Volterra Integral Approach to Impulsive Renewal Systems: Application to Networked Control

We analyze impulsive systems with independent and identically distributed intervals between transitions. Our approach involves the derivation of novel results for Volterra integral equations with positive kernel. We highlight several applications of these results, and show that when applied to the analysis of impulsive systems they allow us to (i) provide necessary and sufficient conditions for mean square stability, stochastic stability and mean exponential stability, which can be equivalently tested in terms of a matrix eigenvalue computation, an LMI feasibility problem, and a Nyquist criterion condition; (ii) assess performance of the impulsive system by computing a second moment Lyapunov exponent. The applicability of our results is illustrated in a benchmark problem considering networked control systems with stochastically spaced transmissions, for which we can guarantee stability for inter-sampling times roughly twice as large as in previous papers.