Non-Gaussian Estimation and Observer-Based Feedback using the Gaussian Mixture Kalman and Extended Kalman Filters

This paper considers the problem of non-Gaussian estimation and observer-based feedback in linear and non-linear settings. Estimation in nonlinear systems with non-Gaussian process noise statistics is important for applications in atmospheric and oceanic sampling. Non-Gaussian filtering is, however, largely problem specific and mostly sub-optimal. This manuscript uses a Gaussian Mixture Model (GMM) to characterize the prior non-Gaussian distribution, and applies the Kalman filter update to estimate the state with uncertainty. The boundedness of error in both linear and nonlinear cases is analytically justified under various assumptions, and the resulting estimate is used for feedback control. To apply GMM in nonlinear settings, we utilize a common extension of the Kalman filter: the Extended Kalman Filter (EKF). The theoretical results are illustrated by numerical simulations.