Content uploaded by Hsin-Han Chiang
Author content
All content in this area was uploaded by Hsin-Han Chiang
Content may be subject to copyright.
Fuzzy Sets and Systems 154 (2005) 182–207 www.elsevier.com/locate/fss
Neural-network-based optimal fuzzy controller design for
nonlinear systems
Shinq-Jen Wua,b,∗, Hsin-Han Chiangb, Han-Tsung Linb, Tsu-Tian Leeb
aDepartment of Electrical Engineering, Da-Yeh University, Chang-Hwa, Taiwan, ROC
bDepartment of Electrical and Control Engineering, National Chiao-Tung University, Hsinchu, Taiwan, ROC
Received 26 May 2004; received in revised form 13 March 2005; accepted 14 March 2005
Available online 25 April 2005
Abstract
Aneural-learning fuzzytechnique isproposed forT–S fuzzy-modelidentification ofmodel-free physicalsystems.
Further, an algorithm with a defined modelling index is proposed to integrate and to guarantee that the proposed
neural-based optimal fuzzy controller can stabilize physical systems; the modelling index is defined to denote
the modelling-error evolution, and to ensure that the training data for neural learning can describe the physical
system behavior very well; the algorithm, which integrates the neural-based fuzzy modelling and optimal fuzzy
controlling process, can implement off-line modelling and on-line optimal control for model-free physical systems.
The neural-fuzzy inference network is a self-organizing inference system to learn fuzzy membership functions and
fuzzy-subsystems’ parameters as data feeding in. Based on the generated T–S fuzzy models for the continuous
mass–spring–damper system and Chua’s chaotic circuit, discrete-time model car system and articulated vehicle,
their corresponding fuzzy controllers are formulated from both local-concept and global-concept fuzzy approach,
respectively. The simulation results demonstrate the performance of the proposed neural-based fuzzy modelling
technique and of the integrated algorithm of neural-based optimal fuzzy control structure.
© 2005 Elsevier B.V.All rights reserved.
Keywords: Riccati equation; Modelling index; Linear T–S fuzzy system; Affine T–S fuzzy system; Exponentially stable
∗Corresponding author. Department of Electrical Engineering, Da-Yeh University, Chang-Hwa, Taiwan, ROC. Tel.:
+8864 8511888x2192; fax: +88662512882.
E-mail addresses: jen@mail.dyu.edu.tw,jen@cn.nctu.edu.tw (S.-J. Wu).
0165-0114/$ -see front matter © 2005 Elsevier B.V.All rights reserved.
doi:10.1016/j.fss.2005.03.011
S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207 183
1. Introduction
Research in fuzzy modelling and fuzzy control has come of age [1,2,5,11,22,23]. There are two main
waysto theoreticallyconstructaT–Sfuzzymodel. Oneisfromlocallinear approximation,whichgenerates
a linear consequent part with a constant term included in each rule; the other is via sector nonlinearity
concept[7,14,15], whichresults ina constant-freelinear consequencefor eachrule. Bothare demonstrated
to be universal approximations to any smooth nonlinear systems [16,20,28]. For simplification, these two
kinds of fuzzy structures are, respectively, denoted as linear and affine T–S fuzzy systems by Tanaka and
Wang [20]. It is noticed that the consequent part of each fuzzy rule in both models are represented by a
linear state equation; the only difference between these two representations is that there exists a constant
singleton in the fuzzy rule consequence for the affine T–S fuzzy model.
The T–S type with no constant term in the local linear consequent part of each rule (linear T–S fuzzy
system) is the most popular fuzzy model for its further intrinsic analysis: T–S model-based fuzzy control
has been successfully applied to many nonlinear systems [15]. The linear matrix inequality (LMI)-based
fuzzy controller is to minimize the upper bound of the performance index [21]. Structure-oriented and
switching fuzzy controller are further developed for more complicated systems [6,12,14]. The optimal
fuzzy control technique is used to minimize the performance index from local-concept or global-concept
approach [25–27]. Recently,Tanaka andWang developed an integrated LMI approach to fuzzy modelling
and controlling a nonlinear system with unknown parameters [10]. Three LMI conditions are derived to
identify the parameters of T–S fuzzy models, and a robust controller is developed to compensate the
identification error.The membership functions and fuzzy rule numbers are chosen as known parameters
in the aforementioned approach. And in order to decrease the computational cost, much research focuses
on rule and consequence order reduction [6,15,17] and on rule switching technique [12]. Advanced
research for fuzzy modelling of more complicated systems is still open. Further, the aforementioned
research is available only for model-based nonlinear systems.
The approach of model-free nonlinear systems to guarantee the proposed fuzzy model under limited
modelling error and the corresponding fuzzy control with desirable implementation is still developing.Yu
and co-workers use a type-1 fuzzy neural network (FNN) with sliding-mode and gradient-decent learning
to control a Duffing system [9]. Wai uses FNN to mimic a perfect control law and compensate the error by
another compensator [19]. Lin and co-workers use FNN to approximate nonlinear functions and develop
adaptive laws to attenuate approximation errors and external disturbance [4]. Hu and Liu fuzzy model
a time-delay system analytically, then use adaptive RBF NN to approximate fuzzy modelling error and
adoptH∞controlto compensatetheerror [8].Wangand co-workersuse type-1 FNNwith adaptiveupdate
law to approximate an optimal controller [24]. Most of them describe systems with fuzzy rules and use
FNN to control the systems. There was no direct approach to identify T–S fuzzy systems of model-free
nonlinear systems.
In this work, we propose a neural fuzzy network (NFN) to achieve identification of a linear T–S
fuzzy model for model-based or model-free systems, which can self-learn the Gaussian-type membership
functions and fuzzy subsystems’ parameters of each rule consequence. The generated linear T–S fuzzy
model can be used to develop fuzzy controllers such as an LMI-based fuzzy controller, structure-oriented
andswitchingfuzzy controllers.In ordertofurther ensurethat the generatedfuzzy modelcanapproximate
the original physical system and more to control the model-free system well, we propose an integrated
algorithm,whichintegratestheproposedneuralfuzzy networkand previouslyproposednonlinearoptimal
fuzzy controller, to guarantee the generated fuzzy system can describe the physical system behavior and
184 S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207
the closed-loop neural-based optimal fuzzy control system is stable. The proposed structure is applied
to fuzzy modelling and optimal controlling of a mass–spring–damper system, a chaotic Chua’s circuit
system, a model car system and an articulated vehicle system.
