Optimal state estimation in the presence of communication costs and packet drops

Consider a first order, linear and time-invariant discrete time system driven by Gaussian, zero mean white process noise, a pre-processor that accepts noisy measurements of the state of the system, and an estimator. The pre-processor and the estimator are not co-located, and, at every time-step, the pre-processor sends either a real number or an erasure symbol to the estimator. We seek the pre-processor and the estimator that jointly minimize a cost that combines three terms; the expected estimation error and a communication cost. The communication cost is zero for erasure symbols and a pre-selected constant otherwise. We show that the optimal pre-processor follows a symmetric threshold policy, and that the optimal estimator is a Kalman-like filter that updates its estimate linearly in the presence of erasures. Other existing work has adopted such a Kalman-like structure, but this paper is the first to prove its optimality.