## Stability of Kalman filtering with Markovian packet losses.(English)Zbl 1261.93083

Summary: 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 $$\{P_{t},\,t\geq 1\}$$ 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.

### MSC:

 93E11 Filtering in stochastic control theory 60G35 Signal detection and filtering (aspects of stochastic processes) 62M20 Inference from stochastic processes and prediction

### Keywords:

networked systems; packet losses; Kalman filtering; stability
Full Text:

### References:

 [1] Anderson, B.D.O.; Moore, J.B., Optimal filtering, (1979), Prentice-Hall Englewood Cliffs, NJ · Zbl 0758.93070 [2] Chong, C.-Y.; Kumar, S.P., Sensor networks: evolution, opportunities, and challenges, Proceedings of IEEE, 91, 1247-1256, (2003) [3] Elliott, E.O., Estimation of error rates for codes on burst-noise channels, Bell system technical journal, 42, 1977-1997, (1963) [4] Fletcher, A. K., Rangan, S., & Goyal, V. K. (2004). Estimation from lossy sensor data: Jump linear modeling and Kalman filtering. In Proceedings of the third ACM/IEEE international symposium on information processing in sensor networks (pp. 251-258). Berkeley, CA. [5] Freedman, D., Markov chains, (1983), Springer New York · Zbl 0212.49801 [6] Gilbert, E.N., Capacity of burst-noise channels, Bell system technical journal, 39, 1253-1265, (1960) [7] Gupta, V., Spanos, D., Hassibi, B., & Murray, R. M. (2005). Optimal LQG control across packet-dropping links. In Proceedings of American control conference (pp. 360-365). Portland, OR. · Zbl 1137.90379 [8] Hadidi, M.T.; Schwartz, S.C., Linear recursive state estimation under uncertain observations, IEEE transactions on automatic control, 24, 944-948, (1979) · Zbl 0416.93087 [9] Hadjicostis, C. N., & Touri, R. (2002). Feedback control utilizing packet dropping network links. In Proceedings of IEEE conference on decision and control (pp. 1205-1210). Las Vegas, NV. [10] Jaffer, A.G.; Gupta, S.C., Recursive Bayesian estimation with uncertain observation, IEEE transactions on information theory, 17, 614-616, (1971) · Zbl 0222.62042 [11] Kailath, T.; Sayed, A.H.; Hassibi, B., Linear estimation, (2000), Prentice-Hall Upper Saddle River, NJ [12] Kalman, R.E., A new approach to linear filtering and prediction problems, Trans. ASME—journal of basic engineering series D, 82, 35-45, (1960) [13] Ling, Q., & Lemmon, M. (2003). Soft real-time scheduling of networked control systems with dropouts governed by a Markov chain (pp. 4845-4550). Denver, CO. [14] Liu, X., & Goldsmith, A. J. (2004). Kalman filtering with partial observation losses. In Proceedings of IEEE conference on decision control (pp. 1413-1418). Paradise Island, Bahamas. [15] Nahi, N.E., Optimal recursive estimation with uncertain observation, IEEE transactions on information theory, 15, 457-462, (1969) · Zbl 0174.51102 [16] Sinopoli, B.; Schenato, L.; Franceschetti, M.; Poolla, K.; Jordan, M.I.; Sastry, S.S., Kalman filtering with intermittent observations, IEEE transactions on automatic control, 49, 1453-1464, (2004) · Zbl 1365.93512 [17] Smith, S.C.; Seiler, P., Estimation with lossy measurements: jump estimators for jump systems, IEEE transactions on automatic control, 48, 2163-2171, (2003) · Zbl 1364.93785 [18] Tugnait, J.K., Asymptotic stability of the MMSE linear filter for systems with uncertain observations, IEEE transactions on information theory, 27, 247-250, (1981) · Zbl 0469.93076 [19] Varaiya, P., Smart cars on smart roads: problems of control, IEEE transactions on automatic control, 38, 195-207, (1993) [20] Zhang, H., Moura, J. M. F., & Krogh, B. (2005). Estimation in sensor networks: A graph approach. In Proceedings of the fourth international symposium on information processing in sensor networks (IPSN) (pp. 203-209). Los Angeles. [21] Zhao, F.; Shin, J.; Reich, J., Information-driven dynamic sensor collaboration, IEEE signal processing magazine, 19, 61-72, (2002)
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