pith. sign in

arxiv: 2604.02510 · v1 · submitted 2026-04-02 · 🧮 math.DS

A Structurally Flat Triangular Form for Three-Input Systems

Georg Hartl , Conrad Gst\"ottner , Markus Sch\"oberl This is my paper

Pith reviewed 2026-05-13 20:15 UTC · model grok-4.3

classification 🧮 math.DS
keywords x-flat systemstriangular formstatic feedback equivalencecontrol-affine systemsstructural flatnessthree inputsflat outputsinput prolongations
0
0 comments X

The pith

Three-input x-flat control-affine systems can be transformed into a structurally flat triangular form under regularity conditions.

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

The authors define a triangular form for x-flat systems with three inputs that ensures structural flatness when certain regularity conditions hold. They derive necessary and sufficient conditions for static feedback equivalence to this form for a system with a given x-flat output. They also give sufficient conditions for achieving equivalence through a finite number of input prolongations in general x-flat three-input systems. This provides a structured approach to handling flatness in nonlinear control systems with multiple inputs. A reader would care because it simplifies trajectory planning and feedback design for such systems.

Core claim

We present a broadly applicable structurally flat triangular form for x-flat control-affine systems with three inputs. Building on recent results for the derivative structure of flat outputs, we define the triangular form together with regularity conditions that guarantee structural flatness, and derive necessary and sufficient conditions for a system with a given x-flat output to be static feedback equivalent to this form. Further, we present sufficient conditions under which general x-flat three-input systems can be rendered static feedback equivalent to the proposed triangular form after a finite number of input prolongations.

What carries the argument

The structurally flat triangular form for three-input systems, which together with regularity conditions on the derivative structure of the x-flat output guarantees that the system is structurally flat and allows static feedback equivalence.

Load-bearing premise

The control-affine system is x-flat and possesses a flat output whose derivatives satisfy the regularity conditions needed for the triangular structure.

What would settle it

A three-input x-flat system whose flat output derivatives violate the regularity conditions and cannot reach the triangular form via static feedback or any finite number of input prolongations would disprove the sufficiency claims.

read the original abstract

We present a broadly applicable structurally flat triangular form for x-flat control-affine systems with three inputs. Building on recent results for the derivative structure of flat outputs, we define the triangular form together with regularity conditions that guarantee structural flatness, and derive necessary and sufficient conditions for a system with a given x-flat output to be static feedback equivalent to this form. Further, we present sufficient conditions under which general x-flat three-input systems can be rendered static feedback equivalent to the proposed triangular form after a finite number of input prolongations.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces a structurally flat triangular form for x-flat control-affine systems with three inputs. It defines this form together with regularity conditions that guarantee structural flatness, derives necessary and sufficient conditions for a system with a given x-flat output to be static-feedback equivalent to the form, and states sufficient conditions under which general x-flat three-input systems become equivalent to the form after finitely many input prolongations. The argument proceeds from the derivative structure of the flat output via explicit coordinate changes and feedback constructions.

Significance. If the derivations hold, the triangular form supplies a concrete canonical representation that can simplify controller synthesis and trajectory planning for a broad class of three-input flat systems. The necessary-and-sufficient equivalence conditions are a clear strength, as they move beyond pure existence statements and tie directly to verifiable rank conditions on the flat-output derivatives. The prolongation result usefully extends the scope without claiming necessity.

minor comments (3)
  1. [Abstract and §1] The abstract and introduction should explicitly state the dimension of the state space (presumably n=3m or similar) and the precise class of control-affine systems considered, to avoid ambiguity for readers outside the immediate flatness literature.
  2. [§2] Notation for the successive derivatives of the flat output (e.g., y, ẏ, ÿ, …) and the associated distribution ranks should be introduced once in a dedicated notation paragraph or table, rather than being redefined inline in each section.
  3. [§3] The paper would benefit from a short remark clarifying whether the regularity conditions are assumed to hold identically or only on a dense open set; this affects the practical verifiability of the necessary-and-sufficient conditions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of the necessary-and-sufficient equivalence conditions, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds from the given x-flat output's derivative structure through explicit coordinate changes and feedback constructions to establish necessary and sufficient conditions for static feedback equivalence to the defined triangular form. No equation or condition reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation chain; the regularity conditions and equivalence criteria are stated independently and verified via direct computation on the system dynamics. Reliance on prior flatness results is external and does not substitute for the paper's internal arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard differential-geometric assumptions for control-affine systems and on recent results about the derivative structure of flat outputs; no new free parameters or invented entities are introduced.

