Main Approaches to Nonlinear Estimation

Abstract

To treat the filtering problem for nonlinear dynamics in financial systems the Extended Kalman Filter is an established approach. However, since this is based on approximate linearization of the system’s dynamics and in the truncation of higher order terms in the associated Taylor series expansion, the Unscented Kalman Filter is frequently used in its place. The latter fitler performs state estimation by averaging on state vectors which are selected at iteration of the filter algorithm according to the columns of the estimation error vector covariance matrix. Moreover, to handle the case of non-Gaussian noises in the filtering procedure the particle filter has been proposed. A number of potential state vector values (particles) is updated in time through elitism criteria and out of this set the estimate of the state vector is computed. Moreover, to treat the distributed filtering and state estimation one can apply established methods for decentralized state estimation, such as the Extended Information Filter (EIF) and the Unscented Information Filter (UIF). EIF stands for the distributed implementation of the Extended Kalman Filter while UIF stands for the distributed implementation of the Unscented Kalman Filter. Moreover, to obtain a distributed filtering scheme in this chapter the Derivative-free Extended Information Filter (DEIF) is implemented. This stands for the distributed implementation of a differential flatness theory-based filtering method under the name Derivative-free distributed nonlinear Kalman Filter. The improved performance of DEIF comparing to the EIF and UIF is confirmed both in terms of improved estimation accuracy and in terms of improved speed of computation. Finally, one can note distributed filtering with the use of the distributed Particle filter. It consists of multiple Particle filters running at distributed computation units while a concensus criterion is used to fuse the local state estimates.