A Robust Tracking Control for Chaotic Chua s Circuits via Fuzzy Approach

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1 IEEE RANSACIONS ON CIRCUIS AND SYSEMS I: UNDAMENAL HEORY AND APPLICAIONS, VOL. 48, NO. 7, JULY [] A. Brambilla and D. D Amore, Energy-based control of numerical errors in time-domain simulation of dynamic circuits, IEEE rans. Circuits Syst., to be published. [2] A. Brambilla, D. D Amore, and E. Dallago, A new numerical method for steady-state circuit analysis, in Proc. th Eur. Conf. Circuit heory and Design, Davos, Switzerland, Aug. Sept., 99, pp [] A. Brambilla, D. D Amore, and M. Santomauro, Simulation of autonomous circuits in the time domain, in Proc. ECCD 95, Istanbul, urkey, Aug. 28 Sept., 995, pp [4] R. elichevesky, K. Kundert, I. Elfadel, and J. White, ast simulation algorithms for R circuits, in Proc. CICC 96, May 996, pp [5] K. Kundert, Simulation methods for R integrated circuits, in Proc. ICCAD 97, Nov. 997, pp. 4. [6] H. Jokinen, Computation of the steady-state solution of nonlinear circuits with time-domain and large-signal-small-signal analysis methods, Acta Polytech. Scand., Electr. Eng. Ser., vol. 87, 997. A Robust racking Control for Chaotic Chua s Circuits via uzzy Approach Yeong-Chan Chang Abstract his paper addresses the problem of designing robust tracking controls for nonlinear chaotic Chua s circuits involving plant uncertainties and external disturbances. A hybrid adaptive-robust tracking control scheme which is based upon a combination of the tracking theory, variable structure system (VSS) control algorithm, and adaptive fuzzy control is developed such that all the states and signals of the closed-loop system are bounded and an tracking control from the tracking error to the external disturbance is guaranteed. In contrast to the previous investigations of controlling Chua s circuits the controller developed here can be extended to handle a broader class of uncertain nonlinear chaotic Chua s circuits. Index erms Chaotic Chua s circuits, tracking control, hybrid adaptive-robust design, Riccati-like matrix equation. I. INRODUCION he control of nonlinear chaotic Chua s circuits is an important topic for numerous practical applications since this circuit exhibits a wide variety of nonlinear dynamic phenomena such as bifurcations and chaos [5], [6], [8]. his chaotic circuit possesses the property of simplicity and universality, and has become a standard prototype for investigation of chaos. A significant research attention has been paid toward studying the control of such circuits in the past few years [2] [6], [], [4], [5], [7], [8]. In most of these previous investigations, the plant dynamics is assumed to be available for implementation or external disturbances are neglected. However, in practical applications, plant uncertainties and external disturbances which may affect the tracking performance are inevitable. herefore, development of an alternative control approach to efficiently treat the robust tracking control of chaotic Chua s circuits involving a large class of uncertainties and variations is highly desirable. uzzy control has recently found extensive applications for a wide variety of industrial systems and many adaptive fuzzy control Manuscript received May, 2; revised ebruary 5, 2. his paper was recommended by Associate Editor. Saito. he author is with the Department of Electrical Engineering, Kun-Shan University of echnology, Yung-Kang, ainan Hsien, aiwan, R.O.C. ( ycchang@mail.ksut.edu.tw). Publisher Item Identifier S ()59-4. ig.. Diagram of nonlinear chaotic Chua s circuit. schemes have been developed [], [7], [2], [], [6]. Specially, some fuzzy-based control schemes are also developed to treat the control of chaotic circuits [2], [5], [5]. In [2] a simple fuzzy logic based intelligent mechanism was developed for predicting and controlling an uncertain chaotic system to a desired target. Chen and Dong [5] applied fuzzy inference systems to identify and control chaotic systems. anaka et al. [5] developed a unified