The graph metric for unstable plants and robustness estimates for feedback stability

In this paper, a "graph metric" is defined that provides a measure of the distance between unstable multivariable plants. The graph metric induces a "graph topology" on unstable plants, which is the weakest possible topology in which feedback stability is robust. Using the graph metric, it is possible to derive estimates for the robustness of feedback stability without assuming that the perturbed and unperturbed plants have the same number of RHP poles. If the perturbed and unperturbed systems have the same RHP poles, then it is possible to obtain necessary and sufficient conditions for robustness with respect to a given class of perturbations. As an application of these results, the design of stabilizing controllers for unstable singularly perturbed systems is studied. Finally, the relationship of the graph metric to the "gap metric" introduced by Zames and El-Sakkary is studied in detail. In particular, it is shown that the robustness results of Zames and El-Sakkary do not enable one to conclude the causality, of the perturbed system, whereas the present results do.