axioms (2)
  • standard math Smoothness of the vector fields defining the control-affine system
    Required for the existence of Lie brackets and for the definition of flat outputs.
  • domain assumption Existence of an x-flat output whose derivatives satisfy the stated regularity conditions
    Central to the equivalence statements.

pith-pipeline@v0.9.0 · 5381 in / 1163 out tokens · 60791 ms · 2026-05-13T20:15:59.009529+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 1 internal anchor

  1. [1]

    Flatness and defect of non-linear systems: introductory theory and examples,

    M. Fliess, J. L ´evine, P. Martin, and P. Rouchon, “Flatness and defect of non-linear systems: introductory theory and examples,”International Journal of Control, vol. 61, no. 6, pp. 1327–1361, 1995

    work page 1995

  2. [2]

    A Lie-B ¨acklund approach to equivalence and flatness of nonlinear systems,

    ——, “A Lie-B ¨acklund approach to equivalence and flatness of nonlinear systems,”IEEE Trans. Autom. Control, vol. 44, no. 5, pp. 922–937, May 1999

    work page 1999

  3. [3]

    On linearization of control sys- tems,

    B. Jakubczyk and W. Respondek, “On linearization of control sys- tems,” vol. 28, no. 9-10, pp. 517–522, 1980

    work page 1980

  4. [4]

    Feedback linearization and driftless systems,

    P. Martin and P. Rouchon, “Feedback linearization and driftless systems,”Mathematics of Control, Signals, and Systems, vol. 7, no. 3, pp. 235–254, Sep. 1994

    work page 1994

  5. [5]

    Flatness of two-input control-affine systems linearizable via a two-fold prolongation,

    F. Nicolau and W. Respondek, “Flatness of two-input control-affine systems linearizable via a two-fold prolongation,” in2016 IEEE 55th Conference on Decision and Control (CDC), Las Vegas, NV , Dec. 2016, pp. 3862–3867

    work page 2016

  6. [6]

    Necessary and sufficient conditions for the linearisability of two-input systems by a two- dimensional endogenous dynamic feedback,

    C. Gst ¨ottner, B. Kolar, and M. Sch ¨oberl, “Necessary and sufficient conditions for the linearisability of two-input systems by a two- dimensional endogenous dynamic feedback,”International Journal of Control, vol. 96, no. 3, pp. 800–821, 2023

    work page 2023

  7. [7]

    Flatness of multi-input control-affine systems linearizable via one-fold prolongation,

    F. Nicolau and W. Respondek, “Flatness of multi-input control-affine systems linearizable via one-fold prolongation,”SIAM Journal on Control and Optimization, vol. 55, no. 5, pp. 3171–3203, Jan. 2017

    work page 2017

  8. [8]

    A structurally flat triangular form based on the extended chained form,

    C. Gst ¨ottner, B. Kolar, and M. Sch ¨oberl, “A structurally flat triangular form based on the extended chained form,”International Journal of Control, vol. 95, no. 5, pp. 1144–1163, 2022

    work page 2022

  9. [9]

    A triangular normal form for x-flat control-affine two-input systems,

    ——, “A triangular normal form for x-flat control-affine two-input systems,” in2024 28th International Conference on Methods and Models in Automation and Robotics (MMAR), Miedzyzdroje, Poland, Aug. 2024, pp. 298–303

    work page 2024

  10. [10]

    A triangular canonical form for a class of 0-flat nonlinear systems,

    S. Bououden, D. Boutat, G. Zheng, J.-P. Barbot, and F. Kratz, “A triangular canonical form for a class of 0-flat nonlinear systems,” International Journal of Control, vol. 84, no. 2, pp. 261–269, 2011

    work page 2011

  11. [11]

    Normal forms for multi-input flat systems of minimal differential weight,

    F. Nicolau and W. Respondek, “Normal forms for multi-input flat systems of minimal differential weight,”International Journal of Robust and Nonlinear Control, vol. 29, no. 10, pp. 3139–3162, Jul. 2019

    work page 2019

  12. [12]

    A flat triangular form for nonlinear systems with two inputs: Necessary and sufficient conditions,

