Passivity-Sliding Mode Control of Uncertain Chaotic Systems with Stochastic Disturbances

This paper is concerned with the stabilization problem of uncertain chaotic systems with stochastic disturbances. A novel sliding function is designed, and then a sliding mode controller is established such that the trajectory of the system converges to the sliding surface in a finite time. Using a virtual state feedback control technique, sufficient condition for the mean square asymptotic stability and passivity of sliding mode dynamics is derived via linear matrix inequality (LMI). Finally, a simulation example is presented to show the validity and advantage of the proposed method. 1. Introduction Chaos control has received a great deal of interest in the last three decades. Many techniques for chaos control are used, such as adaptive control [1, 2], backstepping control [3], fuzzy control [4], and sliding mode control [5–7]. In the above methods, the chaotic system model has a deterministic differential equation; there is no random parameter or random excitation on the system governing equation. But the chaotic system may be affected by stochastic disturbances due to environmental noise [8, 9]. Stochastic chaotic systems appear in many fields such as chemistry [10], physics and laser science [11], and economics [12]. So it is necessary to study such systems. There have been some meaningful results about chaos system with stochastic disturbances [13–15]. In [13], the ergodic theory and stochastic noise are considered. Time-delay feedback control of the Van der Pol oscillator under the influence of white noise is investigated in [14]. In [15], an adaptive control of of chaotic systems with white Gaussian noise is considered. On the other hand, sliding mode control is a very effective approach for the robust control systems. It has many attractive features such as fast response, good transient response, and insensitivity to variations. Some related results have been presented [16, 17]. In [16], control of stochastic chaos via sliding mode control is investigated. In [17], chaos synchronization of nonlinear gyros with stochastic excitation is considered by using sliding mode control. Since then, from a practical point of view, many systems need to be passive in order to attenuate noises. So it is necessary to investigate the passivity of chaotic systems via sliding mode control. However, there have been few results in this respect. Motivated by the above reasons, passivity-sliding mode control problem of uncertain chaotic systems with stochastic disturbances is considered. Stochastic disturbance is a standard Wienner process, the derivative of which