New Algorithms in Real Time Solution of the Nonlinear Filtering Problem

Abstract

It is well known that the filtering theory has important applications in both military and commercial industries. The Kalman-Bucy filter has been used in many areas such as navigational and guidance systems, radar tracking, solar mapping, and satellite orbit determination. However, the Kalman-Bucy filter has limited applicability because of the linearity assumptions of the drift term and observation term as well as the Gaussian assumption of the initial value. Therefore there has been an intensive interest in solving the nonlinear filtering problem. The central problem of nonlinear filtering theory is to solve the DMZ equation in real time and memoryless way. In this paper, we shall describe three methods to solve the DMZ equation: Brockett-Mitter estimation algebra method, direct method, and new algorithm method. The first two methods are relatively easy to implement in hardware and can solve a large class of nonlinear filtering problems. We shall present the recent advance in the third method which solves all the nonlinear filtering problems in a real-time manner in theory. 1. Introduction. It is well known that the filtering theory has important ap- plications in both military and commercial industries. The Kalman-Bucy filter has been used in many areas such as navigational and guidance systems, radar tracking, solar mapping, and satellite orbit determination. However, the Kalman-Bucy filter has limited applicability because of the linearity assumptions of the drift term and observation term as well as the Gaussian assumption of the initial value. Therefore there has been an intensive interest in solving the nonlinear filtering problem. The nonlinear filtering problem involves the estimation of a stochastic process x = {xt} (called the signal or state process) that cannot be observed directly. Information con- taining x is obtained from observations of a related process y = {yt} (the observation process). The goal of nonlinear filtering is to determine the conditional density �(t,x) of xt given the observation history of {ys: 0 6 s 6 t}. In the late 1960s, Duncan (Du), Mortensen (Mo) and Zakai (Za) independently derived the Duncan-Mortensen-Zakai (DMZ) equation for the nonlinear filtering theory which the conditional probability density �(t,x) must satisfy. The central problem of nonlinear filtering theory is to solve the DMZ equation in real time and memoryless way. In this paper, we shall describe three methods to solve the DMZ equation: Brockett-Mitter estimation al- gebra method, direct method, and new algorithm method. The first two methods are relatively easy to implement in hardware and can solve a large class of nonlinear filtering problems. We shall present the recent advance in the third method which solves all the nonlinear filtering problems in a real-time manner theoretically.