Compact sets in the graph topology and applications to approximation of system design

The graph topology plays a central role in characterizing the robustness of feedback systems. In particular, it provides necessary and sufficient conditions for the continuity properties of the transfer matrices of stabilized closed-loop systems. It is possible to derive stronger conclusions by confining our attention to a compact set of controllers. Specifically, if a family of plants is stabilized by each controller belonging to a compact set of controllers, then the closed-loop transfer matrix is uniformly continuous. However, at present a precise characterization of compactness in the graph topology is not available. That is the topic of the present paper. In general it appears difficult to give a necessary and sufficient conditions for a set to be compact. Hence we give a necessary condition and a sufficient condition, and discuss the gap between the two. The necessary condition is standard, while the proof of the sufficient condition is based on three major theorems in analysis: the Baire category theorem, Montel's theorem on normal families of analytic functions, and the corona theorem for H
^∞
. Finally, it is shown how the notion of a compact set of controllers can be applied to the problem of approximate design and performance estimation for sampled-data control systems.