Abstract This paper studies the problem of synchronization in a dynamical network, where we identify the nodes of the undirected graph as identical Lur’e-type nonlinear systems and the edges as heterogeneous and asymptotically stable linear systems, i.e., both of the nodes and edges are dynamical systems. It is shown that sufficiently fast convergence rate of each edge guarantees the dynamic edges degenerate into their associated static maps, equivalent to assigning the constant weights to the edges (as in the usual synchronization problems). Thus, in effect, the dynamical network becomes the static one characterized by the Laplacian matrix of the graph with the weights being the gains of the static maps. Then, the large DC gains of the edges, i.e., the large weights, are shown to yield the synchronization of the state variables of the Lur’e-type systems in the static network. The synchronization in the dynamical network is at last guaranteed by the combination of these two results. The results in this paper are applicable to the synchronization of the terminal voltages of the Lur’e-type circuits that are interconnected with each other via the lines having their associated line admittances.

中文翻译：

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收敛速度快、直流增益大的线性动力网络中Lur'e型非线性系统的同步

摘要 本文研究了动态网络中的同步问题，其中我们将无向图的节点识别为相同的Lur'e型非线性系统，将边识别为异构且渐近稳定的线性系统，即节点和边是动态系统。结果表明，每条边足够快的收敛速度保证了动态边退化为它们相关的静态图，相当于将恒定权重分配给边（如通常的同步问题）。因此，实际上，动态网络变成了静态网络，其特征是图的拉普拉斯矩阵，权重是静态图的增益。然后，边缘的大 DC 增益，即大权重，显示产生静态网络中 Lur'e 型系统状态变量的同步。这两个结果的结合最终保证了动态网络中的同步。本文的结果适用于通过具有相关线路导纳的线路相互互连的 Lur'e 型电路的端电压同步。