Pith. sign in

REVIEW 2 major objections 2 minor 36 references

Multi-Partite Output Regulation of Multi-Agent Systems

T0 review · 2 major / 2 minor · reviewed 2026-05-23 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read Multi-partite output regulation extends cooperative regulation so agents in any chosen grouping track scaled copies of one reference signal.

desk verdict The paper generalizes CORP to arbitrary k-partitions via a transformation and supplies two design strategies, with the second relying on an implication under a mild graph condition. read the letter →

arxiv 2503.02313 v3 submitted 2025-03-04 eess.SY cs.SYmath.OC

Kürşad Metehan Gül , Selahattin Burak Sars{i}lmaz This is my paper
classification eess.SYcs.SYmath.OC
keywords multi-agentsystemsoutputregulationmulti-partitedistributedcontrolheterogeneouslineargraphpartitioningfeedforwardcooperative
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 defines a k-partition transformation that lets any grouping of nodes be imposed on a multi-agent system. It then states the multi-partite output regulation problem, in which followers belonging to the same part of the partition must asymptotically track a scalar multiple of an external reference while the closed-loop system remains internally stable. Necessary and sufficient solvability conditions for a feedforward-based distributed controller are shown to reduce directly to those of the cooperative output regulation problem, producing an explicit but partition-dependent design procedure. Under a mild structural condition on the communication graph, the authors prove that this partition-dependent condition is implied by a simpler partition-independent version, which supplies a second, more scalable design procedure.

What carries the argument

The k-partition transformation, which maps any desired partition of the node set into the multi-partite output regulation problem.

What would settle it

A communication graph obeying the mild structural condition together with a choice of partitions for which the partition-dependent condition fails to hold when only the partition-independent condition is satisfied.

Watch

Extended reading notes

Core claim

The necessary and sufficient conditions for solvability of the multi-partite output regulation problem with a feedforward-based distributed control law follow from the cooperative output regulation problem and produce the first design strategy for the control parameters. This condition is implied by its partition-independent version under a mild structural condition on the communication graph, which yields a second, more scalable design strategy.

Load-bearing premise

The communication graph satisfies a mild structural condition that makes the partition-dependent solvability condition follow from the partition-independent version.

Editorial extensions

If this is right

  • The multi-partite output regulation problem contains the cooperative output regulation problem and the bipartite output regulation problem as special cases.
  • A feedforward-based distributed controller solves the problem when the stated conditions derived from cooperative output regulation hold.
  • The first design strategy requires checking a condition that depends on the specific partition chosen.
  • The second design strategy replaces the partition-dependent check with a partition-independent one whenever the mild structural condition on the graph is met.
  • An experiment on physical hardware and two numerical examples confirm that both strategies achieve the required tracking while preserving internal stability.

Reading between the lines

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

  • The same partition transformation could be reused to formulate regulation objectives that change over time by switching among several fixed partitions.
  • Because the controller structure itself does not change with the partition, the framework may reduce the number of separate controller designs needed when a network must support several different grouping patterns.
  • The scalability gain from the partition-independent condition suggests the method could be applied to networks whose size makes enumerating all partitions impractical.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The paper introduces a k-partition transformation to formulate the multi-partite output regulation problem (MORP) for heterogeneous linear multi-agent systems, generalizing the cooperative output regulation problem (CORP) and bipartite output regulation problem (BORP). It derives necessary and sufficient solvability conditions for a feedforward-based distributed control law from CORP results, yielding a first (partition-dependent) design strategy for control parameters; it then proves that this condition is implied by a partition-independent version under a mild structural condition on the communication graph, enabling a second, more scalable design strategy. The claims are supported by two numerical examples comparing the strategies and one experiment demonstrating flexibility.

Significance. If the implication holds under the stated mild condition, the work provides a unified, graph-independent perspective on output regulation for arbitrary node partitions in MAS, extending beyond cooperation and bipartition with two concrete design strategies. The derivation from established CORP results is a strength, as is the explicit comparison of scalability between strategies in the examples. This could enable more flexible controller synthesis in applications involving signed or partitioned networks.

