Optimality, approximation, and complexity in set membership H∞ identification

Investigates the set membership identification of time-invariant, discrete-time, exponentially stable, possibly infinite-dimensional, linear systems from time or frequency-domain data, corrupted by deterministic noise. The aim is to deliver not a single model, but a set of models whose size in H∞ norm measures the uncertainty in the identification. The main focus of the note is on the optimality properties for finite data and on the tradeoff between optimality and complexity of approximated low order model sets. A method is given for evaluating convergent and computationally efficient inner and outer approximations of the value set for a given frequency. Such approximations allow one to compute, within any desired accuracy, the identification error of any identified model, and to evaluate an optimal model at any given number of frequencies. By suitably approximating these values, model sets with nominal models in RH∞ are then derived, whose order is selected by trading off between model set complexity and identification accuracy degradation. This degradation is evaluated by computing the optimality level, defined as the ratio between the reduced model identification error and the optimal one. A numerical example demonstrates the effectiveness of the presented results.