Observer-Based Control for Discrete-Time Hidden Semi-Markov Jump Systems

In this note, the control problem for discrete-time hidden semi-Markov jump systems is investigated under the situation that the system modes are unavailable directly. Compared with the previous results, the case when both system modes and states are immeasurable simultaneously is fully considered and an observer-based control scheme is established for discrete-time hidden semi-Markov jump systems. The random parameters of the systems are subject to a two-layer stochastic process and the underlying stochastic process is assumed to be subject to a semi-Markov process. Furthermore, a novel switched Lyapunov function is used, which may be monotonically increasing or decreasing at the instant of nonmodal switching. Moreover, by means of the stochastic stability theory and semi-Markov kernel method, some novel sufficient conditions ensuring $\sigma$ -error mean-square stability of the closed-loop system are obtained by a power term reduction method. Finally, an illustrative example is introduced to verify the feasibility of the proposed control method.