Analysis and synthesis of discrete nonlinear passive systems via affine TS fuzzy models
ABSTRACT The article considers the analysis and synthesis problem for the discrete nonlinear systems, which are represented by the discrete affine Takagi–Sugeno (T–S) fuzzy models. The state feedback fuzzy controller design methodology is developed to guarantee that the affine T–S fuzzy models achieve Lyapunov stability and strict input passivity. In order to find a suitable fuzzy controller, an Iterative Linear Matrix Inequality (ILMI) algorithm is employed in this article to solve the stability conditions for the closed-loop affine T–S fuzzy models. Finally, the application of the proposed fuzzy controller design methodology is manifested via a numerical example with computer simulations.
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ABSTRACT: This paper discusses the control problem for a class of Takagi-Sugeno (T-S) fuzzy bilinear systems with time-varying state and input delays. Utilizing the parallel distributed compensation (PDC), an overall fuzzy controller is designed to stabilize the T-S fuzzy bilinear systems with time-varying state and input delays. In addition, based on a Lyapunov-Krasoviskii function, the delay-dependent stabilization conditions are proposed in terms of linear matrix inequality to guarantee the asymptotic stabilization of T-S fuzzy bilinear systems with time-varying state and input delays. Finally, a numerical example is illustrated to demonstrate the feasibility and effectiveness of the proposed schemes.01/2011; - [Show abstract] [Hide abstract]
ABSTRACT: In this article, Takagi–Sugeno (T–S) fuzzy control theory is proposed as a key tool to design an effective active queue management (AQM) router for the transmission control protocol (TCP) networks. The probability control of packet marking in the TCP networks is characterised by an input constrained control problem in this article. By modelling the TCP network into a time-delay affine T–S fuzzy model, an input constrained fuzzy control methodology is developed in this article to serve the AQM router design. The proposed fuzzy control approach, which is developed based on the parallel distributed compensation technique, can provide smaller probability of dropping packets than previous AQM design schemes. Lastly, a numerical simulation is provided to illustrate the usefulness and effectiveness of the proposed design approach.International Journal of Systems Science 04/2011; 2011(pp. 1–17). · 1.31 Impact Factor - [Show abstract] [Hide abstract]
ABSTRACT: This paper deals with the problem of observer-based robust passive fuzzy control for uncertain Takagi-Sugeno (T-S) fuzzy model with multiplicative noises. For describing the stochastic behaviors of the system, the stochastic differential equation is used to structure the stochastic T-S fuzzy model. And, the uncertainties of the controlled system are considered in this paper for dealing with molding errors and varying parameters. Furthermore, using the Lyapunov function and passivity theory, the sufficient stability conditions can be derived in term of Linear Matrix Inequality (LMI) via two-step procedure. Finally, the numerical simulations are proposed to show the effectiveness and usefulness of this paper.01/2011;
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International Journal of Systems Science
Vol. 39, No. 8, August 2008, 809–821
Analysis and synthesis of discrete nonlinear passive systems via affine T–S fuzzy models
Wen-Jer Chang*, Cheung Chieh Ku and Wei Chang
Department of Marine Engineering, National Taiwan Ocean University, Keelung 202, Taiwan, Republic of China
(Received 29 April 2006; final version received 6 January 2008)
The article considers the analysis and synthesis problem for the discrete nonlinear systems, which are represented
by the discrete affine Takagi–Sugeno (T–S) fuzzy models. The state feedback fuzzy controller design
methodology is developed to guarantee that the affine T–S fuzzy models achieve Lyapunov stability and strict
input passivity. In order to find a suitable fuzzy controller, an Iterative Linear Matrix Inequality (ILMI)
algorithm is employed in this article to solve the stability conditions for the closed-loop affine T–S fuzzy models.
Finally, the application of the proposed fuzzy controller design methodology is manifested via a numerical
example with computer simulations.
Keywords: affine Takagi–Sugeno fuzzy model; passive systems; S-procedure; iterative linear matrix inequality
1. Introduction
Passivity
Trentrlman 2002a,b; Hill and Moylan 1976; Byrnes
and Lin 1994, 1995) gives a framework for analysis of
the nonlinear or linear systems using an input-output
description based on energy-related consideration. The
main ideal behind this is that many important physical
systems have certain input-output properties related to
the conservation, dissipation and transport of energy.
Using the energy concept to interpret the behaviours
and properties of system are described with Lyapunov
function and passivity theory. As for a storage function
(Mohseni, Yaz and Olejniczak 1998; Tan, Soh and Xie
1999; Kaneko and Fujii 2000; Lozano, Brogliato,
Egeland and Maschke 2000; Li, Wang and Shao 2002;
Shao and Feng 2004) measures the amount for energy
stored in the internal of the system, so it is naturally
described by using internal variables of the system, that
is, state variables. Lyapunov function is represented as
a quadratic function of state variables in this article.
The passivity theory is intimately related to Lyapunov
stability theory. The passive property has many types
for expressing total energy and input energy of systems
via power supply. So, how to take account of choosing
the power supply (Byrnes and Lin 1993, 1995; Wang
et al. 2002; Willems, and Trentrlman 2002a,b) is a very
interesting task for the system. In the literature (Byrnes
and Lin 1993, 1995; Mohseni et al. 1998; Tan et al.
1999; Kaneko and Fujii 2000; Li et al. 2002; Shao and
Feng 2004), the passivity theory is employed to analyse
the stability of the discrete systems. The authors of
Lozano et al. (2000) indicate that the discrete systems
theory (Willems1971;Willemsand
can also be the passive system except for the strictly
causal discrete system. And the discrete passive systems
have the counterpart to most of the results in the
continuous case. In this article, an affine T–S fuzzy
model is employed to describe the discrete nonlinear
system, which is not a strictly causal discrete system.
