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arxiv: 2605.17715 · v1 · pith:6IZGFWJKnew · submitted 2026-05-18 · 📡 eess.SY · cs.SY· math.DS

Observer-Based Stabilization for Linear Multi-Agent Dynamical Systems Using Generalized Frequency Variables

G. Q. Bao Tran , Yutaka Hori , Shinji Hara This is my paper

Pith reviewed 2026-05-19 22:37 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.DS
keywords multi-agent systemsobserver-based controlseparation principlegeneralized frequency variablesoutput feedbacklinear MIMO agentsnetwork stabilityhomogeneous networks
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The pith

A separation principle lets observers and controllers be designed independently for stable output feedback in homogeneous multi-agent networks.

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

The paper develops networked controllers and observers for homogeneous linear MIMO agents that stabilize both the system states and the estimation errors. It uses generalized frequency variables to analyze and guarantee stability, then shows that the observer and controller can be designed separately and later combined without losing stability. This separation matters because many multi-agent applications have only partial measurements available, so full-state feedback is impractical. The approach is demonstrated on an unstable network of pendulums on carts, and the work also derives necessary conditions for controllability and observability from agent dynamics and network connections.

Core claim

For homogeneous networks of linear MIMO agents, generalized frequency variables allow the stability of the closed-loop system and the error dynamics to be analyzed separately, establishing a separation principle under which the observer and controller can be designed independently and then combined to achieve a stable output-feedback system.

What carries the argument

Generalized frequency variables that decouple stability analysis of the closed-loop dynamics from the error dynamics.

If this is right

  • Observer and controller can be designed independently and combined for stable output feedback.
  • Necessary conditions for controllability and observability follow from agent properties and network structure.
  • The method applies to highly unstable oscillatory networks such as locally actuated pendulums on carts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The separation may reduce computational effort in large networks by avoiding joint redesign of observer and controller gains.
  • If homogeneity holds only approximately, the method could still provide a useful starting point for nearby heterogeneous cases.
  • The controllability and observability conditions suggest that network topology can be chosen to ensure the separation principle applies.

Load-bearing premise

The network consists of homogeneous linear MIMO agents for which generalized frequency variables can be defined and applied to the stability analysis.

What would settle it

A case in which separately designed observer and controller, each using the generalized frequency variable method, produce an unstable combined output-feedback system on the pendulum network.

Figures

Figures reproduced from arXiv: 2605.17715 by G. Q. Bao Tran, Shinji Hara, Yutaka Hori.

Figure 3
Figure 3. Figure 3: Left: Stability region from h(s) (blue) and σ(A − BK). Right: Exponentially stabilized trajectories. 4. ON CONTROLLABILITY AND OBSERVABILITY We discuss the controllability and observability of sys￾tem (5), which correspond to those of the pairs (A, B) and (A, C), respectively, and are sufficient conditions for con￾troller and observer design. Recall that in the MIMO agent case (m > 1), (Trumpf and Trentelm… view at source ↗
Figure 1
Figure 1. Figure 1: Left: Stability region Λs from h(s) (blue) and σ(A). Right: Unstable trajectories of uncontrolled network. We consider a non-collocated sensor–actuator configura￾tion, which is typically more difficult than the collocated case. The control input enters the first agent and the mea￾surement is taken from the third one, i.e., B = (1, 0, 0, 0) and C = (0 0 1 0). Each SISO agent is minimal, and since (A, B) and… view at source ↗
Figure 2
Figure 2. Figure 2: Left: Stability region Λs from h(s) (blue) and σ(A − LC). Right: Estimation results and errors. Simulation results in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
read the original abstract

We address the conditions and design of controllers and observers for homogeneous networks of linear MIMO agents. We develop networked controllers and observers that ensure the stability of both the system state and the estimation error, leveraging the concept of generalized frequency variables. A separation principle for networks is then established, showing that the observer and controller can be designed independently and combined to achieve a stable output feedback. Our results are illustrated via a highly unstable, oscillatory network of locally actuated pendulums on carts. Finally, necessary conditions for controllability and observability -- derived from agent properties and network structure -- are established and discussed.

