Unified Set Membership theory for identification, prediction and filtering of nonlinear systems

The problem of making inferences from data measured on nonlinear systems is investigated within a Set Membership (SM) framework and it is shown that identification, prediction and filtering can be treated as specific instances of the general presented theory. The SM framework presents an alternative view to the Parametric Statistical (PS) framework, more widely used for studying the above specific problems. In particular, in the SM framework, a bound only on the gradient of the model regression function is assumed, at difference from PS methods which assume the choice of a parametric functional form of the regression function. Moreover, the SM theory assumes only that the noise is bounded, in contrast with PS approaches, which rely on noise assumptions such as stationarity, uncorrelation, type of distribution, etc. The basic notions and results of the general inference making theory are presented. Moreover, some of the main results that can be obtained for the specific inferences of identification, prediction and filtering are reviewed. Concluding comments on the presented results are also reported, focused on the discussion of two basic questions: what may be gained in identification, prediction and filtering of nonlinear systems by using the presented SM framework instead of the widely diffused PS framework? why SM methods could provide stronger results than the PS methods, requiring weaker assumptions on system and on noise?