Synchronization of unidirectional coupled chaotic systems with unknown channel time-delay: Adaptive robust observer-based approach

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1 Chaos, Solitons and Fractals (5) Synchronization of unidirectional coupled chaotic systems with unknown channel time-delay: Adaptive robust observer-based approach Jui-Sheng Lin a,b, Teh-Lu Liao a, *, Jun-Juh Yan b, Her-Terng Yau b a Department of Engineering Science, National Cheng Kung University, Tainan 71, Taiwan, ROC b Department of Electrical Engineering, Far-East College, Hsin-Shih Town, Tainan 744, Taiwan, ROC Accepted 8 January 5 Communicated by Y. Aizawa Abstract In this paper, an adaptive robust observer-based scheme for the synchronization of unidirectional coupled chaotic systems with unknown channel time-delay and system uncertainties is proposed. The effects of time-delay arise from the physical characteristics of coupled channel, while the system uncertainties arise due to unknown but bounded external disturbances and parametric perturbations. By appropriately selecting the observer controller and adaptation mechanism, the master slave chaotic synchronization can be guaranteed by Lyapunov approach. Finally, the ChuaÕs circuit is used as an illustrative example, where simulation results are given to demonstrate the effectiveness of the proposed scheme. Ó 5 Elsevier Ltd. All rights reserved. 1. Introduction Over the past decades, control and synchronization of chaotic systems has become more and more interesting topics to engineering and science communities since the pioneering work of Ott et al. [1]. Nowadays, chaos and its synchronization have found a lot of useful applications in many fields of engineering and science such as in secure communications, chemical reactions, power converters, biological systems, and information processing, etc. [ 4]. The typical configuration of chaotic synchronization consists of master and slave systems. The master system drives the slave system via a scalar signal transmitted through the coupled channel. In recent years, several adaptive and switching methods have been proposed and sufficient conditions are derived to synchronize the master slave chaotic systems [5 1]. Besides, several observer-based methods have also been extensively applied to the chaotic synchronization and control problems [11 15]. The slave system is designed as an observer, and all of the state variables of the slave system are constructed by the transmitted scalar signal from the master system. However, without channel time-delay, * Corresponding author. Tel.: x3337; fax: address: tlliao@mail.ncku.edu.tw (T.-L. Liao) /$ - see front matter Ó 5 Elsevier Ltd. All rights reserved. doi:1.11/j.chaos.5..5

2 97 J.-S. Lin et al. / Chaos, Solitons and Fractals (5) the above researches have shown that the synchronization between unidirectional coupled chaotic systems can be ensured under a suitable observer design. Nevertheless, from the viewpoint of engineering applications and characteristics of channel, a delay time always exists. When the unavoidable delay is taken into account, the synchronization problem of master slave systems is then considered as that of a class of coupled nonlinear time-delayed systems. For this problem, several theoretical analyses of phase sensitivity and destroying synchronization [1], and bifurcation phenomena [17] have been studied, as well as an experimental result has confirmed the synchronization of two coupled circuits with a channel delay [18]. Furthermore, the work [19] derived some sufficient conditions for synchronizing the master slave systems based on the Lyapunov Krasovskii approach. More recently, the work [] re-defined the chaotic synchronization in such a way that the state of slave system at time t asymptotically synchronizes that of the master system at time t s, that is lim kxðt sþ ^xðtþk ¼ t!1 where x and ^xðtþ are the state of master and slave systems, respectively, and s is a finite channel time-delay, which is often unknown. Inspired by the previous works, this paper addresses this practical issue in the synchronization of unidirectional coupled master slave chaotic systems with the unavoidable signal propagation time-delay. Based on Lyapunov stability theory, an adaptive robust observer-based slave system is designed to synchronize the master system in the presence of channel time-delay, even when the master system is perturbed by both unknown parametric uncertainties and external disturbances. Furthermore, we consider the ChuaÕs circuit as an illustrative example, and its numerical simulation results show that, in spite of different initial conditions and system uncertainties, the proposed adaptive robust observer-based synchronization scheme can be successfully applied to unidirectional coupled chaotic systems with channel time-delay. The organization of this paper is as follows. The system formulation of the master slave chaotic synchronization with channel time-delay and uncertainties is presented in Section. An adaptive robust observer-based slave system and the main results of this paper are derived in Section 3. The illustrative example and its numerical simulations that demonstrate the effectiveness of the proposed scheme are given in Section 4. Finally, conclusions are made in Section 5. Throughout this paper, it is noted that k(w) denotes an eigenvalue of W and k max (W) represents the max [k i (W)],i =1,...,n, and kwk represents the Euclidean norm when W is a vector or the induced norm when W is a matrix. However, sign(s) is the sign function of s, ifs >, sign(s) = 1; if s =, sign(s) = ; and if s <, sign(s)= 1.. System formulation In general, chaotic systems are described by a set of nonlinear differential equations. Moreover, in many cases, the dynamical equations can be separated into a linear dynamics with respect to state variables and a nonlinear feedback part with respect to the system states. Here we will consider the chaotic system subjected to unknown parametric perturbation and external disturbances, therefore, the chaotic dynamics can be described by the following equation: _xðtþ ¼ AxðtÞþ f ðxðtþþ þ B/ðxðtÞ; dðtþþ ð1aþ yðtþ ¼CxðtÞ ð1bþ where x R n is the state vector, y R m is the output vector. f(x) R n 1 represents the continuous nonlinear function. d R p 1 is a bounded vector of external disturbances. /(x,d) R m 1 represents a nonlinear vector field that may include parametric perturbations and external disturbances. A and C are two constant matrices of appropriate dimensions, and B R n m is the injection map of the unknown dynamics. Remark 1. A wide variety of chaotic systems can be represented by the form of (1) without the term of /(x,d). For examples, Rössler system [1], the MCK circuit [], ChauÕs circuit [3], Duffing-Holmes system [4], Lorzen system [5], Lü system [], Chen chaotic dynamical system [7] and the oscillators reported in [8] all belong to the class of systems defined by Eq. (1a). For the system (1), we make the following assumptions. Assumption A.1. There is a bounded region C R n 1 containing the whole attractor of system (1) such that every orbit of system (1) never leaves it. Thus k/ðxðtþ; dðtþþk < b < 1 ðaþ where b >, is an unknown and sufficiently large constant.

