Adaptive Synchronization of Unknown Chaotic Systems Using Mamdani Fuzzy Approach

Adaptive Synchronization of Unknown Chaotic Systems Using Mamdani Fuzzy Approach

G. El-Ghazaly (University of Genova, Italy)
Copyright: © 2012 |Pages: 17
DOI: 10.4018/ijsda.2012070104
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Abstract

The synchronization problem for a general-class of unknown chaotic systems is addressed. Fuzzy systems in Mamdani type are employed to provide an approximate model of the master chaotic system using an adaptive approach. Within this, a number of fuzzy systems are utilized to approximate the unknown nonlinear functions of the master system. An approximate model similar to the master system is constructed which is the slave system. The error dynamics between the master and slave chaotic systems is used to build a suitable control input and fuzzy systems’ parameters adaptive laws to force the slave system to be synchronized with master system. The stability of the overall synchronization system is derived based on Lyapunov stability theory. Its shown that under appropriate assumptions, the proposed approach guarantees the boundness and asymptotic convergence of synchronization errors to a small neighborhood of origin. An extensive simulation study is performed on both Duffing and Rössler chaotic systems to show the effectiveness of the proposed scheme.
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1. Introduction

Dynamic systems with chaotic behavior are complex nonlinear systems with that widely appear in both nature and man-made systems. However, in some practical situations, this chaotic behavior is not desirable and needed to be controlled. On the other hand, this behavior founds a wide number of applications and mostly those applications can be achieved by synchronizing two chaotic systems. Due to the complex nonlinear dynamic behavior of chaotic systems, the problems of chaos control and synchronization is always difficult and challenging.

Synchronization of chaotic systems has received a significant attention and plays an important role in nonlinear science during the last two decades (Pecora, 1990; Agiza, 2004; Nayfeh, 1995; Takeo, 2005; Yau & Sheih, 2008). Recently, chaos and its synchronization have found several useful applications in many fields of engineering and science, such as in secure communication, biological systems, power converters, chemical reactions, and information systems, etc. (Chen & Dong, 1998). Many approaches have been presented for synchronization of chaotic systems such as periodic parametric perturbation method (Astakhov et al., 1997), drive-response synchronization method (Yang et al., 1999), adaptive control method (Wang et al., 2004; Chua et al., 1996; Liao, 1996; Lian et al., 2002; Wu et al., 1996), variable structure (sliding mode) control method (Fang et al., 1999; Yau, 2004), and backstepping control method (Wang & Ge, 2001; Bowong & Kakmeni, 2004; Lu & Zhang, 2004).

Since introduced by Zadeh (1965), fuzzy systems have received more attention and have been successfully adopted in a wide variety of engineering disciplines including control systems and signal processing applications (Wang, 1994). The key features of fuzzy logic systems behind its great success are that it can incorporate linguistic information from human experts and provides effective and systematic framework to handle nonlinear systems especially complex and ill-defined systems. Several fuzzy-based schemes are used for chaos synchronization and control. In Yau and Sheih (2008), classical fuzzy logic is used to design a robust controller, where the fuzzy rules are subject to a common Lyapunov function such that the error dynamics satisfies stability in Lyapunov sense. Also, many T-S fuzzy based-observer and controller methods which are based on parallel distributed compensation (PDC) are found in Tanaka et al. (1998), Kim et al. (2005), Ting (2005), and Hayon et al. (2006). However, satisfactory results are obtained; these approaches are based on the assumption that the system dynamics must be known.

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