arxiv: 2604.16908 · v1 · submitted 2026-04-18 · 📡 eess.SY · cs.SY

Recognition: unknown

End-to-End ILC for Repetitive Untrackable Tasks: A Cooperative Game Perspective

Hongfeng Tao, Rodrigo A. Gonz\'alez, Tom Oomen, Wojciech Paszke, Zhihe Zhuang

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Pith reviewed 2026-05-10 07:12 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords iterative learning controlcooperative gameuntrackable tasksend-to-end ILCnorm-optimal ILCrepetitive controlclosed-loop systems

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The pith

For untrackable repetitive tasks, jointly updating both reference and ILC feedforward input yields lower cost than standard norm-optimal ILC under a sufficient condition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Standard iterative learning control assumes a perfect tracking input exists, yet many practical tasks are untrackable because of inherent system limits. The paper develops an end-to-end ILC scheme that updates the reference signal and the feedforward input together from trial to trial using only measurement data. By casting the resulting closed-loop behavior as a two-player cooperative game whose players share aligned objectives, the authors derive a sufficient condition guaranteeing that this joint-update strategy produces a lower cost than conventional one-player norm-optimal ILC. The approach is illustrated on a numerical example of a repetitive untrackable task. Readers should care because it offers a concrete way to improve performance precisely when perfect tracking is impossible.

Core claim

The paper discovers a sufficient condition under which the two-player end-to-end ILC has a lower cost than the one-player norm-optimal ILC for repetitive untrackable tasks. The reference input and the ILC feedforward input are updated simultaneously on the basis of measurement data, and the closed-loop dynamics are analyzed as a cooperative game in which the two updaters act as players with aligned objectives.

What carries the argument

The cooperative-game representation of the closed-loop ILC dynamics in which the reference updater and the feedforward updater are treated as players whose objectives are aligned.

Load-bearing premise

The closed-loop ILC dynamics admit a cooperative-game representation in which the reference updater and feedforward updater have aligned objectives, and the derived sufficient condition remains valid under real disturbances and model mismatch.

What would settle it

A simulation or experiment in which the stated sufficient condition holds yet the measured cost of the two-player end-to-end ILC exceeds that of the one-player norm-optimal ILC.

Figures

Figures reproduced from arXiv: 2604.16908 by Hongfeng Tao, Rodrigo A. Gonz\'alez, Tom Oomen, Wojciech Paszke, Zhihe Zhuang.

**Figure 2.** Figure 2: The 30th tracking outputs of the end-to-end ILC [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Cost convergence of the end-to-end ILC compared [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

read the original abstract

An inherent assumption of perfect tracking in iterative learning control (ILC) is that there exists an ILC input such that the generated output can track the desired trajectory reference. This assumption may fail in practice, which gives rise to desired but untrackable tasks. This paper gives an end-to-end ILC design for repetitive untrackable tasks in closed-loop systems. The reference input is trial-to-trial updated together with the ILC feedforward input based on the measurement data. This two-player behavior of the closed-loop ILC system is investigated from a cooperative game perspective. A sufficient condition for the two-player end-to-end ILC to have a lower cost than the one-player norm optimal ILC (NOILC) is discovered. Finally, a numerical example is given to verify the effectiveness of the developed method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper frames joint reference and feedforward updates in ILC as a cooperative game for untrackable tasks and claims a sufficient condition for lower cost than standard NOILC, but the edge appears limited to nominal no-disturbance cases.

read the letter

The main takeaway is that by viewing the reference updater and feedforward updater as cooperative players, the authors derive a sufficient condition under which their two-player end-to-end ILC achieves lower cost than one-player norm-optimal ILC for repetitive tasks that cannot be perfectly tracked. This is the concrete new angle they bring to the table. Standard ILC literature assumes perfect trackability, so relaxing that by jointly tuning the reference from data is a practical step forward. The numerical example is presented to confirm the cost reduction holds when the condition is satisfied. They earn credit for directly addressing a common real-world shortfall in ILC applications and for keeping the design data-driven and closed-loop. The soft spot is that the sufficient condition is derived under the usual ILC assumptions of perfect model knowledge and zero disturbances. Once model mismatch or external disturbances enter, the strict cost advantage is likely to vanish, and the example does not test sensitivity to those factors. The abstract gives no proof sketch, so the full paper needs to show whether the condition is robust or merely holds in the ideal case. This work is aimed at ILC researchers in systems and control who face untrackable repetitive tasks. A reader interested in game-theoretic reformulations of iterative methods would find value in the framing. The claim is specific and the example provides a starting point for verification, so the paper deserves a serious referee rather than a desk reject. I would recommend sending it out for peer review.

