H/sub ∞/ identification and model quality evaluation

Abstract

Set membership H 1 identification is investigated using time-domain data and mixed parametric and nonparametric models as well as supposing power bounded measurement errors. The problem of optimally estimating the unknown parameters and evaluating the minimal worst case identification error, called radius of information, is solved. For classes of models affine in the parameters, the radius of information is obtained as function of the H1 norm of the unmodeled dynamics. A method is given for estimating this norm from the available data and some general a priori information on the unmodeled dynamics, thus allowing the actual evaluation of the radius of information. The radius represents a measure of the "predictive ability" of the considered class of models, and it is then used for comparing the quality of different classes of models and for the order selection of their parametric part. The effectiveness of the proposed procedure is tested on some numerical examples and compared with standard statistical criteria for model order selection.Index Terms-Identification for robust control, model order selection, set membership identification, uncertainty model identification.

Comparing different approaches to model error modeling in robust identification

“…Remark. The use of residual data in set membership identification for evaluating the worst-case H ∞ norm of the unmodeled dynamics has been introduced in [10], for standard least-squares nominal models. The above set membership model error modeling strategy can be seen as a general framework in which the structures of the nominal model and model error model, and the corresponding identification algorithms must be chosen by the user according to the specific problem (a priori knowledge, noise bound, error norm, etc.…”

“…More recently, the presence of model errors has been explicitly accounted for in several works and different settings (see e.g. [30,10,9]). …”

On Model Error Modeling in Set Membership Identification

“…The use of residual data in set membership identification for evaluating the worstcase H ∞ norm of the unmodeled dynamics has been introduced in [11], for standard least-squares nominal models. The above set membership model error modeling strategy can be seen as a general framework in which the structures of the nominal and error models, and the corresponding identification algorithms must be chosen by the user according to the specific problem (a priori knowledge, noise bound, error norm, etc.)…”

“…Model errors were explicitly accounted for in [9,10]. Estimation of the size of the model error from residual data has been adopted in the set membership context in [11], for nominal models obtained via least-squares identification. Separation of unmodeled dynamics and noise in a setting including both stochastic and deterministic uncertainties has been studied in [12].…”

Validation of state-space models with time-varying parameter uncertainty

“…The model quality estimation problem has been considered from different viewpoints, both in a stochastic [1], [2] and a deterministic [3], [4], [5], [6] setting. In these references, uncertainty in both H ∞ and L 1 is considered, but the underlying assumptions seems to be that the "true process", and consequently also the perturbation, is a linear time-invariant system of high order.…”