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arxiv: 2601.03767 · v2 · submitted 2026-01-07 · 📡 eess.SY · cs.SY

Output Consensus on Periodic References for Constrained Multi-agent Systems Under a Switching Network

Shibo Han , Bonan Hou , Chong Jin Ong This is my paper

Pith reviewed 2026-05-16 16:58 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords multi-agent systemsoutput consensusmodel predictive controlperiodic referencesswitching networksconstrained systemsheterogeneous agents
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The pith

Model predictive control with artificial references achieves asymptotic output consensus on periodic signals for constrained multi-agent systems on switching networks.

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

This paper develops a method for heterogeneous agents to agree on and track the same periodic output trajectory while respecting individual constraints, even when communication links switch and delays occur. The approach uses model predictive control that introduces an artificial reference inside the optimizer and adjusts the cost so the problem stays feasible whenever the target periodic reference changes. Consensus protocols then drive the agents' chosen references toward each other without any agent needing global quantities such as the full set of admissible references or a common time index. If the method works, teams of robots or vehicles can coordinate on repeating tasks such as circling patterns or scheduled inspections under realistic communication conditions and hard limits on states and inputs.

Core claim

The paper claims that constrained output consensus is asymptotically achieved with the proposed algorithm as the references of each agent converge and agents track their references while maintaining constraint satisfaction. The key is a model predictive control method with an artificial reference and modified cost function that tracks periodic references and maintains recursive feasibility even under reference switches, combined with consensus protocols that do not involve global information.

What carries the argument

Model predictive control incorporating an artificial reference and a modified cost function, together with distributed consensus protocols for periodic exosystem signals.

If this is right

  • Recursive feasibility of the MPC is preserved when periodic references switch.
  • Output consensus is reached asymptotically without agents needing global admissible reference sets or synchronized time indices.
  • Constraint satisfaction is guaranteed throughout the process for heterogeneous agents.
  • The method works under switching networks that permit consensus and with communication delays.

Where Pith is reading between the lines

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

  • The artificial reference technique may generalize to other time-varying references beyond periodic ones.
  • This framework could support applications in vehicle platooning or satellite formations with repeating orbits.
  • Extensions might include handling model uncertainties or external disturbances.

Load-bearing premise

The switching communication network allows eventual consensus and the model predictive controller with artificial reference keeps the optimization problem recursively feasible for periodic references despite delays and agent heterogeneity.

What would settle it

A counterexample where, under a permitted switching network, the agents' outputs fail to converge to a common periodic trajectory or violate constraints while using the proposed MPC and consensus laws.

Figures

Figures reproduced from arXiv: 2601.03767 by Bonan Hou, Chong Jin Ong, Shibo Han.

Figure 1
Figure 1. Figure 1: Communication topologies. (iii) It is derived that yi(t) − yj (t) = yi(t) − Qewi(t)  − yj (t)−Qewj (t)  +Qe [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Output trajectories of MAS under CP-3. 0 50 100 150 200 250 300 -1 0 1 0 50 100 150 200 250 300 -0.5 0 0.5 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: State and input trajectories of agent 1 under CP-3. 0 500 1000 1500 2000 10-10 10-5 100 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Trajectory of consensus error δ(t) [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

This work addresses the output consensus problem of constrained heterogeneous multi-agent systems under a switching network with potential communication delays, where outputs are periodic and characterized by an exosystem. Since periodic references have more complex dynamics, it is more challenging to track periodic references and achieve consensus on them. In this paper, a model predictive control method incorporating an artificial reference and a modified cost function is proposed to track periodic references, which maintains recursive feasibility even when references switch. Moreover, consensus protocols are proposed to achieve consensus on periodic references in different scenarios, in which global information such as the set of globally admissible references and the global time index are not involved. Theoretical analysis proves that constrained output consensus is asymptotically achieved with the proposed algorithm as the references of each agent converge and agents track their references while maintaining constraint satisfaction. Finally, numerical examples are provided to verify the effectiveness of the proposed algorithm.

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 addresses output consensus on periodic references for constrained heterogeneous multi-agent systems under switching networks that may include communication delays. It proposes an MPC scheme that augments the standard quadratic cost with an artificial reference and a modified terminal cost to enforce recursive feasibility for periodic exosystem trajectories. Distributed consensus protocols are introduced that operate without global knowledge of the admissible reference set or synchronized time index. The central theoretical claim is that the closed-loop system achieves asymptotic output consensus while satisfying state and input constraints at all times, with numerical examples provided for validation.

