Speed Loop — S2: ADRC

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

S2 (Active Disturbance Rejection Control) makes the load torque disturbance explicit: an Extended State Observer (ESO) estimates the total disturbance (load torque + model error) in real time and the control law cancels it. This gives dramatically faster load-step rejection than integral action alone, with only two tuning parameters (\(\omega_c\), \(\omega_o\)). LQI handles the disturbance only implicitly through its integral state; ADRC cancels it before the control law sees it.

Operates exclusively in the 1 kHz outer handler. Requires \(K_t\) and \(J\) from mechanical RLS.

Prerequisites

See documentation/theory/foc-plant-models.md §2 for the speed plant derivation.

Mathematical Foundation

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

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

The unknown load torque \(T_L/J\) enters as a state perturbation. S2 estimates it in real time with an Extended State Observer and cancels it before the control law sees it.

S2 — Active Disturbance Rejection Speed Control (ADRC)

Extended plant: Lump total disturbance \(f\) (absorbing both \(T_L\) and model error) into the state:

\[ \frac{d\omega_m}{dt} = b_0\, u + f, \qquad b_0 = \frac{K_t}{J} \]

ESO (second-order, bandwidth \(\omega_o\), binomial gain placement):

\[ L = [\beta_1,\; \beta_2]^T, \quad \beta_1 = 2\omega_o,\quad \beta_2 = \omega_o^2 \]

Discrete update at \(T_s^o\):

\[ e_{obs}[k] = \omega_m[k] - \hat{\omega}_m[k] \] \[ \hat{\omega}_m[k+1] = \hat{\omega}_m[k] + T_s^o\,(\beta_1 e_{obs}[k] + \hat{f}[k] + b_0 u[k]) \] \[ \hat{f}[k+1] = \hat{f}[k] + T_s^o \cdot \beta_2 e_{obs}[k] \]

Control law:

\[ u_0[k] = \omega_c\,(\omega_m^*[k] - \hat{\omega}_m[k]) \] \[ \boxed{u[k] = \frac{u_0[k] - \hat{f}[k]}{b_0}} \]

Tuning (two parameters): - \(\omega_c\): desired closed-loop speed bandwidth (rad/s). Sets command-tracking rise time. - \(\omega_o\): ESO bandwidth. Rule of thumb: \(\omega_o = 3\)–\(10 \times \omega_c\). - \(b_0 = K_t/J\) from RLS. Tolerates ±50% error without stability loss.

Performance: Under a step load torque, the ADRC suppresses the speed dip in approximately \(1/\omega_o\) seconds — typically an order of magnitude faster than integral action alone.

Relation to toolbox: ActiveDisturbanceRejectionControl<float,1> implements this ESO and control law — the template parameter is the plant order, and the extended state carrying \(\hat{f}\) is added on top of it. Compute(ω_ref, ω_meas) is called once per 1 kHz handler cycle.

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.
• ESO estimates \(\hat{f}\) with a lag proportional to \(1/\omega_o\). Disturbances faster than \(\omega_o\) are not fully cancelled.
• High \(\omega_o\) amplifies speed measurement noise. Use a velocity smoother upstream if needed.
• The ESO is discretized with forward Euler, whose poles leave the unit circle for \(\omega_o T_s^o > 0.83\). The toolbox enforces the conservative precondition \(\omega_o T_s^o < 0.5/n\); the configured observer bandwidth is clamped to \(\omega_o \leq 0.4 / T_s^o\) to stay inside it, which caps the usable \(\omega_o/\omega_c\) ratio at 1 kHz.
• The clipped \(i_q^*\) is fed back into the ESO each cycle, so a sustained current clip does not bias \(\hat{f}\).
• Speed estimate must be smooth. A noisy encoder derivative degrades all speed controllers equally.

References

1. Krishnan, R. — Permanent Magnet Synchronous and Brushless DC Motor Drives, CRC Press, 2010.
2. Han, J. — “From PID to Active Disturbance Rejection Control”, IEEE Trans. Ind. Electron., 56(3):900–906, 2009.
3. Gao, Z. — “Scaling and Bandwidth-Parameterization Based Controller Tuning”, Proc. ACC, 2003.