Exponential Stability of Markovian Jumping Systems via Adaptive Sliding Mode Control

In this paper, the exponential stability in mean square for Markovian jumping systems (MJSs) is discussed. A new dynamic model, which involves parameters uncertainties, nonlinearities, and Lévy noises, is proposed. Moreover, an adaptive sliding mode controller is built to study the stability of such a complex model. First, an integral-type sliding mode surface (SMS) is established to obtain the sliding mode motion dynamics of MJSs. By the generalized Itô formula and the Lyapunov stability theory, some sufficient conditions are obtained to make sure the exponential stability in mean square for the sliding mode motion dynamics. Second, an adaptive sliding mode control law is provided to assure the reachability of the specified SMS. Furthermore, corresponding parameters of the sliding mode controller and the SMS can be got by solving the convex optimization problem. Finally, the validity of the stability results obtained is illustrated by a numerical simulation and a practical simulation.