Impulsive Systems triggered by Superposed Renewal Processes

We consider impulsive systems with several reset maps triggered by independent renewal processes, i.e., the intervals between jumps associated with a given reset map are identically distributed and independent of the other jump intervals. Considering linear dynamic and reset maps, we establish that mean exponential stability is equivalent to the spectral radius of an integral operator being less than one. The result builds upon a stochastic Lyapunov function approach which allows for providing stability conditions in the general case where the dynamic and the reset maps are non-linear. We also prove that the origin of an impulsive system with non-linear dynamic and reset maps is stable with probability one if the linearization about zero equilibrium is mean exponentially stable, which justifies the importance of studying the linear case. The application of these results is illustrated in the context of networked control systems. The results in this paper permit the analysis of scenarios in which sensors transmit through independent communication links, which introduce independent and identically distributed intervals between transmissions, instead of sharing a single communication link, as considered in a previous work.