Synchronization of a modified Chua’s circuit system via adaptive sliding mode control

https://doi.org/10.1016/j.chaos.2006.06.008Get rights and content

Abstract

This study addresses the adaptive synchronization of a modified Chua’s circuit system with both unknown system parameters and the nonlinearity in the control input. An adaptive switching surface is newly adopted such that it becomes easy to ensure the stability of the error dynamics in the sliding mode. Based on this adaptive switching surface, an adaptive sliding mode controller (ASMC) is derived to guarantee the occurrence of the sliding motion, even when the system is undergoing input nonlinearity. This method can also be easily extended to a general class of Chua’s circuits. An illustrative example is given to show the applicability of the proposed ASMC design.

Introduction

Chaos synchronization has received increasing attention over the last decade [1]. Chaos synchronization can be applied in the vast areas of physics and engineering systems such as in chemical reactions, power converters, biological systems, information processing, especially in secure communication [2], [3], [4], [5], [6]. Many deep theories have been developed to achieve chaos synchronization. For example, adaptive control [7], [8], variable structure control [9], [10], [11], optimal control [12], digital redesign control [13], backstepping control [14], [15], etc.

The Chua’s circuit system is one of the paradigms of chaos since it exhibits a wide variety of nonlinear dynamics phenomena such as bifurcations and chaos. It contains three energy-store elements (an inductor, and two capacitors), a linear resistor and a single nonlinear resistor. Aguilar-Ibanez et al. [16] applied the differential flatness approach for controlling of the Chua’s system. Hegazi et al. [17] used the Lyapunov direct method to achieve the adaptive synchronization of Chua’s circuit systems. Yassen [18] proposed an adaptive control law to achieve synchronization of two identical modified Chua’s circuit systems. Many studies for modified Chua’s circuit systems can also be found in [14], [19]. Unfortunately, all the above-mentioned works on the chaos synchronization concentrate on overall systems with a ‘linear input’. However, due to physical limitations and external disturbances, there always exists nonlinearity in the control input [20]. Their existence may lead to serious degradation of system performance and might cause chaotic perturbations to original regular behavior if the controller is not well designed.

This paper aims to the development of an ASMC for synchronizing the state trajectories of two identical modified Chua’s circuit systems. It is assumed that the system parameters are unknown and the control input is subjected to a nonlinearity raised from physical limitations and disturbances. A novel adaptive switching surface, which makes it easy to guarantee the stability of the error dynamics in the sliding mode, is first proposed. And then, based on this adaptive switching surface, an adaptive sliding mode controller (ASMC) is derived to guarantee the occurrence of the sliding motion. Finally, we present the numerical simulation results to illustrate the effectiveness of the proposed control scheme.

Section snippets

Adaptive synchronization via sliding mode control

In this section, we consider the robust synchronization of two identical modified Chua’s system with an adaptive sliding mode controller.

Numerical example

In this section, simulation results are presented to demonstrate and verify the performance of the present design. The parameters p and q chosen p = 10 and q=1007 [18] in the simulation to ensure the existence of chaos for the derive system (2). The initial states of the derive system (2) are x1(0) = 0.65, y1(0) = 0, z1(0) = 0 and initial states of the response system (3) are x2(0) = 0, y2 (0) = 0.5, z2(0) = −0.3. The chaotic attractor of the system (2) in the xy plane is shown in Fig. 2. The nonlinear

Conclusions

This paper has proposed an adaptive sliding mode controller design for synchronization of the modified Chua’s circuit system with both unknown system parameters and the nonlinearity in the control input. By the novel adaptive switching surface, it is found the stability of the error dynamics in the sliding mode is easily ensured. An adaptive sliding mode controller has also been proposed to guarantee the occurrence of the sliding motion, even with unknown system parameters and nonlinear control

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