It is well known that the T–S fuzzy model [Tanaka
and Sano (1994); Wang, Tanaka and Griffin (1996);
Chang and Sun (2003)] provides a useful tool for
stability analysis and fuzzy controller design for
complex nonlinear systems. The T–S fuzzy model is
described by IF-THEN rules and the corresponding
fuzzy controller design is developed based on Parallel
Distribution Compensation (PDC) technique (Tanaka
and Wang 2001; Wang et al. 1996). One can approach
the trajectory of an original nonlinear system via
combination of the membership function and the linear
subsystems of the T–S fuzzy model. The pioneers, Kim
and his colleagues (Kim and Kim 2001) have provided
a sufficient condition for the stability analysis of the
affine T–S fuzzy models in sense of Lyapunov
inequality. The affine T–S fuzzy model means the
T–S fuzzy model of which consequent part is affine and
which has a constant bias term. The affine T–S fuzzy
model is more natural and appealing to human beings
than the homogeneous one. The homogeneous T–S
fuzzy model is usually studied by the researchers due to
the ease of analysis. In this article, both the stability
analysis and controller synthesis of discrete-time
affine T–S fuzzy systems are considered. In general,
the stability conditions of affine T–S fuzzy systems are
most Bilinear Matrix Inequality (BMI) problems
*Corresponding author. Email: wjchang@mail.ntou.edu.tw
ISSN 0020–7721 print/ISSN 1464–5319 online
? 2008 Taylor & Francis
DOI: 10.1080/00207720801902580
http://www.informaworld.com
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which were discussed in Kim and Kim (2001) and
Chang and Chang (2005, 2006). The BMI problem
may be difficult to solve by numerically convex
optimisation technique (Boyd, Ghaoui, Feron and
Balakrishnan 1994) and cannot be calculated via
Linear Matrix Inequality (LMI) technique. For this
reason, the Iterative Linear Matrix Inequality (ILMI)
algorithm is developed in this article to solve the BMI
conditions and to find the feasible solutions of
fuzzy controller design problems for affine T–S
fuzzy models.
In order to analyse the stability of the nonlinear
systems with external disturbances, the affine T–S
fuzzy model and passivity theory are employed in this
article. In Clacev, Gorez and Neyer (1998) and Clacev
(1998), the authors used the Mamdani fuzzy model to
simulate the trajectories of the nonlinear system and
apply the passivity theory to analyse the stability.
Besides, for describing the behaviours of the nonlinear
system clearly and for attenuating the external
disturbance energy, the authors proposed the passive
controller design technique for homogeneous T–S
fuzzy model in Li, Zhang and Liao (2005, 2006).
Based on the results of Li et al. (2005, 2006) and Uang
(2005), the homogeneous T–S fuzzy model can achieve
passivity performance and stability via the designed
fuzzy controller. For affine T–S fuzzy model with
external disturbances, the authors of Chang and Chang
(2006) use the H1control method to achieve system
stabilityandH1
Unfortunately, there is not any literature to discuss
the fuzzy controller design with passivity for the affine
T–S fuzzy systems. For controlling accurately and
attenuating the external disturbance energy, the
stability conditions are derived by passivity theory to
design the fuzzy controller to achieve the performance
for affine T–S fuzzy model in this article.
The principle structure of this article is described as
follows. The descriptions of discrete affine T–S fuzzy
models and passivity properties are presented in
Section 2. Based on PDC concept, the stability analysis
and solutions of fuzzy controller design problems are
presentedfor discrete affine
in Section 3. Applications of the proposed design
approach via numerical simulations are shown in
Section 4. Finally, concluding remarks are made in
Section 5.
performance constraints.
T–Sfuzzymodel
2. System descriptions and stability analysis
with passivity
The passivity and Lyapunov stability conditions
of discrete affine T–S fuzzy systems are discussed
based on Lyapunov criterion in this section. Besides,
the S-procedure (Boyd et al. 1994; Kim and Kim 2001)
is used to ensure the analysis and synthesis results.
The discrete-time affine T–S fuzz system considered
in this article is described as follows:
Plant part:
RulePlanti: IF x1(k) is Mi1and x2(k) is Mi2and...and
xp(k) is MipTHEN
xðk þ 1Þ ¼ AixðkÞ þ BuiuðkÞ þ BwiwðkÞ þ ai
ð1aÞ
yðkÞ ¼ CixðkÞ þ DiwðkÞ,
where Ai2 <nx?nx, Bui2 <nx?nu, Bwi2 <nx?ny, ai2 <nx,
Ci2 <ny?nxand Di2 <ny?nyare constant matrices and
x k ð Þ 2 <nxis the state vector, y k ð Þ 2 <nyis the output
vector, w k ð Þ 2 <nyis the square-integrable exogenous
input vector, u k ð Þ 2 <nuis the control input vector, Mij
is the fuzzy set, p is the premise variable number and r
is number of fuzzy model rules. Given the pair
of ðxðkÞ,uðkÞÞ, the closed-loop of the fuzzy model (1)
can be referred as follows:
Xr
i ¼ 1,...,r
ð1bÞ
xðkþ1Þ¼
i¼1!iðxðkÞÞ AixðkÞþBuiuðkÞþBwiwðkÞþai
Xr
ðÞ
i¼1!iðxðkÞÞ
ð2aÞ
yðkÞ ¼
Xr
i¼1!iðxðkÞÞ CixðkÞ þ DiwðkÞ
Xr
ðÞ
i¼1!iðxðkÞÞ
,i ¼ 1,...,r
ð2bÞ
or
xðk þ 1Þ ¼
X
? AixðkÞ þ BuiuðkÞ þ BwiwðkÞ þ ai
r
i¼1
hiðxðkÞÞ
ðÞð3aÞ
yðkÞ ¼
X
r
i¼1
hiðxðkÞÞ CixðkÞ þ DiwðkÞðÞ,i ¼ 1,...,r
ð3bÞ
where
!iðxðkÞÞ ¼ ?
p
j¼1MijðxjðkÞÞð4Þ
hiðxðkÞÞ ¼
!iðxðkÞÞ
Xr
i¼1!iðxðkÞÞ
ð5Þ
and Mij(xj(k)) is the grade of membership function of
xj(k)inMij.Itisassumed
Pr
that
i¼1hiðxðkÞÞ ¼ 1 and
!iðxðkÞÞ ? 0,
i¼1!iðxðkÞÞ40, hiðxðkÞÞ ? 0,Pr
i ¼ 1,...,r for all x(k).
810W.-J. Chang et al.
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For the nonlinear plant provided in (2) or (3), the
fuzzy controller is designed to share the same fuzzy set
with the plant. The fuzzy controller design is developed
based on the PDC technique, which can be referred to
[Wang et al. (1996)]. The discrete-time T–S fuzzy
controller has the following form.
Controller part:
RuleController i: IF x1(k) is Mi1 and x2(k) is Mi2
and...and xp(k) is MipTHEN
u k ð Þ ¼ ?FixðkÞ,i ¼ 1,...,r
ð6Þ
Based on PDC concept, the fuzzy controller can be
represented via summation as follows:
uðkÞ ¼
X
r
i¼1
hiðxðkÞÞð?FixðkÞÞ,i ¼ 1,...,r
ð7Þ
Substituting (7) into (3a), one can obtain the corre-
sponding closed-loop system such as
xðk þ 1Þ ¼
X
? AixðkÞ ? BuiFjxðkÞ þ BwiwðkÞ þ ai
X
?
X
i5j
ð
Ai? BuiFj
2
?
