Robust synchronization of drive–response chaotic systems via adaptive sliding mode control

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Robust synchronization of drive–response chaotic systems via

adaptive sliding mode control

Wang-Long Li

a

, Kuo-Ming Chang

b,*

a_{Institute of Nanotechnology and Microsystems Engineering, National Cheng Kung University, Tainan 701, Taiwan} b_{Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung 807, Taiwan}

Accepted 21 June 2007

Abstract

A robust adaptive sliding control scheme is developed in this study to achieve synchronization for two identical cha-otic systems in the presence of uncertain system parameters, external disturbances and nonlinear control inputs. An adaptation algorithm is given based on the Lyapunov stability theory. Using this adaptation technique to estimate the upper-bounds of parameter variation and external disturbance uncertainties, an adaptive sliding mode controller is then constructed without requiring the bounds of parameter and disturbance uncertainties to be known in advance. It is proven that the proposed adaptive sliding mode controller can maintain the existence of sliding mode in ﬁnite time in uncertain chaotic systems. Finally, numerical simulations are presented to show the eﬀectiveness of the proposed con-trol scheme.

1. Introduction

Since Pecora and Carroll[1]introduced a PC method to synchronize two identical chaotic systems with different initial conditions in 1990, chaos synchronization has received increasing attention and been investigated widely due to its potential application in wide areas of physics and engineering sciences, such as secure communication, informa-tion processing, biological systems, and chemical reacinforma-tion[2–4]. Many effective control methods[5–8]have been pro-posed to achieve chaos synchronization, such as linear and nonlinear feedback controls. However, most researches on chaos synchronization focused on certain chaotic systems without consideration of parameter variation uncertainty and external disturbance perturbation. But in practical situations, many chaotic systems are inevitably affected by parameter variations and external disturbances. Moreover, some or all of the system parameters and external disturbance uncer-tainties are unknown or variable from time to time. Therefore, investigation of system parameter variations and exter-nal disturbance perturbations in synchronization between drive and response chaotic systems has become an interesting and important research topic in recent years. In the work[9], an adaptive control law with single-state variable feedback was derived and applied to achieve the state synchronization of two identical Lorenz systems. An active sliding mode control was proposed by Zhang and Ma[10]to synchronize chaotic systems with parameter perturbation. Zhang et al.

*_{Corresponding author.}

E-mail address:[email protected](K.-M. Chang).

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Please cite this article in press as: Li W-L, Chang K-M, Robust synchronization of drive–response chaotic systems via ..., Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.06.067

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[11]also presented a sliding mode control to resolve the conquer synchronization problem in noise-perturbed chaotic systems. In the work[12], an intermittent parametric adaptive control method was studied to synchronize two logistic maps, and the corresponding suﬃcient conditions for synchronization are drawn. Based on Lyapunov stabilization the-ory, Huang et al.[13]proposed an adaptive controller with parameters identiﬁcation for synchronizing a class of cha-otic systems with unknown parameters. Park[14] developed a nonfragile controller using the Lyapunov functional technique combined with LMI technique to achieve synchronization problem of a class of chaotic systems with control-ler gain variations.

Owing to physical limitations, there always exist nonlinear effects in the control actuators. Neglecting the effect of input nonlinearities usually results in degeneration upon system performance. In addition to the problem of parameter variation uncertainty and external disturbance perturbation, the nonlinear input effect is also discussed in this paper. Sliding mode control is a well-known approach with its robustness against uncertainties, disturbances, and unmodeled dynamics. Therefore, in this paper, an adaptive sliding mode control is proposed to achieve the synchronization prob-lem for two identical chaotic systems in the presence of uncertain system parameter variation, external disturbance per-turbation, and nonlinear control inputs. Based on the Lyapunov stability theory, an adaptation law is given to estimate the upper-bound values of system uncertainties recursively. Consequently, a robust adaptive sliding mode controller can be constructed by applying the adaptation law: the existence of sliding mode can be maintained even without know-ing the upper-bounds of parameter and disturbance uncertainties. Finally, numerical simulations are presented to show the effectiveness and robustness of the proposed control scheme.

