Remote State Estimation With Communication Costs for First-Order LTI Systems

Consider a first order linear time-invariant discrete time system driven by 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 time-step, the pre-processor transmits either a real number or a free 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 free symbols induce zero cost. This paper proves 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 free symbols, are jointly optimal for our problem.