2. Neural-based fuzzy model and optimal controller
2.1. Neural-based fuzzy inference structure
As we know, the T–S fuzzy model is basically a locally linearized fuzzy model, which describes global
behavior by fuzzily blending linear subsystems. Most T–S fuzzy models are identified by, respectively,
local linear approximation and sector nonlinearity concept [7,15], which fuzzily blends the bounded
values of each nonlinear term to achieve global or semi-global effect. Accordingly, two kinds of T–S
fuzzy system representations, affine T–S fuzzy model and linear T–S fuzzy model, are generated. The
difference between these two representations is that a singleton is included in the fuzzy subsystems of
the affine T–S fuzzy model. Both fuzzy models are demonstrated to be universal approximations of any
smooth nonlinear system to any desired accuracy. However, these two modelling techniques (local linear
approximation and sector nonlinearity concept) are available for model-based systems only. Besides,
since the controller design for linear T–S fuzzy model has been developed very well, it is important to
propose a modelling technique to construct a linear T–S fuzzy system not only for model-based but also
for model-free nonlinear physical systems.
Juang and Lin proposed a neural-fuzzy inference network with self-learning ability (SONFIN) [3],
though an affine T–S fuzzy system can be obtained via this network by regarding external inputs as aug-
mented state variables. However, the singleton in each fuzzy rule is the key consequence for learning and
the state-dependent terms are just optional generated for compensation. The learning process will always
diverge by just deleting the singleton from the rule consequence of Juang’s algorithm directly. In other
words, basic SONFIN structure will learn type-1 fuzzy system basically. We modified this neural fuzzy
network such that the input- and state-dependent terms initially exist and the corresponding parameters
are adapted by the gradient method; in other words, the learning process will focus on generating input-
and state-dependent terms.
We here name the modified NFN to be linear-NFN and Juang’s to be affine-NFN to denote the con-
structed fuzzy models to be linear T–S type and affine T–S type, respectively. Notice that even these two
structures are similar in representation but the learning spirit is totally different. That is, the singleton is
the key term and state-dependent terms are optional generated for compensation in the rule consequences
of affine type, but the input- and state-dependent terms are now the key terms in those of linear type.
Fig. 1 describes the proposed six-layer linear NFN structure for realizing a linear T–S fuzzy model.
This structure is similar to Juang’s except for the rule representation in the fifth layer. Each node in the
structurepossesses finiteweighted fan-inconnectionstothelast-layer nodesand fan-outconnectionsto the
next-layer nodes.An integration function is associated with the fan-in operation to integrate information,
activation and evidence; in other words, the integration function is the net input of a node. For example,
for the ith node in the kth layer, we have
net −inputk
i=f(u
k
1i,u
k
2i,...,u
k
pi,w
k
1i,w
k
2i,...,w
k
pi),
S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207 185
Fig. 1. linear-SONFIN structure.
where uk
1i,u
k
2i,...,u
k
piare the inputs of the ith node and wk
1i,w
k
2i,...,w
k
piare the associated weights.
The output operation ok
iis then proceeded by an activation function a(·),
ok
i=a(net −inputk
i).
We now briefly describe the proposed six-layer NFN structure as follows.
186 S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207
Layer 1: Each node in this layer is correspondent to one input variable and transmits the input variable
to the next layer directly; that is,
f=u1
i,a
1=f,
where the linking weight w1
iis unity in this layer. Layer 2: Each node in this layer denotes a linguistic
label; that is, the input variables are fuzzified in this layer. We choose Gaussian distribution as the
membership function and the operation is performed as
f(u
2
ij )=−
(u2
i−mij )2
2
ij
,a
2(f ) =ef,
where mij and ij are the mean and the standard deviation of the Gaussian membership function of the
jth term of the ith input variable u2
i.
Layer 3: The fuzzily blending operation is performed in this layer and hence each node represents one
fuzzy logic rule; that is,
f(u
3
i)=
n
i=1
u3
i=e−[Di(x−mi)]t[Di(x −mi)],a
3(f ) =f,
where nis the number of Layer-2 nodes joining with the ith rule precondition, Di=diag(1/i1,
1/i2,...,1/in)and mi=(mi1,m
i2,...,m
in). Notice that the weighted factor for each fan-in stream
is unity in this layer and the node output is in fact the firing strength of the corresponding fuzzy rule.
Layer 4: This layer is for normalization and hence generalizes the normalized fire strength of the fuzzy
rule in the following way:
f(u
4
i)=
i
u4
i,a
4(f ) =u4
i
f,
where all weighted factors are unity in this layer.
Layer 5: This layer is the consequence layer. Each node in Layer 4 has its corresponding node. Notice
that the node outputs in Layer 4 are the key consequences of the fuzzy rules in Juang’s NFN; but now,
they are only the basic node inputs to store the fire strength information. Not only the input variables in
Layer 1 but also the external inputs of the physical system are included as the node inputs to generate
the consequence condition for the corresponding fuzzy rule. In other words, the activity function in this
layer is
f=
j
ajixj+
m
bmi um,a
5(f ) =f·u5
i.
Layer6:Eachnode inthis layeriscorrespondent toone systemoutputvariable.This layer isto integrate
the actions from Layer 5, and hence to perform the defuzzifier operation for the fuzzy logic system. In
other words,
f(u
6
i)=
i
u6
i,a
6(f ) =f.
S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207 187
Hence, the input variables xjare fuzzified as fuzzy variables whose corresponding term sets Tji have
Gaussian membership function with mean mji and standard deviation ji; the corresponding output for
the neural network is
SX (t ) =AiX(t) +Biu(t), i =1,...,r.
In other words, the proposed NFN structure is in fact a neural-based linearT–S fuzzy modelling structure.
Vianeurallearningtechnique,thisstructurewill proceedthe structureand parameterlearning concurrently
and generate the following linear T–S fuzzy system:
Ri:If x1is T1i(m1i,1i),..., x
nis Tni (mni ,ni ), then Y(t) =CX(t)
SX (t ) =AiX(t) +Biu(t), i =1, ...,r, (1)
where Ridenotes the ith rule of the fuzzy model; x1, ...,x
nare system states; Tji(mji,ji),j=
1,...,n, is the fuzzy term of the input fuzzy variable xjin the ith rule with mji and ji being the mean
and standard deviation of the Gaussian membership function; SX(t) denotes ˙
X(t) for the continuous case
andX(t+1)forthediscrete case;X(t) =[x1, ...,x
n]t∈
nisthestate vector, Y(t) =[y1, ...,y
n]t∈
nis the system output vector, and u(t) ∈
mis the system input (i.e., control output); and Ai,Biand
Care, respectively, n×n,n×mand n×nmatrices.
Structure learning includes both precondition and consequence identification of a fuzzy IF–THEN
rule. Precondition identification (input-space partition) is formulated as the combinational optimization
problem to minimize the number of generated rules and the number of fuzzy term sets for each input
fuzzy variable, where the input space is partitioned in a flexible way via the aligned clustering-based
algorithm. Consequence identification is to decide the significant terms (states and inputs) to be added via
projected-based correlation measure of each rule. The combined precondition and consequence structure
identification scheme can set up an economical and dynamically growing network automatically. In
other words, this NFN structure possesses the self-construction ability to generate its rule nodes, term set
nodes and linking weights between nodes. As for the parameter learning, based on the supervised learning
algorithm,theleastmeansquare algorithmis adoptedto adjustthe parametersin theruleconsequence,and
the back-propagation algorithm for minimizing a given cost function is adopted to adjust the parameters
in the rule precondition.
