On the Semi-Decidability of Remote State Estimation and Stabilization via Noisy Communication Channels

TL;DR

This paper investigates the fundamental limits of remote state estimation and stabilization over noisy channels, revealing semi-decidability results that depend on system disturbance and feedback, with implications for autonomous system design.

Contribution

It connects control theory with computability and information theory, analyzing the decidability of stabilization problems using recent results on channel capacities.

Findings

Decidability depends on whether the plant is disturbed by noise.

Results show semi-decidability in various scenarios with and without feedback.

Fundamental limits are established for computer-aided control systems.

Abstract

We consider the task of remote state estimation and stabilization of disturbed linear plants via noisy communication channels. In 2007 Matveev and Savkin established a surprising link between this problem and Shannon's theory of zero-error communication. By applying very recent results of computability of the channel reliability function and computability of the zero-error capacity of noisy channels by Boche and Deppe, we analyze if, on the set of linear time-invariant systems paired with a noisy communication channel, it is uniformly decidable by means of a Turing machine whether remote state estimation and stabilization is possible. The answer to this question largely depends on whether the plant is disturbed by random noise or not. Our analysis incorporates scenarios both with and without channel feedback, as well as a weakened form of state estimation and stabilization. In the broadest sense, our results yield a fundamental limit to the capabilities of computer-aided design and autonomous systems, assuming they are based on real-world digital computers.