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research-article

Kalman Filter and its Modern Extensions for the Continuous-time Nonlinear Filtering Problem

[+] Author and Article Information
Amirhossein Taghvaei

Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign
taghvae2@illinois.edu

Jana de Wiljes

Institut für Mathematik, Universität Potsdam
wiljes@uni-potsdam.de

Prashant Mehta

Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign
mehtapg@illinois.edu

Sebastian Reich

Institut für Mathematik, Universität Potsdam and Department of Mathematics and Statistics, University of Reading
sereich@uni-potsdam.de

1Corresponding author.

ASME doi:10.1115/1.4037780 History: Received February 14, 2017; Revised July 31, 2017

Abstract

This paper is concerned with the filtering problem in continuous-time. Three algorithmic solution approaches for this problem are reviewed: (i) the classical Kalman-Bucy filter which provides an exact solution for the linear Gaussian problem, (ii) the ensemble Kalman-Bucy filter (EnKBF) which is an approximate filter and represents an extension of the Kalman-Bucy filter to nonlinear problems, and (iii) the feedback particle filter (FPF) which represents an extension of the EnKBF and furthermore provides for a consistent solution in the general nonlinear, non-Gaussian case. The common feature of the three algorithms is the gain times error formula to implement the update step (to account for conditioning due to observations) in the filter. In contrast to the commonly used sequential Monte Carlo methods, the EnKBF and FPF avoid the resampling of the particles in the importance sampling update step. Moreover, the gain times innovation feedback structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also addresses the issue of non-uniqueness of the filter update formula and formulates a novel approximation algorithm based on ideas from optimal transport and coupling of measures. Performance of this and other algorithms is illustrated for a numerical example.

Copyright (c) 2017 by ASME
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