Neural-network-based optimal fuzzy controller design for
Transcript Of Neural-network-based optimal fuzzy controller design for
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
A neural-learning fuzzy technique is proposed for T–S fuzzy-model identification of model-free physical systems. 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.: +886 4 8511888x2192; fax: +886 6 2512882.
E-mail addresses: [email protected], [email protected] (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 ways to theoretically construct a T–S fuzzy model. One is from local linear approximation, which generates a linear consequent part with a constant term included in each rule; the other is via sector nonlinearity concept [7,14,15], which results in a constant-free linear consequence for each rule. Both are 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 and Wang 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 adopt H∞ control to compensate the error [8]. Wang and co-workers use type-1 FNN with adaptive update 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 and switching fuzzy controllers. In order to further ensure that the generated fuzzy model can approximate the original physical system and more to control the model-free system well, we propose an integrated algorithm, which integrates the proposed neural fuzzy network and previously proposed nonlinear optimal fuzzy controller, to guarantee the generated fuzzy system can describe the physical system behavior and
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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 augmented 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 inputand state-dependent terms.
We here name the modified NFN to be linear-NFN and Juang’s to be affine-NFN to denote the constructed 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 structure possesses finite weighted fan-in connections to the last-layer nodes and fan-out connections to 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 − inputki = f (uk1i , uk2i , . . . , ukpi , w1ki , w2ki , . . . , wpki ),
S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182 – 207
185
Fig. 1. linear-SONFIN structure.
where
u
k 1
i
,
uk2
i
,
.
.
.
,
ukp
i
are
the
inputs
of
the
ith
node
and
w1ki , w2ki , . . . , wpk i
are
the
associated
weights.
The output operation oik is then proceeded by an activation function a(·),
oik = a(net − inputki ).
We now briefly describe the proposed six-layer NFN structure as follows.
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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 = u1i , a1 = f,
where the linking weight wi1 is 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 (u2 ) = − (u2i − mij )2 ,
ij
2
ij
a2(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 u2i .
Layer 3: The fuzzily blending operation is performed in this layer and hence each node represents one fuzzy logic rule; that is,
n
f (u3i ) = u3i = e−[Di (x−mi )]t [Di (x−mi )],
i=1
a3(f ) = f,
where n is the number of Layer-2 nodes joining with the ith rule precondition, Di = diag(1/ i1, 1/ i2, . . . , 1/ in) and mi = (mi1, mi2, . . . , min). 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 (u4i ) = u4i ,
i
a4(f ) = u4i , 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 = aji xj + bmi um,
j
m
a5(f ) = f · u5i .
Layer 6: Each node in this layer is correspondent to one system output variable. This layer is to integrate the actions from Layer 5, and hence to perform the defuzzifier operation for the fuzzy logic system. In other words,
f (u6i ) = u6i ,
i
a6(f ) = f.
S.-J. Wu et al. / Fuzzy Sets and Systems 154 (2005) 182 – 207
187
Hence, the input variables xj are 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 linear T–S fuzzy modelling structure. Via neural learning technique, this structure will proceed the structure and parameter learning concurrently and generate the following linear T–S fuzzy system:
Ri : If x1 is T1i(m1i, 1i), . . . , xn is Tni(mni, ni), then Y (t) = CX(t)
SX(t) = AiX(t) + Biu(t), i = 1, . . . , r,
(1)
where Ri denotes the ith rule of the fuzzy model; x1, . . . , xn are system states; Tji(mji, ji), j = 1, . . . , n, is the fuzzy term of the input fuzzy variable xj in 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 and X(t +1) for the discrete case; X(t) = [x1, . . . , xn]t ∈ n is the state vector, Y (t) = [y1, . . . , yn ]t ∈
n is the system output vector, and u(t) ∈ m is the system input (i.e., control output); and Ai, Bi and C are, respectively, n × n, n × m and n × n matrices.
