4.39)
Step 2: Taking into account that
a0 1 , a1
3
2 , a2
a3
and using (8.4.10), we obtain
A0
ª 0 a1 º
«0 a »
3¼
¬
ª0 a0 º
«1 a »
¬
2¼
ª0 2 º
« 0 2 » , A1
¬
¼
ª 0 1º
« 1 3» .
¬
¼
(8.4.40)
Step 3: In this case, the equality (8.4.32) has the form
ª b 0 z 2 b1 z b12j º
ª¬1 z 2 º¼ « 01 j 2 11 j
2 »
«¬b2 j z b2 j z b2 j »¼
ncj 4 z 4 ncj 3 z 3 ncj 2 z 2 ncj1 z ncj 0 ,
(8.4.41)
j 1, 2,
and
n14c 1, n13c 0, n12c
c 2, n20
c 2.
n21
2, n11c
1, n10c
c
1, n24
c
0, n23
c
1, n22
1,
1
2, b21
0, b210
1, b122
2, b121
2,
From (8.4.41), we have
b112
0
12
1, b111
2
22
1, b110
1
22
b
0, b
1, b
B0
ª0 0 º
«1 0 » , B1
¬
¼
0, b212
0
22
1, b
0
and
ª1 2 º
«0 1 » , B 2
¬
¼
ª1 2º
«2 1» , C
¬
¼
>0 1@ .
(8.4.42)
The desired positive minimal realisation (8.4.3) of (8.4.37) is given by (8.4.38),
(8.4.40) and (8.4.42).
456
Polynomial and Rational Matrices
Remark 8.4.2.
Note that the role of the delays in the control and output of the system can be
interchanged.
8.5 Realisation Problem for Positive Continuous-time Systems
with Delays
8.5.1 Problem Formulation
Consider the multi-variable continuous-time system with h delays in state and q
delays in control
x (t )
q
h
¦ A x(t id ) ¦ B u(t jd ),
i
i 0
y (t )
j
(8.5.1)
j 0
Cx(t ) Du (t ),
where x(t) n, u(t) m, y(t) p are the state, input and output vectors,
respectively, and Ai nun, i = 0,1,…,h, Bj num, j = 0,1,…,q, C pun, D pum and
d > 0 is a delay.
The transfer matrix of the system (8.5.1) is given by
T( s, w)
C [I m s A 0 A1w ! A h wh ]1
u[B 0 B1w ! B q wq ] D, w
(8.5.2)
e hs .
Let Mn be the set of nun Metzler matrices.
Definition 8.5.1. The matrices
A 0  M n , A i  nun , i 1,.., h, B j  num , j
C  pun , D  pum
0,1,..., q,
(8.5.3)
are called a positive realisation of a given transfer matrix T(s, w) if they satisfy the
equality (8.5.2). A realisation is called minimal if the dimension nun of matrices
Ai, i = 0,1,…,h, is minimal among all realisations of T(s, w)
The positive realisation problem can be stated as follows. Given a proper
transfer matrix T(s, w), find a positive realisation (8.5.3) of T(s, w).
Sufficient conditions for solvability of the problem will be established and a
procedure for the computation of a positive minimal realisation will be proposed
below.
Positive Linear Systems with Delays
457
8.5.2 Problem Solution
The transfer matrix (8.5.2) can be rewritten in the form
T( s, w)
C Adj H ( s, w) B 0 B1w " B q wq
det H ( s, w)
D
(8.5.4)
N( s, w)
D,
d ( s, w)
where
H ( s, w) [I m s A 0 A1w ! A h wh ],
(8.5.5)
N(s, w) C Adj H(s, w) B 0 B1w " B q w ,
q
d ( s, w) det H(s, w).
(8.5.6)
From (8.5.4), we have
D
lim T( s, w) ,
(8.5.7)
s of
since lim H 1 ( s, w) 0.
s of
The strictly proper part of T(s, w) is given by
Tsp ( s, w)
T( s, w) D
N( s, w)
.
d ( s, w)
(8.5.8)
Therefore, the positive realization problem has been reduced to finding
matrices
A 0  M n , A k  m um , k
1,...,q, B j  m , j 1, ! , q, C  pun (8.5.9)
for a given strictly proper transfer matrix (8.5.8).
Lemma 5.8.1. If
A0
ª0
«1
«
«0
«
«#
«0
¬
0 ! 0
0 ! 0
a00 º
a01 »»
1 ! 0 a02 » , A i
»
# % #
# »
0 ! 1 a0 n1 »¼
ª0
«0
«
«#
«
«¬0
0 " 0
0 " 0
ai 0 º
ai1 »»
, i 1,..., h, (8.5.10)
# % #
# »
»
0 " 0 ai n1 »¼
458
Polynomial and Rational Matrices
then
d ( s,w)
det[I n s A 0 A1w " A h wh ]
(8.5.11)
s n d n1s n1 d n2 s n2 ! d1s d 0
where
dj
d j ( w)
ah , j wh ah1, j wh1 ! a1 j w a0 j , j
0,1,..., n 1 . (8.5.12)
Proof. Expansion of the determinant with respect to the n-th column yields
s
det [I n s A 0 A1w ! A h wh ]
0
0
1
s
"
"
d 0
0
d1
0
1 "
0
d 2
#
0
0
#
0
0
% #
#
" s
d n 2
" 1 s d n1
s n d n1s n1 d n2 s n2 ! d1s d 0 .
Ŷ
Remark 8.5.1.
There exist many different matrices A0,A1,…,Ah giving the same desired
polynomial d(s, w) [164,168,166,171, 173].
Remark 8.5.2.
The matrix A0 is a Metzler matrix and the matrices A1,…,Ah have nonnegative
entries if and only if the coefficients aij of the polynomial d(s, w) are nonnegative,
except a0,n-1, which can be arbitrary.
Remark 8.5.3.
The dimension nun of matrices (8.5.10) is the smallest possible one for the given
d(s, w).
Lemma 8.5.2. If the matrices Ai, i=0,1,…,h, have the form (8.5.10), then the n-th
row of the adjoint matrix Adj H(s, w) has the form
R n ( s ) [1 s ! s n1 ] .
Proof. Taking into account that
Adj H ( s, w) H ( s, w)
I n d ( s, w) ,
(8.5.13)
Positive Linear Systems with Delays
459
it is easy to verify that
R n ( s ) H ( s, w) [0 ! 0 1] d ( s, w) .
