Model class optimality in set membership identification

Model class optimality in set membership identification is investigated. Taking into account that the model selection has to be made using finite and noisy measurements, the n-width concept generalizes to the conditional radius of information concept. The aim of this paper is to show that model selection based on these two concepts may be drastically different. In particular it is shown how to compute the conditional radius of information in the H² approximate identification of exponentially stable systems by means of FIR model class, which is the optimal one in the n-width sense. The “peaking effect” presented by the radius of information with respect of the FIR order is demonstrated, in opposition of the monotone decreasing behaviour of n-width. This implies the existence of a model class guaranteeing a lower worst-case identification error than the one provided by the optimal model class in the n-width sense. Numerical examples show that the improvements in using the optimal FIR model in the radius sense may be quite relevant even for moderate level of noise