3 J.-S. Lin et al. / Chaos, Solitons and Fractals (5) Assumption A.. The nonlinear function f(x(t)) with f() = satisfies the Lipschitz condition, and there exists a positive constant c such that kf ðxðt 1 ÞÞ f ðxðt ÞÞk ckxðt 1 Þ xðt Þk ðbþ Assumption A.3. There exist a gain matrix L and two positive definite matrices P and Q such that the following conditions hold: ða LCÞ T P þ PðA LCÞ ¼ Q B T P ¼ C c < k minðqþ k max ðpþ ð3þ ð4þ ð5þ Remark. (i) Since the states of chaotic systems are bounded and all external disturbances are bounded, the uncertainty /(x, d) is then bounded. Hence, the Assumption A.1 is satisfied. (ii) Eqs. (3) and (4) will hold if the linear system: _e ¼ðA LCÞe þ Bu; y ¼ Ce is a strictly positive real system. Herein, taking the channel time-delay into account, the delayed signal y(t s) will drive the slave system. An observer-based slave system is then designed as follows: _^xðtþ ¼A^xðtÞþf ð^xðtþþ þ Lðyðt sþ ^yðtþþ þ uðtþ ^yðtþ ¼C^xðtÞ ðaþ ðbþ where ^x indicates the dynamic estimate of the state x, and s is unknown but constant channel time-delay, and the constant vector L R n m is observer gain such that (A LC) is an exponentially stable matrix, which is possible since the pair (C, A) is observable. While u(t) is a feedback control law, which will be appropriately designed to achieve the synchronization. The main aim of this paper is to design a suitable control law u(t) using the only available delayed signal y(t s) and the output of observer-based slave system ^yðtþ so that x(t s) of the master system will asymptotically synchronize with ^xðtþ of the observer-based slave system even under the effect of unknown external disturbances and parametric perturbations, that is kxðt sþ ^xðtþk! as time t!1. 3. Adaptive robust observer design and main results If the upper bound of the uncertainty /(x,d) is known, i.e. b is known in priori, then a robust control law u(t)in(a) can be designed to overcome the effect of unknown parametric perturbations and unknown external disturbances, and further to achieve synchronization. The main problem is that the b is unknown. Therefore, an adaptive robust control law for the observer-based slave system is needed. Herein, a robust control law is designed as follows: uðtþ ¼w^bðtÞB½signðyðt sþ ^yðtþþš ð7þ where w is an arbitrarily chosen parameter such that w > 1, and ^b is an estimate of b. However, the estimated ^b is updated on-line by the following adaptation law: _^bðtþ ¼q 1 kyðt sþ ^yðtþk; ^bðþ ¼^b ð8þ where ^b > is the bounded initial values of ^bðtþ, while q is a positive constant specified by the designer. Fig. 1 depicts the structure of unidirectional coupled chaotic systems proposed in this work. For further analysis of stability and synchronization, let us define the state error vector between the master system at time t s and the observer-based slave system at timet as follows: eðtþ ¼xðt sþ ^xðtþ ð9þ Now, we replace t with t s in (1), and then obtain the master system equation at time t s as follows: _xðt sþ ¼Axðt sþþf ðxðt sþþ þ B/ðxðt sþ; dðt sþþ ð1þ

