  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
mixture filters and particle filters were introduced to handle non-Gaussianity
and nonlinearity. However, these methods have seen only limited success as most
mixture filters attempt to fix the number of mixture modes during estimation
process, and the particle filters suffer from the curse of dimensionality. In
this paper, we propose a particle based Gaussian mixture 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 estimation performance of the
proposed filtering approach is demonstrated through several test cases.
