H Synchronization of Chaotic Systems via Delayed Feedback Control

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1 International Journal of Automation and Computing 7(2), May 21, DOI: 1.17/s H Synchronization of Chaotic Systems via Delayed Feedback Control Li Sheng 1, 2 Hui-Zhong Yang 1 1 Institute of Measurement and Process Control, Jiangnan University, Wuxi , PRC 2 Institute for Systems Research, University of Maryland, College Park 274, USA Abstract: The H synchronization problem for a class of delayed chaotic systems with external disturbance is investigated. A novel delayed feedback controller is established under which the chaotic master and slave systems are synchronized with a guaranteed H performance. Based on the Lyapunov stability theory, a delay-dependent condition is derived and formulated in the form of linear matrix inequality (LMI). A numerical simulation is also presented to validate the effectiveness of the developed theoretical results. Keywords: Chaotic systems, H synchronization, delayed feedback control, Lyapunov theory, linear matrix inequality (LMI). 1 Introduction Since the pioneering work of Pecora and Carroll in 199 1, the synchronization of coupled chaotic systems has been intensively investigated due to its potential applications in various fields such as biology 2, motion control 3, secure communication 4,5, and so on. Based on various control theories, a number of synchronization schemes such as impulsive control 6, time-delay feedback approach 7, adaptive active control 8, and sliding mode control design 9 have been proposed in the literature. In remote master-slave synchronization schemes, time delays are often encountered due to the information transmission between the transmitter and receiver. Time delay always appears in transmission-reception systems owing to the finite speed of information processing, and it may destroy the synchronization if it is beyond a certain limit. Recently, there have been some research efforts to investigate the delay effect on master-slave synchronization. In 1, some delay-dependent criteria and a delayed feedback control method were proposed to investigate the master-slave synchronization problem for chaotic systems. In 11, some new delay-dependent synchronization criteria for Lur e systems were developed without employing any model transformation based on a more general Lur e-postnikov Lyapunov functional. On the other hand, some noises or disturbances always exist in real systems that may cause instability and poor performance. Therefore, the effect of the noises or disturbances must be reduced in the synchronization process for chaotic systems. Recently, Hou et al. 12 firstly adopted the H control concept to reduce the effect of disturbance for the chaotic synchronization problem. In 13, 14, the authors designed several new controllers for the H synchronization of the chaotic systems with external disturbance. However, in these recent publications, the transmission delay was not taken into account in the synchronization Manuscript received December 5, 28; revised May 27, 29 This work was supported by National Natural Science Foundation of China (No ). scheme. With the above motivation, the aim of this paper is to investigate the H synchronization problem for the delayed chaotic systems with disturbance. Based on the Lyapunov stability theory and LMI technique, the novel delayed feedback controller is designed which not only guarantees the synchronization of both master and slave systems but also reduces the effect of external disturbance on an H norm constraint. Throughout this paper, R n and R n m denote, respectively, the n-dimensional Euclidean space and the set of all n m real matrices. For real symmetric matrices X and Y, the notation X Y (X > Y ) means that the matrix X Y is positive semi-definite (positive definite, respectively). Let L 2, ) be the space of square-integrable vector functions over, ) with the norm ϕ(t) 2 = ϕ T (t)ϕ(t)dt. The symbol within a matrix represents the symmetric term of the matrix. 2 Problem formulation and some preliminaries Consider the master-slave synchronization scheme of delayed chaotic systems as follows: ẋ(t) = A x(t) + A 1f(x(t))+ M : A 2f(x(t τ)) p(t) = Hx(t) ẏ(t) = A y(t) + A 1f(y(t)) + u(t)+ S : A 2f(y(t τ)) + Dw(t) q(t) = Hy(t) C : u(t) = K(x(t) y(t)) + (1) (2) M(p(t τ) q(t τ)) (3)

