Modified projective synchronization of chaotic systems with disturbances via active sliding mode control

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Abstract

We apply the active sliding mode control technique to realize the modified projective synchronization of the chaotic systems. The disturbances are considered both in the drive system and the response system. The sufficient conditions for the modified projective synchronization both the non-identical and identical chaotic systems are presented. The corresponding numerical simulations are provided to illuminate the effectiveness of the proposed active sliding mode controllers.

Introduction

Chaos synchronization has attracted a lot of attention among scientists from a variety of research fields since the seminal work of Pecora and Carroll [1]. From then on, chaos synchronization has been developed extensively and intensively due to its potential applications in many fields [2], [3], [4]. The idea of synchronization is to use the output of the drive system to control the response system so that the output of the response system follows the output of the drive system asymptotically. Many techniques have been developed to realize chaos synchronization such as OGY method [5], linear and nonlinear feedback control method [6], [7], active control method [8], backstepping method [9], adaptive control method [10], and sliding mode control method [11], etc.

However, most of researches mentioned above have concentrated on studying the complete synchronization. In the practical applications, the complete synchronization is difficult to occur except under ideal conditions. Recently, an interesting synchronization phenomenon has been discovered, called the modified projective synchronization [12], which the states of the drive and response systems synchronize up to a constant scaling matrix. The complete synchronization, anti-synchronization and projective synchronization belong to the special cases of the modified projective synchronization. So modified projective synchronization is worth studying because of its importance. And some research results for the modified projective synchronization have been obtained in recent years [12], [13], [14]. These research results are realized without any disturbance. However, the noise disturbance is inevitable in the practical situations. So synchronization of concrete models is unavoidably subject to noise disturbance. Therefore investigation of synchronization for the chaotic systems by the impact of disturbance has become an important research topic. Zhang et al. [15] considered the synchronization of the unified chaotic system with disturbance using the sliding mode control. Li et al. [16] investigated the synchronization for the chaotic systems with uncertain parameter, disturbances and nonlinear control input, based on the adaptive control and sliding mode control methods. Recently, a new method called “active sliding mode control technique” is proposed to realize chaos synchronization [17], [18], [19], based on the advantages of the active control and the sliding mode control method.

Motivated by the above analysis, in this paper, we consider the modified projective synchronization for non-identical and identical chaotic systems with fully and partial disturbances. Based on the active sliding mode control technique, the sufficient conditions are given to assure the modified projective synchronization occurs. The corresponding numerical simulations are provided to illuminate the effectiveness of the active sliding mode controllers.

Section snippets

Active sliding mode controller design

Consider the following chaotic systemsx˙=A1x+f1(x),y˙=A2y+f2(y).where x = [x1, x2, …, xn]T and y = [y1, y2, …, yn]T denote state variables, A1 and A2 are constant matrices, f1, f2 are the continuous nonlinear functions. If there exists a constant matrix J = diag{j1, j2, …, jn}, such that limt→∞ ∥x − Jy∥ = 0, then we call the systems (1), (2) achieve the “modified projective synchronization”, and J denotes a “scaling matrix”.

Remark 1

If A1 ≠ A2, f1(·) ≠ f2(·), then the systems (1), (2) are non-identical chaotic systems.

Remark 2

Numerical simulations

In this section, we apply two simulation examples to illustrate the effectiveness of the proposed approaches. The modified projective synchronization is simulated for the non-identical and identical chaotic systems, respectively.

  • Case I:

    When the drive system and response system are non-identical chaotic systems, the Lorenz system and Chen system are considered as drive system and response system, respectively. They are described as followsx˙1=-10x1+10x2+d11,x˙2=28x1-x2-x1x3+d12,x˙3=-8/3x3+x1x2+d13.andy˙

Conclusions

In this paper, the modified projective synchronization for the identical and non-identical chaotic systems with fully and partially disturbances is investigated. Combining the advantages of the active control and sliding mode control method, the sufficient conditions of modified projective synchronization are obtained. The numerical simulations are given to demonstrate the effectiveness of the proposed active sliding mode controllers although the chaotic systems influenced by the disturbances.

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