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SI: Conference Collection IMETI2020
Measurement and Control
2022, Vol. 55(3-4) 119–125
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DOI: 10.1177/00202940221083583
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Stable and quadratic-optimal
parallel-distributed-compensation controller
design for time-varying Takagi–Sugeno fuzzy
model System: A complementary
computational approach
Fu-I Chou
1
and Wen-Hsien Ho
2,3,4
Abstract
A complementary computational approach is proposed for the time-varying Takagi–Sugeno fuzzy model system (TVTSFMS).
The proposed approach integrates orthogonal-functional approach (OFA), hybrid Taguchi genetic algorithm (HTGA), and a
stabilizability condition (SC) for use in designing stable and quadratic-optimal parallel-distributed-compensation (SQOPDC)
controllers for optimal control problems. First, the SC was set according to linear matrix inequalities (LMIs). Next, OFA was
used to derive an algorithm that only required algebraic computation to solve the TVTSFMS. Finally, The HTGA could be used
to search the SQOPDC controller for the TVTSFMS. The SQOPDC controller obtained by the proposed complementary
computational approach was evaluated in a case study of a vibratory pendulum design; the successful design verified the
usability of the proposed hybrid intelligent computing method.
Keywords
Takagi-Sugeno fuzzy model, orthogonal-functional approach, hybrid Taguchi-genetic algorithm, time-varying, parallel-
distributed-compensation controller
Date received: 3 February 2021; accepted: 7 February 2022
Introduction
The plant in an actual control system is mostly nonlinear.
Therefore, scholars have developed many different approaches
to designing controllers in nonlinear control systems in order
to solve controller design problems in actual control systems.
1
In the Takagi–Sugeno fuzzy model, each fuzzy rule for the
local linear control system is represented by a linear dynamic
equation.
2–7
The overall nonlinear control system is then
formalized by combining the fuzzy rules. Therefore, linear
control theory is generally used in controller design. For
nonlinear time-varying systems, it is difficult to design the
controllers. Using Takagi–Sugeno fuzzy model (TSFM), the
overall nonlinear time-varying system can be formalized by
combining the fuzzy rules such that the linear control theory
can be used in the controller design facilitating the controller
easy implementation in practical situation. However, an im-
portant research issue is the design of stable and quadratic-
optimal parallel-distributed-compensation (SQOPDC) con-
trollers for a time-varying Takagi–Sugeno fuzzy model system
(TVTSFMS) with a minimized performance index. Though
the PI/PD/PID controller can be easily implemented in
practical situation, it is very difficult to design the PI/PD/PID
controller for the time-varying Takagi–Sugeno fuzzy model
system (TVTSFMS) with a minimized performance index. In
general, linear matrix inequalities (LMIs) are applied to design
the SQOPDC controller of time-invariant Takagi–Sugeno
fuzzy model systems. However, the LMIs are not directly
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1
Department of Electrical Engineering, National Kaohsiung University of
Science and Technology, Kaohsiung, Taiwan
2
Department of Healthcare Administration and Medical Informatics,
Kaohsiung Medical University, Kaohsiung, Taiwan
3
Department of Medical Research, Kaohsiung Medical University Hospital,
Kaohsiung, Taiwan
4
Department of Mechanical Engineering, National Pingtung University of
Science and Technology, Pingtung, Taiwan
Corresponding author:
Wen-Hsien Ho, Department of Healthcare Administration and Medical
Informatics, Kaohsiung Medical University, 100 Shin-Chuan 1st Road,
Kaohsiung 807, Taiwan.
Email: whho@kmu.edu.tw
applicable for solving the problem of designing an SQOPDC
controller for a TVTSFMS.
8–14
Therefore, the objective of this
study was to develop a complementary computational ap-
proach in which orthogonal-functional approach (OFA) is used
to convert the SQOPDC controller design problem of the
TVTSFMS into an algebraic computation problem.
