Synchronization of Chaotic Systems Using Linear and Nonlinear Feedback Control
This paper presents an algorithm for the synchronization of chaotic systems using feedback control method in which the control input is defined as a sum of a linear and a non linear term. The nonlinear term eliminates the nonlinear part in the error syste
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Abstract This paper presents an algorithm for the synchronization of chaotic systems using feedback control method in which the control input is defined as a sum of a linear and a non linear term. The nonlinear term eliminates the nonlinear part in the error system and the linear term stabilizes the resulting linear system. The effectiveness of the proposed method is tested and some simulation results are provided on a complex chaotic system.
1 Introduction The synchronization of chaotic systems is a topic that has received much attention since the work of Pecora and Carroll published in 1990. It is due to its potential technological applications in fields such as: chaos control, information processing, secure communication and so on [1–4]. Pecora and Carroll [5] suggest that the phenomenon of chaos synchronism may to serve as the basis for new ways to achieve secure communication. Since different techniques such as: delay feedback method [6], adaptive control [7], sliding mode control [8], active control [9, 10], backstepping control [11, 12], and so on [13–17]. The synchronization consists to apply on a slave system a control function, such that the error between this one and a master system tends to zero. The problem of synchronization can then be expressed as a problem of control that consists in minimizing the error between the master and the slave by applying a control law.
A. Ikhlef () N. Mansouri Laboratoire d’Automatique et Robotique, Departement d’Electronique, facult´e des sciences de l’Ing´enieur, Universit´e Mentouri Constantine, Constantine, Algeria e-mail: [email protected]; nor [email protected] S.G. Stavrinides et al. (eds.), Chaos and Complex Systems, DOI 10.1007/978-3-642-33914-1 40, © Springer-Verlag Berlin Heidelberg 2013
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A. Ikhlef and N. Mansouri
In this paper, we are interested with the synchronization of complex chaotic systems by introducing an algorithm based on linear and nonlinear control method in which the control input is defined as a sum of a linear and a non linear term. The designed control functions are simple, easy to implement and allow the synchronization of chaotic systems in a short time. The paper is organized as follows: the nonlinear and linear feedback control algorithm is developed in Sect. 2. In Sect. 3, the designed controller is applied in simulation for the synchronization of complex chaotic systems, represented by two satellite chaotic systems. We give in Sect. 4, the numerical simulation results obtained with this example and the paper is concluded by some remarks on the approach performances.
2 Controller Design Generally, a chaotic system is defined as follows: 8 xP 1 D f1 .x1 ; x2 / ˆ ˆ ˆ < xP 2 D f2 .x1 ; x2 ; x3 / :: ˆ ˆ : ˆ : xP n D fn .x1 ; x2 ; : : : ; xn / C Dx .t/
(1)
where f1 is a linear function, fi .i D 2; 3 : : : ; n/ represent nonlinear functions and Dx .t/ a nonlinear term. For some systems, Dx .t/ D 0. The system (1) is taken as master and the following system as slave: 8 yP1 D f1 .y1 ; y2 / C u1 ˆ ˆ ˆ < yP2 D f2 .y1 ; y2 ; y3
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