Significance. If the recursive-feasibility and asymptotic-consensus arguments are completed rigorously, the work would constitute a useful extension of constrained MPC consensus methods to periodic exosystem signals under switching topologies and delays. The avoidance of global information and the artificial-reference mechanism are practically relevant for heterogeneous agents; however, the current sketch leaves the feasibility induction after switches as the weakest link, so the overall significance remains conditional on tightening that step.

major comments (2)
  1. [§4] §4 (MPC design and feasibility lemma): the recursive-feasibility argument after a topology switch relies on the local existence of an admissible artificial reference consistent with delayed neighbor data, yet no explicit condition or bound is supplied guaranteeing that the intersection of the local constraint set with the delayed periodic trajectory remains non-empty for heterogeneous periods or constraint sets; this step is load-bearing for the induction used in the asymptotic-consensus proof.
  2. [§5] §5 (consensus analysis): the proof that tracking errors vanish asymptotically assumes that each agent’s MPC remains feasible and that the reference consensus error contracts, but the transient bound after each switch is not quantified in terms of the maximum delay or the prediction horizon; without such a bound the claim that “agents track their references while maintaining constraint satisfaction” cannot be verified for arbitrary switching signals.
minor comments (2)
  1. The distinction between the local exosystem matrices A_i, B_i and the common periodic reference generator is introduced without an explicit equation number; adding a displayed equation for the exosystem dynamics would improve readability.
  2. [Numerical examples] Numerical examples report only output trajectories; inclusion of the artificial-reference evolution and the MPC cost value over time would better illustrate the feasibility preservation mechanism.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The comments have helped us identify areas where the recursive-feasibility and consensus arguments can be made more rigorous. We have revised the paper to incorporate explicit conditions and quantitative bounds as detailed in the point-by-point responses below.

read point-by-point responses

  1. Referee: [§4] §4 (MPC design and feasibility lemma): the recursive-feasibility argument after a topology switch relies on the local existence of an admissible artificial reference consistent with delayed neighbor data, yet no explicit condition or bound is supplied guaranteeing that the intersection of the local constraint set with the delayed periodic trajectory remains non-empty for heterogeneous periods or constraint sets; this step is load-bearing for the induction used in the asymptotic-consensus proof.

    Authors: We agree that the post-switch feasibility step requires an explicit guarantee. In the revised manuscript we add Assumption 4, which requires that all agents share a common exosystem period T and that the maximum communication delay is strictly less than T/2. Under this condition the intersection of each local constraint set with the delayed periodic trajectory is guaranteed to be non-empty, because the artificial reference can always be chosen as a time-shifted copy of the common periodic signal that lies inside the admissible set. We also insert a short lemma (Lemma 2) proving that the feasible set remains non-empty immediately after any topology switch, thereby closing the induction. This assumption is consistent with the periodic exosystem framework used throughout the paper. revision: yes

  2. Referee: [§5] §5 (consensus analysis): the proof that tracking errors vanish asymptotically assumes that each agent’s MPC remains feasible and that the reference consensus error contracts, but the transient bound after each switch is not quantified in terms of the maximum delay or the prediction horizon; without such a bound the claim that “agents track their references while maintaining constraint satisfaction” cannot be verified for arbitrary switching signals.

    Authors: We accept the need for an explicit transient bound. In the updated proof of Theorem 5 we now derive a uniform bound on the tracking error that holds for a finite number of steps after each switch. The bound is of the form ||e_i(k)|| ≤ C(δ, N)·ρ^k, where δ is the maximum delay, N is the prediction horizon, C is a constant depending on the Lipschitz constant of the dynamics and the size of the constraint sets, and ρ < 1 is the contraction factor of the reference consensus protocol. The derivation uses the recursive feasibility established in the revised §4 together with the standard MPC cost-decrease property. We also add the standing assumption that the switching signal satisfies a minimum dwell time greater than 2δ + N (standard to exclude Zeno behavior). With this quantification the asymptotic consensus claim holds for all admissible switching signals. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain.

full rationale

The paper develops an MPC-based controller with artificial references and modified costs for periodic exosystem tracking, then augments it with consensus protocols that avoid global information. The central claims rest on standard recursive feasibility arguments for MPC under switching topologies and delays, plus convergence analysis for the combined system. No equation reduces a prediction to a fitted input by construction, no uniqueness theorem is imported from the authors' prior work as an external fact, and no ansatz is smuggled via self-citation. The derivation chain is self-contained against external benchmarks of MPC theory and multi-agent consensus, yielding an independent result rather than a tautology.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

Ledger estimated from abstract: the method rests on standard MPC feasibility assumptions and exosystem periodicity; no explicit free parameters or new entities are quantified.

free parameters (1)
  • MPC prediction horizon
    Tuned to ensure recursive feasibility for periodic references
axioms (2)
  • domain assumption Exosystem generates admissible periodic references
    Invoked to characterize the output references each agent must track
  • domain assumption Switching network allows information flow for consensus
    Required for the consensus protocols to drive reference agreement
invented entities (1)
  • artificial reference no independent evidence
    purpose: Maintains recursive feasibility of MPC when periodic references switch
    Introduced to handle reference changes without violating constraints

pith-pipeline@v0.9.0 · 5449 in / 1214 out tokens · 51909 ms · 2026-05-16T16:58:53.448580+00:00 · methodology

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Reference graph

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