major comments (2)
  1. [Section deriving the second design strategy] The section proving the implication (abstract and the section deriving the second design strategy): the claim that the partition-dependent condition is implied by its partition-independent version under a mild structural condition is load-bearing for the scalability advantage of the second strategy, yet the manuscript does not explicitly define the mild structural condition or provide verifiable key steps of the proof, preventing assessment of whether the implication holds for the graphs used in the numerical examples.
  2. [Section on MORP solvability conditions] Section on the first design strategy and CORP implication: while the necessary and sufficient conditions are stated to follow from CORP, the explicit mapping from CORP parameters to the MORP feedforward law (including how the k-partition affects the regulator equations) is not detailed enough to confirm the claimed drawback in scalability or to reproduce the parameter design.
minor comments (2)
  1. The definition and properties of the k-partition transformation would benefit from an early illustrative example with a small graph to clarify how it achieves arbitrary partitions before the MORP formulation.
  2. Notation for the partition sets and the associated transformation matrix should be made consistent across the problem statement and the control law equations to avoid ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment below and will make the indicated revisions to improve clarity and reproducibility.

read point-by-point responses

  1. Referee: [Section deriving the second design strategy] The section proving the implication (abstract and the section deriving the second design strategy): the claim that the partition-dependent condition is implied by its partition-independent version under a mild structural condition is load-bearing for the scalability advantage of the second strategy, yet the manuscript does not explicitly define the mild structural condition or provide verifiable key steps of the proof, preventing assessment of whether the implication holds for the graphs used in the numerical examples.

    Authors: We agree that an explicit definition of the mild structural condition and inclusion of key proof steps are necessary for verification. In the revised manuscript, we will clearly define the mild structural condition and provide the key steps of the proof (in the main text or an appendix) to enable assessment for the numerical examples. revision: yes

  2. Referee: [Section on MORP solvability conditions] Section on the first design strategy and CORP implication: while the necessary and sufficient conditions are stated to follow from CORP, the explicit mapping from CORP parameters to the MORP feedforward law (including how the k-partition affects the regulator equations) is not detailed enough to confirm the claimed drawback in scalability or to reproduce the parameter design.

    Authors: We agree that more explicit details on the parameter mapping would aid reproducibility and better illustrate the scalability drawback. In the revision, we will expand the relevant section to include the explicit mapping from CORP parameters to the MORP feedforward law and detail the effect of the k-partition on the regulator equations. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation from CORP and graph implication proof are independent

full rationale

The paper derives necessary and sufficient conditions for MORP solvability directly from the established CORP framework and presents a mathematical proof that a partition-dependent condition is implied by its partition-independent version under a mild structural graph condition. No self-definitional constructs, fitted parameters renamed as predictions, or load-bearing self-citations that reduce claims to inputs by construction are present. The central results rely on external CORP literature and an explicit proof step, making the derivation self-contained against the listed circularity patterns.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

The paper builds on standard linear MAS assumptions from CORP and introduces one new mathematical object (k-partition transformation) without independent evidence outside the formulation itself.

assumptions (1)
  • domain assumption Standard assumptions of linear multi-agent systems including stabilizability, detectability, and the existence of solutions to regulator equations from the CORP.
    These are invoked implicitly when stating that MORP conditions follow from CORP.
invented entities (1)
  • k-partition transformation
    purpose: To achieve any desired partition of the nodes in a graph for formulating the MORP.
    New concept introduced to generalize beyond cooperation and bipartition; no independent evidence provided.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Multi-Partite Output Regulation of Multi-Agent Systems." pith.science (2026). https://pith.science/paper/2503.02313

@misc{pith2026250302313,
  author       = {Pith},
  title        = {Pith review of: Multi-Partite Output Regulation of Multi-Agent Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2503.02313}},
  note         = {Machine review of arXiv:2503.02313}
}
abstract

This article proposes a simple, graph-independent perspective on partitioning the node set of a graph and provides multi-agent systems (MASs) with objectives beyond cooperation and bipartition. Specifically, we first introduce the notion of $k$-partition transformation to achieve any desired partition of the nodes. Then, we use this notion to formulate the multi-partite output regulation problem (MORP) of heterogeneous linear MASs, which comprises the existing cooperative output regulation problem (CORP) and bipartite output regulation problem (BORP) as subcases. The goal of the MORP is to design a distributed control law such that each follower that belongs to the same set in the partition asymptotically tracks a scalar multiple of the reference while ensuring the internal stability of the closed-loop system. It is shown that the necessary and sufficient conditions for the solvability of the MORP with a feedforward-based distributed control law follow from the CORP and lead to the first design strategy for the control parameters. However, it has a drawback in terms of scalability due to a partition-dependent condition. We prove that this condition is implied by its partition-independent version under a mild structural condition. This implication yields the second design strategy that is much more scalable than the first one. Finally, an experiment is conducted to demonstrate the MORP's flexibility, and two numerical examples are provided to illustrate its generality and compare both design strategies regarding scalability.

