Characterization of Information Channels for Asymptotic Mean Stationarity and Stochastic Stability of Non-stationary/Unstable Linear Systems

TL;DR

This paper establishes precise conditions under which unstable non-stationary linear systems can be stabilized over noisy communication channels, linking channel capacity to system stability and ergodic properties.

Contribution

It provides tight necessary and sufficient conditions for stochastic stabilizability of non-stationary linear systems over channels with memory and feedback, using a novel drift-based proof approach.

Findings

Channel capacity must exceed the sum of logs of unstable poles for stability.

Necessary and sufficient conditions are derived for various stochastic stability notions.

Conditions for finite second moments under unbounded noise are also established.

Abstract

Stabilization of non-stationary linear systems over noisy communication channels is considered. Stochastically stable sources, and unstable but noise-free or bounded-noise systems have been extensively studied in information theory and control theory literature since 1970s, with a renewed interest in the past decade. There have also been studies on non-causal and causal coding of unstable/non-stationary linear Gaussian sources. In this paper, tight necessary and sufficient conditions for stochastic stabilizability of unstable (non-stationary) possibly multi-dimensional linear systems driven by Gaussian noise over discrete channels (possibly with memory and feedback) are presented. Stochastic stability notions include recurrence, asymptotic mean stationarity and sample path ergodicity, and the existence of finite second moments. Our constructive proof uses random-time state-dependent stochastic drift criteria for stabilization of Markov chains. For asymptotic mean stationarity (and thus sample path ergodicity), it is sufficient that the capacity of a channel is (strictly) greater than the sum of the logarithms of the unstable pole magnitudes for memoryless channels and a class of channels with memory. This condition is also necessary under a mild technical condition. Sufficient conditions for the existence of finite average second moments for such systems driven by unbounded noise are provided.