Referee Report

2 major / 1 minor

Summary. The paper proposes an end-to-end ILC design for repetitive untrackable tasks in closed-loop systems. Both the reference input and ILC feedforward input are updated trial-to-trial from measurement data. The closed-loop behavior is recast as a two-player cooperative game with aligned objectives, yielding a sufficient condition under which this two-player scheme achieves strictly lower cost than standard one-player norm-optimal ILC (NOILC). A numerical example is included to illustrate effectiveness.

Significance. If the sufficient condition is rigorously derived and the cooperative-game representation is faithful, the work supplies a principled way to handle untrackable references by jointly optimizing reference and feedforward updates. The game-theoretic framing is novel within ILC and could generalize to other multi-component learning schemes. The numerical example provides concrete verification under nominal conditions, which is a positive step toward reproducibility.

major comments (2)

[Main Results / Cooperative Game Formulation] The derivation of the sufficient condition (presumably the central theorem in the main-results section) is presented under the standard ILC assumptions of perfect model knowledge and zero disturbances. The manuscript must explicitly state whether the inequality remains valid under bounded model mismatch or additive disturbances; if the proof relies on exact cancellation of the plant inverse, the condition is not load-bearing for practical untrackable tasks.
[Numerical Example] The numerical example only demonstrates the nominal case. To support the claim that the two-player scheme outperforms NOILC, the example should include at least one run with realistic disturbances or parametric uncertainty and report the realized cost ratio relative to the derived sufficient-condition threshold.

minor comments (1)

[Abstract / Problem Formulation] Notation for the two players' cost functions and the joint objective should be introduced once and used consistently; the current abstract leaves the precise definition of 'aligned objectives' implicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the major comments point by point below, agreeing to revisions that clarify the scope of the results and strengthen the numerical validation while preserving the nominal theoretical contribution.

read point-by-point responses

Referee: [Main Results / Cooperative Game Formulation] The derivation of the sufficient condition (presumably the central theorem in the main-results section) is presented under the standard ILC assumptions of perfect model knowledge and zero disturbances. The manuscript must explicitly state whether the inequality remains valid under bounded model mismatch or additive disturbances; if the proof relies on exact cancellation of the plant inverse, the condition is not load-bearing for practical untrackable tasks.

Authors: We agree that the sufficient condition is derived under the standard assumptions of perfect model knowledge and zero disturbances. The cooperative-game recasting and the proof of the cost inequality use the nominal lifted-system representation of the closed-loop dynamics; this does involve operations equivalent to the nominal plant inverse in the lifted domain. Consequently, we cannot assert that the inequality holds under bounded model mismatch or additive disturbances without further analysis. In the revised manuscript we will explicitly state these assumptions in the theorem and add a dedicated remark clarifying that the result is nominal, while noting that robustness extensions constitute future work. This accurately delimits the contribution without overstating its practical scope. revision: yes
Referee: [Numerical Example] The numerical example only demonstrates the nominal case. To support the claim that the two-player scheme outperforms NOILC, the example should include at least one run with realistic disturbances or parametric uncertainty and report the realized cost ratio relative to the derived sufficient-condition threshold.

Authors: We acknowledge that the present numerical example is confined to the nominal case. To address the concern, the revised manuscript will incorporate an additional simulation that includes bounded additive disturbances and parametric uncertainty. We will report the realized costs for both the proposed end-to-end ILC and standard NOILC, compute the cost ratio, and compare it against the threshold supplied by the sufficient condition. This will provide concrete evidence of relative performance beyond the ideal setting. revision: yes

Circularity Check

0 steps flagged

No circularity: sufficient condition derived from independent game reformulation

full rationale

The visible abstract and reader's summary indicate that the central result is a sufficient condition obtained after recasting closed-loop ILC dynamics as a cooperative game with aligned player objectives. No equations, fitted parameters, or self-citations are shown that would make the inequality hold by construction or reduce the prediction to the input data. The derivation therefore retains independent mathematical content under the stated nominal assumptions, consistent with a self-contained analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the modeling choice that the reference and ILC updates constitute a cooperative two-player game whose cost can be compared directly to single-player NOILC.

axioms (1)

domain assumption The closed-loop ILC system admits a cooperative-game representation between reference updater and feedforward updater
Invoked to justify the two-player formulation and the cost comparison.

pith-pipeline@v0.9.0 · 5448 in / 1141 out tokens · 43297 ms · 2026-05-10T07:12:38.038458+00:00 · methodology

discussion (0)

Reference graph

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