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An Implicit Algorithm of Solving Nonlinear Filtering Problems

Published online by Cambridge University Press:  03 June 2015

Feng Bao*
Affiliation:
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA
Yanzhao Cao*
Affiliation:
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA School of Mathematics and Computational Sciences, Sun Yat-sen University, China
Xiaoying Han*
Affiliation:
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA
*Corresponding
Corresponding author.Email:yzc0009@auburn.edu
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Abstract

Nonlinear filter problems arise in many applications such as communications and signal processing. Commonly used numerical simulation methods include Kalman filter method, particle filter method, etc. In this paper a novel numerical algorithm is constructed based on samples of the current state obtained by solving the state equation implicitly. Numerical experiments demonstrate that our algorithm is more accurate than the Kalman filter and more stable than the particle filter.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Alspach, D. and Sorenson, H.Nonlinear bayesian estimation using gaussian sum approximations. IEEE Trans. Automatic Control, 17: 439448, 1972.CrossRefGoogle Scholar
[2]Bar-Shalom, Y. and Fortmann, T. E.Tracking and data association, in: Mathematics in Science and Engineering, volume 179. Academic Press Inc., San Diego, CA, 1988.Google Scholar
[3]Bolic, M., Djuric, P. M., and Hong, S.Resampling algorithms and architectures for distributed particle filters. IEEE Trans. Signal Process., 53(7): 24422450, 2005.CrossRefGoogle Scholar
[4]Chen, Z.Bayesian filtering: from kalman filters to particle filters, and beyond. Unpublished manuscript, 2011.Google Scholar
[5]Crisan, D. and Obanubi, O.Particle filters with random resampling times. Stochastic Process. Appl., 122(4): 13321368, 2012.CrossRefGoogle Scholar
[6]Crisan, D.Exact rates of convergence for a branching particle approximation to the solution of the Zakai equation. Ann. Probab., 31(2): 693718, 2003.Google Scholar
[7]Crisan, D. and Doucet, A.A survey of convergence results on particle filtering methods for practitioners. IEEE Trans. Sig. Proc., 50(3): 736746, 2002.CrossRefGoogle Scholar
[8]Dunik, J., Simandl, M., and Straka, O.Unscented Kalman filter: aspects and adaptive setting of scaling parameter. IEEE Trans. Automat. Control, 57(9): 24112416, 2012.CrossRefGoogle Scholar
[9]Einicke, G. A.Asymptotic optimality of the minimum-variance fixed-interval smoother. IEEE Trans. Signal Process., 55(4): 15431547, 2007.CrossRefGoogle Scholar
[10]Gobet, E., Pages, G., Pham, H., and Printems, J.Discretization and simulation of the Zakai equation. SIAM J. Numer. Anal., 44(6): 25052538, 2006.CrossRefGoogle Scholar
[11]Gordon, N. J., Salmond, D. J., and Smith, A. F. M.Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE PROCEEDING-F, 140(2): 107113, 1993.Google Scholar
[12]Hu, Y., Kallianpur, G., and Xiong, J.An approximation for the Zakai equation. Appl. Math. Optim., 45(1): 2344, 2002.CrossRefGoogle Scholar
[13]Jazwinski, A. H.Stochastic Processing and Filtering Theory, volume 64. Academic Press, New York, 1973.Google Scholar
[14]Julier, S. J. and LaViola, J. J. Jr.On Kalman filtering with nonlinear equality constraints. IEEE Trans. Signal Process., 55(6, part 2): 27742784, 2007.CrossRefGoogle Scholar
[15]Xiong, K., Liu, L., and Liu, Y.Robust extended Kalman filtering for nonlinear systems with multiplicative noises. Optimal Control Appl. Methods, 32(1): 4763, 2011.Google Scholar
[16]Kalman, R. E. and Bucy, R. S.New results in linear filtering and prediction theory. Trans. ASME Ser. D. J. Basic Engrg., 83: 95108, 1961.CrossRefGoogle Scholar
[17]Luo, X., Moroz, I. M., and Hoteit, I.Scaled unscented transform Gaussian sum filter: theory and application. Phys. D, 239(10): 684701, 2010.CrossRefGoogle Scholar
[18]Masreliez, C. J.Approximate non-gaussian filtering with linear state and observation relations. IEEE Trans. Automatic Control, 20: 107110, 1975.CrossRefGoogle Scholar
[19]Masreliez, C. J. and Martin, R. D.Robust Bayesian estimation for the linear model and ro-bustifying the Kalman filter. IEEE Trans. Automatic Control, AC-22(3): 361371, 1977.CrossRefGoogle Scholar
[20]Nagata, M. and Sawaragi, Y.Error analysis of the Schmidt-Kalman filter. Internat. J. Systems Sci., 7(7): 769778, 1976.CrossRefGoogle Scholar
[21]Orguner, U. and Gustafsson, F.Target tracking with particle filters under signal propagation delays. IEEE Trans. Signal Process., 59(6): 24852495, 2011.CrossRefGoogle Scholar
[22]Price, C. F.An analysis of the divergence problem in the Kalman filter. IEEE Trans. Automatic Control, AC-13: 699702, 1968.CrossRefGoogle Scholar
[23]Printems, J.On the discretization in time of parabolic stochastic partial differential equations. M2AN Math. Model. Numer. Anal., 35(6): 10551078, 2001.CrossRefGoogle Scholar
[24]Terejanu, G., Singla, P., Singh, T., and Scott, P. D.Adaptive Gaussian sum filter for nonlinear Bayesian estimation. IEEE Trans. Automat. Control, 56(9): 21512156, 2011.CrossRefGoogle Scholar
[25]Ulker, Y. and Gunsel, B.Multiple model target tracking with variable rate particle filters. Digit. Signal Process., 22(3): 417429, 2012.CrossRefGoogle Scholar
[26]Zakai, M.On the optimal filtering of diffusion processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 11: 230243, 1969.CrossRefGoogle Scholar
[27]Zhang, H. and Laneuville, D.Grid based solution of zakai equation with adaptive local refinement for bearing-only tracking. IEEE Aerospace Conference, 2008.CrossRefGoogle Scholar
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