Filtering and control for wide bandwidth noise and ‘nearly’ linear systems

Abstract

Typically, modern stochastic control theory uses ideal white noise driven systems (Itô equations), and if the observed data is corrupted by noise, that noise is usually assumed to be ‘white Gaussian’. If the models are linear, a Kalman-Bucy filter is then used to estimate the state, and a control based on this estimate is computed. Actually, the noise processes are rarely ‘white’, and the system is only approximated in some serse by a diffusion. But, owing to lack of ‘computable’ alternatives, one still uses the above procedure. Then the ‘filter’ estimates and associated control might be quite far from being optimal. We examine the sense in which such estimates and/or control are useful, in order to justify the the use of the commonly used procedure. For the filtering problem where the signal is a ‘near’ Gauss-Markov process and the observation noise is wide band, it is shown that the usual filter is ‘nearly optimal’ with respect to a very natural class of alternative data processors. The asymptotic (in time and bandwidth) problem is treated, as is the conditional Gaussian case. Similar results are obtained for the combined filtering and control problem, where it is shown that good controls for the ‘ideal’ model are also good for the actual physical model, with respect to a natural class of alternative controls, for control over a finite time interval and the average cost per unit time problem.

The paper is an outline of some of the work reported in [9].