Exact minimum number of bits to stabilize a linear system

TL;DR

This paper establishes the exact minimum data rate needed to stabilize an unstable linear stochastic system, demonstrating that more than the system gain in bits is necessary and sufficient for stability.

Contribution

It provides a new proof that the minimum number of messages exceeds the system gain for stability and introduces an efficient quantization scheme with a matching converse proof.

Findings

M > a is necessary and sufficient for stability

A zoom-in/zoom-out quantizer achieves data rate efficiency

The results extend to vector systems and noisy channels

Abstract

We consider an unstable scalar linear stochastic system, $X_{n+1}=a X_n + Z_n - U_n$, where $a \geq 1$ is the system gain, $Z_n$'s are independent random variables with bounded $\alpha$-th moments, and $U_n$'s are the control actions that are chosen by a controller who receives a single element of a finite set $\{1, \ldots, M\}$ as its only information about system state $X_i$. We show new proofs that $M > a$ is necessary and sufficient for $\beta$-moment stability, for any $\beta < \alpha$. Our achievable scheme is a uniform quantizer of the zoom-in / zoom-out type that codes over multiple time instants for data rate efficiency; the controller uses its memory of the past to correctly interpret the received bits. We analyze its performance using probabilistic arguments. We show a simple proof of a matching converse using information-theoretic techniques. Our results generalize to vector systems, to systems with dependent Gaussian noise, and to the scenario in which a small fraction of transmitted messages is lost.