Anti-windup denotes the design of a compensator which augments an existing controller on a plant subject to input saturation. In particular, in the anti-windup problem statement it is assumed that a controller has already been designed disregarding the saturation effect, so that the corresponding closed-loop behaves very desirably for small enough signals. The goal of anti-windup is then to recover as much as possible that unconstrained performance (and, at the very least, stability) also for large signals, for which the saturation nonlinearity operates in its nonlinear region. We proposed a solution to the antiwindup problem for linear dead-time systems that is applicable to any linear control system, including the case where the unconstrained controller contains internal time delays (such as in the case where it arises from a Smith predictor design). Moreover, a global solution is given to the problem under the (necessary) property that the plant state matrix has eigenvalues with non positive real part, thus extending previous results to the case of poles on the imaginary axis.