Sliding mode control for state-delayed Markov jump systems with partly unknown transition probabilities

In this paper, we probe the $H_{\infty }$ control problem of state-delayed Markov jump systems with partly unknown transition probabilities by using sliding mode control approach. The $H_{\infty }$ performance index is to improve the system performance of the control system. The exact information of transition probabilities is assumed to be partially known, and the bounds of nonlinear function are unknown. Firstly, a sliding mode observer is designed to estimate the unmeasured state. Secondly, an integral sliding mode surface is constructed such that the reduced-order sliding motion is insensitive to all admissible uncertainties, nonlinearities and external disturbances. Thirdly, we design an adaptive sliding mode controller to maintain the state trajectories in the sliding mode surface. The sufficient condition of stochastic stability for the closed-loop system is derived via Lyapunov stability theory. Finally, a numerical example is given to show the effectiveness of the proposed results.