X
i¼j
þ BwiwðkÞ þ aiÞ
X
? GijxðkÞ þ NijwðkÞ þ ~ aij
Gij¼ ððAi? BuiFjÞ þ ðAj? BujFiÞÞ=2;
Nij¼ ðBwiþ BwjÞ=2 and ~ aij¼ ðaiþ ajÞ=2.
Assumption 1:Theregion
nx-dimensional convex polyhedron, which is known as a
cell. The cell indices are denoted by~I and union of all
cells X ? [i2~IXiis referred to as the partition. Let^I ?~I
r
i¼1
X
r
j¼1
hiðxðkÞÞhjðxðkÞÞ
??
¼
r
i¼1
X
Ai? BuiFj
r
X
?
Bwiþ Bwj
2
r
j¼1
hiðxðkÞÞhjðxðkÞÞ
?xðkÞ þ BwiwðkÞ þ ai
hiðxðkÞÞhjðxðkÞÞ
???
¼ 2
i¼1
?
Þ
r
j¼1
?
?þ Aj? BujFi
wðkÞ þaiþ aj
??
??
xðkÞ
þ
?
2
?
þ
r
i¼1
ðÞ
h2
iðxðkÞÞ Ai? BuiFi
ðÞxðkÞð
?
r
i¼1
X
r
j¼1
hiðxðkÞÞhjðxðkÞÞ
??
ð8Þ
where
Xi? <nx
is
be the set of indices for the fuzzy rules that contain the
origin and others be the set of indices for the fuzzy rules
that do not contain the origin. Here, the origin x(0)¼0 is
the equilibrium point of the discrete affine T–S fuzzy
model and it is assumed ai¼ 0 for i 2^I.
The concept of the passivity theory proposes a very
useful tool in the stability analysis and controller
synthesis for linear or nonlinear dynamic systems. In
the passivity theory, the power supply is an important
role, which has the definition as follows.
Definition 1:
disturbance input w(k) and output y(k) is said to be
strictly passive (Mohseni et al. 1998; Lozano et al.
2000) if
Thesystem(3)withexternal
2
X
kq
k¼0
yTk ð Þw k ð Þ4
X
kq
k¼0
yTk ð ÞYy k ð Þ þ ?wTk ð Þw k ð Þ
??
ð9Þ
where Y 2 <ny?ny, ? ? 0 and kq40.
In the left-hand
Pkq
hand side of the (9) represents the output energy
Pkq
parameters for determining the dissipation properties.
In physical systems, the power supply can be asso-
ciated with the concept of stored energy (Hill and
Moylan 1976) or dissipated energy (Willems 1971).
Based on these concepts, one can see that if the passive
inequality (9) is held that means the stored or
dissipated energy is bigger than the energy of external
disturbance and output. Then, one can say that the
passive system can resist the effect of the external
disturbance. Throughout this article, we are only
concerned about a strictly input passive system such
as Y¼0 and ? ?0. Thus, the inequality (9) can be
represented as
side of the(9), the term
k¼0yTðkÞwðkÞ represents the power supply to the
system from external disturbance input. And the right-
k¼0yTðkÞYyðkÞ and external disturbance energy
Pkq
k¼0?wTðkÞwðkÞ, in which Y and ? are the adjustable
2
X
kq
k¼0
yTk ð Þw k ð Þ4
X
kq
k¼0
?wTk ð Þw k ð Þð10Þ
There are many other types of passive inequality,
which can be referred to in Kaneko and Fujii (2000).
The passivity analysis problem can also be considered
as an H1analysis problem via some assumptions that
can be found in Byrnes and Lin (1993, 1994) and
Mohseni et al. (1998). The passivity conditions for the
discrete-time affine T–S fuzzy system (8) and (3b) can
be characterised by the following theorem.
Theorem 1:
P>0, scalars ? ?0 and "ijq?0 for satisfying the
If there exist a positive definite matrix
International Journal of Systems Science811
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following conditions, then the discrete-time affine T–S
fuzzy model (3) driven by fuzzy controller (7) is strictly
input passive and Lyapunov stable.
"
!11
!T
21
!21
!22
#
50,i 2^I
ð11Þ
!11?
X
!21
p
q¼1
"ijqTijq
!T
21
GT
ijP~ aij?
X
ijP~ aij
p
q¼1
"ijqnijq
!22
NT
~ aT
ijPGij?
X
p
q¼1
"ijqnT
ijq~ aT
ijPNij
~ aT
ijP~ aij?
X
p
q¼1
"ijqvijq
2
6
6
6
6
4
6
6
6
6
3
7
7
7
7
5
7
7
7
7
50,
i62^I
where
ð12Þ
!11¼ GT
ijPGij? P
ð13Þ
!21¼ NT
ijPGij? Ci
ð14Þ
!22¼ NT
ijPNij? Di? DT
iþ ?I
ð15Þ
and I is the compatible dimension identity matrix.
Besides, Tijq2 <nx?nx, nijq2 <nxand vijq2 < are defined
such as
?ijqðxðkÞÞ ¼ xTðkÞTijqxðkÞ þ 2xTðkÞnijqþ vijq? 0 ð16Þ
Proof:First, we select the case of fuzzy rules i 62^I
in following proof. Applying the Lyapunov function
VðxðkÞÞ ¼ xTðkÞPxðkÞ represents the storage function
of the passive theory for the closed-loop system (8).
By evaluating the first forward difference of VðxðkÞÞ
along the trajectories of (8), we have
?VðxðkÞÞ ¼ xTðk þ 1ÞPxðk þ 1Þ ? xTðkÞPxðkÞ
X
? ðxðkÞÞhzðxðkÞÞ GijxðkÞ þ NijwðkÞ þ ~ aij
? P GszxðkÞ þ NszwðkÞ þ ~ asz
¼
r
i¼1
X
r
j¼1
X
r
s¼1
X
r
z¼1
hiðxðkÞÞhjðxðkÞÞhs
??T
ðÞ ? xTðkÞPxðkÞ
ð17Þ
Nsz¼
where
ðBwsþ BÞwz=2 and ~ asz¼ ðasþ azÞ=2.
Choose the performance function such as
Gsz¼ ððAs? BusFzÞ þ ðAz? BuzFsÞÞ=2,
JD¼
X
kq
k¼0
?wTðkÞwðkÞ ? 2yTðkÞwðkÞ
??
ð18Þ
with zero initial condition for all w(k). Then, for any
nonzero w(k) one has
JD?
X
X
X
kq
k¼0
kq
?wTk ð ÞwðkÞ ? 2yTðkÞwðkÞ
?
?
?þ VðxðkÞÞ
¼
k¼0
kq
?wTðkÞwðkÞ ? 2yTðkÞwðkÞ þ ?VðxðkÞÞ
?
