International Journal of Approximate Reasoning 6 (00) 9±44 www.elsevier.com/locate/ijar Fuzzy control of a class of multivariable nonlinear systems subject to parameter uncertainties: model reference approach H.K. Lam *, F.H.F. Leung, P.K.S. Tam Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Received February 000; received in revised form August 000; accepted November 000 Abstract This paper presents the fuzzy control of a class of multivariable nonlinear systems subject to parameter uncertainties. The nonlinear plant tacled in this paper is an nthorder nonlinear system with n inputs. If the input matrix B inside the fuzzy plant model is invertible, a fuzzy controller can be designed such that the states of the closed-loop system will follow those of a user-de ned stable reference model despite the presence of parameter uncertainties. A numerical example will be given to show the design procedures and the merits of the proposed fuzzy controller. Ó 00 Elsevier Science Inc. All rights reserved. Keywords: Fuzzy control; Fuzzy plant model; Model reference; Multivariable nonlinear system; Parameter uncertainties; Stability. Introduction Fuzzy control is one of the useful control techniques for uncertain and ill-de ned nonlinear systems. Control actions of the fuzzy controller are described by some linguistic rules. This property maes the control algorithm to be understood easily. The early design of fuzzy controllers is heuristic. It * Corresponding author. E-mail address: hl@eie.polyu.edu.h (H.K. Lam). 0888-63X/0/$ - see front matter Ó 00 Elsevier Science Inc. All rights reserved. PII: S 0 8 8 8-63X(00)00068-
30 H.K. Lam et al. / Internat. J. Approx. Reason. 6 (00) 9±44 incorporates the experience or nowledge of the designer into the rules of the fuzzy controller, which is ne tuned based on trial and error. A fuzzy controller implemented by neural networ was proposed in [3,4]. Through the use of tuning methods, fuzzy rules can be generated automatically. These methodologies mae the design simple; however, the design does not guarantee the system stability, robustness and good performance. To facilitate the formal analysis of fuzzy control systems and the design of fuzzy controllers, Taagi±Sugeno (TS) fuzzy plant model was proposed in [,3]. The TS model represents a nonlinear system as a weighted sum of some linear systems. Based on this structure, fuzzy controllers comprising a number of sub-controllers were proposed. State feedbac controllers were proposed as the sub-controllers in [,5±7,7,8]. The closed-loop system is guaranteed to be asymptotically stable if there exists a common solution for a number of linear matrix inequalities (LMI). Other stability conditions can be found in [0,,3]. The LMI-based design of fuzzy controllers can also be found in [4±6]. The problem will become more complex if the fuzzy plant models are subject to parameter uncertainties. When A i and B i ; which are the systems matrices of the fuzzy plant model, are subject to parameter uncertainties, e.g., DA i and DB i ; robustness analysis can be found in the literature. In [9,0], only the case that A i contains parameter uncertainty DA i was considered. Lyapunov stability theory was employed to nd stability conditions and ranges of the elements of DA i such that the nonlinear plant connected with a fuzzy state feedbac controller is stable. In [5,7,8], both A i and B i were considered to have parameter uncertainties. Stability conditions had been derived based on H theory [5] and by estimating the matrix measures of the system matrices of the fuzzy plant model [7,8]. It can be seen in [5,7±0] that their prime objective and the analysis results were on the system stability; the system performance is not considered at all. In [], a sliding mode controller was proposed as the subcontroller. To compensate the e ect caused by the parameter uncertainties to the system stability, a high gain switching component was used. Adaptive fuzzy controllers were also