Particle Gaussian Mixture (PGM) filters

Recursive estimation of nonlinear dynamical systems is an important problem that arises in several engineering applications. Consistent and accurate propagation of uncertainties is important to ensuring good estimation performance. It is well known that the posterior state estimates in nonlinear problems may assume non-Gaussian multimodal densities. In the past, Gaussian filters and particle filters were introduced to handle non-Gaussianity and nonlinearity. However, these methods have seen only limited success as most filters attempt to fix the number of modes during the estimation process, and the particle filters suffer from the curse of dimensionality. In this paper, we propose a particle based Gaussian filtering approach for the general nonlinear estimation problem that is free of the particle depletion problem inherent to most particle filters. We employ an ensemble of randomly sampled states for the propagation of state probability density. A Gaussian mixture model of the propagated uncertainty is then recovered by clustering the ensemble. The posterior density is obtained subsequently through a Kalman measurement update of the mixture modes. We prove the weak convergence of the PGM density to the true filter density assuming exponential forgetting of initial conditions by the true filter. The performance of the proposed filtering approach is demonstrated through several test cases and is extensively compared to other nonlinear filters.