Continuous-time Nonlinear Estimation Filters Using UKF-aided Gaussian Sum Representations

An approximate nonlinear estimation method for continuous-time systems with discrete-time measurements is developed. The approach evaluates the Gaussian sum approximation of the a priori probability density function (pdf) by solving the Fokker-Planck equation numerically. Approximate evaluation of the a posteriori pdf is achieved by using Gaussian sums, a priori pdf and measurements in Bayes rule. Mean and covariance values of Gaussians are chosen by the help of an Unscented Kalman Filter (UKF), with respect to a region where a priori and a posteriori pdfs are approximated. Weights of the Gaussians are updated using the deterministically chosen grid points in the specified domains. UKF here acts as a one step look ahead mechanism to determine the high probability regions where a priori and a posteriori pdfs can reside. The a priori and a posteriori pdfs are approximated around these high probability regions. The developed approach is compared with UKF and Particle Filter in a one dimensional nonlinear system.