Stability of Kalman filtering with Markovian packet losses

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. Based on the sojourn time of each visit to the failure or successful packet reception state, we analyze the behavior of the estimation error covariance matrix and introduce the notion of peak covariance, as an estimate of filtering deterioration caused by packet losses, 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, and obtain a sufficient and necessary condition for the scalar case. Finally, the relationship between two different types of stability notions is discussed.

[1]  Bruno Sinopoli,et al.  Kalman filtering with intermittent observations , 2004, IEEE Transactions on Automatic Control.

[2]  Amin G. Jaffer,et al.  Recursive Bayesian estimation with uncertain observation (Corresp.) , 1971, IEEE Trans. Inf. Theory.

[3]  Peter Seiler,et al.  Estimation with lossy measurements: jump estimators for jump systems , 2003, IEEE Trans. Autom. Control..

[4]  V.K. Goyal,et al.  Estimation from lossy sensor data: jump linear modeling and Kalman filtering , 2004, Third International Symposium on Information Processing in Sensor Networks, 2004. IPSN 2004.

[5]  A. Goldsmith,et al.  Kalman filtering with partial observation losses , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[6]  E. O. Elliott.  Estimates of error rates for codes on burst-noise channels , 1963 .

[7]  Qiang Ling,et al.  Soft real-time scheduling of networked control systems with dropouts governed by a Markov chain , 2003, Proceedings of the 2003 American Control Conference, 2003..

[8]  Mohamed T. Hadidi,et al.  Linear recursive state estimators under uncertain observations , 1978 .

[9]  Richard M. Murray,et al.  Optimal LQG control across packet-dropping links , 2007, Syst. Control. Lett..

[10]  E. Gilbert.  Capacity of a burst-noise channel , 1960 .

[11]  Chee-Yee Chong,et al.  Sensor networks: evolution, opportunities, and challenges , 2003, Proc. IEEE.

[12]  José M. F. Moura,et al.  Estimation in sensor networks: a graph approach , 2005, IPSN 2005. Fourth International Symposium on Information Processing in Sensor Networks, 2005..

[13]  Feng Zhao,et al.  Information-driven dynamic sensor collaboration , 2002, IEEE Signal Process. Mag..

[14]  Nasser E. Nahi,et al.  Optimal recursive estimation with uncertain observation , 1969, IEEE Trans. Inf. Theory.

[15]  Petros G. Voulgaris,et al.  On optimal ℓ∞ to ℓ∞ filtering , 1995, Autom..

[16]  Christoforos N. Hadjicostis,et al.  Feedback control utilizing packet dropping network links , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[17]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[18]  Jitendra K. Tugnait.  Asymptotic stability of the MMSE linear filter for systems with uncertain observations , 1981, IEEE Trans. Inf. Theory.

[19]  D. Vere-Jones.  Markov Chains , 1972, Nature.

[20]  Pravin Varaiya,et al.  Smart cars on smart roads: problems of control , 1991, IEEE Trans. Autom. Control..