Second Order Extended Ensemble Filter for Non-linear Filtering

Abstract

Whenever the state of a system must be estimated from noisy information, 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 observation 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. It replaces linearization of non-linearities with Monte Carlo methods. The particle filter formed a basis for Ensemble Kalman Filter (EnKF) an extension of Kalman filter to non-linear filtering. The EnKF reduced the computational cost but its innovation process does 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 and Matlab. We used Lorenz 63 system of ordinary equations and differential Matlab to test the performance of the new filter. 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). SoEEKF performs much better than the other four filters.