On the equivalence of proportional-integral and proportional-resonant controllers with anti-windup

It is shown that proportional-integral (PI) control in the synchronously rotating (d, q)-reference frame and proportional-resonant (PR) control in the stationary ({\alpha}, \b{eta})-reference frame, both with anti-windup, are equivalent if and only if their implementation is done correctly in state space and the controller parameters and the initial values are identical. It is shown that an equivalence in the frequency domain does only hold if simplifying assumptions are satisfied. As consequence of the equivalence, both closed- loop control performances are identical with respect to closed-loop dynamics and steady-state accuracy. The control performance will only differ if their implementation is not done correctly or the time delay induced by the voltage source inverter becomes significant. To the best knowledge of the author, equivalence of PR and PI controllers with anti-windup has not been shown before (in particular not in state space).