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Stability of Open Multi-agent Systems over Dynamic Signed Digraphs

Pelin Sekercioglu, Angela Fontan, Dimos V. Dimarogonas

TL;DR

This work addresses the synchronization problem in open multi-agent systems (OMAS) containing both cooperative and antagonistic interactions, and shows that the OMAS exhibit a more general form of synchronization, including trivial consensus, bipartite consensus and containment.

Abstract

We address the synchronization problem in open multi-agent systems (OMAS) containing both cooperative and antagonistic interactions. In these systems, agents can join or leave the network over time, and the interaction structure may evolve accordingly. To capture these dynamical structural changes, we represent the network as a switched system interconnected over a dynamic and directed signed graph. Additionally, the network may contain one or multiple leader groups that influence the behavior of the remaining agents. In general, we show that the OMAS exhibit a more general form of synchronization, including trivial consensus, bipartite consensus and containment. Our approach uses the signed edge-based agreement protocol, and constructs strict Lyapunov functions for signed networks described by signed edge-Laplacian matrices containing multiple zero eigenvalues. Numerical simulations validate our theoretical results.

Stability of Open Multi-agent Systems over Dynamic Signed Digraphs

TL;DR

This work addresses the synchronization problem in open multi-agent systems (OMAS) containing both cooperative and antagonistic interactions, and shows that the OMAS exhibit a more general form of synchronization, including trivial consensus, bipartite consensus and containment.

Abstract

We address the synchronization problem in open multi-agent systems (OMAS) containing both cooperative and antagonistic interactions. In these systems, agents can join or leave the network over time, and the interaction structure may evolve accordingly. To capture these dynamical structural changes, we represent the network as a switched system interconnected over a dynamic and directed signed graph. Additionally, the network may contain one or multiple leader groups that influence the behavior of the remaining agents. In general, we show that the OMAS exhibit a more general form of synchronization, including trivial consensus, bipartite consensus and containment. Our approach uses the signed edge-based agreement protocol, and constructs strict Lyapunov functions for signed networks described by signed edge-Laplacian matrices containing multiple zero eigenvalues. Numerical simulations validate our theoretical results.
Paper Structure (9 sections, 5 theorems, 19 equations, 3 figures)

This paper contains 9 sections, 5 theorems, 19 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Signed digraphs
  4. Stability of cooperative OMAS
  5. Model and Problem Formulation
  6. Lyapunov Equation for Directed Signed Edge Laplacians
  7. Main Results
  8. Numerical Example
  9. Conclusion

Key Result

Theorem 1

(xue2022stability) Consider the system app1 with the switching signal $\sigma(t)$ on $[t_0,t_f],$$0\leq t_0 < t_f < + \infty$. Assume that, for any two consecutive modes $\phi$, $\hat{\phi} \in \mathcal{P}$, where $\hat{\phi}$ precedes $\phi$, there exist class $\mathcal{K}_{\infty}$ functions $\und where $\gamma_{\phi} > 0$ and $\Omega_{\phi, \hat{\phi}}>1$. Moreover, asssume that $\sigma(t)$ sat

Figures (3)

  • Figure 1: A signed digon sign-symmetric digraph containing $3$ leader groups, where the black edges represent cooperative interactions and the dashed red edges represent antagonistic interactions. The first leader group is a SB-rooted SCC containing the leader nodes $\nu_1, \nu_2, \nu_3,$ and $\nu_4$, the second leader group is a SUB-rooted SCC containing the leader nodes $\nu_5, \nu_6,$ and $\nu_7$, and the third leader group is a root (leader) node, $\nu_8$. The node $\nu_9$ is the follower node.
  • Figure 2: The black edges represent cooperative interactions, and the dashed red edges represent antagonistic interactions.
  • Figure 3: Evolution of the trajectories. (a): Agents \ref{['FO']}-\ref{['CL']}. (b): Edges \ref{['edgestates']}. (c): Synchronization errors \ref{['dyn_edge1']} when $\tau$ satisfies \ref{['cond']}. (d): synchronization errors \ref{['dyn_edge1']} when $\tau$ does not satisfy \ref{['cond']}.

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Remark 1
  • Theorem 2
  • proof
  • Theorem 3
  • Remark 2
  • proof
  • Theorem 4
  • ...and 4 more