Stability and stabilization of discrete-time semi-Markov jump linear systems with delay in controller mode switching

This paper is concerned with the asynchronous control problem for a class of discrete-time semi-Markov jump linear systems via the semi-Markov kernel approach, where “asynchronous control” refers to a mismatch of the modes between the system and the mode-dependent controller, which results from one-step time delays in the mode switching of the controller. By employing a Lyapunov function depending on both the sojourn time and system mode, sufficient conditions on the existence of the desired mode-dependent controller are developed such that the closed-loop semi-Markov jump linear system is σ-error mean square stable. Finally, a numerical example is provided to illustrate the effectiveness of the proposed control strategy.