Exact Minimum Number of Bits to Stabilize a Linear System

• 成果类型: Article
• 署名作者: Kostina, Victoria; Peres, Yuval; Ranade, Gireeja; Sellke, Mark
• 署名单位: California Institute of Technology; University of California System; University of California Berkeley; Stanford University
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2021.3126679
• 发表日期: 2022
• 页码: 5548-5554
• 关键词: Data rate theorem; linear stochastic control; source coding

摘要

We consider an unstable scalar linear stochastic system, Xn+1 = aX(n) + Z(n) - U-n, where a >= 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, ..., 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 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.