Almost Sure Stability of Stochastic Linear Systems with Ergodic Parameters

This paper deals with the notion of almost sure stability for linear stochastic systems whose dynamic matrix depends on an ergodic process. It is shown that such systems are exponentially almost surely stable if and only if their transition matrix is averagely contractive over a finite, yet unknown, time interval. In order to test this condition, an efficient computational procedure based on a Monte Carlo strategy is proposed. Moreover, an H∞ condition ensuring exponential almost sure stability is proven. The condition requires that the mean square value of the ergodic process is less than a constant involving the H∞-norm of an appropriate transfer function. The applicability of the proposed methods is illustrated by means of a numerical example.