approach to controlling chaos via an linear matrix inequality (LMI)-based fuzzy control system design. his paper addresses the robust tracking control for a large class of nonlinear chaotic Chua s circuits in the presence of plant uncertainties and external disturbances. Nonlinear H tracking theory, variable structure system (VSS) control algorithm and fuzzy control design are combined together to construct a VSS indirect adaptive fuzzybased H tracking controller such that the resulting closed-loop circuit system guarantees a satisfactorily transient and asymptotic performance in the sense that the tracking error can be made as small as possible in terms of L boundedness and H tracking performance. In order to implement the developed controller, knowledge of the dynamic model of nonlinear Chua s circuits doesn t require for feedback and moreover only a linear Riccati-like equation must be solved. Consequently, the control scheme developed in this study can be easily implemented from the viewpoint of practical applications. In Section II, the tracking control problem of nonlinear chaotic Chua s circuits is presented. Section III develops a hybrid adaptive-robust fuzzy-based controller. Section IV presents a simulation example. inally, the conclusion is given in Section V. II. PRELIMINARY A. Problem Statement he chaotic Chua s circuit, as shown in ig., is a simple electronic system that consists of one inductor L, two capacitors C ;C 2, one linear resistor R, and one nonlinear resistor g. he dynamic equations of Chua s circuit with control inputs are described by [5], [8] _v c = C R (v c v c ) g(v c )+u + d () _v c = (vc vc )+il + u2 + d2 (2) C 2 R _i L = L (v c + u )+d () where i L current through the inductor L; v c,v c voltages across C and C 2, respectively; g() current through the nonlinear resistor; u ;u 2,u control inputs; d ;d 2 and d external disturbances. Let x(t) =[v c (t); v c (t); i L (t)] be the state variable /$. 2 IEEE

2 89 IEEE RANSACIONS ON CIRCUIS AND SYSEMS I: UNDAMENAL HEORY AND APPLICAIONS, VOL. 48, NO. 7, JULY 2 Give a desired reference signal x d (t) =[x d (t); x d (t); x d (t)] which is assumed to be bounded and continuously differentiable, i.e., there is a compact set d such that x d (t) 2 d for all t. he objective of this paper is to design an adaptive-robust tracking controller for nonlinear chaotic Chua s circuits () () involving plant uncertainties and external disturbances such that the resulting closed-loop system guarantees that all the states and signals are bounded and the tracking error should be as small as possible. aking e(t) = x(t) x d (t) = [v c (t) xd (t); v c (t) x d (t); i L (t) x d (t)], the tracking error dynamic equation can be expressed as ig. 2. he basic configuration of a fuzzy logic system. _e = (x) +G(x)u + d _x d (4) where u =[u ;u 2 ;u ] ;d =[d ;d 2 ;d ], and (x) = G(x) = f (x) f 2(x) f (x) = C C 2 L C R (v c v c ) g(v c ) C 2 R (vc : vc )+il L (vc ) It is clear that if the function (x) and the matrix G(x) are well-known, by employing the technique of feedback linearization [9] there exists a suitable control law such that the resulting closed-loop system can be shown to achieve a satisfactorily tracking performance. However, in practical circuit systems, the inductor, capacitor, and resistor may have uncertain variations around their nominal values due to heating. he parameters and characteristics of electronic elements may vary while the circuit system has operated for a long time, and the circuit system may receive unpredictable interference from the environment where it resides, etc. herefore, in order to efficiently control the chaotic Chua s circuit it is necessary to consider the effects due to plant uncertainties and external disturbances. In this study, the values of the inductor L, capacitors C and C 2, and resistors R and g() are assumed to be unknown. herefore, the nonlinear dynamics (x) and G(x) are also unknown and cannot be available directly in the robust control design. he philosophy of our tracking design