?
k¼0
Lðx,w,kÞð19Þ
According to (17) and (19), one has
Lðx,w,kÞ ¼ ?VðxðkÞÞ þ ?wTðkÞwðkÞ ? 2yTðkÞwðkÞ
X
? GijxðkÞ þ NijwðkÞ þ ~ aij
? P GijxðkÞ þ NijwðkÞ þ ~ aij
? xTðkÞPxðkÞ þ ?wTðkÞwðkÞ ? 2yTðkÞwðkÞ
X
xðkÞ
wðkÞ
1
~ aT
2
4
Applying the S-procedure (Boyd et al. 1994; Kim and
Kim 2001), one can add the LMI region of the x(k)
where each rule is activated. In the Appendix 1, the
structures of the matrices of S-procedure have been
proposed. The computation details for the correspond-
ing matrices of S-procedure can be referred to
(KimandKim2001).
S-procedure technique, the (20) can be satisfied if
there exist "ijq?0 such that
?
r
i¼1
X
r
j¼1
hiðxðkÞÞhjðxðkÞÞ
??T
?
X
2
4
xðkÞ
wðkÞ
?
¼
r
i¼1
r
j¼1
hiðxðkÞÞhjðxðkÞÞ
3
5
3
5
?
67
T
!11
!T
21
GT
ijP~ aij
NT
ijP~ aij
~ aT
ijP~ aij
!21
!22
ijPGij
~ aT
ijPNij
2
6
6
4
3
7
7
5
?
1
67
ð20Þ
Throughapplyingthe
Lðx,w,kÞ ?
X
r
i¼1
X
r
j¼1
hiðxðkÞÞhjðxðkÞÞ
xðkÞ
wðkÞ
1
3
7
2
6
6
4
3
7
xðkÞ
wðkÞ
1
7
5
T
?
!11
!T
21
GT
ijP~ aij
!21
!22
NT
ijP~ aij
~ aT
ijPGij
~ aT
ijPNij
~ aT
ijP~ aij
2
6
6
6
4
7
7
5
2
6
6
4
3
7
7
5
?5
X
p
q¼1
"ijq?ijqðxðkÞÞð21Þ
812W.-J. Chang et al.
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where ?ijqðxðkÞÞ 2 < is defined in (16). Due to "ijq?0
and ?ijqðxðkÞÞ ? 0, the (21) can be rewritten as
Obviously, if inequality (12) is held for all xðkÞ 2 Xi
and i 62^I thenPkq
JD50
k¼0Lðx,w,kÞ50. In this case, one has
ð23Þ
or
2
X
kq
k¼0
yTðkÞwðkÞ4
X
kq
k¼0
?wTðkÞwðkÞð24Þ
Since (24) is equivalent to (10), it is easy to show that
the system is strictly input passive with external
disturbance input and output.
Now, we have to show that the discrete-time affine
T–S fuzzy model (8) is Lyapunov stable. From (22),
iftheinequality (12)
Lðx,w,kÞ50. By assuming the external disturbance
input w(k)¼0, one has
isheld,thus,onehas
?V x k ð Þ ðÞ50
ð25Þ
Thus, the discrete-time affine T–S fuzzy model (3) is
Lyapunov stable and strictly input passive driven by
the fuzzy controller (7). Besides, the proof of condition
(11) of rules i 2^I is similar to (12) with setting ~ ai¼ 0
and assuming ?ijqðxðkÞÞ be zero. The proof is
completed.
In order to achieve the sufficient conditions of (11)
and (12), one must find suitable positive definite
common matrix P ¼ PT40, scalars "ijp? 0 and
? ? 0. The forms of the conditions (11) and (12) are
the BMI ones, which cannot be calculated by LMI
approach for obtaining the solution to satisfy the
conditions directly. Hence, the ILMI algorithm is
developed to solve the problem of BMI and to find the
feasible solution for the conditions (11) and (12).
By applying some transform techniques, the BMI
conditions can be transformed into LMI conditions via
the applications of auxiliary variables in the next
section.
3. Fuzzy controller design by using ILMI algorithm
In this section, an ILMI algorithm is applied to obtain
fuzzy controller to achieve the conditions of Theorem 1
for discrete-time affine T–S fuzzy model (3). In order
to solve the BMI conditions of Theorem 1, we first
hold some auxiliary variables as given matrices and
then converting it to a LMI problem. After that, the
fuzzy controller design problem can be solved by LMI
toolbox of MATLAB.
Theorem 2:
P40, X40 and scalars ?51, ? 40, "ijq? 0 for
satisfying the following conditions, then the discrete-time
affine T–S fuzzy model (3) is strictly input passive and
Lyapunov stable with fuzzy controller (7).
If there exist positive definite matrices
)ij50
XTPX ? X ? 0
(
,i 2^I
ð26Þ
and
^)ij50
XTPX ? X ? 0
(
,i 62^I
ð27Þ
where
)ij¼
??P
Gij
GT
ij
GT
ijX?1Nij? CT
0
i
?X
0
NT
ijX?1Gij? Ci
NT
ijX?1Nij? Di? DT
iþ ?I
2
6
6
4
3
7
7
5,
Lðx,w,kÞ ?
X
r
i¼1
X
r
j¼1
hiðxðkÞÞhjðxðkÞÞ
xðkÞ
wðkÞ
1
2
6
6
4
3
7
7
5
T
!11?
X
!21
p
q¼1
"ijqTijq
!T
21
GT
ijP~ aij?
X
ijP~ aij
p
q¼1
"ijqnijq
!22
NT
~ aT
ijPGij?
X
p
q¼1
"ijqnT
ijq
~ aT
ijPNij
~ aT
ijP~ aij?
X
p
q¼1
"ijqvijq
2
6
6
6
6
4
6
6
6
6
3
7
7
7
7
5
7
7
7
7
xðkÞ
wðkÞ
1
2
6
6
4
3
7
7
550
ð22Þ
International Journal of Systems Science 813
Page 6
Downloaded By: [Chang, Wen-Jer] At: 12:38 24 May 2008
Proof:
nique, the first condition of (27), i.e.^)ij50, can be
rewritten as
Using the Schur complement converting tech-
Obviously, if the decay rate ?51 and P40
are held, then inequality (28) is negative definite.
Multiplying (28) by ½ðxTðkÞwTðkÞ1? and ½xTðkÞwTðkÞ1?T,
respectively, and rearranging it gives
xTðkÞGT
ijX?1GijxðkÞ þ xTðkÞGT
ijX?1NijwðkÞ ? xTðkÞCT
iwðkÞ
þ xTðkÞGT
ijX?1~ aijþ wTðkÞNT
ijX?1GijxðkÞ
? wTðkÞCixðkÞ þ wTðkÞNT
ijX?1NijwðkÞ
þ ?wTðkÞwðkÞ ? wTðkÞDT
iwðkÞ ? wTðkÞDiwðkÞ
þ wTðkÞNT
ijX?1~ aijþ ~ aT
ijX?1GijxðkÞ þ ~ aT
ijX?1NijwðkÞ
þ ~ aT
ijX?1~ aij? xTðkÞPxðkÞ ?