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, the least mean square algorithm is adopted to adjust the parameters in the rule consequence, 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 stabilize the corresponding linear T–S fuzzy system once each fuzzy subsystem is completely 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 neurallearning-based T–S fuzzy system is consistent with the real nonlinear system. For measuring the modelling
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error, we define a modelling index as
IM(t ) = YLcslonfin(t ) + ,
(2)
Y cl(t) +
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 system very well. Therefore, we further integrate the neural-fuzzy modelling process and the optimal 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 y1 is S1i, . . . , yn is Sn i, then u(t) = ri(t), i = 1, . . . , ,
(3)
where y1, . . . , yn are the elements of output vector Y (t), S1i, . . . , Sn i are 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 m space. 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 ∈ n and a rule-based fuzzy controller in Eq. (3), find the individual optimal control law, ri∗(·), i = 1, . . . , , such that the composed optimal controller, u∗(·), can minimize the quadratic cost functional, J (u(·)), over all possible inputs
u(·).
∞
J (u(·)) = [Xt (t)L(t)X(t) + ut (t)u(t)] dt (continuous),
(4)
t0 ∞
J (u(·)) = [Xt (t)L(t)X(t) + ut (t)u(t)] (discrete-time),
(5)
t =t0
where Xt (t)L(t)X(t) is state-trajectory penalty with L(t) belonging to a symmetric positive semi-definite n × n matrix and ut (t)u(t) is fuel consumption.
For the local approach, we first adopt the principles of dynamic programming to transform the quadratic
optimization problem into a successively ongoing dynamic problem with regard to the state resulting from
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 ) = (Xlt LlXl + utl ul) dl, t ∈ [t0, ∞) (continuous),
(6)
t
∞
Jt (ut ) = [Xlt LlXl + utl ul], t ∈ [t0, ∞) (discrete)
(7)
t
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189
Fig. 2. Integrated neural-fuzzy modelling and controlling (INFMC) algorithm.
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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,
X˙ l = Ail Xl + Bil ril , l ∈ [t, ∞), i = 1, . . . , r (continuous),
(8)
Xl+1 = Ail Xl + Bil ril , l ∈ [t, ∞), i = 1, . . . , r (discrete)
(9)
with the initial state resulting from the previous decision, i.e., X0t = Xt∗, (1) find the optimal local decision at time instant t, ri∗t , for minimizing the cost functional,
∞
Jt (rit ) = (Xlt LlXl + ritl ril ) dl, t ∈ [t0, ∞) (continuous),
(10)
t
∞
Jt (rit ) = (Xlt LlXl + ritl ril ], t ∈ [t0, ∞) (discrete);
(11)
t
(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=1
hi
(Xt∗
)ri∗t
.
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, Bi, C, L be given constant matrices.
If (Ai, Bi) is c.c. and (Ai, C) is c.o. for i = 1, . . . , r, then
(1) there exists a unique n × n symmetric positive semi-definite solution
i ∞
of
the
steady-state
Riccati
equation (SSRE)
AtiK + KAi − KBiBit K + L = 0;
(12)
(2) the asymptotically local optimal fuzzy control law is
ri∗(t) = −Bit i∞X∗(t),
i = 1, . . . , r,
(13)
and their “blending” global minimizer u∗(t) =
n i=1
hi
(X∗)ri∗(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
n
X˙ ∗(t) = hi(X∗)(Ai − BiBit i∞)X∗(t)
(14)
i=1
is exponentially stable.
Proof. From the inference in the above, we can get local optimal fuzzy control law ri∗(t) in Eq. (13) and the local feedback fuzzy subsystem, X˙ ∗(t) = (Ai − BiBit i∞)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, Bi, C, L be given constant matrices. If (Ai, Bi) 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 + AtiV (∞)[In + BiBit V (∞)]−1Ai,
(15)
V (∞) = L + AtiV (∞)Ai − AtiV (∞)Bi[In + Bit V (∞)Bi]−1Bit V (∞)Ai;
(16)
(2) the asymptotically local optimal fuzzy control law is
ri∗(t) = −[In + Bit i(∞)Bi]−1Bit i(∞)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 + BiBit i(∞)]−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 rulebased 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],
A1 (t )
B1 (t )
r1 (t )
A(t) = ... , B(t) = ... , R(t) = ...
