Stability analysis and stabilization of semi-Markov jump linear systems with unavailable sojourn-time information

This study addresses the stability and stabilization problems of discrete-time semi-Markov jump linear systems (S-MJLSs) with unavailable sojourn-time information. The sojourn-time probability mass functions (S-TPMFs) of discrete-time semi-Markov chains are no longer confined to the geometric distribution, and it is difficult to obtain accurate and comprehensive information for S-TPMFs in practice. This is because S-TPMFs are usually deduced from the statistical characteristics according to the sampled-data, while adequate samples are often costly and time consuming. In this study, when the S-TPMFs for semi-Markov chains are assumed to be unavailable, the σ-error mean square stability is investigated for discrete-time S-MJLSs with some widely used assumptions; for semi-Markov chains, only the transition probability matrix of the embedded chain is used. In addition, the existence conditions of the effective controller are provided for closed-loop systems without using the information of S-TPMFs. Numerical examples are presented to illustrate the validity of the obtained theoretical results.