Figures

Figures reproduced from arXiv: 2503.02313 by the authors.

Figure 1
Figure 1. The trajectories of the MAS. Here, yij denotes the jth entry of follower i’s output for i = 1, 2, 3 and y0j denotes the jth entry of v while “×” and “o” marks yij at the initial and final times for i = 0, 1, 2, 3, respectively. Example 1: As indicated in Remark 2, the MORP includes the BORP. To make the MORP comparable with the existing solutions to the BORP, we force k-partition transformations to be gauge transfor… view at source ↗
Figure 3
Figure 3. Elapsed times of both design strategies with respect [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    H. Cai, Y . Su, and J. Huang, Cooperative control of multi-agent sys- tems: Distributed observer and distributed internal model approaches. Springer, 2022

    work page 2022

  2. [2]

    Consensus problems on networks with antago nistic interac- tions,

    C. Altafini, “Consensus problems on networks with antago nistic interac- tions,” IEEE Trans. Autom. Control , vol. 58, no. 4, pp. 935–946, 2013

    work page 2013

  3. [3]

    Leader- follower and leaderless bipartite formation-consensus ov er undirected coopetition networks and under proximity and collision-av oidance con- straints,

    P . S ¸ ekercio˘ glu, I. Sarras, A. Lor´ ıa, E. Panteley, andJ. Marzat, “Leader- follower and leaderless bipartite formation-consensus ov er undirected coopetition networks and under proximity and collision-av oidance con- straints,” Int. J. Control , vol. 98, no. 3, pp. 642–656, 2025

    work page 2025

  4. [4]

    Unmanned aircraft systems : Roles, missions, and future concepts,

    K. M. Sayler and M. E. DeVine, “Unmanned aircraft systems : Roles, missions, and future concepts,” Congressional Research Service , 2022

    work page 2022

  5. [5]

    Cooperative output regulation of lin ear multi-agent systems,

    Y . Su and J. Huang, “Cooperative output regulation of lin ear multi-agent systems,” IEEE Trans. Autom. Control , vol. 57, no. 4, pp. 1062–1066, 2012

    work page 2012

  6. [6]

    Cooperative output regulation with application to multi-agent consensus under switching network,

    ——, “Cooperative output regulation with application to multi-agent consensus under switching network,” IEEE Trans. Syst., Man, Cybern. B. Cybern. , vol. 42, no. 3, pp. 864–875, 2012

    work page 2012

  7. [7]

    The adaptive dist ributed observer approach to the cooperative output regulation of l inear multi- agent systems,

    H. Cai, F. L. Lewis, G. Hu, and J. Huang, “The adaptive dist ributed observer approach to the cooperative output regulation of l inear multi- agent systems,” Automatica, vol. 75, pp. 299–305, 2017

    work page 2017

  8. [8]

    Cooperative output regulation of linea r multi-agent systems by a novel distributed dynamic compensator,

    M. Lu and L. Liu, “Cooperative output regulation of linea r multi-agent systems by a novel distributed dynamic compensator,” IEEE Trans. Autom. Control, vol. 62, no. 12, pp. 6481–6488, 2017

    work page 2017

Show all 36 references
  1. [9]

    Cooperative output regulation subject to uncertain exosystems and its application: A full y distributed adaptive observer approach,

    L. Zhang, M. Lu, F. Deng, and J. Chen, “Cooperative output regulation subject to uncertain exosystems and its application: A full y distributed adaptive observer approach,” IEEE Control Netw. Syst. , to be published, doi: 10.1109/TCNS.2025.3543667

    2025 doi

  2. [10]

    Distributed outpu t regulation of heterogeneous uncertain linear agents,

    S. Kawamura, K. Cai, and M. Kishida, “Distributed outpu t regulation of heterogeneous uncertain linear agents,” Automatica, vol. 119, p. 109094, 2020

    2020

  3. [11]

    A distributed control a pproach for heterogeneous linear multiagent systems,

    S. B. Sarsilmaz and T. Y ucelen, “A distributed control a pproach for heterogeneous linear multiagent systems,” Int. J. Control , vol. 94, no. 5, pp. 1402–1414, 2021

    2021

  4. [12]

    Coopera- tive output regulation of heterogeneous multiagent system s: A global distributed control synthesis approach,

