Discrete-Time Sliding-Mode in Stochastic Systems

In this paper, we study the discrete-time sliding mode control for continuous-time linear systems with stationary stochastic disturbances. We study both cases of disturbances with bounded and unbounded power spectra. In the first case, the disturbance is assumed to satisfy an Ito type stochastic differential equation. The optimal filtering problem is solved to minimize the deviation from the sliding manifold in the mean square sense. Since the system under control is continuous with a sampled controller, the corresponding filtering problem is of mixed continuous-discrete type. Its solution provides optimal estimates as conditional expectation of the discrete-time disturbances, given the ¿-algebra generated by the continuous-time random process.