Reference-Governed Distributed Safe Gradient Flow for Safe Optimal Output Agreement of Multi-Agent Systems

Bo Yang; Wei Xiao; Xinping Guan; Zhanglin Shangguan

REVIEW 1 major objections 2 minor 30 references

Reference-Governed Distributed Safe Gradient Flow for Safe Optimal Output Agreement of Multi-Agent Systems

T0 review · 1 major / 2 minor · reviewed 2026-07-03 · grok-4.3

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Pith's one-line read A reference-governed two-layer architecture ensures safe optimal output agreement without changing the steady-state optimum.

desk verdict The two-layer reference-governed architecture with first-order CBF filtering and DSMs from the internal-model Lyapunov function is the main new piece, but optimality preservation still rests on DSM-compatibility conditions that look assumed rather than guaranteed by the construction. read the letter →

arxiv 2607.02192 v1 pith:BATJZVE5 submitted 2026-07-02 eess.SY cs.SY

Zhanglin Shangguan , Wei Xiao , Bo Yang , Xinping Guan This is my paper · ORCID

classification eess.SYcs.SY

keywords safegradientflowmulti-agentsystemscontrolbarrierfunctionsoutputagreementdynamicsafetymarginsdistributedoptimizationLyapunovsmall-gain

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

This paper proposes a method to achieve safe optimal output agreement for nonlinear multi-agent systems that have output safety constraints. It uses a reference-governed two-layer architecture that separates the lower-layer output regulation from the upper-layer distributed optimization. The upper layer filters the reference gradient flow using first-order control barrier function constraints to make tuning easier and keep the steady-state optimality. The lower layer uses an internal-model-based output regulator to build dynamic safety margins that certify transient safety. The paper proves that this setup maintains forward invariance, preserves the optimal solution under certain conditions, and converges using a Lyapunov small-gain argument. A sympathetic reader would care because it offers a way to coordinate multiple agents safely without the problems of high-order barrier functions that can mess up the final solution.

What carries the argument

The reference-governed two-layer architecture that applies first-order control barrier functions to the reference gradient flow and derives dynamic safety margins from a reference-dependent Lyapunov function in the internal-model-based regulator.

What would settle it

Finding a case where the system reaches a steady-state different from the unconstrained optimal solution despite satisfying the DSM-compatibility conditions would falsify the preservation of optimality.

Watch

Extended reading notes

Core claim

The central claim is that the reference-governed distributed safe gradient flow, implemented via a two-layer architecture, filters the gradient flow with first-order CBFs in the upper layer and constructs DSMs from the lower-layer regulator to ensure safety and optimality preservation under DSM-compatibility conditions, with convergence shown by Lyapunov small-gain.

Load-bearing premise

The DSM-compatibility conditions hold so that the constructed dynamic safety margins do not alter the steady-state optimal solution of the original agreement problem.

Editorial extensions

If this is right

Forward invariance of the safe output sets is proven.
The original optimal solution is preserved if DSM-compatibility conditions hold.
Convergence to the optimal agreement is guaranteed by the small-gain argument.
Simulations show better performance than HOCBF methods and ability to escape spurious equilibria.

Reading between the lines

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

This separation of layers could simplify implementation in physical systems by allowing independent tuning of safety and optimization.
The approach might apply to other multi-agent tasks like formation control or resource allocation with safety needs.
Using dynamic safety margins could lead to less conservative safety enforcement compared to static barriers.

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Signed reviews

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main thing to know is that this paper puts forward a reference-governed two-layer split for safe optimal output agreement in nonlinear multi-agent systems. The upper layer runs distributed gradient flow filtered by first-order CBFs on the reference, while the lower layer uses an internal-model regulator whose reference-dependent Lyapunov function supplies the dynamic safety margins. This is meant to sidestep the tuning problems and extra equilibria that come with feeding gradient flow straight through the plant via HOCBFs.

What is actually new is the explicit separation plus the construction of DSMs directly from that Lyapunov function, together with the claim that this keeps the steady-state optimum intact under DSM-compatibility conditions. The abstract also flags a Lyapunov small-gain argument for convergence and mentions simulations that compare against HOCBF methods and handle nonconvex obstacles via adaptive tangential shaping.

The paper does a straightforward job identifying the practical drawbacks of direct HOCBF-based feedback optimization. The two-layer idea is a reasonable engineering response to those drawbacks.

The soft spot is exactly where the stress-test points: optimal-solution preservation is stated to hold only under DSM-compatibility conditions, yet the abstract gives no indication that the DSM construction itself forces those conditions to hold for general output constraints or nonconvex objectives. If compatibility is an extra requirement rather than a consequence of the architecture, the preservation result is conditional in a way that needs explicit verification. The abstract asserts existence of proofs for invariance and convergence but supplies no derivation details, assumptions, or quantitative simulation results, so the strength of the arguments is hard to judge from what is shown.

