Worst-case conditional system identification in a general class of norms

Abstract

We consider optimality of identification of linear systems by reduced complexity models which belong to a subspace (of possibly low dimension) of a system space. Identification is based on the ‘a priori’ knowledge that an unknown system belongs to a certain class, and ‘a posteriori’ information given by its nonexact output measurements. Measurement errors are assumed unknown but norm bounded. The analysis is not restricted to specific norms in system and measurement spaces; generic norms are considered. We derive tight upper and lower bounds on the minimal local worst-case identification error, and define an algorithm whose error is within the derived bounds. The results are specified for product norms (which include l p norms, 1≤p≤+∞) in the system space. The bounds are expressed in terms of the model error which measures a quality of representing systems by a model subspace, and the diameter of information reflecting the size of a model uncertainty set. The model error is further splitted into truncated model error and residual error which controls “tail” properties of the unknown system. Examples of model and input selection in some special cases are given, and numerical experiments showing the behavior of the estimates reported.