This article addresses the problem of anti-windup design for feedback linearizable non-linear systems with input constraints, subject to dynamic controllers with linear dynamics and non-linear gains. A method for designing the controller gains and a non-linear 'observer-based' anti-windup modification is proposed, which allows reducing the behaviour of the closed-loop system under the dynamic feedback controller to that under a given static feedback linearizing controller, arbitrarily fast. The proposed anti-windup method allows, thus, to attenuate the effect of windup arbitrarily fast and guarantee that the region of closed-loop asymptotic stability under the dynamic controller is limited only by the corresponding region under the static feedback linearizing controller. The application and advantages of the method are illustrated through a simulation study on a methyl-methacrylate polymerization reactor.