Fast Consensus Seeking in First Order Multi-Agent Systems with Antagonistic Interactionst

It is well known that based on consensus protocols, all agents in a multi-agent system can asymptotically converge to a common value. The convergence rate of a multi-agent system depends on the smallest non-zero eigenvalue of the Laplacian matrix $L$ . More specifically, the convergence rate of consensus is determined by the magnitude of $\lambda_{2}$ ( $L$ ). In this paper, we introduce a superposition system, and study how to change the convergence rate without destroying the connectivity of the undirected graphs. By using the inequality of eigenvalues, the problem of how the convergence rate changes is studied. When there is an eigenvector $x$ of $\lambda_{2}(L)$ satisfying $\tilde{L}x \neq 0$ , the multiplicity of $\lambda_{2}(L)$ is proved to be changed. Finally, the simulations are given to demonstrate the effectiveness of the theoretical results.