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arxiv: 2604.13394 · v1 · submitted 2026-04-15 · 📡 eess.SY · cs.SY

Recognition: unknown

Distributed Resilient Fixed-Time Control for Cooperative Output Regulation of MASs over Directed Graphs under DoS Attacks

Wenji Cao , Lu Liu , Dan Zhang , Gang Feng
Authors on Pith no claims yet

Pith reviewed 2026-05-10 13:26 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords fixed-time controlmulti-agent systemscooperative output regulationdenial-of-service attacksdirected graphsresilient controldistributed observer
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The pith

A distributed resilient controller drives regulated outputs of linear multi-agent systems to zero in fixed time over directed graphs despite denial-of-service attacks.

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

The paper constructs a distributed resilient fixed-time controller made of an observer and a per-agent control law that together achieve cooperative output regulation for linear multi-agent systems. The design operates on general directed communication graphs and continues to function when denial-of-service attacks temporarily sever links, provided recovery intervals exist. Convergence of the regulated outputs occurs within a time bound fixed by controller parameters alone and therefore independent of the agents' starting states. Prior fixed-time resilient methods had required symmetric Laplacians or strong connectivity; this approach removes those restrictions. If the construction holds, regulation performance remains guaranteed under intermittent communication loss without retuning for each new initial condition or attack pattern.

Core claim

The paper proposes a distributed resilient fixed-time controller comprising a distributed resilient fixed-time observer and a distributed resilient fixed-time control law for each agent. This controller enables the regulated outputs of linear multi-agent systems over directed graphs to converge to zero in a fixed time whose upper bound does not depend on initial states, even under DoS attacks. The design avoids assumptions of Laplacian symmetry, strong connectivity, or detail-balanced graphs required by prior methods.

What carries the argument

The distributed resilient fixed-time observer combined with the per-agent control law, which estimates system states and enforces output regulation resiliently to DoS interruptions without requiring graph symmetry.

If this is right

  • Regulated outputs reach zero within a time determined solely by design parameters, independent of initial conditions.
  • The controller applies to arbitrary directed graphs without symmetry or strong-connectivity requirements.
  • Agents tolerate temporary link failures as long as sufficient attack-free intervals remain for convergence.
  • A simulation example confirms that the outputs converge as predicted under the attack model.

Where Pith is reading between the lines

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

  • The fixed-time guarantee could support real-time certification of performance in networks where attack durations are bounded but unknown in advance.
  • Relaxing the linearity assumption to Lipschitz nonlinear dynamics would require only local modifications to the observer gain design.
  • The same resilience structure might transfer to event-triggered implementations that further reduce communication load during attack-free periods.
  • Similar recovery-interval arguments could be tested on other attack classes such as jamming or replay that also produce intermittent connectivity loss.

Load-bearing premise

The multi-agent system must be linear with known dynamics, and denial-of-service attacks must leave enough recovery intervals for the observer and controller to complete their fixed-time tasks.

What would settle it

Run the closed-loop system with the stated observer and controller under a sequence of DoS attacks whose total active duration exceeds the recovery intervals assumed in the stability analysis; if the regulated outputs fail to reach and stay at zero by the predicted fixed-time bound, the claim is falsified.

Figures

Figures reproduced from arXiv: 2604.13394 by Dan Zhang, Gang Feng, Lu Liu, Wenji Cao.

Figure 1
Figure 1. Figure 1: The adjacency weights are set to aij = 1 if (j, i) ∈ E, and aij = 0 otherwise. One can observe that Assumption 2 is satisfied. Agent i is modeled as an inverted pendulum on a cart [9]. The linearized model of agent i is given by (2), where Ai =    0 1 0 0 0 0 g 0 0 0 0 1 0 fi liM1i (M1i+M2i)g liM1i − fi M1i   , Bi =    0 0 0 1 liM1i   , Ei =    0 0 χi1+χi2 M1i 0 0 0 χi2 liM1i 0   , Ci = … view at source ↗
Figure 1
Figure 1. Figure 1: Communication graph G. Choose K = 1.78·I5, which guarantees that HT K +KH− 2I5 ≥ 0. Over the time interval [0, 160], the parameters for DoS attacks are selected as follows: νd = 0.2 and pd = 4.9. The design parameters of the observer (6) are specified as µ1 = 7.5, µ2 = 7, µ3 = 11, α = 0.7, β = 1.45. Then, one can calculate that c1 = 21.4662, c2 = 10.3531, c3 = 21.8649, c4 = 10.2899, c5 = 0.5727, cˆ1 = 0.07… view at source ↗
Figure 3
Figure 3. Figure 3: Profiles of kη˜(t)k for the fixed-time observer (6) (left) and the exponentially converging observer adapted from [18] (right). Table II: Comparison of convergence performance. α β µ1, µ2, µ3 Time (sec) 0.7 1.45 7.5, 7, 11.5 6.3057 8.5, 9.5, 12.5 6.2414 9.5, 13, 14.5 6.1877 0.75 1.35 9.9, 18.6, 18.6 6.1599 1.45 6.1583 1.55 6.1564 0.85 1.35 8, 11.2, 15 6.3111 0.8 6.2715 0.75 6.2391 Furthermore, several simu… view at source ↗
read the original abstract

This paper addresses the problem of fixed-time cooperative output regulation for linear multi-agent systems over directed graphs under denial-of-service attacks. A novel distributed resilient fixed-time controller is developed that comprises a distributed resilient fixed-time observer taking general directed graphs into consideration, and a distributed resilient fixed-time control law for each agent. The proposed controller neither depends on Laplacian symmetry nor requires strong connectivity and detail-balanced condition, in contrast to existing distributed resilient fixed-time controllers. Under the proposed controller, the regulated outputs converge to zero in a fixed time with its upper bound independent of the initial states of the multi-agent system. Ultimately, the efficacy of the proposed controller is demonstrated via a simulation example.

