The semi-Markov jump linear system (S-MJLS) is more general than the Markov jump linear system (MJLS) in modeling some practical systems. Unlike the constant transition rates in the MJLS, the transition rates of the S-MJLS are time varying. This paper focuses on the robust stochastic stability condition and the robust control design problem for the S-MJLS with norm-bounded uncertainties. The infinitesimal generator for the constructed Lyapunov function is first derived. Numerically solvable sufficient conditions for the stochastic stability of S-MJLSs are then established in terms of linear matrix inequalities. To reduce the conservativeness of the stability conditions, we propose to incorporate the upper and lower bounds of the transition rate and meanwhile apply a new partition scheme. The robust state feedback controller is accordingly developed. Simulation studies and comparisons demonstrate the effectiveness and advantages of the proposed methods.Consider the following unforced continuous-time S-MJLS with norm-bounded uncertainties:Following the same techniques used in Theorem 2 and Corollary 1, Theorem 4 and Corollary 2 can be readily proved, and hence, the proof is omitted here.
The semi-Markov jump linear system is more general than the classic Markov jump linear system. In the semi-Markov jump linear systems, the governing stochastic process is not a Markov process, but a semi-Markov process. Instead of the exponential distribution for the sojourn-time in each mode in the jump linear system, the Weibull distribution is considered in this paper. By deriving the infinitesimal generator for the Lyapunov function of the semi-Markov jump linear system, the numerically testable sufficient conditions for stochastic stability of semi-Markov jump linear systems are obtained. Numerical examples are provided to validate the proposed sufficient stochastic stability conditions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.