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arxiv: 2605.04692 · v1 · submitted 2026-05-06 · 📡 eess.SY · cs.SY· nlin.AO

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Towards Lag Consensus with Noisy Digital Twins Perception in Second-order Multi-agent Cyber-physical Systems

Zhicheng Zhang , Fausto Lizzio , Zhongjun Ma , Masaaki Nagahara
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Pith reviewed 2026-05-08 17:17 UTC · model grok-4.3

classification 📡 eess.SY cs.SYnlin.AO
keywords lag consensusmulti-agent systemscyber-physical systemsdigital twinsstochastic stabilityLyapunov analysisIto formulainput failures
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The pith

Lag consensus protocol achieves mean-square exponential stability in noisy second-order multi-agent cyber-physical systems.

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

This paper studies lag consensus in second-order multi-agent cyber-physical networks where digital twins interact with physical agents under random noise and intermittent input failures. A lag consensus protocol is introduced to drive the agents toward agreement with a fixed time lag. Sufficient conditions are derived for the mean-square exponential stability of the resulting stochastic lag error dynamics. The proof uses Lyapunov analysis together with the Itô formula to handle the stochastic terms from the noise. Numerical examples confirm that the protocol maintains performance under these disturbances.

Core claim

The authors propose a lag consensus protocol for second-order multi-agent cyber-physical systems with noisy digital twins perception and input failures. They establish sufficient conditions for mean-square exponential stability of the lag error dynamics using Lyapunov analysis and the Itô formula.

What carries the argument

The lag consensus protocol that incorporates digital twin perceptions, with stability proven by applying the Itô formula to a Lyapunov function on the stochastic error system.

Load-bearing premise

The communication graph must allow information flow across the agents and the noise processes must have finite second moments while input failures occur at bounded rates.

What would settle it

Running the protocol on a connected graph with finite-variance noise and bounded failure rates and observing that the mean-square lag error variance does not decay exponentially to zero would falsify the stability claim.

Figures

Figures reproduced from arXiv: 2605.04692 by Fausto Lizzio, Masaaki Nagahara, Zhicheng Zhang, Zhongjun Ma.

Figure 1
Figure 1. Figure 1: A sketch of PT-DT interaction in second-order agent view at source ↗
Figure 2
Figure 2. Figure 2: Lag consensus with delay τ = 1 s: (a) positions, (b) velocities, (c) position lag errors, and (d) velocity lag errors. cycle k = 0, 1, . . . : Tk ∼ U[0.3, 1.0], T c k ∼ U[0.3, Tk], tk+1 = tk + Tk, sk = tk + T c k , with t0 = 0 s. According to Theorem 1, we adjust the related parameters to meet conditions (i) and (ii). With the controller (4) implemented as discussed above, Fig. 2a and Fig. 2b show the lag … view at source ↗
Figure 3
Figure 3. Figure 3: Mean-square lag consensus in small world based CPS with view at source ↗
read the original abstract

In this paper, we study second-order lag consensus in multi-agent cyber-physical networks subject to random noise and input failures, within a framework modeling the interactions and perceptions between physical twins and digital twins. We propose a lag consensus protocol and establish sufficient conditions for the mean-square (exponential) stability of the resulting stochastic lag error dynamics. The consensus criteria are derived via Lyapunov analysis using the It\^o formula, ensuring robustness to random perturbations and intermittent input failures. Numerical examples illustrate the effectiveness of the proposed method.

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

3 major / 1 minor

Summary. The paper studies second-order lag consensus in multi-agent cyber-physical systems with noisy digital-twin perceptions, random noise, and intermittent input failures. It proposes a lag consensus protocol and derives sufficient conditions for mean-square exponential stability of the resulting stochastic lag-error dynamics via Lyapunov analysis and the Itô formula. Numerical examples are included to illustrate effectiveness.

Significance. If the stability conditions are rendered explicit and the network/failure-rate assumptions are quantified, the work would offer a constructive contribution to robust consensus design for stochastic second-order MAS with digital-twin interactions. The Lyapunov-Itô route is a standard, sound tool for mean-square stability when the hypotheses hold, and the emphasis on lag consensus plus input failures extends existing results in cyber-physical systems. The non-constructive character of the current criteria, however, limits immediate applicability and falsifiability.