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

2 major / 2 minor

Summary. The paper develops conditions and designs for networked controllers and observers for homogeneous networks of linear MIMO agents, using generalized frequency variables to ensure stability of both the closed-loop system state and the estimation error. It establishes a separation principle allowing independent design of the observer and controller for stable output feedback. Results are illustrated on a network of locally actuated pendulums on carts, and necessary conditions for controllability and observability are derived from agent properties and network structure.

Significance. If the derivations hold, the work provides a practical framework for observer-based stabilization in multi-agent systems by decoupling stability analysis via generalized frequency variables. The separation principle is a key strength, enabling independent design, and the explicit controllability/observability conditions tied to network structure add value for applications. The pendulum example demonstrates handling of unstable oscillatory dynamics.

major comments (2)
  1. [separation principle derivation] The separation principle section: the decoupling of closed-loop state stability from error dynamics relies on homogeneity of the linear MIMO agents and the specific definition of generalized frequency variables; it is unclear whether this extends without additional assumptions on the network Laplacian or agent transfer functions when the agents are MIMO rather than SISO.
  2. [controllability/observability section] Controllability and observability conditions: the necessary conditions derived from agent properties and network structure appear to assume a specific form of the interconnection; a counter-example or explicit statement of the graph assumptions (e.g., connectedness) would strengthen the claim that these conditions are both necessary and sufficient for the overall network.
minor comments (2)
  1. [example section] In the pendulum example, the specific values of the cart-pendulum parameters, the network adjacency matrix, and the chosen generalized frequency variable should be tabulated to allow direct reproduction of the simulation results.
  2. [preliminaries] Notation for the generalized frequency variable should be introduced with a clear definition early in the paper, including how it differs from standard frequency-domain variables in the MIMO case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the revisions we will make to improve clarity.

read point-by-point responses

  1. Referee: The separation principle section: the decoupling of closed-loop state stability from error dynamics relies on homogeneity of the linear MIMO agents and the specific definition of generalized frequency variables; it is unclear whether this extends without additional assumptions on the network Laplacian or agent transfer functions when the agents are MIMO rather than SISO.

    Authors: The separation principle derivation relies on the homogeneity assumption, under which all agents share an identical MIMO transfer function matrix, combined with the algebraic properties of the generalized frequency variable transformation. This allows the closed-loop system to be decomposed into independent modal subsystems indexed by the eigenvalues of the network Laplacian. The decoupling between state and error dynamics follows from the block structure of the closed-loop matrix and holds for MIMO agents via the same matrix operations used in the SISO case; no SISO-specific restrictions are invoked. The Laplacian is assumed to be symmetric (undirected graph) with standard spectral properties, but no further restrictions on its form or on the agent transfer functions are required beyond homogeneity and properness. We will revise the section to explicitly state these points and add a clarifying remark on the MIMO extension. revision: yes

  2. Referee: Controllability and observability conditions: the necessary conditions derived from agent properties and network structure appear to assume a specific form of the interconnection; a counter-example or explicit statement of the graph assumptions (e.g., connectedness) would strengthen the claim that these conditions are both necessary and sufficient for the overall network.

    Authors: We agree that the graph assumptions merit explicit statement. The necessary conditions are derived under the standing assumption of an undirected connected graph, which guarantees that the Laplacian has a simple zero eigenvalue and that the network cannot be partitioned into independent subsystems. Under this assumption the overall controllability/observability reduces to conditions on the agent dynamics evaluated at the nonzero Laplacian eigenvalues. If the graph is disconnected the network decomposes and each component must separately satisfy the conditions. We will revise the relevant section to state the connectedness assumption clearly, note its necessity for the claimed necessity and sufficiency, and briefly discuss the disconnected case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives a separation principle for observer-based stabilization in homogeneous linear MIMO multi-agent networks by applying generalized frequency variables to decouple closed-loop state stability from error dynamics. Controllability and observability conditions are explicitly derived from agent properties and network structure rather than assumed or fitted. No load-bearing steps reduce by construction to self-definitions, renamed empirical patterns, or unverified self-citation chains; the central claims rest on independent derivations from the stated homogeneity and frequency-variable framework.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5634 in / 1060 out tokens · 29672 ms · 2026-05-19T22:37:11.438021+00:00 · methodology

