Fundamental Limits for Sensor-Based Control via the Gibbs Variational Principle

2026-03-18

AI Summary

This work addresses a critical limitation in existing methods for evaluating the fundamental performance limits of sensor-based feedback control, which typically rely on open-loop assumptions and consequently yield loose or even vacuous lower bounds precisely when feedback is most effective—such as under low sensor noise. By applying the Gibbs variational principle to the joint path measure of states and observations, the authors derive a tight lower bound on the expected cost achievable by any causal feedback controller in partially observable systems. A key innovation is a self-consistent refinement mechanism: the controller implicitly constrains the state distribution, thereby limiting the information extractable by the sensor and dynamically tightening the bound, leading to a unique fixed-point equation. Leveraging path-space probabilistic analysis, free energy minimization, and convexity verification, the bound is efficiently computed via bisection. In nonlinear Dubins vehicle tracking tasks, this self-consistent bound closely approximates the optimal cost across varying sensor noise levels, markedly outperforming conventional open-loop bounds that degrade or fail entirely.

Technology Category

• Intelligent Robots: State Estimation
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

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

Abstract

Fundamental limits on the performance of feedback controllers are essential for benchmarking algorithms, guiding sensor selection, and certifying task feasibility -- yet few general-purpose tools exist for computing them. Existing information-theoretic approaches overestimate the information a sensor must provide by evaluating it against the uncontrolled system, producing bounds that degrade precisely when feedback is most valuable. We derive a lower bound on the minimum expected cost of any causal feedback controller under partial observations by applying the Gibbs variational principle to the joint path measure over states and observations. The bound applies to nonlinear, nonholonomic, and hybrid dynamics with unbounded costs and admits a self-consistent refinement: any good controller concentrates the state, which limits the information the sensor can extract, which tightens the bound. The resulting fixed-point equation has a unique solution computable by bisection, and we provide conditions under which the free energy minimization is provably convex, yielding a certifiably correct numerical bound. On a nonlinear Dubins car tracking problem, the self-consistent bound captures most of the optimal cost across sensor noise levels, while the open-loop variant is vacuous at low noise.

Problem

• fundamental limits
• sensor-based control
• feedback control
• information theory
• performance bounds

Innovation

• Gibbs variational principle
• fundamental limits
• sensor-based control
• self-consistent bound
• partial observability