Gap Metrics for Linear Time-Varying Systems

Abstract

The relationship between two gap metrics is investigated for a class of time-varying linear systems. Specifically, 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 $\nu$-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 $\nu$-gap metric induces the same topology as an apposite adaptation of Feintuch's time-varying gap metric for discrete-time systems defined over finite-energy signals of uniformly lower-bounded support. The two generalized metrics are therefore qualitatively equivalent in robust stability analysis for linear time-varying feedback interconnections. Quantitatively, the $\nu$-gap between a given pair of systems is never larger in value than the Feintuch gap.