Chaos, Solitons and Fractals

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1 Chaos, Solitons an Fractals xxx (00) xxx xxx Contents lists available at ScienceDirect Chaos, Solitons an Fractals journal homepage: Impulsive stability of chaotic systems represente by T-S moel Xiaohong Zhang a, *, Anmar Khara c, Dong Li b, Dan Yang a a School of Software Engineering, Chongqing University, Chongqing , PR China b College of Mathematics & Physics, Chongqing University, Chongqing , PR China c Department of Mathematics, University of British Columbia, Vancouver BC, Canaa V6T 1Z article info abstract Article history: Accepte 30 July 00 Available online xxxx Communicate by Prof. Ji-Huan He In this paper, a novel an unifie control approach that combines intelligent fuzzy logic methoology with impulsive control is evelope for controlling a class of chaotic systems. We first introuce impulses into each subsystem of T-S fuzzy IF-THEN rules an then present a unifie T-S impulsive fuzzy moel for chaos control. Base on the new moel, a simple an unifie set of conitions for controlling chaotic systems is erive by Lyapunov metho techniques an a esign proceure for estimating bouns on control matrices is also given. These results are shown to be less conservative than those existing ones in the literature an several numerical examples are presente to illustrate the effectiveness of this metho. Ó 00 Elsevier Lt. All rights reserve. 1. Introuction In recent years, it has become quite popular to aopt the so-calle Takagi Sugeno (T-S) type fuzzy moels [1 7,17 0] to represent or approximate a nonlinear system. These fuzzy moels are escribe by a family of fuzzy IF-THEN rules which represent local linear input output relations of a nonlinear system. The overall fuzzy moels are achieve by smoothly blening these local linear moels together through fuzzy membership functions. As a result, it has been feasible to apply conventional linear system theory to analyze nonlinear control systems [1,3,]. In particular, the stability issue of chaotic systems represente by T-S fuzzy moel has been investigate extensively in a unifie way [,3,5,]. Uner the framework of T-S fuzzy moel, we foun that the existing controllers of the chaotic system represente by T-S moel ha almost employe PDC an LMI techniques [,3,5,], where PDC was use for converting T-S fuzzy rules into close-loop systems while LMI was employe to fin common positive efinite matrices. However, PDC is a continuous input control metho an not available for the evelopment of igital control evices. On the other han, impulsive control is an attractive alternative because it not only allows the stabilization of a nonlinear system using only small control impulses but also offers a irect metho for moulating igital information onto a chaotic carrier signal for sprea spectrum applications. Moreover, impulsive control may also offer a simple an efficient metho to eal with systems base on the evelopment of igital control evices which generate control impulses at iscrete moments [9 16]. But the existing impulsive control theory lacks the unifie way for ealing with ifferent chaotic systems. Therefore, it is necessary to evelop a new technique for controlling chaos in such a way that impulsive control characterizes fuzzy intelligence. It is important to point out that combining impulsive control with fuzzy moeling for chaotic system has been rarely iscusse until now. This paper is evote to proviing an alternative an novel approach for controlling T-S fuzzy chaotic systems base on impulsive control techniques. That is, uner the framework of T-S fuzzy rules, impulses are introuce into every subsystem of T-S fuzzy chaotic systems where these impulses are viewe as the corresponing control inputs that nee to be esigne. In this case, the overall fuzzy moels become impulsive ifferential equations obtaine by smoothly blening these local * Corresponing author. Tel.: aress: xhongz@yahoo.com.cn (X. Zhang) /$ - see front matter Ó 00 Elsevier Lt. All rights reserve. oi: /j.chaos

