Constrained multimodal ensemble Kalman filter based on Kullback–Leibler (KL) divergence

Abstract

The aim of this study is to incorporate inequality constraints in the state estimation problem for nonlinear systems. In particular, we consider the case where the posterior density is multimodal. To this end, we propose a Gaussian mixture model-based ensemble Kalman filter (GMM-EnKF) in which the probability density function of the state is approximated using a Gaussian mixture distribution. To handle inequality constraints in the recursive filtering framework for multimodal distributions, we propose to project the unconstrained GMM-EnKF into the constrained region. This can be accomplished by determining the constrained posterior density function such that the Kullback–Leibler divergence between the unconstrained and constrained GMM is minimized. Since the resulting optimization problem is non-convex, we propose to solve a two-step convex optimization problem in the update step of the state estimation problem. Two demonstrative case studies are presented to illustrate the effectiveness of the proposed constrained GMM-EnKF algorithm.