Abstract
This work presents a filter for estimating the state of nonlinear dynamic systems. It is based on optimal recursive approximation the state densities by means of Dirac mixture functions in order to allow for a closed form solution of the prediction and filter step. The approximation approach is based on a systematic minimization of a distance measure and is hence optimal and deterministic. In contrast to non-deterministic methods we are able to determine the optimal number of components in the Dirac mixture. A further benefit of the proposed approach is the consideration of measurements during the approximation process in order to avoid parameter degradation.
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References
Huber, M., Brunn, D., Hanebeck, U.D.: Closed-Form Prediction of Nonlinear Dynamic Systems by Means of Gaussian Mixture Approximation of the Transition Density. In: International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2006), Heidelberg, Deutschland. (2006) 98–103
Doucet, A., Freitas, N.D., Gordon, N.: Sequential Monte Carlo Methods in Practice. Springer-Verlag, New York (2001)
Geweke, J.: Bayesian Inference in Econometric Models using Monte Carlo Integration. Econometrica 24 (1989) 1317–1399
Gordon, N.: Bayesian Methods for Tracking. PhD thesis, University of London (1993)
Julier, S., Uhlmann, J.: A New Extension of the Kalman Filter to Nonlinear Systems. In: Proceedings of SPIE AeroSense, 11th International Symposium on Aerospace/Defense Sensing, Simulation, and Controls, Orlando, FL. (1997)
Alspach, D.L., Sorenson, H.W.: Nonlinear Bayesian Estimation Using Gaussian Sum Approximation. IEEE Transactions on Automatic Control AC–17 (1972) 439–448
Hanebeck, U.D., Briechle, K., Rauh, A.: Progressive Bayes: A New Framework for Nonlinear State Estimation. In: Proceedings of SPIE. Volume 5099., Orlando, Florida (2003) 256–267 AeroSense Symposium.
Schrempf, O.C., Brunn, D., Hanebeck, U.D.: Dirac Mixture Density Approximation Based on Minimization of the Weighted Cramér–von Mises Distance. In: Proceedings of the International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2006), Heidelberg, Germany. (2006) 512–517
Schrempf, O.C., Hanebeck, U.D.: Recursive Prediction of Stochastic Nonlinear Systems Based on Dirac Mixture Approximations. In: Proceedings of the American Control Conference (ACC ’07), New York City, USA. (2007)
Schrempf, O.C., Hanebeck, U.D.: A State Estimator for Nonlinear Stochastic Systems Based on Dirac Mixture Approximations. In: 4th Intl. Conference on Informatics in Control, Automation and Robotics (ICINCO 2007). Volume SPSMC., Angers, France (2007) 54–61
Kullback, S., Leibler, R.A.: On Information and Sufficiency. Annals of Mathematical Statistics 22 (1951) 79–86
Boos, D.D.: Minimum Distance Estimators for Location and Goodness of Fit. Journal of the American Statistical association 76 (1981) 663–670
Bucy, R.S.: Bayes Theorem and Digital Realizations for Non-Linear Filters. Journal of Astronautical Sciences 17 (1969) 80–94
Schrempf, O.C., Brunn, D., Hanebeck, U.D.: Density Approximation Based on Dirac Mixtures with Regard to Nonlinear Estimation and Filtering. In: Proceedings of the 45th IEEE Conference on Decision and Control (CDC’06), San Diego, California, USA. (2006)
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Schrempf, O.C., Hanebeck, U.D. (2009). Dirac Mixture Approximation for Nonlinear Stochastic Filtering. In: Filipe, J., Cetto, J.A., Ferrier, JL. (eds) Informatics in Control, Automation and Robotics. Lecture Notes in Electrical Engineering, vol 24. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85640-5_22
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DOI: https://doi.org/10.1007/978-3-540-85640-5_22
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