15
To
simplify the controller design problem, the proposed approach
also integrates hybrid Taguchi genetic algorithm (HTGA)
16–19
and LMIs in the design process to ensure that the TVTSFMS can
be stabilized in a closed loop. Here it should be noticed that, for
the TVTSFMS, the OFA or the LMI technique cannot be applied
alone to findtheSQOPDCcontrollers,anditisalsoverydifficult
to apply the genetic algorithms alone to find the SQOPDC
controllers. So, in this paper, we complementarily fuse the OFA,
the HTGA and the LMI technique to solve the design problem to
be studied. The proposed integrative method fusing the OFA, the
HTGA and the LMI technique belongs to the hard-computing-
assisted soft-computing category, where the OFA and the LMI
technique belong to the hard computing constituents and the
HTGA is one of the soft computing constituents. That is, the main
contribution of this paper is to integrate the OFA, the HTGA, and
an LMI-based stabilizability condition (SC) for use in designing
SQOPDC controllers for optimal control problems. A system
block for the closed-loop system is shown in Figure 1.
Problem statement
A TVTSFMS can be expressed as follows:
~
Ri:IFz1ðtÞis Mi1and …and zgðtÞis Mig
THEN _
xðtÞ¼AiðtÞxðtÞþBiðtÞuðtÞ, (1)
with the initial state vector xð0Þ, where
~
Riði¼1, 2,…,NÞ
denotes the i-th implication, Nis the number of fuzzy rules,
xðtÞ¼½x1ðtÞ,x2ðtÞ,…,xnðtÞTdenotes the n-dimensional
state vector, uðtÞ¼½u1ðtÞ,u2ðtÞ,…,upðtÞTdenotes the p-
dimensional input vector, ziðtÞði¼1, 2,…,gÞare the premise
variables, AiðtÞand BiðtÞði¼1, 2,…,NÞare, respectively,
the n×nand n×pconsequent time-varying matrices, and Mij
ði¼1, 2,…,Nand j¼1, 2,…,gÞare the fuzzy sets.
The TVTSFMS in equation (1) has the following two
characteristics: (i) time-varying elements aijk ðtÞin AiðtÞ
belonging to ½aijk ,aijk , where aijk and aijk are given constants
and (ii) time-varying elements bijk ðtÞin BiðtÞbelonging to
½bijk ,bijk , where bijk bijk are given constants.
Equation (1) can be rewritten as
_
xðtÞ¼X
N
i¼1
hiðzðtÞÞðAiðtÞxðtÞþBiðtÞuðtÞÞ (2)
in which zðtÞ¼½z1ðtÞ,z2ðtÞ,…,xgðtÞTdenotes the g-di-
mensional premise vector, hiðzðtÞÞ ¼ wiðzðtÞÞ=P
N
i¼1
wiðzðtÞÞ,
wiðzðtÞÞ ¼ ∏g
j¼1MijðziðtÞÞ, and Mij ðzjðtÞÞ are the grades of
membership of zjðtÞin the fussy sets Mij ði¼1, 2,…,N
and j ¼1, 2,…,gÞ. It can be seen that, for all t,hiðzðtÞÞ ≥0
and P
N
i¼1
hiðzðtÞÞ ¼ 1:Here, we consider uðtÞ¼P
N
i¼1
hiðzðtÞÞ
FiðxðtÞÞ is the SQOPDC controller
20
and where Fiis the local
feedback gain matrices, which are expressed as
_
xðtÞ¼X
N
i¼1X
N
j¼1
hiðzðtÞÞhjðzðtÞÞAiðtÞBiðtÞFjxðtÞ:(3)
In equation (3), the stabilizability problem is whether Fiof
the SQOPDC controller meets the pre-specified SC. There-
fore, the following sections propose an SC based on LMI to
formalize the instances of equation (3) that can be stabilized.
