How Much Sensing Information Is Needed to Control an Unstable Linear System?

Abstract

Modern control systems increasingly rely on sensing to infer the system state before control actions can be taken. Yet a given observation mechanism may fail to preserve sufficient information about the unstable modes, regardless of the downstream estimator or controller. This paper asks how much sensing information is needed to estimate and control an unstable linear system, whose measurements are generated by a prescribed, possibly nonlinear and non-Gaussian, observation law p(y_t|x_t). To address this question, we first quantify sensing information using directed information, thereby accounting for causal feedback. We then establish necessary and sufficient information rate conditions for estimating and controlling this linear system. For necessity, keeping either the estimation error or the closed-loop state bounded in mean square requires a directed information rate of at least the open-loop expansion rate R_exp. This lower bound remains valid under additive process noise. Since this rate is difficult to evaluate, we derive computable bounds for nonlinear observations with additive noise. An upper bound below R_exp certifies infeasibility, whereas a lower bound above R_exp + R_NG certifies sufficiency under posterior covariance regularity. For linear Gaussian observations, the tight upper bound is determined by the steady-state Riccati equation. For sufficiency, the posterior non-Gaussianity rate R_NG measures the divergence rate from the covariance-matched Gaussian. Under uniform posterior covariance regularity, a rate above R_exp + R_NG guarantees mean-square convergence of the estimation error. For a stabilizable plant, certainty-equivalence feedback also guarantees mean-square convergence of the closed-loop state. Finally, verifiable curvature conditions on the likelihood and prior make R_NG vanish, so the sufficient threshold equals R_exp.