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ISSN(Print) 1975-0102
ISSN(Online) 2093-7423
J Electr Eng Technol.2017; 12(?): 1921-718
http://doi.org/10.???/JEET.2017.12.3.1921
Fuzzy Model-Based Output-Tracking Control for 2 Degree-of-Freedom
Helicopter
Wook Chang*, Ji Hyun Moon** and Ho Jae Lee†
Abstract – This paper addresses the control problem of a laboratory-level 2 degree-of-freedom
helicopter. The exact fuzzy model in a Takagi – Sugeno form is constructed by the sector nonlinearity
technique, and is then represented as a set of uncertain linear systems. Output-tracking controller is
designed in terms of linear matrix inequalities and the closed-loop stability is rigorously analyzed.
Experimental evaluation shows that the proposed method is of benefit to many real industrial plants.
Keywords: Experimental helicopter, fuzzy model, output tracking, stability
2. Modeling
2.1 System Dynamics
As shown in Fig. 1, the experiment system consists of a
body and a supporting base. The main and the tail rotors
are driven by two voltage inputs to control the pitch and
the yaw angles of the body, respectively. Observing Fig. 2,
the dynamics is derived as:
Pr
oo
f
re
in
Various control schemes have been developed for
practical nonlinear plants, among which a successful one
can be referred to fuzzy control in a Takagi–Sugeno (T–
S) form. Achievements from theoretical standpoint in
this area are actually too many to cite. To the contrary, its
practical usefulness has not been sufficiently revealed.
One can exemplify the applications to chaos [1-3], power
systems [4], buck converters [5], and gas furnace [6]. We
note that their effectivenesses are evaluated through
numerical simulations.
Motivated by the above, we address the output-tracking
control problem of a laboratory-level experimental helicopter
with 2 degree-of-freedom (DOF) [7]. Although the apparatus
is not an actual but a simplified one, it exhibits main
characteristics of the real helicopter, i.e., intrinsic instability,
nonlinearity, and the cross-coupled dynamics [8, 9]. Hence,
the problem of interest herein is yet practically valuable.
In this paper, T–S fuzzy modeling of the 2-DOF
helicopter is first performed by the aid of the sector
nonlinearity technique [10]. For nonlinear output-tracking
(regulation), a partial differential Francis–Isidori–Byrnes
(F–I–B) equation should be involved. When a system is
restricted to the T–S form, the F–I–B equation is reduced
to a nonlinear ordinary differential form, which is still
difficult or impossible to solve [11, 12]. The reason lies in
the interaction among the fuzzy rules. In order to ease that
burden, the fuzzy model is converted to a set of uncertain
linear systems [13]. Based on that, the output-tracking
controller to guarantee a uniform ultimate boundedness
(UUB) is synthesized, where the exosystem state is
regarded as disturbance. Since it is actually a deterministic,
rather than unpredictable, input, the refined stability is
investigated. Experiment results are included to demonstrate the effectiveness of the proposed method.
Notations An ellipsis is adopted for long symmetric
matrix expressions, e.g.,
g
1. Introduction
Corresponding Author: Dept. of Electronic Engineering, Inha
Univerity, Korea. ([email protected])
*
Kitronyx, Korea.([email protected])
** Dept. of Electronic Engineering, Inha Univerity, Korea.
([email protected])
Received: January 20, 2016; Accepted: November 4, 2017
(1)
†
Fig. 1. Experimental 2-DOF helicopter
Copyright ⓒ The Korean Institute of Electrical Engineers
licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
1921
Fuzzy Model-Based Output-Tracking Control for 2 Degree-of-Freedom Helicopter
where
is the pitch angle;
is the yaw angle.
The parameters are summarized in Table 1.
Remark 1: The rotation of the main rotor causes a load
on the rotor shaft that is in turn seen at the
torque
yaw axis. It is known as the parallel axis theorem [14].
Similarly, the rotation of the tail rotor causes a force
acting on the body at a distance
from the yaw
axis as well as a torque
and
Let the input and the state be
. Then the state-space equation for (1) is
constructed by
with slight abuse on the notation for the output
, where
and
.
2.2 Fuzzy modeling
To represent (2) as a T–S form, consider the following
convex combination
Solving this yields
(2)
in
g
From the fact that
oo
f
re
is determined,
where we set
rad, due to the structural restriction
on the hinge between the body and the base.
