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Algorithm overview

This note maps the code to the method in the paper (references.bib → glac2018optimal). For the full derivation see the paper and its reference [6] (eldeeb2017unified).

Problem

For a synchronous machine in the rotor-oriented dq frame, find the stator current setpoint (i_sd*, i_sq*) that produces a requested torque m_ref while

  • minimising stator Joule losses (maximum torque per current, MTPC/MTPA),
  • respecting the current limit |i_s| ≤ I_sfn, and
  • respecting the voltage limit |u_s| ≤ Û set by the DC link.

Each of these conditions is a quadric (conic) in the (i_sd, i_sd) plane. The optimal setpoint always lies on the intersection of two of them; which two depends on the operating regime (below/above base speed, current- or voltage-limited, MTPV).

The five quadrics

All quadrics use the form Q(d,q) = x11·d² + 2·x12·d·q + x22·q² + 2·x1·d + 2·x2·q + x0 = 0 in per-unit currents (d = i_sd/I_sfn). They are assembled in FFIS_matrixInit_Par (speed-independent) and FFIS_matrixInit_M (speed/torque dependent):

Quadric Symbol in code Meaning
MTPC / MTPA hyperbola FFIS_MTpC max torque per current
maximum current circle FFIS_MCurr `
torque hyperbola FFIS_NTorq constant torque = m_ref
voltage ellipse FFIS_NVolt `
MTPV hyperbola FFIS_MTpV max torque per voltage (flux)

Online regime selection

The key contribution of the paper is that the regime is chosen online, every control step, rather than from a precomputed speed schedule. This lets the controller react immediately to changing parameters (saturation, thermal drift). In FFIS_optimal_control:

  1. Re-normalise the (possibly time-varying) parameters and the operating point (Normalize_Par, Normalize_M) and rebuild the quadric coefficients.
  2. Intersect the relevant quadric pairs with FFIS_intersection, which reduces the problem to a single quartic and solves it with the poly34f solver.
  3. Compare the requested torque against the torque available on the voltage ellipse (Mintersect) and the maximum feasible torque (Mmaxfeas) to pick the operating regime, then return the setpoint on the appropriate intersection.

Numerical considerations (paper Sec. IV)

The implementation targets a single-precision DSP, so it normalises everything to per-unit to keep magnitudes ≈ O(1), and:

  • uses an experimentally tuned tolerance ZERO_TOL when deciding how to scale the pencil of quadrics (paper's ε);
  • clamps the speed used in the voltage ellipse below a threshold ω_min = 0.8·ω_N to avoid an ill-conditioned ellipse at low speed;
  • (in this port) guards the cube-root helper against non-finite arguments and falls back to a zero reference if a degenerate input would otherwise yield a non-finite setpoint. See the source comments and THIRD_PARTY_NOTICES.md.

Differences from the originally archived firmware source

This repository fixes a few issues so the code builds and runs correctly on a host PC; each is commented at the site in the source:

  • Torque hyperbola scaling — NTorq.x0 is normalised by Usfn·Isfn (the per-unit current convention), not Usfn/Isfn.
  • Per-cycle re-normalisation — optimal_control now calls Normalize_Par, so the controller reacts to the commanded speed each step (without it wk_N kept its init-time value).
  • Low-speed clamp — a stray ; after the clamp condition (which made the clamp unconditional) was removed.
  • Robustness — the root3 infinite-loop guard and the non-finite fallback described above.