    H. B. Silveira, P. S. Pereira Da Silva, and P. Rouchon, “A flat triangular form for nonlinear systems with two inputs: Necessary and sufficient conditions,”European Journal of Control, vol. 22, pp. 17–22, Mar. 2015

    work page 2015

  13. [13]

    On an implicit triangular decom- position of nonlinear control systems that are 1-flat—A constructive approach,

    M. Sch ¨oberl and K. Schlacher, “On an implicit triangular decom- position of nonlinear control systems that are 1-flat—A constructive approach,”Automatica, vol. 50, no. 6, pp. 1649–1655, Jun. 2014

    work page 2014

  14. [14]

    On triangular forms for x- flat control-affine systems with two inputs,

    G. Hartl, C. Gst ¨ottner, and M. Sch ¨oberl, “On triangular forms for x- flat control-affine systems with two inputs,” in2025 29th International Conference on System Theory, Control and Computing (ICSTCC), Cluj-Napoca, Oct. 2025, pp. 161–168

    work page 2025

  15. [15]

    Multi-input control-affine systems static feedback equivalent to a triangular form and their flatness,

    S. Li, F. Nicolau, and W. Respondek, “Multi-input control-affine systems static feedback equivalent to a triangular form and their flatness,”International Journal of Control, vol. 89, no. 1, pp. 1–24, 2016

    work page 2016

  16. [16]

    On a flat triangular form based on the extended chained form,

    C. Gst ¨ottner, B. Kolar, and M. Sch ¨oberl, “On a flat triangular form based on the extended chained form,”IFAC-PapersOnLine, vol. 54, no. 9, pp. 245–252, 2021

    work page 2021

  17. [17]

    A flat triangular structure based on a multi-chained form,

    G. Hartl, C. Gst ¨ottner, and M. Sch ¨oberl, “A flat triangular structure based on a multi-chained form,” Nov. 2025, arXiv:2510.26632, Accepted for ECC 2026. [Online]. Available: http://arxiv.org/abs/2510.26632

    work page arXiv 2025

  18. [18]

    On the Linearization of Flat Multi-Input Systems via Prolongations

    G. Hartl, C. Gst ¨ottner, and M. Sch ¨oberl, “On the linearization of flat multi-input systems via prolongations,” Feb. 2026, arXiv:2602.17562. [Online]. Available: https://arxiv.org/abs/2602.17562

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  19. [19]

    Properties of flat systems with regard to the parameterization of the system variables by the flat output,

    B. Kolar, M. Sch ¨oberl, and K. Schlacher, “Properties of flat systems with regard to the parameterization of the system variables by the flat output,”IFAC-PapersOnLine, vol. 49, no. 18, pp. 814–819, 2016

    work page 2016

  20. [20]

    The role of symmetry in constructing geometric flat outputs for free-flying robotic systems,

    J. Welde, M. D. Kvalheim, and V . Kumar, “The role of symmetry in constructing geometric flat outputs for free-flying robotic systems,” in2023 IEEE International Conference on Robotics and Automation (ICRA), London, United Kingdom, May 2023, pp. 12 247–12 253

    work page 2023

  21. [21]

    On the exact linearization of minimally underactuated configuration flat Lagrangian systems in generalized state representation,

    G. Hartl, C. Gst ¨ottner, B. Kolar, and M. Sch ¨oberl, “On the exact linearization of minimally underactuated configuration flat Lagrangian systems in generalized state representation,”IFAC-PapersOnLine, vol. 58, no. 17, pp. 244–249, 2024

    work page 2024

  22. [22]

    Kolar,Contributions to the Differential Geometric Analysis and Control of Flat Systems, ser

    B. Kolar,Contributions to the Differential Geometric Analysis and Control of Flat Systems, ser. Modellierung und Regelung komplexer dynamischer Systeme. Aachen: Shaker, 2017, vol. 35

    work page 2017

  23. [23]

    Gst ¨ottner,Analysis and Control of Flat Systems by Geometric Methods, ser

    C. Gst ¨ottner,Analysis and Control of Flat Systems by Geometric Methods, ser. Modellierung und Regelung komplexer dynamischer Systeme. D ¨uren: Shaker Verlag, 2023, vol. 59. APPENDIXI PROOF OFTHEOREM2 Necessity:It suffices to verify that the sequence (15) for any system of the form (GT F 3) with flat output φ= (z 1 1, z1 2, z1 3)consists solely of integr...

    work page 2023