reported in [8,9] to deal with systems with unnown parameters. The value of each unnown parameter was estimated by the adaptive fuzzy controller. However, by introducing adaptivity to the fuzzy controller, the structure of the controller becomes more complex. As mentioned above, some fuzzy state-feedbac controllers guarantee only the system stability. Although stability is one essential aspect to be considered, other aspects including the system performance, are also important. In order to have a simple fuzzy state feedbac controller which not only guarantees the system stability but also gives good robustness property and system performance, a design methodology will be proposed in this paper. Under the proposed design methodology, the system states of the closed-loop system will follow those of a stable user-de ned reference model. A class of multivariable nonlinear system, an nth-order-n-input nonlinear system subject to parameter
H.K. Lam et al. / Internat. J. Approx. Reason. 6 (00) 9±44 3 uncertainties, will be considered in this paper. This system will be represented by a fuzzy plant model. The input matrix B inside the fuzzy plant model is required to be invertible. This class of systems can be found in the real world. For instance, h r manipulators [], two-wheeled mobile robots [] and twolin robot arms [3] are examples of this class of systems. At the end of this paper, a numerical example will be given to show the design procedure and the merits of the proposed fuzzy controller.. Reference model, fuzzy plant model and fuzzy controller An nth-order-n-input nonlinear plant subject to parameter uncertainties will be considered. This plant is represented by a fuzzy plant model that expresses the plant as a weighted sum of some linear systems with parameter uncertainty information. A fuzzy controller is to be designed to close the feedbac loop such that the system states of the closed-loop system will follow those of a stable reference model... Reference model A reference model is a stable linear system given by, _^x t ˆH m^x t B m r t ; where H m R nn is a constant stable system matrix, B m R nn a constant input matrix, _^x R n the system state vector of this reference model and r t R n is the bounded reference input... Fuzzy plant model with parameter uncertainty information Let p be the number of fuzzy rules describing the uncertain nonlinear plant. The ith rule is of the following form, Rule i : IF x t is M i and... and x n t is M i n THEN _x t ˆ A i DA i x t B i u t ; where M i is a fuzzy set of rule i corresponding to the state x t ; ˆ ; ;...; n; i ˆ ; ;...; p; A i R nn and B i R nn are the nown system and input matrices, respectively; DA i R nn is the parameter uncertainties of A i within nown ranges; x t R n is the system state vector and u t R n is the input vector. The plant dynamics is described by, _x t ˆXp x t A i DA i x t B i u t Š; 3
3 H.K. Lam et al. / Internat. J. Approx. Reason. 6 (00) 9±44 where x t ˆ ; x t 0; Š for all i 4 is a nown nonlinear function of x t and l M i x t ˆ P p jˆ l M j x t l M i x l M i n x n x t l M j x t l M j n x n t ; l M i x t is the grade of membership of the fuzzy set M i..3. Fuzzy controller 5 A fuzzy controller having p fuzzy rules is to be designed for the plant. The jth rule of the fuzzy controller is of the following format: Rule j : IF x t is M i and... and x n t is M i n THEN u t ˆu j t ; 6 where u j t R n ; j ˆ ; ;...; p; is the output of the jth rule controller that will be de ned in the next section. The global output of the fuzzy controller is given by u t ˆXp w j x t u j t : 7 jˆ 3. Design of the fuzzy controller In this section, we describe the design of the fuzzy controller, i.e., u j t for j ˆ ; ;...; p; such that the closed-loop system behaves lie the stable reference model. From (3), (4) and (7), writing x t as, we have, _x t ˆ Xp ˆ ˆ Xp Xp ˆ Xp " A i A i A i DA i x t B i! DA i x t DA i x t B Xp jˆ Xp jˆ w j u j t w j u j t # B i! jˆ w j u j t A i DA i x t Bu i t Š; 8!
