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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,
Urbana, IL 61801
e-mail: taghvae2@illinois.edu

Jana de Wiljes

Institut für Mathematik,
Universität Potsdam,
Potsdam D-14476, Germany
e-mail: wiljes@uni-potsdam.de

Prashant G. Mehta

Department of Mechanical Science
and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: mehtapg@illinois.edu

Sebastian Reich

Institut für Mathematik,
Universität Potsdam,
Potsdam D-14476, Germany;
Department of Mathematics and Statistics,
University of Reading,
Reading RG6 6AX, UK
e-mail: sereich@uni-potsdam.de

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received February 14, 2017; final manuscript received July 31, 2017; published online November 8, 2017. Assoc. Editor: Puneet Singla.

J. Dyn. Sys., Meas., Control 140(3), 030904 (Nov 08, 2017) (11 pages) Paper No: DS-17-1087; doi: 10.1115/1.4037780 History: Received February 14, 2017; Revised July 31, 2017

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 the 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 feedback control structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also discusses the issue of nonuniqueness 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.

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Figures

Grahic Jump Location
Fig. 1

Innovation error-based feedback structure for the (a) Kalman filter and (b) nonlinear FPF

Grahic Jump Location
Fig. 2

The exact solution to the Poisson equation using the formula (13). The density ρ is the sum of two Gaussians N(−1,σ2) and N(+1,σ2), and h(x) = x. The density is depicted as the shaded curve in the background.

Grahic Jump Location
Fig. 3

Constant gain approximation in the FPF

Grahic Jump Location
Fig. 4

Comparison of the gain function approximations obtained using Galerkin (a), kernel (c), and the optimal coupling (e) algorithms. The exact gain function is depicted as a solid line and the density ρ is depicted as a shaded region in the background. (b), (d), and (e) Depict the Monte Carlo estimate of the empirical error (26) as a function of the number of particles N. The Monte Carlo estimate is obtained by averaging the empirical error over 100 simulations. (a) Galerkin gain approximation for N = 100, (b) Galerkin gain approx. error as a function of N, (c) Kernel-based gain approximation for N = 100, (d) Kernel-based gain approx. error as a function of N, (e) optimal coupling gain approximation for N = 100, and (f) optimal coupling gain approx. error as a function of N.

Grahic Jump Location
Fig. 5

Comparison of the Monte Carlo estimate of the empirical error (26) as a function of the parameter ε. The number of particles N = 200.

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