Speed Loop — S3: Two-DOF

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

S3 (Two-DOF speed control) resolves a fundamental limitation of single-DOF speed controllers: tightening the feedback loop for faster setpoint following degrades load stiffness and vice versa. A reference pre-filter \(F(z)\) in the forward path shapes command tracking independently from the feedback loop designed for disturbance rejection. This is particularly valuable for servo speed control where both fast tracking and high stiffness are required simultaneously.

Operates exclusively in the 1 kHz outer handler. No mechanical RLS required.

Prerequisites

See documentation/theory/foc-plant-models.md §2 for the speed plant derivation. See documentation/theory/speed-loop-pi.md for PI gain design.

Mathematical Foundation

All speed controllers operate on the discrete mechanical speed plant from documentation/theory/foc-plant-models.md §2:

\[ \omega_m[k+1] = A_d^o \cdot \omega_m[k] + B_d^o \cdot u[k] \]

S3 adds a reference pre-filter \(F(z)\) in the forward path that shapes tracking independently from the feedback loop designed for disturbance stiffness.

S3 — Two-DOF Speed Control

Structure: The PI feedback controller receives a pre-filtered setpoint:

\[ e_{fb}[k] = F(z)\,\omega_m^*[k] - \omega_m[k] \]

The pre-filter is a first-order low-pass parameterised by \(\tau_{ff}\):

\[ F(z) = \frac{1 - e^{-T_s^o/\tau_{ff}}}{z - e^{-T_s^o/\tau_{ff}}} \]

Decoupled transfer functions:

\[ T_{track}(z) = F(z)\,C_{fb}(z)\,G(z)\,(1 + C_{fb}(z)\,G(z))^{-1} \] \[ T_{dist}(z) = G(z)\,(1 + C_{fb}(z)\,G(z))^{-1} \]

\(\tau_{ff}\) shapes tracking speed; PI gains shape disturbance rejection. Neither constrains the other.

Relation to toolbox: an ExponentialMovingAverage realises \(F(z)\) and an incremental PI provides \(C_{fb}(z)\), with \(\alpha = 1 - e^{-T_s^o/\tau_{ff}}\) mapping the pre-filter time constant onto the filter.

Tuning: - PI gains (\(K_p\), \(K_i\)): design for disturbance rejection bandwidth \(\omega_{bw}\) as in standard PI (see documentation/theory/speed-loop-pi.md). - \(\tau_{ff}\): set \(\leq 1/\omega_{bw}\) for tracking at least as fast as the feedback loop. Set larger to deliberately smooth abrupt setpoint changes.

Numerical Properties

Limitations & Assumptions

• Treats the current loop as ideal (\(i_q \approx i_q^*\)). Valid when current-loop bandwidth is at least 10× the speed-loop bandwidth.
• \(\tau_{ff}\) must be smaller than the closed-loop mechanical time constant to avoid pre-filter lag causing overshoot.
• If the incoming speed reference is already smooth, \(\tau_{ff}\) can equal the PI time constant, which degenerates the Two-DOF to a standard PI.
• Speed estimate must be smooth. A noisy encoder derivative degrades all speed controllers equally.

References

1. Åström, K.J. & Wittenmark, B. — Computer-Controlled Systems, 3rd ed., Prentice Hall, 1997.
2. Åström, K.J. & Hägglund, T. — Advanced PID Control, ISA, 2006. (Two-DOF PID structure, reference pre-filter design.)