2.2. Neural-fuzzy-based optimal fuzzy controller
Though the proposed NFN structure can obtain the linear T–S fuzzy model for the model-free systems,
the critical issue is how to ensure the training data sufficiently enough for describing the system behavior
effectively. As we know, once the designed optimal fuzzy controller u∗(t) is applied to a real physical
system, then the deviation between real and estimated output comes from modelling error and controlling
error.Via our previous papers [25–27], we know the proposed optimal fuzzy controller can exponentially
stabilizethecorresponding linearT–Sfuzzy systemonce eachfuzzysubsystem iscompletely controllable
(c.c.) and completely observable (c.o.). In other words, the closed-loop real system compensated with the
optimal fuzzy controller is exponentially stable in the case of zero modelling error; that is, the neural-
learning-basedT–Sfuzzy systemis consistentwith thereal nonlinearsystem. Formeasuringthemodelling
188 S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207
error, we define a modelling index as
IM(t) =Ycl
Lsonfin(t ) +
Ycl(t ) +,(2)
where YLsonfin(t ) is the output of the proposed neural-learning-based T–S fuzzy closed-loop system and
Y(t) is the output of the real physical closed-loop system; is a small constant to ensure a nonzero
denominator.Accordingly to the stability of the optimal fuzzy closed-loop system [25–27], we know the
index must approach unity as time goes to infinity once the fuzzy model can approximate the real physical
systemvery well.Therefore,wefurtherintegrate theneural-fuzzy modellingprocessandtheoptimal fuzzy
controlling design scheme into an integrated neural-fuzzy modelling and controlling (INFMC) algorithm
in Fig. 2. Via this INFMC algorithm, we can guarantee that the proposed neural-learning-based T–S fuzzy
models can describe the real physical systems well and obtain the corresponding optimal fuzzy controller.
In the rest of this subsection, the adopted local- and global-based optimal fuzzy controllers are described
briefly as follows.
Based on the generated T–S fuzzy model from Section 2.1, we assume all desired controllers are in the
form of
Ri:If y1is S1i, ..., y
nis Sni,then u(t) =ri(t), i =1, ...,,(3)
where y1, ...,y
nare the elements of output vector Y(t),S1i, ...,S
niare the input fuzzy terms in the
ith control rule, and the plant input (i.e., control output) vector u(t) or ri(t) is in mspace. Our quadratic
optimal fuzzy control problem is then described as follows:
Problem 1. Given the rule-based fuzzy system in Eq. (1) with X(t0)=X0∈
nand a rule-based fuzzy
controller in Eq. (3), find the individual optimal control law, r∗
i(·),i=1,...,, such that the composed
optimal controller, u∗(·), can minimize the quadratic cost functional, J (u(·)), over all possible inputs
u(·).
J (u(·)) =∞
t0
[Xt(t)L(t)X(t ) +ut(t )u(t )]dt(continuous), (4)
J (u(·)) =
∞
t=t0
[Xt(t)L(t)X(t ) +ut(t )u(t )](discrete-time), (5)
whereXt(t)L(t )X(t) is state-trajectorypenalty withL(t ) belongingto asymmetricpositivesemi-definite
n×nmatrix and ut(t)u(t) is fuel consumption.
Forthe localapproach,we firstadopt theprinciplesof dynamicprogramming totransformthe quadratic
optimizationprobleminto asuccessivelyongoingdynamic problemwith regardtothe stateresultingfrom
the previous decision. Then, based on the additive property of energy, we know that, at any time-step t,
if we can find the optimal local decision (optimal control law) for minimizing
Jt(ut)=∞
t
(Xt
lLlXl+ut
lul)dl, t ∈[t0,∞)(continuous), (6)
Jt(ut)=
∞
t
[Xt
lLlXl+ut
lul],t∈[t0,∞)(discrete)(7)
S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207 189
Fig. 2. Integrated neural-fuzzy modelling and controlling (INFMC) algorithm.
190 S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207
with regard to a fuzzy subsystem, the composed global decision can be a global minimizer of total cost
with regard to a fuzzy system. In other words, based on the local viewpoint of the global optimal fuzzy
control, we know that solving the quadratic optimal control problem is to find only one corresponding
optimal solution of the fuzzy controller for each rule of the fuzzy model. Thereupon, both the fuzzy
model and admissible fuzzy controller have, more precisely, the same input variables and same input
space partition, and there exists only one optimal fuzzy control rule for each fuzzy subsystem described
by a fuzzy rule in the fuzzy model. In short, the local-concept optimization technology is first adopted to
rewrite the quadratic optimization problems into the following successively ongoing dynamic problems
with regard to the state resulting from the previous decision [25].
Problem 1.1. Given the fuzzy subsystem,
˙
Xl=AilXl+Bilril,l∈[t,∞), i =1,...,r (continuous), (8)
Xl+1=AilXl+Bilril,l∈[t,∞), i =1,...,r (discrete)(9)
with the initial state resulting from the previous decision, i.e., X0t=X∗
t,
(1) find the optimal local decision at time instant t,r∗
it, for minimizing the cost functional,
Jt(rit)=∞
t
(Xt
lLlXl+rt
ilril)dl, t ∈[t0,∞)(continuous), (10)
Jt(rit)=
∞
t
(Xt
lLlXl+rt
ilril],t∈[t0,∞)(discrete);(11)
(2) obtain the optimal global decision at time instant t,u∗
t, for minimizing the cost functional Jt(ut)in
Eqs. (6) and (7), by fuzzily blending each local decision, i.e., u∗
t=r
i=1hi(X∗
t)r∗
it.
Since the local fuzzy system (i.e., fuzzy subsystem) is linear, its quadratic optimization problem is
the same as the general linear quadratic issue. Therefore, it is realizable that solving the optimal control
problem for a fuzzy subsystem can be achieved by simply generalizing the classical theorem from the
deterministic case to the fuzzy case. Hence, we have the following corresponding local-concept optimal
continuous fuzzy controller design schemes.
Proposition 1 (Local-concept continuous Wu and Lin [25]). For a continuous fuzzy controller,respec-
tively,in Eq.(3) and the continuous fuzzy system in Eq. (1), let Ai,B
i,C,Lbe given constant matrices.
