Limits on Achievable Robustness Against Coprime Factor Uncertainty

We consider the problem of robustness optimization against normalized coprime factor uncertainty in single-input, single-output systems. We show that loop shapes known from classical analysis to be inconsistent with closed-loop robust stability will tend to have poor optimal robustness. Such loop shapes include those with a high crossover frequency relative to a nonminimum phase zero, a low crossover frequency relative to an unstable pole, or a rapid rolloff raw near gain crossover. Our results consist of a set of lower bounds on the optimal cost of the robustness opdmization problem, each lower bound being appropriate to one of these three problematic loop shapes. The lower bounds are derived using the Poisson integrl, and display the qualitative relationship between the loopshape and the level of optimal robustness.