Estimation of the soil-dependent time-varying parameters of the hopper sedimentation model: The FPF versus the BPF

Wiljes

Journal of Dynamic Systems, Measurement, and Control

et al. 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 an 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 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.

Section: Constant Gain Approximationmentioning

confidence: 99%

Kalman Filter and Its Modern Extensions for the Continuous-Time Nonlinear Filtering Problem

Wiljes

Journal of Dynamic Systems, Measurement, and Control

et al. 2017

“…This approximation reduces to the Kalman gain in the linear Gaussian case. For the general case, this approximation often suffices in practice particularly so when the conditional distribution is unimodal [56], [7], [47].…”

Section: Feedback Particle Filter With Concentrated Distributionsmentioning

confidence: 99%

“…For the ease of taking the expectation, we convert (47) to its Itô form (see Theorem 1.2 in [52]): For real-valued continuous semi-martingales A, B,C,…”

Section: Appendix a Proof Of Propositionmentioning

confidence: 99%

Feedback particle filter on matrix lie groups

Zhang

2016 American Control Conference (ACC)

2016

This paper is concerned with the problem of continuous-time nonlinear filtering for stochastic processes on a connected matrix Lie group. The main contribution of this paper is to derive the feedback particle filter (FPF) algorithm for this problem. In its general form, the FPF is shown to provide a coordinate-free description of the filter that automatically satisfies the geometric constraints of the manifold. The particle dynamics are encapsulated in a Stratonovich stochastic differential equation that retains the feedback structure of the original (Euclidean) FPF. The implementation of the filter requires a solution of a Poisson equation on the Lie group, and two numerical algorithms are described for this purpose. As an example, the FPF is applied to the problem of attitude estimation -a nonlinear filtering problem on the Lie group SO(3). The formulae of the filter are described using both the rotation matrix and the quaternion coordinates. Comparisons are also provided between the FPF and some popular algorithms for attitude estimation, namely the multiplicative EKF, the unscented quaternion estimator, the left invariant EKF, and the invariant ensemble Kalman filter. Numerical simulations are presented to illustrate the comparisons.

“…In numerical evaluations and comparisons, it is often found that the control-based algorithms exhibit smaller simulation variance and better scaling properties with the problem dimension (number of state variables). For example, several research groups have reported favorable comparisons for the FPF algorithm as compared to the traditional particle filter algorithms; cf., [24], [23], [2], [25], [29]. However, there is no theoretical justification/understanding of this.…”

Section: Introductionmentioning

confidence: 99%

Error Analysis for the Linear Feedback Particle Filter

2018 Annual American Control Conference (ACC)

2018

This paper is concerned with the convergence and the error analysis for the feedback particle filter (FPF) algorithm. The FPF is a controlled interacting particle system where the control law is designed to solve the nonlinear filtering problem. For the linear Gaussian case, certain simplifications arise whereby the linear FPF reduces to one form of the ensemble Kalman filter. For this and for the more general nonlinear non-Gaussian case, it has been an open problem to relate the convergence and error properties of the finite-N algorithm to the mean-field limit (where the exactness results have been obtained). In this paper, the equations for empirical mean and covariance are derived for the finite-N linear FPF. Remarkably, for a certain deterministic form of FPF, the equations for mean and variance are identical to the Kalman filter. This allows strong conclusions on convergence and error properties based on the classical filter stability theory for the Kalman filter. It is shown that the error converges to zero even with finite number of particles. The paper also presents propagation of chaos estimates for the finite-N linear filter. The error estimates are illustrated with numerical experiments.