Lyapunov-Based Controller for a Class of Stochastic Chaotic Systems

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