2 X. Zhang et al. / Chaos, Solitons an Fractals xxx (00) xxx xxx linear impulsive moels together through fuzzy membership functions. In other wors, the intelligent fuzzy logic methoology will be combine with simple impulsive control techniques so as to evelop a more powerful controller that can efficiently stabilize complex chaotic systems by using only small impulses. It will be emonstrate that this novel control moel not only provies a unifie approach for controlling a class of chaotic systems but also possesses the control merits of impulsive control techniques. The rest of this paper is organize as follows. Section escribes a fuzzy moeling methoology for T-S fuzzy chaotic systems an presents impulsive fuzzy control moel for controlling complex chaotic systems. In Section 3, theoretical stability analysis of impulsive fuzzy chaotic systems is evelope, while, in Section 4, several numerical simulations of chaotic systems are carrie out base on the propose metho. Finally, in Section 5, some concluing remarks are mae.. Impulsive control moels for fuzzy chaotic systems It is well known that many chaotic systems, such as Lorenz, Chen s, Chua s an Rossler systems, can be exactly represente by a unifie fuzzy T-S moel propose by Takagi an Sugeno []. In orer to emphasize the avantages of our methos over the conventional T-S moel, we initially consier the case when the ith rules of the T-S fuzzy moels for the chaotic systems are of the following forms [1,3]: Plant Rule i: IFz 1 (t) ism i1 an... an z p (t) ism ip THEN _xðtþ ¼A i xðtþ; i ¼ 1; ;...; r; ð1þ where M ij is the fuzzy set an r is the number of IF-THEN rules, x(t) R m is the state vector, A i R nm are known constant matrices, z 1 (t),...,z p (t) are the premise variables. Remark 1. Uner the framework of (1), the various approaches to controlling chaos have been investigate base on LMI an PDC techniques. Up to now, we foun that the existing methos for controlling chaos base on fuzzy T-S moel have solely use PDC technique, in which LMI algorithm has been use for fining common symmetry positive matrices. It shoul be pointe out, however, that the existing methos belong to continuous input control. From the above iscussion, we shall introuce a iscrete input control into system (1) an propose a novel moel so as to exten orinary T-S moel. To construct an impulsive or iscrete control plant from a fuzzy T-S moel employing a chaotic system, consier first a iscrete set {s j } of time instants, where 0 < s 1 < s < < s j < s j+1 <, s j? 1, as j? 1. Let Dxj s¼sj xðs þ j Þ xðs j Þ be the jump in the state variable at the time instant s j, where xðs Þ¼xðs j jþ. The impulsive control structure of the nonlinear systems represente by T-S fuzzy moel is then efine by Plant Rule i: IFz 1 (t) ism i1 an... an z p (t) ism ip THEN _xðtþ ¼A ixðtþ t 6¼ s j ; i ¼ 1; ;...; r; j ¼ 1; ;... ðþ where M ij, A i R nm, x(t) R m, z 1 (t),...,z p (t) an r have the same meanings as in (1), an K i,j R nm enotes the control gain of the jth impulsive instant s j. The efuzzifie output of the impulsive T-S fuzzy system is inferre as follows: _xðtþ ¼ Pr h i ðzðtþþa i xðtþ t 6¼ s j ; ð3þ >: Dx ¼ Pr h i ðzðtþþk i;j x t ¼ s j ; where z(t)=[z 1 (t),z (t),...,z p (t)] T, w i ðzðtþþ ¼ Q p j¼1 M P ijðz j ðtþþ, r w iðzðtþþ > 0; w i