If (Ai,B
i)is c.c. and (Ai,C)is c.o. for i=1, ...,r,then
(1) there exists a unique n×nsymmetric positive semi-definite solution i
∞of the steady-state Riccati
equation (SSRE)
At
iK+KAi−KBiBt
iK+L=0;(12)
(2) the asymptotically local optimal fuzzy control law is
r∗
i(t) =−Bt
ii
∞X∗(t), i =1, ...,r, (13)
and their “blending”global minimizer u∗(t) =n
i=1hi(X∗)r∗
i(t) minimizes J (u(·)) in Eq.(4);
S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207 191
(3) and the optimal global feedback fuzzy subsystem
˙
X∗(t) =
n
i=1
hi(X∗)(Ai−BiBt
ii
∞)X∗(t ) (14)
is exponentially stable.
Proof. From the inference in the above, we can get local optimal fuzzy control law r∗
i(t) in Eq. (13)
and the local feedback fuzzy subsystem, ˙
X∗(t) =(Ai−BiBt
ii
∞)X∗(t ), is exponentially stable. We
demonstrate the stability of the composed global feedback fuzzy subsystem in Eq. (14) in the Appendix.
For the discrete-time system, we have the following corresponding local-concept optimal discrete-time
fuzzy controller design schemes.
Proposition 2 (Local-concept discrete-time,Wu and Lin [25]). For the discrete-time fuzzy controller in
Eq.(3) and the discrete-time fuzzy system in Eq. (1), let Ai,B
i,C,L be given constant matrices. If
(Ai,B
i)is stabilizable and (Ai,C)is detectable for i=1, ...,r, then
(1) there exists a unique symmetric positive semi-definite solution i(∞)of the following algebraic
SSRE,
V(∞)=L+At
iV(∞)[In+BiBt
iV(∞)]−1Ai,(15)
V(∞)=L+At
iV(∞)Ai−At
iV(∞)Bi[In+Bt
iV(∞)Bi]−1Bt
iV(∞)Ai;(16)
(2) the asymptotically local optimal fuzzy control law is
r∗
i(t) =−[In+Bt
ii(∞)Bi]−1Bt
ii(∞)AiX∗(t), t =t0, ...,N −1,(17)
and the resultant global controller u∗(t ) minimizes J (u(·)) in Eq.(5);
(3) moreover,the optimal local feedback fuzzy subsystem,
X∗(t +1)=[In+BiBt
ii(∞)]−1AiX∗(t), (18)
is asymptotically and exponentially stable.
As for the global-concept technique, since each penalty term in the performance index is with regard
to the entire fuzzy system and controller, we fuzzily blend the distributed fuzzy subsystems and rule-
based fuzzy controller into the entire fuzzy system and entire fuzzy controller formulations, and unify
the individual matrices into synthetical matrices to form a linear-like global system representation of a
fuzzy system,
SX (t) =H (X (t))A(t )X(t ) +H (X (t))B (t)W (Y (t ))R (t),
Y(t)=C (t )X(t), (19)
where H(X(t)) =[h1(X(t))In... hr(X(t ))In],W(Y(t)) =[w1(Y (t))Im... w(Y (t ))Im],
A(t) =
A1(t)
.
.
.
Ar(t)
, B(t ) =
B1(t)
.
.
.
Br(t)
, R(t ) =
r1(t)
.
.
.
r(t)
192 S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207
with Inand Imdenoting the identity matrices of dimension nand m, respectively. And, hi(X(t )) and
wi(Y (t )) are the normalized fire strengths for the ith fuzzy rule in the fuzzy system and in the fuzzy
controller, respectively. Furthermore, a multistage-decomposition approach is adopted to transform the
optimal control problem into an ongoing stage-by-stage dynamic issue [26,27].
Notice that the formulation and simplification of a quadratic optimal fuzzy control problem is achieved
byfuzzilymerging distributedrule-based T–Stypefuzzy subsystemsintoan entirefuzzy system.This can
initiate and activate the research in global optimal fuzzy controller design. The unification of individual
matrices (Ai(k) and Bi(k),i=1,...,r) and normalized membership functions (hi(X(k)),i=1,...,r,
and wi(Y (k)),i=1,...,) into synthetical matrices (A(k),B(k),H(X(k)) and W(Y(k))) generates
alinear-like global system representation of a fuzzy system with the value of each element of the non-
linear terms (H(X(k)) and W(Y(k))) being located in segment [0,1]. This linear-like representation
motivates us to develop the design scheme of a global optimal fuzzy controller in the way of general
linear quadratic approach, i.e., calculus-of-variation method. Moreover, the multistage-decomposition
approach is to transform the optimal control problem into an ongoing stage-by-stage dynamic issue; that
is, the optimal solutions can be resolved from Nsegmental nonlinear TPBVP instead of the nonlinear
TPBVP for the entire horizon. This decomposition operation can speed up numerical solution and keep
the global optimality at the same time. Furthermore, ¯
Ndenotes the number of stages at which mem-
bership functions can be assumed to be invariant during the whole single stage and is assumed to make
the backward recursive Riccati-like equation available. This avoids the high computational complexity
of the collocation method at the expense of approximate optimality due to the time-invariant assump-
tion. Furthermore, a procedure including a dynamical decomposition algorithm is proposed to justify the
time-invariant assumption in practice [26].
According to the derivation above, we can obtain the global-concept-based optimal fuzzy controller
for both continuous and discrete-time fuzzy systems as follows.