2 L. Sheng and H. Z. Yang / H Synchronization of Chaotic Systems via Delayed Feedback Control 231 with master system M, slave system S, and controller C, where the slave system S is subject to an input noise. The master and slave systems are delayed chaotic systems with state vectors x(t), y(t) R n, and the output vectors p(t), q(t) R l, respectively. w(t) R m is a disturbance term belonging to L 2, ). The matrices A, A 1, A 2 R n n, D R n m, and H R l n are known constant matrices. τ is a constant time delay. f( ) = f 1( ), f 2( ),, f n( ) T and f i( ) : R R (i = 1, 2,, n) is a nonlinear function, and we assume that it is globally Lipschitz continuous (g.l.c): (H) f i(ξ 1) f i(ξ 2) L i ξ 1 ξ 2, ξ 1, ξ 2 R n (4) where L i >, i = 1, 2,, n. Let L = diag(l 1, L 2,, L n). Remark 1. The function f i(ξ 2) : R R (i = 1, 2,, n) is said to be locally Lipschitz continuous (l.l.c) 15 in R if for any ξ 2 R, there exists a positive constant L iξ2 > and some δ > such that f i(ξ 1) f i(ξ 2) L iξ2 ξ 1 ξ 2, ξ 1 R satisfying ξ 1 ξ 2 δ, i = 1, 2,, n. If δ = + and L iξ2 > is independent of ξ 2, then the l.l.c function f i(ξ 2) becomes a g.l.c function. Obviously, the l.l.c functions are more general than the g.l.c functions. If the nonlinear function f i( )(i = 1, 2,, n) is l.l.c instead of g.l.c and the synchronization error is sufficiently small, we can derive some local synchronization conditions for the chaotic systems. Therefore, the local or global synchronization depends on each particular problem. The scheme aims at synchronizing the master system M to the noise-perturbed slave system S with a guaranteed H performance by applying the delayed feedback controller u(t) R n. K R n n and M R n l are the feedback gain matrices. Let e(t) = y(t) x(t) be the synchronization error; then the error dynamics between (1) and (2) can be expressed by ė(t) = (A + K)e(t) + A 1g(e(t))+ A 2g(e(t τ)) MHe(t τ) + Dw(t) z(t) = He(t) where z(t) = q(t) p(t), g(e(t)) = f(y(t)) f(x(t)), and g(e(t τ)) = f(y(t τ)) f(x(t τ)). According to the condition (4), g( ) = g 1( ), g 2( ),, g n( ) T possesses the following property: (5) g i(e i(t)) L i e i(t) i = 1, 2,, n. (6) To give our main results, we need the following definition and lemmas. Definition Let γ > be a given scalar for the guaranteed H disturbance attenuation level. The masterslave synchronization scheme (1) (3) synchronizes, and the error system (5) is globally asymptotically stable with guaranteed H performance, if the following conditions are satisfied: 1) With zero disturbance, the synchronization error system (5) is globally asymptotically stable. 2) With zero initial condition, the following condition holds: J = ( i.e., z T (t)z(t) γ 2 w T (t)w(t) dt, sup w,w L 2, z(t) 2 w(t) 2 γ ). (7) Remark 2. It is noted that, as K = and τ =, the above synchronization system reduces to the one studied in 12. In this case, the difference between this work and 12 is the time-delay in the outputs, i.e., p(t τ) and q(t τ) instead of p(t) and q(t), respectively. With M =, the H synchronization of chaotic systems has been studied in 13 and 14. Lemma (Schur complement) Given constant matrices Ω 1, Ω 2, and Ω 3, where Ω 1 = Ω T 1 and < Ω 2 = Ω T 2, then Ω 1 + Ω T 3 Ω 1 2 Ω3 < if and only if Ω 1 Ω T 3 Ω 3 Ω 2 < or Ω 2 Ω 3 Ω T 3 Ω 1 <. Lemma For real matrices Σ >, Π, and a vector ξ(t) with appropriate dimensions, the integral inequality holds. β α ė T (s)σė(s)ds β (β α)ξ T (t)π T Σ 1 Πξ(t) + 2ζ T (t)π T ė(s)ds 3 Main results In this