ðzðtþþ P 0, h i ðzðtþþ ¼ w i ðzðtþþ= P r P w iðzðtþþ, r h iðzðtþþ ¼ 1; h i ðzðtþþ P 0, i =1,,...,r. Note that system (3) will be calle impulsive fuzzy moel hereafter. Remark. There exist some obvious ifferences between the local subsystems of (1) an (). In fact, the former is a linear system, while the latter is an impulsive one. Therefore we ten to think of () as being an extension of (1). Remark 3. The stability of system (3) can be analyze by esigning a set of control matrices K i, j an impulsive control intervals. Moreover, this unifie impulsive control strategy is available for a class of chaotic systems ue to the fact that (3) has the ability to unify a whole class of chaotic systems, such as Lorenz, Chen s an Chua s systems. 3. Impulsive stability of fuzzy chaotic systems In this section, we shall focus our attention on the stability properties of impulsive fuzzy chaotic systems an erive sufficient conitions leaing to the impulsive control of these systems. Theorem 1. Let P be an n n symmetric an positive efinite matrix an k max (P) an k min (P) be the largest an smallest eigenvalues of P, respectively. If there exist constant scalars j > 0c > 0, b j (j N), n > 1, such that

3 X. Zhang et al. / Chaos, Solitons an Fractals xxx (00) xxx xxx 3 ð1þ PA i þ A T i P ji < 0 ði ¼ 1; ;...; rþ; c ¼ j=k minðpþ; ð4þ ðþ b j ¼ k maxðpþ kði þ K i;j Þk; k min ðpþ ð5þ ð3þ lnðnb j Þþcr j 6 0; ð6þ where 0 < r j = s j s j 1 < 1 is the impulsive interval, then the impulsive fuzzy control system in Eq. (3) is globally asymptotically stable at the origin. Proof. Consier a Lyapunov function of the form VðxÞ ¼ 1 xt Px. The time erivative of V(x)along the solution of Eq. (3) is _VðxÞ ¼ 1 Xr xt h i ðzðtþþðpa i þ A T i PÞx tðt i 1; t i Š; where k i > 0 for h i (z(t)) [0 1]. By (4), we have _VðxÞ ¼ 1 h i ðzðtþþx T ½ðPA i þ A T i PÞŠx 6 1 jkxk 6 j k min ðpþ VðxðtÞÞ: ¼ cvðxðtþþ t ðt j 1 ; t j Š ði ¼ 1; ;...Þ: ð7þ On the other han, it follows from the secon equation in (3) an the properties of h i (z(t)) that Vðxðs þ j ÞÞ ¼ 1 h i ðzðtþþ½ði þ K i;j Þxðs j ÞŠ T P Xr h i ðzðtþþði þ K i;j Þxðs j Þ " #" # 6 1 k maxðpþ Xr h i ðzðtþþði þ K i;j Þ T x T ðs j Þ h i ðzðtþþði þ K i;j Þxðs j Þ ¼ 1 k maxðpþ Xr h i ðzðtþþði þ K i;j Þxðs j Þ 6 1 k maxðpþ Xr 6 k maxðpþ k min ðpþ ¼ b j Vðxðs j ÞÞ j N kði þ K i;j Þkkxðs j Þk kði þ K i;j ÞkVðxðs j ÞÞ ðþ By letting j = 1 in inequality (7), we obtain for any t (t 0,t 1 ] VðxðtÞÞ 6 Vðxðt 0 ÞÞ expðcðt t 0 ÞÞ; which implies that Vðxðt 1 ÞÞ 6 Vðxðt 0 ÞÞ expðcðt 1 t 0 ÞÞ an Vðxðt þ 1 ÞÞ 6 b 1Vðxðt 1 ÞÞ 6 b 1 Vðxðt 0 ÞÞ expðcðt 1 t 0 ÞÞ. Similarly for any t (t 1,t ] VðxðtÞÞ 6 Vðxðt þ 1 ÞÞ expðcðt t 1ÞÞ 6 b 1 Vðxðt 0 ÞÞ expðcðt t 0 ÞÞ: In general for any t (t j,t j+1 ], VðxðtÞÞ 6 Vðxðt 0 ÞÞb 1 b b j expðcðt t 0 ÞÞ: From inequality (6), we may conclue that b j expðcr j Þ 6 1=n; j N: Thus for t (t j,t j+1 ], j N VðxðtÞÞ 6 Vðxðt 0 ÞÞb 1 b b j expðcðt t 0 ÞÞ ¼ Vðxðt 0 ÞÞ½b 1 expðcr 1 ÞŠ½b j expðcr j ÞŠ expðcðt t j ÞÞ 6 Vðxðt 0 ÞÞ 1 n j expðcr jþ1 Þ: Then it follows from the above inequality that the trivial solution of system (3) is globally asymptotically stable. In general, since A i, i =1,,...,r, of many chaotic systems are simple, we can irectly employ the eigenvalues of the matrices to obtain the stability conitions for the impulsive fuzzy system. h

