Deciding the Problem of Remote State Estimation via Noisy Communication Channels on Real Number Signal Processing Hardware

Abstract

We consider a decision problem associated to the task of estimating the state of a dynamic plant remotely via a noisy communication channel: given the characteristics of some unstable linear time-invariant (LTI) plant and some discrete memoryless channel (DMC), does there exist an encoder/decoder pair that allows for the remote tracking of the plant’s state with bounded error? Questions of this kind are becoming increasingly important in communication technologies, since future communication networks are expected to incorporate distributed control and decision-making. Analytically, this problem has been shown to involve the zero-error capacity of the DMC. Starting from this result, we approach the problem from the view of theoretical computer science, with an explicit treatment of the underlying machine Model. In particular, we prove that for every pair of a finite channel input alphabet and a finite channel output alphabet, there exists a Blum-Shub-Smale (BSS) algorithm that computes the zero-error capacity in dependence of the channel matrix. Based on this, we devise a BSS algorithm that solves the above decision problem given the plant’s and DMC’s characteristics. BSS machines are a promising candidate for a universal model of real number processing hardware, comparable to the Turing machine in the digital domain. Recently, we observe an increased interest in research and development towards real number and/or analog computing hardware, usually referred to by the term "neuromorphic computing".