Proposition 3 (Global-concept continuous, Wu and Lin [26]). Consider the time-invariant fuzzy system
inEq.(1)andfuzzycontrollerinEqs.(3); ifN> ¯
N,(Ai,B
i)isc.c. and(Ai,C)isc.o.,foralli=1,...,r,
then
(X∗
∞(t), R∗
∞(t)) =(Xi∗
∞(t), Ri∗
∞(t)), ∀t∈[ti
0,ti
1],t1
0=t0,tN
1=∞,i =1,...,N, (20)
where Ri∗
∞(t) is the ith-stage asymptotically optimal control law,
Ri∗
∞(t) =−Wt
i[WiWt
i]−1BtHt
ii
∞Xi∗
∞(t), t ∈[ti
0,∞), (21)
which minimizes Ji
∞(R(·)) =∞
ti
0[Xt(t)LX(t ) +Rt(t )W t
iWiR(t)]dt,and Xi∗
∞(t) is the corresponding
asymptotically optimal trajectory that satisfies
˙
Xi∗
∞(t) =(HiA−HiBBtHt
ii
∞)Xi∗
∞(t), t ∈[ti
0,∞), (22)
where i
∞is the unique symmetric positive semidefinite solution of the SSRE,
AtHt
iK+KHiA−KHiBBtHt
iK+CtC=0.(23)
Proposition 4 (Global-concept discrete-time,Wu and Lin [27]). Consider the time-invariant fuzzy sys-
tem and fuzzy controller described,respectively,by Eqs.(1) and (3) with L=CtC.If there exists ¯
Nsuch
S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207 193
that if N> ¯
N,(Ai,B
i)is c.c. and (Ai,C)is c.o.,i=1,...,r,then,for each stage,(X∗
∞(k), R∗
∞(k)) =
Xi∗
∞(k), Ri∗
∞(k)),∀k∈[ki
0,k
i
1−1],k
1
0=k0,k
N
1=∞,where the ith-stage asymptotically optimal control
law,
Ri∗
∞(k) =−Wt
i[WiWt
i]−1BtHt
ii
∞[In+HiBBtHt
ii
∞]−1HiAXi∗
∞(k), k ∈[ki
0,∞), (24)
which minimizes Ji
∞(R(·)) =∞
k=ki
0[Xt(k)LX(k) +Rt(k )W t
iWiR(k)];Xi∗
∞(k) is the corresponding
asymptotically optimal trajectory,
Xi∗
∞(k +1)=[In+HiBBtHt
ii
∞]−1HiAXi∗
∞(k), k ∈[ki
0∞), (25)
wherei
∞istheunique symmetricpositive semidefinite solutionof thediscrete-time algebraicRiccati-like
equation,
K=L+AtHt
iK[In+HiBBtHt
iK]−1HiA, (26)
K=L+AtHt
iKHiA−AtHt
iKHiB[In+BtHt
iKHiB]−1BtHt
iKHiA. (27)
3. Physical system modelling and controlling
In this section, we shall generate the T–S fuzzy models and design the optimal controllers for four
complicated nonlinear physical systems. The INFMC algorithm is adopted to integrate the neural-fuzzy
modellingand optimalfuzzy controllingprocess, andmore toguarantee thatthe proposedneural-learning-
basedT–Sfuzzy modelscanapproximate theoriginalphysical systemsverywell. Theneural-fuzzy-based
optimal fuzzy controller are designed from both local and global concept, respectively. Simulation results
show that the proposed optimal fuzzy controllers can effectively drive the physical systems to the target
points in a short time.
3.1. Neural-based T–S fuzzy modelling
In this section, we shall use the proposed linear NFN structure to generate the corresponding linear
T–S fuzzy models for the mass–spring–damper system [11], the chaotic Chua’s circuit system [21], the
model car system [13] and the articulated vehicle system [18], respectively.
A mass–spring–damper system can be formulated as [11]
¨x=−0.1˙x3−0.02x−0.67x3+u, (28)
where x∈[−1.51.5]and ˙x∈[−1.51.5].
It is not necessary to train the input/output pattern repeatedly in the learning process. There initially
exists no rule in the neural-fuzzy structure. As on-line feeding in the training data, the following opera-
tions are done simultaneously: the input/output spaces are partitioned, the fuzzy rules are generated, the
consequent structure and the parameters in the structure are identified optimally.The training results are
shown in Fig. 3 and the neural-learning-based T–S fuzzy model for the mass–spring–damper system is
194 S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207
Fig. 3. Neural-based fuzzy modelling (solid line) for a continuous mass–spring–damper system (dashed line).
as follows:
Ri:If x1is T1i(m1i,1i)and x2is T2i(m2i,2i),
then ˙
X(t) =AiX(t) +Biu(t), i =1,...,5,(29)
where fuzzy term sets T11(−0.4158,0.6545),T21(0.3982,0.5249),T12(−0.597,0.7889),
T22(−0.8596,0.6376),T13(0.1681,0.4798),T23(0.3514,0.6428),T14(−0.5881,0.7827),
T24(−1.1486,0.6783),T15(−0.5881,0.7827),T25(1.1379,1.0588);
A1=0.3718 0.6995
10
,A
2=0.0014 1.3836
10
A3=0−0.1848
10
,
A4=0−2.7786
10
,A
5=−0.741 −1.5384
10
,
Bi=1
0,i=1,...,5;X(t) =x(t)
˙x(t) .
According to the controllability and observability analysis in [25], we know the generated fuzzy model
in Eq. (29) is c.c. and c.o.
S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207 195
We next consider a more complex continuous nonlinear chaotic system, Chua’s circuit, which is an
electronicsystemwith oneinductor (L),twocapacitors (C1,C2),one linearresistor (R) andone piecewise
linear or nonlinear resistor (g) included. The dynamic behavior of Chua’s circuit can be described as [21]
˙vC1=1
C11
R(vC2−vC1−g(vC1)),
˙vC2=1
C21
R(vC1−vC2)+iL,
˙
iL=1
L(−vC2−R0iL), (30)
where vC1and vC2are the voltage of capacitors and iLis the instant current of the inductor; the nonlinear
resistor is characterized as g(vC1)=GbvC1+1
2(Ga−Gb)(|vC1+E|−|vC1−E|)with parameters
Ga,G
b<0. We denote the state variable X=[vC1,vC2,iL]tand choose R=10/7, R0=0, L=1/7,
C1=0.1, C2=2, Ga=−4, Gb=−0.1 and E=1. After successful training in Fig. 4, the generated
T–S fuzzy model is
Ri:If x1is T1i(m1i,1i), then ˙
X(t) =AiX(t), i =1,...,4,(31)
Fig. 4. Neural-based fuzzy modelling (solid line) for a continuous Chua’s chaotic system (dashed line).
196 S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207
where the fuzzy term sets T11(−0.105,1.556),T12(7.45,5.525),T13(0.031,5.004),T14(−8.939,7.622);
A1=
68.85 7 0
0.35 −0.35 0.5
0−70
,A
2=
−2.395 7 0
0.35 −0.35 0.5
0−70
,A
3=
12.06 7 0
0.35 −0.35 0.5
0−70
,
A4=
−2.508 7 0
0.35 −0.35 0.5
0−70
.
We now step for fuzzy modelling the discrete-time nonlinear systems, the car model system [13],
x1(k +1)=x1(k) +vt
ltan(u(k)) ,
x2(k +1)=x2(k) +vt sin(x1(k)),
x3(k +1)=x3(k) +vt cos(x1(k)), (32)
where x1(k),x2(k) and x3(k) are, respectively, the angle of the car, the vertical and horizontal position
of the rear end of the car; u(k) is the steering angle, lis the length of the car, tis the sampling time and
vis the constant speed. The parameters were chosen as l=2.8m,v=1.0m/s and t=1.0s. After
neuro-fuzzy modelling in Fig. 5, we have
Ri:If xi(k) is T1i(m1i,1i), then X(k +1)=AiX(k) +Biu(k), i =1,...,5,(33)
Fig. 5. Neural-based fuzzy modelling (solid line) for a discrete-time model car system (dashed line).
S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207 197
where fuzzy term sets T11(0.001,0.224),T12(−0.57,0.353),T13(0.567,0.345),T14(−1.562,0.618),
T15(1.561.0.619);X(k) =[x1(k ), x2(k)]t;
A1=10
1.01 0 ,A
2=10
0.969 1 A3=10
0.974 1 ,
A4=10
0.686 1 ,A
5=10
0.672 1 ,B
1=0.377
−0.003,B
2=0.39
0.01,B
3=0.388
0.011,
B4=0.394
−0.104 ,B
5=0.403
−0.15,
which is c.c. and c.o.
