Stability and stabilization of a class of stochastic switching systems with lower bound of sojourn time

This paper is concerned with the stability and stabilization issues for a family of discrete-time stochastic switching systems with bounded sojourn time. The stochastic switching systems are modeled by semi-Markov jump linear systems and the semi-Markov kernel approach is employed to handle the stability and stabilization problems. The sojourn time of each system mode is considered to have both upper and lower bounds, which is more general than the scenarios in previous literature that only consider the upper bound of sojourn time. The concept of σ-error mean square stability is put forward in a new form by taking into account the lower bounds of sojourn time for all system modes. By virtue of a Lyapunov function that not only depends on the current system mode but also on the elapsed time the system has been in the current mode, together with certain techniques eliminating powers of matrices, numerically testable stability and stabilization criteria in the sense of the proposed σ-error mean square stability are obtained for the closed-loop stochastic switching system. Finally, a numerical example and a practical example of a DC motor are utilized to demonstrate the effectiveness of the proposed control strategy and the superiority of allowing for the lower bound of sojourn time.