Filtering and control for wide bandwidth noise driven systems

Much of modern stochastic control theory uses ideal white noise driven models (Itô equations). If the observed data are corrupted by noise, then the noise is usually assumed to be white Gaussian. Typically, if the underlying models are linear, one uses a Kalman-Bucy filter to get an estimate of the state, and then bases the control on this estimate. In practice, the noises are rarely white, and the reference signals and the systems are only approximations in some sense to a diffusion. Nevertheless, owing to lack of viable alternatives, one still uses the Kalman-Bucy filter, etc. Then the estimates are not optimal and, indeed, might be quite far from being optimal. The same is true for the corresponding control. (Examples are given to illustrate this.) The sense in which the estimates and/or control is useful needs to be examined in order to justify the use of the commonly used procedure. The issue is much deeper than mere robustness in the usual sense, since basic questions of interpretation of the results are involved. This paper deals with these questions. For the filtering problem where the signal is a near Gauss-Markov process and the observation noise wide band, it is shown that the usual method is nearly optimal with respect to a class of alternative data processors. This alternative class is rather natural and includes the data processors which one would normally want to use. It is unlikely that the class can be enlarged very much in general. The asymptotic (in time and bandwidth) problem is treated, as is the (much harder) conditional Gaussian case, and a case where the observations are nonlinear. The basic techniques are those of weak convergence theory. 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 case of alternative controls. The control problem over a finite interval as well as the average cost per unit time problem are considered.