×

zbMATH — the first resource for mathematics

Extended Kalman filtering with stochastic nonlinearities and multiple missing measurements. (English) Zbl 1257.93099
Summary: In this paper, the extended Kalman filtering problem is investigated for a class of nonlinear systems with multiple missing measurements over a finite horizon. Both deterministic and stochastic nonlinearities are included in the system model, where the stochastic nonlinearities are described by statistical means that could reflect the multiplicative stochastic disturbances. The phenomenon of measurement missing occurs in a random way and the missing probability for each sensor is governed by an individual random variable satisfying a certain probability distribution over the interval [0,1]. Such a probability distribution is allowed to be any commonly used distribution over the interval [0,1] with known conditional probability. The aim of the addressed filtering problem is to design a filter such that, in the presence of both the stochastic nonlinearities and multiple missing measurements, there exists an upper bound for the filtering error covariance. Subsequently, such an upper bound is minimized by properly designing the filter gain at each sampling instant. It is shown that the desired filter can be obtained in terms of the solutions to two Riccati-like difference equations that are of a form suitable for recursive computation in online applications. An illustrative example is given to demonstrate the effectiveness of the proposed filter design scheme.

MSC:
93E11 Filtering in stochastic control theory
93E10 Estimation and detection in stochastic control theory
93C10 Nonlinear systems in control theory
93C55 Discrete-time control/observation systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Basin, M.; Shi, P.; Calderon-Alvarez, D., Central suboptimal \(H_\infty\) filter design for nonlinear polynomial systems, International journal of adaptive control and signal processing, 23, 10, 926-939, (2009) · Zbl 1298.93148
[2] Basin, M.; Shi, P.; Calderon-Alvarez, D., Approximate finite-dimensional filtering for polynomial states over polynomial observations, International journal of control, 83, 4, 724-730, (2010) · Zbl 1209.93149
[3] Bishop, A.N.; Savkin, A.V.; Pathirana, P.N., Vision-based target tracking and surveillance with robust set-valued state estimation, IEEE signal processing letters, 17, 3, 289-292, (2010)
[4] Calafiore, G., Reliable localization using set-valued nonlinear filters, IEEE transactions on systems, man, and cybernetics-part A: systems and humans, 35, 2, 189-197, (2005)
[5] Cheng, T.M.; Malyavej, V.; Savkin, A.V., Decentralized robust set-valued state estimation in networked multiple sensor systems, Computers & mathematics with applications, 59, 8, 2636-2646, (2010) · Zbl 1193.93160
[6] Chen, W.; Zheng, W., Exponential stability of nonlinear time-delay systems with delayed impulse effects, Automatica, 47, 5, 1075-1083, (2011) · Zbl 1233.93080
[7] Horn, R.A.; Johnson, C.R., Topic in matrix analysis, (1991), Cambridge University Press New York
[8] Hounkpevi, F.O.; Yaz, E., Robust minimum variance linear state estimators for multiple sensors with different failure rates, Automatica, 43, 7, 1274-1280, (2007) · Zbl 1123.93085
[9] Hounkpevi, F.O.; Yaz, E., Minimum variance generalized state estimators for multiple sensors with different delay rates, Signal processing, 87, 4, 602-613, (2007) · Zbl 1186.94148
[10] James, M.R.; Petersen, I.R., Nonlinear state estimation for uncertain systems with an integral constraint, IEEE transactions on signal processing, 46, 11, 2926-2937, (1998)
[11] Kallapur, A.G.; Petersen, I.R.; Anavatti, S.G., A discrete-time robust extended Kalman filter for uncertain systems with sum quadratic constraints, IEEE transactions on automatic control, 54, 4, 850-854, (2009) · Zbl 1367.93648
[12] Kluge, S.; Reif, K.; Brokate, M., Stochastic stability of the extended Kalman filter with intermittent observations, IEEE transactions on automatic control, 55, 2, 514-518, (2010) · Zbl 1368.93717
[13] Li, P.; Lam, J., Disturbance analysis of nonlinear differential equation models of genetic SUM regulatory networks, IEEE-ACM transactions on computational biology and bioinformatics, 8, 1, 253-259, (2011)
