Current Loop — A1: Decoupled PID + Feedforward

Theory document: Explains the mathematical and engineering principles behind a component or algorithm. This document is descriptive — it records the why and how at a scientific level, independent of any specific implementation. Equations use KaTeX (\(inline\) and \[block\]). Block diagrams use Mermaid or ASCII art.

Overview

A1 augments the baseline PI current controller with an explicit feedforward that cancels the cross-axis coupling terms \(\omega_e L_s i_q\) and \(\omega_e L_s i_d\) and the back-EMF disturbance \(\omega_e \psi_f\). At high electrical speed \(\omega_e\), these coupling voltages become large relative to the RL drive voltage, reducing effective PI bandwidth and introducing cross-axis torque ripple. Explicit cancellation restores full PI bandwidth independently of speed.

Operates exclusively in the 20 kHz FOC ISR. Requires electrical RLS estimates (\(L_s\), \(R_s\)) and the motor constant \(\psi_f\).

Prerequisites

See documentation/theory/foc-plant-models.md for all base symbols. See documentation/theory/current-loop-pi.md for the baseline PI design and gain normalisation.

Mathematical Foundation

All current controllers operate on the decoupled per-axis RL plant derived in documentation/theory/foc-plant-models.md §1. After feedforward decoupling, each axis reduces to an independent first-order system:

\[ i[k+1] = A_d^i \cdot i[k] + B_d^i \cdot v'[k], \qquad A_d^i = e^{-R_s T_s^i / L_s},\quad B_d^i = \frac{1 - A_d^i}{R_s} \]

A1 derives the feedforward law that recovers this decoupled plant from the coupled PMSM voltage equations and then applies a standard PI.

A1 — Decoupled PID + Feedforward

Motivation: Plain PI controllers treat the cross-coupling terms \(\omega_e L_s i_q\) and \(-\omega_e L_s i_d\) as disturbances. At high electrical speed \(\omega_e\), the coupling voltage becomes large relative to the RL drive voltage, reducing effective PI bandwidth and introducing cross-axis torque ripple. Explicit cancellation restores full PI bandwidth independently of speed.

Feedforward law: The total applied voltage cancels coupling before presenting the decoupled RL plant to the PI regulator:

\[ \boxed{v_d = v_d^{PI} - \omega_e L_s i_q} \] \[ \boxed{v_q = v_q^{PI} + \omega_e L_s i_d + \omega_e \psi_f} \]

After applying these voltages, both axes reduce to the independent RL plant and the PI controllers are designed identically to the standard case.

Normalisation: Feedforward terms are scaled to the same normalised unit as the PI output (normalisation scale \(\sqrt{3}/V_{dc}\)):

\[ f_d = -\omega_e L_s i_q \cdot \frac{\sqrt{3}}{V_{dc}}, \qquad f_q = (\omega_e L_s i_d + \omega_e \psi_f) \cdot \frac{\sqrt{3}}{V_{dc}} \]

Gain design: PI gains are identical to the pole-zero cancellation design in documentation/theory/current-loop-pi.md. The feedforward is additive and does not change the closed-loop transfer function.

RLS dependency: \(L_s\) from the electrical RLS estimator. \(\psi_f\) is a motor constant calibrated during alignment. \(V_{dc}\) is measured dynamically.

Performance: At \(\omega_e = 1000\) rad/s with \(L_s = 0.5\) mH and \(i_q = 5\) A, the uncorrected coupling is \(\omega_e L_s i_q = 2.5\) V — 10% of a 24 V bus. This is the dominant current control error at speed for an uncompensated PI.

Output Voltage Limit

The output voltage vector is subject to the circular SVM constraint:

\[ \sqrt{(v_d')^2 + (v_q')^2} \leq 1 \]

See documentation/theory/current-loop-pi.md — Output Voltage Limit — for the full derivation and the interaction with integral anti-windup.

The limit is applied once, to the sum of the PI output and the decoupling feedforward:

\[ v'[k] = \Pi\bigl(\tilde{v}'_{PI}[k] + v'_{ff}[k]\bigr) \]

Limiting the PI output on its own first and the sum again afterwards reserves no headroom for the feedforward, so decoupling degrades exactly at the high speeds where the cross-coupling and back-EMF terms it cancels are largest. The share committed back to the PI’s integrator is the applied vector minus the feedforward, so the feedforward is never integrated as if the loop had asked for it.

Numerical Properties

Limitations & Assumptions

• Feedforward accuracy is proportional to \(L_s\) accuracy: 20% \(L_s\) error leaves 20% residual coupling.
• Requires \(\psi_f\) for back-EMF cancellation on the q-axis. \(\psi_f\) is set at alignment and not updated online.
• Assumes balanced three-phase operation (\(i_a + i_b + i_c = 0\)).
• Requires accurate electrical angle \(\theta_e\). Angle errors couple directly into the dq frame.
• Output subject to the circular SVM constraint — see Output Voltage Limit above.

References

1. Krishnan, R. — Permanent Magnet Synchronous and Brushless DC Motor Drives, CRC Press, 2010.
2. Åström, K.J. & Wittenmark, B. — Computer-Controlled Systems, 3rd ed., Prentice Hall, 1997.