Learning complex systems from data: the Set Membership approach

Abstract

In the paper the problem of making inferences on unknown nonlinear system (e.g. identification, prediction, smoothing, filtering, control design, decision making, fault detection, etc.) based on finite and noise-corrupted measurements is considered. Inferences are usually obtained by means of models of the system, estimated from measurements within a finitely parametrized model class describing the functional form of involved nonlinearities, whose proper choice is realized by a search, from the simplest to more complex ones (linear, bilinear, polynomial, neural networks, etc.). In this paper an alternative approach, recently developed by the authors is presented. The approach, based on a Set Membership framework, does not need assumptions on the functional form of the regression function describing the system, but requires only some information on its regularity, given by bounds on the derivatives. In this way, the problem of considering approximate functional forms is circumvented. Moreover, noise is assumed to be bounded, in contrast with statistical methods, which rely on assumptions such as stationarity, ergodicity, uncorrelation, type of distribution, etc., whose validity may be difficult to be reliably tested and is lost in presence of approximate modeling. In this paper some of the main results developed by the authors are presented within a unifying framework. In particular, necessary and sufficient conditions for checking the assumptions validity are given and optimal and almost optimal algorithms are presented for the cases that the desired inferences are identification and prediction.