Stability Analysis of Discrete-Time Semi-Markov Jump Linear Systems

This article discusses the stability and stabilization problems for the discrete-time semi-Markov jump linear systems (S-MJLSs) with bounded sojourn times. By using the method of multiple-Lyapunov functions and the concept of discrete-time semi-Markov kernel, we provide the sufficient conditions for σ-error mean square stability of S-MJLSs. In such conditions, the transition probability matrix of embedded Markov chain is considered, which is more general than the scenarios in previous works that only consider the sojourn time of each subsystem mode. Based on such conditions, the stabilization controller design method is also proposed for the resulting closed-loop systems. Two examples are given to illustrate the effectiveness of our results.