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

1 major / 2 minor

Summary. The paper develops a distributed resilient fixed-time observer and controller for linear multi-agent systems over directed graphs under DoS attacks to achieve cooperative output regulation. The design avoids Laplacian symmetry, strong connectivity, and detail-balance assumptions. It claims that regulated outputs converge to zero in a fixed time whose upper bound is independent of initial states, with efficacy shown via simulation.

Significance. If the fixed-time stability analysis holds with the claimed independence from initial conditions, the result strengthens resilient MAS control by extending fixed-time methods to general directed graphs and intermittent DoS without strong connectivity, providing a priori bounded convergence times useful for safety-critical applications.

major comments (1)
  1. [Stability analysis / Theorem on fixed-time convergence] The central fixed-time claim requires that the convergence-time bound T_max derived from the piecewise Lyapunov analysis (during attack-free and recovery intervals) is independent of x(0). The manuscript must explicitly show how attack frequency and duration parameters are absorbed into gain-dependent terms only, without residual dependence that would make the practical bound vary with attack statistics (see skeptic note on decay rates dominating growth).
minor comments (2)
  1. [Problem formulation] Clarify the precise DoS attack model (e.g., bounds on frequency and duration) and how the resilient observer recovers during communication intervals.
  2. [Simulation] In the simulation example, provide quantitative comparison of convergence times across different initial conditions to illustrate independence from x(0).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive review. We have revised the manuscript to explicitly strengthen the fixed-time stability analysis as requested.

read point-by-point responses

  1. Referee: [Stability analysis / Theorem on fixed-time convergence] The central fixed-time claim requires that the convergence-time bound T_max derived from the piecewise Lyapunov analysis (during attack-free and recovery intervals) is independent of x(0). The manuscript must explicitly show how attack frequency and duration parameters are absorbed into gain-dependent terms only, without residual dependence that would make the practical bound vary with attack statistics (see skeptic note on decay rates dominating growth).

    Authors: We agree that an explicit derivation is needed to confirm independence from initial conditions. In the revised proof of Theorem 3, we start from the piecewise Lyapunov derivative: during attack-free intervals, we have V̇ ≤ −k1 V^α − k2 V^β (0<α<1<β) which yields the standard fixed-time bound T1 = (1/(k1(1−α))) + (1/(k2(β−1))) after integration; during DoS intervals, V̇ ≤ k3 V with k3 depending on the attack-induced perturbation. Under the standard DoS assumption that the total attack duration up to time t satisfies ∫_0^t χ(τ)dτ ≤ ηt + Td (with known bounds η<1, Td), the number and length of attack intervals within any finite window are bounded a priori. Substituting these bounds into the integrated inequality produces an overall upper bound T_max that is a function solely of the controller gains k1,k2,k3 and the fixed DoS parameters η,Td; the initial value V(0) cancels out because of the fractional powers. The gain-selection conditions are then stated explicitly so that k1,k2 are chosen sufficiently large relative to k3(η,Td) to guarantee that the net decay dominates any growth, thereby absorbing the attack statistics into the gain terms. Once the gains are fixed, T_max is a numerical constant independent of x(0) and of any particular realization of the attack sequence (provided the sequence obeys the assumed bounds). We have added the full chain of inequalities and the resulting closed-form expression for T_max in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain; fixed-time bound derived from standard Lyapunov analysis.

full rationale

The paper designs a distributed resilient fixed-time observer and controller for linear MASs over directed graphs under DoS attacks, claiming regulated outputs converge in fixed time independent of initial states. This follows from piecewise Lyapunov analysis between attack intervals using standard fixed-time stability lemmas (not self-cited as load-bearing). No self-definitional steps, fitted parameters renamed as predictions, or ansatz smuggled via self-citation appear in the abstract or described derivation. The bound depends on gains, system matrices, and attack frequency/duration bounds as external parameters, without reducing to its own inputs by construction. The result is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the approach relies on standard assumptions from linear MAS control theory and graph theory. No free parameters, invented entities, or ad-hoc axioms are identifiable from the given text.

axioms (1)
  • domain assumption The multi-agent system dynamics are linear and the communication graph is directed.
    Implicit in the problem statement for cooperative output regulation.

pith-pipeline@v0.9.0 · 5416 in / 1245 out tokens · 45138 ms · 2026-05-10T13:26:05.118395+00:00 · methodology

discussion (0)

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