A Kalman-Bucy Filtering Theory for Affine Hereditary Differential Equations

Abstract

There have been a number of different approaches to the filtering problem for delay systems, for example, Kwakernaak [12], Kushner and Barnea [11], Kailath [9] and Lindquist [13). The main theoretical contribution to the problem is by Lindquist, who proves a duality theorem between estimation and control for stochastic systems with time delay, using the (nonrandom) theory of linear functional differential equations as expounded by Halanay, Hale, Banks et al neatly avoiding the Riccati equation which occurs in the Kalman-Bucy theory. This paper incorporates a more direct approach and generalizes the Kalman-Bucy filtering theory for a class of linear delay equations, along the lines of Kwakernaak in [12]. This is done by formulating the problem as one in the abstract Hilbert space M² introduced by Delfour and Mitter in their theory of affine hereditary differential equations in [7], and using a similar approach to that in “Infinite Dimensional Filtering” [4].