[14] Li, P.; Lam, J.; Shu, Z., \(H_\infty\) positive filtering for positive linear discrete-time systems: an augmentation approach, IEEE transactions on automatic control, 55, 10, 2337-2342, (2010) · Zbl 1368.93719
[15] Mao, X., Stochastic differential equations and applications, (2007), Horwood Publishing Chichester
[16] NaNacara, W.; Yaz, E., Recursive estimators for linear and nonlinear systems with uncertain observations, Signal processing, 62, 2, 215-228, (1997) · Zbl 0908.93061
[17] Pathirana, P.N.; Ekanayake, S.W.; Savkin, A.V., Fusion based 3D tracking of mobile transmitters via robust set-valued state estimation with RSS measurements, IEEE communications letters, 15, 5, 554-556, (2011)
[18] Reif, K.; G√ľnther, S.; Yaz, E.; Unbehauen, R., Stochastic stability of the discrete-time extended Kalman filter, IEEE transactions on automatic control, 44, 4, 714-728, (1999) · Zbl 0967.93090
[19] Rotstein, H., Sznaier, M., & Idan, M. (1994). \(H_2 / H_\infty\) filtering-theory and an aerospace application. In Proceedings of the 1994 American control conference. 2, (pp. 1791-1795).
[20] Sahebsara, M.; Chen, T.; Shah, S.L., Optimal \(H_2\) filtering with random sensor delay, multiple packet dropout and uncertain observations, International journal of control, 80, 2, 292-301, (2007) · Zbl 1140.93486
[21] Shi, P.; Mahmoud, M.; Nguang, S.K.; Ismail, A., Robust filtering for jumping systems with mode-dependent delays, Signal processing, 86, 1, 140-152, (2006) · Zbl 1163.94387
[22] Sun, S.; Xie, L.; Xiao, W.; Soh, Y.C., Optimal linear estimation for systems with multiple packet dropouts, Automatica, 44, 5, 1333-1342, (2008) · Zbl 1283.93271
[23] Theodor, Y.; Shaked, U., Robust discrete-time minimum-variance filtering, IEEE transactions on signal processing, 44, 2, 181-189, (1996)
[24] Wang, Z.; Ho, D.W.C.; Liu, X., Variance-constrained filtering for uncertain stochastic systems with missing measurements, IEEE transactions on automatic control, 48, 7, 1254-1258, (2003) · Zbl 1364.93814
[25] Wang, Z.; Liu, X.; Liu, Y.; Liang, J.; Vinciotti, V., An extended Kalman filtering approach to modelling nonlinear dynamic gene regulatory networks via short gene expression time series, IEEE/ACM transactions on computational biology and bioinformatics, 6, 3, 410-419, (2009)
[26] Wei, G.; Wang, Z.; Shu, H., Robust filtering with stochastic nonlinearities and multiple missing measurements, Automatica, 45, 3, 836-841, (2009) · Zbl 1168.93407
[27] Wu, L.; Zheng, W., Weighted \(H_\infty\) model reduction for linear switched systems with time-varying delay, Automatica, 45, 1, 186-193, (2009) · Zbl 1154.93326
[28] Xie, L.; Lu, L.; Zhang, D.; Zhang, H., Improved robust \(H_2\) and \(H_\infty\) filtering for uncertain discrete-time systems, Automatica, 40, 5, 873-880, (2004) · Zbl 1050.93072
[29] Xie, L.; Soh, Y.C.; de Souza, C.E., Robust Kalman filtering for uncertain discrete-time systems, IEEE transactions on automatic control, 39, 6, 1310-1314, (1994) · Zbl 0812.93069
[30] Xiong, J.; Lam, J., Fixed-order robust \(H_\infty\) filter design for Markovian jump systems with uncertain switching probabilities, IEEE transactions on signal processing, 54, 4, 1421-1430, (2006) · Zbl 1373.94736
[31] Xiong, K.; Liu, L.; Liu, Y., Robust extended Kalman filtering for nonlinear systems with multiplicative noises, Optimal control applications and methods, 32, 1, 47-63, (2011) · Zbl 1213.93192
[32] Xiong, K.; Wei, C.; Liu, L., Robust extended Kalman filtering for nonlinear systems with stochastic uncertainties, IEEE transactions on systems, man, and cybernetics-part A: systems and humans, 40, 2, 399-405, (2010)
[33] Yaz, E., On the optimal state estimation of a class of discrete-time nonlinear systems, IEEE transactions on circuits and systems, 34, 9, 1127-1129, (1987) · Zbl 0633.93064
[34] Yaz, E.; Yaz, Y., State estimation of uncertain nonlinear stochastic systems with general criteria, Applied mathematics letters, 14, 5, 605-610, (2001) · Zbl 0976.93078
[35] Yaz, Y.; Yaz, E., A new formulation of some discrete-time stochastic parameter state estimation problems, Applied mathematics letters, 10, 6, 13-19, (1997) · Zbl 0887.93063
[36] Yue, D.; Han, Q., Network-based robust \(H_\infty\) filtering for uncertain linear systems, IEEE transactions on signal processing, 54, 11, 4293-4301, (2006) · Zbl 1373.93111
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.