Theorem: To stabilize SQOPDC controller Fiof the
TVTSFMS in equation (3), there exists a symmetric positive
definite matrix Pthat enables the controller to satisfy the
following LMI
UT
ijl PþPUijl < 0, (4)
where
Uijl ¼X
n
α¼0X
n
β¼0
εiαβðtÞEj
iαβ X
n
α¼0X
p
β¼0
ηiαβðtÞLijαβ
εiαβðtÞ¼εiαβ or εiαβ
ηiαβðtÞ¼ηiαβ or ηiαβ
(5)
areconstantmatricesinwhichEj
iαβ ¼Eiαβ denotes an n×n
constant matrix with 1 in the αβ-th entry and 0 elsewhere and
in which ηiαβ ¼ðbiαβ biαβ Þ=2, ηiαβ ¼ðbiαβ biαβ Þ=2,
Lijαβ ¼ViαβFj,andViαβ denote an n×pconstant matrix with 1
in the αβ-thentryand0elsewhere.
Proof: See Appendix.
Therefore, the question considered in this study is how to
specify a value for Fiof the SQOPDC controller such that (i)
equation (3) meets the requirements of the LMIs in equation
(4) and (ii) Control performance under the initial state
conditions can be optimized by minimizing the following
performance index
J¼X
q1
k¼0Zðkþ1Þtf
ktfxTðtÞQxðtÞþxTðtÞQxðtÞdt, (6)
where tfis the shortest time interval.
Figure 1. Stable and quadratic-optimal parallel-distributed-
compensation controller of the time-varying Takagi–Sugeno fuzzy
model system.
120 Measurement and Control 55(3-4)
Therefore, the SQOPDC controller design process is
specifying Fiand then executing the following two steps to
optimize stability and control:
Step 1: Verify that the SC of the LMI in equation (4) is the
constraint condition.
Step 2: For the TVTSFMS, minimize Jin equation (6).
That is, the design problem of the SQOPDC controller for
the TVTSFMS is a constrained-optimization problem. In the
next section, we will complementarily fuse the OFA, the
HTGA and the presented LMI-based SC to solve the
SQOPDC controller design problem of the TVTSFMS in
equation (1), where the performance index Jin equation (6)
subject to the constraint of SC in equation (4) is considered to
be directly minimized, where if there is no solution for
equation (4), a penalty is given in the HTGA.
SQOPDC controller design
First, we define
t¼kt
fþη, (7)
and
xk¼xktf:(8)
Next, an orthogonal function (OF) can be used to obtain
xðtÞ¼X
m1
s¼0
xðkÞ
sPsðtÞ¼~
xðkÞPðtÞ, (9)
AiðtÞ¼X
m1
s¼0
AðkÞ
is PsðtÞ, (10)
and
BiðtÞ¼X
m1
s¼0
BðkÞ
is PsðtÞ, (11)
where PðtÞ¼½P0ðtÞ,P1ðtÞ,…,Pm1ðtÞTrepresents the ba-
sis vector of OF, PiðtÞrepresents OF, and xðkÞ
s,AðkÞ
is and BðkÞ
is
represent the coefficient matrices.
By substituting uðtÞ¼P
N
i¼1
hiðzðtÞÞFixðtÞand xðtÞ¼
~
xðkÞPðtÞinto equation (6), performance index Jcan be re-
written as
J¼X
q1
k¼0
trace"Wð~
xðkÞÞTðQþX
N
i¼1X
N
j¼1
hiðzkÞhjðzkÞFT
iRFjÞð~
xðkÞÞ#,
(12)
where hiðzkÞ¼hiðzðkt
fÞÞ, and where constant matrix
W¼Rðkþ1Þtf
ktfPðtÞPTðtÞdt .