(8.5.14)
Ŷ
The strictly proper matrix Tsp(s, w) can always be written in the form
Tsp ( s, w)
ª N1 ( s, w) º
«
»
« d1 ( s, w) »
«
»,
#
«
»
« N p ( s, w) »
« d ( s, w) »
¬ p
¼
d k ( s, w)
s nk d nk 1s nk 1 ! d1s d 0 , k
(8.5.15)
where
di
ahi ii whi ! a1ii w a0i i , i
d i ( w)
1,..., p,
(8.5.16)
0,1,..., nk 1
is the least common denominator of the k-th row of Tsp(s, w) and
N k ( s, w) [nk 1 ( s, w),..., nkm ( s, w)], k
nk 1 nk 1
kj
nkj ( s, w)
i
kj
n
iq
kj
n
s
0
0j
! a w a ,
i1
kj
q
1
1j
i0
kj
n w ! n w n , i
1,..., p,
j
(8.5.17)
0,1,..., m,
0,1,..., nk 1.
By Lemma 8.5.1 we may associate to the polynomial (8.5.16) the matrices
ª0 0
«
«1 0
A k 0 «0 1
«
«# #
«0 0
¬
k 1,.., p, i
k
º
a00
k »
! 0 a01 »
k »
! 0 a02
, A ki
»
% #
# »
! 1 a0k nk 1 »¼
1,..., hk ,
! 0
ª0
«
«0
«#
«
¬«0
aik0 º
»
aik1 »
,
# % #
# »
»
0 " 0 aiknk 1 ¼»
0 " 0
0 " 0
(8.5.18)
satisfying the condition
d k ( s, w)
det [I nk s A k 0 A k 1w ! A khk whk ], k
1,.., p .
(8.5.19)
460
Polynomial and Rational Matrices
Let
A0
block diag [ A10 ! A p 0 ]  nun ,
Ai
block diag [ A1i ! A pi ]  nun ( n
Bk
C
ª b11k " b1km º
«
» k
« # % # » , bij
k »
«bpk1 " bpm
¬
¼
ª bijk1 º
« »
« # », k
«bijkni »
¬ ¼
block diag[c1 ! c p ], c
k
(8.5.20)
n1 " n p ),
0,1,..., q; i 1,..., p; j 1,.., m, (8.5.21)
[0 ! 0 1]  1unk , k
1,..., p .
The number of delays q in control is equal to the degree of the polynomial
matrix N(s, w) in variable w.
From (8.5.8), (8.5.17), (8.5.22), and (8.5.24)( 8.5.26), we obtain for the j-th
column of Tsp(s, w)
Tspj (s, w) CH 1 (s, w)[B 0 B1w ! Bq w q ] j
^
1
block diag[c1 ! c p ] §¨ block diag ª¬I n1 s A10 A11w ! A1h1 w h1 º¼ ,...
©
ª b10j b11j w ! b1qj w q º
«
»
h
#
...,[I np s A p 0 A p1w ! A php w p ]1 «
»
«bpj0 b1pj w ! bpjq w q »
¬
¼
`
­° 1
½
1
n 1 °
block diag ®
[1 s ! s n1 1 ],!,
[1 s ! s p ]¾
d p ( s, w )
¯° d1 (s, w)
¿°
ª b10j b11j w ! b1qj w q º
«
»
u«
#
»
«bpj0 b1pj w ! bpjq w q »
¬
¼
ª (b1qnj 1 w q ! b11nj 1 w b10jn1 )s n1 1 ! b1qj1w q ! b111j w b101j º
«
»
d1 (s, w)
«
»
«
»
#
«
»
« (bqnp w q ! b1n p w b 0 n p )s n p 1 ! b q1w q ! b11w b01 »
pj
pj
pj
pj
pj
« pj
»
d p (s, w)
«¬
»¼
ª n1 j (s, w) º
«
»
« d1 (s, w) »
«
» , j 1,..., m,
#
«
»
« n pj (s, w) »
« d (s, w) »
¬ p
¼
(8.5.23)
Positive Linear Systems with Delays
461
and nij(s, w) are given by (8.5.17).
A comparison of the coefficients at the same powers of s and w of the equality
(8.5.23) yields
b101j
n100j , b111j
! , b10jn1
n101j ,..., b1qj1
n1n1j 1,0 , b11nj 1
n10jq ,...
n1n1j 1,1 ,..., b1qnj 1
n1n1j 1,q
"""""""""""""""""""
b
01
pj
!, b
00
pj
11
pj
n ,b
0 n1
pj
n
01
pj
n ,..., b
n p 1,0
pj
1n p
pj
,b
q1
pj
n
(8.5.24)
0q
pj
n ,...
n p 1,1
pj
qn
,..., bpj p
n 1,q
n pjp
for j = 1,…,m.
Theorem 8.5.1. There exists a positive realisation (8.5.3) of T(s, w) if
1.
T( f )
2.
lim T( s, w)  pum ,
(8.5.25)
s of
the coefficients of dk(s, w) k = 1,…,p are nonnegative, except a0 nk 1 ,
k = 1,…,p, i.e.,
aijk t 0, i 1,..., hk ;
3.
j
0,1,..., nk 1, k
1,..., p ,
(8.5.26)
the coefficients of Nj(s, w), j =1,…,m are nonnegative, i.e.,
nijk t 0, for i 1,..., p; j 1,..., m; k
0,1,..., q .
(8.5.27)
Proof. The condition (8.5.25) implies D +pum. If the conditions (8.5.26) are
satisfied, then the matrices (8.2.18) have nonnegative entries except a0,n k 1,
k = 1,…,p, which can be arbitrary. In this case, A0Mn and Ai +nun, i = 1,…,h. If
additionally the conditions (8.5.27) are satisfied, then from (8.5.24) it follows that
Bk +num, k=0,1,…,q. The matrix C of the form (8.5.22) is independent of T(s, w)
and always has nonnegative entries.
Ŷ
Theorem 8.5.2. The realisation (8.5.3) of T(s, w) is minimal if the polynomials
d1(s, w),…,dp(s, w) are relatively prime (coprime).
Proof. If the polynomials d1(s, w),…,dp(s, w) are relatively prime, then
d(s, w) = d1(s, w),…,dp(s, w) and by Remark 8.5.3 the matrices (8.5.20) have
minimal dimensions.
Ŷ
462
Polynomial and Rational Matrices
If the conditions of Theorem 8.5.2 are satisfied, then a positive minimal
realisation (8.5.3) of T(s, w) can be found by the use of the following procedure.
Procedure 8.5.1.
Step 1: Using (8.5.7) and (8.5.8) find the matrix D and the strictly proper matrix
Tsp(s, w).
Step 2: Knowing the coefficients of dk(s, w), k = 1,…,p, find the matrices (8.5.18)
and (8.5.20).
Step 3: Knowing the coefficients of Nj(s, w), j = 1,…m, and using (8.5.24), (8.5.21)
find the matrices Bi, i=0,1,…,q, and the matrix C.
Example 8.5.1.
Using above procedure find a positive realisation (8.5.3) of the transfer matrix
T( s, w)
ª s 2 ( w2 w 2) s w2 w
,
« 2
2
2
« s ( w 2) s (2w w 1)
«
w2 1
,
«
s 2 w2 w 1
¬
º
s 2 3s (2w2 1)
»
s ( w2 2) s (2 w2 w 1) » (8.5.28)
.
