A control feedback system with saturation nonlinearities is said to have robustness preserving characteristics if the constrained system is as robust as its linear counterpart. Such characteristics can be desirable. It has been proved for some special cases that the anti-windup version of the Internal Model Control architecture can offer such characteristics for first-order plants with delays against norm-bounded (not only LTI) uncertainty. This paper provides general conditions expressed in the frequency domain which allow to test for the preservation of robustness for plants of any order. In addition, a class of robustness preserving controllers is characterised in terms of the Zames-Falb conditions. The test is shown to be easily implementable and exploited for anti-windup tuning