FOC Plant Models — Discretization and State-Space Foundations

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

This document derives the discrete-time state-space plant models used by all advanced FOC controllers. It is a shared prerequisite for: - the current, speed and position loop controllers indexed in documentation/theory/advanced-controllers.md

The reader is assumed familiar with Clarke/Park transforms, SVM, and PI current control as described in documentation/theory/foc.md.

Prerequisites

Mathematical Foundation

1. PMSM Current Loop Plant (dq Frame)

The PMSM voltage equations in the rotor-synchronous dq frame (from documentation/theory/foc.md Section 4), assuming a surface-mounted motor with \(L_d = L_q = L_s\):

\[ v_d = R_s i_d + L_s \frac{di_d}{dt} - \omega_e L_s i_q \] \[ v_q = R_s i_q + L_s \frac{di_q}{dt} + \omega_e L_s i_d + \omega_e \psi_f \]

Rearranged as explicit state equations:

\[ \frac{di_d}{dt} = -\frac{R_s}{L_s} i_d + \omega_e i_q + \frac{1}{L_s} v_d \] \[ \frac{di_q}{dt} = -\frac{R_s}{L_s} i_q - \omega_e i_d - \frac{\psi_f \omega_e}{L_s} + \frac{1}{L_s} v_q \]

The cross-coupling terms \(+\omega_e i_q\) (on d) and \(-\omega_e i_d\) (on q), and the back-EMF disturbance \(-\psi_f \omega_e / L_s\) (on q), prevent the two axes from being treated as independent RL plants. Advanced current controllers either cancel these terms via feedforward or treat them as bounded disturbances.

Decoupled Per-Axis Plant

After applying the feedforward voltages defined in the current-loop controllers document, each axis reduces to an independent first-order RL system with decoupled input \(v'\):

\[ \frac{di}{dt} = -\frac{R_s}{L_s}\, i + \frac{1}{L_s}\, v' \]

Discrete-Time Current Plant

ZOH discretization at \(T_s^i\) gives per-axis matrices:

\[ A_d^i = e^{-R_s T_s^i / L_s}, \qquad B_d^i = \frac{1 - A_d^i}{R_s} \]

This exact pair is what the implementation uses, evaluated with a real exponential — not the Euler approximation below.

For \(R_s T_s^i / L_s \ll 1\) (typical: \(< 0.1\)), the Euler approximation is accurate:

\[ A_d^i \approx 1 - \frac{R_s T_s^i}{L_s}, \qquad B_d^i \approx \frac{T_s^i}{L_s} \]

Note that \(B_d^i\) must be taken from the same discretization as \(A_d^i\). Pairing the exact \(A_d^i = e^{-R_s T_s^i/L_s}\) with the Euler \(B_d^i = T_s^i/L_s\) is not a ZOH model of anything; the matching exact input matrix is \((1 - A_d^i)/R_s\).

The per-axis discrete plant:

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

RLS estimates \(\hat{R}_s\) and \(\hat{L}_s\) are substituted at configuration time.

2. Mechanical (Speed) Plant

The mechanical rotor dynamics driven by electromagnetic torque:

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

The simulated plant splits \(T_L\) into two terms. A load opposes motion whatever its direction, so it changes sign with \(\omega_m\) and vanishes at rest, like friction. An external torque keeps its sign whatever the rotor does, like gravity on an arm or a step applied by a test to disturb the loop, and it is the term a position controller has to hold against. Both are additive on the right-hand side:

\[ T_L = T_{load} \cdot \operatorname{sgn}(\omega_m) + T_{ext} \]

Treating the current loop as ideal (\(i_q \approx i_q^*\)), the outer-loop control input is \(u = i_q^*\):

\[ \frac{d\omega_m}{dt} = -\frac{B_f}{J} \omega_m + \frac{K_t}{J} u - \frac{T_L}{J} \]

This is a first-order plant (state: \(\omega_m\), input: \(i_q^*\), disturbance: \(T_L/J\)).

Discretization at \(T_s^o\). The exact ZOH pair is:

\[ A_d^o = e^{-B_f T_s^o / J}, \qquad B_d^o = \frac{K_t}{B_f}\left(1 - A_d^o\right) \]

The implementation uses the forward-Euler pair instead:

\[ \boxed{A_d^o = 1 - \frac{B_f T_s^o}{J}, \qquad B_d^o = \frac{K_t T_s^o}{J}} \]

Both matrices come from the same approximation, which is the part that matters: the exact \(A_d^o\) paired with the Euler \(B_d^o\) would be a model of no system at all. The Euler pair is valid while \(B_f T_s^o / J \ll 1\), which holds comfortably at 1 kHz for the inertias and frictions the mechanical identification returns, and unlike the exact pair it stays defined for a frictionless load, where \(K_t (1 - A_d^o) / B_f\) is \(0/0\).

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

RLS estimates \(\hat{J}\) and \(\hat{B}_f\) from documentation/theory/friction-inertia-estimation.md are substituted at configuration time.