is expected that the fuzzy approximator equipped with an adaptive algorithm is introduced first to learn the uncertain dynamics. Next, two additional robust control algorithms, i.e., VSS algorithm and nonlinear H control algorithm, are employed to efficiently attenuate the effects on the tracking error due to the fuzzy approximation error and the external disturbance. wo cases with respect to different knowledge of the input gain matrix will be proposed, sequentially. One is assumed that the values of C ;C 2 and L can be split into a nominal part plus an uncertain part and the other is assumed that these three values are nonlinear time-varying and unknown. B. Description of uzzy Systems he basic configuration of the fuzzy system shown in ig. 2 is constructed from the fuzzy If hen rules using some specific inference, fuzzification, and defuzzification strategies [], [6]. he fuzzy system performs a mapping from U R N to V R. he fuzzifier maps a crisp point in U into a fuzzy set in U. he fuzzy rule base consists of a collection of fuzzy If hen rules such as R (l) : If x is l ; ;x N is l N ; hen y is G l (5) in which x = (x ;...;x N ) 2 U and y 2 V R are the input and output of the fuzzy system, respectively, i l, i =; 2;...;Nand G l are fuzzy sets, and l =;...;Mwhere M denotes the number of fuzzy If hen rules. he fuzzy inference engine performs a mapping from fuzzy sets in U to fuzzy sets in V, based upon the fuzzy If hen rules and the compositional rule of inference. he defuzzifier maps a fuzzy set in V to a crisp point in V. According to the universal approximation theorem [], [6] the fuzzy system with center-average defuzzifier, product inference and singleton fuzzifier is of the following form: M l l= y(x) = M l= N (x i ) i= N i= (x i ) where () is the numbership function of the fuzzy set i l and l is the point at which G achieves its maximum value. Moreover, for any given real continuous function f (x) on a compact subset U R N and arbitrary >, there exists a fuzzy system y(x) in the form of (6) such that maxx2u kf(x) y(x)k <. III. CONROLLER DESIGN AND SABILIY ANALYSIS irst, we shall focus on the case that L; C and C 2 can be partitioned as L = L + L, C = C + C and C 2 = C 2 + C 2 where L ;C and C 2 denote the known nominal values and L; C and C 2 denote the unknown nonlinear time-varying perturbations. After some simple manipulations, the input matrix G(t; x) can be expressed as G(t; x) = G + G(t; x) where G = diag[=c ; =C 2 ; =L ] and G = diag[(c =C C ); (C 2=C 2C 2); (L=L L)]. ake the universal approximation system ^ (x; 2) with x 2 U x for some compact set U x R, to approximate the uncertain term (x) where 2 contains the tunable approximation parameters. Let ^f i (x; 2i); i=; 2; denote the ith component of ^ (x; 2) and use to approximate f i(x). rom (6) ^f i(x; 2 i) can be expressed as (6) ^f i (x; 2i) = i (x)2i; i =; 2; (7)

3 IEEE RANSACIONS ON CIRCUIS AND SYSEMS I: UNDAMENAL HEORY AND APPLICAIONS, VOL. 48, NO. 7, JULY 2 89 where 2 i =[i ;...; im ] 2 R m for some m i >, is a parameter vector and i (x) =[ i (x);...; im (x)] is a regressive vector with the regressor il (x) defined as il (x) = m l= j= j= (x j) (x j) ; l =;...;m i : (8) Here, the linearly parametrized fuzzy model [], [2], [], [6] is employed in the approximation procedure. hat is, the membership functions (x j ) for l m i and j are specified beforehand and the number m i is given so that ^f i (x; 2 i ) can approximate f i (x) as best as possible. Consequently, the fuzzy system ^ (x; 2) can be expressed as ^ (x; 2) = (x) 2 (x) (x) = Y (x)2 (9) where Y (x) 2 R 2m is a basis matrix and 2 =[2 ; 22 ; 2 ] 2 R m with m = m + m 2 + m. According to the universal approximation theorem, there exists an optimal approximation parameter 2 [], [], [6] such that ^ (x; 2 ) can approximate (x) as best as possible. As in many previous adaptive fuzzy designs, 2 will be learned by using an adaptive algorithm. Suppose the constrained region of 2 is chosen to be a convex hypercube. hat is, consider = f2jbij ij c ij ; j m i ; i g