⚡ electrical engineering

Consensus Tracking of Perturbed Open Multi-Agent Systems with Repelling Antagonistic Interactions

This paper addresses the consensus tracking problem in open multi-agent systems subject to migration-induced perturbations and repelling antagonistic interactions by modeling them as perturbed multi-mode multi-dimensional systems and proving that ultimately bounded or asymptotic tracking can be achieved under specific network switching conditions.

Original authors: Mengqi Xue, Yuchao Xiong, Yue Song

Published 2026-05-26

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CC BY 4.0

Original authors: Mengqi Xue, Yuchao Xiong, Yue Song

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a flock of birds, a swarm of drones, or a line of self-driving cars. In a perfect world, they all move together in harmony, following a leader. This is what engineers call a "Multi-Agent System."

But this paper focuses on a much messier, more realistic version: an Open Multi-Agent System (OMAS). Think of this like a busy highway or a crowded dance floor. Agents (the cars or dancers) are constantly joining the group and leaving it. Sometimes a new car merges in; sometimes one crashes out. Because the group size is always changing, the "network" connecting them is constantly switching on and off.

The researchers in this paper asked a tough question: What happens if this chaotic, changing group is also being attacked or disturbed?

They looked at two specific types of trouble:

Repelling Antagonistic Interactions: Imagine if, instead of helping each other, some agents started pushing each other away. In a network, this is like a "bad connection" or a "sabotage link" that tries to pull the group apart rather than keep them together.
Persistent Perturbations: Imagine the agents are constantly being bumped by external forces—like wind gusts, road bumps, or digital noise—that never quite go away.

The Core Problem: The "Push and Pull"

The authors found that if the "pushing apart" (repelling) forces become stronger than the "holding together" (cooperative) forces, the whole system becomes unstable. It doesn't matter if the network is connected or not; if the bad connections dominate, the group will naturally fall apart or go wild.

The Solution: A "Tug-of-War" Strategy

So, how do you keep a group of constantly changing, bumping, and sometimes fighting agents from falling apart?

The authors propose a set of rules for how the network should switch between different states. They use a concept called Piecewise Average Dwell Time.

Here is a simple analogy:
Imagine the agents are playing a game of Tug-of-War.

Good Teams (Stable Modes): These are moments when the network is healthy, with mostly positive connections. The rope is being pulled toward the leader.
Bad Teams (Unstable Modes): These are moments when the network is disrupted, with too many "repelling" connections. The rope is being pulled away from the leader, or the team is falling apart.

The paper proves that you can still win the game (achieve Consensus Tracking, meaning everyone stays close to the leader) even if the Bad Teams are strong, IF you follow two rules:

The Ratio Rule: The Good Teams must pull for a long enough time compared to the Bad Teams. You can't let the Bad Teams pull for too long, or the rope snaps.
The Switching Rule: You can't switch between Good and Bad teams too frantically. You need to stay in a "Good Team" mode for a minimum amount of time to let the group recover and stabilize before switching again.

The Results: Two Scenarios

The paper shows two different outcomes based on how "loud" the disturbances are:

Scenario A: The Noisy, Bumpy World (Non-Vanishing Perturbations)
If the external bumps and the "pushing" forces never stop completely, the agents will never get perfectly close to the leader. However, the paper proves they will stay within a safe, predictable distance. They might jitter and jump around, but they won't run away. They achieve what is called "ultimately bounded consensus."
- Analogy: Think of a group of people trying to walk in a straight line while being constantly nudged by a crowd. They won't walk in a perfect straight line, but they will stay in a tight cluster and not get lost.
Scenario B: The Calming Down World (Vanishing Perturbations)
If the external bumps eventually stop and the "pushing" forces fade away, the agents can achieve perfect, asymptotic consensus. They will eventually line up perfectly with the leader.
- Analogy: Once the crowd stops nudging them, the group naturally smooths out and walks in a perfect line.

Why This Matters (According to the Paper)

The authors built a new mathematical framework (called an M3D system) to handle this specific mix of changing sizes, bad connections, and constant noise. They showed that even in a chaotic environment where the network is broken or disconnected at times, as long as the "Good" connections dominate the "Bad" ones over time, the group can still function and follow the leader.

In short: Even if your team is constantly changing, getting bumped, and occasionally fighting, you can still win the race as long as you spend enough time working together and not too long fighting each other.

