Cenker BİÇER, Esin KÖKSAL BABACAN, Levent ÖZBEK

Stability of the adaptive fading extended Kalman filter with the matrix forgetting factor

The extended Kalman filter is extensively used in nonlinear state estimation problems. As long as the system characteristics are correctly known, the extended Kalman filter gives the best performance. However, when the system information is partially known or incorrect, the extended Kalman filter may diverge or give biased estimates. An extensive number of works has been published to improve the performance of the extended Kalman filter. Many researchers have proposed the introduction of a forgetting factor, both into the Kalman filter and the extended Kalman filter, to improve the performance. However, there are 2 fundamental problems with this approach: the incorporation of the optimal forgetting factor into the (extended) Kalman filter and the selection of the optimal forgetting factor. These problems have not yet been fully resolved and are still open problems in the field. In this study, we propose a new adaptive fading extended Kalman filter with a matrix forgetting factor, and 2 methods are analyzed for the selection of the optimal forgetting factor. The stability properties of the proposed filter are also investigated. Results of the stability analysis show that the proposed filter is an exponential observer for nonlinear deterministic systems. Additionally, the convergence speed of the filter is simulated.

Anahtar Kelimeler:

Adaptive fading Kalman filter, extended Kalman filter, adaptive fading extended Kalman filter, forgetting factor, stability analysis

Stability of the adaptive fading extended Kalman filter with the matrix forgetting factor

Keywords:

Adaptive fading Kalman filter, extended Kalman filter, adaptive fading extended Kalman filter, forgetting factor, stability analysis,

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Y. Bar-Shalom, X.R. Li, T. Kirubarajan, Estimation with Applications to Tracking and Navigation, New York, Wiley Interscience, 2001.
B.D.O Anderson, J.B. Moore, Optimal Filtering, New Jersey, Prentice Hall, 1979.
A. Gelb, Applied Optimal Estimation, Cambridge, MIT Press, 1984.
E.K. Babacan, L. Ozbek, M. Efe, “Stability of the extended Kalman Şlter when the states are constrained”, IEEE Transactions on Automatic Control, Vol. 53, pp. 2707-2711, 2008.
K.H. Kim, J.G. Lee, C.G. Park, G.I. Jee, “The stability analysis of the adaptive fading extended Kalman Şlter”, th IEEE International Conference on Control, pp. 982-987, 2007.
Q. Xia, M. Rao, Y. Ying, X. Shen, “Adaptive fading Kalman Şlter with an application”, Automatica, Vol. 30, pp. 1338, 1994.
L. ¨Ozbek, F. ¨Ozt¨urk, F. Aliev, “Kalman Şltresinde kayıpları ¨onlemek i¸cin bir y¨ontem”, Otomatik Kontrol Ulusal Toplantısı, ˙Istanbul, Yayın No: 588, pp. 31-38, 1996.
L. ¨Ozbek, F.A. Aliev, “Comments on ‘Adaptive fading Kalman Şlter with an applications”’, Automatica, Vol. 34, pp. 1163-1164, 1998.
C. Hide, T. Moore, M. Smith, “Adaptive Kalman Şltering for low-cost INS/GPS”, Journal of Navigation, Vol. 56, pp. 143-152, 2003.
Y. Geng, J. Wang, “Adaptive estimation of multiple fading factors in Kalman Şlter for navigation applications”, GPS Solutions, Vol. 12, pp. 273-279, 2008.
L. Ozbek, M. Efe, “An adaptive extended Kalman Şlter with application to compartment models”, Communication in Statistics - Simulation and Computation, Vol. 3, pp. 145-158, 2003.
K.H. Kim, G.I. Jee, C.G. Park, J.G. Lee, “The stability analysis of the adaptive fading extended Kalman Şlter using the innovation covariance”, International Journal of Control, Automation, and Systems, Vol. 7, pp. 49-56, J.N. Yang, S. Lin, H. Huang, L. Zhou, “An adaptive extended Kalman Şlter for structural damage identiŞcation”, Structural Control and Health Monitoring, Vol. 13, pp. 849-867, 2006.
M. Boutayeb, H. Rafaralahy, M. Darouach, “Convergence analysis of the extended Kalman Şlter used as an observer for nonlinear deterministic discrete-time systems”, IEEE Transactions on Automatic Control, Vol. 42, pp. 581-586, K. Reif, R. Unbehauen, “The extended Kalman Şlter as an exponential observer for nonlinear systems”, IEEE Transactions on Signal Processing, Vol. 47, pp. 2324-2328, 1999.
L. ¨Ozbek, E.K. Babacan, M. Efe, “Stochastic stability of the discrete-time constrained extended Kalman Şlter,” Turkish Journal of Electrical Engineering & Computer Sciences, Vol. 18, pp. 211-223, 2010.
V. Lakshmikantham, D. Trigiante, Theory of Diﬀerence Equations: Numerical Methods and Applications, London, Academic Press, 1988.
F.L. Lewis, Optimal Estimation, New York, Wiley, 1986.
N.M. McClamroch, State Models of Dynamic Systems, New York, Springer-Verlag, 1980.
F. ¨Ozt¨urk, L. ¨Ozbek, Matematiksel Modelleme ve Sim¨ulasyon, Ankara, Gazi Kitabevi, 2004.