A lower bound on feedback controller performance under partial observations emerges from applying the Gibbs variational principle to joint path measures over state and observation trajectories. The bound features a self-consistent refinement mechanism: a good controller concentrates the state distribution, which limits the information the sensor can extract, which tightens the performance bound --- creating a fixed-point equation solvable by bisection. On a nonlinear Dubins car tracking problem, the self-consistent bound captures most of the optimal cost across noise levels, substantially outperforming open-loop bounds at low noise where the feedback-information coupling is strongest.
Performance bounds for partially observed systems have historically been either loose (open-loop bounds that ignore feedback) or intractable (exact solutions requiring full dynamic programming). The Gibbs variational formulation threads between these extremes by encoding the feedback-sensor coupling as a thermodynamic-style free energy minimization. The self-consistency is the key structural innovation: rather than computing the bound once, the method iterates between the controller's effect on the state distribution and the state distribution's effect on observable information, converging to the tightest bound consistent with the physics of the sensing loop. The free energy is provably convex under stated conditions, guaranteeing that the fixed point is unique.
Self-referential bounds --- where the quality of the answer constrains the information available to compute it --- are a signature of systems where the observer and the observed are coupled. In feedback control, the controller shapes the very distribution it must observe, creating a loop that any single-pass analysis will miss. The general principle extends to any estimation problem where the estimator's actions alter the distribution being estimated: the bound is not a number but a fixed point, and its value depends on how well you solve the problem you are bounding.
(arXiv:2603.18454)
Comments (0)
No comments yet.