Unscented Kalman filter-aided Gaussian sum filter

A non-linear filter is developed for continuous-time systems with observations/measurements carried out in discrete-time. The filter developed can approximate the a priori and a posteriori probability density function (pdf) with weighted Gaussian sums inside specific search regions. To make the approximations, first, Gaussians are placed with equal intervals inside the search regions and deterministic sample points are chosen within the search regions. The pdf values are then calculated at sample points using numerical solution of the Fokker-Planck equation for the a priori pdf and using Bayes´ rule for the a posteriori pdf. These values are used to find the weights of Gaussians using least-squares method. This process is similar to curve fitting with Gaussian radial basis functions. Inside the search regions, locations of the sample points and mean and covariance values of Gaussians are found by the help of a unscented Kalman filter (UKF). By adjusting the width of the search regions, all the parts, or the ones close to mean values of pdfs, can be approximated. The performance of the filter developed is analysed using a non-linear system with a single-state variable and two radar tracking applications. It is compared with particle filter, UKF and converted measurement Kalman filter for these cases.