Nonlinear Bayesian estimation based on mixture of Gram Charlier Series

Optimal estimation of the states of a nonlinear continuous dynamic system with discrete measurements can be obtained by solving Fokker Planck Kolmogorov Equation (FPKE), along with the Bayes´ formula. However, solving the FPKE is formidable in most cases especially for multidimensional systems. In this paper we propose a nonlinear Bayesian filtering algorithm based on mixture of Gram Charlier Series (GCS) approximation for state Probability Density Function (PDF) conditioned on measurements known as Bayes´ a posteriori PDF. GCS is based on set of orthogonal polynomials i.e., Hermite polynomials and Gaussian PDF. In this paper an earlier work on third order GCS approximations for Bayes´ a posteriori PDFs by Culver [1] has been improved by adapting a similar approach suggested by Alspach & Sorenson in Gaussian Sum Filtering (GSF) [2]. Application on tracking of satellite or Exoatmospheric Re-entry Vehicle (ERV) is performed. The simulations results for tracking of an ERV, or satellite using radar measurements shows that the presented filter can outperform single GCS filter, EKF and GSF.