Adaptive sliding mode control of chaotic dynamical systems with application to synchronization

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Abstract

We address the problem of control and synchronization of a class of uncertain chaotic systems. Our approach follows techniques of sliding mode control and adaptive estimation law. The adaptive algorithm is constructed based on the sliding mode control to ensure perfect tracking and synchronization in presence of system uncertainty and external disturbance. Stability of the closed-loop system is proved using Lyapunov stability theory. Our theoretical findings are supported by simulation results.

Introduction

Dynamic systems described by nonlinear differential equations can be extremely sensitive to initial conditions. This phenomenon is known as deterministic chaos, which means that, although the system mathematical description is deterministic, its behavior is still unpredictable [14].

Chaos phenomenon is an interesting subject in the nonlinear systems. The concept of controlling and synchronization of chaotic systems have attracted many interests, since the evolutionary work on chaos control was first presented by Ott et al. in 1990 [24], followed by the Pyragas time-delayed auto-synchronization control scheme [28]; and the pioneering work on the synchronization of identical chaotic systems evolving from different initial conditions was first introduced by Pecora and Carroll [26], the same year. The possible applications of chaos control and synchronization arouse many research activities in recent years. This fact has motivated researchers to seek for various effective methods to achieve these goals [3], [7], [22], [33], [36]. So, the chaos control and synchronization as well as their application have become the hotspot in nonlinear fields.

In literature, from the viewpoint that control theory is the origin of the techniques of chaos control, synchronization of chaos has evolved somewhat in its own right [18]. From the other point of view, there are some works which deal with the problem of chaos synchronization in the framework of nonlinear control theory [5], [16], [23]. So, the study of chaos control and chaos synchronization unify due to this fact.

In recent years, several strategies to control and synchronize chaos have been developed, such as linear and nonlinear feedback control [6], [17], [29], adaptive control [11], [15], [34], [35], [37], sliding mode control [9], [20]. A widely considered controlling method consists in adding an input control signal to attempt to stabilize an unstable equilibrium point or an unstable periodic orbit. This input control signal can be constructed using linear state feedback or nonlinear state feedback [12].

In the past several decades, the sliding mode control (SMC) has been effectively applied to control the systems with uncertainties because of the intrinsic nature of robustness of sliding mode [27]. However, the SMC suffers from the problem of chattering, which is caused by the high-speed switching of the controller output in order to establish a sliding mode. The undesirable chattering may excite the high-frequency system response and result in unpredictable instabilities [32].

The adaptive techniques have been widely applied to control and synchronize chaotic systems [15], [31]. Recently, researchers have utilized the adaptive techniques together with the sliding mode control for many engineering systems to smooth the output from a sliding mode controller and alleviate the chattering in the pure SMC [8], [27].

In this paper, we address chaos control and synchronization using sliding mode theory. In the second section, dynamics of the system is described and an appropriate sliding surface is selected. In the third section, the adaptive sliding mode control (ASMC) scheme is briefly introduced. The proposed scheme is fairly simple in comparison with other works [21], [38] and decreases the cost and complexity of the closed-loop system. Besides, this controller reduces the chattering phenomenon and guarantees some properties, such as the robust performance and stability properties in presence of parameter uncertainties and external disturbance. Then, stability of the proposed scheme is analyzed. In Section 5, Genesio system [13] is considered to verify the validity of proposed control scheme by a computer simulation, respectively.

Many approaches have been presented for the synchronization of chaotic systems, but in most of them have been assumed that the master and slave system are the same [7], [39]. Hence, the synchronization of two different chaotic systems plays a significant role in practical applications [10], [25], [30], [31]. This problem becomes more difficult in presence of environmental disturbance, measurement noise, or if the two chaotic systems have some uncertainties. In Section 4, synchronization of two different chaotic systems via the ASMC is investigated. Theoretical results are verified via simulating the synchronization of Genesio system [13] and Arneodo system [1]. At the end, conclusion is presented.

Section snippets

System description for uncertain chaotic system with ASMC

Generally, the nonlinear differential equations are only an approximate description of the actual plant due to the presence of various uncertainties. Let the chaotic dynamical systems be represented in the Brunovsky form [2] by the following differential equations:x˙i=xi+1,1in1,x˙n=f0(X,t)+Δf(X,t)+d(t)+u(t)X=[x1,x2,...,xn]TRn,where X(t)=[x1(t),x2(t),,xn(t)]T=[x(t),x˙(t),,x(n-1)(t)]TRn is the state vector, f0(X,t) is given as nonlinear function of X and t, Δf(X,t) is time-varying, not

Stability analysis

Theorem 1

Consider the system (1) is controlled by u(t) in (7). Then the condition ss˙<0 is guaranteed and the error trajectories converge to the sliding surface (5).

Proof Using (1), (5), direct differentiation of s yields:s˙=e˙n+i=1n1cie˙iMultiplying both sides of Eq. (11) with s yields:ss˙=se˙n+i=1n1cie˙i=sf0(X,t)+Δf(X,t)+d(t)+ueq+urxd(n)(t)+i=1n1ciei+1=s[Δf(X,t)+d(t)+Ksgn(s)]αs+βs+Ks=[K(α+β)]s.If we select K < −(α + β) in Eq. (12), one can conclude that the reaching condition (ss˙<0) is always

Synchronization via adaptive sliding mode controller

Next, we consider the chaos synchronization, from a dynamical control perspective. In this case, the chaos synchronization can be regarded as a model-tracking problem, in which the response system, can track the drive system asymptotically. We consider the slave system (1) to follow the master chaotic system with the following dynamics:y˙i=yi+1,1in1,y˙n=f0(Y,t)+Δfm(Y,t)+dm(t)Y=[y1,y2,,yn]TRn,then states of the master and slave system will be synchronized. It is worth to notice that Y(t)=[y1

Simulation results

This section of the paper presents two illustrative examples to verify and demonstrate the effectiveness of the proposed control scheme. The simulation results are carried out using the MATLAB software. The fourth order Runge–Kutta integration algorithm was performed to solve the differential equations. A time step size 0.001 was employed.

Consider the Genesio chaotic system as follows:x˙1=x2x˙2=x3x˙3=cx1bx2ax3+x12,where x1, x2, x3 are state variables, and a, b and c are the positive real

Conclusion

This work presents the control and synchronization of chaos by designing the adaptive sliding mode controller. In the proposed approach, by applying appropriate control signal based on adaptive update law, a continuous control signal is achieved and stability of the system is guaranteed. Asymptotic stability of the closed-loop nonlinear system is guaranteed by means of Lyapunov stability theory. As the simulations show, the new controller could track and stabilize the desired trajectory within

References (39)

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    It has been demonstrated that two or more chaotic systems can synchronize by linking them with mutual coupling or with a common signal. Many methods have been presented for the control and synchronization of chaotic systems such as adaptive control [1–4], robust control [5–8], sliding mode control [9–12], etc. The centrifugal flywheel governor is a particularly interesting nonlinear dynamical system, and it plays an important role which automatically controls the speed of an engine and prevents the damage caused by a sudden change in load torque.

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