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

A recent linear time-varying (LTV) generalisation of Vinnicombe’s ν-gap measure of the difference between openloop systems from a closed-loop perspective, is defined in terms of normalised strong representations of system graphs and a Fredholm index constraint on a family of Toeplitz-Wiener-Hopf operators, parametrised by start time. It is shown that the LTV definition of the ν-gap is invariant with respect to choice of strong normalised graph representations and that it is a metric. For a class of time-invariant systems, the generalised ν-gap is shown to reduce to the original frequency-domain definition.