X
p
q¼1
"ijq?ijqðxðkÞÞ
5ð? ? 1ÞxTðkÞPxðkÞð29Þ
or
GijxðkÞ þ NijwðkÞ þ ~ aij
? xTðkÞPxðkÞ þ ?wTðkÞwðkÞ ? 2yTðkÞwðkÞ
X
??TX?1GijxðkÞ þ NijwðkÞ þ ~ aij
??
?
p
q¼1
"ijq?ijqðxðkÞÞ5ð? ? 1ÞxTðkÞPxðkÞð30Þ
If there exist P40 such that XTPX ? X ? 0 is held,
one can obtain the following inequality according to
(30).
GijxðkÞ þ NijwðkÞ þ ~ aij
? xTðkÞPxðkÞ þ ?wTðkÞwðkÞ ? 2yTðkÞwðkÞ
X
??TP GijxðkÞ þ NijwðkÞ þ ~ aij
??
?
p
q¼1
"ijq?ijqðxðkÞÞ5ð? ? 1ÞxTðkÞPxðkÞð31Þ
or
^)ij¼
??P ?
X
Gij
p
q¼1
"ijqTijq
GT
ij
GT
ijX?1Nij? CT
i
GT
ijX?1~ aij?
X
p
q¼1
"ijqnijq
?X
0
00
NT
ijX?1Gij? Ci
NT
ijX?1Nij? Di? DT
iþ ?INT
ijX?1~ aij
~ aT
ijX?1Gij?
X
p
q¼1
"ijqnT
ijq
0
~ aT
ijX?1Nij
~ aT
ijX?1~ aij?
X
p
q¼1
"ijqvijq
2
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
5
:
GT
ijX?1Gij? P ?
X
p
q¼1
"ijqTijq
GT
ijX?1Nij? CT
i
GT
ijX?1~ aij?
X
p
q¼1
"ijqnijq
NT
ijX?1Gij? Ci
NT
ijX?1Nij? Di? DT
iþ ?INT
ijX?1~ aij
~ aT
ijX?1Gij?
X
p
q¼1
"ijqnT
ijq
~ aT
ijX?1Nij
~ aT
ijX?1~ aij?
X
p
q¼1
"ijqvijq
2
6
6
6
6
4
6
6
6
6
3
7
7
7
7
5
7
7
7
7
5
ð? ? 1ÞP
0
00
00
000
2
6
6
4
3
7
7
5
ð28Þ
xðkÞ
wðkÞ
1
2
6
6
4
3
7
7
5
T
!11?
X
!21
p
q¼1
"ijqTijq
!T
21
GT
ijP~ aij?
X
ijP~ aij
p
q¼1
"ijqnijq
!22
NT
~ aT
ijPGij?
X
p
q¼1
"ijqnT
ijq
~ aT
ijPNij
~ aT
ijP~ aij?
X
p
q¼1
"ijqvijq
2
6
6
6
6
4
6
6
6
6
3
7
7
7
7
5
7
7
7
7
xðkÞ
wðkÞ
1
2
6
6
4
3
7
7
5v5
xðkÞ
wðkÞ
1
2
6
6
4
3
7
7
5
T
ð? ? 1ÞP
0
00
00
000
2
6
6
4
3
7
7
5
xðkÞ
wðkÞ
1
2
6
6
4
3
7
7
5
ð32Þ
814W.-J. Chang et al.
Page 7
Downloaded By: [Chang, Wen-Jer] At: 12:38 24 May 2008
Obviously,
!11?
X
!21
p
q¼1
"ijqTijq
!T
21
GT
ijP~ aij?
X
ijP~ aij
p
q¼1
"ijqnijq
!22
NT
~ aT
ijPGij?
2
6
Since P40 and ?51, one has
2
6
6
4
X
p
q¼1
"ijqnT
ijq
~ aT
ijPNij
~ aT
ijP~ aij?
X
p
q¼1
"ijqvijq
2
6
6
6
6
4
6
6
6
6
3
7
7
7
7
5
7
7
7
7
5
ð? ? 1ÞP
0
00
00
000
6
4
3
7
7
5
ð33Þ
!11?
X
!21
p
q¼1
"ijqTijq
!T
21
GT
ijP~ aij?
X
ijP~ aij
p
q¼1
"ijqnijq
!22
NT
~ aT
ijPGij?
X
p
q¼1
"ijqnT
ijq
~ aT
ijPNij
~ aT
ijP~ aij?
X
~ p
q¼1
"ijqvijq
6
6
6
6
6
6
6
3
7
7
5
7
7
7
7
7
7
7
50
ð34Þ
The inequality (34) is equivalent to (12). Hence, in the
case of i 62^I, if condition (27) is satisfied, then the
discrete-time affine T–S fuzzy model is strictly input
passive and Lyapunov stable. The proof of (26) for rule
including the origin (i 2^I) is similar to (27) by setting ~ ai
and ?ijqbe all zero. Hence, the proof of (26) is omitted
here and the proof of this theorem is completed.
In order to achieve the conditions (26) and (27), an
ILMI algorithm is developed to interactively search for
P Fi, ?, "ijqand to update the auxiliary variable X until
?<1. The operation process of the present ILMI
algorithm is proposed as follows:
ILMI algorithm
Step 1:
follows:
Define the iterative auxiliary variable as
Xt ð Þ¼ P?1
t?1
ðÞ
ð35Þ
where t is iterative index. Set t¼1 and the initial P(0)
can be calculated by the following Riccati equation.
^A
TP0 ð Þ^A ? P0 ð Þ?
þ Q ¼ 0
where^A ¼Pr
^A
TP0 ð Þ^B
??
1 þ^B
TP0 ð Þ^B
???1
^B
TP0 ð Þ^A
??
ð36Þ
i¼1Ai=r,^B ¼Pr
i¼1Bui=r and Q>0. The
matrix Q is assigned by the designers. Through the
initial auxiliary variable X(1), the ILMI algorithm can
be started as following steps.
Step 2:
optimisation problem for P(t), Fi(t)and "ijq(t)from (26)
and (27) subject to minimising ?(t). That is,
Use the auxiliary variable X(t)to solve the
Minimise ?t ð Þ
Subject to Pt ð Þ40, "ijq t ð Þ40, XT
? Xt ð Þ? 0 ð26Þ for i 2^I and ð27Þ for i 62^I:
If ?t ð Þ51, then P(t), Fi(t)and "ijq t ð Þobtained in Step 2
are the feasible solutions for the conditions (26) and
(27). The iterative manner is stopped in this step.