2. System description and problem formulation

Consider the following chaotic system described by

_x¼ Ax þ f ðxÞ þ gðtÞ; ð1Þ

where xðtÞ 2 Rn_{is a n-dimensional state vector of the system, A}_{2 R}nn_{is the matrix of the system parameter, gðtÞ 2 R}n_is

the external input signal and fðÞ : Rn_{! R}n _{is a continuous nonlinear function vector satisfying the following global}

Lipschitz condition,

kf ðxÞ f ðyÞk 6 qkx yk; 8x; y 2 Rn_;

wherekÆk denotes the Euclidean norm and q is the Lipschitz constant. As we know, many chaotic systems investigated are based on form(1), such as Murali–Lakshmanan–Chua (MLC) system[15]and Genesio system[16]. The problem discussed in this paper concerns the synchronization problem of system(1)using the drive–response conﬁguration. Here, the system(1)is considered as a drive system. A response system with the same form of Eq.(1)with the nonlinear control input vector nðÞ : Rn_{! R}n _{and external disturbance dðtÞ 2 R}n _{is introduced as follows:}

_y¼ A1yþ f ðyÞ þ gðtÞ þ nðuÞ þ dðtÞ; ð2Þ

where y2 Rn _{is the state vector of the response system, A}

12 Rnn is an unknown constant system parameter matrix,

nðuÞ ¼ ½ n1ðu1Þ n2ðu2Þ nnðunÞ T

2 Rn _is _a _continuous _nonlinear _function _vector _with _{n(0) = 0,} _u_¼

½ u1 u2 un T

2 Rn _{is a control vector, dðtÞ 2 R}n _{is an external disturbance. The response system has the same}

structure as the drive system, except that it is subjected to unknown parameter variations and external disturbance uncertainties. Before proceeding with the main results of this paper, the following assumptions, which specify the class of uncertain response systems are made.

Assumption 1. Nonlinear input functions niðuiÞ, i ¼ 1; . . . ; n, are sector bounded by ui, i¼ 1; . . . ; n, respectively. It yields

positive constants ci1, i¼ 1; . . . ; n and ci2, i¼ 1; . . . ; n, such that the following conditions are satisﬁed. ci16niðuuiiÞ

6ci2,

for i¼ 1; . . . ; n.

Assumption 2. Uncertain system parameter error matrix and external disturbance vector are norm-bounded. It means that there exists two positive constants l1and l2large enough such thatkA1 Ak 6 l1andkdðX ; tÞk 6 l2.

FromAssumption 1, it can be obtained straightforward that ci1u2iðtÞ 6 uiðtÞniðuiðtÞÞ 6 ci2u2iðtÞ; i¼ 1; . . . ; n:

Then, we have

c11u21ðtÞ þ þ cn1u2nðtÞ 6 u1ðtÞn1ðu1ðtÞÞ þ þ unðtÞnnðunðtÞÞ 6 c12u21ðtÞ þ þ cn2u2nðtÞ: 2 W.-L. Li, K.-M. Chang / Chaos, Solitons and Fractals xxx (2007) xxx–xxx

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troller works well for the synchronization of drive–response chaotic systems with system parameter variations and external disturbance uncertainties.

5. Conclusions

In this paper, the design problem of adaptive sliding mode controller for synchronization of a class of drive–response chaotic systems with parameter variation uncertainty, external disturbance perturbation, and control input nonlinearity is studied using the Lyapunov method. The proposed control scheme can be implemented without requiring the bounds of unknown parameter and disturbance uncertainties to be known in advance. Numerical results show that the pro-posed control scheme is very eﬀective and robust against system uncertainties.

References

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[12] Dai D, Ma XK. Chaos synchronization by using intermittent parametric adaptive control method. Phys Lett A 2001;288:23–8. [13] Huang L, Wang M, Feng R. Parameters identiﬁcation and adaptive synchronization of chaotic systems with unknown

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[14] Park JH. Synchronization of a class of chaotic dynamic systems with controller gain variations. Chaos, Solitons & Fractals 2006;27:1279–84.

[15] Murali K, Lakshmanan M. Synchronization through compound chaotic signal in Chua’s circuit and Murali–Lakshmanan–Chua circuit. Int J Bifurc Chaos 1997;7:415–21.

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Fig. 4. Time responses of switching functions.

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