Stability of Optimal Estimator with Unobservable Measurement Loss: The Unstable System Case

We study the stability of the optimal estimator (OE) for an unstable system with unobservable measurement loss (UML). The results are twofold. (i) We obtain a necessary and sufficient stability condition: For an unstable UML system, there is a critical value such that the OE is stable with probability 1, if and only if the measurement-arrival rate is greater than this value. This value is identical to the critical value that determines the stability of the OE with observable measurement loss (OML). (ii) As a byproduct, we obtain a property on estimation performance: When the system matrix is non-singular, the performance of the OE with UML converges with probability 1 to that of the OE with OML. These two results suggest that the absence of measurement-loss observability does not affect the stability and performance of the OE.