Gap metrics for linear time-varying systems

Abstract

The relationship between two gap metrics is investigated for a class of time-varying linear systems. Speci°Cally, systems are considered to be causal linear maps between finite-energy continuous-time signals of nonuniformly lower-bounded support, and by definition, the corresponding input-output graphs admit normalized so-called strong-right and strong-left representations. First, a time-varying generalization of Vinnicombe's ν-gap is revisited to rectify an omission in the original developments and to verify compliance with the axioms of a metric. Subsequently, it is shown that this generalized ν-gap metric induces the same topology as an apposite adaptation of Feintuch's time-varying gap m..