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    2013 , issn =

    Biochemical oscillations in delayed negative cyclic feedback: Existence and profiles , journal =. 2013 , issn =

    work page 2013

  2. [2]

    and Aihara, K

    Chen, L. and Aihara, K. , journal=. Stability of genetic regulatory networks with time delay , year=

    work page

  3. [3]

    Nature , volume=

    A synthetic oscillatory network of transcriptional regulators , author=. Nature , volume=. 2000 , publisher=

    work page 2000

  4. [4]

    2011 , note =

    Existence criteria of periodic oscillations in cyclic gene regulatory networks , journal =. 2011 , note =

    work page 2011

  5. [5]

    Stability analysis of linear systems with generalized frequency variables and its applications to formation control , year=

    Shinji Hara and Tomohisa Hayakawa and Hikaru Sugata , booktitle=. Stability analysis of linear systems with generalized frequency variables and its applications to formation control , year=

    work page

  6. [6]

    Stability analysis of systems with generalized frequency variables , year=

    Hara, Shinji and Tanaka, Hideaki and Iwasaki, Tetsuya , journal=. Stability analysis of systems with generalized frequency variables , year=

    work page

  7. [7]

    Differential dissipativity theory for dominance analysis , year=

    Forni, Fulvio and Sepulchre, Rodolphe , journal=. Differential dissipativity theory for dominance analysis , year=

    work page

  8. [8]

    Nature Communications , volume=

    Automated optogenetic feedback control for precise and robust regulation of gene expression and cell growth , author=. Nature Communications , volume=. 2016 , publisher=

    work page 2016

  9. [9]

    Automatica , volume =

    Robust stability analysis for. Automatica , volume =. 2019 , issn =

    work page 2019

  10. [10]

    SICE Journal of Control, Measurement, and System Integration , year = 2011, month = jan, volume =

    work page 2011

  11. [11]

    , journal=

    Luenberger, D. , journal=. Canonical forms for linear multivariable systems , year=

    work page

  12. [12]

    Cooperative gain output feedback stabilization for multi-agent dynamical systems , year=

    Hara, Shinji and Kanno, Masaaki and Tanaka, Hideaki , booktitle=. Cooperative gain output feedback stabilization for multi-agent dynamical systems , year=

    work page

  13. [13]

    and Glover, Keith , title =

    Zhou, Kemin and Doyle, John C. and Glover, Keith , title =. 1996 , isbn =

    work page 1996

  14. [14]

    , journal=

    Trumpf, Jochen and Trentelman, Harry L. , journal=. Controllability and stabilizability of networks of linear systems , year=

    work page

  15. [15]

    , journal=

    Shih-Ho Wang and Davison, E. , journal=. On the stabilization of decentralized control systems , year=

    work page

  16. [16]

    2014 , issn =

    Decentralized control: Status and outlook , journal =. 2014 , issn =

    work page 2014

  17. [17]

    Multi-agent system for rogue drone interception , year=

    Souli, Nicolas and Kolios, Panayiotis and Ellinas, Georgios , journal=. Multi-agent system for rogue drone interception , year=

    work page

  18. [18]

    2018 , publisher=

    Linear systems theory , author=. 2018 , publisher=

    work page 2018

  19. [19]

    Basic study on wheel speed based driving force control for multi-motor electric vehicles with glocal stability analysis , year=

    Nguyen, Binh-Minh and Ueno, Takumi and Hosomi, Yuki and Sato, Tona and Hara, Shinji and Fujimoto, Hiroshi , booktitle=. Basic study on wheel speed based driving force control for multi-motor electric vehicles with glocal stability analysis , year=

    work page

  20. [20]

    Tran, Gia Quoc Bao and Daniel Liberzon and Hyungbo Shim , title =

    work page