Nonconservative Stability Criteria for Semi-Markovian Impulsive Switched Systems

SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1783-1808, June 2024.
Abstract. This paper proposes criteria for establishing the asymptotic moment stability of semi-Markovian impulsive switched systems. Under some mild assumptions, we formulate an auxiliary linear time-delayed system based on the Lyapunov characterizations of the subsystems and impulses, as well as the properties of the underlying semi-Markovian impulsive switching signal. Our main result provides an upper bound on the moment, which is directly related to a solution of the aforementioned linear time-delayed system. Specifically, the semi-Markovian impulsive switched system is asymptotically moment stable if the auxiliary linear time-delayed system is asymptotically stable. In situations where the mode-dependent sojourn time distributions of the underlying impulsive switching signals are all exponential, uniform, or trigonometric, we deduce explicit formulae for the auxiliary linear time-delayed systems. To prove the main result, we compute the expected gain function, which requires formulating a generalized renewal equation. Finally, we test our stability criteria on a numerical example in different scenarios and show that our stability results are nonconservative compared to the statistically obtained average of state-norms and state-norm-squares.