www.elsevier.com/locate/fss
Neural-network-based optimal fuzzy controller design for
nonlinear systems
Shinq-Jen Wu
a,b,∗, Hsin-Han Chiang
b, Han-Tsung Lin
b, Tsu-Tian Lee
baDepartment 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 ofthe integrated algorithm ofneural-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 ofElectrical Engineering, Da-Yeh University, Chang-Hwa, Taiwan, ROC. Tel.:
+886 4 8511888x2192; fax: +886 6 2512882.
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
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 offuzzy structures are, respectively, denoted as linear and affine T–S fuzzy systems by Tanaka and Wang [20]. It is noticed that the consequent part ofeach 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 ofeach 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 adoptH∞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 ofthem 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
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 ofeach nonlinear term to achieve global or semi-global effect. Accordingly, two kinds ofT–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 ofaffine type, but the input- and state-dependent terms are now the key terms in those oflinear 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
Fig. 1. linear-SONFIN structure.
whereuk1i, uk2i, . . . , ukpi are the inputs ofthe ith node andw1ki, wk2i, . . . , wpik are the associated weights. The output operationoki is then proceeded by an activation functiona(·),
oki = a(net − inputki).
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, a1 = f,
where the linking weightw1i 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 ij) = −(u 2 i − mij)2 2 ij , a2(f ) = ef,
wheremij andij are the mean and the standard deviation ofthe Gaussian membership function ofthe
jth term ofthe ith input variableu2i.
Layer 3: The fuzzily blending operation is performed in this layer and hence each node represents one
fuzzy logic rule; that is,
f (u3 i) = n i=1 u3 i = e−[Di(x−mi)]t[Di(x−mi)], a3(f ) = f,
where n is the number ofLayer-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 (u4 i) = i u4 i, a4(f ) = u 4 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 ofthe 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 ofthe 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 bmium, 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 (u6 i) = i u6 i, a6(f ) = f.
Hence, the input variablesxj are fuzzified as fuzzy variables whose corresponding term setsTjihave Gaussian membership function with meanmjiand standard deviationji; 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 x1isT1i(m1i,1i), . . . , xnisTni(mni,ni), then Y (t) = CX(t)
SX(t) = AiX(t) + Biu(t), i = 1, . . . , r, (1)
where Ri denotes the ith rule ofthe 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 withmji andjibeing the mean and standard deviation ofthe Gaussian membership function;SX(t) denotes ˙X(t) for the continuous case andX(t +1) for the discrete case; X(t) = [x1, . . . , xn]t ∈ nis the state vector,Y (t) = [y1, . . . , yn ]t ∈
n
is the system output vector, andu(t) ∈ mis the system input (i.e., control output); andAi,Bi and
C are, respectively,n × n, n × m and n × n matrices.
Structure learning includes both precondition and consequence identification ofa fuzzy IF–THEN rule. Precondition identification (input-space partition) is formulated as the combinational optimization problem to minimize the number ofgenerated rules and the number offuzzy 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 ofeach 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 controlleru∗(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 neural-learning-based T–S fuzzy system is consistent with the real nonlinear system. For measuring the modelling
error, we define a modelling index as
IM(t) = Y cl
Lsonfin(t) +
Ycl(t) + , (2)
whereYLsonfin(t) is the output ofthe proposed neural-learning-based T–S fuzzy closed-loop system and
Y (t) is the output ofthe real physical closed-loop system; is a small constant to ensure a nonzero denominator. Accordingly to the stability ofthe 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 ofthis 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 isS1i, . . . , yn isSn i, then u(t) = ri(t), i = 1, . . . ,, (3)
wherey1, . . . , yn are the elements ofoutput vectorY (t), S1i, . . . , Sn iare the input fuzzy terms in the
ith control rule, and the plant input (i.e., control output) vectoru(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) withX(t0) = X0 ∈ nand 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(·)) = ∞ t0 [X t(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-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 ofdynamic 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-stept, ifwe can find the optimal local decision (optimal control law) for minimizing
Jt(ut) = ∞ t (X t lLlXl + utlul) dl, t ∈ [t0, ∞) (continuous), (6) Jt(ut) = ∞ t [XltLlXl+ utlul], t ∈ [t0, ∞) (discrete) (7)
with regard to a fuzzy subsystem, the composed global decision can be a global minimizer oftotal cost with regard to a fuzzy system. In other words, based on the local viewpoint ofthe 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 = Xt∗,
(1) find the optimal local decision at time instant t,rit∗, for minimizing the cost functional,
Jt(rit) = ∞ t (X t lLlXl + ritlril) dl, t ∈ [t0, ∞) (continuous), (10) Jt(rit) = ∞ t (XltLlXl+ riltril], t ∈ [t0, ∞) (discrete); (11)
(2) obtain the optimal global decision at time instant t,u∗t, for minimizing the cost functionalJt(ut) in Eqs. (6) and (7), by fuzzily blending each local decision, i.e.,u∗t =ri=1hi(X∗t)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), letAi, 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 uniquen × n symmetric positive semi-definite solutioni∞of the steady-state Riccati equation (SSRE)
AtiK + KAi− KBiBitK + L = 0; (12)
(2) the asymptotically local optimal fuzzy control law is
ri∗(t) = −Biti∞X∗(t), i = 1, . . . , r, (13)
(3) and the optimal global feedback fuzzy subsystem
˙X∗(t) =n
i=1
hi(X∗)(Ai− BiBiti∞)X∗(t) (14)
is exponentially stable.