H.K. Lam et al. / Internat. J. Approx. Reason. 6 (00) 9±44 33 where, B ˆ Xp B i : Note that B is a nown function of x. From () and (8) and the property that let, x t ˆ ; _e ˆ _x t _^x t ˆXp ˆ Xp ˆ Xp A i DA i x t Bu i t Š H m^x t B m r t A i DA i x t Bu i t Š Xp h A i H m^x t B m r t 9 i DA i x t Bu i t H m^x t B m r t ; 0 where e t ˆx t ^x t is an error vector. We de ne the following Lyapunov function, V ˆ e t T Pe t ; where T denotes the transpose of a vector or matrix, and P R nn is a symmetric positive de nite matrix. Then, by di erentiating (), we have, _V ˆ _e t T Pe t e t T P_e t : From (0) and (), we have, _V ˆ ( h A i e t T P Xp i ) T DA i x t Bu i t H m^x t B m r t Pe t A i DA i x t Bu i t H m^x t B m r t Š: 3 We design u i t ; i ˆ ; ;...; p; as follows, 8! B e t He t A i x t H m^x t B m r t e t P DA i max x t >< u i ˆ if e t 6ˆ0; >: B A i x t H m^x t B m r t if e t ˆ0; 4
34 H.K. Lam et al. / Internat. J. Approx. Reason. 6 (00) 9±44 where denotes the l norm for vectors and l induced norm for matrices, DA i 6 DA i max ; H R nn is a stable matrix to be designed. A bloc diagram of the closed-loop system is shown in Fig.. In (4), it is assumed that B exists (B is nonsingular). In the latter part of this section, we shall provide a way to chec if the assumption is valid. From (3), (4) and assuming that e t 6ˆ0, wehave ( _V ˆ e t T P Xp " #) e t e t He t DA i x t P DA T i max x t Pe t " # e t e t He t DA i x t P DA i max x t ˆ e t T H T P PH e t! Xp e t T PDA i x t e t T Pe t e t PDA i max x t 6 e t T H T P PH e t Xp e t P 6 e t T Qe t Xp DA i x t! e t T Pe t e t PDA i max x t e t P DA i DA i max x t ; 5 where Q ˆ H T P PH is a symmetric positive de nite matrix. As DA i DA i max 6 0, from (5), we have, _V 6 e t T Qe t 6 0: 6 Fig.. A bloc diagram of the closed-loop system.
H.K. Lam et al. / Internat. J. Approx. Reason. 6 (00) 9±44 35 If e t ˆ0; _V ˆ 0: Hence, we can conclude that e t!0 as t!. In the following, we shall derive a su cient condition to chec the existence of B : From (9), and considering the following dynamic system, _z t ˆBz t ˆXp B i z t : 7 If the nonlinear system of (7) is asymptotically stable, it implies that B exists. To ensure the asymptotic stability, consider the following Lyapunov function, V z ˆ z t T P z z t ; 8 where P z R nn is a symmetric positive de nite matrix. Then, by di erentiating (), we have, _V z ˆ _z t T P z z t z t T P z _z t : 9 From (7) and (9), we have, _V z ˆ T 3 X 4 B i z t! p P z z t z t T P z B i z t 5 ˆ z t T B T i P z P z B i z t ˆ z t T Q i z t ; 0 where Q i ˆ B T i P z P z B i : If Q i < 0 for all i ˆ ;...; p; then, from (0), we have, _V ˆ z t T Q i z t 6 0: The nonlinear system of (7) is then asymptotically stable and B exists. On the other hand, if we let B ˆ B and B i ˆ B i ; and consider _z t ˆBz t ˆ P p B i z t. It can be shown that B exists if there exist B T i P z P z B i < 0 for all i ˆ ; ;...; p. The existence of B implies the existence of B : The results of this section can be summarized by the following lemma. Lemma. The fuzzy control system of (8) subject to plant parameter uncertainties is guaranteed to be asymptotically stable, and its states will follow those of a stable reference model of (), if the following two conditions satisfy;