13
From the orthogonal condition and the properties of OF,
the product of any two OFs can be acquired and can be
expressed by the following formula
15
PaðtÞPbðtÞ¼X
m1
s¼0
ξabsPsðtÞ:(13)
The consequent output only requires inference in an ex-
tremely short time interval. Therefore, integrating equation
(2) yields
xðtÞxktf¼X
N
i¼1
hiðzkÞ"Zt
ktf
AiðtÞxðtÞdt þZt
ktf
BiðtÞuðtÞdt#:
(14)
The coefficient matrix of equation (14) can then be ex-
pressed as
~
xðkÞ½xk,0, 0,…,0¼X
N
i¼1X
N
j¼1
hiðzkÞhjðzkÞ~
yðkÞ
iH
þX
N
i¼1X
N
j¼1
hiðzkÞhjðzkÞ~
vðkÞ
ij H:
(15)
Given a set of local feedback gain matrices
fF1,F2,…,FNg,~
xðkÞcan then be acquired by the algebraic
calculation.
The above algebraic calculation clearly indicated that
specifying a fF1,F2,…,FNgenables the calculation of ~
xðkÞ,
which in turn facilitates the use of equation (12) to calculate
the performance index. Therefore, this control problem can
be converted into the following a static-constrained opti-
mization problem
minimize J¼Gf111,f112 ,…,fNpn :(16)
If (i) constraints jfijk j≤Cijk , are met and if (ii) the LMI of
equation (4) is established (where Cijk is a given positive real
number based on actual engineering considerations), the
Figure 2. Vibratory pendulum system.
Chou and Ho 121
OFA and LMI can be used to obtain a constrained algebraic
equation that represents the stability of the TVTSFMS and
the SQOPDC control design problem. This representation
substantially simplifies the SQOPDC control problem. The
HTGA can then be used to determine the optimal solution for
equation (16).
16–19
Remark 1: From the results mentioned above, we can see
that, by using the OFA, we can transform the SQOPDC
control problem for the TVTSFMS into a static optimization
problem represented by the algebraic equations. Then, by
incorporating the LMI-based SC for this TVTSFMS, the
static optimization problem becomes a static constrained-
optimization problem represented by the algebraic equations
with constraint of LMI-based SC. This means that the main
characteristic and contribution of this technique, bringing the
OFA and the LMI-based SC together, is that it reduces the
mixed H
2
/LMI PDC controllers design problem to that of
solving a static constrained-optimization problem of alge-
braic form; thus greatly simplifying the mixed H
2
/LMI PDC
controllers design problem and facilitating the work of ap-
plying genetic algorithms to solve the SQOPDC controllers
design problem. The proposed integrative method, that
complementarily fuses the OFA, the HTGA and the LMI-
based SC, belongs to the hard-computing-assisted soft-
computing category.
Remark 2: For the problem of designing the SQOPDC
controllers of the TVTSFMS under the criterion of mini-
mizing a performance index. But, the LMI-based
approaches
8–14
cannot be applied to find the SQOPDC
controllers of the TVTSFMS under the criterion of directly
minimizing a performance index. In addition, though the
genetic algorithms can solve the complex static-optimization
problems that are not easy to analyze mathematically, it is
also very difficult to only use the genetic algorithms to solve
the dynamic-optimization problem of designing the SQOPDC
controllers of the TVTSFMS. Hence, summing up the above
statements and reasons, we can see that it is worth while to present
a complementarily integrative approach fusing the HTGA, the
LMI-based SC to find the SQOPDC controllers of the TVTSFMS
under the criterion of minimizing a performance index.
Illustrative examples
Consider a vibratory pendulum system (Figure 2),
21
which
can be represented by a two-rule TVTSFMS:
~
R1:IFz1ðtÞis M11ðz1ðtÞÞ ,
THEN _
xðtÞ¼A1ðtÞxðtÞþB1ðtÞuðtÞ(17)
~
R2:IFz1ðtÞis M21ðz1ðtÞÞ ,
THEN _
xðtÞ¼A2ðtÞxðtÞþB2ðtÞuðtÞ(18)
Table 1. Results of performance comparison of hybrid Taguchi genetic algorithm and traditional genetic algorithm.
