H synchronization of time-delayed chaotic systems

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Abstract

This paper considers H synchronization of a class of time-delayed chaotic systems with external disturbance. Based on Lyapunov theory and linear matrix inequality (LMI) formulation, the dynamic feedback controller is established to not only guarantee synchronization between derive and response systems, but also reduce the effect of external disturbance to an H norm constraint. Then, a criterion for existence of the controller is given in terms of LMIs. Finally, a numerical simulation is presented to show the effectiveness of the proposed chaos synchronization scheme.

Introduction

During the last decades, chaotic systems have attracted a lot of interests. Many fundamental characteristics have been found in a chaotic system, such as excessive sensitivity to initial conditions and fractal properties of the motion in phase space. In particular, chaos synchronization, first proposed by Fujisaka and Yamada in 1983 [1], did not received great attention until 1990 [2]. From then on, chaos synchronization has been extensively investigated due to the potential application in various fields such as biology, economics, signal generator design, secure communication, and so on [3], [4], [5], [6]. Based on various control theories, a number of synchronization schemes have been proposed in the literature [7], [8], [9], [10], [11], [12], [13], [14], [15].

In real physical systems, some noise or disturbances always exist that may cause instability and poor performance. Therefore, the effect of the noises or disturbances must be also reduced in synchronization process for chaotic systems. In this regards, recently, Hou et al. [16] firstly adopted the H control concept [17], [18] to reduce the effect of the disturbance for chaotic synchronization problem of a class of chaotic systems. On the other hand, since Mackey and Glass [19] first found chaos in time-delay system, there has been increasing interest in time-delay chaotic systems [20], [21]. The synchronization problem for time-delayed chaotic systems is also investigated by some researchers [22], [23], [24]. However, the H synchronization for time-delayed chaotic systems with disturbances has not been studied yet.

In this paper, we consider the problem of H chaos synchronization for time-delayed chaotic system with disturbance. Dynamic feedback controller for the synchronization between derive and response systems is proposed. By the feedback control scheme, the closed-loop error system is asymptotically stable and the H-norm from the disturbance to controlled output is reduced to a prescribed level. Based on the Lyapunov method and LMI framework, an existence criterion for such controller is represented in terms of LMIs. The LMIs can be easily solved by various convex optimization algorithms developed recently [25].

Notation: Rn denotes the n-dimensional Euclidean space, and Rm×n is the set of m×n real matrix. |·| represents the absolute value. · refers to the Euclidean vector norm and the induced matrix norm. I denotes the identity matrix with appropriate dimension. AT means the transpose of the matrix A. We denote the positive (non-negative) definiteness of by A>0(A0) and the negative (non-positive) definiteness of A by A<0(A0). A>B means A-B is a positive definite matrix. diag{} denotes the block diagonal matrix. represents the elements below the main diagonal of a symmetric matrix. λmin(A) denotes the smallest eigenvalue of A.

Section snippets

Problem formulation

Consider a class of time-delayed chaotic systems described byx˙(t)=A0x(t)+A1f(x(t))+A2f(x(t-h)),zx(t)=Cx(t),where xRn is the state variable, zxRq is the output, the matrices A0,A1,A2Rn×n and CRq×n are known constant matrices, and f(x(t))Rn is a nonlinear function vector satisfying the global Lipschitz condition: i.e,|f(x1)-f(x2)||F(x1-x2)|,x1,x2Rn,for a known constant matrix F.

The synchronization problem of system (1) is considered using the drive-response configuration [2]. This is, if

Main result

The main result for achieving H synchronization is stated in the following theorem.

Theorem 1

For given η, α, ϵ, and γ, there exist a dynamic feedback controller (6) for the error system (5), if there exist any matrices A^,B^,C^, a positive scalar β, and positive-definite matrices D^, S and Y satisfying the following LMIs:Φ=Φ1Φ2A1A200DΦ4Φ3SA1SA200SD0-αI00000-βI0000-ϵI+βFTF000-Q00-γ2I0Φ5<0,andYIIS>0,whereΦ1=A0Y+YA0T+BC^+C^TBT+D^+ηY,Φ2=A0+A^T+ϵY++αYFTF+YCTC+ηI,Φ3=SA0+A0TS+B^+B^T+

Conclusions

The problem of H synchronization for a class of time-delayed chaotic systems with disturbances has been presented. Based on Lyapunov theory and LMI formulation, A dynamic feedback control scheme has proposed to guarantee synchronization for drive and response systems and reduce the H-norm from the disturbance input to the output error within a prescribed level. Furthermore, a model of Hopfield neural network is given to illustrate the effectiveness of the proposed control scheme.

References (26)

  • P.K. Das et al.

    Phys. Lett. A

    (1991)
  • C.C. Wang et al.

    Chaos, Solitons Fractals

    (2004)
  • X.S. Yang et al.

    Chaos, Solitons Fractals

    (2002)
  • Ju H. Park et al.

    Chaos, Solitons Fractals

    (2005)
  • Ju H. Park et al.

    Phys. Lett. A

    (2007)
  • Ju H. Park

    Chaos, Solitons Fractals

    (2005)
  • Ju H. Park

    Chaos, Solitons Fractals

    (2005)
  • Ju H. Park et al.

    Chaos, Solitons Fractals

    (2007)
  • Yi-You Hou et al.

    Phyisca A

    (2007)
  • Y. Tian et al.

    Physica D

    (1998)
  • W. Zhu et al.

    Chaos, Solitons Fractals

    (2008)
  • J. Cao et al.

    Phys. Lett. A

    (2006)
  • H.T. Lu

    Phys. Lett. A

    (2002)
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