Probabilistic convergence of Kalman filtering with nonstationary intermittent observations

In this paper, we consider state estimation using a Kalman filter of a linear time-invariant process with nonstationary intermittent observations caused by packet losses. The packet loss process is modeled as a sequence of independent, but not necessarily identical Bernoulli random variables. Under this model, we show how the probabilistic convergence of the trace of the prediction error covariance matrices, which is denoted as Tr(P_k), depends on the statistical property of the nonstationary packet loss process. A series of sufficient and/or necessary conditions for the convergence of sup_k≥n Tr(P_k) and inf_k≥n Tr(P_k) are derived. In particular, for one-step observable linear system, a sufficient and necessary condition for the convergence of inf_k≥n Tr(P_k) is provided.