Proof. From the inference in the above, we can get local optimal fuzzy control lawri∗(t) in Eq. (13) and the local feedback fuzzy subsystem, ˙X∗(t) = (Ai − BiBiti∞)X∗(t), is exponentially stable. We demonstrate the stability ofthe 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, Wuand 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 solutioni(∞) of the following algebraic
SSRE,
V (∞) = L + AtiV (∞)[In+ BiBitV (∞)]−1Ai, (15)
V (∞) = L + AitV (∞)Ai− AtiV (∞)Bi[In+ BitV (∞)Bi]−1BitV (∞)Ai; (16) (2) the asymptotically local optimal fuzzy control law is
ri∗(t) = −[In+ Biti(∞)Bi]−1Biti(∞)AiX∗(t), t = t0, . . . , N − 1, (17)
and the resultant global controlleru∗(t) minimizes J (u(·)) in Eq. (5); (3) moreover, the optimal local feedback fuzzy subsystem,
X∗(t + 1) = [In+ BiBiti(∞)]−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 ofa 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)
whereH (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)
with In andIm denoting the identity matrices ofdimensionn and 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 ofa quadratic optimal fuzzy control problem is achieved by fuzzily merging distributed rule-based T–S type fuzzy subsystems into an entire fuzzy 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, andwi(Y (k)), i = 1, . . . ,) into synthetical matrices (A(k), B(k), H (X(k)) and W(Y (k))) generates a linear-like global system representation ofa fuzzy system with the value ofeach element ofthe 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 ofa global optimal fuzzy controller in the way ofgeneral 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 fromN segmental nonlinear TPBVP instead ofthe nonlinear TPBVP for the entire horizon. This decomposition operation can speed up numerical solution and keep the global optimality at the same time. Furthermore, ¯N denotes the number ofstages 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 ofthe collocation method at the expense ofapproximate 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
in Eq. (1) and fuzzy controller in Eqs. (3); ifN > ¯N, (Ai, Bi) is c.c. and (Ai, C ) is c.o., for all i = 1, . . . , r,
then
(X∞∗ (t), R∞∗ (t)) = (Xi∞∗(t), R∞i∗(t)), ∀t ∈ [t0i, t1i], t01 = t0, t1N = ∞, i = 1, . . . , N, (20)
whereR∞i∗(t) is the ith-stage asymptotically optimal control law,
R∞i∗(t) = −Wit[WiWit]−1BtHiti∞Xi∞∗(t), t ∈ [t0i, ∞), (21)
which minimizesJ∞i (R(·)) = t∞i
0 [X
t(t)LX(t) + Rt(t)Wt
iWiR(t)] dt, and X∞i∗(t) is the corresponding asymptotically optimal trajectory that satisfies
˙Xi∗
∞(t) = (HiA − HiBBtHiti∞)Xi∞∗(t), t ∈ [t0i, ∞), (22)
wherei∞is the unique symmetric positive semidefinite solution of the SSRE,
AtHitK + KHiA − KHiBBtHitK + CtC = 0. (23)
Proposition 4 (Global-concept discrete-time, Wuand Lin[27]). Consider the time-invariant fuzzy
that ifN > ¯N, (Ai, Bi) is c.c. and (Ai, C ) is c.o., i = 1, . . . , r, then, for each stage, (X∞∗ (k), R∗∞(k)) = Xi∗
∞(k), R∞i∗(k)), ∀k ∈ [ki0, k1i−1], k 1
0 = k0, kN1 = ∞, where the ith-stage asymptotically optimal control
law, R∞i∗(k) = −Wit[WiWit]−1BtHiti∞[In+ HiBBtHiti∞]−1HiAXi∞∗(k), k ∈ [k0i, ∞), (24) which minimizes J∞i (R(·)) = ∞k=ki 0[X t(k)LX(k) + Rt(k)Wt iWiR(k)]; X∞i∗(k) is the corresponding asymptotically optimal trajectory,
X∞i∗(k + 1) = [In+ HiBBtHiti∞]−1HiAXi∞∗(k), k ∈ [ki0∞), (25)
wherei∞is the unique symmetric positive semidefinite solution of the discrete-time algebraic Riccati-like equation,
K = L + AtHitK[In+ HiBBtHitK]−1HiA, (26)
K = L + AtHitKHiA − AtHitKHiB[In+ BtHitKHiB]−1BtHitKHiA. (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 modelling and optimal fuzzy controlling process, and more to guarantee that the proposed neural-learning-based T–S fuzzy models can approximate the original physical systems very well. The neural-fuzzy-neural-learning-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)
wherex ∈ [−1.5 1.5] and ˙x ∈ [−1.5 1.5].