»
2 s 2 w2 2
»
s 2 w2 w 1
¼
2
It is easy to verify that the assumptions of Theorem 8.5.2 are satisfied.
Using Procedure 8.5.1, we obtain the following.
Step 1: From (8.5.7) and (8.5.8), we have
D
lim T( s, w)
s of
ª1 1 º
«0 2 »
¬
¼
(8.5.29)
and
Tsp ( s, w)
T( s, w) D
ª
ws w2 1
« s 2 ( w2 2) s (2 w2 w 1)
«
«
w2 1
«
s 2 w2 w 1
¬
º
( w2 1) s w
s ( w 2) s (2w2 w 1) »» (8.5.30)
.
»
2( w2 w)
»
s 2 w2 w 1
¼
2
Step 2: Taking into account that
d1 ( s, w)
s 2 ( w2 2) s (2 w2 w 1),
d 2 ( s, w)
s 2w2 w 1,
and using (8.5.18) and (8.5.20), we obtain
2
Positive Linear Systems with Delays
A0
A2
ª A10
« 0
¬
ª0 1 0 º
«
»
«1 2 0 » , A1
«0 0 1»
¬
¼
ª0 2 0 º
«
»
«0 1 0 » .
«0 0 2»
¬
¼
0 º
A 20 »¼
ª A 21
« 0
¬
0 º
A 22 »¼
ª A11
« 0
¬
0 º
A 21 »¼
ª0 1 0º
«
»
«0 0 0» ,
«0 0 1 »
¬
¼
463
(8.5.31)
Step 3: In this case,
n11 ( s, w)
ws w2 1, n12 ( s, w)
n22 ( s, w)
2( w2 w).
( w2 1) s w,
Using (8.5.24) and (8.5.21), we obtain
B0
ª b1101
« 02
«b11
01
« b21
¬
B2
ª b1121
« 22
«b11
« b2121
¬
b1201 º
»
b1202 »
01 »
b22
¼
21
b22 º
»
b1222 »
b2221 »¼
ª1 0 º
«0 1 » , B
«
» 1
«¬1 0 »¼
ª b1111
« 12
«b11
11
« b21
¬
ª1 0 º
«0 1 » and C
«
»
«¬1 2 »¼
b1211 º
»
b1212 »
11 »
b22
¼
ª0 1 º
«1 0 » ,
«
»
«¬ 0 2 »¼
(8.5.32)
ª0 1 0º
«0 0 1 » .
¬
¼
The desired positive realisation of (8.5.3) of (8.5.28) is given by (8.5.29),
(8.5.31) and (8.5.32). The realisation is minimal, since the polynomials d1(s, w),
d2(s, w) are relatively prime.
8.6 Positive Realisations for Singular Multi-variable Discretetime Systems with Delays
8.6.1 Problem Formulation
Consider the discrete-time linear system with q state delays and q input delays
described by the equations
q
Ex(i 1)
¦
A j x(i j ) B j u (i j ) ,
(8.6.1a)
j 0
y (i )
Cx(i ) i  ' ,
(8.6.1b)
464
Polynomial and Rational Matrices
where x(i) n, u(i) m, y(i) p are the state, input (control) and output vectors
respectively, and E, Ak nun, Bk num, k = 0,1,…,q, C pun.
It is assumed that det E = 0 and
det ª¬ Ez q 1 A 0 z q A1 z q 1 ! A q º¼ z 0 for some z  (the field of complex numbers).
(8.6.2)
The initial conditions for (8.6.1a) are given by
x(i )  n , u (i )  m for i
(8.6.3)
0,1,..., q.
Let us assume that the matrices E, A0, A1, B0, B1, C have the following
canonical forms [81, 127]
ªI n1 0 º
n un
E block diag ª¬E1 , E2 ,..., E p º¼  nun , Ei «
» i i,
«¬ 0 0 »¼
p
i 1,..., p, n
¦n ,
i
i 1
Aj
a ji
A qi
aqi
Bj
bilj
C
Ci
block diag ª¬ A j1 , A j 2 ,..., A jp º¼  nun , A ji
ª a ji º
ni
ni 1
« a ni »  , a ji  , j
¬ ji ¼
ª 0
º
aqi »  ni uni , aqi
«I
«¬ ni 1
»¼
ª a1qi º
«
»
ni 1
« # »  , j 1,..., q,
« aqini 1 »
¬
¼
j
ª b11 " b1jm º
«
»
num
j
#
«
»  , bii
j »
«bpj1 " bpm
¬
¼
ª¬ 0 a ji º¼  ni uni ,
1,..., q 1; i 1,..., p,
ª aqi º
ni
« a ni »  ,
qi
¬
¼
ª bilj º
« l ni » ,
¬bil ¼
ª bilj º
« »
« # » , i 1,..., p; l 1,..., n,
«bill ni »
¬ ¼
block diag ª¬C1 C2 ! C p º¼  pun ,
>0 0 ! 1@  1uni , i 1,..., p.
(8.6.4)
Positive Linear Systems with Delays
465
Definition 8.6.1. The system (8.6.1) is called (internally) positive if for every
x(k) +n, u(k) +m, k = 0,1,…,q, and all inputs u(i) +m, i +, we have
x(i) +n and y(i) +p for i +
Theorem 8.6.1. The system (8.6.1) with matrices of the forms (8.6.4) is positive if
and only if
akil t 0 for k
ni
ki
a
0,1,..., q; i 1,..., p; l
ni
qi
0, a ! 0 for k
0,1,..., ni ,
(8.6.5a)
0,1,..., q 1; i 1,..., p,
bijk  ni for i 1,..., p; j 1,..., m; k
(8.6.5b)
0,1,..., q.
Proof. Let
xk (i )
ª xk (i ) º
nk
nk 1
« x (i ) »  , x (i )  , i  ' , k
kn
¬ k ¼
1,..., p
(8.6.6a)
be the k-th (k = 1,…,p) subvector of x(i) corresponding to the k-th block of (8.6.4)
and
A qk
ª 0
«I
¬« nk 2
nk
jk
jnk
k1
b
º
0 »  ( nk 1)u( nk 1) , B jk
¼»
jnk
km
ª¬b , ... , b
º¼ , enk
j
j
¬ªbk 1 , ... , bkm ¼º ,
1u( nk 1)
>0, ... , 0 1@  (8.6.6b)
.
Using (8.6.1a), (8.6.4) and (8.6.6), we may write
q
x k (i 1)
q
A qk x (i q) ¦ a jk x jnk (i j ) ¦ B jk u(i j ) ,
j 0
aqknk xknk (i q )
(8.6.7a)
j 0
q
enk xk (i q ) ¦ b njkk u (i j ) .
(8.6.7b)
j 0
If the conditions (8.6.5) are satisfied, then using (8.6.7a), for i=0,1,…,q, and the
initial conditions (8.6.3), we may compute
xk (i )  nk 1 , for i 1,..., q 1.
