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, and uniform a priori estimate of the performance can be given. 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 condition 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 two major theorems in analysis: Montel's theorem on normal families of analytic functions, and the corona theorem for coprimeness in 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.

Inspec keywords: graph theory; set theory; closed loop systems; feedback; approximation theory; matrix algebra; sampled data systems; stability

Other keywords: sufficient condition; Montel's theorem; approximate design problem; graph topology; corona theorem; closed-loop transfer matrix; continuity properties; necessary condition; feedback systems; compact sets; stabilized closed-loop systems; uniformly continuous; analytic functions; performance estimation; sampled-data control systems; uniform a priori estimate

Subjects: Combinatorial mathematics