The aim of this paper is to give a novel solution to the full order anti-windup (AW) compensation problem for stable systems with input saturation. The solution is obtained by "completing the square'' in three steps and requires the solution to a single bounded-real Riccati equation, characterized by the open-loop plant's H-infinity norm. The Riccati equation plays the role of the LMIs usually found in anti-windup synthesis, but, in addition to its numerical advantages, it yields a family of anti-windup compensators with the same L-2 performance. This family of compensators is parameterized by a matrix which is intimately linked with both the poles of the anti-windup compensator and the robustness properties of the closed-loop saturated system. Thus, this matrix allows a robust anti-windup problem to be solved in a straightforward and intuitive manner. The effectiveness of the proposed technique is demonstrated on a simple example.