4 974 J.-S. Lin et al. / Chaos, Solitons and Fractals (5) uncertainties φ( x, d(t)) master system y(t) unknown delay y( t τ ) adaptation mechanism βˆ(t) ŷ(t) + + slave system u(t) xˆ ( t) x( t τ ) synchronization control law u(t) Structure of the observer Fig. 1. The unidirectional coupled chaotic systems with channel delay. Then, from Eqs. () (8), the following error equation is obtained: _eðtþ ¼ðA LCÞeðtÞþfðxðt sþþ f ð^xðtþþ þ B/ðxðt sþ; dðt sþþ w^bðtþb½signðyðt sþ ^yðtþþš Now, the main result of this paper is stated by the following theorem. Main Theorem. Consider the master chaotic system (1) satisfying the Assumptions A.1 A.3. If the robust observer-based slave system is designed by () and (7) with the adaptation law (8), then the synchronization of unidirectional coupled chaotic systems with channel time-delay is achieved, i.e. keðtþk ¼ kxðt sþ ^xðtþk ¼ as t!1. Before proceeding to the proof of our main theorem, the following Barbalat lemma is necessary. Lemma 1 (Barbalat lemma, [9]). If w: R! R + is a uniformly continuous positive function for t P and if the limit of the integral lim t!1 Z t wðkþdk exists and is finite, then lim wðtþ ¼ ð13þ t!1 Proof of main theorem. Consider the following Lyapunov function candidate V ðtþ ¼e T ðtþpeðtþþqðb ^bðtþþ ð14þ where P is the positive definite solution of Eq. (3). It can easily verify that V(t) is a non-negative function and that it is radically unbounded, i.e. V(t)! + 1 as e(t) and ðb ^bðtþþ! þ1. Taking the time derivative of V(t) along with the dynamics (8) and (11) yields _V ðtþ ¼e T ðtþ½ða LCÞ T P þ PðA LCÞŠeðtÞþe T ðtþp½f ðxðt sþþ f ð^xðtþþš þ e T ðtþpb½/ðxðt sþ; dðt sþþ ð11þ ð1þ w^bðtþsignðyðt sþ ^yðtþþš þ qðb ^bðtþþð _^bðtþþ ð15þ By noting that e T ðtþp½f ðxðt sþþ f ð^xðtþþš ck max ðpþkeðtþk ð1þ

5 J.-S. Lin et al. / Chaos, Solitons and Fractals (5) and e T PB ¼ e T ðtþc T ¼ ðyðt sþ ^yðtþþ T e T ðtþpb½/ðxðt sþ; dðt sþþ w^bðtþsignðyðt sþ ^yðtþþš þ qðb ^bðtþþð _^bðtþþ ð17þ kðyðt sþ ^yðtþþkðb ^bðtþþ ðw 1Þ^bðtÞkðyðt sþ ^yðtþþk qðb ^bðtþþð _^bðtþþ ¼ ðw 1Þ^bðtÞkðyðt sþ ^yðtþþk in which Eq. (a) has been applied, then we can further obtain _V ðtþ ðk min ðqþ ck max ðpþþkeðtþk ¼ wðtþ where w(t)=(k min (Q) ck max (P))ke(t)k. Integrating the above equation from zero to t yields V ðþ P V ðtþþ Z t wðkþdk P Z t wðkþdk As t goes infinite, the above integral is always less than or equal to V(). However, V() is positive and finite, thus according to Lemma 1, we obtain lim wðtþ ¼lim ðk minðqþ ck max ðpþþkeðtþk ¼ ð1þ t!1 t!1 Furthermore, since (k min (Q) ck max (P)) >, Eq. (1) implies lim eðtþ ¼. It is obvious that the asymptotic stability of the overall system is guaranteed, and the synchronization of unidirectional coupled systems (1) and () (8) is con- t!1 sequently achieved. This completes the proof. h ð18þ ð19þ ðþ 4. An illustrative example: Chua s circuit In order to show the effectiveness of the proposed scheme, the ChuaÕs circuit is used as an illustrative example. The nonlinear master system with uncertainties is given by 8 >< _x 1 ¼ 1ðx x 1 f ðx 1 ÞÞ þ dðtþ _x ¼ x 1 x þ x 3 ðþ >: _x 3 ¼ 15x :385x 3 where f(x 1 )=bx 1 +.5(a b) Æ (jx 1 +1j jx 1 1j) denote a three-segment piecewise linear function in which a and b are two negative real constants, It has been shown that if a and b are appropriately chosen, then the system () possesses a chaotic behavior. Herein, we set a = 1.8 and b =.9 and define y = x 1, then Eq. () can be rewritten as _x ¼ x þ 4 5ð:9x 1 þ :95ðjx 1 þ 1j jx 1 1jÞ þ dðtþþ ¼ Ax þ B/ðx; dðtþþ ð3aþ 15 :385 y ¼ x 1 ¼ ½1 Šx ¼ Cx ð3bþ where the uncertainty /(x,d) arose from the unknown constants a and b or from fluctuations of constants a and b, and d(t) =.(x 1 + x ) +.1 denotes external disturbance. As shown in Fig., the system (3) processes a chaotic phenomenon with bounded double scroll attractor. Therefore, there exists a sufficiently positive constant b < 1 such that the Assumption A.1 is satisfied. Now to establish the slave system, we first select L T ¼ ½ 9 1:883 :44 Š ð4þ It can be shown that (A,B) is a controllable pair and (C,A) is an observable pair. Also the eigenvalues of matrix A LC are.9737 and.534 ± j Moreover, the following symmetric and positive-definite matrices P ¼ 4 15:37 :958 5; Q ¼ 4 5 ð5þ :958 1:91

