Speed Loop — PI (Baseline)

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

The baseline speed controller is a discrete incremental PI regulator that outputs a \(q\)-axis current reference \(i_q^*\) for the inner current loop. Pole-zero cancellation against the mechanical speed plant places the closed-loop speed bandwidth at \(\omega_{bw}^s\) directly. No mechanical RLS convergence is required — datasheet values for \(J\) and \(B_f\) are sufficient, and the integral term corrects steady-state error from model mismatch.

Operates exclusively in the 1 kHz outer handler.

Prerequisites

See documentation/theory/foc-plant-models.md for all base symbols and ZOH derivations.

Mathematical Foundation

Plant Model

Treating the current loop as ideal (\(i_q \approx i_q^*\)), the mechanical speed dynamics are:

\[ J \frac{d\omega_m}{dt} = K_t\, i_q^* - B_f\, \omega_m - T_L \]

This is a first-order plant with a single real pole. The continuous-time transfer function from \(i_q^*\) to \(\omega_m\) (ignoring load disturbance \(T_L\)) is:

\[ G_\omega(s) = \frac{K_t}{J s + B_f} = \frac{K_t/J}{s + B_f/J} \]

ZOH-discretized at \(T_s^o\) (from documentation/theory/foc-plant-models.md §2):

\[ \omega_m[k+1] = A_d^o\, \omega_m[k] + B_d^o\, u[k], \qquad A_d^o = e^{-B_f T_s^o / J},\quad B_d^o = \frac{K_t T_s^o}{J} \]

PI Gain Design — Pole-Zero Cancellation

The mechanical plant has a single pole at \(s = -B_f/J\). A PI controller:

\[ C(s) = K_p \cdot \frac{s + K_i/K_p}{s} \]

achieves pole-zero cancellation by placing the PI zero at the plant pole:

\[ \frac{K_i}{K_p} = \frac{B_f}{J} \]

The open-loop gain is then \(K_p K_t / (Js)\), giving a first-order closed loop with bandwidth \(\omega_{bw}\). Solving for gains:

\[ K_p = \frac{J\, \omega_{bw}}{K_t}, \qquad K_i = \frac{B_f\, \omega_{bw}}{K_t} \]

What the implementation uses

The derivation above is the textbook one; PidSpeedController::ApplyGains departs from it in two ways, and both matter when reading the code against this chapter.

\[ \boxed{K_p = \frac{2\, J\, \omega_{bw}}{K_t}, \qquad \frac{K_i}{K_p} = \max\!\left(\frac{B_f}{J},\ \frac{\omega_{bw}}{4}\right)} \]

The factor 2 on \(K_p\) is not produced by the derivation, which places crossover at \(\omega_{bw}\) exactly. It is an empirical margin carried by the implementation; this chapter records it rather than justifying it, because no derivation here yields it.

The integral zero is floored at \(\omega_{bw}/4\), so the pole-zero cancellation this section is named after happens only when \(B_f/J\) is already the larger of the two. For the identified parameters of a small PMSM it is not — \(B_f/J\) is typically well under \(\omega_{bw}/4\) — so in practice the floor sets the zero and the plant pole is not cancelled. The floor exists so that \(K_i\) stays non-zero when the mechanical identification returns \(B_f = 0\), which would otherwise leave a pure proportional law with a standing error.

Where the floor sits decides load rejection. With the factor 2 on \(K_p\) the loop closes as \(s^2 + 2\omega_{bw}s + 2\omega_{bw} z\), where \(z\) is the integral zero. Its slow root sits near \(z\) and is what works a load torque off: the speed error after a torque step \(\tau\) decays as \(\frac{\tau}{J\,(p_2 - p_1)}\left(e^{p_1 t} - e^{p_2 t}\right)\). A floor of \(\omega_{bw}/10\) put that root near \(\omega_{bw}/10\) — about 10 rad/s at a 100 rad/s bandwidth, a time constant near 100 ms — so a 0.5 N·m step on the reference rotor (7.06 µkg·m²) was still 10.4 rad/s off a 20 rad/s setpoint 0.4 s later and needed nearly half a second to return inside 3 rad/s. At \(\omega_{bw}/4\) the slow root moves to about \(0.29\,\omega_{bw}\) and the same recovery takes under 200 ms. The zero also enters the reference path, so it trades against step overshoot: at \(\omega_{bw}/4\) it nearly cancels the slow root and a step from rest overshoots by under 10 % on the same rotor; at \(\omega_{bw}/2\) the overshoot passes 15 %.

Both gains are then divided by \(I_{q,max}\) before they reach the PID, and its output is multiplied by \(I_{q,max}\) again, so the controller works in normalised units and the scaling cancels.

Practical limit: speed bandwidth must satisfy the cascade separation principle: \(\omega_{bw}^s \leq \omega_{bw}^i / 10\), where \(\omega_{bw}^i\) is the current-loop bandwidth. Typical: \(\omega_{bw}^s = 100\)–\(500\) rad/s at a 1 kHz control rate.

Incremental Form and Anti-Windup

The implementation uses the incremental PI to avoid initialisation issues:

\[ \Delta u[k] = K_p\,\bigl(e[k] - e[k-1]\bigr) + K_i\, T_s^o\, e[k] \]

The \(T_s^o\) factor is folded into the gain the implementation stores, so the ki in the code is the \(K_i T_s^o\) of this expression, not the \(K_i\) of the continuous design above. \[ u[k] = \mathrm{clamp}\bigl(u[k-1] + \Delta u[k],\ -I_{q,max},\ I_{q,max}\bigr) \]

where \(e[k] = \omega_m^*[k] - \omega_m[k]\). Clamping to \(\pm I_{q,max}\) limits the current demand to the hardware envelope and provides inherent anti-windup — no separate back-calculation is needed.

Numerical Properties

Limitations & Assumptions

• Treats the current loop as ideal (\(i_q \approx i_q^*\)). Valid when the current-loop bandwidth is at least 10× the speed bandwidth.
• Pole-zero cancellation accuracy depends on knowledge of \(B_f/J\). Model mismatch shifts the cancellation but integral action removes any resulting steady-state error.
• Single-DOF structure — cannot independently shape command tracking and disturbance rejection. Use S3 (Two-DOF) when both are required simultaneously.
• Speed estimate must be smooth. A noisy encoder derivative degrades this controller equally to all speed controllers; an alpha-beta filter or Kalman smoother is recommended upstream.

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.
3. Åström, K.J. & Hägglund, T. — Advanced PID Control, ISA, 2006.