Stability of Kalman filtering with Markovian packet losses

Abstract

Abstract-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 {Pt, 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.

Kalman filtering over unreliable communication networks with bounded Markovian packet dropouts

“…Bernoulli losses and showed that a critical value for the arrival rate exists such that the mean filtering error covariance is bounded for any arrival rate greater than that critical value. The stability analysis of Kalman filtering with binary Markovian packet losses can be found in [13,14] where the notion of peak covariance stability was first introduced in [15]. It is noticed in [13,15] that the given sufficient condition for the peak covariance stability involves an estimation of an upper bound of the solution of standard Riccati recursion.…”

“…We note that the stability studied in [13,14,15] is in fact in terms of mean filtering error covariance, and thus the term -peak covariance stability in the mean sense would be more precise. We still use peak covariance stability here for the consistency of the present paper with the aforementioned ones.…”

“…We still use peak covariance stability here for the consistency of the present paper with the aforementioned ones. In the sequel, the sojourn times of loss cycle(a period of successive packet dropouts) and normal cycle(a period of successive packet receipts) are defined similarly to the binary dropout case [12,13]. The main results of the present paper include: 1) a less conservative method for estimating the upper bound of the solution of the standard Riccati recursion and 2) a sufficient condition for the peak covariance stability which is related to the observability index and the dynamics of the underlying system as well as the probability transition matrix of the Markov chain that describes the packet-loss process.…”

Optimal Estimation and Control for Lossy Network: Stability, Convergence, and Performance

“…Here, what we consider is their homogeneous parts (32) and (33). Clearly, in the homogeneous parts there is no process noise ω k and the estimation error e s k is set to be zero as well, which is equivalent to setting their covariances to be zeros, that is, Q = S k = 0.…”

“…Its stability was studied in the pioneering work [25] where it is pointed out that there exists a critical value which determines the boundedness of the expected estimation error covariance (EEC), i.e., E[P k ]. Following [25], various aspects have been further researched, including the bound for the critical value [32], the distribution for EEC [27,28], the Markov packet losses case [26,[33][34][35]. In [29,30], the authors pointed out that P({P k ≤ M }) is a better evaluation for the estimation performance than the quantity E[P k ], and then obtained the lower and upper bounds for P({P k ≤ M }).…”