Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller

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Abstract

In this paper, a robust adaptive sliding mode controller (RASMC) is proposed to realize chaos synchronization between two different chaotic systems with uncertainties, external disturbances and fully unknown parameters. It is assumed that both master and slave chaotic systems are perturbed by uncertainties, external disturbances and unknown parameters. The bounds of the uncertainties and external disturbances are assumed to be unknown in advance. Suitable update laws are designed to tackle the uncertainties, external disturbances and unknown parameters. For constructing the RASMC a simple sliding surface is first designed. Then, the RASMC is derived to guarantee the occurrence of the sliding motion. The robustness and stability of the proposed RASMC is proved using Lyapunov stability theory. Finally, the introduced RASMC is applied to achieve chaos synchronization between three different pairs of the chaotic systems (Lorenz–Chen, Chen–Lorenz, and Liu–Lorenz) in the presence of the uncertainties, external disturbances and unknown parameters. Some numerical simulations are given to demonstrate the robustness and efficiency of the proposed RASMC.

Research highlights

 RHtriangle Practical synchronization of two different chaotic systems.  RHtriangle Consideration of the effects of uncertainties, disturbances and unknown parameters.  RHtriangle Design of a robust adaptive sliding mode controller.  RHtriangle Verifying theoretical results by numerical simulations.

Introduction

In the past few decades, much attention has been gained for the study of nonlinear science, especially chaos. Chaos is a particular case of nonlinear dynamics that has some specific characteristics such as extraordinary sensitivity to initial conditions and system parameter variations, broad Fourier transform spectra and fractal properties of the motion in the phase space. Due to these especial properties, chaos has been used in many practical engineering areas such as chemical reactions, power converters, secure communications, information processing, biological systems and mechanical systems [1], [2], [3] and many various control techniques have been proposed for controlling and synchronizing of chaotic systems, including sliding mode control [4], [5], [6], optimal control [7], [8], [9], adaptive control [10], [11], nonlinear feedback control [12], backstepping method [13], [14], passive control [15], H∞ approach [16], fuzzy logic control [17], PID control [18], etc.

In addition to the control and stabilization of chaos, synchronization of chaotic systems is a fascinating concept which has been received considerable interest among nonlinear scientists in recent times. For chaos synchronization there are two chaotic systems called the master (drive) system and slave (response) system. The objective of the designed controller for synchronization is to make the output of the master system follows the output of the slave system asymptotically.

Unfortunately, most of the above mentioned works on chaos synchronization have focused on chaotic systems without model uncertainties and external disturbances in both master and slave systems. However, in practical applications, due to the modeling errors, structural variations of the systems and un-modeled dynamics uncertainties are present in the chaotic system dynamics. Moreover, in practical situations, chaotic systems are unavoidably affected by external disturbances such as environment and measurement noises. So, synchronization of chaotic systems with uncertainties and external disturbances is effectively significant in applications. In this regard, some researchers have proposed a number of techniques for synchronization of uncertain chaotic systems that includes nonlinear feedback control [19], sliding mode control [20], [21], [22], [23], [24], backstepping procedure [25], linear state feedback control [26], [27], active control [28] and some other methods.

However, all of the mentioned above works have a common serious drawback: they have concentrated on the synchronization of two identical chaotic systems. But, the method of the synchronization of two different chaotic systems is far from being straightforward. Also, in many real world applications, there are no exactly two identical chaotic systems. Therefore, the problem of chaos synchronization between two different uncertain chaotic systems is an important research issue. And a few researchers have developed some techniques for it that includes sliding mode control [29], [30], [31] and neural fuzzy control [32].

Nevertheless, the previous methods have studied chaotic systems with fully (or partially) known parameters. While, in practice, it is hard to exactly determine the values of the system parameters in priori. Therefore, synchronization of chaotic systems with unknown parameters is essential and useful in real-life applications. Consequently, some approaches, such as sliding mode control [33], [34], finite-time based control [35], adaptive control [36], [37], [38], [39], optimal control [40], [41], fuzzy control [42], [43], [44], have been developed for synchronization of two identical chaotic systems with unknown parameters and some methods, such as adaptive control [45], [46], [47], [48], [49], sliding mode control [50] and backstepping method [51], have been proposed for synchronization of two different chaotic systems with unknown parameters.

