The necessity and sufficiency of anytime capacity for stabilization of a linear system over a noisy communication link Part I: scalar systems

TL;DR

This paper demonstrates that traditional Shannon capacity is insufficient for stabilizing unstable scalar linear systems over noisy channels, and introduces the concept of anytime capacity as both necessary and sufficient for stabilization under various conditions.

Contribution

It establishes the necessity of anytime capacity for stabilization and extends the sufficiency of this concept to complex scenarios including noisy observations and delayed feedback.

Findings

Anytime capacity is necessary for stabilization.

Adequate anytime capacity can be sufficient for stabilization.

Results extend to continuous-time systems and complex communication scenarios.

Abstract

We review how Shannon's classical notion of capacity is not enough to characterize a noisy communication channel if the channel is intended to be used as part of a feedback loop to stabilize an unstable scalar linear system. While classical capacity is not enough, another sense of capacity (parametrized by reliability) called ``anytime capacity'' is shown to be necessary for the stabilization of an unstable process. The required rate is given by the log of the unstable system gain and the required reliability comes from the sense of stability desired. A consequence of this necessity result is a sequential generalization of the Schalkwijk/Kailath scheme for communication over the AWGN channel with feedback. In cases of sufficiently rich information patterns between the encoder and decoder, adequate anytime capacity is also shown to be sufficient for there to exist a stabilizing controller. These sufficiency results are then generalized to cases with noisy observations, delayed control actions, and without any explicit feedback between the observer and the controller. Both necessary and sufficient conditions are extended to continuous time systems as well. We close with comments discussing a hierarchy of difficulty for communication problems and how these results establish where stabilization problems sit in that hierarchy.