and = f2jbij ij c ij + ; j m i ; i g where the values of b ij; c ij and > can be arbitrarily specified by the designer. Give a smooth function 8 =[8 ; 8 2 ; 8 ] where 8 2 R m 2 ; R m 2 and 8 2 R m 2 should be specified later. herefore, the smooth projection algorithm with respect to can be expressed as [] Proj[8; 2] = ij ; if ( ij >c ij and ij > ) ij ; if ( ij <b ij and ij < ) ij ; otherwise, j m i ; i () where ij =(+(cij ij =)) ij, ij =(+(ij b ij =)) ij and ij denotes the j element of 8 i. Let (x) = (x) ^ (x; 2 ) be the minimum approximation error and f i be the i element of. hroughout this section we need the following assumptions. A) here exists a constant < < such that j i(gg )j, 8 i. A2) here exists a constant f > such that jf i j f, 8 i. A) here exists a constant M d > such that kd(t)k 2 dt M d, i.e., d 2 L 2 [; ). he assumption A) is equivalent to jc =C j; jc 2=C 2j and jl=lj, and so it always holds for small parameter perturbations. According to the universal approximation theorem the assumption A2 also holds. he assumption A implies the external disturbance is of finite-energy. his kind of disturbance usually appears in a physically operating circuit. heorem : Consider the nonlinear chaotic Chua s circuit () (). Suppose that Assumptions A) A) are satisfied. If there exists a symmetric positive matrix P = P > satisfying the Riccati-like equation K P PK + Q + P 2 I 2 ( )R P = () where > is a prescribed attenuation level, Q = Q > is a prescribed weighting matrix, K =diag[k ;k 2;k ] for some constants k i > is a state feedback control gain, and R = diag[r ;r 2;r ] for some constants r i > is an H control gain, then the following VSS adaptive fuzzy-based control law with u = G (Y (x)2 + _x d Ke + u h + u s ) (2) u h = 2 R Pe () u s = f + M e sgn(pe) (4) _2 =Proj[8; 2] (5) where M e = ky 2 + _xd Kek, sgn(pe) = [sgn((pe) ); sgn((pe) 2); sgn((pe) )], 8 = Y Pe, and denotes the adaptive gain, guarantees that i) all the variables of the closed-loop system () (), (2) (5) are bounded and ii) an H tracking performance from d(t) to e(t) is achieved, i.e., ke(t)k 2 Q dt 2V () + 2 kd(t)k 2 dt 8 < (6) where V () denotes the initial conditions. Proof: aking into account the minimum approximation error and the control law (2), the error dynamic equation (4) can be rewritten as _e = ^ (x; 2 )+ (x) +G u +G(t; x)u + d _x d = Ke Y (x) ~ 2+ (x) +G(t; x)u + u h + u s + d where ~ 2=2 2. Choose a Lyapunov function as (7) V = 2 e Pe+ 2 2 ~ 2: ~ (8) By completing the squares and using the control input u h in (), the time derivative V _ along the error trajectory (7) is _V = 2 e (K P PK)e + uh (I +GG ) Pe + d Pe+ us (I +GG ) Pe +( +GG (Y 2+ _x d Ke)) Pe 2 ~ Y Pe+ 2 _ 2 ~ = 2 e K P PK +P 2 I R (I +GG ) P e Pe d Pe d us (I +GG ) Pe+( +GG 2 2 d d (Y 2+ _x d Ke)) Pe ~ 2 Y Pe+ _ 2 ~ 2: (9)

4 892 IEEE RANSACIONS ON CIRCUIS AND SYSEMS I: UNDAMENAL HEORY AND APPLICAIONS, VOL. 48, NO. 7, JULY 2 rom the update law (5) which is a standard smooth projection algorithm [] we can guarantee that = _ 2 ~ 2 ~ 2 Y Pe and 2(t) 2 for all t if 2() 2. Moreover, by u s in (4) and the assumptions A) A2), we get us (I +GG ) Pe+( +GG (Y 2+ _x d Ke)) Pe ( f + M e ) i= j(pe) i j + i= j( +GG (Y 2+ _x d Ke)) i jj(pe) i j: (2) Summarily, using the Riccati-like equation () the derivative _ V can be bounded as _V 2 e 2 e K P PK + P 2 I R (I +GG ) P e d d K P PK + P I 2 ( )R P e d d 2 e Qe d d: (2) Integrating the above inequality from t =to t = yields 2V ( )+ ke(t)k 2 Q dt 2V () + 2 kd(t)k 2 dt (22) for all <, i.e., the H performance in (6) is achieved []. Moreover, from V (t) in (8) and A) it is clear that e(t) 2 e = feje Pe 2V () + 2 M d g and so x(t) 2 U x = fxje(t) 2 e; x d (t) 2 d g8 t. his implies all the variables are bounded. Remark : he proposed controller (2) consists of three parts: the adaptive fuzzy system, ^ (x; 2) = Y (x)2 equipped with (5), which is used to learn the unknown dynamics (x), the VSS controller, u s,to eliminate the effect of the approximation error, and the robust H controller, u h, to achieve the