On a limiting procedure in linear recursive estimation theory

Abstract

Kalman [I] gave the equations of an optimal recursive filter for a linear discrete system. These equations are valid for the case in which some of the measurements contain correlated noise or no noise, i.e., the case in which the correlation matrix of white noise in the measurements (usually denoted by R) is singular. But while deriving results for a continuous system through a limiting procedure, the assumption is made that R is nonsingular. Here it is shown how the limiting procedure may be carried out when R is singular. This leads to the results obtained by Bryson and Johansen [2] and Bucy [3] using different techniques. The present approach has the advantage of being straightforward. It also gives more insight into the relationship between the discrete and the continuous cases. The corresponding results for smoothing are also presented. (“Filtering” involves making an estimate at a time t, using measurements made before tl; “smoothing” involves making an estimate at a time t1 using measurements made both before and after t, .)