New approaches to optimal filter design.

The filtering problem is a particular case of the so-called generalized estimation problem, whose aim is to infer, at each time instant, the value assumed by an internal signal of a dynamic system, driven by both known and unknown inputs, using past and possibly current samples of the known inputs and the available outputs. Most of the existent available filter design techniques assume the availability of a mathematical model of the dynamic system generating the data. However, in many practical applications, this assumption does not hold and, typically, a two-step procedure is followed, where a model of the system is identified from experimental data and then a filter is designed on the basis of this model. In this Ph.D. Thesis new approaches to filter design are investigated. First, an approach to filter design for Linear Time-Varying (LTV) systems is presented, for the case of norm bounded disturbances, the optimal filter is selected within the set of all Finite Impulse Response (FIR) systems with fixed length and assigned exponential decay. Then, a novel approach to filter design when the system under study is (partially) unknown is presented. This approach uses the available experimental data, not for the identification of a model of the system dynamics, but for the direct design of the filter. The filter design from data procedure is investigated in different contexts, for linear time invariant (LTI) and linear parameter-varying (LPV) systems. A Set Membership formulation is followed and, for classes of filters with exponentially decaying impulse response, approximating sets are determined that guarantee to contain all the solutions to the optimal filtering problem. Methods are proposed for designing almost-optimal filters with finite impulse response, whose worst-case filtering error is at most twice the lowest achievable one. In the case of LTI H-infinity SISO filters, an efficient technique is derived, able to provide convergent bounds on the guaranteed worst-case estimation error of the designed filter. In the case of LPV systems, nonlinear identification techniques are used to approximate the optimal filter and the resulting filter is shown to be optimal. Numerical examples, taken from literature and automotive applications, illustrate the effectiveness of the proposed solutions.