desired H tracking performance. Hence, in practice this controller is a hybrid adaptive-robust controller. Remark 2: i) Result in heorem indicates that the control design relies only on the solution of an algebraic Riccati-like equation (). By setting R =(=() 2 )I 2 and Q be a diagonal matrix, a simple solution is obtained as P =(=2)K Q. Since a small perturbation G(t) implies the bounded value is smaller, there is a trade-off between the magnitude of G(t) and the control gain in u h. ii) Since the value f may be made arbitrarily small by increasing the number of fuzzy If hen rules, from (4) there is a trade-off between the number of fuzzy rules as well as the control gain in u s. Remark : rom (5) it is shown that b ij ij c ij + for j m i and i. So, it can be concluded that b i ^f i (x; 2 i ) c i where b i = minjm (b ij ) and c i =maxjm (c ij + ). m Actually, since each ij(x) 2 (; ] and j= ij(x) =, we get b i m j= (bij )ij ^f i(x; 2 i) m j= (cij + )ij ci. Based on this property if the upper and lower bounds of f i (x) (e.g., f i f i (x) f i ) are well-known, then we can simply set b ij = f i;c ij = f i for all j =;...;m i. Since the nonsmoothness of sgn(pe) results in the VSS control algorithm (4) with discontinuities between the sliding surfaces, the VSS adaptive fuzzy-based controller developed above may lead to control chattering and could excite high-frequency unmodeled modes [8]. o avoid this discontinuity we use the exponential modification to replace u s in (4). heorem 2: Consider the nonlinear chaotic Chua s circuit () (). Under the same conditions as in heorem, if the VSS type control u s in (4) is modified to the continuous control u s = M E(x) M E (x)pe km E (x)pek + et (2) where M E (x) = p f + M e (x) and > ; > are positive constants, then i) if d 2 L 2 [; ), anh tracking performance is achieved; ii) if d 2 L[; ), the tracking error is uniformly ultimately bounded; iii) if d 2 L 2 [; ) \ L[; ), then lim t! e(t) =; and iv) all the variables are bounded. Proof: ollowing the proof in heorem and using () (), (5) it can be shown that _V 2 e Qe d d + us (I +GG ) Pe +( +GG (Y 2+ _x d Ke)) Pe: (24) By u s in (2) and the assumptions A) A2) we get us (I +GG ) Pe +( +GG (Y 2+ _x d Ke)) Pe (MEPe) (I +GG ) (M EPe) ( )(km EPek + et + km E Pek ) e t : (25) Consequently, the derivative _ V can be bounded as _V 2 e Qe d d + e t : (26) Integrating the above inequality yields ke(t)k 2 Q dt 2V () + 2 ( e )+ 2 kd(t)k 2 dt: (27) his implies an H tracking performance is achieved. Next, suppose d is bounded, i.e., there is an d > such that kdk d. he inequality (26) can be rewritten as _ V (=2) q kek 2 + (=2) 2 2 d + where q denotes the minimum eigenvalue of Q. hen, for any given > there is a choice of q such that _ V kek 2 < 8ke(t)k >for some >. his implies that there is a > such that ke(t)k for all t []. Moreover, if d() 2 L 2 \ L[; ), from the closed-loop error system it is clear that _e(t) is uniformly bounded. his implies that e(t) is uniformly continuous. Based on the Barbalat s lemma [] and the inequality (26), it can be concluded that lim t! e(t) =. inally, as in the proof of heorem we can conclude that all the states and signals of the closed-loop system are bounded. Remark 4: he term (2=)( e ) in (27) is yielded owing to the smooth modification of u s and can be viewed as an external disturbance. he smaller the value and the larger the value, the less smooth is the robust controller u s and the smaller is the term (2=)( e ).As =, this term is equal to zero and this H tracking performance is reduced to (6). In the above analyses, the input gain matrix G(t; x) is assumed to be expressed as a nominal part plus a small perturbation. However, in practical applications G(t; x) may be completely unknown and cannot be available directly in the control design. Without loss of generality, assume the values of C ;C 2 and L are nonlinear timevarying, dependent on the state variable, and unknown. or simplicity of notation, let g (t; x) ==C (t; x), g 2(t; x) ==C 2(t; x) and g (t; x) ==L(t; x).

