Robust, fragile, or optimal?

We show by examples that optimum and robust controllers, designed by using the H₂, H_∞, l¹, and μ formulations, can produce extremely fragile controllers, in the sense that vanishingly small perturbations of the coefficients of the designed controller destabilize the closed-loop control system. The examples show that this fragility usually manifests itself as extremely poor gain and phase margins of the closed-loop system. The calculations given here should raise a cautionary note and draw attention to the larger issue of controller sensitivity which may be important in other nonoptimal design techniques as well