On the Solvability of Quasi-Regulator Equations in Non-smooth Output Regulation

Daniele Astolfi; Giordano Scarciotti; Zirui Niu

REVIEW 1 major objections 1 minor 3 references

On the Solvability of Quasi-Regulator Equations in Non-smooth Output Regulation

T0 review · 1 major / 1 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read A non-smooth non-resonance condition is necessary and sufficient for solvability of quasi-regulator equations when relative degree requirements hold.

desk verdict The paper claims a necessary and sufficient non-smooth non-resonance condition for quasi-regulator equations via DAE reformulation, but the abstract supplies no proof steps or examples. read the letter →

arxiv 2605.28665 v1 pith:HZSF534E submitted 2026-05-27 eess.SY cs.SYmath.OC

Zirui Niu , Daniele Astolfi , Giordano Scarciotti This is my paper · ORCID

classification eess.SYcs.SYmath.OC

keywords quasi-regulatorequationsnon-smoothnon-resonanceconditionoutputregulationdifferential-algebraicrelativedegreelinearsystemssolvabilitynon-periodicsignals

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper determines when the quasi-regulator equations admit solutions for linear systems driven by non-smooth non-periodic exogenous signals. It rewrites those equations as differential-algebraic equations and shows that the system's relative degree plays a decisive role in solvability. The central result is a non-smooth non-resonance condition that gives an exact if-and-only-if test under stated relative-degree restrictions. This test matters because the output regulation problem for irregular signals cannot be solved unless the equations themselves are solvable. Without such a test, regulators for real-world non-smooth references cannot be reliably designed.

What carries the argument

The non-smooth non-resonance condition obtained from the differential-algebraic reformulation of the quasi-regulator equations.

What would settle it

A concrete linear system meeting the relative-degree requirements for which the non-smooth non-resonance condition holds yet the quasi-regulator equations have no solution, or the reverse.

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Extended reading notes

Core claim

We reformulate the quasi-regulator equations as differential-algebraic equations and propose a non-smooth non-resonance condition that, under specific relative degree requirements, provides a necessary and sufficient characterization of the solvability of the quasi-regulator equations.

Load-bearing premise

The reformulation of the quasi-regulator equations as differential-algebraic equations is valid and the relative degree satisfies the requirements that make the non-smooth non-resonance condition necessary and sufficient.

Editorial extensions

If this is right

The quasi-regulator equations possess solutions precisely when the non-smooth non-resonance condition is satisfied, provided the relative-degree requirements are met.
Output regulation for non-smooth non-periodic signals is possible exactly when this solvability condition holds.
The approach supplies an algebraic test that can be checked before attempting to construct a regulator.
The characterization applies to linear systems whose relative degree meets the paper's stated requirements.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same differential-algebraic view might be used to derive analogous conditions for systems whose relative degree falls outside the stated requirements.
Numerical verification of the condition on given state-space matrices would turn the theoretical test into a practical design step.
The result opens a route to compare solvability across different classes of non-smooth signals by varying only the exogenous-signal generator.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The core contribution is a characterization of when the quasi-regulator equations admit solutions for linear systems driven by non-smooth exogenous signals. They recast the equations as a differential-algebraic system and tie solvability to a new non-smooth non-resonance condition once the plant meets stated relative-degree requirements. That condition is presented as the main novelty.

The approach builds directly on classical output-regulation techniques. Rewriting the regulator equations as DAEs is a reasonable move when the signals lose smoothness, and the emphasis on relative degree aligns with how that quantity governs the existence of feedforward solutions in the smooth case. If the derivation holds, it gives a concrete test that practitioners could check without solving the full regulator problem first.

The soft spot is the absence of any derivation, explicit statement of the condition, or verification that the DAE form is equivalent to the original equations. The abstract asserts necessity and sufficiency under the relative-degree requirements, yet supplies no steps to confirm that all solutions are preserved or that no extra restrictions appear. Without those details it is difficult to assess whether the result is as general as claimed or whether counter-examples exist outside the stated assumptions.

The work targets researchers already working on output regulation for linear systems. A reader comfortable with Francis equations and DAE theory will follow the argument without much extra background. It is worth sending to peer review because the claim is precise and sits inside an established subfield; a referee can check the missing steps and either confirm or tighten the conditions.

Referee Report

1 major / 1 minor

Summary. The paper addresses solvability of quasi-regulator equations arising in output regulation of linear systems driven by non-smooth, non-periodic exogenous signals. It reformulates the equations as differential-algebraic equations, stresses the role of relative degree, and proposes a non-smooth non-resonance condition that is asserted to be necessary and sufficient for solvability under specific relative-degree requirements.

