Sliding mode control of uncertain unified chaotic systems

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Abstract

This paper investigates the chaos control of the uncertain unified chaotic systems by means of sliding mode control. A proportional plus integral sliding surface is introduced to obtain a sliding mode control law. To confirm the validity of the proposed method, numerical simulations are presented graphically.

Introduction

Chaos is a periodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions [1]. The fundamental characteristics of chaotic behavior come from the internal structure of the systems, and chaotic behaviors are more complicated than limit cycle behaviors. Today, chaos has been seen to have many useful applications in many engineering systems such as in chemical reactors, genetic control systems, power converters, lasers, biological systems, and secure communication systems [1], [2]. Chaos can be useful in propagation of mixing in processes, such as in convective heat transfer. However, chaotic behavior may lead to undesirable effects as well, such as uncontrolled oscillations in a power grid, and may need to be regulated [2].

After chaos control was introduced in [3], it has turned out to be an important area of nonlinear science, and various control approaches have been proposed. The Ott–Grebogi–Yorke (OGY) method [3], variable structure control [4], nonlinear feedback control [5], and some other methods [6], [7] have been successfully applied to chaotic systems. The sliding mode control (SMC) scheme is one of these methods [8], [9], and recently there has been a great deal of attention given to using SMC for controlling chaos. SMC is an effective methodology for controlling systems with variable structures and provides a systematic approach to the problem of maintaining stability and consistent performance in the face of modeling imprecision [10], [11], [12]. SMC is particularly preferred due to its capability to tolerate disturbances and dynamic model uncertainties.

Chaotic systems include nonlinearities and often some parameters which cannot be exactly defined [13], [14]. Therefore, a robust control method such as SMC would have the advantage of the capability to handle such uncertainties. In this study, a proportional plus integral (PI) sliding surface is introduced, and by satisfying the reachability condition, a suitable sliding mode control law is developed for chaos control of uncertain unified chaotic systems which was recently introduced by Lü et al. [15]. The unified chaotic system that has a single adjustable parameter exhibits all chaotic behaviors of Lorenz [16], Chen [17] and Lü [18] chaotic systems which all have three parameters. Lorenz type systems, or the unified chaotic systems, can be seen in atmospheric sciences, laser devices, and some other systems related to convection.

This work researches chaos control of the uncertain chaotic systems by means of sliding mode control. In the Section 2, chaos control of an uncertain unified chaotic system is presented. In Section 3, numerical simulations are provided to confirm the validity of the method, and finally, conclusions are given.

Section snippets

Sliding mode control of uncertain unified chaotic systems

The unified chaotic system is introduced by Lü et al. [15] with a single parameter: {ẋ1=(25α+10)(x2x1)ẋ2=(2835α)x1+(29α1)x2x1x3ẋ3=x1x2(8+α)x3/3 where x1, x2, x3 are state variables and α[0,1] is the system parameter. When α[0,0.8), system (1) is called the generalized Lorenz chaotic system. When α=0.8, system (1) is called the Lü chaotic system, and when α(0.8,1], system (1) is called the generalized Chen chaotic system.

The controlled unified chaotic system can be rewritten as

Numerical simulations

Here the numerical results are given to confirm the validity of the proposed method. In the numerical simulations, the gain matrix K is selected as K=diag(1,1,1) such that KB=diag(1,1,1) is nonsingular. The desired eigenvalues of the matrix ABL are taken as P=[55.0015.0001], and by using the pole placement method, the gain matrix L is found as L=[5100284.0010002.3334]. As a result, the matrix K(ABL) is computed as K(ABL)=diag(5,5.001,5.0001). The PI switching surfaces are obtained as {

Conclusions

This study proposes a sliding mode control method for chaos control of the uncertain unified chaotic systems. A sliding mode control law is developed by using a PI switching surface, and the reachability condition is satisfied. I believe that this method will be generalized.

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