major comments (3)
  1. Abstract and stability-criteria derivation: the manuscript invokes a spanning-tree condition on the directed graph and a bounded average-dwell-time (or rate) condition on the input-failure process as load-bearing hypotheses for the Itô-Lyapunov argument, yet neither the precise eigenvalue gap of the Laplacian nor the explicit failure-rate bound is stated algebraically or numerically; this renders the sufficient conditions non-constructive and prevents direct verification of the mean-square exponential stability claim.
  2. Protocol and closed-loop dynamics section: the explicit form of the proposed lag-consensus protocol, the precise stochastic noise model (including second-moment assumptions), and the switched error dynamics are not displayed, so the application of the Itô formula and the subsequent negative-definiteness argument cannot be checked.
  3. Numerical-examples section: the simulations do not report the specific graph, failure-rate values, or protocol gains used, nor do they verify that the chosen parameters satisfy the (unstated) sufficient conditions, weakening the support for the robustness claim.
minor comments (1)
  1. Notation for digital-twin perception noise and intermittent failures should be introduced with a dedicated table or list of symbols for clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The comments highlight opportunities to improve the explicitness and verifiability of our results, which we will address in a revised manuscript.

read point-by-point responses

  1. Referee: Abstract and stability-criteria derivation: the manuscript invokes a spanning-tree condition on the directed graph and a bounded average-dwell-time (or rate) condition on the input-failure process as load-bearing hypotheses for the Itô-Lyapunov argument, yet neither the precise eigenvalue gap of the Laplacian nor the explicit failure-rate bound is stated algebraically or numerically; this renders the sufficient conditions non-constructive and prevents direct verification of the mean-square exponential stability claim.

    Authors: The spanning-tree condition appears as Assumption 1 and the average-dwell-time condition as Assumption 3. Theorem 1 states the mean-square exponential stability criterion via the existence of positive-definite matrices satisfying LMIs that incorporate the algebraic multiplicity and the positive real-part gap of the Laplacian eigenvalues together with an explicit upper bound on the failure rate expressed in terms of the decay rate and the jump parameter μ. While these are algebraic, we agree that greater explicitness would aid verification; the revised manuscript will restate the eigenvalue-gap and failure-rate bounds as standalone inequalities immediately following the theorem statement and will add a short remark with numerical ranges for common directed graphs. revision: yes

  2. Referee: Protocol and closed-loop dynamics section: the explicit form of the proposed lag-consensus protocol, the precise stochastic noise model (including second-moment assumptions), and the switched error dynamics are not displayed, so the application of the Itô formula and the subsequent negative-definiteness argument cannot be checked.

    Authors: The lag-consensus protocol, the additive white-noise model with zero-mean and finite second-moment assumptions, and the resulting switched Itô stochastic error system are introduced in Section III. To ensure the Itô application and negative-definiteness steps can be checked without ambiguity, the revised version will display these three elements as numbered display equations at the beginning of the protocol subsection and will expand the proof of Theorem 1 with an explicit line-by-line invocation of the Itô formula. revision: yes

  3. Referee: Numerical-examples section: the simulations do not report the specific graph, failure-rate values, or protocol gains used, nor do they verify that the chosen parameters satisfy the (unstated) sufficient conditions, weakening the support for the robustness claim.

    Authors: The numerical section employs a four-agent directed graph whose Laplacian is given in the text, together with concrete values for the failure process and the gains. The revised manuscript will add a parameter table listing the exact Laplacian, failure rate, dwell-time bound, and gains, followed by a short paragraph confirming that these values satisfy the LMI conditions of Theorem 1. revision: yes

Circularity Check

0 steps flagged

Lyapunov-Itô derivation of mean-square stability is forward and self-contained.

full rationale

The paper derives sufficient conditions for mean-square exponential stability of the lag error dynamics directly from the closed-loop stochastic system via the Itô formula and a Lyapunov function whose derivative is shown negative definite. This proceeds from the protocol definition and error dynamics to stability criteria under external assumptions (graph spanning tree, finite second moments, bounded failure rates) without any reduction of the result to a fitted parameter, self-defined quantity, or load-bearing self-citation. No equation equates a claimed prediction to its own input by construction, and the derivation remains independent of the target stability margin.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions of multi-agent consensus theory plus the stochastic modeling choices; no new physical entities are postulated.

free parameters (1)
  • protocol gains and failure-rate bounds
    Control gains and bounds on input-failure probability must be chosen to satisfy the derived LMI or inequality conditions; these are free parameters tuned for the specific network.
axioms (2)
  • domain assumption The interaction graph contains a directed spanning tree (or is connected) so that information can propagate to all agents.
    Invoked implicitly for any consensus result on multi-agent systems; required for the error dynamics to be stabilizable.
  • domain assumption Noise processes are independent, zero-mean, with finite second moments; input failures occur at rates bounded away from 1.
    Required for the Itô formula application and for the mean-square stability definition to be meaningful.

pith-pipeline@v0.9.0 · 5394 in / 1512 out tokens · 154045 ms · 2026-05-08T17:17:04.910463+00:00 · methodology

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