A valid mathematical model for approximate nonlinear minimal-variance filtering

Abstract

The problem considered herein is the continuous estimation of the state of a physical system undergoing random motion, where the knowledge of the motion is given by a known function of the state plus noise. The physical problem is abstracted to the extent that the random motion is considered to be a Markov process, and the noise is assumed to be white noise; consequently, the resulting mathematical problem is formulated in terms of stochastic differential equations. The optimal estimate for a minimal-variance criterion is known to be the conditional expectation of the state of the system given the measurements; the filter derived herein is a stochastic differential equation for an approximation to the optimal estimate. A formal derivation of a stochastic differential equation for an approximate nonlinear filter was presented by Bass et al. [l], based on an approach suggested by Kushner [2] and developed partially by Bucy [3]. It was later shown by Schwartz and Bass [4] that inherent in the approach is the assumption that the conditional probability density function is nearly completely contained in a neighborhood of the conditional expectation, and a new approximation was derived that replaced that assumption with the assumption that the conditional density is nearly Gaussian. The same filter was independently derived by Fisher [5].