Convergence and compactness of families of proper plants in the graph topology

The graph topology plays a central role in characterizing the robustness of feedback systems. In particular, it provides necessary and sufficient conditions for the transfer matrix of a stabilized closed-loop system to be continuous with respect to the controller. If in addition we confine our attention to a compact set of controllers, we can draw much stronger conclusions, for example, the uniform continuity of the closed-loop transfer matrix. This motivates a detailed study of convergence in the graph topology, and a characterization of compactness in this topology. Readily verifiable necessary and sufficient conditions for a set (of controllers) to be compact the graph topology are not available at present. Following our preliminary results in an earlier paper, the present paper gives a simple characterization of convergence in the graph topology in the field of fractions associated with the disc algebra. The result is then applied to characterize compact sets in the graph topology in the field of fractions associated with the disc algebra. An application to the problem of approximate system design is also discussed.