In this article, by analyzing the eigenvalues and eigenvectors of Laplacian $L$ , we investigate the controllability of multiagent systems under equitable partitions. Two classes of nontrivial cells are defined according to the different numbers of links between them, which are completely connected nontrivial cells (CCNCs) and incompletely connected nontrivial cells. For the system with CCNCs, a necessary condition for controllability is found to be choosing leaders from each nontrivial cell, the number of which should be one less than the cardinality of the cell. It is shown that the controllability is affected by three factors: 1) the number of the links between nontrivial cells; 2) the rank of the connection matrix; and 3) the odevity of the capacity of the nontrivial cells. In the case of nontrivial cells under the equitable partition, there are automorphisms of interconnection graph $\mathcal {G}$ , which induce the eigenvectors of $L$ with zero entries. For the system with automorphisms, by taking advantage of the property of eigenvectors associated with $L$ , we propose several graphical necessary conditions for controllability. In addition, by the PBH rank criterion, the controllable subspaces of the system with different classes of nontrivial cells are compared. Finally, a necessary and sufficient condition for controllability under minimum inputs is given.

中文翻译：

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公平划分下多智能体系统可控性的图解条件

在本文中，通过分析拉普拉斯算子的特征值和特征向量 $L$ ，我们研究了公平分区下多智能体系统的可控性。根据它们之间的链接数量不同，定义了两类非平凡细胞，它们是完全连接的非平凡细胞（CCNC）和不完全连接的非平凡细胞。对于具有 CCNC 的系统，发现可控性的必要条件是从每个非平凡单元中选择领导者，其数量应小于单元的基数。结果表明，可控性受三个因素影响：1）非平凡细胞之间的链接数；2）连接矩阵的秩；和 3) 非平凡细胞的能力的 odevity。在公平划分下的非平凡单元格的情况下，存在互连图的自同构 $\mathcal {G}$ ，这导致了特征向量 $L$ 零条目。对于具有自同构的系统，通过利用与 $L$ ，我们为可控性提出了几个图形化的必要条件。此外，通过PBH秩准则，比较了具有不同类别非平凡单元的系统的可控子空间。最后，给出了最小输入下可控性的充要条件。