Current Loop — A3: Sliding-Mode

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

A3 (Sliding-Mode) is robust to model uncertainty — the most critical early in operation before RLS has converged and during thermal transients that shift \(R_s\). The SMC drives the current error onto a stable sliding surface regardless of parameter mismatch, within a defined gain margin. Unlike A1 and A2, A3 deliberately omits the A1 feedforward and treats residual coupling and noise as a bounded disturbance covered by the switching gain \(K_{sw}\).

Operates exclusively in the 20 kHz FOC ISR. Partial RLS dependency — \(A_d^i\), \(B_d^i\) are used but the controller remains stable over a wide range of mismatch.

Prerequisites

See documentation/theory/foc-plant-models.md for ZOH derivations.

Mathematical Foundation

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

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

A3 deliberately omits the A1 feedforward and treats the residual coupling and noise as a bounded disturbance covered by the switching gain \(K_{sw}\); that robustness is the reason to select it before RLS has converged.

A3 — Sliding-Mode Current Control

Sliding surface: For each axis independently, defined on the tracking error \(e = i - i^*\):

\[ s_d = i_d - i_d^*, \qquad s_q = i_q - i_q^* \]

The surface \(s = 0\) is the desired zero-error manifold.

Equivalent control and the reaching law: The error propagates as \(e[k+1] = A_d^i e[k] + B_d^i u[k] + (A_d^i - 1) i^*[k]\). The law chooses its voltage so that the error obeys the exponential reaching law

\[ e[k+1] = \lambda\, e[k] - K_{sw}\,\mathrm{sat}\!\left(\frac{e[k]}{\phi}\right), \qquad \lambda = e^{-\omega_{bw} T_s} \]

which needs

\[ \boxed{u[k] = -\frac{A_d^i - \lambda}{B_d^i}\, e[k] - \frac{K_{sw}}{B_d^i}\,\mathrm{sat}\!\left(\frac{e[k]}{\phi}\right) + \frac{1 - A_d^i}{B_d^i}\, i^*[k]} \]

where \(\mathrm{sat}(x) = \mathrm{clamp}(x,-1,1)\). Because \(B_d^i = (1 - A_d^i)/R_s\), the last term is exactly \(R_s i^*[k]\) — the resistive drop the reference current requires. All terms carry a leading minus sign on the error because it is measured as \(i - i^*\): a current below its reference gives \(e < 0\) and must command a positive voltage. The boundary layer \(\phi > 0\) replaces the discontinuous sign function with saturation to prevent chattering.

The toolbox’s SlidingModeControl computes its equivalent term as \(-(S B)^{-1} S A\, e\), i.e. it cancels whatever the state matrix it is given predicts. It is handed \(A_d^i - \lambda\), not \(A_d^i\). Handing it \(A_d^i\) — the form this law shipped with — makes the equivalent term deadbeat: it drives the next error to zero on its own, and the switching term then pushes the error \(K_{sw}\) past zero every time it leaves the boundary layer. From rest a step of \(\phi\) overshot by \(K_{sw}/\phi\), 40 % with the shipped defaults, and oscillated with ratio \(-K_{sw}/\phi\) after that; the SIL measured 46–50 % on a 0.5 A step.

Parameter choices: - \(\lambda\) follows the current-loop bandwidth, so large steps approach the surface at the loop’s own rate rather than at the fixed \(K_{sw}\) per sample. - \(K_{sw}\): must exceed the worst-case disturbance. Because \(u_{sw}\) is obtained by dividing by \(B_d^i\), the gain carries the same unit as the sliding surface (A). Sizing it so the switching term commands at most 30% of the maximum phase voltage \(V_{dc}/\sqrt{3}\) gives the starting point: \(K_{sw} = 0.3 \cdot B_d^i \cdot V_{dc} / \sqrt{3}\). - \(\phi\): boundary layer width (A). Typical: \(0.1\)–\(0.5\) A. Smaller gives tighter tracking but more high-frequency actuation.

Discrete stability constraint — this bounds the parameters together and is not optional. Inside the boundary layer the error obeys \(e[k+1] = (\lambda - K_{sw}/\phi)\, e[k]\). It contracts monotonically, without crossing the surface, when \(0 \le \lambda - K_{sw}/\phi < 1\), i.e. \(K_{sw} \le \lambda\,\phi\); it still converges, alternating in sign, while \(K_{sw} < (1 + \lambda)\,\phi\). The shipped defaults are \(K_{sw} = 0.2\) A and \(\phi = 0.5\) A; at the default bandwidth \(\lambda \approx 0.73\), so the ratio \(0.4\) leaves a monotonic approach.

Robustness: Insensitive to \(R_s/L_s\) mismatch as long as mismatch is bounded by \(K_{sw}\). Covers \(\pm 50\%\) thermal variation in \(R_s\) and the initial RLS convergence transient.

Relation to toolbox: SlidingModeControl<float,1,1> implements equivalent + switching with boundary-layer saturation on the error alone; the equilibrium term \(R_s i^*\) is added by SlidingModeCurrentController. Plant matrices \(A_d^i, B_d^i\) are constructed from RLS estimates.

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. A3 has no integral action, so it is unaffected by windup at the voltage limit.

Numerical Properties

Limitations & Assumptions

• Boundary layer \(\phi\) introduces a steady-state band proportional to \(\phi / K_{sw}\). Set \(\phi\) small enough that this band is below ADC resolution.
• \(K_{sw}\) must exceed the worst-case disturbance; undersizing prevents reaching the sliding surface.
• Stability constraint: \(K_{sw} < \phi\) is mandatory for discrete convergence.
• 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. Utkin, V. — Sliding Modes in Control and Optimization, Springer, 1992. (Equivalent control; boundary-layer chattering suppression.)
3. Åström, K.J. & Wittenmark, B. — Computer-Controlled Systems, 3rd ed., Prentice Hall, 1997.