We further concern ourselves with a multi-dimensional and more complicated discrete-time nonlinear
articulated vehicle [18],
x1(k +1)=x1(k) +vt
ltan(u(k)),
x2(k) =x1(k) −x3(k),
x3(k +1)=x3(k) +vt
Lsin(x2(k)),
x4(k +1)=x4(k) +vtcos(x2(k)) sinx3(k +1)+x3(k)
2,
x5(k +1)=x5(k) +vtcos(x2(k)) cosx3(k +1)+x3(k)
2,(34)
where u(k) is the steering angle; x1(k),x2(k),x3(k),x4(k) and x5(k) are the angle of truck, the angle
difference between truck and trailer, and the angle of trailer, the vertical and horizontal position of the
rear end of the trailer, respectively. We set l=0.2m,L=0.32 m, v=−0.1m/s, t=0.5. After
neuro-fuzzy modelling in Fig. 6, we obtain the following corresponding linear T–S fuzzy model:
Ri:If x2(k) is T2i(m2i,2i),x
3(k) is T3i(m3i,3i)and x4(k) is T4i(m4i,4i),
then X(k +1)=AiX(k) +Biu(k), i =1,...,4,(35)
where fuzzy term sets T21(0.178,0.182),T31(1.316,0.188),T41(0.63,0.18),T22(0.809,0.123),
T32(0.532,0.801),T42(−0.07,0.578),T23(0.809,0.123),T33(183.6,254.1)T
43(−1.03,0547),
T24(−0.043,1.19),T34(−1.614,0.975),T44(0.409,0.89),T25(−0.043,1.19),T35(0.935,2.059),
T45(−1.44,2.32);
A1=
−0.388 0.155 0.42
−2.848 0.215 3.039
−5.085 −0.889 3.476
,A
2=
−0.232 0.133 −0.051
−1.246 −0.167 0.114
−2.688 −6.848 1.669
,
A3=
0.062 0.641 −0.242
−1.076 2.063 −2.07
−11.16 9.032 −0.596
,A
4=
3.808 −0.562 0.057
−0.129 0.995 0.000
−0.504 0.069 0.99
,B
1=
1.602
3.662
5.345
,
198 S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207
Fig. 6. Neural-based fuzzy modelling (solid line) for a discrete-time articulated vehicle system (dashed line).
A5=
2.674 −0.348 0.029
−0.17 1.004 −0.0003
−0.366 0.042 0.992
,B
2=
1.329
0.391
−23.31
,B
3=
1.162
12.56
128.5
,B
4=
−2.395
−0.017
0.404
,
B5=
−1.433
0.012
0.281
;X(k) =
x1(k)
x2(k)
x3(k)
x4(k)
t
,
which is also c.c. and c.o.
3.2. Optimal fuzzy controlling
Based on the proposed T–S fuzzy model in Eq. (29) for the continuous mass–spring–damper sys-
tem, the fuzzy model in Eq. (31) for continuous Chua’s circuit, the fuzzy model in Eq. (33) for the
discrete-time model-car system and the fuzzy model in Eq. (35) for the discrete-time articulated vehi-
cle system, we can now obtain the corresponding optimal fuzzy controllers, which can achieve global
minimum effect under quadratic performance consideration defined on the entire fuzzy system and fuzzy
controller.
Fig. 7 shows the simulation results for the mass–spring–damper system in Eq. (28) at the initial
conditions, X(0)=(−1,−1)t,(−1,1)t,(1,−1)tand (1,1)t, and the designed local-concept
S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207 199
1.5
1
0.5
0
-0.5
-1
-1.5
1.5
0.5
-0.5
-1
-1.5
velocity
109876543210 time
109876543210 time
109876543210 time
0
1
position
4
3.5
3
2.5
2
1.5
1
0.5
0
-0.5
1
u*(t)
(1,1)
(1,-1)
(-1,-1)
(-1,1)
(1,1)
(1,-1)
(-1,-1)
(-1,1)
Fig. 7. (a) The state responses of the continuous mass–spring–damper system with the designed local-concept optimal controller
at the four initial conditions: X(0)=(−1,−1)t,(−1,1)t,(1,−1)tand (1,1)t; (b) the designed local-concept optimal
controller with X(0)=(−1,−1)t.
optimal controller with X(0)=(−/2,10,0)t.As for the automaton chaotic system, in order to control
the chaotic behavior, the external forces are imposed on the Chua’s circuit in Eq. (30); and hence the
corresponding automaton forced-free fuzzy model in Eq. (31) is then rewritten as the following forced
200 S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207
15
10
5
0
-5
-10
-15
x1
0 50 100 150 200 250
time 0 50 100 150 200 250
time
200 201 202 203 204 205 206 207 208 209 210
time
time
-25
-20
-15
-10
-5
0
5
u*
0 50 100 150 200 250
25
-25
20
15
10
5
0
-5
-10
-20
-15
x3
6
4
2
0
-2
-4
-8
x2
x1 v.s. t x2 v.s. t
x3 v.s. t u* v.s. t
Fig. 8. The state responses of the continuous Chua’s chaotic system with initial conditions X(0)=(0,1,0)t, actuated by the
designed global-concept optimal controller at t=200.
fuzzy model,
Ri:If x1is T1i,then ˙
X(t) =AiX(t) +BiU(t), i =1,...,3,(36)
where
U(t) =
u1(k)
u2(k)
u3(k)
S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207 201
3
2
1
0
-1
-2
-3
x1
010203040506070
time-step
010203040506070
time-step
-10 10 20 30 40 50 60
time-step
x 2 v.s. x3
15
10
5
0
-5
-10
-15
x2
15
10
5
0
-5
-10
-15
x2
x1 v.s. t
x2 v.s. t
(p i/2, -10, 0)
(-p i/2, -10, 0)
(-p i/2, -10, 5)
(p i/2, -10, 5)
(p i/2, -10, 0) (-p i/2, -10, 5)
(-p i/2, -10, 0)
(p i/2, -10, 5)
(-p i/2, -10, 0)
(p i/2, -10, 0)
(p i/2, -10, 5)
(-p i/2, -10, 5)
0
Fig. 9. The state responses and trajectories of the discrete-time model car system with the designed local-concept optimal
controller at the four initial conditions: X(0)=(−/2,10,0)t,(/2,10,5)t,(/2,−10,0)tand (−/2,−10,5)t.
