On the Gap Metric for Finite-Dimensional Linear Time-Varying Systems in Continuous Time

Abstract

It is well-known that the gap metric, and its variants, provide a natural framework for studying the robustness of feedback interconnections [1–4]. In fact, for arbitrary linear systems, it is known that the gap metric induces the weakest topology in which both feedback stability and closed-loop performance are both robust properties [5]. In this article, we study the gap metric for the class of finite-dimensional, linear time-varying systems, in continuous-time. In particular, it is shown that the gap between two systems is equal to the norm of an operator composed of normalised left and right representations of the system graphs. The development relies on tools familiar from the time-invariant setting. This is in contrast to the abstract results which underpin the discretetime generalisations of the gap framework to the time-varying setting [6,7]. Using ideas from [8,9], an alternative characterisation of the gap is also established in terms of a standard linear fractional synthesis problem from robust control theory. This characterisation appears to be useful for approximation in the gap, and might be extendable to other classes of system.