Signalling and Control in Nonlinear Stochastic Systems: An Information State Approach with Applications

TL;DR

This paper develops an information-theoretic framework for optimal signaling and control in nonlinear stochastic systems, introducing the concept of control-coding capacity and analyzing strategies involving multiple Kalman filters.

Contribution

It introduces the control-coding capacity for nonlinear systems and characterizes optimal strategies using information states and Kalman filters, extending control theory with an information-theoretic perspective.

Findings

Characterization of control-coding capacity in nonlinear systems

Finite-dimensional sufficient statistics for signaling and control strategies

Explicit optimal control strategies derived via Riccati equations

Abstract

We consider optimal signalling and control of discrete-time nonlinear partially observable stochastic systems in state space form. In the first part of the paper, we characterize the operational {\it control-coding capacity}, $C_{FB}$ in bits/second, by an information theoretic optimization problem of encoding signals or messages into randomized controller-encoder strategies, and reproducing the messages at the output of the system using a decoder or estimator with arbitrary small asymptotic error probability. Our analysis of $C_{FB}$ is based on realizations of randomized strategies (controller-encoders), in terms of information states of nonlinear filtering theory, and either uniform or arbitrary distributed random variables (RVs). In the second part of the paper, we analyze the linear-quadratic Gaussian partially observable stochastic system (LQG-POSS). We show that simultaneous signalling and control leads to randomized strategies described by finite-dimensional sufficient statistics, that involve two Kalman-filters, and consist of control, estimation and signalling strategies. We apply decentralized optimization techniques to prove a separation principle, and to derive the optimal control part of randomized strategies explicitly in terms of a control matrix difference Riccati equation (DRE).