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arxiv: 2603.10836 · v4 · pith:W5GWZTTEnew · submitted 2026-03-11 · 📡 eess.SY · cs.SY

Distributed Safety Critical Control among Uncontrollable Agents Using Reconstructed Control Barrier Functions

Yuzhang Peng , Wei Wang , Jiaqi Yan , Mengze Yu This is my paper

Pith reviewed 2026-05-22 10:51 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords multi-agent systemscontrol barrier functionsdistributed controlsafety-critical controlquadratic programmingadaptive observersuncontrollable agents
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The pith

Reconstructed control barrier functions let multi-agent systems enforce safety through local controllers even when some agents cannot be controlled.

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

The paper tries to show how groups of agents can stay safe during collaborative tasks when some members have uncertain and uncontrollable behaviors. Standard control barrier functions create coupled constraints that depend on multiple agents' states at once, which blocks a fully distributed solution. By using a distributed adaptive observer to estimate other agents' states and then adjusting the reconstruction with a prescribed performance parameter, the authors make it possible for each agent to satisfy a local constraint that still implies the original global safety condition. They wrap this in a quadratic programming controller and prove the whole system remains safe. A reader would care because this removes a major obstacle to deploying robot teams or vehicle groups in real settings where perfect control over every participant is impossible.

Core claim

By reconstructing the coupled control barrier function from state estimates supplied by a distributed adaptive observer and modifying it with a prescribed performance adaptive parameter, the authors obtain a local constraint whose satisfaction is sufficient to meet the original coupled constraint; a quadratic programming controller built on this reconstructed function then guarantees safety of the multi-agent system even in the presence of uncontrollable agents with uncertain dynamics.

What carries the argument

Reconstructed control barrier function, formed by combining distributed adaptive observer estimates with a prescribed performance adaptive parameter so that the local version enforces the original coupled safety condition.

If this is right

  • The quadratic programming controller can be solved locally at each controllable agent without requiring full state sharing.
  • Safety is maintained even when uncontrollable agents exhibit uncertain dynamic behaviors.
  • The reconstruction step decouples the original interdependent CBF constraints while preserving their safety guarantee.
  • The overall scheme yields a rigorous proof of forward invariance of the safe set for the entire multi-agent system.

Where Pith is reading between the lines

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

  • The same reconstruction idea might extend to other distributed constraint problems where coupling arises from shared safety specifications.
  • Physical robot experiments with occasional actuator failures would test how well the observer quality assumption holds under real sensor noise.
  • If the observer can be made robust to packet loss, the method could apply to teams communicating over unreliable networks.

Load-bearing premise

The distributed adaptive observer must generate state estimates of high enough quality and the prescribed performance parameter must be chosen so that satisfying the reconstructed constraint automatically satisfies the original coupled constraint.

What would settle it

A simulation or hardware trial in which the observer estimates are intentionally degraded or the performance parameter is deliberately mistuned, producing an unsafe state of the multi-agent system while the reconstructed constraint remains satisfied.

Figures

Figures reproduced from arXiv: 2603.10836 by Jiaqi Yan, Mengze Yu, Wei Wang, Yuzhang Peng.

Figure 2
Figure 2. Figure 2: Trajectories of 4 robots. 0 20 40 60 0 0.5 0 20 40 60 0 0.5 0 20 40 60 0 0.5 0 20 40 60 0 0.5 1 0 20 40 60 0 0.5 1 0 20 40 60 0 0.5 1 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Reconstructed CBF hˆ i and reconstruction error ei, i = 1, . . . , 3. The initial states of the robots are set as fol￾lows: [p1 (0)T , θ1(0)] = [4, 4, 0], [p2 (0)T , θ2(0)] = [4, 3, 0], [p3 (0)T , θ3(0)] = [4, 0.5, −π], [p4 (0)T , θ4(0)] = [3.5, 0.6, −π]. The distributed observers here are used solely for estimating pi and the distributed observers parameters are set as: the initial estimate xˆi,j is set t… view at source ↗
Figure 4
Figure 4. Figure 4: Control input ui, i = 1, . . . , 4. The trajectories of the robots are presented in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

This paper investigates the distributed safety critical control for multi-agent systems (MASs) in the presence of uncontrollable agents with uncertain behaviors. To ensure system safety, the control barrier function (CBF) is employed in this paper. However, a key challenge is that the CBF constraints are coupled when MASs perform collaborative tasks, which depend on information from multiple agents and impede the design of a fully distributed safe control scheme. To overcome this, a novel reconstructed CBF approach is proposed. In this method, the coupled CBF is reconstructed by leveraging state estimates of other agents obtained from a distributed adaptive observer. Furthermore, a prescribed performance adaptive parameter is designed to modify this reconstruction, ensuring that satisfying the reconstructed CBF constraint is sufficient to meet the original coupled one. Based on the reconstructed CBF, we design a safety-critical quadratic programming (QP) controller and prove that the proposed distributed control scheme rigorously guarantees the safety of the MAS, even in the uncertain dynamic environments involving uncontrollable agents. The effectiveness of the proposed method is illustrated through a simulation.

