Exact Minimum Number of Bits to Stabilize a Linear System

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
$\lbrace 1, \ldots, M\rbrace$
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 the performance of our scheme 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.