Approximate solution for nonlinear filtering and identification problems.

Abstract

Filtering and identification problems of partially observable stochastic dynamical systems has been considered. A modification of the extended Kalman filter (EKF) called a modified extended Kalman filter (MEKF) and a theoretical justification for this modification have been investigated. A simple but powerful numerical method for the approximation of the unnormalized conditional (probability) density of filtered diffusion process which satisfies Zakai equation arises from diffusion processes observed in correlated (or uncorrelated) noises and solves the nonlinear filtering problem has been presented. Using Galerkin technique the solution of Zakai equation has been approximated by means of a sequence of nonstandard basis functions given by a parameterized family of Gaussian densities. The spatial domain for the solution of Zakai equation and the completeness of the Gaussian densities have been also investigated. The methods are illustrated by some numerical examples. Techniques of optimal control theory as well as linear filter theory have been utilized in identifying the parameters of linear (partially observable) stochastic differential systems. Using the method of simulated annealing a computational algorithm for identifying the unknown parameters from the available observation has been derived. The results are illustrated by some examples.