Significance. If the claimed necessary-and-sufficient characterization holds, the work would supply a structural criterion for existence of solutions to the quasi-regulator equations, extending classical output-regulation theory to a broader class of reference and disturbance signals that appear in applications. The DAE reformulation itself may also prove useful for subsequent analysis or numerical solution of the regulator equations.

major comments (1)

[Abstract] Abstract: the central claim that a non-smooth non-resonance condition furnishes a necessary and sufficient characterization is stated without any derivation steps, explicit statement of the condition, or the DAE system itself; this absence is load-bearing because the abstract supplies the only visible support for the result.

minor comments (1)

[Abstract] The abstract refers to 'specific relative degree requirements' without indicating what those requirements are; a brief parenthetical or footnote would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report and the opportunity to respond. Below we address the major comment point by point.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that a non-smooth non-resonance condition furnishes a necessary and sufficient characterization is stated without any derivation steps, explicit statement of the condition, or the DAE system itself; this absence is load-bearing because the abstract supplies the only visible support for the result.

Authors: Abstracts are concise summaries and, by standard academic practice, do not contain derivation steps or the full technical statements of conditions and systems; those appear in the body of the manuscript. The DAE reformulation of the quasi-regulator equations is introduced in Section II, the relative-degree requirements are stated there, and the non-smooth non-resonance condition together with its necessity-and-sufficiency proof is given in Section III. The abstract therefore accurately reflects the paper's central result without needing to embed these details. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The derivation rests on reformulating the quasi-regulator equations as DAEs and invoking standard facts about relative degree to obtain a necessary-and-sufficient non-smooth non-resonance condition. No equations, fitted parameters, or predictions are shown that reduce to the paper's own inputs by construction; the central claim is presented as a characterization derived from external DAE theory rather than from self-referential definitions or self-citations. The analysis is therefore self-contained against external mathematical benchmarks.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities. The solvability claim implicitly rests on standard facts about linear DAEs and relative degree that are not derived inside the paper.

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Cite this review

Pith. "Pith review of On the Solvability of Quasi-Regulator Equations in Non-smooth Output Regulation." pith.science (2026). https://pith.science/paper/HZSF534E

@misc{pith2026260528665,
  author       = {Pith},
  title        = {Pith review of: On the Solvability of Quasi-Regulator Equations in Non-smooth Output Regulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HZSF534E}},
  note         = {Machine review of arXiv:2605.28665}
}

read the original abstract

Motivated by the prevalence of non-smooth, possibly non-periodic signals in real-world applications, the output regulation of linear systems subject to non-smooth non-periodic exogenous signals has emerged as a challenging problem. A fundamental prerequisite for solving this problem is the existence of solutions to the so-called ``quasi-regulator equations''. In this paper, we investigate the solvability of these equations. To this end, we reformulate the quasi-regulator equations as differential-algebraic equations and highlight the critical role played by the system's relative degree. We finally propose a ``non-smooth non-resonance condition'' that, under specific relative degree requirements, provides a necessary and sufficient characterization of the solvability of the quasi-regulator equations.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

3 extracted references · 1 canonical work pages

[1]

(2010).Principles of discontinuous dynamical systems

Akhmet, M. (2010).Principles of discontinuous dynamical systems. Springer Science & Business Media. Bhatia, R. and Rosenthal, P. (1997). How and why to solve the operator equation AX - XB = Y.Bulletin of the London Mathematical Society, 29(1), 1–21. Byrnes, C.I. and Isidori, A. (2003). Limit sets, zero dy- namics, and internal models in the problem of non...

2010
[2]

IEEE. Chen, Z. and Huang, J. (2005). Robust output regulation with nonlinear exosystems.Automatica, 41(8), 1447–

2005
[3]

Cortes, J. (2008). Discontinuous dynamical systems.IEEE Control Systems Magazine, 28(3), 36–73. Davison, E. (1976). The robust control of a servomech- anism problem for linear time-invariant multivariable systems.IEEE Transactions on Automatic Control, 21(1), 25–34. Do Carmo, M.P. (1992).Riemannian geometry. Birkh¨ auser Boston. Francis, B.A. (1977). The ...

work page arXiv 2008

Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.

[1] [1]

(2010).Principles of discontinuous dynamical systems

Akhmet, M. (2010).Principles of discontinuous dynamical systems. Springer Science & Business Media. Bhatia, R. and Rosenthal, P. (1997). How and why to solve the operator equation AX - XB = Y.Bulletin of the London Mathematical Society, 29(1), 1–21. Byrnes, C.I. and Isidori, A. (2003). Limit sets, zero dy- namics, and internal models in the problem of non...

2010

[2] [2]

IEEE. Chen, Z. and Huang, J. (2005). Robust output regulation with nonlinear exosystems.Automatica, 41(8), 1447–

2005

[3] [3]

Cortes, J. (2008). Discontinuous dynamical systems.IEEE Control Systems Magazine, 28(3), 36–73. Davison, E. (1976). The robust control of a servomech- anism problem for linear time-invariant multivariable systems.IEEE Transactions on Automatic Control, 21(1), 25–34. Do Carmo, M.P. (1992).Riemannian geometry. Birkh¨ auser Boston. Francis, B.A. (1977). The ...

work page arXiv 2008