    A. T. Koru, S. B. Sarsılmaz, T. Y ucelen, and E. Johnson, “ Coopera- tive output regulation of heterogeneous multiagent system s: A global distributed control synthesis approach,” IEEE Trans. Autom. Control , vol. 66, no. 9, pp. 4289–4296, 2021

    2021

  5. [13]

    Regional eigenvalue assignment in coopera tive linear output regulation,

    A. T. Koru, S. B. Sarsılmaz, T. Y ucelen, J. A. Muse, F. L. L ewis, and B. Ac ¸ıkmes ¸e, “Regional eigenvalue assignment in coopera tive linear output regulation,” IEEE Trans. Autom. Control, vol. 68, no. 7, pp. 4265– 4272, 2023

    2023

  6. [14]

    An internal model approach to coopera tive output regulation over switching graphs,

    A. T. Koru, S. B. Sarsılmaz, Y . Kartal, F. L. Lewis, T. Y uc elen, J. A. Muse, and A. Davoudi, “An internal model approach to coopera tive output regulation over switching graphs,” IEEE Trans. Autom. Control , vol. 69, no. 11, pp. 7980–7987, 2024

    2024

  7. [15]

    Biparti te tracking of homogeneous and heterogeneous linear multi-agent syste ms,

    Q. Jiao, H. Zhang, S. Xu, F. L. Lewis, and L. Xie, “Biparti te tracking of homogeneous and heterogeneous linear multi-agent syste ms,” Int. J. Control, vol. 92, no. 12, pp. 2963–2972, 2019

    2019

  8. [16]

    Biparti te output regulation of multi-agent systems with antagonistic inter actions,

    H. Dehghani Aghbolagh, M. Zamani, and Z. Chen, “Biparti te output regulation of multi-agent systems with antagonistic inter actions,” in Proc. Asian Control Conf. , 2017, pp. 321–325

    2017

  9. [17]

    Bipartite output consensus for h eterogeneous multi-agent systems via output regulation approach,

    T. Han and W. X. Zheng, “Bipartite output consensus for h eterogeneous multi-agent systems via output regulation approach,” IEEE Trans. Cir- cuits Syst. II, Exp. Briefs , vol. 68, no. 1, pp. 281–285, 2021

    2021

  10. [18]

    Robust bipartite output regulat ion of linear uncertain multi-agent systems,

    D. Liang and J. Huang, “Robust bipartite output regulat ion of linear uncertain multi-agent systems,” Int. J. Control , vol. 95, no. 1, pp. 42– 49, 2022

    2022

  11. [19]

    Cluster consensus in networks of agen ts with weighted cooperative–competitive interactions,

    J. Zhan and X. Li, “Cluster consensus in networks of agen ts with weighted cooperative–competitive interactions,” IEEE Trans. Circuits Syst. II, Exp. Briefs , vol. 65, no. 2, pp. 241–245, 2018

    2018

  12. [20]

    Consensus for cluster s of agents with cooperative and antagonistic relationships,

    G. De Pasquale and M. E. V alcher, “Consensus for cluster s of agents with cooperative and antagonistic relationships,” Automatica, vol. 135, p. 110002, 2022

    2022

  13. [21]

    Fully distrib uted dynamic event-triggered pinning cluster consensus control for het erogeneous multiagent systems with cooperative–competitive interac tions,

    L. Ji, X. Liu, C. Zhang, S. Y ang, and H. Li, “Fully distrib uted dynamic event-triggered pinning cluster consensus control for het erogeneous multiagent systems with cooperative–competitive interac tions,” IEEE Trans. Syst., Man, Cybern. Syst. , vol. 53, no. 1, pp. 394–404, 2023

    2023

  14. [22]

    Scaled consensus,

    S. Roy, “Scaled consensus,” Automatica, vol. 51, pp. 259–262, 2015

    2015

  15. [23]

    Scaled consensus of secon d-order multi-agent systems based on edge-event driven control,

    M. Cao, Q. Zhou, and Y . Cheng, “Scaled consensus of secon d-order multi-agent systems based on edge-event driven control,” J. Frank. Inst., vol. 359, no. 14, pp. 7319–7336, 2022

    2022

  16. [24]

    Kernel design a nd distributed, self-triggered control for coordination of a utonomous multi- agent configurations,

    L. DeVries, A. Sims, and M. D. M. Kutzer, “Kernel design a nd distributed, self-triggered control for coordination of a utonomous multi- agent configurations,” Robotica, vol. 36, no. 7, p. 1077–1097, 2018

