Ergodicity Conditions For Controlled Stochastic Non-Linear Systems Under Information Constraints

TL;DR

This paper establishes fundamental lower bounds on the communication channel capacity needed to ensure ergodic stability in controlled stochastic nonlinear systems, extending classical results to more general nonlinear cases.

Contribution

It introduces a new lower bound on channel capacity for nonlinear systems using a modified invariance entropy and ergodic theorems, generalizing linear system results.

Findings

Lower bounds on channel capacity are derived for ergodic stability.

The bounds are expressed via the log-determinant of the linearization.

Results extend classical linear system formulas to nonlinear systems.

Abstract

Consider a stochastic nonlinear system controlled over a possibly noisy communication channel. An important problem is to characterize the largest class of channels for which there exist 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. We prove this bound by introducing a modified version of invariance entropy and utilizing the almost sure convergence of sample paths guaranteed by the pointwise ergodic theorem. The fundamental bounds obtained generalize well-known formulas for linear systems, and are in some cases more refined than those obtained for nonlinear systems via information-theoretic methods.