Next from (8.6.7b)
xknk (q 1)  and from (8.6.7a)
466
Polynomial and Rational Matrices
xk (q 2)  nk 1.
Continuing the procedure we may find
xk (i )  nk , for i
' and k
1,..., p
and from (8.6.1b) y(i) = Cx(i) +p for i +.
The necessity follows immediately from the arbitrariness of the initial
conditions (8.6.3) and of the input u(i) and can be shown in a similar way as for
systems without delays [127].
Ŷ
Remark 8.6.1.
Using (8.6.6b) we may eliminate xn k from (8.6.7a) and (8.6.1b) and we obtain a
standard positive system with delays and advanced arguments in control.
The transfer matrix of (8.6.1) is given by
T( z )
C[Ez A 0 A1 z 1 ! A q z q ]1 (B 0 B1 z 1 ! B q z q )
C[E z q1 A 0 z q A1 z q 1 ! A q ]1 (B 0 z q B1 z q 1 ! B q ).
(8.6.8)
Definition 8.6.2. Matrices (8.6.4) satisfying (8.6.5a) are called a positive
realisation of the transfer matrix T(z) if they satisfy (8.6.8). The realisation is
called minimal if the dimension nun of E, Ak, k = 0,1 is minimal among all
realisations of T(z).
The positive minimal realisation problem can be stated as follows. Given an
improper transfer matrix T(z), find a positive (minimal) realisation of T(z)
Solvability conditions for the positive (minimal) realizstion problem will be
established and a procedure for computation of a positive (minimal) realisation of
T(z) will be presented.
8.6.1 Problem Solution
To solve the positive realisation problem we shall use the following two lemmas.
Lemma 8.6.1. If the matrix E k has the form (8.6.4) and
Positive Linear Systems with Delays
" 0
A0k
ª0
«
«0
«#
«
«0
«
¬«0
aqk º
»
" 0 a2 qk 1 »
% #
# » , A1k
»
" 0 ank 1 »
»
0 ¼»
" 0
0 " 0
0 " 0
A qk k
ª0
«1
«
«0
«
«#
«
«0
«0
¬
ª0
«
«0
«#
«
«0
«
¬« 0
467
" 0 aqk 1 º
»
" 0 a2 qk »
% #
# » , ... ,
»
" 0 ank 2 »
»
0 ¼»
" 0
(8.6.9)
a0
aqk 1
º
»
»
1 " 0
a2( qk 1) »
»,
»
# % #
#
»
0 " 0 a( nk 2)( qk 1) »
0 " 1
1 »¼
then
dk ( z)
det ª¬Ek z qk 1 A 0 k z qk ! A qk ,k º¼
z nk ank 1 z nk 1 ! a1 z a0 , k
where nk
(8.6.10)
1,..., p.
(nk 1)(qk 1).
Proof. Expansion of the determinant with respect to the ni-th column yields
det[Ek z qk 1 A 0 k z qk ! A qk ,k ]
z qk 1
1
!
0
aqk z qk aqk 1 z qk 1 ! a0
z qk 1 !
0
a2 qk 1 z qk a2 qk z qk 1 ! aqk 1
0
#
0
#
0
0
0
nk
z ank 1 z
%
#
qk 1
! z
!
nk 1
#
qk
ank 1 z ank 2 z
1
! a1 z a0 , k
qk 1
! a( nk 2)( qk 1)
1
1,..., p.
Ŷ
Lemma 8.6.2. If the matrix Ek has the form (8.6.4) and the matrices Aik,
i=0,1,…,q, have the forms (8.6.9), then the nk-th row Rn k (z) of the adjoint matrix
Adj [Ek z qk 1 A 0 k z qk ! A qk ,k ]
468
Polynomial and Rational Matrices
has the form
R nk ( z ) [1 z qk 1 ! z nk ], k
1,..., p .
(8.6.11)
Proof. Taking into account that
Adj ª¬Ek z qk 1 A 0 k z qk ! A qk ,k º¼ ª¬E k z qk 1 A 0 k z qk ! A qk ,k º¼
I nk d k ( z ),
it is easy to verify that
R nk ( z ) ª¬Ek z qk 1 A 0 k z qk ! A qk ,k º¼
[0 ! 0 1] d k ( z ) .
Ŷ
Let a given improper transfer matrix have the form
ª n11 ( z )
n ( z) º
, ... , 1m
«
»
d1 ( z ) »
« d1 ( z )
«
»,
#
«
»
n pm ( z ) »
« n p1 ( z )
« d ( z ) , ... , d ( z ) »
p
¬ p
¼
T( z )
where
t
t
nkjkj z kj ! n1kj z nkj0
nkj ( z )
rk
dk ( z)
z akrk 1 z
rk 1
k
1,..., p; j 1,..., m ,
! ak1 z ak 0 .
(8.6.13a)
(8.6.13b)
The number of delays q is equal to
1,..., p) ,
q
max (tk rk ) (k
tk
max tkj , j 1,..., m .
k
(8.6.14)
where
j
If the matrices Ek Ajk have the forms (8.6.4), then the minimal nk is given by the
formula
Positive Linear Systems with Delays
nk t
tk 1
, k
tk rk 1
1,..., p .
469
(8.6.15)
The formula (8.6.15) can be justified as follows. If the matrix Ek has a
canonical form then
1,..., p .
(nk 1)(tk rk 1) t rk , k
(8.6.16)
Solving (8.6.16) with respect to nk, we obtain (8.6.15).
Knowing the coefficients of the denominators d1(z),…,dp(z) of (8.6.12), we may
find the matrices Aji of the forms (8.6.4) such that (8.6.10) hold.
Let (B0zq + … + Bq)j and Tj(z), j = 1,…,m, be the j-th column of the matrix
q
B0z + … + Bq and T(z), respectively.
Using (8.6.8), (8.6.9) and (8.6.10), we obtain
C [E z q1 A 0 z q ! A q ]1 (B 0 z q ! B q ) j
Tj ( z )
^
1
block diag[C1 ! C p ] block diag ª¬E1 z q1 A 01 z q ! A q1 º¼ ,!
...,[E p z
q 1
q
1
A 0 p z ! A qp ]
`
ªb1jj z q ... b1qj º
«
»
#
u«
»
«bpj0 z q ... bpjq »
¬
¼
­ 1
ª¬1 z q1 1 ! z ( q1 1)( n1 1) º¼ ,...
block diag ®
¯ d1 ( z )
ª b j z q ... b1qj º
½ « 1j
1 ª
»
( q p 1)( n p 1) °
q p 1
º¾ «
! z
#
...,
1 z
»
¼
dp ( z) ¬
°¿ «b0 z q ... b q »
pj ¼
¬ pj
ª b10jn1 z t1 j b11nj 1 z t1 j 1 ... b1qj1,1 z b1qj1 º ª n1 j ( z ) º
«
» «
»
d1 ( z )
«
» « d1 ( z ) »
«
» « # » , j 1,..., m,
#
«
» «
»
« b 0 n p z t pj b1n p z t pj 1 ... b q 1,1 z b q1 » « n pj ( z ) »
pj
pj
pj
pj
«
» «
d ( z) »
d p ( z)
«¬
¼» ¬ p ¼
(8.6.17)
where nij(z), i=1,…,p, are defined by (8.6.13a).
