Optimal robustness in the gap metric

It is shown that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations. The proof of this result involves showing that a ball of uncertainty in the gap metric, of a given radius, is equal to a ball of uncertainty, of the same radius, defined by perturbations of a normalized right coprime fraction, provided the radius is sufficiently small for a controller to be found to stabilize the ball. The relationship between balls of uncertainty defined by perturbations of left fractions with those defined by perturbations of right fractions is investigated. It is shown 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. The problem of combined plant and controller uncertainty is also addressed. The plant and controller are shown to play reciprocal roles with respect to uncertainty in the gap. The minimal amount of combined plant and controller uncertainty that may cause instability is determined. A detailed summary of the main properties of the gap metric is given, and a dual metric called the T-gap metric is introduced. The extension of the results to infinite dimensional systems is discussed