Technical Summary: Consensus Tracking of Perturbed Open Multi-Agent Systems with Repelling Antagonistic Interactions

Problem Statement
This work addresses the consensus tracking problem for Open Multi-Agent Systems (OMASs) subject to migration-induced adversities. Unlike traditional multi-agent systems, OMASs feature agents that dynamically join or leave the network, resulting in a naturally switching and size-varying topology. The specific challenges investigated include:

Non-vanishing perturbations: Both continuous dynamics/state perturbations and impulsive state jumps caused by agent migrations.
Repelling antagonistic interactions: Interactions that can destabilize the network, modeled via signed digraphs where negative edges (repelling) may dominate cooperative ones.
Intermittently disconnected signed digraphs: The network topology is allowed to be directed and may lack connectivity at certain times.

The core difficulty lies in the fact that when repelling interactions dominate cooperative ones, the system creates unstable subsystems regardless of network connectivity, complicating the achievement of consensus tracking for a reference leader.

Methodology
The authors model the OMAS as a perturbed multi-mode multi-dimensional (M³D) system. This framework integrates dimension variations (due to agents joining/leaving) and non-vanishing impulsive perturbations into a unified structure.

Key methodological steps include:

System Formulation: The agent dynamics are described by linear equations with a consensus tracking protocol. The network topology is represented by a switching signed digraph $\tilde{G}_{\sigma(t)}$ . The authors define a "repelling Laplacian" matrix, noting that if negative edges outnumber positive ones, the matrix possesses eigenvalues with negative real parts, leading to instability.
Error Dynamics: The consensus tracking problem is transformed into the stability analysis of a tracking error system. This error system is characterized as a switched system with:
- Continuous dynamics governed by matrices $\tilde{A}_{\sigma(t)}$ .
- Discrete jumps at switching instants $t_k$ caused by agent migrations and external impulses.
Stability Analysis: The authors extend stability theory for M³D systems. They classify switching modes into stable sets ( $P_s$ $P_{s}$ ) and unstable sets ( $P_u$ $P_{u}$ ).
- Modes in $P_s$ correspond to topologies with only positive edges and a directed spanning tree rooted at the leader.
- Modes in $P_u$ include topologies with only positive edges but no spanning tree, or topologies where repelling interactions dominate.
Switching Conditions: To ensure stability despite unstable subsystems and non-vanishing perturbations, the authors derive conditions based on:
- Piecewise Average Dwell Time (ADT): A constraint on the frequency of switching.
- Activation Time Ratio: A constraint on the ratio of time spent in stable modes versus unstable modes. Specifically, the stabilizing digraphs must activate long enough to neutralize the destabilizing effects of the repelling interactions.

Key Contributions

Problem Extension: The work extends previous OMAS research (e.g., Xue et al., 2022) by addressing leaderless consensus under state perturbations to consensus tracking under both non-vanishing perturbations and antagonistic interactions.
Analysis of Repelling Interactions: The paper provides a theoretical characterization showing that if repelling interactions dominate cooperative ones, the OMAS exhibits instability regardless of network connectivity. This fills a gap in existing literature which often assumes purely cooperative or opposing (bipartite) interactions.
Novel M³D Analysis Method: A new analysis method is proposed that accommodates non-vanishing perturbations and destabilizing repelling interactions. Unlike previous works, this method:
- Does not require the "alternating switching" condition.
- Applies to signed graphs that are directed and intermittently disconnected.
- Handles both state-independent and state-dependent impulses.

Results
Theoretical analysis yields the following results:

Ultimate Boundedness: Under non-vanishing perturbations and state-independent impulses, the system achieves ultimately bounded consensus tracking. The ultimate bound depends on the magnitude of the perturbations and the switching parameters.
Asymptotic Tracking: If perturbations are vanishing (i.e., no external disturbances and state-dependent impulses that vanish as the error vanishes), asymptotic consensus tracking is achieved.
Switching Conditions: The results establish that consensus is achievable if the switching signal satisfies specific piecewise ADT and activation time ratio conditions. These conditions ensure that the stabilizing modes activate sufficiently to counteract the destabilizing modes and perturbations.

Significance
The paper claims theoretical and practical significance in focusing on migration-induced adversities in OMASs. It highlights that agent migrations can introduce abrupt disturbances and convert cooperative interactions into antagonistic ones (e.g., due to cyber-physical attacks like Trojan devices). By demonstrating that consensus can be maintained even when repelling interactions dominate, provided specific switching conditions are met, the work offers a robust framework for controlling flexible networks in adverse environments, such as mobile ad-hoc networks or systems subject to dynamic resource exhaustion. The simulation examples verify that the derived conditions allow the system to achieve bounded tracking in the presence of persistent disturbances and asymptotic tracking when disturbances vanish.

The Core Problem: The "Push and Pull"

The Solution: A "Tug-of-War" Strategy

The Results: Two Scenarios

Why This Matters (According to the Paper)

Technical Summary: Consensus Tracking of Perturbed Open Multi-Agent Systems with Repelling Antagonistic Interactions

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