Stabilization of hidden semi-Markov jump systems: Emission probability approach

In this paper, the issues of stability and stabilization for a class of discrete-time hidden semi-Markov jump linear systems (HS-MJLSs) are addressed. A stabilizing controller dependent on both the observed mode and the elapsed time is constructed. By virtue of the emission probability, the controller can stabilize the underlying system when a mode mismatch between the true system mode and its observed one occurs in the system operation. A Lyapunov function, which depends not only on the current system mode, but also on the time that the system has elapsed within the current mode is proposed for less conservativeness. In the sense of σ-error mean square stability (σ-MSS), numerically testable criteria on the basis of semi-Markov kernel (SMK) and emission probability of HS-MJLSs are obtained, with the aid of certain techniques that can eliminate the terms containing products of matrices are employed. A numerical example is provided to illustrate the effectiveness of the proposed control strategy and the advantage of considering the observed-mode and elapsed-time-dependent in the proposed control scheme for HS-MJLSs.