Kalman filtering with Markovian packet losses and stability criteria

We consider Kalman filtering in a network with packet losses, and use a two state Markov chain to describe the normal operating condition of packet delivery and transmission failure. We analyze the behavior of the estimation error covariance matrix and introduce the notion of peak covariance, which describes the upper envelope of the sequence of error covariance matrices {P_t, t > 1} for the case of an unstable scalar model. We give sufficient conditions for the stability of the peak covariance process in the general vector case; for the scalar case we obtain a sufficient and necessary condition, and derive upper and lower bounds for the tail distribution of the peak variance. For practically verifying the stability condition, we further introduce a suboptimal estimator and develop a numerical procedure to generate tighter estimate for the constants involved in the stability criterion.