Comparing the coefficients at the same powers of z of numerators of (8.6.17),
we obtain
b10jn1
t
n11jj , bij1n1
t 1
n11jj , ... , b1qj1,1
n11 j , b1qj1
n10j
............................................................................., j 1,..., p .
b
0np
pj
t pj
pj
1n p
pj
n , b
n
t pj 1
pj
, ... , bpjq 1,1
n1pj , bpjq1
n0pj
(8.6.18)
470
Polynomial and Rational Matrices
Theorem 8.6.2. There exists a positive realisation of (8.6.12) if
1. the coefficients of denominators (8.6.13b) are nonnegative, i.e.,
aki t 0, for k
2.
1,..., p; i
0,1,..., rk 1,
(8.6.19)
the coefficients of numerators (8.6.13a) are nonnegative, i.e.,
t
nkjkj t 0, for k
(8.6.20)
1,..., p; j 1,..., m.
Proof. If the conditions (8.6.20) are satisfied, then from (8.6.18) it follows that
Bj +nun, for j = 0,1,…,q. Additionally, if the condition (8.6.19) is satisfied, then
by Theorem 8.6.1 the realisation is positive.
Ŷ
Theorem 8.6.3. The realization of T(z) is minimal if the denominators
di(z),…,dp(z) are relatively prime (coprime).
Proof. If the denominators are relatively prime, then
d ( z)
det [Ez q 1 A 0 z q ! A q ]
d1 ( z ) ... d p ( z )
and the matrices E, Aj, j = 0,1,…,q, have minimal possible dimensions.
Ŷ
If the conditions of Theorem 8.6.2 are satisfied, then a positive (minimal)
realisation of (8.6.12) can be found by the use of the following procedure.
Procedure 8.6.1.
Step 1: Knowing the degrees tk of the numerators nij(z) and rk of the denominators
dk(z) and using (8.6.14), find the number of delays q and from (8.6.15) the
minimal nk for k = 1,…,p.
Step 2: Using the coefficients of dk(z) k = 1,…,p, find the matrices Aj j = 0,1,…,q,
E and C.
Step 3: Using (8.6.18), find the matrices Bj j = 0,1,…,q.
Remark 8.6.2.
The matrices E and C have the canonical forms (8.6.4) and their dimensions
depend only on T(z).
Example 8.6.1.
Find a positive realisation of the transfer matrix
Positive Linear Systems with Delays
ª z3 2z 2 z 3
« z2 2z 1
« 3
« z 2z2 z
«¬
z 2 3z
T( z )
3z 3 2 z 2 º
z 2 2 z 1 »»
.
z 2 3z »
z 2 3 z »¼
471
(8.6.21)
It is easy to check that the transfer matrix (8.6.21) satisfies the conditions
(8.6.19) and (8.6.20).
Using the above procedure we obtain the following.
Step 1: In this case, t1 = t2 = 3, r1 = r2 = 2. Hence
q
max(tk rk ) 1
k
and from (8.6.15), we obtain n1 = n2 = 2.
Step 2: Taking into account that d1(z) = z3 – 2z - 1 and d1(z) = z2 – 3z, we obtain
E
ª E1 0 º
«0 E »
2¼
¬
A1
ª A11
« 0
¬
ª1
«
«0
«0
«
¬0
0 º
A12 »¼
0 0 0º
0 0 0 »»
,
0 1 0»
»
0 0 0¼
ª0 1 0
«1 1 0
«
«0 0 0
«
¬0 0 1
A0
ª A 01
« 0
¬
0º
0 »»
, C
0»
»
1¼
0 º
A 02 »¼
ªC1 0 º
«0 C »
2¼
¬
ª0
«
«0
«0
«
¬0
2 0 0º
0 0 0 »»
,
0 0 3»
»
0 0 0¼
(8.6.22)
ª0 1 0 0 º
«0 0 0 1 » .
¬
¼
Step 3: Using (8.6.18), we obtain
B0
ª b1101 b1201 º
« 02
02 »
«b11 b12 »
01
« b21
b 01 »
« 02 22
»
02
«¬b21 b22 »¼
ª1
«1
«
«0
«
¬1
2º
3 »»
, B1
3»
»
0¼
ª b1111 b1211 º
« 12 12 »
«b11 b12 »
11
« b21
b11 »
« 12 22
»
12
«¬b21 b22 »¼
ª3
«2
«
«1
«
¬2
2º
0 »»
.
0»
»
1¼
(8.6.23)
The desired realisation of (8.6.21) is given by (8.6.22) and (8.6.23). It is a
positive minimal realization, since the polynomials d1(z) = z3 – 2z - 1 and
d1(z) = z2 – 3z are relatively prime.
Remark 8.6.3.
Note that if
(nk 1)(qk 1) ! rk , for some k  [1,..., p],
(8.6.24)
472
Polynomial and Rational Matrices
then the numerator and the denominator of the k-th row of the transfer matrix
(8.6.12) should be multiplied by
z vk , where vk
( nk 1)(qk 1) rk .
Otherwise the obtained Aj, j=0,1,…,q, do not belong to a positive realisation of
(8.6.12). For example, if the given transfer matrix (8.6.12) has the form
T( z )
ª z3 2z 2 z 3
« z2 2z 1
«
« z2 2z 1
«¬
z 3
3z 3 2 z 2 º
z 2 2 z 1 »»
,
z 3
»
»¼
z 3
(8.6.25)
then for k = 2, we have n2 = 2 q2 = 1 r2 = 1 and v2 = (n2 – 1)(q2 + 1) – r2 = 1. In this
case, by multiplying the numerator and denominator of the second row of (8.6.25)
by z, we obtain the transfer matrix (8.6.21).
The matrices A20 and A12 for the second row of (8.6.25) have the forms
A 02
ª0 1º
«0 0 » , A12
¬
¼
ª0 3º
«1 0 » ,
¬
¼
and they do not belong to a positive realisation of (8.6.25).
Appendix
Selected Problems of Controllability and Observability
of Linear Systems
A.1 Reachability
Consider the following discrete-time linear system
xi 1 Axi Bui , i
yi Cxi Dui ,
0, 1, ... ,
where xi n is the state vector, ui
A nun, B num, C pun, D pum.
(A.1a)
(A.1b)
m
the input vector, yi
p
the output vector;
Definition A.1. The system (A.1) (or the pair (A,B)) is called reachable, if for
every vector xf n there exists an integer q > 0 and a sequence of inputs {ui,
i=0,1,…,q1} such that for x0 = 0, xq = xf.