Ar (t)
Br (t)
r (t)
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
A neural-learning fuzzy technique is proposed for T–S fuzzy-model identification of model-free physical systems. 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.: +886 4 8511888x2192; fax: +886 6 2512882.
E-mail addresses: [email protected], [email protected] (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 ways to theoretically construct a T–S fuzzy model. One is from local linear approximation, which generates a linear consequent part with a constant term included in each rule; the other is via sector nonlinearity concept [7,14,15], which results in a constant-free linear consequence for each rule. Both are 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 and Wang 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 adopt H∞ control to compensate the error [8]. Wang and co-workers use type-1 FNN with adaptive update 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 and switching fuzzy controllers. In order to further ensure that the generated fuzzy model can approximate the original physical system and more to control the model-free system well, we propose an integrated algorithm, which integrates the proposed neural fuzzy network and previously proposed nonlinear optimal 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 augmented 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 inputand state-dependent terms.
We here name the modified NFN to be linear-NFN and Juang’s to be affine-NFN to denote the constructed 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 structure possesses finite weighted fan-in connections to the last-layer nodes and fan-out connections to 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 − inputki = f (uk1i , uk2i , . . . , ukpi , w1ki , w2ki , . . . , wpki ),
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185
Fig. 1. linear-SONFIN structure.
where
u
k 1
i
,
uk2
i
,
.
.
.
,
ukp
i
are
the
inputs
of
the
ith
node
and
w1ki , w2ki , . . . , wpk i
are
the
associated
weights.
The output operation oik is then proceeded by an activation function a(·),
oik = a(net − inputki ).
We now briefly describe the proposed six-layer NFN structure as follows.
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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 = u1i , a1 = f,
where the linking weight wi1 is 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 (u2 ) = − (u2i − mij )2 ,
ij
2
ij
a2(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 u2i .
Layer 3: The fuzzily blending operation is performed in this layer and hence each node represents one fuzzy logic rule; that is,
n
f (u3i ) = u3i = e−[Di (x−mi )]t [Di (x−mi )],
i=1
a3(f ) = f,
where n is the number of Layer-2 nodes joining with the ith rule precondition, Di = diag(1/ i1, 1/ i2, . . . , 1/ in) and mi = (mi1, mi2, . . . , min). 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 (u4i ) = u4i ,
i
a4(f ) = u4i , 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 = aji xj + bmi um,
j
m
a5(f ) = f · u5i .
Layer 6: Each node in this layer is correspondent to one system output variable. This layer is to integrate the actions from Layer 5, and hence to perform the defuzzifier operation for the fuzzy logic system. In other words,
f (u6i ) = u6i ,
i
a6(f ) = f.
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187
Hence, the input variables xj are 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 linear T–S fuzzy modelling structure. Via neural learning technique, this structure will proceed the structure and parameter learning concurrently and generate the following linear T–S fuzzy system:
Ri : If x1 is T1i(m1i, 1i), . . . , xn is Tni(mni, ni), then Y (t) = CX(t)
SX(t) = AiX(t) + Biu(t), i = 1, . . . , r,
(1)
where Ri denotes the ith rule of the fuzzy model; x1, . . . , xn are system states; Tji(mji, ji), j = 1, . . . , n, is the fuzzy term of the input fuzzy variable xj in 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 and X(t +1) for the discrete case; X(t) = [x1, . . . , xn]t ∈ n is the state vector, Y (t) = [y1, . . . , yn ]t ∈
n is the system output vector, and u(t) ∈ m is the system input (i.e., control output); and Ai, Bi and C are, respectively, n × n, n × m and n × n matrices.
