Exact minimum number of bits to stabilize a linear system

We consider an unstable scalar linear stochastic system, X
_n
+
₁
= aXn + Z
_n
- U
_n
.; where a ≥ 1 is the system gain, Zn's are independent random variables with bounded α-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, ..., M} as its only information about system state X
_i
. We show that M = [a] + 1 is necessary and sufficient for ß- moment stability, for any ß <; a. Our achievable scheme is a uniform quantizer of the zoom-in / zoom-out type. We analyze its performance using probabilistic arguments. We prove 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.