Otherwise, go to Step 3.
t ð ÞPt ð ÞXt ð Þ
Step 3:
for P(t), Fi(t)and "ijq t ð Þby using ?(t)and X(t)determined
in Step 2.
Resolve the following optimisation problem
Minimise trace ðPt ð ÞÞ
Subject to Pt ð Þ40, "ijq t ð Þ40, XT
? Xt ð Þ? 0 ð26Þ for i 2^I and ð27Þ for i 62^I:
If kX?1
small value, then the conditions (26) and (27) are not
feasible and then stop the iterative manner. Otherwise,
set t¼tþ1 to update the auxiliary variable X(t)by
Xt ð Þ¼ P?1
back to the Step 2.
t ð ÞPt ð ÞXt ð Þ
t ð Þ? Pt ð Þk5?, where ? is a pre-determined
t?1
ðÞwith the obtained P(t?1)in Step 3 and go
According to the above ILMI algorithm, a numerical
exampleis presentedto
of proposed fuzzy controller design approach for
discrete-time affine T–S fuzzy models. Using the
proposed fuzzy controller design procedure, the
feasible solutions P>0, Fiand "ijq40 can be solved
by the LMI toolbox of MATLAB to satisfy conditions
(26) and (27) of Theorem 2.
showtheapplications
4. A numerical example
In this example, a computer simulated truck-trailer
system is considered to illustrate the applications
of proposed fuzzy controller design approach. The
dynamic equations of the nonlinear truck-trailer
system have been discussed in Tanaka and Sano
(1994) such as
x1ðk þ 1Þ ¼ 1:3636x1ðkÞ ? 0:7143uðkÞ þ 0:1wðkÞ ð37aÞ
x2ðk þ 1Þ ¼ x2ðkÞ ? 0:3636x1ðkÞð37bÞ
x3ðk þ 1Þ ¼ x3ðkÞ ? 2sin x2ðkÞ ? 0:1818x1ðkÞ½?ð37cÞ
yðkÞ ¼ x1ðkÞ þ wðkÞð37dÞ
International Journal of Systems Science815
Page 8
Downloaded By: [Chang, Wen-Jer] At: 12:38 24 May 2008
where x1(k) is the angle difference between truck and
trailer (note that x1(k) corresponds to two ‘jackknife’
positions, 90?and ?90?); x2(k) is the angle of the
trailer; x3(k) is the vertical position of rear end of the
trailer; w(k) is the external disturbance input provided
by a random zero-mean white noise with variance 0.1.
It can be found that the equilibrium point of (37) is
x1ðkÞ ¼ x2ðkÞ ¼ x3ðkÞ ¼ uðkÞ ¼ wðkÞ ¼ 0. In addition
to the equilibrium point, we choose other two
operating points to obtain the linearised subsystems
for the truck-trailer system (37). Thus, these three
operating points are given as follows:
ðx?,u?,w?Þoper1¼ ð?88o? 90o0j0 0 j Þð38aÞ
ðx,u,wÞoper2¼ ð0o0o0j0 0 j Þ
ð38bÞ
ðxþ,uþ,wþÞoper3¼ ð88o90o0j0 0 j Þ
The structure of affine T–S fuzzy model for the
nonlinear truck-trailer system (37) can be constructed
by linearisation method through the above three linear
operating points with the membership functions
described in Figures 1 and 2. Thus, one can represent
the discrete affine T–S fuzzy model, which is composed
by the nine fuzzy rules, as follows.
ð38cÞ
Plant part:
RulePlant1: IF x1(k) is M11and x2(k) is M12THEN
xðk þ 1Þ
¼
1:3636
?0:3636
0:1002
00
10
?0:5513 1
2
4
6
6
2
4
3
5xðkÞ
7
7
3
5wðkÞ þþ
?0:7143
0
0
3
5uðkÞ þ
?xðkÞ þ wðkÞ
7
0:1
0
0
2
4
6
0
0
1:2105
2
4
6
3
5
7
yðkÞ ¼ 1
RulePlant2: IF x1(k) is M21and x2(k) is M12THEN
2
4
?0:7143
0
6
00
?
ð39aÞ
xðk þ 1Þ ¼
1:3636
?0:3636
0
2
4
00
10
0
3
5uðkÞ þ
?xðkÞ þ wðkÞ
1
6
3
5xðkÞ
7
þ
0
7
0:1
0
0
2
4
6
3
5wðkÞ þ
7
0
0
2
2
4
6
3
5
7
yðkÞ ¼ 1
RulePlant3: IF x1(k) is M31and x2(k) is M12THEN
2
6
?0:7143
0
6
00
?
ð39bÞ
xðkþ1Þ¼
1:363600
?0:3636
?0:1002 0:5513 1
2
4
10
6
4
3
7
0:1
6
7
5xðkÞ
2
6
þ
0
6
3
7
7
5uðkÞþ
0
0
4
3
7
7
5wðkÞþ
0
0
2:9424
2
6
6
4
3
7
7
5
yðkÞ¼ 1 0 0
RulePlant4: IF x1(k) is M11and x2(k) is M22THEN
2
6
?0:7143
0
6
??xðkÞþwðkÞð39cÞ
xðkþ1Þ¼
1:3636 00
?0:3636
0:3495
2
4
10
?1:9225 1
3
7
6
4
3
7
7
5xðkÞ
0:1
7
þ
0
67
5uðkÞþ
0
0
2
6
6
4
3
7
5wðkÞþ
0
0
?0:0144
2
6
6
4
3
7
7
5
yðkÞ¼ 1 0 0
??xðkÞþwðkÞð39dÞ
1
0
M12
M32
M22
x2(k)
−90°
−45°
90°
45°
Figure 2. Membership function of x2(k).
M11
M31
M21
x1(k)
−88°
−50°
88°
50°
1
0
Figure 1. Membership function of x1(k).
816W.-J. Chang et al.
