Robust chaos synchronization of noise-perturbed chaotic systems with multiple time-delays

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Abstract

The aim of this paper is to propose an output coupling and feedback scheme, which is not only to guarantee the asymptotic synchronization between the master and the slave chaotic systems with multiple time-delays but also to attenuate the effects of noise perturbation on the overall error system to a prescribed level in terms of the performance index H-norm. The output coupling and feedback gain is derived on the basis of the Lyapunov theory and the linear matrix inequality (LMI) technique. Some numerical examples are given to demonstrate the effectiveness of the main results.

Introduction

During the past 20 years, many famous chaotic systems such as Lorenz system [1], Chua’s circuit [2], Chen system [3], Rössler system [4] and so on have been proposed and their complex behaviors have also been widely studied. Pecora and Carroll [5] first introduced a synchronization methodology for two unidirectional coupled chaotic systems. After that, many efficient schemes and techniques for controlling chaos synchronization such as adaptive control [6], fuzzy control [7], digital control [8], state feedback control [9], sampled driving signals [10], time-delay feedback control [11], and observer-based approach [12], etc have been developed. Furthermore, some attractive results of synchronization and applications in secure communication for a class of chaotic systems have been reported [13], [14], [15], [16].

For dynamic systems, the effect of time-delay frequently causes system instability and decreases system performance. The stability analysis and control design of time-delay systems has been of great interest to many scientists and engineers over the past years. For chaotic systems with time-delay, several works have proposed the problem for various chaotic systems in the literature [17], [18], [19], [20], [21]. In Refs. [17], [18], the design method of guaranteed control for stabilizing time-delay chaotic systems has been proposed. In Ref. [19], a time-delay master–slave chaos synchronization of two coupled systems using the unidirectional linear error feedback scheme has been developed. Controlling chaos of the Chen–Lee system with multiple time-delays has been studied in Ref. [20]. Based on the Lyapunov exponent and the Galerkin projection technique, the stability and chaos control of multiple time-delays Rössler system was analyzed in Ref. [21]. In recent years, the H control has been widely applied to stabilize a class of uncertain time-delay systems dealing with external noise and disturbance in Refs. [22], [23], [24], [25]. The H control was also proposed to reduce the effect of the noise/disturbance input to the overall system error state or the regulated output to within a prescribed level [26]. More recently, the H synchronization problem for a general class of chaotic systems without time-delays in the state had been developed in Ref. [27]. However, to the best of our knowledge, the H control problem for a class of chaotic systems with multiple time-delays had not been considered in the literature. The purpose of this study is to design a robust output coupling and feedback scheme, which is not only to asymptotically synchronize the master and the slave systems with multiple time-delays but also to guarantee the attenuation performance of noise perturbation within a prescribed index γ. Two illustrative examples, the Rössler and the Chen chaotic systems with multiple time-delays are presented to demonstrate the effectiveness of the proposed approach.

Throughout this paper, I denotes the identity matrix of appropriate dimensions. For a real matrix A, its transpose and spectral norm are denoted by AT and A, respectively. Q=QT>0(Q=QT<0) implies that Q is a symmetric positive (negative) definite matrix. The notation * in symmetric block matrices or long matrix expressions throughout the paper represents an ellipsis for terms that are induced by symmetry, e.g. [BCD]=[BCCTD]. For a vector x, x(t) means the Euclidean vector norm at time t, while x20x(t)2dt. If x2<, then x(t)L2[0,) where L2[0,) stands for the space of square integral functions on [0,).

Section snippets

Main result

Consider the master and slave chaotic systems with multiple time-delays described by the following differential Eqs. (1a), (1b), respectively. ẋm(t)=Axm(t)+i=1NAixm(tτi)+Bg(xm(t),ym(t))+dym(t)=Cxm(t) and ẋs(t)=Axs(t)+i=1NAixs(tτi)+Bg(xs(t),ym(t))+d+Dw(t)+L(ym(t)ys(t))ys(t)=Cxs(t), where xmn and xsn are the master system’s state and slave system’s state, respectively. wl is the external noise perturbation, d denotes the constant input, and B,C,D are constant matrices with

Examples and simulation results

To demonstrate the validity of the proposed synchronization approach, we consider the following two noise-perturbed chaotic systems with multiple time-delays in this section.

Example 1

Consider the noise-perturbed chaotic Ro¨ssler system with two time-delays as follows [21]: ẋ1=x2x3+a1x1(tτ1)+a2x1(tτ2)ẋ2=x1+zx2ẋ3=b+x3x1cx3. For instance, the parameters a1=0.2, a2=0.4, z=0.25, b=0.3, c=5, τ1=1, τ2=2. The chaotic behavior of Rössler system with two time-delays is shown in Fig. 1. In order to

Conclusion

An output coupling and feedback scheme for H synchronization with noise perturbation attenuation γ of a class of noise-perturbed chaotic systems with multiple time-delays has been studied. Based on the Lyapunov theory and the LMI optimization technique, the output coupling and feedback gain for the slave systems has been derived not only to guarantee the asymptotic synchronization but also to ensure a prescribed noise perturbation attenuation performance. Finally, we have showed the

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