3. Position Plant (Double Integrator)

Adding the kinematic relation \(\dot{\theta}_m = \omega_m\) yields a two-state plant:

\[ \begin{pmatrix} \dot{\theta}_m \\ \dot{\omega}_m \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 0 & -B_f/J \end{pmatrix} \begin{pmatrix} \theta_m \\ \omega_m \end{pmatrix} + \begin{pmatrix} 0 \\ K_t/J \end{pmatrix} u \]

Discretized at \(T_s^o\) with the same forward-Euler \(A_d^o\) and \(B_d^o\) as the speed plant:

\[ A_d^p = \begin{pmatrix} 1 & T_s^o \\ 0 & A_d^o \end{pmatrix}, \qquad B_d^p = \begin{pmatrix} 0 \\ B_d^o \end{pmatrix} \]

The \([1,\, T_s^o]\) row approximation holds when \(B_f T_s^o / J \ll 1\). For high-friction systems, the exact matrix exponential should be used.

Block Diagrams

Current Loop Plant Structure

Speed and Position Plant Cascade

Numerical Properties

Reference Motors

The plant model is exercised, in the unit tests, the simulator and the software-in-the-loop suite, with two motors whose parameters are published by their vendors and by the evaluation kits built around them. The values below are per phase for a wye winding, which is what the model and the identification services use. The headers under motor_parameters/ carry exactly these numbers.

How the numbers were obtained:

• Line-to-line to per-phase. Datasheets quote terminal values; for a wye winding \(R_s = R_{LL}/2\) and \(L_s = L_{LL}/2\). The Teknic sheet gives 0.72 Ω and 0.40 mH line-to-line. The Anaheim sheet gives 1.20 mH line-to-line (0.60 mH per phase); the 0.64 mH and 0.405 Ω in the header are TI’s InstaSPIN identification of the same motor, which agrees with the sheet to within the spread expected between a six-step datasheet rating and a sinusoidal measurement.
• Flux linkage from back-EMF. With a line-to-line peak constant \(K_{e}\) per 1000 rpm (\(\omega_m = 104.7\) rad/s, \(\omega_e = p\,\omega_m\)): \(\psi_f = K_e / (\sqrt{3}\,\omega_e)\). Teknic: \(4.64 / (\sqrt{3} \cdot 418.9) = 6.40\) mWb. TI reports the Anaheim flux as \(0.03416\) V/Hz, a peak phase voltage per electrical hertz, so \(\psi_f = 0.03416 / 2\pi = 5.44\) mWb.
• Inertia. \(1\ \text{oz·in·s}^2 = 7.0616 \times 10^{-3}\ \text{kg·m}^2\): 0.00068 oz·in·s² (Anaheim) and 0.001 oz·in·s² (Teknic).
• Viscous friction. Neither vendor publishes a viscous term or a no-load current to derive one from. The values are chosen so that friction torque at rated speed is about 3 % of rated torque (4.2 mN·m at 419 rad/s for the Anaheim, 9.4 mN·m at 628 rad/s for the Teknic), and the identification scenarios treat them as plant truth, not as a claim about the real machines.
• Drive rating. maxSupportedCurrent is the inverter’s limit, not the motor’s: 10 A and 20 A are what a small drive for each class supports and what the injection-limit check in every calibration procedure compares against. The 15 % DC resistance test draws about 5.5 A (Anaheim, 24 V) and 10 A (Teknic, 40 V), inside both ratings.

The two motors bracket the plant in the directions that matter to the estimators: the Teknic has the shorter electrical time constant (11 control ticks at 20 kHz, still well inside the Euler stability region), and its \(R_s T_s / L_s\) of 0.18 at the 10 kHz identification rate makes the ZOH bias of the inductance estimator visible (see resistance-inductance-estimation.md), where the Anaheim’s 0.063 keeps it small.

Sources:

• Anaheim Automation, BLY17 Series Product Sheet (L010228) and the BLY172S-24V-4000 product page: 24 V, 4000 rpm, 53 W, 18 oz·in, 1.20 mH line-to-line, 3.35 V/krpm, 0.00068 oz·in·s².
• Texas Instruments MotorWare user.h, motor Anaheim_BLY172S: \(R_s = 0.4051\) Ω, \(L_s = 0.6399\) mH, rated flux 0.03416 V/Hz, 4 pole pairs.
• Teknic, Industrial-Grade NEMA 23 Motors datasheet (N23_Industrial_Grade_Motors_v6.3.pdf): M-2310P-LN-04K, 40 V, 6000 rpm, 0.72 Ω and 0.40 mH line-to-line, 4.64 Vpk/krpm, 0.001 oz·in·s².
• NXP, MCUXpresso SDK FOC user guide, Teknic motor, which restates the same electrical values for the FRDM-MC kits, and the TI LVSERVOMTR page for the 2MTR-DYNO.

References

1. Krishnan, R. — Permanent Magnet Synchronous and Brushless DC Motor Drives, CRC Press, 2010.
2. Åström, K.J. & Wittenmark, B. — Computer-Controlled Systems: Theory and Design, 3rd ed., Prentice Hall, 1997. (Chapter 3: ZOH discretization.)