Optimality in SM Identification of Nonlinear Systems

Abstract

In the paper the problem of identifying discrete time nonlinear systems from finite and noise-corrupted measurements is considered. Most methods in the literature are based on the estimation of a model within a finitely parametrized model class describing the functional form of involved nonlinearities. A key problem in these methods is the proper choice of the model class, typically realized by a search, from the simplest to more complex ones (linear, bilinear, polynomial, neural networks, etc.). In this paper an alternative approach, based on a Set Membership framework is presented, not requiring assumptions on the functional form of the regression function describing the relations between measured input and output, but assuming only some information on its regularity, given by bounds on its gradient. 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. Necessary and sufficient conditions for assumptions validation are given. An optimal interval estimate of the regression function is obtained, providing its uncertainty range for any assigned regressor values. The set estimate allows to derive an optimal identification algorithm, giving estimates with minimal guaranteed Lp error on the overall domain of the regressors. The properties of the optimal estimate are investigated and its worst-case Lp identification error is evaluated.