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 manuscript proposes a reconstructed control barrier function (CBF) method for distributed safety-critical control of multi-agent systems (MAS) containing uncontrollable agents with uncertain dynamics. A distributed adaptive observer provides state estimates of neighboring agents; these estimates are used to reconstruct the coupled CBF, which is further modified by a tunable prescribed performance adaptive parameter so that satisfaction of the reconstructed constraint is claimed to imply satisfaction of the original coupled CBF. A quadratic-programming (QP) safety filter is then synthesized, and the authors assert a rigorous proof that the resulting distributed controller guarantees safety of the MAS. Effectiveness is illustrated by a single simulation example.

Significance. If the central sufficiency argument between the reconstructed and original CBF holds for arbitrary uncertain inputs, the work would meaningfully extend CBF-based safety control to partially observed, distributed MAS settings with uncontrollable agents. The combination of adaptive observers and performance functions directly targets the coupling and uncertainty obstacles that currently limit fully distributed implementations. The result would be of interest to the safety-critical control community provided the error-bound derivation is made explicit and independent of unknown signals.

major comments (2)
  1. [Section on reconstructed CBF and sufficiency proof (preceding the main safety theorem)] The proof that any control satisfying the reconstructed CBF constraint also satisfies the original coupled CBF (the step that justifies the subsequent QP safety guarantee) requires that the observer error be ultimately bounded by a quantity that the prescribed performance adaptive parameter can dominate. Because the uncontrollable agents are described only as having 'uncertain behaviors,' their dynamics act as unknown inputs to the observer error system. The manuscript does not appear to supply an explicit ultimate bound on this error that is independent of the unknown input; without such a bound the reconstruction step does not rigorously close. This issue is load-bearing for the central safety claim.
  2. [QP controller design and safety proof] The QP controller is shown to enforce the reconstructed CBF constraint, but the safety conclusion for the original system inherits the same gap identified above. An explicit statement of the ultimate bound on observer error (or a design procedure that selects the performance parameter after the bound is known) is needed before the distributed safety guarantee can be considered established.
minor comments (2)
  1. Notation for the reconstructed CBF and the performance function should be introduced with a clear table or diagram showing the relationship to the original coupled CBF; current presentation makes it difficult to track which quantities are estimates versus true states.
  2. [Simulation results] The simulation section would benefit from reporting the actual observer error norms and the chosen performance parameter values so that readers can assess whether the error was indeed dominated as required by the proof.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments correctly identify that the sufficiency argument linking the reconstructed CBF to the original coupled CBF is central to the safety guarantee and requires an explicit, input-independent ultimate bound on the observer error. We address each major comment below and will incorporate the necessary clarifications and derivations in the revised manuscript.

read point-by-point responses

  1. Referee: [Section on reconstructed CBF and sufficiency proof (preceding the main safety theorem)] The proof that any control satisfying the reconstructed CBF constraint also satisfies the original coupled CBF (the step that justifies the subsequent QP safety guarantee) requires that the observer error be ultimately bounded by a quantity that the prescribed performance adaptive parameter can dominate. Because the uncontrollable agents are described only as having 'uncertain behaviors,' their dynamics act as unknown inputs to the observer error system. The manuscript does not appear to supply an explicit ultimate bound on this error that is independent of the unknown input; without such a bound the reconstruction step does not rigorously close. This issue is load-bearing for the central safety claim.

