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

TL;DR

This work develops an information-state framework to quantify and design signaling and control for discrete-time nonlinear partially observable stochastic systems, introducing the control-coding capacity $C_{FB}(\kappa)$ as the maximal CC rate under an average-payoff constraint. It shows that the problem can be recast via sufficient statistics (information states) and randomized inputs, with directed information $I(A^n\rightarrow Y^n)$ governing capacity; for LQG-POSS, it establishes Gaussian-input optimality and a decentralized separation principle comprising two Kalman filters and a control Riccati equation, enabling explicit Riccati-based design and a matrix-ARE/ DRE representation. The asymptotic limit exists under time-invariance assumptions and reduces to a log-det objective constrained by AREs, unifying stochastic control with information-channel theory and recovering prior results as special cases. Overall, the paper provides a rigorous, generalizable bridge between control and communications in systems with memory and partial observability, with practical implications for coded control and communication over noisy, stateful channels.

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).

Theorems & Definitions (22)

• Theorem II.1
• proof
• Lemma II.1
• proof
• Theorem II.2
• proof
• Theorem II.3
• proof
• Lemma III.1
• proof