[Paper Review] On the equivalence of proportional-integral and proportional-resonant controllers with anti-windup

Christoph M. Hackl|arXiv (Cornell University)|Oct 23, 2016

Numerical methods for differential equations13 citations

TL;DR

This paper establishes the state-space equivalence of proportional-integral (PI) and proportional-resonant (PR) controllers with anti-windup in both the synchronously rotating (d, q) and stationary (α, β) reference frames, provided controller parameters and initial conditions are identical. The equivalence holds regardless of time-varying electrical angular velocity, proving that both controllers yield identical closed-loop dynamics and steady-state accuracy when implemented correctly.

ABSTRACT

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).

Motivation & Objective

• To resolve ambiguity in the equivalence of PI and PR controllers under time-varying electrical angular velocity, particularly in machine and grid-side control applications.
• To formally establish equivalence in state space rather than relying on frequency-domain assumptions with constant angular frequency.
• To demonstrate that both controllers produce identical closed-loop performance when implemented with identical parameters and initial conditions.
• To clarify that differences in performance arise only from incorrect implementation or significant inverter time delays, not from the controller type itself.
• To provide a rigorous foundation for controller selection in applications like PMSM drives and weak grid-connected inverters where frequency variations are common.

Proposed method

• Derives the state-space formulation of a PI controller with anti-windup in the (d, q)-reference frame using the Park transformation and time-derivative rules.
• Derives the state-space formulation of a PR controller with anti-windup in the (α, β)-reference frame, incorporating time-varying angular velocity.
• Applies the inverse Park transformation to relate the (α, β) and (d, q) frame dynamics, enabling cross-frame comparison.
• Uses matrix identities involving the rotation matrix Tp(φk), the skew-symmetric matrix J, and their commutative properties to simplify time-derivative expressions.
• Compares the resulting differential equations of both controllers under identical parameter and initial value conditions.
• Demonstrates that the closed-loop dynamics and control input equations are mathematically identical when controller parameters and initial states are matched.

Experimental results

Research questions

• RQ1 Under what conditions is a PI controller in the (d, q)-frame equivalent to a PR controller in the (α, β)-frame with anti-windup?
• RQ2 Does the equivalence hold when the electrical angular velocity is time-varying, rather than constant?
• RQ3 What role do initial conditions and controller parameter matching play in achieving equivalence?
• RQ4 Why do frequency-domain analyses fail to capture the full equivalence when angular velocity varies?
• RQ5 Can the equivalence be rigorously proven in state space without simplifying assumptions?

Key findings

• PI and PR controllers with anti-windup are equivalent in both the (d, q) and (α, β) reference frames if and only if their controller parameters and initial values are identical.
• The equivalence is proven in state space, making it valid for time-varying electrical angular velocity, which invalidates classical frequency-domain equivalence proofs.
• Closed-loop dynamics and steady-state accuracy are identical for both controllers when implemented correctly with matching parameters and initial conditions.
• Performance differences only arise from incorrect implementation or significant voltage source inverter time delays, not from the controller type.
• The PR controller's state equation in the (α, β)-frame and the PI controller's state equation in the (d, q)-frame are mathematically identical under parameter and initial value matching.
• The paper provides the first rigorous state-space proof of equivalence between PI and PR controllers with anti-windup, resolving prior ambiguities in the literature.