Mean performance
index
Median performance
index
Standard deviation in
performance index
Minimal performance
index
Maximal
performance index
HTGA TGA HTGA TGA HTGA TGA HTGA TGA HTGA TGA
2.8213 3.1634 2.8210 2.9947 0.0011 0.3952 2.8201 2.8385 2.8240 4.4668
HTGA: hybrid Taguchi genetic algorithm; TGA: traditional genetic algorithm
Figure 3. Convergence performance comparison of hybrid
Taguchi genetic algorithm and traditional genetic algorithm.
Figure 4. The x1ðtÞ(rad) (solid line) and x2ðtÞ(rad/s) (dashed line)
obtained by the stable and quadratic-optimal parallel-distributed-
compensation controller in response to control input uðtÞ(N-m)
(dashed-dotted line).
122 Measurement and Control 55(3-4)
where A1ðtÞ¼2
6
6
6
6
4
01
g
l
ω2Y
l
cos ωt!0
3
7
7
7
7
5
,A2ðtÞ¼
2
6
6
6
6
4
01
2
π g
l
ω2Y
l
cos ωt!0
3
7
7
7
7
5
,B1ðtÞ¼B2ðtÞ¼2
6
6
4
0
1
ml2
3
7
7
5
,
m¼1kg, l¼1m, ω¼1Hz, Y¼0:01 m, and
g¼9:8m=s2:
In equation (17), the proposed complementary compu-
tational approach is used to design the SQOPDC controller,
to obtain a symmetric positive definite matrix Pin equation
(4), and to minimize Jin equation (16). The OF type con-
sidered in this case was the Legendre function.
Table 1 lists the average, median, minimum, maximum, and
standard deviation of the performance indices obtained in 100
computations performed using the proposed HTGA. For com-
parison, the table also displays the results for 100 computations
performed using a traditional genetic algorithm (TGA).
22
Tab le 1 and Figure 3 show the results of the performance
comparisons of the HTGA and TGA. Compared to the TGA, the
HTGA had (i) superior average and median performance indices,
(ii) smaller SDs in performance indices, and (iii) superior
convergence. Accordingly, HTGA yielded and more stable
solutions. Therefore, compared with TGA, the HTGA is more
effective for routine use in designing an SQOPDC controller.
Figure 4 shows the x1ðtÞ,x2ðtÞand uðtÞresponses of the
vibratory pendulum system obtained after implementation
of the SQOPDC controller designed using the HTGA ðF1¼
½0:2661 1:0889 and F2¼½0:2528 1:1583 ). Its
symmetric positive definite matrix was
P¼"15:7691 0:5082
0:5082 1:8590 #:Figure 5 also illustrates the x1ðtÞ
and x2ðtÞresponses of the vibratory pendulum system
without the SQOPDC controller.
Figures 4 and 5and the above results confirm that, by
integrating OFA, HTGA, and LMI, the proposed comple-
mentary computational approach effectively stabilized the
TVTSFMS, minimized the performance index, and achieved
a satisfactory control effect.
Conclusions
This study developed an algebraic algorithm for using OFA
to solve a TVTSFMS controller design problem. An alge-
braic algorithm was integrated in HTGA to design an
SQOPDC controller for a TVTSFMS constrained by LMIs to
minimize the performance index. The OFA was also used to
convert the performance index to algebraic form. Application
of the proposed approach in the design of a vibratory pen-
dulum system verified the effectiveness of the approach for
designing the SQOPDC controller for the TVTSFMS.
