On the equivalence of proportional-integral and proportional-resonant controllers with anti-windup
This resolves a theoretical ambiguity for control engineers designing grid-connected converters, though the result is incremental as it formalizes known practical equivalence.
The paper proves that proportional-integral (PI) controllers in the synchronous reference frame and proportional-resonant (PR) controllers in the stationary reference frame, both with anti-windup, are equivalent when correctly implemented in state space, yielding identical closed-loop dynamics and steady-state accuracy. Differences arise only from incorrect implementation or significant inverter time delay.
It is shown that proportional-integral (PI) control in the synchronously rotating (d, q)-reference frame and proportional-resonant (PR) control in the stationary (α, \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).