It results in the two-rule affine T–S fuzzy model of
Fig. 2. Free body diagram of the 2-DOF helicopter
Pr
Table 1. System parameters of the experimental setup
1922 │ J Electr Eng Technol.2017; 12(1): 1921-718
Wook Chang, Ji Hyun Moon and Ho Jae Lee
where
,
are known of compatible dimensions,
where ,
is an unknown function matrix with
.
and
The model (4) is then rewritten as
(6)
3. Design
Consider the exosystem
g
To deal with the non-vanishing perturbation
, we introduce a new state
.
Defining
, the augmented dynamics becomes
in
is the reference for
where
of generality, we assume
following transform
(3)
where
to track. Without loss
. Consider the
(7)
oo
f
2.3 Alternative representation
re
and take the controller of the form
Pr
To overcome strong nonlinear interactions among the
fuzzy rules, (3) is represented as a set of uncertain linear
systems [13]. Define an indicate function
(8)
and
where
equation
are solutions to the following matrix
The closed-loop system is given by
is represented as a set of uncertain
then (3) with
linear systems in the form of
(9)
(4)
where
(5)
in (5) is used not to
Remark 2: Matrix
represent real uncertainties but to lump all interactions
among the local models in (3) in terms of uncertainty.
Remark 3: Matrix (5) can be structured as follows:
where
. Due to Remark 3,
.
can be decomposed in the form of
,
Theorem 1: If there exist
, and
s.t.
also
,
(10)
http://www.jeet.or.kr │ 1923
Fuzzy Model-Based Output-Tracking Control for 2 Degree-of-Freedom Helicopter
then
in (6) tracks
with UUB via (8), where
.
Proof: Define a Lyapunov function by
. Then
along (9) excluding
is negative definite for all
exosystems (i.e., fast-varying ) by using the approach of
Filippov [17], where the trajectory of (3) is defined by a
solution of the differential inclusion
stands for the convex hull of a set. The
where
methodology to conduct this generalization will be
if
4. Further Analysis
is the deterministic external input, rather than
Since
an unpredictable disturbance, the more refined stability can
be investigated.
with some
Assumption 1: Given any frozen
,
s.t.
bounded
.
,
Theorem 2: Theorem 1 guarantees that
s.t., for
re
in
g
Congruence transforming the foregoing inequality with
, changing variable
, and using the Schur
complement, we obtain (10). If this is true, we have
where
and
oo
f
along (9). Moreover
Pr
is chosen so as to be
. Define
. One can easily agree that
as long
as
, where
is a compact set.
According to the standard Lyapunov theorem, there exists a
finite time
s.t.
enters
at
and
remains for all
. Thus is bounded and ultimately
converges to
. This implies that the claim follows.
● Model (9) is represented as a state-dependent switched
system, while (8) generates a time-driven reference signal.
One may say that (9) contains time-dependent switching.
In fact, it is not easy to distinguish between state- and timedependent switching, because trajectories under statedependent switching are also those under a suitably defined
time-dependent one [15]. In view of that observation, (9)
can be regarded as a time-dependent switched system. We
know that stability of the time-dependent switching among
stable systems can be determined via such as an average
dwell time [16], which can be guaranteed if the exosystem
is slowly varying).
is suitably defined (i.e,
● It is also possible to cover the more general
1924 │ J Electr Eng Technol.2017; 12(1): 1921-718
Proof: Let
to write
(11)
By expansion, we compute
where
(please see Appendix for details). Hence, (11) is rewritten
as
(12)
Wook Chang, Ji Hyun Moon and Ho Jae Lee
We know that
, where
s.t.
,
along
. Thereby
Let
and
Let
Straightforward calculation of
) results in
.
along (12) (rather than
and
then the foregoing inequality can be majorized by
and vanishes at
,
g
is in
in
Since
s.t.
that
It now holds
Combining this with (15) results in
which is contradictory, so we conclude
. We will
does not exist. From the
oo
f
Continuity says that
for
and
Lyapunov, we have
re
(13)
and
Pr
(14)
By continuity again,
, as
so (13) is
(15)
s.t.
