The Filter Design From Data (FD2) Problem: Parametric-Statistical Approach

A large body of literature exists on the filter design problem, assuming that the system to be filtered is known. However, in most practical situations, the system is not known, but a set of measured data is available. In such situations, a two-step procedure is typically adopted: a model is identified from this data set, and a filter is designed based on the identified model. In this paper, we consider an alternative approach, which uses the available data not for the identification of a model, but for the direct design of the filter. Such a direct design is investigated within a parametric-statistical framework for both the cases of linear time-invariant and nonlinear systems. The noise is assumed to be stochastic, and optimality refers to minimizing the estimation error variance. It is shown that the direct design has superior features with respect to the two-step design, especially in the presence of modeling errors. Another relevant advantage of the direct design over the two-step procedure is that minimum variance (Kalman) filters for nonlinear systems are, in general, difficult to derive and/or to implement. On the contrary, the direct approach allows for a very efficient filter design. To demonstrate the effectiveness of the proposed direct design, two examples are presented: the first is related to estimation of the Lorentz chaotic attractor; the second, involving real data, is related to estimation of vehicle yaw rate.