On the metric property of an LTV generalisation of the ν-gap

A recent linear time-varying (LTV) generalisation of Vinnicombe´s ν-gap metric is defined in terms of normalised strong representations of system graphs and a Fredholm index constraint on a family of Toeplitz-Wiener-Hopf operators, parameterised by start time. It is shown in this paper that the LTV definition is invariant with respect to the choice of graph representations and that it is indeed a metric. For a class of time-invariant systems, the generalised ν-gap reduces to the original frequency-domain definition.