Construct a comparison system
Therefore
Integrating the both sides of (13) from
It remains to show that
whenever
. From the theorem, we know
majorized to
to
gives
which can be shown to be asymptotic stable, by using a
generalization of the invariance arguments in [18, Lemma
. Then the comparison
5.3.71] under the property
as
.
principle says that
5. Experimental Results
The parameters for the experimental setup are tabulated
in Table 1. Variables and , and their time-derivatives
are measured through the encoders attached on the vertical
and horizontal axis, respectively. The periods for analog to
http://www.jeet.or.kr │ 1925
Fuzzy Model-Based Output-Tracking Control for 2 Degree-of-Freedom Helicopter
digital conversion and vice versa are
s, which is
small enough to neglect the effects of the sampled data.
Experiments are carried out according to the exosystem
parameterized by
for
and
for
, signifying that
is zero and sufficiently
small in an average sense in the respective time intervals.
. By Theorem 1 with
The initial state is set to
, we obtain
in
g
with
oo
f
re
Fig. 3. Time responses of p and y by the proposed method:
actual (solid) and desired (dashed)
Pr
s
Time responses are shown in Fig. 3. Before
),
and are well guided to the horizontal
(i.e.,
s (i.e.,
), all
attitude. After
angles of the controlled 2-DOF helicopter that have
sustained the equilibrium quickly follow the desired
sinusoidal trajectories with a bounded deviation, despite
the nonlinear dynamic behavior. This result coincides with
the stability analysis that we conclude in Theorem 2. Fig. 4
shows the control voltage inputs for the main rotor and the
tail rotor.
6. Conclusions
We discussed the modeling and the output-tracking
control of the experimental 2-DOF helicopter. The fuzzy
model in T--S form was derived and rearranged as a set of
uncertain linear systems to exclude the interaction among
fuzzy rules. The design condition was formulated in the
linear matrix inequality format. The closed-loop stability
was further analyzed. The experimental results have
convincingly demonstrated the feasibility of the developed
technique.
1926 │ J Electr Eng Technol.2017; 12(1): 1921-718
Fig. 4. Time responses of
method
and
by the proposed
Wook Chang, Ji Hyun Moon and Ho Jae Lee
Reminding of (11), the chain rule gives
[3]
[4]
[5]
Integrating the foregoing derivative with respect to
$\theta$, we have
[6]
in
[7]
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g
Appendix
[9]
re
[10]
Pr
oo
f
[11]
[12]
[13]
Acknowledgements
This research was supported by Basic Science Research
Program through the National Research Foundation of Korea
(NRF) funded by the Korea government (Ministry of Science,
ICT &amp; Future Planning) (No. 2014R1A2A2A01005664).
[14]
[15]
[16]
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[18]
http://www.jeet.or.kr │ 1927
Fuzzy Model-Based Output-Tracking Control for 2 Degree-of-Freedom Helicopter
in
Ji Hyun Moon He received his B.S.
and M.S. degrees from the Department
of Electronic Engineering, Inha University, Incheon, Korea, in 2012, 2014,
respectively. He is currently pursuing a
Ph.D. degree at the same university.
His research interests include fuzzy
control systems, multi-agent systems
and an underwater glider, and their applications.
g
Wook Chang He received his B.S.,
M.S., and Ph. D. degrees form the
Department of Electrical and Electronic Engineering, Yonsei University,
Seoul, Korea, in 1994, 1996, and 2001,
respectively. In 2001, he was a member
of research staff at Samsung Advanced
Institute of Technology. In 2010, he
founded Sensible UI and successfully sold it to a UK
company in 2012. In 2014, he founded another company
Kitronyx, Inc. where he has been working as a Chief
Executive Officer. His research interests include fuzzy
control systems, digital redesign, multi-touch technologies,
and pressure sensing technologies.
Pr
oo
f
re
Ho Jae Lee He received his B.S., M.S.,
and Ph. D. degrees from the Department
of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea,
in 1998, 2000, and 2004, respectively.
In 2005, he was a Visiting Assistant
Professor with the Department of
Electrical and Computer Engineering,
University of Houston, Houston, TX. Since 2006, he has
been with the School of Electronic Engineering, Inha
University, Incheon, Korea, where he is currently an
Associate Professor. His research interests include fuzzy
control systems, hybrid dynamical systems, large-scale
systems, and digital redesign.
1928 │ J Electr Eng Technol.2017; 12(1): 1921-718