section, by using the Lyapunov functional method, a delay-dependent synchronization criterion is proposed for the stability of the error system (5), and thus, the slave system (2) can robustly track the master system (3) with a guaranteed H performance γ. Theorem 1. Under condition (4), for given ε, τ, and γ, the master system (1) and the slave system (2) are synchronized with a guaranteed H performance γ, if there exist matrices P >, Q >, R >, S >, T >, X = X 1X 2 X 6, U, F, and G such that the following LMI holds Θ 11 Θ 12 Θ 13 Θ 14 Θ 15 Θ 16 τx 1 Θ 22 X 3 X 4 Θ 25 X 6 τx 2 S A T 1 U τx 3 T A T 2 U τx 4 Θ 55 UD τx 5 γ 2 I τx 6 τ R where α < (8)

3 232 International Journal of Automation and Computing 7(2), May 21 Θ 11 = X 1 + X T 1 Θ 12 = X T 1 + Q εua εf εa T U T εf T + LSL + H T H Θ 13 = X 3 εua 1 Θ 14 = X 4 εua 2 + X 2 + εgh Θ 15 = P + X 5 + εu Θ 16 = X 6 εud Θ 22 = Q + LT L X 2 X T 2 Θ 25 = X 5 + H T G T Θ 55 = U + U T + τr. Moreover, the feedback gain matrices K and M can be designed as K = U 1 F and M = U 1 G. Proof. Since LMI (8) implies that Θ 55 = U +U T +τr <, together with τ > and R >, we know U + U T <. Therefore, the matrix U is nonsingular. The proof of this theorem is divided into two parts. First, we will show the global asymptotic stability of the error system (5) with w(t) =. Second, the inequality (7) will be proven under the given condition. When w(t) =, system (5) is described by ė(t) = (A + K)e(t) + A 1g(e(t)) + A 2g(e(t τ)) MHe(t τ). (9) Consider the Lyapunov functional V (t) = e T (t)p e(t) + e T (s)qe(s)ds + t τ τ t+θ ė T (s)rė(s)dsdθ. (1) Computing the time derivative of V (t) along the solution of (5), we have ė T (s)rė(s)ds τζ T (t) ˆX T R 1 ˆXζ(t) + t τ From (6), we have gi 2 (e i(t)) 2ζ T (t) ˆX T (e(t) e(t τ)). (12) L 2 i e 2 i (t) gi 2 (e i(t τ)) L 2 i e 2 i (t τ). Then, we can easily obtain the following inequalities: e T (t)lsle(t) g T (e(t))sg(e(t)) (13) e T (t τ)lt Le(t τ) The equation (9) ensures that g T (e(t τ))t g(e(t τ)). (14) ė(t) 2 e T (t)y 1 + ė T (t)y 2 (A + K)e(t) A 1g(e(t)) A 2g(e(t τ)) + MHe(t τ) where Y 1 and Y 2 are any appropriate matrices. To facilitate the design of the feedback gain matrices K and M, Y 1 and Y 2 are chosen as εu and U, respectively, with ε being a constant scalar. Then, we have ė(t) 2 e T (t)εu + ė (t)u T (A + K)e(t) A 1g(e(t)) A 2g(e(t τ)) + MHe(t τ). Substituting (12) (15) into (11) yields (15) V (t) 2e T (t)p ė(t)+e T (t)qe(t) e T (t τ)qe(t τ) + τė T (t)rė(t) + τζ T (t) ˆX T R 1 ˆXζ(t) + V (t) = 2e T (t)p ė(t) + e T (t)qe(t) e T (t τ)qe(t τ) + τė T (t)rė(t) ė T (s)rė(s)ds. (11) t τ Define ζ T (t) = e T (t) e T (t τ) g T (e(t)) g T (e(t τ)) ė T (t). According to Lemma 2, for any matrix ˆX = X 1 X 2 X 3 X 4 X 5 with appropriate dimensions, we have the following inequality where 2ζ T (t) ˆX T (e(t) e(t τ)) + e T (t)lsle(t) g T (e(t))sg(e(t)) + e T (t τ)lt Le(t τ) g T (e(t τ))t g(e(t τ)) + 2 e T (t)εu + ė T (t)u ė(t) (A + K)e(t) A 1g(e(t)) A 2g(e(t τ)) + MHe(t τ) = ζ T (t) Ψ 1 + τ ˆX T R 1 ˆX ζ(t) (16)

4 L. Sheng and H. Z. Yang / H Synchronization of Chaotic Systems via Delayed Feedback Control 233 Υ 11 Υ 12 Θ 13 Θ 14 Θ 15 Θ 22 X 3 X 4 Υ 25 Ψ 1 = S A T 1 U T A T 2 U Υ 11 = X 1 + X T 1 Υ 12 = X T 1 Θ 55 + Q εu(a + K) ε(a + K) T U T + LSL + X 2 + εumh Υ 25 = X 5 + H T M T U T. It follows from the LMI (8) that Θ 11 Θ 12 Θ 13 Θ 14 Θ 15 τx 1 Θ 22 X 3 X 4 Θ 25 τx 2 S A T 1 U τx 3 <. (17) T A T 2 U τx 4 Θ 55 τx 5 τ R Applying the changes of variables K = U 1 F, M = U 1 G and considering Lemma 1, the LMI (17) is equivalent to Ψ 1+τ ˆX T R 1 ˆX <. Then, V (t) < for all ζ(t). According to the Lyapunov theory, the system (5) is globally asymptotically stable. Next, we will prove that (7) is satisfied under zero initial condition for all nonzero w(t) L 2, ). Define a cost function as follows: J(t) = z T (t)z(t) γ 2 w T (t)w(t) dt. (18) Considering V (t) in (1), we have V () = and V (t) for t > under the zero