Edge-based Synchronization over Signed Digraphs with Multiple Leaders

TL;DR

New spectral properties of signed edge-Laplacian matrices containing multiple zero eigenvalues are presented and global exponential stability of the synchronization errors are established and the equilibrium to which all edge states converge is computed.

Abstract

We address the edge-based synchronization problem in first-order multi-agent systems containing both cooperative and antagonistic interactions with one or multiple leader groups. The presence of multiple leaders and antagonistic interactions means that the multi-agent typically does not achieve consensus, unless specific conditions (on the number of leaders and on the signed graph) are met, in which case the agents reach a trivial form of consensus. In general, we show that the multi-agent system exhibits a more general form of synchronization, including bipartite consensus and containment. Our approach uses the signed edge-based agreement protocol for signed networks described by signed edge-Laplacian matrices. In particular, in this work, we present new spectral properties of signed edge-Laplacian matrices containing multiple zero eigenvalues and establish global exponential stability of the synchronization errors. Moreover, we compute the equilibrium to which all edge states converge. Numerical simulations validate our theoretical results.

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. (a) $\mathcal{G}_1$ is SUB and has $\nu_5$ as a root node. (b) $\mathcal{G}_2$ is SB ($\mathcal{V}_1 = \{ \nu_1, \nu_3, \nu_5, \nu_9 \}$ and $\mathcal{V}_2 = \{ \nu_2, \nu_4, \nu_6, \nu_7, \nu_8 \}$) and has $\nu_7$ and $\nu_9$ as root nodes. (c) $\mathcal{G}_3$ is SB ($\mathcal{V}_1 = \{ \nu_1, \nu_3, \nu_5, \nu_7, \nu_8, \nu_9 \}$ and $\mathcal{V}_2 = \{ \nu_2, \nu_4, \nu_6 \}$) and contains $\nu_9$ as a root node and a SB-rooted SCC formed by $\nu_1$, $\nu_8$, and $\nu_9$. (d) $\mathcal{G}_3$ is SUB and contains $\nu_9$ as a root node and a SUB-rooted SCC formed by $\nu_1$, $\nu_8$, and $\nu_9$.
• Figure 3: Evolution of the trajectories of agents $x$\ref{['FO']}-\ref{['CL']}, edges $e$\ref{['edgestates']} and synchronization errors $\bar{e}$\ref{['dyn_edge1']}. (a): $\mathcal{G}_1$. (b): $\mathcal{G}_2$. (c): $\mathcal{G}_3$. (d): $\mathcal{G}_4$.

Theorems & Definitions (26)

• Remark 1
• Remark 2
• Remark 3
• Lemma 1
• proof
• Lemma 2
• Lemma 3
• Lemma 4
• proof
• Lemma 5