202 S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207
1.2
1
0.8
-0.8
0.6
-0.6
0.4
-0.4
0.2
-0.2
0
x1
0 50 100 150
time-step
0 50 100 150
time-step
0 50 100 150
time-step
-2
-1.5
-1
-0.5
0
0.5
1
x3
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
x2
(-45, -65, 0.59)
(-45, -65, 0.59)
(-45, -65, 0.59)
(-1.61, -116, -1.71)
(-1.61, -116, -1.71)
(-1.61, -116, -1.71)
(12.4, 73.7, 0.59)
(12.4, 73.7, 0.59)
(12.4, 73.7, 0.59)
(5, 55, -1.71)
(5, 55, -1.71)
(5, 55, -1.71)
Fig. 10. The state responses of the discrete-time articulated vehicle system with the designed global-concept optimal controller
at the four initial conditions: X(0)=(−86.1◦,12.4◦,73.7◦,0.59,−0.41)t,(−110◦,−45◦,−65◦,0.59,−0.61)t,
(−118◦,−1.61◦,−116◦,−1.71,−0.41)tand (60◦,5◦,55◦,−1.71,−0.61)t.
S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207 203
1
0.5
0
-0.5
-1
-1.5
1.5
-2
1
x3
0.5
0
-0.5
-1
-1.5
u*(t)
0 50 100 150
time-step
0 50 100 150
time-step
(-45, -65, 0.59)
(-1.61, -116, -1.71)
(12.4, 73.7, 0.59)
(5, 55, -1.71)
Fig. 11. (a) The trajectory of the discrete-time articulated vehicle system with the designed global-concept optimal controller
at the four initial conditions: X(0)=(−86.1◦,12.4◦,73.7◦,0.59,−0.41)t,(−110◦,−45◦,−65◦,0.59,−0.61)t,
(−118◦,−1.61◦,−116◦,−1.71,−0.41)tand (60◦,5◦,55◦,−1.71,−0.61)t; (b) the designed global-concept optimal
controller with X(0)=(−86.1◦,12.4◦,73.7◦,0.59,−0.41)t.
is the imposed external input and Bi(t),i=1,...,3, is chosen as the identity matrix with dimension
3×3. Fig. 8 shows the state responses of the continuous Chua’s chaotic system with initial conditions
X(0)=(0,1,0)t, controlled by the designed global-concept optimal controller applied at t=200.
As for discrete-time system, the steering angle of the model car is restricted to u(k) < /2. Hence, we
assume the controller output for the model car system is u(k) < 1.2. Based on the proposed fuzzy model
and the corresponding local-concept fuzzy controller, we have the simulation results for four initial con-
ditions, X(0)=(−/2,10,0)t,(/2,10,5)t,(/2,−10,0)tand (−/2,−10,5)t, in Fig. 9. Fig. 10
shows the state response of the articulated vehicle closed-loop system controlled by the proposed global-
concept optimal fuzzy controller at initial conditions, X(0)=(−86.1◦,12.4◦,73.7◦,0.59,−0.41)t,
(−110◦,−45◦,−65◦,0.59,−0.61)t,(−118◦,−1.61◦,−116◦,−1.71,−0.41)tand(60◦,5◦,55◦,
−1.71,−0.61)t. Fig. 11 is the trajectory at initial conditions, X(0)=(−86.1◦,12.4◦,73.7◦,0.59,
−0.41)t,(−110◦,−45◦,−65◦,0.59,−0.61)t,(−118◦,−1.61◦,−116◦,−1.71,−0.41)t
204 S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207
Fig. 12. The modelling index IM(t) for local-concept optimal fuzzy controller actuated mass–spring–damper system with initial
condition X(0)=[−1,−1]t, where the dashed line denotes output Y(t)=x(t) and the solid line denotes Y(t) =˙x(t).
and (60◦,5◦,55◦,−1.71,−0.61)t, and the proposed global-concept optimal controller with X(0)=
(−86.1◦,12.4◦,73.7◦,0.59,−0.41)t.
3.3. Performance and stability
The modelling index IMin Eq. (2) not only provides an index to integrate the neural-fuzzy modelling
and controlling for both the model-based and model-free physical system due to the stability properties
mentionedinour previouspaper,but alsoserves asamodelling-error indexfor theneural-fuzzy modelling
process in both transient and infinite-time states. Fig. 12 shows the modelling index evolution of the
proposed neural-learning-based T–S fuzzy model in Eq. (29) for the mass–spring–damper system in
Eq. (28) with initial condition set to be X(0)=[−1,−1]t. The index is nearly coincident with one
except in some trivial points; in other words, the modelling error approaches zero in the large. Hence,
with the INFMC algorithm, we can self-organize a T–S fuzzy model of a physical system under limited
modelling error.
Furthermore, since the proposed neural-based T–S fuzzy model can approximate the real physical very
well, the properties of the closed-loop system (the real physical system compensated with the proposed
optimal fuzzy controller) are the same as those of the closed-loop fuzzy system (the T–S fuzzy model
compensated with the proposed optimal fuzzy controller). For the mass–spring–damper system, since
each fuzzy system in Eqs. (29) is c.c. and c.o., we know the local-concept closed-loop fuzzy systems
and the global-concept closed-loop fuzzy system, and then their corresponding closed-loop real physical
S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207 205
systems are exponentially stable [25–27]. The same properties can be found in Chua’s circuit, model-car
and articulated vehicle closed-loop systems.
4. Conclusions
A neural-learning-based fuzzy inference network, which emphasizes physical system input- and state-
dependence consequences in each fuzzy rule, is proposed to achieve the linearT–S fuzzy modelling. Both
the local-concept and global-concept optimal fuzzy controller design scheme are adopted to stabilize the
nonlinear system. Furthermore, based on the guaranteed stability properties, an INFMC algorithm with
defined modelling index included is proposed to integrate the neural-fuzzy modelling and optimal fuzzy
controlling. Via the proposed INFMC algorithm, the neural-based fuzzy model for a nonlinear system is
ensured; hence, the intrinsic properties of the closed-loop physical system can be captured by those of
the corresponding closed-loop fuzzy system. Two continuous and two discrete-time physical systems are
concerned in implementation of the modified neural-fuzzy structure and the proposed INFMC algorithm.
Simulation results demonstrate that the proposed NFN can self-organize the linear T–S fuzzy models
for those real systems with limited modelling errors and that the proposed neural-based optimal fuzzy
controller can drive the physical systems to desired targets in a short time.
Appendix A.
Proof of Proposition 1. Via the converse theorem, we know the stability of the resultant feedback fuzzy
system concurs with that of the linearized fuzzy system (with respect to Xo)
˙
X∗(t) =
r
i=1
hi(Xo)[Ai−BiBT
ii
∞]X∗(t). (37)
For clarity, we introduce the notation Aci to denote the local feedback system matrix. Then, as we know
each feedback fuzzy subsystem is exponentially stable, which means the spectrum of Aci,i=1,...,r,
denoted by [Aci], is located in the open left-half plane of the complex space, Co
−, i.e., [Aci]⊂C◦
−,
i=1,...,r. Accordingly, we have [hi(Xo)Aci]⊂Co
−,i=1,...,r, via the spectral mapping theorem
and hi(Xo)∈[0,1]for all Xo∈
n. Hence, the zero solution of ˙
X(t) =hi(Xo)Aci X(t) on tt0is
exponentially stable; in other words, there exists constants ai>0 and mi>0 such that for all t0∈
+
ehi(Xo)Aci (t−t0)mie−ai(t−t0),∀tt0,i=1,...,r.
