Optimal Robustness in the Gap Metric

This paper explores the use of the gap metric as a tool for the quantitative analysis and design of feedback systems. A key result in the paper is that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations. It is shown that a ball of uncertainty in the gap metric is equal to a ball of uncertainty of equal radius defined by perturbations of a normalized right coprime fraction-provided the radius is smaller than a certain quantity. This quantity is never less than the radius of the maximal gap ball about the plant which can be stabilized by some fixed controller. We study the problem of robust stabilization for normalized coprime factor perturbations and develop operator-theoretic proofs for several results obtained by Glover and McFarlane. We point out a connection between the problem of robust stabilization and the Matrix-Valued Corona Problem. We study the relationship between balls of uncertainty defined by perturbations of left fractions with those defined by perturbations of right fractions. We show that these are, in general, different, although it turns out that a controller stabilizes all plants in the left ball if and only if it stabilizes all plants in the right ball. We consider the problem of stabilization of feedback systems in the presence of combined plant and controller uncertainty. We show that if a given controller stabilizes a ball of plants about a nominal plant, where distances are measured by the gap metric, then the nominal plant stabilizes a ball of controllers about the given controller of the same radius. This reveals a certain reciprocity between the role of plant and controller in a feedback system. We also give necessary and sufficient conditions for robust stability under simultaneous perturbations of the plant and the controller. The paper gives a detailed summary of the main properties of the gap metric and introduces a dual metric, which is called the T-gap metric. The extension of the results to infinite-dimensional systems is discussed. Finally, the question of well-posedness of stability for optimal robustness problems is investigated.