6 97 J.-S. Lin et al. / Chaos, Solitons and Fractals (5) x Fig.. The chaotic behavior of the system (3) in x 1 x plane with initial condition x() = [.1.1.1] T. x 1 x (t ), xˆ (t) 1 τ (a) Time (sec) x (t τ ), xˆ (t) -.5 (b) Time (sec) 4 x (t ), ˆ (t) 3 τ x (c) Time (sec) Fig. 3. State trajectories of x 1 (t s), x (t s), x 3 (t s) (dashed) and ^x 1 ðtþ; ^x ðtþ; ^x 3 ðtþ (solid), respectively. are selected to satisfy the Assumption A.3. By choosing w =>1,^b ¼ 3 >, q =., the adaptive robust observerbased slave system is derived as follows:

7 J.-S. Lin et al. / Chaos, Solitons and Fractals (5) βˆ (t) Time (sec) Fig. 4. The time response of the estimated parameter ^bðtþ _^x ¼ x þ 4 1:83 5ðyðt sþ ^yðtþþ þ ^bðtþ 4 5signðyðt sþ ^yðtþþ 15 :385 :44 with the following update law ðþ _^bðtþ ¼5kyðt sþ ^yðtþk; ^bðþ ¼3 ð7þ Numerical simulations are performed with the following parameters: channel time-delay s =.1 and initial values x 1 () =.1, x () =.1, x 3 () =.1 and ^x 1 ðþ ¼; ^x ðþ ¼ 1; ^x 3 ðþ ¼.Fig. 3 shows the state trajectories of the master system at time t s and that of the observer-based slave system at timet that also verifies the chaotic synchronization under this proposed scheme. The time response of the estimated parameter ^bðtþ is depicted in Fig. 4. It is obvious that the estimated ^bðtþ converges to a constant value. All results demonstrate the effectiveness of the synchronization scheme proposed in this paper. 5. Conclusions This paper has developed an adaptive robust observer-based approach to solve the synchronization problem of a class of unidirectional coupled chaotic systems in the presence of channel time-delay and systemõs uncertainties including parametric perturbations and external disturbances. Given certain structural conditions of the master system, an adaptive robust observer-based slave system was constructed so that the master slave synchronization with some lag due to the channel delay can be achieved. Both synchronization and stability are guaranteed by the Lyapunov stability theory. Numerical simulations are given to verify the theoretical results. References [1] Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 199;4: [] Carroll TL, Pecora LM. Synchronizing chaotic systems. IEEE Trans Circuits Syst I 1991;38:453. [3] Chen G, Dong X. From chaos to order: methodologies, perspectives and applications. Singapore: World Scientific; [4] Nayfeh AH. Applied nonlinear dynamics. New York: Wiley; [5] Wang Y, Guan ZH, Wen X. Adaptive synchronization for Chen chaotic system with fully unknown parameters. Chaos, Solitons & Fractals 4;19: [] Chua LO, Yang T, Zhong GQ, Wu CW. Adaptive synchronization of ChuaÕs oscillators. Int J Bifurcat Chaos 199;(1): [7] Liao TL. Adaptive synchronization of two Lorenz systems. Chaos, Solitons & Fractals 1998;9: [8] Lian KY, Liu P, Chiang TS, Chiu CS. Adaptive synchronization design for chaotic systems via a scalar driving signal. IEEE Trans Circuits Syst I ;49:17 7.

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