Then, the state transition matrix, (t, t0), of the linearized fuzzy system in Eq. (37) is
(t, t0)=er
i=1hi(Xo)Aci (t−t0)
r
i=1
ehi(Xo)Aci (t−t0)
r
i=1
mie−ai(t−t0)me−a(t−t0),
where m
=r
i=1mi>0 and a
=r
i=1ai>0. Therefore, the linearized fuzzy system and also the
feedback fuzzy system are exponentially stable.
206 S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207
References
[1] S.G. Cao, N.W. Rees, G. Feng, Analysis and design for a class of complex control systems, Part I: Fuzzy modelling and
identification, Automatica 33 (6) (1997) 1017–1028.
[2] S.G. Cao, N.W. Rees, G. Feng, Analysis and design for a class of complex control systems, Part II: Fuzzy controller design,
Automatica 33 (6) (1997) 1029–1039.
[3] C.F. Juang, C.T. Lin, An on-line self-constructing neural fuzzy inference network and its applications, IEEE Trans. Fuzzy
Systems 6 (1) (1998).
[4] T.C. Lin, C.H.Wang, H.L. Liu, Observer-based indirect adaptive fuzzy-neural tracking control for nonlinear SISO systems
using VSS and H∞approach, Fuzzy Sets and Systems 143 (2004) 211–232.
[5] X.J. Ma, Z.Q. Sun,Y.Y. He, Analysis and design of fuzzy controller and fuzzy observer, IEEETrans. Fuzzy Systems 6 (1)
(1998) 41–51.
[6] H. Ohtake, K. Tanaka, A construction method of switching Lyapunov function for nonlinear systems, FUZZ-IEEE’02,
Hawaii, 2002, pp. 221–226.
[7] H. Ohtake, K. Tanaka, H.O. Wang, Fuzzy modelling via sector nonlinearity concept, Proc. IFSA/NAFIPS, Canada, 2001,
pp. 127–132.
[8] S. Su,Y. Liu, Robust H∞control of multiple time-delay uncertain nonlinear system using fuzzy model and adaptive neural
network, Fuzzy Sets and Systems 146 (2004) 403–420.
[9] S. Su, X. Yu, Z. Man, A fuzzy neural network approximator with fast terminal sliding mode and its applications, Fuzzy
Sets and Systems 148 (2004) 469–486.
[10] K. Tanaka, T. Hori, K. Yamafuji, H.O. Wang, An integrated fuzzy control system design for nonlinear systems, in:
Proceedings of the Decision and Control, 1999, pp. 4349–4354.
[11] K. Tanaka, T. Ikeda, H.O.Wang, Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic
stabilizability, H∞control theory and linear matrix inequalities, IEEE Trans. Fuzzy Systems 4 (1) (1996) 1–13.
[12] K. Tanaka, M. Iwazaki, H.O. Wang, Switching control of an R/C hovercraft: stabilization and smooth switching, IEEE
Trans. Fuzzy Systems 31 (2001) 853–863.
[13] K. Tanakan, M. Sano, Trajectory stabilization of a model car via fuzzy control, Fuzzy Sets and Systems 70 (1995)
155–170.
[14] K. Tanaka, T. Taniguchi, S. Hori, H.O. Wang, Structure-oriented design for a class of nonlinear systems, in: Proceedings
of the FUZZ-IEEE’01,Australia, 2001, pp. 696–699.
[15] K. Tanaka, T. Taniguchi, H.O. Wang, Generalized Takagi–Sugeno fuzzy systems: rule reduction and robust control, in:
Proceedings of the FUZZ-IEEE’00, SanAntonio, 2000, pp. 688–693.
[16] K. Tanaka, H.O. Wang, Fuzzy Control Systems Design andAnalysis: A Linear Matrix Inequality Approach,Wiley, New
York, 2001.
[17] T.Taniguchi,K.Tanaka,H.Ohtake,H.O.Wang,Modelconstruction,rule reductionand robustcompensation forgeneralized
form of Takagi–Sugeno fuzzy system, IEEE Trans. Fuzzy Systems 9 (4) (2001) 525–538.
[18] M. Tokunaga, H. Ichihashi, Backer-upper control of a trailer trunck by neuro-fuzzy optimal control, in: Proceedings of the
8th Fuzzy System Symposium (in Japanese), 1992, pp. 49–52.
[19] R.J. Wai, Robust fuzzy neural network control for nonlinear motor-toggle servomechanism, Fuzzy Sets and Systems 139
(2003) 185–208.
[20] H.O. Wang, J. Li, D. Niemann, K. Tanaka,T–S fuzzy model with linear rule consequence and PDC controller: a universal
framework for nonlinear control system, in: Proceedings of the FUZZ-IEEE’00, 2000, pp. 549–554.
[21] H.O. Wang, K. Tanaka,An LMI-based stable fuzzy control of nonlinear systems and its application to control of chaos,
in: Proceedings of the FUZZ-IEEE’96, 1996.
[22] H. Wang, K. Tanaka, M. Griffin, Parallel distributed compensation of nonlinear systems by Takagi and Sugeno’s fuzzy
model, in: Proceedings of the FUZZ-IEEE’95,Yokahama, 1995, pp. 531–538.
[23] H.O. Wang, K. Tanaka, M.F. Griffin,An approach to fuzzy control of nonlinear system: stability and design issues, IEEE
Trans. Fuzzy Systems 4 (1) (1996) 14–23.
[24] W.Y. Wang,Y.G. Leu, T.T. Lee, Output-feedback control of nonlinear systems using direct adaptive fuzzy-neural controller,
Fuzzy Sets and Systems 140 (2003) 341–358.
[25] S.J. Wu, C.T. Lin, Optimal fuzzy controller design: local concept approach, IEEE Trans. on Fuzzy Systems 8 (2) (2000)
171–185.
S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182–207 207
[26] S.J. Wu, C.T. Lin, Optimal fuzzy controller design in continuous fuzzy system: global concept approach, IEEE Trans.
Fuzzy Systems 8 (6) (2000) 713–729.
[27] S.J. Wu, C.T. Lin, Discrete-time optimal fuzzy controller design: global concept approach, IEEE Trans. Fuzzy Systems 10
(1) (2002) 21–38.
[28] X.J. Zeng, M.G. Singh, Approximation theory of fuzzy systems-MIMO case, IEEE Trans. Fuzzy Systems 3 (2) (1995)
219–235.
