A Gaussian-inverse Gamma mixture Distributions and Expectation-Maximization Based Robust Kalman Filter

In the study of the state estimation for the systems with unknown time-varying non-Gaussian noises, the existing robust Kalman filters (RKFs) perform well. However, the calculation loads of these RKFs usually are large and their performance is easily affected by the roughly preselected initial process noise covariance matrix (PNCM). To solve the problems, a new RKF is proposed. Firstly, a Gaussian-inverse Gamma mixture distribution is developed to model the inaccurate noises and a simple hierarchical Gaussian (HG) model is constructed. Then, the expectation-maximization (EM) method is applied to realize the adaptive adjustment of the prior scale matrix of the prediction error covariance. Based on the HG model and EM, a robust KF is derived, where the variational Bayesian (VB) approach is used to jointly estimate model parameters and an alternate iteration method is employed to reduce the computation time. Finally, our filter performance is tested. Compared with the existing state-of-the-art robust filters, the proposed filter has slightly better estimation accuracy and significantly less computation load. Meanwhile, the filter performance is almost not affected by the selection accuracy of initial PNCM.

I. Introduction

The Kalman filter (KF) has been widely used in many applications since it was proposed, such as navigation, signal processing and target tracking. As an online and recursive state estimator, KF is optimal when the state-space model is linear Gaussian [1], [2]. However, in the case of inaccurate noise statistics and/or non-Gaussian noises, the estimation accuracy of KF will be seriously affected. The non-Gaussian noises cannot be modeled accurately by the uniform probability density function (PDF), and the posterior PDF usually has no analytical solution, it is very difficult to design a non-Gaussian filter [3], [4].