Page 9
Downloaded By: [Chang, Wen-Jer] At: 12:38 24 May 2008
RulePlant5: IF x1(k) is M21and x2(k) is M22THEN
2
6
?0:7143
0
6
xðk þ 1Þ ¼
1:363600
?0:3636
0:3636
2
4
10
?2
3
7
1
6
4
3
7
7
5xðkÞ
2
6
þ
0
67
5uðkÞ þ
0:1
0
0
6
4
3
7
7
5wðkÞ
yðkÞ ¼ 100
??xðkÞ þ wðkÞð39eÞ
RulePlant6: IF x1(k) is M31and x2(k) is M22THEN
2
6
?0:7143
0
6
xðkþ1Þ¼
1:3636 00
?0:3636
0:3495
2
4
10
?1:9225 1
3
7
6
4
3
7
7
5xðkÞ
0:1
7
þ
0
6
7
5uðkÞþ
0
0
2
6
6
4
3
7
5wðkÞþ
0
0
0:0144
2
6
6
4
3
7
7
5
yðkÞ¼ 1 0 0
??xðkÞþwðkÞð39fÞ
RulePlant7: IF x1(k) is M11and x2(k) is M32THEN
2
4
?0:7143
0
6
xðkþ1Þ¼
1:3636
?0:3636
?0:1002 0:5513 1
2
4
?xðkÞþwðkÞ
00
10
6
3
5xðkÞ
0:1
6
7
þ
0
3
5uðkÞþ
7
0
0
2
4
3
5wðkÞþ
7
0
0
?2:9424
2
4
6
3
5
7
yðkÞ¼ 1 0 0
?
ð39gÞ
RulePlant8: IF x1(k) is M21and x2(k) is M32THEN
2
6
?0:7143
0
6
xðk þ 1Þ ¼
1:363600
?0:3636
0
2
4
10
0
3
7
1
6
4
3
7
7
5xðkÞ
þ
0
67
5uðkÞ þ
0:1
0
0
2
6
6
4
3
7
7
5wðkÞ þ
0
0
?2
2
6
6
4
3
7
7
5
yðkÞ ¼ 100
??xðkÞ þ wðkÞð39hÞ
RulePlant9: IF x1(k) is M31and x2(k) is M32THEN
xðkþ1Þ¼
1:363600
?0:3636
0:1002
2
6
10
?0:5513 1
3
7
2
6
6
4
3
7
7
5xðkÞ
0:1
7
þ
?0:7143
0
0
6
4
7
5uðkÞþ
0
0
2
6
6
4
3
7
5wðkÞþ
0
0
?1:2105
2
6
6
4
3
7
7
5
yðkÞ¼ 1 0 0
??xðkÞþwðkÞð39iÞ
The corresponding matrices of S-procedure are pre-
sented as follows:
For Rule 111, ?88o? x1k ð Þ ? ?50o, which means
2
6
?1
6
T111¼
100
000
000
6
2
6
ð
4
3
7
7
5,
n111¼
2?50 ? 88
ðÞ?=180
½?
0
0
4
3
7
7
5and
Þ
v111¼ ?50?=180
Þ ? ?88?=180
ðð40aÞ
For Rule 331, 50o? x1k ð Þ ? 88o, which means
2
6
?1
6
T331¼
100
000
000
6
2
6
ð
4
3
7
7
5,
n331¼
250 þ 88
ðÞ?=180
½?
0
0
4
3
7
Þ
7
5and
v331¼ 50?=180
Þ ? 88?=180
ðð40bÞ
For Rule 112, ?90o? x2k ð Þ ? ?45o, which means
2
6
?1
6
T111¼
100
000
000
6
2
6
ð
4
3
7
7
5,
n111¼
2?45 ? 90
ðÞ?=180
½?
0
0
4
3
7
7
5
and v111¼ ?45?=180
Þ ? ?90?=180
ðÞð40cÞ
International Journal of Systems Science817
Page 10
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For Rule 332, 45o? x2k ð Þ ? 90o, which means
2
6
?1
6
T331¼
100
000
000
6
2
6
ð
4
3
7
7
5,
n331¼
245 þ 90
ðÞ?=180
½?
0
0
4
3
7
Þ
7
5and
v331¼ 45?=180
Þ ? 90?=180
ðð40dÞ
where the notion Rule ijq means the activation region
between RulePlanti and RulePlantj of the plant part on
the xq(k), e.g., Rule 123 means the activation region
between RulePlant1 and RulePlant2 for the x3(k) on the
plant part.
For the truck-trailer system (37), the control
purpose is to realise the backward movement of the
truck-trailer system to guarantee the Lyapunov stabi-
lity and passivity for the closed-loop system. Through
the discrete affine T–S fuzzy model (39), the fuzzy
controller can be designed by applying Theorem 2 and
fuzzy controller design procedure, i.e. ILMI algorithm.
First, let us choose the dissipative rate ? ¼1 and set
the matrix Q as the identity matrix. Then, the initial
matrix P(0)can be solved via Riccati Equation (36)
as follows:
P0 ð Þ¼
51:104
?29:3354
55:9672
8:5445
?29:3354
8:5445
?14:4917
7:3738
?14:4917
2
6
6
4
3
7
7
5
ð41Þ
Thus, the matrix X(1)can be obtained from (35)
and (41).
X1 ð Þ¼ P?1
0 ð Þ¼
0:02820:0129
?0:0074
0:06810:0129 0:0217
?0:00740:06810:2781
2
6
6
4
3
7
7
5
ð42Þ
Using the initial auxiliary variable (42), one can obtain
the minimum ? from Step 2. If ?<1, then one can
obtain the feasible solutions for fuzzy controllers.
Otherwise, if ? ? 1, a new matrix P(1) should be
re-calculated from the Step 3 and then be substituted
into Step 1 to produce new auxiliary parameter X(2).
Repeating this ILMI algorithm until ?<1, one
can obtain the feasible solution of suitable fuzzy
controllers. In this example, the feasible solution is
obtained after six iterations of the ILMI algorithm.
The final ? is 0.9009 and the feasible solutions are
obtained as
2
6
0:60650:1749
6
P ¼
3:6494
?6:0895
18:3963
0:642
?6:0895
0:642
?1:8353
0:27
?1:8353
6
2
6
4
3
7
7
5,
X ¼
?0:2718
1:06520:1749 0:2185
?0:27181:0652 11:5266
4
3
7
7
5,
"111¼ "331¼ 19:8444:
Then, the PDC-based fuzzy controller can be obtained
as follows:
Controller part:
RuleController1: IF x1(k) is M11and x2(k) is M12THEN
?
RuleController2: IF x1(k) is M21and x2(k) is M12THEN
?
RuleController3: IF x1(k) is M31and x2(k) is M12THEN
?
RuleController4: IF x1(k) is M11and x2(k) is M22THEN
?
RuleController5: IF x1(k) is M21and x2(k) is M22THEN
?
RuleController6: IF x1(k) is M31and x2(k) is M22THEN
?
RuleController7: IF x1(k) is M11and x2(k) is M32THEN
?
RuleController8: IF x1(k) is M21and x2(k) is M32THEN
?
RuleController9: IF x1(k) is M31and x2(k) is M32THEN
?