    Authors: We agree that an explicit ultimate bound independent of the unknown inputs is required to close the argument rigorously. The distributed adaptive observer in the manuscript is constructed so that the estimation error is ultimately bounded under the standard assumption of bounded uncertain dynamics. In the revision we will insert a new lemma immediately before the main safety theorem that derives this explicit ultimate bound using the Lyapunov analysis of the observer error system and the adaptive laws. We will then state a clear selection rule for the prescribed performance adaptive parameter that ensures it dominates the derived bound, thereby guaranteeing that satisfaction of the reconstructed CBF constraint implies satisfaction of the original coupled CBF. This addition will be placed in the section preceding the main safety theorem. revision: yes

  2. Referee: [QP controller design and safety proof] The QP controller is shown to enforce the reconstructed CBF constraint, but the safety conclusion for the original system inherits the same gap identified above. An explicit statement of the ultimate bound on observer error (or a design procedure that selects the performance parameter after the bound is known) is needed before the distributed safety guarantee can be considered established.

    Authors: We concur that the safety conclusion inherits the same requirement. Once the new lemma supplies the explicit bound and the corresponding design procedure for the performance parameter, the QP safety filter enforces the reconstructed constraint, which then rigorously implies safety of the original system. In the revised manuscript we will update the statement and proof of the main safety theorem to reference the new lemma and to include the explicit bound and parameter-selection procedure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs a reconstructed CBF from distributed adaptive observer estimates plus a prescribed performance parameter chosen to dominate observer error, then designs a QP controller whose safety follows from the standard CBF invariance condition applied to the reconstructed constraint. This is an explicit engineering design step rather than a self-definition or a fitted quantity renamed as a prediction. No load-bearing step reduces to its own input by construction, and the central safety guarantee does not rely on a self-citation chain or an ansatz smuggled from prior work by the same authors. The derivation therefore remains independent of the target result.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The method depends on the existence and convergence properties of the distributed adaptive observer and on the ability to choose the prescribed performance parameter to bridge the reconstructed and original constraints.

free parameters (1)
  • prescribed performance adaptive parameter
    Designed to modify the reconstruction so that the local constraint implies the original coupled safety constraint.
axioms (1)
  • domain assumption The distributed adaptive observer converges to sufficiently accurate state estimates of other agents despite uncertainties.
    Invoked to justify that the reconstructed CBF can be used in place of the coupled version.
invented entities (1)
  • reconstructed CBF no independent evidence
    purpose: Decouples the originally coupled safety constraints for distributed implementation.
    Constructed from observer estimates and the adaptive parameter.

pith-pipeline@v0.9.0 · 5714 in / 1251 out tokens · 35859 ms · 2026-05-22T10:51:58.006440+00:00 · methodology

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    Multirobot adversa rial resilience using control barrier functions,

    M. Cavorsi, L. Sabattini and S. Gil, “Multirobot adversa rial resilience using control barrier functions,” IEEE Transactions on Robotics , vol. 40, pp. 797–815, 2024

    work page 2024

  2. [2]

    Composite learning a daptive safety critical control with application to adaptive cruis e of intelligent vehicles,

    J. Shen, Y . Liu, W. Wang and Z. Wang, “Composite learning a daptive safety critical control with application to adaptive cruis e of intelligent vehicles,” IEEE Transactions on Industrial Electronics , vol. 72, no. 10, pp. 10793-10803, 2025

    work page 2025

  3. [3]

    Integration of prescribed p erformance with control barrier functions for attitude control and all ocation with reaction wheels,

    H. Y ang, H. Dong and X. Zhao, “Integration of prescribed p erformance with control barrier functions for attitude control and all ocation with reaction wheels,” IEEE Transactions on Aerospace and Electronic Systems, vol. 61, no. 2, pp. 1775–1786, 2025

    work page 2025

  4. [4]

    Real-time obstacle avoidance for manipulat ors and mobile robots,

    O. Khatib, “Real-time obstacle avoidance for manipulat ors and mobile robots,” The International Journal of Robotics Research , vol. 5, no. 1, pp. 90–98, 1986

    work page 1986

  5. [5]

    Robust adaptive control of feedback linearizable mimo nonlinear systems with presc ribed performance,

    C. P . Bechlioulis and G. A. Rovithakis, “Robust adaptive control of feedback linearizable mimo nonlinear systems with presc ribed performance,” IEEE Transactions on Automatic Control , vol. 53, no. 9, pp. 2090–2099, 2008

    work page 2090

  6. [6]

    Nonovershooting Control of str ict- feedback nonlinear systems,

    M. Krstic and M. Bement, “Nonovershooting Control of str ict- feedback nonlinear systems,” IEEE Transactions on Automatic Con- trol, vol. 51, no. 12, pp. 1938–1943, 2006

    work page 1938

  7. [7]

