Optimality of distributed state estimators with communication cost: A case study

Consider a first order, linear and time-invariant discrete time system driven by Gaussian and zero mean white process noise, a pre-processor that accepts causal measurements of the state of the system, and a state estimator. The pre-processor and the state estimator are not co-located, and, at every timestep, 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 two terms; the expected squared state 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 threshold policy, and that the optimal estimator is a Kalman-like filter that updates its estimate linearly in the presence of erasures. Existing work has adopted such a Kalman-like structure, but this paper is the first to prove its optimality.