Synchronization of a modified Chua’s circuit system via adaptive sliding mode control
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 [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 x–y 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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Secure communication with a chaotic system owning logic element
2018, AEU - International Journal of Electronics and CommunicationsCitation Excerpt :For this purpose, various synchronization methods have been developed after the first synchronization method was found by Pecora-Carroll in 1990 [39]. Some of the effective ones are: adaptive control [40], active control [41], feedback control [42], passive control [43], backstepping design [44] and sliding mode control [45–50]. The sliding mode control method makes the synchronized system robust with a simple control algorithm.
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2017, Chaos, Solitons and FractalsCitation Excerpt :Some sliding mode variants such as proportional-integral based adaptive SMC [41], integral SMC [42], and active SMC [43] methods are also used for the original Chua's circuit. However, the synchronization of Chua oscillators is achieved through the SMC method [44,45], but all of them deal with the modified versions of Chua attractors. In this study, focusing on the Chua's circuit, its chaotic behavior is described; its control and synchronization are applied according to the solutions of nonlinear differential equations through the SMC method.
Zero-Hopf bifurcation in a Chua system
2017, Nonlinear Analysis: Real World ApplicationsCitation Excerpt :It was presented by Chua, Komuro and Matsumoto [1] in 1986 and exhibits a rich range of dynamical behavior. There are several different models of Chua’s systems see for instance [2–6]. A zero-Hopf equilibrium is an equilibrium point of a 3-dimensional autonomous differential system which has a zero eigenvalue and a pair of purely imaginary eigenvalues.
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2017, OptikCitation Excerpt :Chaotic system has a complex nonlinear response like random signal, so it can be applied in military secure communication [1–8].
Chaos synchronization for master slave piecewise linear systems: Application to Chua's circuit
2012, Communications in Nonlinear Science and Numerical SimulationCitation Excerpt :On the other hand, a variety of control methods had developed to synchronize Chua’s circuits. Examples include active control [19], adaptive control [20,21], back-stepping control [22], impulsive control [23,24], sliding mode control [25], nonlinear control [26], predictive control [27] and robust control [28]. Recently, synchronization of Chua’s circuits by linear state feedback control has received much attention due to its simple configuration and easy implementation.
Exponential synchronization of chaotic systems subject to uncertainties in the control input
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