Acknowledgements
The authors of this study would like to express their gratitude to
Professor Jyh-Horng Chou (Department of Electrical Engineering,
National Kaohsiung University of Science and Technology,
Kaohsiung, Taiwan) for his constructive suggestions and guidance
in this research. The authors also thank to the “Intelligent
Manufacturing Research Center”(iMRC) from the Featured Areas
Research Center Program within the framework of the Higher
Education Sprout Project by the Ministry of Education (MOE) in
Taiwan, R.O.C.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect
to the research, authorship, and/or publication of this article.
Funding
The author(s) disclosed receipt of the following financial
support for the research, authorship, and/or publication of this
article: This work was supported in part by the Ministry of
Science and Technology, Taiwan, R.O.C., under Grant Number
MOST 109-2221-E-037-005.
ORCID iD
Wen-Hsien Ho https://orcid.org/0000-0001-6194-0563
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Appendix
Proof of Theorem
Since (i) time-varying elements aijk ðtÞin AiðtÞbelong to ½aijk ,aijk ,
and since (ii) time-varying elements bijk ðtÞin BiðtÞbelong to
½bijk ,bijk , the TVTSFMS in equation (3) can be rewritten as
_
xðtÞ¼X
N
i¼1X
N
j¼1
hiðzðtÞÞhjðzðtÞÞ
AiþX
n
α¼1X
n
β¼1
εiαβðtÞEiαβ BiþX
n
α¼1X
p
β¼1
ηiαβðtÞViαβ !Fj!xðtÞ
(19)
where constant matrices Aiand Bicontain elements aijk ¼
ðaijk þaijk Þ=2 and bijs ¼ðbijs þbijs Þ=2, respectively.
The TVTSFMS in equation (19) can then be rewritten as
_
xðtÞ¼X
N
i¼1X
N
j¼1
hiðzðtÞÞhjðzðtÞÞ
X
n
α¼0X
n
β¼0
εiαβðtÞEj
iαβ X
n
α¼0X
p
β¼0
ηiαβðtÞLijαβ !xðtÞ,
(20)
where Ej
i00 ¼Lij00 ¼
~
Aij, and
~
Aij ¼AiBiFj;Ej
iαβ ¼Eiαβ
and Lijατ ¼Viατ Fj:
The Uijl are constant matrices defined as
Uijl ¼X
n
α¼0X
n
β¼0
εiαβðtÞEj
iαβ X
n
α¼0X
p
β¼0
ηiαβðtÞLijαβ εiαβðtÞ¼εiαβ or εiαβ :
ηiαβðtÞ¼ηiαβ or ηiαβ :
(21)
X
n
α¼0X
n
β¼0
εiαβðtÞEj
iαβ X
n
α¼0X
p
β¼0
ηiαβðtÞLijαβ
¼X
c
l¼1
θijlðtÞUijl ,
(22)
where θijl ðtÞ≥0, P
c
l¼1
θijlðtÞ¼1, and c¼2nðnþpÞ:
Equation (19) then becomes
_
xðtÞ¼X
N
i¼1X
N
j¼1
hiðzðtÞÞhjðzðtÞÞ
X
c
l¼1
θijlðtÞUijl !xðtÞ:
(23)
Where VðxðtÞÞ ¼ xTðtÞPxðtÞis a candidate quadratic Lya-
punov function for the system in equation (23)
124 Measurement and Control 55(3-4)
_
VðxðtÞÞ ¼ X
N
i¼1X
N
j¼1X
c
l¼1
hiðzðtÞÞ hjðzðtÞÞ θijlðtÞxTðtÞUT
ijlPþPUijl xðtÞ:
(24)
For the specified feedback gain matrices Fi, if there exists
a symmetric positive definite matrix P
UT
ijlPþPUijl < 0, then
_
VðxðtÞÞ <0, and "xðtÞ≠0. (25)
Therefore, we can conclude that, to design an SQOPDC
controller for the TVTSFMS (equation (3)), a series of Fiin a
symmetric positive definitive matrix Pmust be specified that
simultaneously satisfies the LMI of equation (4). This
completes the verification.
Chou and Ho 125