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
Fig. 3. Neural-based fuzzy modelling (solid line) for a continuous mass–spring–damper system (dashed line).
as follows:
Ri : If x1isT1i(m1i,1i) and x2isT2i(m2i,2i),
then ˙X(t) = AiX(t) + Biu(t), i = 1, . . . , 5, (29) where fuzzy term setsT11(−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 1 0 , A2= 0.0014 1.3836 1 0 A3= 0 −0.1848 1 0 , A4= 0 −2.7786 1 0 , A5 = −0.741 −1.5384 1 0 , 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.
We next consider a more complex continuous nonlinear chaotic system, Chua’s circuit, which is an electronic system with one inductor (L), two capacitors (C1,C2), one linear resistor (R) and one piecewise
linear or nonlinear resistor (g) included. The dynamic behavior ofChua’s circuit can be described as[21]
˙vC1=C1 1 1 R(vC2− vC1− g(vC1)) , ˙vC2=C1 2 1 R(vC1− vC2) + iL , ˙iL=L1(−vC2− R0iL), (30)
wherevC1andvC2are the voltage ofcapacitors andiLis the instant current ofthe inductor; the nonlinear resistor is characterized asg(vC1) = GbvC1 + 21(Ga − Gb)(|vC1+ E| − |vC1 − E|) with parameters
Ga, Gb < 0. We denote the state variable X = [vC1,vC2,iL]t and chooseR = 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 x1isT1i(m1i,1i), then ˙X(t) = AiX(t), i = 1, . . . , 4, (31)
where the fuzzy term setsT11(−0.105, 1.556), T12(7.45, 5.525), T13(0.031, 5.004), T14(−8.939, 7.622); A1= 680.35 −0.35 0.5.85 7 0 0 −7 0 , A2= −2.3950.35 −0.35 0.57 0 0 −7 0 , A3= 120.35 −0.35 0.5.06 7 0 0 −7 0 , A4= −2.5080.35 −0.35 0.57 0 0 −7 0 .
We now step for fuzzy modelling the discrete-time nonlinear systems, the car model system[13],
x1(k + 1) = x1(k) + vt
l tan(u(k)), x2(k + 1) = x2(k) + vt sin(x1(k)),
x3(k + 1) = x3(k) + vt cos(x1(k)), (32)
wherex1(k), x2(k) and x3(k) are, respectively, the angle ofthe car, the vertical and horizontal position
ofthe rear end ofthe car;u(k) is the steering angle, l is the length ofthe car, t is the sampling time and
v is the constant speed. The parameters were chosen as l = 2.8 m, v = 1.0 m/s and t = 1.0 s. 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)
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= 1 0 1.01 0 , A2 = 1 0 0.969 1 A3 = 1 0 0.974 1 , A4 = 1 0 0.686 1 , A5 = 1 0 0.672 1 , B1 = 0.377 −0.003 , B2= 0.39 0.01 , B3= 0.388 0.011 , B4 = 0.394 −0.104 , B5= 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 l tan(u(k)), x2(k) = x1(k) − x3(k), x3(k + 1) = x3(k) + vt L sin(x2(k)), x4(k + 1) = x4(k) + vt cos(x2(k)) sin x 3(k + 1) + x3(k) 2 , x5(k + 1) = x5(k) + vt cos(x2(k)) cos x 3(k + 1) + x3(k) 2 , (34)
whereu(k) is the steering angle; x1(k), x2(k), x3(k), x4(k) and x5(k) are the angle oftruck, the angle
difference between truck and trailer, and the angle oftrailer, the vertical and horizontal position ofthe rear end ofthe trailer, respectively. We setl = 0.2 m, L = 0.32 m, v = −0.1 m/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) , x3(k) is T3i(m3i,3i) and x4(k) is T4i(m4i,4i),
thenX(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) T43(−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 = −0.388−2.848 00.155.215 30.039.42 −5.085 −0.889 3.476 , A2 = −0.232−1.246 −0.1670.133 −0.0510.114 −2.688 −6.848 1.669 , A3 = −1.076 2.063 −2.070.062 0.641 −0.242 −11.16 9.032 −0.596 , A4 = −0.1293.808 −0.562 0.0570.995 0.000 −0.504 0.069 0.99 , B1= 13.602.662 5.345 ,
Fig. 6. Neural-based fuzzy modelling (solid line) for a discrete-time articulated vehicle system (dashed line). A5 = −0.172.674 −0.3481.004 −0.00030.029 −0.366 0.042 0.992 , B2 = 10.329.391 −23.31 , B3= 112.162.56 128.5 , B4 = −2.395−0.017 0.404 , B5 = −1.4330.012 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)t and (1, 1)t, and the designed local-concept
1.5 1 0.5 0 -0.5 -1 -1.5 1.5 0.5 -0.5 -1 -1.5 v elocity 10 9 8 7 6 5 4 3 2 1 0 time 10 9 8 7 6 5 4 3 2 1 0 time 10 9 8 7 6 5 4 3 2 1 0 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 ofthe 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)t and(1, 1)t; (b) the designed local-concept optimal controller withX(0) = (−1, − 1)t.
