Certifying the Optimality of a Distributed State Estimation System via Majorization Theory

Consider a first order linear time-invariant discrete time system driven by process noise, a preprocessor that accepts causal measurements of the state of the system, and a state estimator. The preprocessor and the state estimator are not co-located, and, at every time-step, the pre-processor transmits 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. In our formulation, the transmission of a real number from the pre-processor to the estimator incurs a positive cost while erasures induce zero cost. This paper is the first to prove analytically that a symmetric threshold policy at the pre-processor and a Kalman-like filter at the estimator, which updates its estimate linearly in the presence of erasures, are jointly optimal for our problem.