This paper deals with the design of compensators that provide a guaranteed level of performance, in the sense of L-2 gain, for systems with time-varying structured uncertainty. For stability, the notion of quadratic stability, using a single Lyapunov function, is used. The uncertainty is assumed to be polytopic (e.g., the uncertainty vector enters the state-space representation of the systems affinely). Recent results in the characterization of H-infinity controllers are used to show that quadratic stability with performance is equivalent to a bi-affine problem. A set of conditions, under which the problem becomes convex, is discussed.