Stability and Stabilization of Semi-Markov Jump Linear Systems with Exponentially Modulated Periodic Distributions of Sojourn Time

This paper is concerned with a class of discrete-time semi-Markov jump linear systems (S-MJLSs) subject to exponentially modulated periodic (EMP) probability density function (PDF) of sojourn time, and the problems of stability and stabilization are addressed. Setting a relatively large period, the considered systems are capable of approximating the general S-MJLSs (without any requirements on sojourn-time PDFs) for which numerically testable stability and stabilization conditions are rather difficult to obtain. Necessary and sufficient criterion for mean square stability of the general S-MJLSs is first derived, which involves an infinite number of conditions and as such not checkable. However, the developments lay a foundation to further establish the numerically testable conditions for the systems when the PDF of sojourn time is EMP albeit the sojourn time can tend to infinity. The derivations explicitly depend on the PDF of sojourn time, which circumvents the difficulty in obtaining the memory transition probabilities of S-MJLSs. The adopted Lyapunov function is sojourn-time-dependent (STD), by which the existence conditions of STD controller are developed as well using certain techniques that can eliminate the terms of power of matrices in the stability conditions. Two illustrative examples including a class of population ecological systems are presented to show the validity and applicability of the developed theoretical results.