On robust performance in H^∞

In this paper we consider the robust performance problem in H^∞ with scalar perturbations. We illustrate how the worst case H^∞ performance can be regarded as an H^∞ norm of a function of several complex variables. Alternatively, a method is presented for computing the worst case performance by computing the H^∞ norm of a single system. This system is constructed from the original nominal system with the perturbations replaced by certain all-pass functions. We show that as the order of the all-pass functions goes to infinity, this H^∞ norm converges to the worst case performance. The implications of this characterization for computing the complex structured singular value and robust synthesis are discussed.