Stability analysis and stabilization of discrete-time non-homogeneous semi-Markov jump linear systems: A polytopic approach

Abstract

This paper addresses the issues of stability analysis and stabilizing controller design for two classes of discrete-time non-homogeneous semi-Markov jump linear systems (S-MJLSs). The first class is concerned with the sojourn-time probability mass functions that are independent of jump instants, while the other considers the existence of probability mass functions of sojourn time depending on jump instants. New techniques are developed respectively for both classes of systems by transforming their corresponding time-varying semi-Markov kernels into more tractable forms. Based on the stochastic stability allowing for bounded sojourn time, the derived stability and stabilization criteria containing the probability distribution information of random sojourn time can be numerically tested. In order to reduce the possible conservatism of the obtained results, a novel class of polytopic quadratic Lyapunov function is constructed, which depends on both the current mode and the elapsed time in the current mode. Finally, the proposed control strategies are applied to an illustrative application of automotive electronic throttle valve to show the effectiveness of the theoretical results. The results also reveal the significance of considering the inhomogeneity of S-MJLSs and the importance of our constructed Lyapunov function.