Sensorless Motor Observers¶

Pulsim v1.5 ships two sensorless rotor observers — algorithms that estimate the rotor electrical angle and mechanical speed from stator voltages and currents alone, without any shaft encoder. Both follow the standard MixedDomainBlockChain block interface (.reset() + .update(*, ...) → tuple).

Why sensorless?¶

Industrial AC drives almost universally avoid the shaft encoder: - One less cable, one less point of failure. - No mechanical alignment / hall-sensor placement. - Cheaper BOM, smaller footprint. - A sealed shaft (no encoder feed-through) handles harsh environments.

Modern variable-frequency drives use sensorless control above ~10 % rated speed; the low-speed window is bridged by V/f-ramp startup or high-frequency-injection (HFI) variants — both queued for future Pulsim releases.

Available observers¶

`SlidingModeObserver` — for PMSM¶

Algorithm (Utkin §10 / TI InstaSPIN SPRABQ8):

Sliding-mode current observer in α-β: $$\frac{d\hat{i}_\alpha}{dt} = \frac{1}{L_s}\left(v_\alpha - R_s\hat{i}_\alpha - z_\alpha\right)$$ $z = K_{sl}\cdot\text{sat}(\hat{i} - i, \varepsilon)$ — saturating-sign injection.
Equivalent-control LPF extracts the back-EMF estimate $\hat{e}$ from $z$.
Angle PLL drives the d-axis projection of $\hat{e}$ to zero; outputs $\hat{\theta}_e$ and $\hat{\omega}_e$ directly (the d-component $\hat{e}_\alpha\cos\hat{\theta} + \hat{e}_\beta\sin\hat{\theta}$ equals $-E\sin(\theta_e - \hat{\theta})$, a clean Goodwin-style phase detector).

`FluxMRASObserver` — for induction motor¶

Algorithm (Schauder 1992, with Tajima-Hori 1993 leaky-integrator enhancement):

Reference model — voltage-driven rotor-flux estimate $\psi_r^{\text{ref}}$ via $\int(v - R_s i - \sigma L_s\,di/dt)\,(L_r/L_m)\,dt$. Pulsim replaces the pure integrator with a low-cutoff leaky integrator (default cutoff voltage_model_hpf_omega = 5 rad/s ≈ 0.8 Hz) to reject the DC drift that would otherwise blow up under any input bias.
Adaptive model — current-driven (uses $\hat{\omega}$).
Adaptation loop — Lyapunov-derived PI on the cross-product $\epsilon = \psi_\beta^{\text{ref}}\hat{\psi}_\alpha - \psi_\alpha^{\text{ref}}\hat{\psi}_\beta$.

Quick start — PMSM SMO¶

import pulsim as p

smo = p.SlidingModeObserver(
    Rs=0.5, Ls=200e-6,
    K_sl=20.0,           # > peak back-EMF voltage
    f_init_hz=50.0,      # initial PLL guess (close to nominal helps)
    omega_lpf=2*math.pi*2000.0,
)

# Each step:
theta_hat, omega_hat, e_a, e_b, low_speed_flag = smo.update(
    v_alpha=v_a, v_beta=v_b,
    i_alpha=i_a, i_beta=i_b,
    dt=DT,
)

End-to-end runnable example: examples/scripts/run_pmsm_smo_sensorless.py. Synthetic 200 rad/s mechanical (800 rad/s electrical) trace; SMO locks the angle within 4.2° peak / 4.2° RMS after a 50 ms lock-in window, $\hat{\omega}$ matches $\omega_{e,\text{true}}$ to 7 sig figs.

Quick start — IM MRAS¶

import pulsim as p

# Either explicit:
mras = p.FluxMRASObserver(
    Rs=2.46, Ls=0.0948, Lr=0.0948, Lm=0.0874, Rr=1.23,
    voltage_model_hpf_omega=5.0,
    Kp_mras=10.0, Ki_mras=500.0,
)

# Or pulled from an existing motor handle:
mras = p.FluxMRASObserver.from_motor(motor)

omega_hat, psi_alpha_adj, psi_beta_adj = mras.update(
    v_alpha=v_a, v_beta=v_b,
    i_alpha=i_a, i_beta=i_b,
    dt=DT,
)

Tuning guide¶

SMO (PMSM)¶

Knob	Default	When to adjust
`K_sl`	50 V	Set to ~ 2 × peak back-EMF. Higher = faster lock, more chatter.
`omega_lpf`	500 Hz	~10 × rated electrical frequency. Higher = less LPF delay, more noise pass-through.
`Kp_pll`	200	PI on the phase error. Increase if lock-in is too slow.
`Ki_pll`	1e4	Integral gain — PLL bandwidth ≈ √Ki.
`epsilon`	0.5 A	Boundary-layer width for `tanh()`. Smaller = closer to bang-bang sign.
`f_init_hz`	50 Hz	Initial PLL frequency guess. Match to nominal motor frequency for faster lock.
`low_speed_threshold`	0.05 V	Back-EMF magnitude under which the `low_speed_flag` output goes True.

MRAS (IM)¶

Knob	Default	When to adjust
`Rr`	0.0 → falls back to `0.5·Rs` heuristic	Strongly recommended to set explicitly via `from_motor()`. Wrong `Rr` → proportional speed offset.
`voltage_model_hpf_omega`	5 rad/s	Set to 0 for pure integrator (original Schauder). Increase if voltage-model drift dominates; decrease for cleaner low-frequency tracking.
`Kp_mras`, `Ki_mras`	50, 1e3	PI gains for the cross-product → $\hat{\omega}$ loop. Lyapunov-derived; retune per-machine if oscillation.

Failure modes (honest catalog)¶

SMO at very low speed — back-EMF magnitude collapses toward K_sl, observer can lose lock. Use the low_speed_flag output to hand off to a V/f startup ramp (industry standard).
MRAS at low speed — voltage-model integration drifts because $R_s i_s$ becomes comparable to the integrand; the leaky-integrator mitigates but does not eliminate this.
MRAS bootstrap with $\hat{\omega} = 0$ — the adaptive-model rotor-flux magnitude stays near zero until $\hat{\omega}$ is in the neighbourhood of $\omega_e^{\text{true}}$, so the cross-product error is dominated by amplitude mismatch instead of phase. Practical implementations bootstrap with a V/f-ramp open-loop start, then hand over to the closed-loop MRAS after ~10 % rated speed.
Parameter mismatch (Rs, Lr, Lm, Rr) — all observers are parametric. Plant-parameter drift (temperature, magnetic saturation) shows up as a steady-state speed/angle offset. PSIM / PLECS workarounds include real-time parameter adaptation; not yet shipped in Pulsim.