Settingtheinitial
x 0 ð Þ ¼ 880
states are shown in Figures 3–5. In the simulations,
uðkÞ ¼ ?2:58142:73
?0:2568
?xðkÞð43aÞ
uðkÞ ¼ ?2:67743:0386
?0:2826
?xðkÞð43bÞ
uðkÞ ¼ ?2:91023:0744
?0:3067
?xðkÞð43cÞ
uðkÞ ¼ ?2:70732:8102
?0:2485
?xðkÞð43dÞ
uðkÞ ¼ ?2:70762:8294
?0:2485
?xðkÞð43eÞ
uðkÞ ¼ ?2:70732:8102
?0:2485
?xðkÞð43fÞ
uðkÞ ¼ ?2:91023:0744
?0:3067
?xðkÞð43gÞ
uðkÞ ¼ ?2:67723:0382
?0:2825
?xðkÞð43hÞ
uðkÞ ¼ ?2:58142:73
?0:2568
?xðkÞ
condition
ð43iÞ
as
of
600
5
??T,thesimulation results
818 W.-J. Chang et al.
Page 11
Downloaded By: [Chang, Wen-Jer] At: 12:38 24 May 2008
the sampling time is set as 0.1 second. For checking the
satisfaction of the strictly input passive property, let us
substitute the simulation results into the following
ratio function.
Xkq
k¼0wTðkÞwðkÞ
It is easy to show that the value of the above ratio
function (44) is large than dissipative rate ? ¼1. Hence,
from these simulation responses and (44), one can find
that the closed-loop system is Lyapunov stable and
strictly input passive via the fuzzy controller (43).
2
Xkq
k¼0yTðkÞwðkÞ
¼ 1:8315
ð44Þ
5. Conclusions
In this article, the Lyapunov function associated with
the stored function for passivity property was con-
sidered. Using the properties of the passivity, this
article has discussed stability analysis and controller
synthesis for the discrete affine T–S fuzzy models.
Through applying the affine T–S fuzzy model, the
complex nonlinear system can be controlled accurately.
Based on the auxiliary variables and LMI technique,
the ILMI algorithm was employed to obtain the
feasible solutions of BMI stability conditions. The
nonlinear truck-trailer system has been provided to
demonstratetheeffectiveness
of the proposed design approach in this article.
Through the results of the simulation, one can find
that the nonlinear truck-trailer system can achieve the
strict input passivity and Lyapunov stability via the
proposed fuzzy controller design.
and applicability
Acknowledgements
The authors would like, to express their sincere gratitude to
anonymous reviewers, who gave us some constructive
comments, criticisms and suggestions. This work was
supported by the National Science Council of the Republic
of China under Contract NSC94-2213-E-019-006.
Notes on contributors
Wen-Jer Chang received the the MS
degree in the Institute of Computer
Science and Electronic Engineering
from the National Central University
in 1990, and the PhD degree from the
Institute of Electrical Engineering of
the National Central University in
1995. Since 1995, he has been with
National Taiwan Ocean University,
Keelung, Taiwan, R.O.C. He is currently the Vice Dean of
Academic Affairs, Director of Center for Teaching and
Learning and a full Professor of the Department of Marine
05 1015 202530 3540 4550
−1
0
1
2
3
4
5
Second
X3 (Meter)
Figure 5. Responses of state x3(k).
0510 152025 3035 404550
−10
0
10
20
30
40
50
60
70
80
90
Second
X1 (Angle)
Figure 3. Responses of state x1(k).
0510 1520 253035 40 4550
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Second
X2 (Angle)
Figure 4. Responses of state x2(k).
International Journal of Systems Science 819
Page 12
Downloaded By: [Chang, Wen-Jer] At: 12:38 24 May 2008
Engineering of National Taiwan Ocean University. He is
now a life member of the CIEE, CACS, CSFAT and
SNAME. Since 2003, Dr Chang was listed in the Marquis
Who’s Who in Science and Engineering. In 2003, he also won
the outstanding young control engineers award granted by
the Chinese Automation Control Society (CACS). In 2004,
he won the universal award of accomplishment granted
by ABI of USA. In 2005, he was selected as an excellent
teacher of the National Taiwan Ocean University. His
recent research interests are fuzzy control, robust control,
performance constrained control.
Cheung-Chieh Ku received the BS and
MS degree from the Department
of MarineEngineering
National Taiwan Ocean University,
Taiwan,R.O.C.,
2006, respectively. He is currently
working towards the PhD degree
intheelectrical
the National Taiwan Ocean University,
ofthe
in 2001and
engineeringat
Taiwan R.O.C. His research interests focus on fuzzy control
and control system theory.
Wei Chang received the BS and MS
degree from the
Marine Engineering of the National
Taiwan Ocean University, Taiwan,
R.O.C. in 2003 and 2005, respectively.
Hisresearch interests
fuzzy control and dynamic system
analysis.
Departmentof
focuson
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Appendix 1. S-procedure
In this appendix, the structures of the matrices Tijq, nijqand
vijq of S-procedure defined in (16) are proposed. For
illustrating S-procedure, it is assumed that the range of x
in which the fuzzy rule i is activated (i.e., hix ð Þ 6¼ 0) is
represented by the following system of at most n linear
inequalities, where n is the number of states:
For x1,x1? ?i1or x1? ?i1or ?i1? x1? ?i1
ðA1Þ
For x2,x2? ?i2or x2? ?i2or ?i2? x2? ?i2...
ðA2Þ
For xn,xn? ?inor xn? ?inor ?in? xn? ?in
ðA3Þ
Here, we denote that q ¼ 1,...,n. The range of x can be
represented by the following quadratic inequalities.
. In casexq? ?iq,
iqx þ viq? 0, where
3
7
?T
xq? ?iq,
iqx þ viq? 0, where
3
7
?T
one has
Fiqx ð Þ ? xTTiqx þ 2mT
0
6
?
. Incase
Fiqx ð Þ ? xTTiqx þ 2mT
2
6
?
. In case ?iq? xq? ?iq, i.e., ðxq? ?iqÞðxq? ?iqÞ ? 0,
one has FiqðxÞ ? xTTiqx þ 2mT
0
.
6
6
6
?
Tiq¼ 0n?n¼
???
..
0
......
.
0
???
0
2
6
4
7
5, miq
0
¼ 0
???
1=2
qthelement
???
and viq¼ ??iq
ðA4Þ
onehas
Tiq¼ 0n?n¼
0
???
..
0
...
.
...
0
???
0
6
6
4
7
7
5, miq
???¼ 0
????1=2
qthelement
0and viq¼ ?iq
ðA5Þ
iqx þ viq? 0, where
Tiq¼
..
1
qthelementof
thediagonal
..
.
0
2
6
6
6
?
6
6
4
6
6
3
7
7
7
???
7
7
7
7
7
5
7
7
, miq
¼ 0
????1=2 ?iqþ ?iq
qthelement
?
0
?T
and viq¼ ?iq?iq
ðA6Þ
International Journal of Systems Science821