Sensing-Limited Control of Noiseless Linear Systems Under Nonlinear Observations

2026-01-19

AI Summary

This study investigates the fundamental information-theoretic limits of control and perception in noiseless linear systems under nonlinear observations. By introducing the notion of average directed information rate, it characterizes the coupling between the information flow from system states to observations and the expansion rate of unstable dynamics, extending classical data-rate constraints to nonlinear observation models. Under regularity conditions such as log-concavity, and leveraging tools from differential entropy divergence and information theory, the work establishes necessary and sufficient conditions for mean-square observability and stabilizability. It further derives a lower bound on the minimal information flow required for stabilization and proves the convergence of estimation error. This work reveals the intrinsic limitations imposed by the perception layer on control performance, bridging a key theoretical gap between information-theoretic bounds and estimation accuracy.

Technology Category

• Machine Learning: Information Theory
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• nonlinear observations
• information-theoretic limits
• mean-square observability
• stabilizability
• sensing-limited control

Innovation

• nonlinear observations
• information-theoretic limits
• directed information
• log-concavity
• mean-square stabilizability