4 4 X. Zhang et al. / Chaos, Solitons an Fractals xxx (00) xxx xxx Theorem. Let P be an n n symmetric an positive efinite matrix whose largest an smallest eigenvalues are k max (P) an k min (P), respectively. Let k i be the largest eigenvalue of PA i þ A T i P (i = 1,,...,r), c = max i=1,,...,r{k i }/k min (P) > 0, an b j ¼ kmaxðpþ k min ðpþ P r kði þ K i;jþk ðj ¼ 1; ;...Þ. If there exists a constant n > 1 such that lnðnb j Þþcr j 6 0 ðj ¼ 1; ;...Þ; ð9þ where 0 < r j = s j s j 1 < 1 (j N) is the impulsive interval, then system (3) is globally asymptotically stable at the origin. Proof. Similar to the proof of Theorem 1. h Theorem 1 can be simplifie further by assuming K = K i,j an r = r j for all i, j, as illustrate in the following corollary. Corollary 1. Let P be an n n symmetric an positive efinite matrix an k max (P) an k min (P) be largest an smallest eigenvalues of P, respectively. If there exist constant scalars j > 0c > 0, b, n > 1, such that ð1þ PA i þ A T i P ji < 0 ði ¼ 1; ;...; rþ; c ¼ j=k minðpþ; ð10þ ðþ b ¼ k maxðpþ kði þ KÞk; k min ðpþ ð11þ ð3þ lnðnbþþcr 6 0 ðj NÞ; ð1þ where 0 < r = s j s j 1 < 1(j N) is the impulsive interval, then the impulsive fuzzy control system (3) is globally asymptotically stable at the origin. Similarly, by assuming that P = IK = K i,j an r = r j for all i, j, intheorem, we obtain the following corollary. Corollary. Let k i be the largest eigenvalue of A i þ A T i ði ¼ 1; ;...; rþ, c = max i=1,,...,r {k i }>0 an b = k(i + K)k(j = 1,,...). If there exists a constant n > 1 such that lnðnbþþcr 6 0 ðj ¼ 1; ;...Þ; ð13þ where 0 < r = s j s j 1 < 1 (j N) is the impulsive interval, then the impulsive fuzzy control system (3) is globally asymptotically stable at the origin. Accoring to Theorem 1 an Corollary 1, once we choose the constant impulsive interval r an the common control matrix K for any instant s j, a simple esign strategy for estimating the boun on K can be obtaine as follows. Algorithm. Fin a positive efinite symmetric matrix P an a corresponing scalar c of Theorem 1 by choosing any positive scalar j in avance an solving the linear matrix inequality (4) base on LMI technique. Choose a suitable impulsive interval r accoring to the nees of the actual applications an a constant n > 1. In term of (6) or (1), compute the boun on b an estimate the boun on I + K by using (5) or (11), i.e. by applying the inequality ki þ Kk < k min ðpþ expð crþ=ðk max ðpþnþ: We coul use (14) in orer to fin the control matrix K. For example, we may choose a iagonal matrix K such that (14) hols. Furthermore, notice that the esigne proceure mentione above coul be reconstructe by basing it on Theorem an Corollary as we i before. ð14þ 4. Experimental results In this section, we illustrate the effectiveness of our results by showing several simulation results employing the Lorenz, Chua s an Rossler systems. Example 1. Lorenz s system _x 1 ðtþ ¼ ax 1 ðtþþax ðtþ; _x ðtþ ¼cx 1 ðtþ x ðtþ x 1 ðtþx 3 ðtþ; >: _x 3 ðtþ ¼x 1 ðtþx ðtþ bx 3 ðtþ; where a, b, an c are constants. Assume that x 1 (t) [,] an >0.By() we have the following impulsive fuzzy control moel. _xðtþ ¼A Rule i: If x 1 (t) ism i, then i xðtþ t 6¼ s j ; i ¼ 1; an j N, where x(t)=[x 1 (t),x (t),x 3 (t)] T, 3 3 a a 0 a a A 1 ¼ 4 c 1 5; A ¼ 4 c 1 5; M 1 ðx 1 ðtþþ ¼ 0:5 1 þ x ; an M ðx 1 ðtþþ ¼ 0:5 1 x : 0 b 0 b

5 X. Zhang et al. / Chaos, Solitons an Fractals xxx (00) xxx xxx 5 In this paper, we choose a = 10, b = /3, c =, an = 30. Accoring to Theorem 1 an Corollary 1, we set r = 0.05, n = 1., j = 00. Base on the algorithm of Theorem 1 or Corollary 1, we may euce that P = [14.946,.550, ;.550, 3.73, ; , ,5.17], k max (P) = , k min (P) =.700, c = 73.55, an b = It follows that ki + Kk < Thus we may choose K i,j = iag([ 0.999, 0.999, 0.999]) (j N) such that lnðnbþþcr ¼ 1:77 < 0: Fig. 1a shows a solution trajectory of the Lorenz system in the state space before applying any control strategy, while Fig. 1b shows the control result of fuzzy Lorenz system starting at the initial conition [ 0.,0.6,0.]