optimal controller withX(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
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 ofthe continuous Chua’s chaotic system with initial conditionsX(0) = (0, 1, 0)t, actuated by the designed global-concept optimal controller att = 200.
fuzzy model, Ri : If x1isT1i, then ˙X(t) = AiX(t) + BiU(t), i = 1, . . . , 3, (36) where U(t) = uu12(k)(k) u3(k)
3 2 1 0 -1 -2 -3 x1 0 10 20 30 40 50 60 70 time-step 0 10 20 30 40 50 60 70 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 ofthe 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.
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 ofthe 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,
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 ofthe 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 withX(0) = (−86.1◦, 12.4◦, 73.7◦, 0.59, − 0.41)t.
is the imposed external input andBi(t), i = 1, . . . , 3, is chosen as the identity matrix with dimension 3× 3. Fig. 8 shows the state responses ofthe continuous Chua’s chaotic system with initial conditions
X(0) = (0, 1, 0)t, controlled by the designed global-concept optimal controller applied att = 200. As for discrete-time system, the steering angle of the model car is restricted tou(k) </2. Hence, we assume the controller output for the model car system isu(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 ofthe 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,
Fig. 12. The modelling indexIM(t) for local-concept optimal fuzzy controller actuated mass–spring–damper system with initial conditionX(0) = [−1, − 1]t, where the dashed line denotes outputY (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 withX(0) =
(−86.1◦, 12.4◦, 73.7◦, 0.59, − 0.41)t.
3.3. Performance and stability
The modelling indexIM in 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 mentioned in our previous paper, but also serves as a modelling-error index for the neural-fuzzy modelling process in both transient and infinite-time states. Fig. 12 shows the modelling index evolution ofthe 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 ofthe 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
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 linear T–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 ofthe 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 ofthe 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.
AppendixA.
Proof of Proposition 1. Via the converse theorem, we know the stability of the resultant feedback fuzzy
system concurs with that ofthe linearized fuzzy system (with respect toXo)
˙X∗(t) =r
i=1
hi(Xo)[Ai− BiBiTi∞]X∗(t). (37)
For clarity, we introduce the notationAcito denote the local feedback system matrix. Then, as we know each feedback fuzzy subsystem is exponentially stable, which means the spectrum ofAci,i = 1, . . . , r, denoted by[Aci], is located in the open left-halfplane ofthe 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
andhi(Xo) ∈ [0, 1] for all Xo ∈ n. Hence, the zero solution of ˙X(t) = hi(Xo)AciX(t) on tt0 is
exponentially stable; in other words, there exists constantsai > 0 and mi> 0 such that for all t0∈ +
ehi(Xo)Aci(t−t0) m
ie−ai(t−t0), ∀tt
0, i = 1, . . . , r.
Then, the state transition matrix,(t, t0), ofthe linearized fuzzy system in Eq. (37) is
(t, t0) = e r 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),
wherem = ri=1mi > 0 and a = ri=1ai > 0. Therefore, the linearized fuzzy system and also the feedback fuzzy system are exponentially stable.
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![Fig. 12. The modelling index I M (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).](https://thumb-ap.123doks.com/thumbv2/9libinfo/7688220.142905/23.816.184.633.129.506/modelling-optimal-controller-actuated-initial-condition-denotes-denotes.webp)
