Position Loop — P4: Iterative Learning Control (ILC)

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.

Not implemented. No such algorithm exists in the firmware: foc::PositionAlgorithm offers pid, cascadeP, lqr, lqi and twoDof, and nothing else is selectable over CAN or the CLI. This chapter is design groundwork for a law that has not been built, kept for the theory rather than as a description of shipped behaviour. Read it as a proposal, not as a specification.

Overview

P4 (ILC — Iterative Learning Control) is for repetitive servo tasks — pick-and-place, CNC passes, inspection cycles. The same position tracking error recurs on every trial because the same reference is executed against the same mechanical dynamics. ILC exploits this: it learns the per-sample feedforward correction that cancels the repeating error over successive trials until the residual is near the sensor noise floor. No other controller achieves this tracking accuracy for periodic tasks.

ILC does not replace the feedback controller — it augments any stable position feedback controller (P1, P2, or P3). Operates exclusively in the 1 kHz outer handler.

Prerequisites

See documentation/theory/foc-plant-models.md §3 for the position plant derivation.

Mathematical Foundation

All position-loop controllers operate on the discrete two-state position plant from documentation/theory/foc-plant-models.md §3. P4 augments any stable feedback controller with a per-sample learned feedforward. Friction compensation adds a nonlinear Iq correction outside the linear plant model.

P4 — Iterative Learning Control (ILC)

D-type update law: Let \(e^{(j)}[k] = \theta_m^*[k] - \theta_m^{(j)}[k]\) be the position error on trial \(j\) at sample \(k\):

\[ \boxed{u_{ILC}^{(j+1)}[k] = Q \cdot \Bigl(u_{ILC}^{(j)}[k] + \ell \cdot e^{(j)}[k+1]\Bigr)} \]

where: - \(Q \in (0,\,1]\): robustness filter. \(Q < 1\) suppresses high-frequency noise at the cost of slower convergence. Typical: \(Q = 0.95\). - \(\ell > 0\): learning gain. Convergence guaranteed for \(\ell \in (0,\,2)\) on a unity-gain plant.

Total control signal: Learned correction added as feedforward to any position feedback controller:

\[ u_{total}^{(j)}[k] = u_{fb}^{(j)}[k] + u_{ILC}^{(j)}[k] \]

\(u_{fb}\) is the output of the active position feedback controller (P2, P1, or P3). ILC does not replace the feedback controller — it augments it.

Storage: One full trial of \(N\) samples stored in a bounded array. At 1 kHz, a 5-second trial requires \(N = 5000\) floats (20 kB). \(N\) is fixed at build time; ILC is the only position algorithm whose storage scales with trial duration.

Convergence condition: \(|Q(1 - \ell\,P_{nom})| < 1\) across all frequencies, where \(P_{nom}\) is the nominal position plant. Guaranteed by \(Q < 1\) and \(\ell\) within its stability range.

Tuning: - \(N\): trial length in samples (must exactly match the repetitive task period). - \(Q\): robustness vs. convergence speed tradeoff. - \(\ell\): learning rate. Start at \(0.5\), increase toward \(1.0\).

Numerical Properties

Limitations & Assumptions

• Only valid for strictly repetitive references. A changed reference period or non-repeatable reference (jogging) will cause the learned correction from the previous trial to degrade tracking.
• Trial length \(N\) is fixed at selection time. It cannot change at runtime.
• ILC requires a stable inner feedback controller (P1, P2, or P3). ILC alone does not stabilise position — it only improves tracking of the feedback loop.
• Encoder noise enters position and velocity estimates. A state observer upstream improves robustness.

References

1. Bristow, D.A., Tharayil, M. & Alleyne, A.G. — “A Survey of Iterative Learning Control”, IEEE Control Systems Magazine, 26(3):96–114, 2006.
2. Åström, K.J. & Wittenmark, B. — Computer-Controlled Systems, 3rd ed., Prentice Hall, 1997.