initial conditions. Therefore, J(t) z T (t)z(t) γ 2 w T (t)w(t) dt + V (t) t V () = z T (t)z(t) γ 2 w T (t)w(t) + V (t) dt. (19) Taking the time derivative of V (t) along the solution of (5), and similar to the proof of (16), it is easy to derive that z T (t)z(t) γ 2 w T (t)w(t) + V (t) ξ T (t) Ψ 2 + τx T R 1 X ξ(t) (2) where ξ T (t) = e T (t) e T (t τ) g T (e(t)) g T (e(t τ)) ė T (t) w T (t), Υ 11 Υ 12 Θ 13 Θ 14 Θ 15 Θ 16 Θ 22 X 3 X 4 Υ 25 X 6 S A T 1 U Ψ 2 = T A T 2 U Θ 55 UD γ 2 I Υ 11 = X 1 + X T 1 + Q εu(a + K) ε(a + K) T U T + LSL + H T H. Similarly, by utilizing the changes of variables K = U 1 F, M = U 1 G and using Lemma 1, the LMI (8) is equivalent to Ψ 2 + τx T R 1 X <. Remark 3. Although the H synchronization problem was discussed in 13,14, some of the results were expressed in terms of matrix inequalities, which are still difficult to handle because of their nonlinearity. In addition, the transmission delay was not introduced in the synchronization scheme in 13,14. In this paper, Theorem 1 provides a delay-dependent synchronization criterion which guarantees the H synchronization between the master system and the slave system. Moreover, several free weighting matrices X, U are introduced into the LMI (8) by employing the equations (12) and (15). These free weight matrices are not even required to be symmetric. Therefore, Theorem 1 is less conservative than some existing results due to the increasing freedom of these free weight matrices. The design of suitable delayed feedback gain matrices K and M can be realized by means of a feasible solution to the LMI (8), which can be solved efficiently via Matlab LMI Toolbox. 4 A simulation example In this section, we give an example to demonstrate the H synchronization performance of the proposed control scheme. Example 1. Consider a delayed chaotic neural network as below { ẋ(t) = A x(t) + A 1f(x(t)) + A 2f(x(t τ)) p(t) = Hx(t) with parameters A = A 2 = 1, A 1 = 1 1.3π 2/ π 2/4 (21) 1+π/4 2,.1 1+π/4, H = 1, τ = 1, where x(t) = x 1(t), x 2(t) T is the state vector of the network, and f i(x) = ( x + 1 x 1 )/2 (i = 1, 2) satisfies condition (4) with L = diag(1, 1). The model (21) has a chaotic attractor with the initial condition x 1(s), x 2(s) T =.1,.1 T, s 1,, which can be seen in Fig. 1.

5 234 International Journal of Automation and Computing 7(2), May T = U = F = G = Fig. 1 The phase plot of master system (21) Taking (21) as the master system, the following slave system is designed for synchronization: ẏ(t) = A y(t) + A 1f(y(t)) + A 2f(y(t τ))+ u(t) + Dw(t) q(t) = Hy(t) (22) with D = 1 1 T, where u(t) is the delayed feedback controller, and w(t) is a Gaussian noise with mean = and variance = 1. When the initial condition is chosen as y 1(s), y 2(s) T =.2,.2 T, s 1,, the chaotic behavior of the slave system (22) is shown in Fig. 2. The delayed feedback controller is then given by u(t) = Ke(t) MHe(t τ) (23) where e(t) = y(t) x(t), and the feedback gain matrices are K = U F = M = U G = According to Theorem 1, the synchronization of systems (21) and (22) can be achieved via the delayed feedback controller (23). Without disturbance signal w(t) and by applying the delayed feedback controller (23), the synchronization errors between the master system and the slave system are depicted in Fig. 3. Fig. 2 The phase plot of slave system (22) with disturbance w(t) and without controller u(t) Let ε = 1 and γ =.3. By using Matlab LMI Control Toolbox, we found a possible solution set of the LMI (8) given in Theorem 1: P = Q = R = S = Fig. 3 The time responses of synchronization errors of systems (21) and (22) without disturbance signal w(t) To observe the H performance with disturbance attenuation, the response of the controlled output error z(t) is shown in Fig. 4. The dynamic H controller (23) reduces the effect of the disturbance input w(t) on the controlled output error z(t) and limits it within a prescribed level γ =.3.

