This paper presents a method that reduces the conservatism inherent in the disturbance representation in standard H. theory. It addresses several discrete-time control synthesis problems where the disturbances acting on the system are structured by a convex family of linear filters ('signal polytope') and where the system disturbance attenuation level attained over the signal polytope is ensured to (just) a prescribed probability. This setup enables realistic multi-feature disturbance delimiting while accounting for uncertainty in the disturbance model itself by allowing probability waivers. The core of the probability aspects of the proposed solutions is the search for a truncated signal polytope which provides both the required probability and the best robust disturbance attenuation level. Many examples are given, including one of an aircraft output-feedback control with a polytope of low-pass filters representing different wind phenomena. The examples demonstrate that addressing realistic disturbances results in better control designs (hence better performance) and that a small certainty waiver can yield a large performance gain.