This is for people working on distributed control and safe optimization in robotics or networked systems who want a concrete architecture that separates regulation from optimization. It deserves a serious referee because the separation addresses a real implementation issue even if the compatibility step requires more work. I would send it to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes a reference-governed two-layer architecture for safe optimal output agreement of nonlinear multi-agent systems subject to output safety constraints. The upper layer implements distributed gradient flow filtered by first-order CBF constraints; the lower layer uses an internal-model regulator whose reference-dependent Lyapunov function yields dynamic safety margins (DSMs). The central claims are proofs of forward invariance, preservation of the original optimal solution under DSM-compatibility conditions, and convergence via a Lyapunov small-gain argument, supported by simulations that also illustrate adaptive tangential shaping to escape spurious equilibria.

Significance. If the DSM-compatibility conditions can be shown to hold identically for the constructed margins and the small-gain argument is fully rigorous, the two-layer separation would constitute a useful alternative to direct HOCBF feedback optimization, offering easier tuning while retaining steady-state optimality. The explicit construction of DSMs from the regulator Lyapunov function and the handling of nonconvex obstacles are potentially valuable contributions.

major comments (1)

[Abstract / main theorems section] Abstract and the section stating the main theorems: the optimal-solution preservation result is stated to hold only under DSM-compatibility conditions, yet the manuscript treats these conditions as an external requirement rather than proving that the DSMs constructed from the reference-dependent Lyapunov function satisfy them for general output safety constraints and nonconvex objectives. Because this condition is load-bearing for the claim that the architecture does not alter the steady-state optimum, the preservation theorem remains conditional.

minor comments (2)

[Abstract] The abstract states that simulations validate the claims but reports neither quantitative metrics, baseline comparisons, nor specific parameter values, making it difficult to assess the practical advantage over HOCBF methods.
[Preliminaries / notation section] Notation for the DSM-compatibility condition and the precise definition of the dynamic safety margins should be introduced with explicit equations before the main theorems to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback. We address the single major comment below.

read point-by-point responses

Referee: [Abstract / main theorems section] Abstract and the section stating the main theorems: the optimal-solution preservation result is stated to hold only under DSM-compatibility conditions, yet the manuscript treats these conditions as an external requirement rather than proving that the DSMs constructed from the reference-dependent Lyapunov function satisfy them for general output safety constraints and nonconvex objectives. Because this condition is load-bearing for the claim that the architecture does not alter the steady-state optimum, the preservation theorem remains conditional.

Authors: We agree that the preservation theorem is stated conditionally on DSM-compatibility. The DSMs are explicitly constructed from the reference-dependent Lyapunov function of the internal-model regulator, and the compatibility conditions are introduced precisely to guarantee that these margins do not alter the equilibria of the upper-layer gradient flow. The manuscript does not claim or prove that the constructed DSMs satisfy the conditions identically for arbitrary nonconvex objectives and general output safety constraints, because satisfaction depends on the specific regulator design, the chosen Lyapunov function, and the geometry of the constraints. Instead, the conditions are presented as verifiable design requirements that can be checked for given problem data. Simulations confirm satisfaction in the considered cases. We will revise the main theorems section to add a remark stating sufficient conditions (e.g., sufficiently small DSM scaling) under which compatibility holds and to clarify the scope of the result. This is a partial revision. revision: partial

Circularity Check

0 steps flagged · score 2.0 of 10

DSM-compatibility stated as external assumption; no reduction by construction or self-citation load-bearing

full rationale

The provided abstract and reader summary indicate the central preservation result is explicitly conditioned on DSM-compatibility conditions rather than derived from the DSM construction itself. No equations are shown reducing to fitted inputs or self-citations by construction. The architecture separates layers with Lyapunov small-gain convergence, which is independent of the compatibility assumption. This qualifies as a minor external requirement (score 2) rather than circularity, as the paper does not claim the conditions hold identically or derive them from the result.

Assumptions & free parameters 1 free parameters · 2 assumptions · 1 invented entities

The central claim depends on standard stability theorems and the newly introduced DSM-compatibility conditions; no explicit free parameters are fitted in the abstract, but tuning of CBF gains is implied.

free parameters (1)

CBF gain parameters
Parameters for the first-order control barrier function constraints in the upper layer, described as easier to tune than HOCBFs.

assumptions (2)

standard math Lyapunov small-gain theorem applies to the interconnected optimization and regulation layers
Invoked for the convergence proof.
domain assumption Internal-model principle holds for the output regulator design
Basis for the lower-layer regulator with reference-dependent Lyapunov function.

invented entities (1)

Dynamic Safety Margins (DSMs)
purpose: Time-varying bounds to certify transient output safety
Constructed from the reference-dependent Lyapunov function in the lower layer.