Theorem A.1. The system (A.1) is reachable if and only if one of the following
conditions is met
1.
rank [B, AB,..., A n1B] n ,
(A.2)
rank [I n z A, B] n, for all finite z  ,
(A.3)
2.
474
Appendix
3.
(A.4)
[I n z A] and B are left coprime matrices.
Proof. Using the solution
xi
i 1
A i x0 ¦ A i k 1Buk
k 0
to equation (A.1a) for i = n, x0= 0 and taking into account that xn = xf, we obtain
xf
n 1
¦A
n k 1
Buk
k 0
ªun1 º
«
»
u
ª¬B, A, B,..., A n1B º¼ « n2 » .
«# »
«
»
¬ u0 ¼
(A.5)
From (A.5) it follows that for every xf there exits {ui, i=0,1,…,q1} if and only
if the condition (A.2) is met.
Let v n be a vector such that vTB = 0 and vTA = zvT for a certain complex
variable z. In this case,
vT AB
zvT B
0, vT A 2 B
zvT AB
0, ..., vT A n1B
that is,
vT ª¬B, AB, ..., A n1B º¼
0.
The condition (A.2) thus implies v = 0. Hence from
vT > I n z A, B @ 0
(A.3) follows.
From (A.3) it follows that there exists a unimodular matrix
U
ª U1 U 2 º
n mu( n m )
[ z] ,
«U U »  4¼
¬ 3
such that
> Iz A , B @ U > I n 0 @
0,
Appendix
475
and
>Iz A@ U1 BU3
In .
(A.6)
Thus [Iz – A] and B are left coprime matrices.
Let
u1i
u10i u11i z u12i z 2 u1ni2 z n2 (i 1, ..., n),
u3i
u30i u31i z u32i z 2 u3ni1 z n1
(A.7)
be the i-th columns of the polynomial matrices U1 nun[z] and U3 mun[z],
respectively. Substituting (A.7) into (A.6) and comparing the coefficients by the
same powers of the variable z , we obtain
Bu30i Au10i
1
3i
ei
0
1i
Bu u Au11i
0
for i 1, ..., n ,
..........................................
n2
3i
Bu
n 1
3i
Bu
n 3
1i
Au
n 2
1i
0
u
u
n2
1i
(A.8)
0
where ei is the i-th column of the identity matrix In.
Pre-multiplying the equations in (A.8) successively by A0,A1,A2,….,An1 and
adding them up, we obtain
Bu30i ABu31i A n1Bu3ni1
ei (i 1, ..., n)
and
ªu30i º
« 1 »
u
ª¬B, AB,..., A n1B º¼ « 3i »
« »
«
»
«¬u3ni1 »¼
ei (i 1, ..., n).
(A.9)
(A.9) implies the condition (A.2). The conditions (A.2), (A.3), and (A.4) are thus
equivalent.
476
Appendix
If the system (A.1) is not reachable, then the set of reachable states from the
point x0 = 0 is given by the image of the matrix [B,AB,…,An-1B].
Example A.1.
Show that the pair
A
ª0
«0
«
«#
«
«0
«¬ a0
1
0
#
0
a1
0
1
#
0
a2
" 0 º
" 0 »»
% # », B
»
" 1 »
" an1 »¼
ª0º
«0»
« »
« #»
« »
«0»
«¬1 »¼
(A.10)
is reachable for arbitrary values of the coefficients a0,a1,…,an1.
Using (A.3), we obtain
rank > I n z A, B @
ª z
« 0
«
rank « #
«
« 0
«¬ a0
1
z
#
0
a1
0
1
#
0
a2
"
0
"
0
%
#
"
1
" z an1
0º
0 »»
#»
»
0»
1 »¼
n,
(A.11)
for all finite z .
The last n columns of (A.11) are linearly independent whatever the values of
the coefficients a0,a1,…,an-1.
Now consider the following continuous-time linear system
x
y
§
Ax Bu ¨ x
©
Cx Du ,
dx ·
¸,
dt ¹
(A.12a)
(A.12b)
where x = x(t) n is the state vector, u = u(t) m the input vector, y = y(t)
output vector; A nun, B num, C pun, D pum.
p
the
Definition D.2. The system (A.12) (or the pair (A,B)) is called reachable if for
every vector xf n there exists a time tf > 0 and an input u(t) over the interval
[0, tf] such that for x0 = 0, x(tf) = xf.
Theorem D.2 The system (A.12) is reachable if and only if one of the following
conditions is satisfied:
Appendix
477
1.
rank [B, AB, ..., A n1B] n ,
(A.13)
rank [I n s A, B] n, for all finite s  ,
(A.14)
>I n s A @
(A.15)
2.
3.
and B are left prime.
The proof is similar to that of Theorem A.1.
A.2. Controllability
Definition A.3. The system (A.1) (or the pair (A,B)) is called controllable to zero
if for an arbitrary initial state x0 z 0 there exists an integer q > 0 and a sequence of
inputs {ui, i=0,1,…,q1} such that xq = 0.
Theorem A.3. The system (A.1) is controllable to zero if and only if one of the
following conditions is met:
1.
Im A n  Im [B, AB,..., A n1B] ,
(A.16)
rank [I dA, B]
(A.17)
2.
n, for all finite d  ,
3.
[I dA] and B are left coprime.
(A.18)
Proof. Using the solution to (A.1), for i = n, xn = 0 we obtain
A n x0
n 1
¦ A nk 1Buk
k 0
ªun1 º
«
»
u
ª¬B, AB,..., A n1B º¼ « n2 » .
«# »
«
»
¬ u0 ¼
(A.19)
478
Appendix
From (A.19) it follows that there exists a sequence of inputs {ui, i=0,1,…,q1}
for an arbitrary x0 if and only if the condition (A.16) is met.
Let v n be a vector such that vTB = 0 and vTA = zvT for a certain variable z. In
the same manner as in the proof of Theorem A.1, we obtain vT[B,AB,…,An-1B] = 0.
The condition (A.16) implies
0
vT A n
O n vT and thus O
0 or v
0.
Hence the matrix [Inz – A, B] has full row rank n for all finite z z 0, which is
equivalent to the conditions (A.17).
Analogously to the proof of Theorem A.1 one can show that the condition
(A.17) implies (A.18), and the condition (A.18) in turn implies (A.16).
Remark A.1.
Each of the conditions (A.13), (A.14) and (A.15) for the system (A.1) with singular
A is only a sufficient condition, but not a necessary one for the controllability of
the system. If det A z 0, then these conditions are also necessary conditions for the
controllability of (A.1). For the system (A.1) with nonsingular A, the conditions of
its controllability are equivalent to the conditions of its reachability.
Example A.2.
