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This paper derives a novel information-theoretic lower bound on the expected cost of any causal feedback controller operating under partial observations, using the Gibbs variational principle applied to the joint state-observation path measure. Unlike prior methods that compare against the uncontrolled system, this bound accounts for the information-limiting effect of the controller itself, leading to tighter estimates. The bound is computed by solving a fixed-point equation, which under certain conditions is provably convex, and is demonstrated to be significantly tighter than open-loop variants on a Dubins car tracking problem.
Information-theoretic limits on control performance are now computable even when feedback matters most, thanks to a new bound that self-consistently accounts for the controller's impact on sensor information.
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.
