Delay-dependent and decay-rate-dependent conditions for exponential mean stability and non-fragile controller design of positive Markov jump linear systems with time-delay

Abstract

This paper proposes some new sufficient conditions and necessary conditions for exponential mean stability with prescribed decay rate, and sufficient conditions for non-fragile controller design of both continuous-time and discrete-time positive Markov jump linear systems with time-delay. First, sufficient stability conditions are derived by constructing novel linear stochastic co-positive Lyapunov-Krasovskii functionals. Second, the corresponding necessary conditions are established by applying a model transformation technique and analyzing the relationship between stochastic stability of the transformed systems and exponential mean stability with given decay rate of the original systems. Compared with the existing conditions, the proposed stability conditions are not only delay-dependent but also decay-rate-dependent, and the sufficient conditions plus the necessary conditions can be used to judge the system stability more precisely. Third, based on the new sufficient conditions, non-fragile controllers are designed by solving linear programming problems such that the closed-loop systems are positive and exponentially mean stable with an expected decay rate. Finally, by numerical examples, the effects of both time-delay and decay rate on exponential mean stability are exploited and the validity of the non-fragile controller design conditions is demonstrated.