Anti-windup systems are modified control structures, which are designed to compensate against the detrimental effects of saturations. This manuscript considers primarily the stability robustness of two well-known anti-windup structures. Sufficient conditions for the stability robustness of the anti-windup structures and optimal robustness against structured norm-bounded plant uncertainty are found. A saturated loop is said to be optimally robust if the constrained loop is as robust as its unconstrained counterpart. The robustness condition is shown to be less conservative than existing results on additive uncertainty. Although it is known that the conventional internal model control provides optimal robustness against additive unstructured uncertainty, this is not the case against the more general uncertainty structure