In conclusion, to the best knowledge of the authors, the challenging problem of chaos synchronization between two different chaotic systems in spite of uncertainties, external disturbances and unknown parameters in both master and slave chaotic systems is not studied to this date. Therefore, the main purpose of this paper is to design a robust adaptive sliding mode controller (RASMC) to synchronize two different chaotic systems in the presence of uncertainties, external disturbances and fully unknown parameters in both master and slave chaotic systems. It is assumed that the bounds of the uncertainties and external disturbances are unknown in advance. A simple suitable sliding surface, which includes synchronization errors, is constructed. Appropriate update laws are derived to tackle the uncertainties, external disturbances and unknown parameters. Then, on the basis of the update laws, the RASMC is designed to guarantee the existence of the sliding motion. The stability and robustness of the proposed RASMC is proved using Lyapunov stability theory. Finally, three well-known chaotic systems (Lorenz, Chen, and Liu systems) are used to verify the applicability and efficiency of the introduced RASMC.

The organization of this paper is as follows. In Section 2, system description and problem formulation are presented. In Section 3, the design procedure of the proposed RASMC is given. Simulation results are included in Section 4. Finally, Section 5 ends this paper with some concluding remarks.

Section snippets

System description and problem formulation

In this paper, the n-dimensional master and slave chaotic systems with uncertainties, external disturbances and unknown parameters are given as follows:

Master system:x˙1(t)=f1(x1,x2,,xn)+F1(x1,x2,,xn)θ+Δf1(x1,x2,,xn,t)+d1m(t)x˙2(t)=f2(x1,x2,,xn)+F2(x1,x2,,xn)θ+Δf2(x1,x2,,xn,t)+d2m(t)x˙n(t)=fn(x1,x2,,xn)+Fn(x1,x2,,xn)θ+Δfn(x1,x2,,xn,t)+dnm(t)Slave system:y˙1(t)=g1(y1,y2,,yn)+G1(y1,y2,,yn)ψ+Δg1(y1,y2,,yn,t)+d1s(t)+u1(t)y˙2(t)=g2(y1,y2,,yn)+G2(y1,y2,,yn)ψ+Δg2(y1,y2,,yn,t)+d2s(t)+u2(

Design of robust adaptive sliding mode controller

Sliding mode control [52] is a robust control method which has many interesting features such as low sensitivity to external disturbances and robustness to the plant uncertainties due to structural variations and un-modeled dynamics. The sliding mode controller is composed of an equivalent control part that describes the behavior of the system when the trajectories stay over the sliding surface and a variable structure control part that enforces the trajectories to reach the sliding surface and

Numerical simulations

In this section, some numerical simulations are presented to validate the efficiency and effectiveness of the proposed RASMC. Numerical simulations are carried out using the MATLAB software. The ode45 solver is used for solving differential equations. The Lorenz [54], Chen [55], and Liu [56] systems are three well-known chaotic systems whose nonlinear equations are given byLorenz:x1˙=10(x2-x1)x2˙=28x1-x2-x1x3x3˙=x1x3-8/3x3Chen:y1˙=35(y2-y1)y2˙=28y2-7y1-y1y3y3˙=y1y3-3y3Liu:z1˙=10(z2-z1)z2˙=40z1-z

Conclusions

In this paper, the problem of practical synchronization of chaotic systems is investigated. In real world applications, there are always some uncertainties and external disturbances in the system dynamics. Also, in practical or experimental situations, the system parameters are inevitably disturbed by external inartificial factors, such as environment temperature, voltage fluctuation, mutual interfere between components, and cannot be exactly known in advance. The synchronization may be

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