Current Loop — A2: Deadbeat

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

A2 (Deadbeat) inverts the discrete current plant model to compute the exact voltage that drives the current to its reference in one sampling step — the minimum physically achievable settling time (50 µs at 20 kHz). This maximises servo stiffness: the torque response is instantaneous relative to any outer loop, so load disturbances are rejected before position error can accumulate. A two-step variant reduces noise amplification at the cost of geometric (non-deadbeat) settling.

Operates exclusively in the 20 kHz FOC ISR. Requires tight electrical RLS convergence (\(L_s\), \(R_s\)) and the A1 feedforward for decoupling at speed.

Prerequisites

See documentation/theory/foc-plant-models.md for ZOH derivations of \(A_d^i\), \(B_d^i\). See documentation/theory/current-loop-decoupled-pid.md for the A1 feedforward used alongside A2.

Mathematical Foundation

All current controllers operate on the decoupled per-axis RL plant derived in documentation/theory/foc-plant-models.md §1:

\[ i[k+1] = A_d^i \cdot i[k] + B_d^i \cdot v'[k] \]

A2 inverts this plant model directly and therefore applies the same A1 feedforward to its inversion result — an exact plant inversion is only exact once the coupling it does not model has been cancelled.

A2 — Deadbeat Current Control

One-step control law: Setting \(i[k+1] = i^*\) and solving for \(v'\):

\[ \boxed{v'[k] = \frac{i^* - A_d^i\, i[k]}{B_d^i}} \]

The result is clamped to \([-1,\,1]\) (normalised). When the step demand exceeds available voltage, settling takes more than one sample but remains optimal for the available headroom.

Two-step variant (noise robustness): The one-step law amplifies measurement noise by \(1/B_d^i = L_s/T_s^i\), which is large for small inductances. The two-step law solves for the minimum-norm input sequence over a two-sample horizon and applies its first element, with the reference gain corrected so that the closed loop retains unity DC gain:

\[ v'[k] = \frac{\left((A_d^i)^2 - A_d^i + 1\right) i^* - (A_d^i)^3\, i[k]}{B_d^i\left((A_d^i)^2 + 1\right)} \]

This reduces the state gain — and hence noise amplification — by roughly half.

Receding-horizon behaviour: because only the first element of the sequence is applied and the solution is recomputed every sample, the two-step law does not settle in exactly two samples. The closed-loop pole is \(A_d^i/\left((A_d^i)^2+1\right)\), giving a geometric rather than deadbeat response. Tracking is nonetheless exact in steady state: the reference gain is obtained by solving

\[ B_d^i\, g_{ref} = 1 - A_d^i + B_d^i\, g_{state} \]

which forces unity DC gain. Taking the raw minimum-norm reference gain instead leaves a static error of \(A_d^i/\left((A_d^i)^2 - A_d^i + 1\right)\) — around 1% at \(A_d^i = 0.90\) and 20% at \(0.61\) — which is worst in the small-\(L_s\) regime the variant targets.

The one-step law is unaffected — it places the closed-loop pole at the origin and tracks exactly.

Model accuracy requirement: A 10% \(L_s\) error leaves a 10% residual error after one step. Deadbeat is the natural progression after Decoupled PID has validated RLS convergence.

Decoupling: the inversion above solves the decoupled plant, so the A1 feedforward terms are added to \(v'\) before the modulation-circle limit. Without them \(\omega_e L_s i_q\) enters as an unmodelled disturbance and the one-step settling property is lost at speed.

Normalisation: \(v'[k]\) (physical volts) normalised identically to the PI output: \(v'_{norm}[k] = v'[k] \cdot \sqrt{3}/V_{dc}\).

Tuning: None — fully determined by \(A_d^i, B_d^i\) from RLS. Only design choice: one-step vs. two-step variant.

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. Deadbeat has no integral action, so it is unaffected by windup at the voltage limit.

Numerical Properties

Limitations & Assumptions

• Most sensitive to \(L_s\) error of all current controllers. Do not activate until electrical RLS has converged (typically after the first few seconds of operation under load).
• The two-step variant halves noise amplification and still tracks exactly, but gives up deadbeat settling for a geometric response with pole \(A_d^i/\left((A_d^i)^2+1\right)\). Prefer it for noisy current measurement or small \(L_s\); prefer one-step for maximum servo stiffness.
• 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.