Stability of Open Multi-agent Systems over Dynamic Signed Graphs

TL;DR

The paper tackles bipartite consensus control in open multi-agent systems with dynamic signed graphs, where nodes and edges can be added and interaction signs switch over time. It uses an edge-based formulation and constructs strict Lyapunov functions for signed edge-Laplacians with multiple zero eigenvalues to prove global asymptotic stability of bipartite consensus under switching via a transition-dependent average dwell-time framework. Key contributions include extending Lyapunov equations to edge-Laplacians with multiple zeros, establishing stability of the edge-error dynamics, and characterizing outcomes (bipartite vs trivial consensus) based on the last switching mode SB vs SUB; these results are validated through numerical simulations with mobile robots. The framework enables stable coordination in evolving networks with antagonistic interactions and time-varying topology, and points to extensions to directed signed graphs and multi-leader settings.

Abstract

This paper addresses the bipartite consensus-control problem in open multi-agent systems containing both cooperative and antagonistic interactions. In these systems, new agents can join and new interactions can be formed over time. Moreover, the types of interactions, cooperative or antagonistic, may change. To model these structural changes, we represent the system as a switched system interconnected over a dynamic signed graph. Using the signed edge-based agreement protocol and constructing strict Lyapunov functions for signed edge-Laplacian matrices with multiple zero eigenvalues, we establish global asymptotic stability of the bipartite consensus control. Numerical simulations validate our theoretical results.

Figures (3)

• Figure 1: Black lines represent cooperative interactions, and dashed red lines represent antagonistic interactions. (a) The initial graph $\mathcal{G}_1$ is a SB signed network of $4$ agents, where $\mathcal{V}_1 = \{\nu_1, \nu_3\},\ \mathcal{V}_2 = \{\nu_2, \nu_4\}$. (b) At $t=t_1$, a new node $\nu_5$ is added to the system and $\mathcal{G}_2$ is a SB signed network of $5$ agents, where $\mathcal{V}_1 = \{\nu_1, \nu_3, \nu_5\}, \mathcal{V}_2 = \{\nu_2,\nu_4\}$. (c) At $t=t_2$, a new node $\nu_6$ is added to the system and $\mathcal{G}_3$ is a SB signed network of $6$ agents, where $\mathcal{V}_1 = \{\nu_1, \nu_3, \nu_5\}, \mathcal{V}_2 = \{\nu_2, \nu_4, \nu_6\}$. (d) At $t=t_3$, the sign of the edge $e_2$ changes from cooperation to antagonism, and $\mathcal{G}_4$ is a SUB signed network. (e) At $t=t_4$, the sign of the edge $e_2$ changes back to cooperation and a new node $\nu_7$ is added to the system. $\mathcal{G}_5$ is a SB signed network of $7$ agents, where $\mathcal{V}_1 = \{\nu_1, \nu_3, \nu_5 \}, \mathcal{V}_2 = \{\nu_2, \nu_4, \nu_6 , \nu_7\}$. (f) At $t=t_5$, the sign of the edge $e_7$ changes from cooperation to antagonism, and $\mathcal{G}_5$ is a SB signed network of $7$ agents, where $\mathcal{V}_1 = \{\nu_1, \nu_3, \nu_5, \nu_7 \}, \mathcal{V}_2 = \{\nu_2, \nu_4, \nu_6 \}$.
• Figure 2: Evolution of the trajectories of the agents' positions.
• Figure 3: Evolution of the trajectories of the edges.

Theorems & Definitions (18)

• Definition 1
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Definition 2
• Remark 1
• Theorem 1
• proof