Gaussian kernel quadrature Kalman filter

Abstract

The solution to practical nonlinear filtering problems broadly relies on Gaussian filtering. The Gaussian filtering involves intractable integrals that are numerically approximated during the filtering. The literature witnesses various Gaussian filters with varying accuracy and computational demand, which are developed using different numerical approximation methods. Among them, the quadrature rule based Gaussian filters are known for offering the best accuracy. They apply the univariate Gauss-Hermite quadrature rule for approximating the intractable integrals. For the practical multivariate filtering problems, they additionally apply a univariate-to-multivariate conversion rule. This paper develops a new quadrature rule based Gaussian filter, named Gaussian kernel quadrature Kalman filter (GKQKF). The proposed GKQKF replaces the univariate Gauss-Hermite quadrature rule with the univariate Gaussian kernel quadrature rule and uses the product rule for extending the univariate quadrature rule in the multivariate domain. The Gaussian kernel quadrature rule improves the numerical approximation accuracy, which results in improved estimation accuracy of the proposed GKQKF over the existing quadrature rule based Gaussian filters. As the quadrature rule based Gaussian filters are the most accurate existing Gaussian filters, the proposed GKQKF outperforms the other existing Gaussian filters as well. The improved accuracy of the proposed GKQKF is validated for three different simulation problems.