36 H.K. Lam et al. / Internat. J. Approx. Reason. 6 (00) 9±44 () B is nonsingular. One sufficient condition to guarantee the nonsingularity of B is that there exists P z such that, B T i P z P z B i < 0 for all i or B T i P z P z B i < 0 for all i: () The control laws of fuzzy controller of (7) are designed as, 8! B e t He t A i x t H m^x t B m r t e t P DA i max x t >< u i ˆ if e t 6ˆ0; >: B A i x t H m^x t B m r t if e t ˆ0: From the control laws stated in Lemma, it can be seen that the controller in each fuzzy controller rule can be regarded as a sliding mode controller. B A i x t H m^x t B m r t can be viewed as an equivalent control term and So, e t e t PDA i max x t P e t e t P DA i Pe t ˆ e t DA i max x t : e t max x t e t e t PDA i max x t can be viewed as a nonlinear (switching) term used to compensate the parameter uncertainties. Hence, we may call the proposed controller a fuzzy sliding mode controller. The procedure for nding the fuzzy controller can be summarized as follows. Step (I). Obtain the mathematical model of the nonlinear plant to be controlled. Step (II). Obtain the fuzzy plant model for the system stated in step (I) by means of a fuzzy modeling method. Step (III). Chec if there exists B by nding the P z according to Lemma. If P z cannot be found, the design fails. P z can be found by using some existing LMI tools. Step (IV). Choose a stable reference model. Step (IV). Design the fuzzy controller according to Lemma. 4. Numerical example A numerical example is given in this section to illustrate the procedure of nding the fuzzy controller. The fuzzy plant model of a nonlinear plant is assumed to be available, and we start from step (II) of the design procedure.
Step (II). The nonlinear plant can be represented by a fuzzy model with the following fuzzy rules, Rule i : IF x t is M i AND _x t is M i THEN _x t ˆ A i DA i x t B i u t ; i ˆ ; ; 3; 4; where the membership functions of Ma i ; a ˆ ; ; are shown in Figs. and 3, respectively, l M l M H.K. Lam et al. / Internat. J. Approx. Reason. 6 (00) 9±44 37 x t ˆ l M x t ˆ x t :5 ; l M 3 _x t ˆ l M 3 _x t ˆ _x t x t ˆ l M 4 x t ˆ x t :5 ; x t ˆ l M 4 x t ˆ _x t 6:75 ; 6:75 ; l M x t ˆ x t " ˆ x t # 0 ; A ˆ A ˆ x t _x t 0:0 0 0 A 3 ˆ A 4 ˆ ; B ˆ B 3 ˆ 0:35 :4387 0 B ˆ B 4 ˆ ; 0:563 0 0 DA ˆ DA ˆ DA 3 ˆ DA 4 ˆ ; d t d t 3 ; ; Fig.. Membership functions of the fuzzy plant model: l M x ˆl M x ˆ x =:5 (solid line), l M 3 x ˆl M 4 x ˆ x =:5 (dotted line).
