Time-invariant Prefix Coding for LQG Control

TL;DR

This paper introduces a novel time-invariant data compression scheme for LQG control that nearly attains theoretical lower bounds on communication rate, ensuring consistent long-term performance without time-varying coding.

Contribution

It proves the existence of a fixed source code for quantizations in LQG control that achieves near-optimal long-term communication efficiency, a first in the literature.

Findings

Nearly achieves known DI lower bounds

Quantizations have a limiting distribution

Fixed source code suffices for optimal long-term performance

Abstract

Motivated by control with communication constraints, in this work we develop a time-invariant data compression architecture for linear-quadratic-Gaussian (LQG) control with minimum bitrate prefix-free feedback. For any fixed control performance, the approach we propose nearly achieves known directed information (DI) lower bounds on the time-average expected codeword length. We refine the analysis of a classical achievability approach, which required quantized plant measurements to be encoded via a time-varying lossless source code. We prove that the sequence of random variables describing the quantizations has a limiting distribution and that the quantizations may be encoded with a fixed source code optimized for this distribution without added time-asymptotic redundancy. Our result follows from analyzing the long-term stochastic behavior of the system, and permits us to additionally guarantee that the time-average codeword length (as opposed to expected length) is almost surely within a few bits of the minimum DI. To our knowledge, this time-invariant achievability result is the first in the literature. The originally published version of the supplementary material included a proof that contained an error that turned out to be inconsequential. This updated preprint corrects this error, which originally appeared under Lemma A.7.