Passive stochastic feedback stability Part I. A general theory†

Abstract

The use of passivity concepts is well known in determining the stability of deterministic feedback systems. In this paper (Part I) a parallel theory is developed for feedback systems in which one or more of the sub-systems is stochastic in nature. This theory is compatible with and essentially subsumes one passive stability theory formulation for deterministic systems with a state representation given in recent work by C. A. Desoer and the author. A definition of a stochastic dynamical system is given in the paper. This definition has a generality comparable to many definitions of deterministic dynamical systems which have appeared in the literature. The input and output spaces of such a stochastic system are classes of random processes and its state is intimately connected with Markov concepts. It is shown that, under mild restrictions, a general deterministic dynamical system subject to ensembles of inputs (random processes) also defines a stochastic dynamical system. In this paper the feedback system a...