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

Vincent Pacelli, Evangelos A. Theodorou

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.

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

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.
Paper Structure (11 sections, 8 theorems, 23 equations, 2 figures, 1 algorithm)

This paper contains 11 sections, 8 theorems, 23 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Task-Relevant Free Energy Bound
  4. Tightening by Coupling Task Cost and Sensor Information
  5. Cost-Information Certificates
  6. Tightening via a Fixed-Point Constraint
  7. Convexity of the Free Energy
  8. Numerical Example
  9. Numerical Implementation
  10. Dubins Car Tracking
  11. Conclusion

Key Result

Theorem 1

Let $\bar{I} \geq 0$ be a uniform MI budget. Then, where $F^x_\beta(\boldsymbol{u}) \coloneqq -\frac{1}{\beta}\log \mathds{E}_{\boldsymbol{\xi}}\qty[\exp\!({-\beta\, J_{\mathrm{trj}}^x(\boldsymbol{u};\, \boldsymbol{\xi})})]$ is the free energy of the state cost.

Figures (2)

  • Figure 1: Normalized cost vs. sensor noise for a Dubins car tracking a figure-eight reference. The sensor observes only position; heading is unobserved. All costs are reported as per-step rates $J/T$ normalized by the open-loop rate $J_{\mathrm{ol}}/T$. SC. Self-consistent TRFE bound (\ref{['thm:sc']}). OL-MI. TRFE bound with open-loop MI budget. LQG. Kalman filter + LQR linearized about the reference (upper bound on ${J}^{\star}$). At low noise the OL-MI bound is vacuous while the SC bound remains informative; by $\sigma_v \approx 2$ the SC bound captures over half the LQG cost.
  • Figure : TRFE Bound Computation

Theorems & Definitions (17)

  • Theorem 1: TRFE Bound
  • proof
  • Remark 1
  • Proposition 1
  • proof
  • Lemma 1: State Cost Budget
  • Lemma 2: MI Bound
  • proof
  • Definition 1: Cost-Information Certificate
  • Proposition 2: Nonlinear Sensors
  • ...and 7 more