5 IEEE RANSACIONS ON CIRCUIS AND SYSEMS I: UNDAMENAL HEORY AND APPLICAIONS, VOL. 48, NO. 7, JULY 2 89 (a) (b) (c) (d) ig.. he responses for the nonlinear chaotic Chua s circuit using the developed controller. (a) Capacitor voltage v (t). (b) Capacitor voltage v (t). (c) Inductor current i (t). (d) Control input u (t). ake the fuzzy system ^g i(x; 2 gi) to approximate the uncertain term g i (t; x). rom (6) ^g i (x; 2 gi ) can be expressed as ^g i (x; 2 gi )= gi(x)2 gi ; i =; 2; (28) where 2 gi 2 R m for some m gi > is a parameter vector and gi (x) 2 R m is a regressive vector. Suppose the constrained regions of 2 gi; i = ; 2; are convex hypercubes. hat is, consider gi = f2gi jb gij gij c gij ; j m gi g and = f2gi jb gij gi gij c gij + gi ; j m gi g where the values of b gij; c gij and gi > can be arbitrarily specified. Let 8 gi = ui gi (Pe) i ;i=; 2; where (Pe) i denotes the i element of Pe. herefore, the smooth projection algorithm of 2 gi with respect to is Proj[8 gi ; 2 gi ] = gij ; if ( gij >c gij & gij > ) gij ; if ( gij <b gij & gij < ) (29) gij ; otherwise, j m gi where gij =(+(cgij gij = gi )) gij, gij =(+(gij b gij= gi)) gij and gij denotes the j element of 8 gi. Let 2 gi; i = ; 2; be the optimal approximation parameters. Define the minimum approximation errors g i (t; x) = g i (t; x) ^g i(x; 2 gi) and the estimated errors ~ 2 gi = 2 gi 2 gi. After some simple manipulations, the input matrix G(t; x) is equal to where G(t; x) = ^G(x; 2 g) Y g(x) ~ 2 g +G(t; x) () ^G(x; 2 g)=diag[^g (x; 2 g); ^g 2(x; 2 g2); ^g (x; 2 g)] Y g (x) =diag[g(x); g2(x); g(x)] ~2 g =diag[ 2 ~ g ; 2 ~ g2 ; 2 ~ g ] G(t; x) =diag[g (t; x); g 2(t; x); g (t; x)]: In addition, we make the following assumption. A4) here is a constant < g < such that jg i(t; x)^g i (x; 2 gi)j g; 8 i. heorem : Consider the uncertain nonlinear chaotic Chua s circuit () (). Suppose A2) A4) are satisfied. If there exists a symmetric positive matrix P = P > satisfying the Riccati-like equation K P PK + Q + P 2 I2 ( g)r P = then the following VSS adaptive fuzzy-based control law () u = ^G (x; 2 g )(Y (x)2 + _x d Ke + u h + u s ) (2)

6 894 IEEE RANSACIONS ON CIRCUIS AND SYSEMS I: UNDAMENAL HEORY AND APPLICAIONS, VOL. 48, NO. 7, JULY 2 (e) (f) (g) ig.. (Continued) he responses for the nonlinear chaotic Chua s circuit using the developed controller. (e) Control input u (t). (f) Control input u (t). (g) phase plane trajectory v v. (h) Phase plane trajectory v i. (h) with u h in (), u s = ( f + gm e)sgn(pe)=( g), _ 2 in (5), and _ 2 gi = gi Proj[8 gi ; 2 gi ] in (29) with b gij gi > 8 j m gi ; i where gi > are the adaptive gains, guarantees all the variables of the closed-loop system are bounded and an H tracking performance is achieved. Proof: rom () and (2), the error dynamic equation (4) can be rewritten as _e = Ke Y (x) 2+ ~ (x) +G(t; x)u Y g(x) 2 ~ gu + u h + u s + d: () Choose a Lyapunov function as W = 2 e Pe+ 2 ~ 2 ~ 2+ i= 2 gi ~ 2 gi ~ 2gi : (4) As in the proof of heorem, the derivative _ W can be bounded as _W 2 e Qe d d u i 2 ~ gi gi (Pe) i + i= i= gi _ 2 gi ~ 2gi : (5) It can be shown that (= gi ) _ 2 gi ~ 2 gi u i ~ 2 gi gi (Pe) i and 2 gi(t) 2 8 t. Similarly to the proof of heorem, an H performance is achieved and all the variables are bounded. inally, we shall show that the matrix ^G(x; 2 g) is invertible. Let b gi =minjm (b gij gi) and c gi =maxjm (c gij + gi). As in Remark, it can be concluded that b gi ^g i (x; 2 gi ) c gi, i =; 2;. hen, if we suitably choose b gij and gi such that b gi >, ^G(x; 2 g ) is invertible and so the controller is well-defined. IV. SIMULAION EXAMPLE Consider the tracking control problem of the uncertain chaotic Chua s circuit shown in ig.. or the convenience of simulation, the nominal parameters are given as C =;C 2 =:5and L =, and the perturbations are given as C =:+: cos(t=2); C 2 =: and L = :5. he perturbed resistors are characterized by R = 5 + sin(t=2) and g(v c ) = v c + :2v c. he exogenous disturbances are d =:25 sin(2t) exp(:t), d 2 =:2 cos(2t) exp(:t) and d =:2 sin(2t) exp(:t). Let the desired voltage and current be x d (t) = sin(t), x d (t) = cos(t) and x d (t) = + sin(t). he VSS adaptive fuzzy-based H tracking control law developed in heorem 2 is employed to treat this trajectory planning problem. We divide the design into three steps. Step : Let x = v c ;x 2 = v c ;x = i L and x 4 = v c v c. Define five fuzzy sets for each x i ;i=; 2; ; 4 with labels i (negative large), 2 i (near.5), i (near ), 4 i (near.5), and 5 i (positive large) which are characterized by the membership functions, respectively, (x i )==(+ exp(5(x i + ))); (x i ) = exp(2(x i + :5) 2 ); (x i ) = exp(2x 2 i ); (x i ) = exp(2(x i :5) 2 ), and (x i)==( + exp(5(x i ))).

