Sensing-Limited Control of Noiseless Linear Systems Under Nonlinear Observations

TL;DR

This paper explores the fundamental limits of controlling noiseless linear systems with nonlinear observations, establishing information-theoretic bounds for stability and estimation under broad conditions.

Contribution

It derives necessary and sufficient conditions for observability and stabilizability in nonlinear observation settings, extending classical data-rate limits.

Findings

Directed information rate must exceed system expansion rate

Log-concavity ensures entropy divergence implies error convergence

Fundamental limits extend classical data-rate bounds to nonlinear observations

Abstract

This paper investigates the fundamental information-theoretic limits for the control and sensing of noiseless linear dynamical systems subject to a broad class of nonlinear observations. We analyze the interactions between the control and sensing components by characterizing the minimum information flow required for stability. Specifically, we derive necessary conditions for mean-square observability and stabilizability, demonstrating that the average directed information rate from the state to the observations must exceed the intrinsic expansion rate of the unstable dynamics. Furthermore, to address the challenges posed by non-Gaussian distributions inherent to nonlinear observation channels, we establish sufficient conditions by imposing regularity assumptions, specifically log-concavity, on the system's probabilistic components. We show that under these conditions, the divergence of differential entropy implies the convergence of the estimation error, thereby closing the gap between information-theoretic bounds and estimation performance. By establishing these results, we unveil the fundamental performance limits imposed by the sensing layer, extending classical data-rate constraints to the more challenging regime of nonlinear observation models.