The pair of matrices
A
ª0 a º
«0 0 » , B
¬
¼
ª1 º
«0 »
¬ ¼
(A.20)
is not reachable, since
rank > Iz A, B @
ª z a 1 º
rank «
» 1, for z
¬0 z 0¼
0.
On the other hand, using (A.17), we obtain
rank > I dA, B @
ª1 da 1 º
rank «
»
¬0 1 0 ¼
2 for arbitrary a and d .
The pair (A.20) is thus controllable for arbitrary a.
Note that in this case the state
xf
ª0 º
«1 »
¬ ¼
is not reachable from the state x0 = 0, since x0 does not belong to
Appendix
479
ª1 º
Im[B, AB] Im « » .
¬0¼
On the other hand, the state
x0
ª0 º
«1 »
¬ ¼
can be brought to zero by the zero input sequence u0 = u1 = 0, since A2 = 0 for
arbitrary a.
Definition A.4. The system (A.12) (or the pair (A,B)) is called controllable to zero
if for an arbitrary initial state x0 there exists a time tf > 0 and an input u = u(t) over
the interval [0, tf]] such that x(tf) = 0.
Theorem A.4. The system (A.12) is controllable to zero if and only if one of the
conditions (A.16), (A.17), (A.18) of Theorem A.3 is met.
The proof of this theorem follows similarly to that of Theorem A.3.
Using the solution to (A.12a), for x(0) = x0, x(tf) = 0, we obtain
xf
e
At f
tf
x0 ³e
0
A (t f W )
tf
Bu (W ) dW
0 and x0
³ e AW Bu (W ) dW
0
since eAt is a nonsingular matrix regardless of the matrix A. Hence the
controllability of a continuous-time system is equivalent to its reachability for
every A.
Example A.3.
We choose as the state variable x the voltage uc on the capacity of the electrical
circuit in Fig. A.1, and as the input the source voltage u. Note that the voltage uc on
the capacity is zero for an arbitrary value of the source voltage u. Therefore
changing u we cannot reach any desired nonzero value of the voltage uc = xf z 0.
Thus this circuit is an example of an uncontrollable system.
480
Appendix
Fig. A.1. Uncontrollable electrical circuit
A.3 Observability
First consider the discrete system (A.1).
Definition A.5. The system (A.1) (or the pair (A,C)) is called observable if there
exists an integer q > 0 such that for given sequences of inputs {ui, i=0,1,…,q-1}
and outputs {yi, i=0,1,…,q-1} one can determine the initial state x0 of this system.
Theorem A.5. The system (A.1) is observable if and only if one of the following
conditions is met:
1.
ª C º
« CA »
»
rank «
« # »
«
»
n 1
¬«CA ¼»
n,
(A.21)
2.
ªI z A º
rank « n
»
C ¼
¬
n for all finite z  ,
(A.22)
3.
>I n z A @
and C are right coprime.
Proof. Substituting the solution of (A.1a) into (A.1b), we obtain
(A.23)
Appendix
yic
i 1
yi Dui ¦ CA i k 1Buk
CA i x0 .
481
(A.24)
k 0
Using (A.24), for i
ª y0c º
« c»
« y1 »
« # »
«
»
¬ ync 1 ¼
0,1,..., n 1 , we have
ª C º
«
»
« CA » x .
« # » 0
«
»
n 1
¬«CA ¼»
(A.25)
For the given sequences {ui, i=0,1,…,q1},{yi, i=0,1,…,q1} the sequence {y’i,
i=0,1,…,n1} is known. From (A.25) we can determine x0 if and only if the
condition (A.21) is met.
Equivalence of the remaining conditions can be proved similarly (dually) as in
Theorem A.1.
Example A.4.
Show that the pair
A
ª A1
«A
¬ 2
0º
, C
A 3 »¼
>C1
0@
is not observable for arbitrary submatrices A1rur, A2(n-r)ur, A3(n-r)u (n-r), C1pur.
It is easy to verify that
Ak
ª A1k
«
¬« *
0 º
»,
A 3k ¼»
(A.26)
where * denotes a submatrix insignificant in the following considerations.
Using (A.21) and (A.26), we obtain
ª C º
« CA »
«
»
« # »
«
»
n 1
«¬CA »¼
0º
ª C1
«CA
»
« 1 1 0» .
« #
#»
«
»
n 1
0¼
¬C1A1
(A.27)
From (A.27) it follows that the condition (A.21) is not met for arbitrary
A1,A2,A3 and C1.
482
Appendix
Definition A.6. The system (A.12) (or the pair (A,C)) is called observable if there
exists a time tf > 0 such that for given u(t) and y(t) for 0 d t d tf, one can determine
the initial state x0 of this system.
Theorem A.6. The system (A.12) is observable if and only if one of the conditions
(A.21), (A.22), (A.23) of Theorem A.5 is met.
The proof of this theorem follows similarly to that of Theorem A.5.
Example A.5.
We take as the state variables in the circuit in Fig. A.2 the voltage uC on the
capacity and the current in the coil iL; as the input u we take the source current i,
and as the output y we take the voltage uR on the resistance R, uR = Ri. The circuit
is described by the following equations
d ª uc º
« »
dt ¬iL ¼
ª
«0
«
«1
¬« L
1º
ª1º
C » ª uc º « »
» « » C i, y [0
« »
i
0» ¬ L ¼ ¬ 0 ¼
¼»
ª uc º
0] « » Ri
¬iL ¼
Fig. A.2. Unobservable electrical circuit
The circuit is not observable, since
C [0
ªC º
0] and thus «
»
¬CA ¼
0.
Note that with both the source current i and the voltage uR known, we cannot
determine the initial state
ªuc (0) º
«
»
¬iL (0) ¼
of this circuit.
Appendix
483
It is easily verifiable that if we choose the voltage on the capacity as the output
y, then the circuit is observable.
A.4 Reconstructability
First consider the discrete-time system (A.1).
Definition A.7. The system (A.1) (or the pair (A,C)) is called reconstructable if
there exists an integer q > 0 such that for the two given sequences: input {ui,
i=0,1,…,q1} and output {yi, i=0,1,…,q1} one can determine the state vector xq
of this system.
Theorem A.7. The system (A.1) is reconstructable if and only if one of the
following conditions is met
1.
ª C º
« CA »
»  Ker A n ,
Ker «
« # »
«
»
n 1
¬«CA ¼»
(A.28)
ªI dA º
rank « n
»
C ¼
¬
(A.29)
2.
n for all finite d  ,
3.
> I n dA @
and are right coprime.
(A.30)
Proof of this theorem is analogous (dual) to that of Theorem A.5.
Example A.6.
The pair
A
ª1
«2
¬
1º
, C [1 1]
2 »¼
is not observable, since
(A.31)
484
Appendix
ª Cº
rank «
»
¬CA ¼
ª1
rank «
¬3
1º
1.
3 »¼
One cannot determine the vector x0 = [x01, x02]T with y0 and y1 known for
u0 = u1 = 0, since y0 = Cx0 = x01 + x02, y1 = 3(x01 + x02), that is, we know only the
sum x01 + x02.