    Control b arrier function based quadratic programs for safety critical syst ems,

    A. D. Ames, X. Xu, J. W. Grizzle, and P . Tabuada, “Control b arrier function based quadratic programs for safety critical syst ems,” IEEE Transactions on Automatic Control , vol. 62, no. 8, pp. 3861–3876, 2017

    work page 2017

  8. [8]

    Receding horizon c ontrol with online barrier function design under signal temporal logic specifica- tions,

    M. Charitidou and D. V . Dimarogonas, “Receding horizon c ontrol with online barrier function design under signal temporal logic specifica- tions,” IEEE Transactions on Automatic Control , vol. 68, no. 6, pp. 3545–3556, 2023

    work page 2023

  9. [9]

    Safety barrier cert ificates for collisions-free multirobot systems,

    L. Wang, A. D. Ames and M. Egerstedt, “Safety barrier cert ificates for collisions-free multirobot systems,” IEEE Transactions on Robotics , vol. 33, no. 3, pp. 661–674, 2017

    work page 2017

  10. [10]

    Barrier function b ased collab- orative control of multiple robots under signal temporal lo gic tasks,

    L. Lindemann and D. V . Dimarogonas, “Barrier function b ased collab- orative control of multiple robots under signal temporal lo gic tasks,” IEEE Transactions on Control of Network Systems , vol. 7, no. 4, pp. 1916–1928, 2020

    work page 1916

  11. [11]

    A continuous-time violation-free multi-agent optimizatio n algorithm and its applications to safe distributed control,

    X. Tan, C. Liu, K. H. Johansson and D. V . Dimarogonas, “A continuous-time violation-free multi-agent optimizatio n algorithm and its applications to safe distributed control,” IEEE Transactions on Automatic Control, vol. 70, no. 8, pp. 5114–5128, 2025

    work page 2025

  12. [12]

    Distribut ed safe nav- igation of multi-agent systems using control barrier funct ion-based controllers,

    P . Mestres, C. Nieto-Granda and J. Cort´ es, “Distribut ed safe nav- igation of multi-agent systems using control barrier funct ion-based controllers,” IEEE Robotics and Automation Letters , vol. 9, no. 7, pp. 6760–6767, 2024

    work page 2024

  13. [13]

    On partially control led multi- agent systems,

    R. I. Brafman and M. Tennenholtz, “On partially control led multi- agent systems,” Journal of Artificial Intelligence Research , vol. 4, pp. 477–507, 1996

    work page 1996

  14. [14]

    Signal temporal logic c ontrol synthesis among uncontrollable dynamic agents with confor mal pre- diction,

    X. Y u, Y . Zhao and L. Lindemann, “Signal temporal logic c ontrol synthesis among uncontrollable dynamic agents with confor mal pre- diction,” Automatica, vol. 183, p. 112616, 2026

    work page 2026

  15. [15]

    Distributed resilient observer-ba sed fault- tolerant control for heterogeneous multiagent systems und er actuator faults and DoS attacks,

    C. Deng and C. Wen, “Distributed resilient observer-ba sed fault- tolerant control for heterogeneous multiagent systems und er actuator faults and DoS attacks,” IEEE Transactions on Control of Network Systems, vol. 7, no. 3, pp. 1308–1318, 2020

    work page 2020

  16. [16]

    Control barrier fu nctions for multi-agent systems under conflicting local signal temp oral logic tasks,

    L. Lindemann and D. V . Dimarogonas, “Control barrier fu nctions for multi-agent systems under conflicting local signal temp oral logic tasks,” IEEE Control Systems Letters , vol. 3, no. 3, pp. 757–762, 2019

    work page 2019

  17. [17]

    Robust formation tr acking control for noncooperative heterogeneous multiagent syst ems,

    J. Chen, H. Mei, Z. Shi and Y . Zhong, “Robust formation tr acking control for noncooperative heterogeneous multiagent syst ems,” IEEE Transactions on Cybernetics , vol. 54, no. 10, pp. 5661–5671, 2024

    work page 2024

  18. [18]

    Godsil and G

    C. Godsil and G. F. Royle, Algebraic Graph Theory , vol. 207. New Y ork, NY , USA: Springer, 2001

    work page 2001

  19. [19]

    Command filter adaptive asymptotic tracking o f uncertain nonlinear systems with time-varying parameters and distur bances,

    Y . X. Li, “Command filter adaptive asymptotic tracking o f uncertain nonlinear systems with time-varying parameters and distur bances,” IEEE Transactions on Automatic Control , vol. 67, no. 6, pp. 2973– 2980, 2022

    work page 2022