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, the least mean square algorithm is adopted to adjust the parameters in the rule consequence, 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 stabilize the corresponding linear T–S fuzzy system once each fuzzy subsystem is completely 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 neurallearning-based T–S fuzzy system is consistent with the real nonlinear system. For measuring the modelling
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error, we define a modelling index as
IM(t ) = YLcslonfin(t ) + ,
(2)
Y cl(t) +
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 system very well. Therefore, we further integrate the neural-fuzzy modelling process and the optimal 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 y1 is S1i, . . . , yn is Sn i, then u(t) = ri(t), i = 1, . . . , ,
(3)
where y1, . . . , yn are the elements of output vector Y (t), S1i, . . . , Sn i are 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 m space. 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 ∈ n and a rule-based fuzzy controller in Eq. (3), find the individual optimal control law, ri∗(·), i = 1, . . . , , such that the composed optimal controller, u∗(·), can minimize the quadratic cost functional, J (u(·)), over all possible inputs
u(·).
∞
J (u(·)) = [Xt (t)L(t)X(t) + ut (t)u(t)] dt (continuous),
(4)
t0 ∞
J (u(·)) = [Xt (t)L(t)X(t) + ut (t)u(t)] (discrete-time),
(5)
t =t0
where Xt (t)L(t)X(t) is state-trajectory penalty with L(t) belonging to a symmetric positive semi-definite n × n matrix and ut (t)u(t) is fuel consumption.
For the local approach, we first adopt the principles of dynamic programming to transform the quadratic
optimization problem into a successively ongoing dynamic problem with regard to the state resulting from
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 ) = (Xlt LlXl + utl ul) dl, t ∈ [t0, ∞) (continuous),
(6)
t
∞
Jt (ut ) = [Xlt LlXl + utl ul], t ∈ [t0, ∞) (discrete)
(7)
t
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189
Fig. 2. Integrated neural-fuzzy modelling and controlling (INFMC) algorithm.
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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,
X˙ l = Ail Xl + Bil ril , l ∈ [t, ∞), i = 1, . . . , r (continuous),
(8)
Xl+1 = Ail Xl + Bil ril , l ∈ [t, ∞), i = 1, . . . , r (discrete)
(9)
with the initial state resulting from the previous decision, i.e., X0t = Xt∗, (1) find the optimal local decision at time instant t, ri∗t , for minimizing the cost functional,
∞
Jt (rit ) = (Xlt LlXl + ritl ril ) dl, t ∈ [t0, ∞) (continuous),
(10)
t
∞
Jt (rit ) = (Xlt LlXl + ritl ril ], t ∈ [t0, ∞) (discrete);
(11)
t
(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=1
hi
(Xt∗
)ri∗t
.
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, Bi, C, L be given constant matrices.
If (Ai, Bi) is c.c. and (Ai, C) is c.o. for i = 1, . . . , r, then
(1) there exists a unique n × n symmetric positive semi-definite solution
i ∞
of
the
steady-state
Riccati
equation (SSRE)
AtiK + KAi − KBiBit K + L = 0;
(12)
(2) the asymptotically local optimal fuzzy control law is
ri∗(t) = −Bit i∞X∗(t),
i = 1, . . . , r,
(13)
and their “blending” global minimizer u∗(t) =
n i=1
hi
(X∗)ri∗(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
n
X˙ ∗(t) = hi(X∗)(Ai − BiBit i∞)X∗(t)
(14)
i=1
is exponentially stable.
Proof. From the inference in the above, we can get local optimal fuzzy control law ri∗(t) in Eq. (13) and the local feedback fuzzy subsystem, X˙ ∗(t) = (Ai − BiBit i∞)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, Bi, C, L be given constant matrices. If (Ai, Bi) 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 + AtiV (∞)[In + BiBit V (∞)]−1Ai,
(15)
V (∞) = L + AtiV (∞)Ai − AtiV (∞)Bi[In + Bit V (∞)Bi]−1Bit V (∞)Ai;
(16)
(2) the asymptotically local optimal fuzzy control law is
ri∗(t) = −[In + Bit i(∞)Bi]−1Bit i(∞)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 + BiBit i(∞)]−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 rulebased 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],
A1 (t )
B1 (t )
r1 (t )
A(t) = ... , B(t) = ... , R(t) = ...
Ar (t)
Br (t)
r (t)