Ergodicity Conditions for Controlled Stochastic Nonlinear Systems under Information Constraints

Abstract

Consider a stochastic nonlinear system controlled over a possibly noisy communication channel. An important problem is to characterize the largest class of channels which admit coding and control policies so that the closed-loop system is stochastically stable. In this paper, we consider the stability notion of (asymptotic) ergodicity. We prove lower bounds on the channel capacity necessary to achieve the stability criterion. Under mild technical assumptions, we obtain that the necessary channel capacity is lower bounded by the log-determinant of the linearization, double-averaged over the state and noise space. Our proof uses a modified version of invariance entropy, and utilizes the almost sure convergence of sample paths guaranteed by the pointwise ergodic theorem. Our results generalize those for linear systems, and are in some cases more refined than those obtained for nonlinear systems via information-theoretic methods.