Gap metric on the subgraphs of systems and the robustness problem

The gap metric between the shift invariant subspaces of the graphs, or subgraphs, of systems is investigated. Under certain index conditions, it is shown that the gap metric on the subgraphs shares the same fundamental property of robust stability as those well known metrics such as the gap metric and the ν-gap metric. It is also shown that the ν-gap metric between two systems is the distance, measured by the gap metric, between their respective sets of all subgraphs under certain index conditions