Exact minimum number of bits to stabilize a linear system

We consider an unstable scalar linear stochastic system, X_(n+1) = aX_n + Z_n − U_n , where a ≥ 1 is the system gain, Z_n'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 Xi. We show new proofs that M > a is necessary and sufficient for β-moment stability, for any β < α. 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.