On the Frequency Response Interpretation of an Indexed L2-gap Metric

A new metric on the space of linear, time-invariant, continuous time systems has recently been introduced. This metric is topologically equivalent to the gap metric, and in general smaller than it¿in fact, it is the smallest metric for which a certain robust stabilization result holds. The great advantage of this metric over the gap is that it readily admits of a frequency response interpretation. In this paper we elaborate on this interpretation and generalize the main robust stability result involving this metric to the case where the metric is measured pointwise in the frequency domain. We also discuss how the metric makes sense, quantitatively, as a measure of the importance of the difference between two plants in a feedback context. In the scalar case, this involves viewing the projection of the Nyquist diagram of a plant onto the Riemann sphere. Finally, it is shown how the metric may be extended beyond one, in such a way that the set of all stabilizable plants is an open ball around any stabilizable plant in this extended metric.