Suboptimality evaluation of approximated models in H∞ identification

The problem of identifying complex linear systems from noise corrupted data is investigated, considering that only approximated models can be estimated and the effects of unmodeled dynamics have to be accounted for. Set membership identification theory (SMIT) aims to deliver not only a model of the system to be identified, but also a measure of its identification error in the form of a bound on the distance, in a given norm, between the model and the system. Optimal models, requiring the computation of the Chebicheff center of unfalsified systems set, are often difficult to be computed or may have high complexity. This motivates the interest in estimating simpler models at the expense of some degradation in the identification accuracy. In the paper, it is shown how to evaluate the suboptimality level of a model, defined as the ratio between its identification error and the minimum one. The case of H_∞ identification of LTI discrete-time systems, from noise corrupted measurements in the time and/or the frequency domain, is considered. The assumption on the noise can account for information on its maximum magnitude and possible uncorrelation properties in deterministic sense. The results of the paper show that models obtained by standard interpolatory algorithms, which are known to have suboptimality level not greater than 2, may actually have a significantly lower level. Moreover, models with suboptimality level less than 2 may be derived, having much lower order than those obtained by interpolatory algorithms. Thus, the order of the identified model can be selected by trading between model complexity and suboptimality level