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arxiv: 1907.10897 · v1 · pith:6CP5SGW6new · submitted 2019-07-25 · 📡 eess.SY · cs.SY

Distributed Consensus for Multiple Lagrangian Systems with Parametric Uncertainties and External Disturbances Under Directed Graphs

Jie Mei This is my paper

Pith reviewed 2026-05-24 16:23 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords distributed consensusLagrangian systemsdirected graphsadaptive controlparametric uncertaintiesexternal disturbancesleaderless consensusweighted average consensus
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The pith

Lagrangian systems achieve weighted average consensus under directed graphs with an explicit equilibrium formula.

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

This paper develops distributed control algorithms for leaderless consensus among multiple Lagrangian systems that have parametric uncertainties and external disturbances. Under fixed directed graphs, an integral term in the auxiliary variable design makes it possible to derive the final consensus value explicitly as a weighted average that depends on the graph topology, the agents' initial positions, and the chosen control gains. For switching directed graphs that are uniformly jointly connected over time, a model reference adaptive algorithm ensures that consensus is still reached. Versions of the algorithms are also given that do not require accurate neighbor velocity measurements. These results matter for coordinating teams of robots or mechanical systems where communication is one-way and information about velocities is noisy or unavailable.

Core claim

By introducing an integrate term in the auxiliary variable design, the final consensus equilibrium can be explicitly derived for the case of a fixed directed graph. The agents achieve weighted average consensus, where the final equilibrium depends on the interactive topology, the initial positions of the agents, and the control gains of the proposed control algorithm. For switching directed graphs, a model reference adaptive consensus based algorithm ensures leaderless consensus provided the infinite sequence of switching graphs is uniformly jointly connected. Similar algorithms without using neighbors' velocity information are also proposed.

What carries the argument

The auxiliary variable with an added integrate term that enables explicit equilibrium derivation, combined with robust continuous terms with adaptive varying gains to handle disturbances.

If this is right

  • Agents reach asymptotic consensus without knowledge of disturbance bounds.
  • The equilibrium position is a weighted average determined by graph, initials, and gains.
  • Consensus holds for switching graphs under uniform joint connectivity.
  • Algorithms work using only local neighbor information without common control gains.
  • Versions exist that avoid needing neighbor velocity data.

Where Pith is reading between the lines

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

  • If the graph connectivity condition is met only marginally, the convergence rate may slow significantly.
  • The approach could extend to other Euler-Lagrange systems like manipulators in formation control tasks.
  • Testing on physical robot platforms would verify robustness to real-world disturbances beyond simulations.

Load-bearing premise

The communication graph is either fixed and directed or the sequence of switching graphs is uniformly jointly connected.

What would settle it

A simulation or experiment where the graph is not connected and the agents fail to reach a common position, or where the observed equilibrium deviates from the predicted weighted average based on initials and gains.

Figures

Figures reproduced from arXiv: 1907.10897 by Jie Mei.

Figure 1
Figure 1. Figure 1: The directed graph that characterizes the interaction among the six [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The angles and angle derivatives of the six agents using (9) under [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The angles and angle derivatives of the six agents using (35) under [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The two possible directed graphs that characterizes the interaction [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The angles and angle derivatives of the six agents using (27) under [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
read the original abstract

In this paper, we study the leaderless consensus problem for multiple Lagrangian systems in the presence of parametric uncertainties and external disturbances under directed graphs. For achieving asymptotic behavior, a robust continuous term with adaptive varying gains is added to alleviate the effects of the external disturbances with unknown bounds. In the case of a fixed directed graph, by introducing an integrate term in the auxiliary variable design, the final consensus equilibrium can be explicitly derived. We show that the agents achieve weighted average consensus, where the final equilibrium is dependent on three factors, namely, the interactive topology, the initial positions of the agents, and the control gains of the proposed control algorithm. In the case of switching directed graphs, a model reference adaptive consensus based algorithm is proposed such that the agents achieve leaderless consensus if the infinite sequence of switching graphs is uniformly jointly connected. Motivated by the fact that the relative velocity information is difficult to obtain accurately, we further propose a leaderless consensus algorithm with gain adaptation for multiple Lagrangian systems without using neighbors' velocity information. We also propose a model reference adaptive consensus based algorithm without using neighbors' velocity information for switching directed graphs. The proposed algorithms are distributed in the sense of using local information from its neighbors and using no comment control gains. Numerical simulations are performed to show the effectiveness of the proposed algorithms.

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

0 major / 2 minor

Summary. The paper studies leaderless consensus for multiple Lagrangian systems with parametric uncertainties and bounded external disturbances under directed graphs. For fixed graphs, an integral term is introduced in the auxiliary variable to explicitly derive a weighted-average consensus equilibrium that depends on the graph topology, agents' initial positions, and the local control gains. Robust adaptive terms with varying gains handle disturbances of unknown magnitude. For switching graphs, a model-reference adaptive algorithm is proposed under the assumption of uniform joint connectivity. Velocity-free variants are also developed for both fixed and switching cases. Effectiveness is illustrated via numerical simulations.

Significance. The explicit recovery of the consensus equilibrium via the integral augmentation for fixed directed graphs is a useful clarification, as the dependence on local gains is a direct and often under-emphasized consequence of such designs. The velocity-free extensions and the adaptive handling of unknown disturbance bounds address practical implementation issues common in Lagrangian multi-agent systems. If the Lyapunov arguments and equilibrium derivations are correct, the work strengthens the literature on robust distributed consensus under directed topologies.

minor comments (2)
  1. Abstract, line ~10: 'no comment control gains' appears to be a typographical error for 'no common control gains'.
  2. The abstract states the main results but provides no indication of the specific Lyapunov functions or gain-adaptation laws; the full manuscript should make these explicit in the theorems and proofs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation of minor revision. The referee's summary correctly reflects the paper's contributions on leaderless consensus protocols for uncertain Lagrangian systems under directed graphs, including the explicit weighted-average equilibrium derivation, adaptive disturbance rejection, and velocity-free variants. No major comments were raised in the report.

Circularity Check

0 steps flagged

Derivation is self-contained with no circularity

full rationale

The paper derives consensus protocols for Lagrangian systems via standard Lyapunov stability analysis, adaptive terms for uncertainties, and an auxiliary integral for explicit equilibrium recovery under fixed directed graphs. These steps follow directly from the proposed control laws and conventional graph connectivity assumptions (spanning tree for fixed case, uniform joint connectivity for switching). No load-bearing self-citations, self-definitional reductions, fitted inputs renamed as predictions, or smuggled ansatzes are present; the weighted-average equilibrium dependence on topology, initials, and gains is an explicit outcome of the integral augmentation rather than a circular fit. The analysis is self-contained against external benchmarks in the multi-agent control literature.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; full assumptions, parameters, and proof structure unavailable. Standard domain assumptions on graph connectivity and bounded disturbances are implied.

free parameters (1)
  • control gains
    Explicitly stated to affect the final consensus equilibrium; chosen by designer.
axioms (1)
  • domain assumption The communication graph is directed; for switching case the infinite sequence is uniformly jointly connected.
    Required for the stated consensus results to hold.

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