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Nonlinear Kalman Filters Explained: A Tutorial on Moment Computations and Sigma Point Methods
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-1971-4295
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
2016 (English)In: Journal of Advances in Information Fusion, ISSN 1557-6418, Vol. 11, no 1, 47-70 p.Article in journal (Refereed) Published
Abstract [en]

Nonlinear Kalman filters are algorithms that approximately solve the Bayesian filtering problem by employing the measurement update of the linear Kalman filter (KF). Numerous variants have been developed over the past decades, perhaps most importantly the popular sampling based sigma point Kalman filters.In order to make the vast literature accessible, we present nonlinear KF variants in a common framework that highlights the computation of mean values and covariance matrices as the main challenge. The way in which these moment integrals are approximated distinguishes, for example, the unscented KF from the divided difference KF.With the KF framework in mind, a moment computation problem is defined and analyzed. It is shown how structural properties can be exploited to simplify its solution. Established moment computation methods, and their basics and extensions, are discussed in an extensive survey. The focus is on the sampling based rules that are used in sigma point KF. More specifically, we present three categories of methods that use sigma-points 1) to represent a distribution (as in the UKF); 2) for numerical integration (as in Gauss-Hermite quadrature); 3) to approximate nonlinear functions (as in interpolation). Prospective benefits and downsides are listed for each of the categories and methods, including accuracy statements. Furthermore, the related KF publications are listed.The theoretical discussion is complemented with a comparative simulation study on instructive examples.

Place, publisher, year, edition, pages
International society of information fusion , 2016. Vol. 11, no 1, 47-70 p.
National Category
Signal Processing
Identifiers
URN: urn:nbn:se:liu:diva-129231OAI: oai:DiVA.org:liu-129231DiVA: diva2:936621
Available from: 2016-06-14 Created: 2016-06-14 Last updated: 2017-02-27Bibliographically approved
In thesis
1. Advanced Kalman Filtering Approaches to Bayesian State Estimation
Open this publication in new window or tab >>Advanced Kalman Filtering Approaches to Bayesian State Estimation
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Bayesian state estimation is a flexible framework to address relevant problems at the heart of existing and upcoming technologies. Application examples are obstacle tracking for driverless cars and indoor navigation using smartphone sensor data. Unfortunately, the mathematical solutions of the underlying theory cannot be translated to computer code in general. Therefore, this thesis discusses algorithms and approximations that are related to the Kalman filter (KF).

Four scientific articles and an introduction with the relevant background on Bayesian state estimation theory and algorithms are included. Two articles discuss nonlinear Kalman filters, which employ the KF measurement update in nonlinear models. The numerous variants are presented in a common framework and the employed moment approximations are analyzed. Furthermore, their application to target tracking problems is discussed. A third article analyzes the ensemble Kalman filter (EnKF), a Monte Carlo implementation of the KF that has been developed for high-dimensional geoscientific filtering problems. The EnKF is presented in a simple KF framework, including its challenges, important extensions, and relations to other filters. Whereas the aforementioned articles contribute to the understanding of existing algorithms, a fourth article devises novel filters and smoothers to address heavy-tailed noise. The development is based on Student’s t distribution and provides simple recursions in the spirit of the KF. The introduction and articles are accompanied by extensive simulation experiments.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. 81 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1832
National Category
Signal Processing Control Engineering Computational Mathematics Computer Science Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-134867 (URN)10.3384/diss.diva-134867 (DOI)9789176855782 (ISBN)
Public defence
2017-04-21, Visionen, B-huset, Campus Valla, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2017-03-22 Created: 2017-02-27 Last updated: 2017-03-22Bibliographically approved

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