7 IEEE RANSACIONS ON CIRCUIS AND SYSEMS I: UNDAMENAL HEORY AND APPLICAIONS, VOL. 48, NO. 7, JULY Now, construct the fuzzy approximator ^f (x; 2 ) in (7). 25 fuzzy rules are included R (l) ij : If x is i and x 4 is j 4 ; hen y is l ij; for i; j =;...; 5 l =;...; 25: he fuzzy regressive vector in (7) is chosen to be = [ ; 2 ;...; 25] 2 R 25 with components as (x) = (x ) (x 4)= D ; 2 (x) = (x ) (x 4 )=;...; 24 (x) = (x ) (x 4 )=D, and 25 (x) = (x ) (x 4 )=D where D = 5 i= 5 j= (x) (x4). Hence, f(x) is approximated by ^f (x; 2 ) = (x)2 with 2 =[; 2;...; 25] 2 R 25. Next, 25 fuzzy rules are included in ^f 2 (x; 2 2 ): R (l) ij : If x is i and x 4 is j 4 ; hen y is l ij; for i; j =;...; 5 l =26;...; 5: Choose 2 =[ 2; 22;...; 225] 2 R 25 with 2(x) = (x ) (x 4 )=D 2 ; 22 (x) = (x ) 2 (x 4 )=D 2 ;...; 225 (x) = 5 5 (x ) (x 4 )=D 2, and D 2 = i= j= (x ) (x 4 ). Hence, ^f 2 (x; 2 2 ) =2 (x)2 2 with 2 2 =[2 ; 22 ;...; 225 ] 2 R 25. inally, 5 fuzzy rules are included in ^f (x; 2 ): R (l) i : If x 2 is 2; i hen y is l i for i =;...; 5; l=5;...; 55: Choose = [ ;...; 5 ] 2 R 5 with (x) = (x 2 )= 5 D ;...; 5(x) = (x 2)=D, and D = i= (x2). Hence, ^f (x; 2 ) = (x)2 with 2 =[;...; 5] 2 R 5. Consequently, (x) is approximated by ^ (x; 2) = Y (x)2 with Y (x) =diag[ (x); 2 (x); (x)] and 2 =[2 ; 22 ; 2 ]. Step 2: Select K =2I 2, =:5and Q =2I 2. Set =:2. Consequently, take R = ( ) 2 I 2 = :2I 2 and solving the equation in () yields P =(=2)K Q =:5I 2. Step : Obtain the tracking controller (2), (), (5), (2) with f = :2; = ; = :, = 2; b ij = 5; c ij = 5 8 ij, and =:5. Choose the initial conditions v c () =, v c () =, i L() =, and 2() = 552. he simulation results are shown in ig.. he phase-plane trajectory v c v c converges to a circle of unit radius centered at (; ) and v c i L converges to a circle of unit radius centered at (; ). hese simulation results indicate that the tracking performance is nice and consequently the effects due to parametric uncertainties and external disturbances in chaotic Chua s control circuits can be efficiently diminished by the developed control algorithm. address the problem of controlling Chua s circuits this paper can be extended to handle a broader class of nonlinear chaotic Chua s circuits in the presence of plant uncertainties and external disturbances, and so this design is quite useful from the viewpoint of practical applications. inally, a simulation example is included to confirm the validity and performance of the developed control scheme. REERENCES []. Başar and P. 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