An Asymptotically Optimal Two-Part Fixed-Rate Coding Scheme for Networked Control with Unbounded Noise

TL;DR

This paper introduces a two-part fixed-rate coding scheme for networked control systems with unbounded noise, achieving optimal state second moment convergence and stability under mild conditions.

Contribution

It presents a novel two-part adaptive fixed-rate coding scheme that ensures optimal second moment convergence and ergodicity in unstable linear systems with unbounded noise.

Findings

Achieves state second moment convergence to the classical optimum.

Ensures system ergodicity and stability with high-rate coding.

Uses random-time state-dependent Lyapunov analysis for proofs.

Abstract

It is known that under fixed-rate information constraints, adaptive quantizers can be used to stabilize an open-loop-unstable linear system on $\mathbb{R}^n$ driven by unbounded noise. These adaptive schemes can be designed so that they have near-optimal rate, and the resulting system will be stable in the sense of having an invariant probability measure, or ergodicity, as well as boundedness of the state second moment. Although structural results and information theoretic bounds of encoders have been studied, the performance of such adaptive fixed-rate quantizers beyond stabilization has not been addressed. In this paper, we propose a two-part adaptive (fixed-rate) coding scheme that achieves state second moment convergence to the classical optimum (i.e., for the fully observed setting) under mild moment conditions on the noise process. The first part, as in prior work, leads to ergodicity (via positive Harris recurrence) and the second part ensures that the state second moment converges to the classical optimum at high rates. These results are established using an intricate analysis which uses random-time state-dependent Lyapunov stochastic drift criteria as a core tool.