6 L. Sheng and H. Z. Yang / H Synchronization of Chaotic Systems via Delayed Feedback Control M. Gao, B. T. Cui. Adaptive synchronization in an array of asymmetric coupled neural networks. Chinese Physics B, vol. 18, no. 1, pp , X. Y. Lou, B. T. Cui. Robust adaptive synchronization of chaotic neural networks by slide technique. Chinese Physics B, vol. 17, no. 2, pp , J. Cao, H. X. Li, D. W. C. Ho. Synchronization criteria of Lur e systems with time-delay feedback control. Chaos, Solitons and Fractals, vol. 23, no. 4, pp , Q. L. Han. New delay-dependent synchronization criteria for Lur e systems using time delay feedback control. Physics Letters A, vol. 36, no. 4 5, pp , 27. Fig. 4 The time response of output error z(t) of systems (21) and (22) with disturbance signal w(t) and zero initial condition 5 Conclusions In this paper, the problem of H synchronization for a class of delayed chaotic systems is investigated. Based on Lyapunov theory and LMI formulation, a delayed feedback control scheme has been proposed under which the controlled slave system can track the master system with a guaranteed H performance. Finally, a numerical example is given to show the effectiveness and feasibility of the obtained results. One of the future research topics is to investigate the H lag synchronization of chaotic systems. References 1 L. M. Pecora, T. L. Carroll. Synchronization in chaotic systems. Physical Review Letters, vol. 64, no. 8, pp , J. J. Fox, C. Jayaprakash, D. L. Wang, S. R. Campbell. Synchronization in relaxation oscillator networks with conduction delays. Neural Computation, vol. 13, no. 5, pp , J. Ren, C. W. Li, D. Z. Zhao. CAN-based synchronized motion control for induction motors. International Journal of Automation and Computing, vol. 6, no. 1, pp , Y. Dai, Y. Z. Cai, X. M. Xu. Synchronization and exponential estimates of complex networks with mixed time-varying coupling delays. International Journal of Automation and Compuating, vol. 6, no. 3, pp , T. Yang, L. O. Chua. Impulsive stabilization for control and synchronization of chaotic system: Theory and application to secure communication. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 44, no. 1, pp , L. Sheng, H. Yang. Exponential synchronization of a class of neural networks with mixed time-varying delays and impulsive effects. Neurocomputing, vol. 71, no , pp , J. Sun. Delay-dependent stability criteria for time-delay chaotic systems via time-delay feedback control. Chaos, Solitons and Fractals, vol. 21, no. 1, pp , Y. Y. Hou, T. L. Liao, J. J. Yan. H synchronization of chaotic systems using output feedback control design. Physica A: Statistical Mechanics and Its Applications, vol. 379, no. 1, pp , S. M. Lee, D. H. Ji, J. H. Park, S. C. Won. H synchronization of chaotic systems via dynamic feedback approach. Physics Letters A, vol. 372, no. 29, pp , J. H. Park, D. H. Ji, S. C. Won, S. M. Lee. H synchronization of time-delayed chaotic systems. Applied Mathematics and Computation, vol. 24, no. 1, pp , J. C. Butcher. The Numerical Analysis of Ordinary Differential Equations, Chichester, UK: Wiley, S. Boyd, L. E. Ghaoui, E. Feron, V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory, Philadephia, Egypt: SIAM, H. Huang, G. Feng. Robust H synchronization of chaotic Lur e systems. Chaos, vol. 18, No , pp. 1 1, 28. Li Sheng received the B. Sc. and M. Sc. degrees from Shandong Normal University, PRC. Currently, he is a Ph. D. candidate at Institute of Measurement and Process Control, Jiangnan University, PRC, and a visiting student at the Institute for Systems Research (ISR), University of Maryland, USA. His research interests include neural networks, chaos synchronization, and stability theory. victory829@yahoo.com.cn (Corresponding author) Hui-Zhong Yang received the M. Eng. and D. Eng. degrees in control theory and control engineering from South China University of Technology, PRC in 1989 and East China University of Science and Technology, PRC in 21, respectively. She is currently a professor at School of Communication and Control Engineering, Jiangnan University, PRC. Her research interests include modeling and monitoring of chemical process, optimization control, and data fusion. yhz jn@163.com