    2018

  17. [25]

    A ssignment and distributed control of Laplacian null space in multiagent s ystems,

    D. Tran, T. Y ucelen, D. Kurtoglu, and S. B. Sarsılmaz, “A ssignment and distributed control of Laplacian null space in multiagent s ystems,” Int. J. Control, to be published, doi: 10.1080/00207179.2025.2463568

    2025 doi

  18. [26]

    Data-driven optimal output cluster sy nchronization control of heterogeneous multi-agent systems,

    H. Li and Q. Wei, “Data-driven optimal output cluster sy nchronization control of heterogeneous multi-agent systems,” IEEE Trans. Autom. Sci. Eng., vol. 21, no. 3, pp. 3910–3920, 2024

    2024

  19. [27]

    Group formation t racking of heterogeneous multiagent systems using reinforcement lea rning,

    Y . Wang, Z. Wang, H. Zhang, and H. Y an, “Group formation t racking of heterogeneous multiagent systems using reinforcement lea rning,” IEEE Trans. Control Netw. Syst. , vol. 12, no. 1, pp. 497–509, 2025

    2025

  20. [28]

    Comtet, Advanced combinatorics: The art of finite and infinite expansions

    L. Comtet, Advanced combinatorics: The art of finite and infinite expansions. D. Reidel Pub. Co., 1974

    1974

  21. [29]

    Bipartite consensus of multi-age nt systems over signed graphs: State feedback and output feedback control a pproaches,

    H. Zhang and J. Chen, “Bipartite consensus of multi-age nt systems over signed graphs: State feedback and output feedback control a pproaches,” Int. J. Robust Nonlinear Control , vol. 27, no. 1, pp. 3–14, 2017

    2017

  22. [30]

    Cooperat ive output regulation with disturbance decoupling,

    S. B. Sarsilmaz, K. M. Gul, and B. Ac ¸ıkmes ¸e, “Cooperat ive output regulation with disturbance decoupling,” in Proc. Amer . Control Conf., 2024, pp. 1831–1836

    2024

  23. [31]

    A dis- tributed adaptive control approach to cooperative output r egulation of a class of heterogeneous uncertain multi-agent systems,

    S. B. Sarsılmaz, A. T. Koru, T. Y ucelen, and B. Ac ¸ıkmes ¸ e, “A dis- tributed adaptive control approach to cooperative output r egulation of a class of heterogeneous uncertain multi-agent systems,” i n Control of Autonomous Aerial V ehicles: Advances in Autopilot Design f o...

    2024

  24. [32]

    Huang, Nonlinear output regulation: Theory and applications

    J. Huang, Nonlinear output regulation: Theory and applications. SIAM, 2004

    2004

  25. [33]

    Disturbance rejection in formation keeping control of non holonomic wheeled robots,

    M. Jafarian, E. V os, C. De Persis, J. Scherpen, and A. van der Schaft, “Disturbance rejection in formation keeping control of non holonomic wheeled robots,” Int. J. Robust Nonlinear Control , vol. 26, no. 15, pp. 3344–3362, 2016

    2016

  26. [34]

    Mult ivariable finite time attitude control for quadrotor UA V: Theory and ex perimenta- tion,

    B. Tian, L. Liu, H. Lu, Z. Zuo, Q. Zong, and Y . Zhang, “Mult ivariable finite time attitude control for quadrotor UA V: Theory and ex perimenta- tion,” IEEE Trans. Ind. Electron. , vol. 65, no. 3, pp. 2567–2577, 2018

    2018

  27. [35]

    Disturbance rejection in space applications: Problems and solutions,

    E. Canuto, A. Molano-Jimenez, and C. Perez-Montenegro , “Disturbance rejection in space applications: Problems and solutions,” Acta Astronaut., vol. 72, pp. 121–131, 2012

    2012

  28. [36]

    A decentralized appr oach to formation maneuvers,

    J. Lawton, R. Beard, and B. Y oung, “A decentralized appr oach to formation maneuvers,” IEEE Trans. Robot. Automat. , vol. 19, no. 6, pp. 933–941, 2003. APPENDIX Lemma 3: Suppose Condition 4 holds. The matrix Aµ is Hurwitz if, and only if, µ satisfies the inequality (8). Proof:...

    2003

Pith tools

Reviewed May 23, 2026 · model on record in the stance chip above.