Discrete-Time Nonlinear Filtering Algorithms Using Gauss–Hermite Quadrature

TLDR

The paper develops a new quadrature Kalman filter (QKF) for nonlinear systems and discusses its extensions to specific applications. The authors derive a QKF by linearizing process and measurement functions with Gauss–Hermite quadrature, extend it to discrete‑time nonlinear systems with non‑Gaussian noise, and introduce a Gaussian‑sum QKF that approximates densities with weighted Gaussian mixtures, using residuals for weights and three mixture‑reduction techniques to control complexity. Simulations demonstrate that the GS‑QKF outperforms bootstrap particle filters and Gaussian‑sum EKFs on nonlinear non‑Gaussian filtering problems.

Abstract

In this paper, a new version of the quadrature Kalman filter (QKF) is developed theoretically and tested experimentally. We first derive the new QKF for nonlinear systems with additive Gaussian noise by linearizing the process and measurement functions using statistical linear regression (SLR) through a set of Gauss-Hermite quadrature points that parameterize the Gaussian density. Moreover, we discuss how the new QKF can be extended and modified to take into account specific details of a given application. We then go on to extend the use of the new QKF to discrete-time, nonlinear systems with additive, possibly non-Gaussian noise. A bank of parallel QKFs, called the Gaussian sum-quadrature Kalman filter (GS-QKF) approximates the predicted and posterior densities as a finite number of weighted sums of Gaussian densities. The weights are obtained from the residuals of the QKFs. Three different Gaussian mixture reduction techniques are presented to alleviate the growing number of the Gaussian sum terms inherent to the GS-QKFs. Simulation results exhibit a significant improvement of the GS-QKFs over other nonlinear filtering approaches, namely, the basic bootstrap (particle) filters and Gaussian-sum extended Kalman filters, to solve nonlinear non- Gaussian filtering problems.