When is a controller optimal in the sense of ℋ∞ loop-shaping?

In this paper we characterize the controllers which are possible solutions of a certain ℋ_∞ control design problem, The problem considered is the optimal robustness problem for (weighted) normalized coprime factor/gap metric uncertainty, which is the basis for the Glover-McFarlane ℋ_∞ loop-shaping design method. Given a plant P and a corresponding controller C, we ask if C can be obtained from the optimization procedure for some choice of weighting function. This paper considers single-input/single-output systems and gives necessary and sufficient conditions for optimality which involve right-half plane pole/zero counts and a certain winding number test based on the Nyquist diagram of PC. The results give a characterization of this class of ℋ_∞-optimal designs in the language of classical control