On the optimality of threshold policies in event triggered estimation with packet drops

We consider a remote state estimation problem, where a sensor transmits local state estimates over an independent and identically distributed (i.i.d.) packet dropping link to a remote estimator. At each discrete time instant, the sensor can decide whether to transmit, with each transmission incurring a fixed energy cost. Performance is quantified via an optimization problem that minimizes a convex combination of the expected error covariance at the remote estimator and expected energy usage. For transmission schedules dependent only on the error covariance at the remote estimator, this work establishes that a threshold policy (i.e. transmit if the error covariance exceeds a certain threshold and don't transmit otherwise) is optimal. This provides a rigorous justification for the use of such threshold policies in event triggered estimation. An extension of the result to Markovian packet drops is also outlined.