. Clearly, the time series of all the variables of the Lorenz system converge to zero. Similar control results can be obtaine for Chen s an Lü systems. Example. Chua s oscillator _x 1 ðtþ ¼10ð x 1 ðtþþx ðtþ fðx 1 ðtþþþ; _x ðtþ ¼x 1 ðtþ x ðtþþx 3 ðtþ; >: _x 3 ðtþ ¼14:7x ðtþ; f ðx 1 ðtþþ ¼ 0:6x 1 ðtþ 0:95ðjx 1 ðtþþ1j jx 1 ðtþ 1jÞ: By () we have the following impulsive fuzzy control moel: _xðtþ ¼A Rule i: Ifx 1 (t) ism i, then i xðtþ t 6¼ s j ; i ¼ 1; an j N, where x(t)=[x 1 (t),x (t),x 3 (t)] T, ð 1Þ ð þ 1Þ A 1 ¼ ; A ¼ ; 0 14: :7 0 /ðx 1 ðtþþ ¼ f ðx Þ=x 1 ðtþ x 1 ðtþ 6¼ 0; M 1 ðx 1 ðtþþ ¼ 0:5ð1 /ðx 1 ðtþþ=þ; M ðx 1 ðtþþ ¼ 1 M 1 ðx 1 ðtþþ; 0:7 x 1 ðtþ ¼0; In this paper, we choose =3,r = 0.005, n = 1., j = 00, then P = [.669, 1.610, ; 1.610,1.4114,5.169; ,5.169,13.764], which implies that k max (P) = , k min (P) =.364, c = 4.445, b = (j N). Thus, we have ji þ Kj < 0:007: From the inequality above we can choose K i,j = iag([ 0.999, 0.999, 0.999]) (j N), such that lnðnbþþcr ¼ 1:9509 < 0: Fig. a shows a solution trajectory of the Chua s oscillator in the state space before applying any control strategy, while Fig. b shows the control result of fuzzy Chua s oscillator starting at the initial conition [ 0.,0.6,0.]. Clearly, the time series of all the variables of this system converge to zero. Example 3. Rossler system _x 1 ðtþ ¼ x ðtþ x 3 ðtþ; _x ðtþ ¼x 1 ðtþþax ðtþ; >: _x 3 ðtþ ¼bx 1 ðtþ fc x ðtþgx 3 ðtþ; where a, b, an c are constants. Assume that x 1 (t) [c,c + ] an > 0. Then we can have the following impulsive fuzzy control moel. Fig. 1. (a) Solution trajectory of the Lorenz system an (b) impulsive control result of fuzzy Lorenz system.

6 6 X. Zhang et al. / Chaos, Solitons an Fractals xxx (00) xxx xxx _xðtþ ¼A Rule i: If x 1 (t) ism i, then i xðtþ t 6¼ s j ;, i = 1, an j N, where x(t)=[x 1 (t),x (t),x 3 (t)] T, A 1 ¼ 4 1 a 0 5; A ¼ 4 1 a 0 5; b 0 b 0 M 1 ðx 1 ðtþþ ¼ 0:5 1 þ c x ; M ðx 1 ðtþþ ¼ 0:5 1 c x ; In this paper, a = 0.34, b = 0.4, c = 4.5 an = 10. In orer to impulsively control Rossler system we choose the remaining parameters as r = 0.005, n = 1., j = 00. Thus accoring to the propose algorithm we have P = [ , 7.313, 1.415; 7.313, ,0.4; 1.415,0.4,3.411], which implies that k max (P) = , k min (P) = , c = 5.977, b = 0.033(j N). Thus we have ki + Kk < Base on the inequality above, K i,j = iag([ 0.999, 0.999, 0.999]) (j N) such that lnðnbþþcr ¼ 3:3 < 0: Fig. 3a shows a solution trajectory of the Rossler system in the state space before applying any control strategy, while Fig. 3b shows the control result of fuzzy Rossler system starting at the initial conition [ 0.,0.6,0.]. Clearly, the time series of all the variables of this system converge to zero. In summary, accoring to the above simulation experiments, we think that the propose control moel for chaotic systems an the corresponing control methos can provie an effective metho controlling chaos. 5. Conclusion Fig.. (a) Solution trajectory of fuzzy Chua s oscillator an (b) impulsive control result of fuzzy Chua s oscillator. In this paper, an impulsive T-S fuzzy framework for controlling T-S fuzzy chaotic systems has been propose through combining impulsive control technique with intelligent fuzzy logic methoology. The propose control methos have the many avantages, such as, small impulse control, fuzzy intelligence an unifie control way. Then some new an less conservative criteria have been erive base on Lyapunov metho to guarantee the global asymptotic stability of the impulsive Fig. 3. (a) Solution trajectory of fuzzy Rossler system an (b) impulsive control result of fuzzy Rossler system.

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