The pair (A.31) is reconstructable, since
ª I dA º
rank « n
»
C ¼
¬
ª1 d
rank «« 2d
«¬ 1
d
º
1 2d »»
1»¼
2, for all finite d  .
From the equation
x2
A 2 x0
ª 3 y0 º
« »
¬ 2 y1 ¼
we can compute x2 with y0 and y1 known.
Remark A.2.
Each of the conditions (A.21), (A.22), (A.23) for the system (A.1) with A singular
is only a sufficient condition and not a necessary one for the reconstructability of
this system. If det Az 0, then these conditions are also necessary ones of the
reconstructability of the system (A.1). For the system (A.1) with A nonsingular, the
conditions of observability are equivalent to those of reconstructability.
Definition A.8. The system (or the pair (A,C)) is called reconstructable if there
exists a time tf > 0 such that with u(t) and y(t) given for 0 d t d tf one can determine
the state vector xf = x(tf) of this system.
Theorem A.8. The system (A.12) is reconstructable if and only if one of the
conditions (A.21), (A.22), (A.23) of Theorem A.5 is satisfied.
The proof of this theorem is analogous to that of Theorem A.5.
The reconstructability of the continuous-time system (A.12) is equivalent to its
observability.
Appendix
485
A.5 Dual System
Definition A.9 The system
xi 1
yi
AT xi CT ui ,
(A.32)
BT xi
is called dual with respect to the system
xi 1
yi
Axi Bui ,
(A.33)
Cxi .
By virtue of Theorem A.1 and A.5 the following result ensues.
Theorem A.9. The system (A.33) is reachable (observable) if and only if its dual
(A.32) is observable (reachable).
The same theorem applies to the continuous-time system (A.12).
6 Stabilizability and Detectability
Consider the discrete-time system (A.1). Let z1,z2,…,zn be the eigenvalues of the
matrix A of this system.
Definition A.10. The eigenvalue zi of the system (A.1) is called controllable if
rank > I n zi A, B @
n (i 1, ..., n) ,
(A.34)
The system (A.1) is reachable if and only if all the eigenvalues z1,z2,…,zn are
controllable.
Definition A.11. The system (A.1) is called stabilizable if all the unstable
eigenvalues |zi| t 1 of this system are controllable.
Theorem A.10. The system (A.1) is stabilisable if and only if
rank > I n z A, B @
n, for all z t 1.
The proof of this theorem is analogous to that of Theorem A.1.
(A.35)
486
Appendix
Definition D.12. An eigenvalue zi is called observable if
ªI z A º
rank « n i
»
C ¼
¬
n (i 1,..., n) .
(A.36)
The system (A.1) is observable if and only if all the eigenvalues z1,z2,…,zn are
observable.
Definition A.13. The system (A.1) is called detectable if all the unstable
eigenvalues (|zi| t 1) of this system are observable.
Theorem A.11. The system (A.1) is detectable if and only if
ªI z A º
rank « n
»
C ¼
¬
n, for all z t 1.
(A.37)
The proof of this theorem is analogous to that of Theorem A.5.
Note that reachability (observability) always implies stabilisability
(detectability) of the system (A.1).
With merely slight modifications, the foregoing considerations apply to
continuous-time systems of the form (A.12).
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Index
Algebraic matrix equation 358
Asymptotic stability 423
Canonical form
Frobenius 45
Jordan 45,231
McMillan 152
Smith 32
Computation of
cyclic realization 231
equivalent standard systems 272
Frobenius canonical form 45
fundamental matrices 276
general
solution
of
polynomial
equations 319
Jordan canonical form 45, 48
minimal deree solution 322
minimal realisation for singular linear
systems 367
normal transfer matrix 244
particular solution of polynomial
equations 313
rational solution 332
similarity transformation matrices 50
Computing
greatest common divisors 77, 79
smallest common multiplication 79
Controllability 475
Cyclic pairs 255
Cyclic realization 220
existence 224
computation 226
Cyclicity 264, 267
Division on polynomial matrices 9
Decomposition
Kalman 291
normal matrices 182
rational function 116
rational matrices 128
regular pencil 87
singular pencil 95
singular systems 299
structural 185, 305
Weierstrass 91
Weierstrass–Kronecker 299
Dual system 482
Electrical circuit 200, 286
fourth order 210
general case 210
RC 288
RL 286
second-order 200
third-order 203
Elementary
divisors 37
operation 20
operations method 54
Eigenvector method 57
Equivalence 42
Equivalent standard systems 279
Eigenvalues of matrix polynomial 345
502
Index
Fraction description of
normal matrices 170
rational matrices 136
Functional observers 391
Normality of matix 164
Kronecker
indices 102
product 340
Perfect observers
for standard systems 384
for systems with unknown inputs 400
full-order 375
of singular systems 367
reduced-order 375, 378, 408
2D systems 396
Polynomial 1
Polynomial operations 5
Polynomial matrix equations 313, 336
bilinear with two unknown 325
rational solution 332
unilateral with two variables 313
Polynomial matrices
division 9
equivalents 27
first degree 42
greatest common divisors 75
inverse matrix 132
lowest common divisors 75
pairs 75
rank 23
reduction 32
relative prime 84
simple 68
upon 20
zeros 37, 39
Problem of realisation 219
Observability 478
Output-Feedback 197
Operations on
polynomial 5
Generalised Bezoute identity 84, 86
Generalization of Sylvester equation rational function 107
rational matrices 124
357
Lyapunov equation 361
Linear independence 23
Matrices
column reduced 30
cyclic 68, 69
decomposition of regular pencil 87
diagonalisation 60, 62
Frobenius canonical form 45
irreducible transfer 305
Jordan canonical form 45
left equivalent 27
normal 163
normalisation 191
rational normal 168
right equivalent 27
row reduced 30
simple 68
simple structure 60
Matrix
arbitrary square 65
equation 313
normal inverse 175, 180, 257
normal transfer 260
pair method 50
variable elements 65
Minimum energy control 435, 440
Normal matrices
fraction description 170
product 175
sum 175
Normal systems
Cyclic 255
singular 255
Reachability 264, 435
Realisation
minimal 220
cyclic 220
Realisation problem for
positive discrete-time systems 444
positive continuous-time systems 453
singular multi-variable discrete-time
systems with delays 461
Index
Rational
System
Reachability 471
continuous-time 282, 422
discrete-time 419
function 107
matrices 107, 124
linear singular 272, 367
Reconstructability 480
positive linear with delays 419
singular discrete-time 255, 272
Robust stability 432
Similarity 42
Synthesis of regulators 155
Space basis 23
Stability of positive linear discrete-time Theorem
systems with delay 423
Bezoute16
Structural stability 244
Cayley–Hamilton 16
Sylvester equation 347
Weierstrass–Kronecker 95
503