Algorithms for worst case identification in \(\mathcal H_{\infty}\) and in the \(\nu\)-gap metric
This paper considers two robustly convergent algorithms for the identification of a linear system from (possibly) noisy frequency response data. Both algorithms are based on the same principle; obtaining a good worst case fit to the data under a smoothness constraint on the obtained model. However they differ in their notions of distance and smoothness. The first algorithm yields an FIR model of a stable system and is optimal, in a certain sense for a finite model order. The second algorithm may be used for modelling unstable plants and yields a real rational approximation in the \( L_2\)-gap. Given a model and a controller stabilizing the true plant, a procedure for winding number correction is also suggested. The approach presented here differs from most conventional algorithms in two important aspects: 1) Provided the frequency response of the model transfer function is sufficiently smooth and under reasonable (qualitative) assumptions that the frequency response of the true plant is also smooth and the measurement noise does not have a periodic component, the worst case fit between the noisy frequency response samples and the frequency response of the model over a sufficiently dense grid of measurement frequencies is a good indication of the `true' distance between the plant and the model. The algorithms presented here offer an explicit trade-off between a quantity related to smoothness of the frequency response of the model and the worst case fit achieved over the given frequency response samples. Two different algorithms presented differ in their notions of the worst case distance and smoothness. 2) In most conventional untuned algorithms, the model order is a function of length of data. This restriction is removed in the work presented here, obtaining robustly convergent algorithms one of which is optimal, in a certain sense, for a finite model order.
scientific article; zbMATH DE number 2117912