38 H.K. Lam et al. / Internat. J. Approx. Reason. 6 (00) 9±44 Fig. 3. Membership functions of the fuzzy plant model: l M _x ˆl M 3 _x ˆ _x =6:75 (solid line), l M _x ˆl M 4 _x ˆ_x =6:75 (dotted line). where d t and d t are the parameter uncertainties. Practically, they are unnown values within given bounds. In this example, they are de ned as timevarying functions to illustrate the robustness of the controller. d t ˆdU dl d t ˆdU dl c L c L du dl du dl cos t so that d t d L ; du Š; cos t so that d t d L ; du Š; 4 5 d L ˆ 0:5; du ˆ 0:5; dl ˆ 0:; du ˆ 0:: Step (III). We choose 39:7945 :695 P z ˆ :695 4:9997 such that Q i ˆ B T i P z P z B i < 0 for i ˆ ; ;...; 4: Hence, we can guarantee the existing of B : Step (IV). The stable reference model is chosen as follows, _^x t ˆH m^x t B m r t ; 6 where H m ˆ 0 ; B m ˆ 0 : 7
H.K. Lam et al. / Internat. J. Approx. Reason. 6 (00) 9±44 39 Step (V). The rules of the fuzzy controller are designed as follows, Rule j : IF x t is M j AND _x t is M j THEN u t ˆu j ; j ˆ ; ; 3; 4; 8 where 8! B e t e t >< He t A i x t H m^x t B m r t PDA max x t u j ˆ if e t 6ˆ0 >: B A i x t H m^x t B m r t if e t ˆ0 for j ˆ ; ; 3; 4. We choose H ˆ 0 4 4 which is a stable matrix, :5000 0:5000 P ˆ 0:5000 :000 and DA max ˆ 0:5099 (from (4) and (5)). Figs. 4 and 5 show the system responses of the fuzzy control system without (solid lines) and with (dash lines) parameter uncertainties, and the reference model (dotted lines) under r t ˆ0; x 0 ˆ :5 0Š T and ^x 0 ˆ 0:5 0Š T. Figs. 6 and 7 show the system responses of the fuzzy control system without (solid lines) and with (dash lines) parameter uncertainties, and the reference model Fig. 4. Responses of x t of the fuzzy control system without (solid line) and with parameter uncertainties (dash line), and the refence model (dotted line) under r t ˆ0; x 0 ˆ :5 0Š T and ^x 0 ˆ 0:5 0Š T :
40 H.K. Lam et al. / Internat. J. Approx. Reason. 6 (00) 9±44 Fig. 5. Responses of x t of the fuzzy control system without (solid line) and with parameter uncertainties (dash line), and the refence model (dotted line) under r t ˆ0; x 0 ˆ :5 0Š T and ^x 0 ˆ 0:5 0Š T : Fig. 6. Responses of x t of the fuzzy control system without (solid line) and with parameter uncertainties (dash line), and the refence model (dotted line) under r t ˆ; x 0 ˆ :5 0Š T and ^x 0 ˆ 0:5 0Š T. (dotted lines) under r t ˆ; x 0 ˆ 00Š T and ^x 0 ˆ 0:5 0Š T. Figs. 8 and 9 show the system responses of the fuzzy control system without (solid lines) and with (dash lines) parameter uncertainties, and the reference model (dotted
H.K. Lam et al. / Internat. J. Approx. Reason. 6 (00) 9±44 4 Fig. 7. Responses of x t of the fuzzy control system without (solid line) and with parameter uncertainties (dash line), and the refence model (dotted line) under r t ˆ; x 0 ˆ :5 0Š T and ^x 0 ˆ 0:5 0Š T. Fig. 8. Responses of x t of the fuzzy control system without (solid line) and with parameter uncertainties (dash line), and the refence model (dotted line) under r t ˆsin 0t ; x 0 ˆ :5 0Š T and ^x 0 ˆ 0:5 0Š T. lines) under r t ˆsin 0t ; x 0 ˆ :5 0Š T and ^x 0 ˆ 0:5 0Š T. It can be seen from the simulation results that the system states of the nonlinear system follow those of the reference model. The responses of the fuzzy control system
4 H.K. Lam et al. / Internat. J. Approx. Reason. 6 (00) 9±44 Fig. 9. Responses of x t of the fuzzy control system without (solid line) and with parameter uncertainties (dash line), and the refence model (dotted line) under r t ˆsin 0t ; x 0 ˆ :5 0Š T and ^x 0 ˆ 0:5 0Š T. with parameter uncertainties are better than that of the fuzzy control system without parameter uncertainties. This is because an additional control signal, i.e., e t e t PDA max x t is used. The reason can also be seen from (5), i.e., _V 6 e t T Qe t e t P DA the term e t P DA DA max x t maes e t approach zero at a faster rate. DA max x t 5. Conclusion Model reference fuzzy control of a class of nth-order n-input nonlinear systems subject to parameter uncertainties has been discussed. A design procedure of the fuzzy controller has been presented. The closed-loop system will
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