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Cite this review

Pith. "Pith review of Reference-Governed Distributed Safe Gradient Flow for Safe Optimal Output Agreement of Multi-Agent Systems." pith.science (2026). https://pith.science/paper/BATJZVE5

@misc{pith2026260702192,
  author       = {Pith},
  title        = {Pith review of: Reference-Governed Distributed Safe Gradient Flow for Safe Optimal Output Agreement of Multi-Agent Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BATJZVE5}},
  note         = {Machine review of arXiv:2607.02192}
}

read the original abstract

This paper studies safe optimal output agreement for nonlinear multi-agent systems with output safety constraints. Existing safe feedback optimization methods often implement gradient-flow dynamics directly through the plant input, which may require high-order control barrier functions (HOCBFs). The resulting derivative-chain design is tuning-sensitive and can introduce additional equilibrium conditions that alter the steady-state optimal solution. We propose a reference-governed two-layer architecture that separates lower-layer output regulation from upper-layer distributed optimization. The upper layer filters the reference gradient flow through first-order control barrier function constraints, which are easier to tune and preserve the steady-state optimality structure of the original agreement problem. The lower layer uses an internal-model-based output regulator with a reference-dependent Lyapunov function, from which dynamic safety margins (DSMs) are constructed to certify transient output safety. We prove forward invariance, optimal-solution preservation under DSM-compatibility conditions, and convergence via a Lyapunov small-gain argument. Simulations validate safe convergence, show advantages over HOCBF-based feedback optimization, and demonstrate adaptive tangential objective shaping for escaping spurious equilibria induced by nonconvex obstacles.

Figures

Figures reproduced from arXiv: 2607.02192 by the authors.

**Figure 2.** Trajectories of the SGF-HOCBF baseline under [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 5.** Average distance to the constrained optimal solution [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

Figures from the paper (2 more)

**Figure 6.** Figure 6: Safety diagnostics of the proposed DSM reference [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: Trajectories with a nonconvex circular obstacle: DSM [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Reviewed July 3, 2026 · model on record in the stance chip above.

[1] [1]

Online optimization as a feedback controller: Stability and tracking.IEEE Transac- tions on Control of Network Systems, 7(1):422–432, 2019

Marcello Colombino, Emiliano Dall’Anese, and An- drey Bernstein. Online optimization as a feedback controller: Stability and tracking.IEEE Transac- tions on Control of Network Systems, 7(1):422–432, 2019

work page 2019

[2] [2]

Feed- back optimization of nonlinear strict-feedback sys- tems.Journal of Systems Science and Complexity, 38(2):717–738, 2025

Tong Liu, Tengfei Liu, and Zhong-Ping Jiang. Feed- back optimization of nonlinear strict-feedback sys- tems.Journal of Systems Science and Complexity, 38(2):717–738, 2025

work page 2025

[3] [3]

Event-triggered and periodic event- triggered extremum seeking control.Automatica, 174:112161, 2025

Victor Hugo Pereira Rodrigues, Tiago Roux Oliveira, Liu Hsu, Mamadou Diagne, and Miroslav Krstic. Event-triggered and periodic event- triggered extremum seeking control.Automatica, 174:112161, 2025

work page 2025

[4] [4]

Global optimal con- sensus for higher-order multi-agent systems with bounded controls.Automatica, 99:301–307, 2019

Yijing Xie and Zongli Lin. Global optimal con- sensus for higher-order multi-agent systems with bounded controls.Automatica, 99:301–307, 2019

work page 2019

[5] [5]

Multi-robot target monitoring and encirclement via triggered dis- tributed feedback optimization.IEEE Transactions on Robotics, 2026

Lorenzo Pichierri, Guido Carnevale, Lorenzo Sforni, and Giuseppe Notarstefano. Multi-robot target monitoring and encirclement via triggered dis- tributed feedback optimization.IEEE Transactions on Robotics, 2026

work page 2026

[6] [6]

Real- time feedback-based optimization of distribution grids: A unified approach.IEEE Transactions on Control of Network Systems, 6(3):1197–1209, 2019

Andrey Bernstein and Emiliano Dall’Anese. Real- time feedback-based optimization of distribution grids: A unified approach.IEEE Transactions on Control of Network Systems, 6(3):1197–1209, 2019

work page 2019

[7] [7]

Distributed control to steer dynamical systems to the generalized nash equilibria for monotone aggregative games with 14 operational constraints.Automatica, 185:112794, 2026

Ming Li, Zhaojian Wang, Mengshuo Jia, Feng Liu, Bo Yang, and Xinping Guan. Distributed control to steer dynamical systems to the generalized nash equilibria for monotone aggregative games with 14 operational constraints.Automatica, 185:112794, 2026

work page 2026

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Distributed optimization for control.Annual Review of Control, Robotics, and Autonomous Systems, 1(1):77–103, 2018

Angelia Nedi´ c and Ji Liu. Distributed optimization for control.Annual Review of Control, Robotics, and Autonomous Systems, 1(1):77–103, 2018

work page 2018