SECOND ORDER EXTENDED ENSEMBLE FILTER (SoEEF) FOR NON-LINEAR FILTERING

Abstract

Whenever the state of a system is estimated from information that is character ized with errors, a state estimator is employed to fuse the data with the model to produce an accurate estimate of the state. When the system dynamics and obser vation models are linear, the Kalman Filter, which is optimal, is used. However, in most applications of interest the system dynamics and observations equations are not-linear and suitable extensions of the Kalman Filter have been developed; for example, the Extended Kalman Filter(EKF). The Extended Kalman Filter is based on linearization by the Taylor series expansion about the mean of the state. This filtering process is however computationally expensive especially in high dimensional data. The cause for this is the high cost of integrating the equation of evolution of covariances. Due to this complexity in integration, new methods were sought known as the particle filters. They replace linearisation of non-linearities with Monte Carlo methods. They also formed a basis for Ensemble Kalman Filter (EnKF) an exten sion of Kalman filter to non-linear models. The EnKF reduced the computational cost but its innovation process did not capture information sufficiently hence there is need to improve its performance. This study has developed a new filter, Second order Extended Ensemble Filter (SoEEF). We derived it from stochastic state models by expansion of expected values to the second order by use of Taylor series together with Monte Carlo method. We used Lorenz 63 system of ordinary differential equations to test the performance of the new filter using the MATLAB. Then we compared its performance with four other filters like Bootstrap Particle Filter (BPF), First order Kalman Bucy Filter (FoEKBF), Second order Kalman Bucy Filter (SoKBF) and First order Extended Ensemble Filter (FoEEF). The performance of SoEEKF improves with the increase in ensemble size.