Stability analysis for impulsive switched systems with bounded sojourn time intervals

This paper introduces novel methods for analyzing the stability of impulsive switched systems with bounded sojourn time intervals. We contend that existing stability results may be overly conservative when sojourn times are treated as stochastic rather than arbitrary over intervals. This motivates our investigation into stochastic stability criteria for impulsive switched systems, where the sojourn times of impulsive switching signals obey uniform distributions over intervals. Our first theoretical contribution offers an exact formula for the component-wise first moment of the state for linear impulsive switched systems, expressed in terms of a solution to an auxiliary linear time-delay system. We extend this approach to nonlinear systems by leveraging a multiple Lyapunov function assumption. Consequently, the exponential mean stability of such impulsive switched systems can be determined by assessing the stability of the auxiliary systems. Nevertheless, this method fails to establish exponential mean stability when the impulsive switched system has unstable subsystems. To address this limitation, we conduct an in-depth frequency domain analysis, revealing a common occurrence of pole-zero cancellation. Leveraging these insights, we propose a non-conservative approach for stability analysis. Simulations demonstrate that our proposed stability criteria effectively determine stability in scenarios where all subsystems are unstable, or instability when all subsystems are stable.