Applied Computational Intelligence and Soft Computing

Volume 2014 (2014), Article ID 613463, 9 pages

http://dx.doi.org/10.1155/2014/613463

## Lyapunov-Based Controller for a Class of Stochastic Chaotic Systems

Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz 5166614776, Iran

Received 25 July 2014; Accepted 4 November 2014; Published 10 December 2014

Academic Editor: Zhang Yi

Copyright © 2014 Hossein Shokouhi-Nejad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This study presents a general control law based on Lyapunov’s direct method for a group of well-known stochastic chaotic systems. Since real chaotic systems have undesired random-like behaviors which have also been deteriorated by environmental noise, chaotic systems are modeled by exciting a deterministic chaotic system with a white noise obtained from derivative of Wiener process which eventually generates an Ito differential equation. Proposed controller not only can asymptotically stabilize these systems in mean-square sense against their undesired intrinsic properties, but also exhibits good transient response. Simulation results highlight effectiveness and feasibility of proposed controller in outperforming stochastic chaotic systems.

#### 1. Introduction

In last two decades, the problem of control of chaotic systems has been widely investigated by many researchers due to the existence of chaos in real practical systems [1–10]. A chaotic system has some inherent characteristics, such as excessive sensitivity to initial conditions, fractal properties of the motion in phase space, and board spectrums of the frequency response; hence, it is usually difficult to accurately predict the future behavior of the chaotic system, which can end up in performance degradation and restriction on the operating range of dynamic systems.

Considering aforesaid, developing strategies for controlling chaos phenomenon based on the features of chaotic motion is highly important; therefore, many nonlinear techniques for chaos control were proposed, such as feedback control [1, 2] and sliding mode control [3–5]. To exploit their advantages, these approaches are integrated as a complex control algorithm such as adaptive sliding mode [6, 7], adaptive fuzzy sliding mode control [8], and predictive feedback control [9]. However, in practice, real systems are usually affected by external perturbations which, in many cases, are of great uncertainty and hence may be treated as random; therefore, controlling chaos in such concrete systems needs to be regarded by stochastic concepts. Stochastic chaotic systems appear in many fields of science and engineering such as mechanical engineering [10], biology systems [11, 12], chemistry [13], physics and laser science [14], and financial systems [15]. For modeling stochastic chaotic systems, an Ito stochastic differential form is utilized by using the derivative of a Wiener process which creates a white Gaussian noise [16].

In [17], the sliding mode control is used for controlling stochastic chaos toward desired unstable periodic orbits of the deterministic chaotic system but the sliding mode suffers from the deficiency (chatter), which is caused by the sign function switch term in the control input. As a result, the convergence of the stochastic states to the desired equilibrium point cannot be completely achieved and state variance converges to a bound around equilibrium point. In other words, they cannot satisfy asymptotical stability condition for stochastic chaotic systems in mean-square sense.

According to the above, the main contribution of this paper is design of a very simple controller for a group of stochastic chaotic systems on the basis of Lyapunov’s direct method which guarantees the asymptotically stabilizing of these systems in mean-square sense. Therefore, by adding stochastic terms to the generalized form of a well-known group of chaotic systems, control problem of these systems is investigated. The Lyapunov direct method which is a useful tool for designing globally stabilizing control schemes is utilized in this work. One of the important advantages of this method is stabilizing systems globally without any linearization. Numerical simulations show that the proposed method can easily eliminate undesired characteristics of chaos phenomenon and stabilize the system.

This paper is organized as follows: general description of a group of chaotic systems is presented in Section 2. In Section 3, Lyapunov-based control law is obtained and stability of the proposed scheme is analyzed. In Section 4, numerical simulation results are shown. Finally, conclusion is addressed in Section 5.

*Notation*. In this paper, is the space of square-integrable vector function over , stands for the usual norm, and is a complete probability space with a filtration satisfying the usual conditions (i.e., filtration contains all P-null sets and is right continuous) [18].

#### 2. System Description

Consider a class of three-dimensional chaotic system described as [6] where , and are state variables and are nonnegative known constants. All of four functions , and are considered as smooth functions, which belong to space and .

*Remark 1. *Note that, in [6], half of recent published chaotic systems are organized by (1). Adding financial system, Table 1 illustrates these chaotic models.

Taking the consideration of control input vector and stochastic terms, the system can be expressed by
where is a nonlinear and sufficiently smooth function and , are zero-mean scalar Wiener process (Brownian motion) on with a natural filtration and are white Gaussian noise. Here, each white Gaussian noise is independent of other ones. Due to some technicalities and restrictions used in definition of [16], system (2) has to be rewritten as follows: