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Necessary and sufficient conditions for the controllability of complex networks with path topology
CHAO Yongcui, JI Zhijian , WANG Yaowei, DONG Jie
School of Automation Engineering, Qingdao University, Qingdao 266071, China
Abstract: The controllability of complex networks is analyzed in the paper for path topology. With adjacency matrix of the system being decomposed into submatrices, the relationship between eigenvalues and eigenvectors is revealed for the partitioned submatrices. Furthermore, necessary and sufficient conditions are derived by taking advantage of the PBH(Popov-Belevitch-Hautus) criteria. In particular, a method is proposed to determine path controllability when the controlled nodes are any single or multiple nodes, as well as the concept of uncontrollable eigenvalues is presented. The expressions for uncontrollable eigenvalues are provided as well. Two theorems in this paper is verified by examples and the results of examples are in agreement with the conclusion of the theorems.
Key words: complex networks     controllability     graph theory     topology     linear time-invariant systems     eigenvalue     eigenvectors     control system

1 预备知识

2 主要结论

2.1 邻接矩阵的分解

B矩阵的具体表达与一系列的控制节点相对应，如果Ic={i1,i2,…,im}，则B=[ei1 ei2eim]。由引理1可知，路图不可控意味着存在非零向量v满足Av=λv，且BTv=0。矩阵定义如下：

Av=λv可得

ξ1=0时，ξ2=ξ3=…=ξn=0，即ξ为零向量，这与ξ是特征向量矛盾。同理可证ξn≠0。矩阵Mν的情况同样证明。

1)矩阵Mν的特征值的形式为

2)矩阵Mν的特征值必是矩阵Mαν+α－1的特征值，其中α

2.2 路图的可控性

erTv=0时，路图是不可控的。由erTv=0得

1)当控制节点为r∈{2,3,…,n－1}时，路图可控的充分必要条件为

G(r,n－r+1)=1

2)当控制节点集Ic={i1,i2,…,im}⊂{1,2,…,n}时，路图可控的充分必要条件为

G(i1,i2i1,…,imim－1,nim+1)=1

G(r,n－r+1)>1

2.3 实例分析

 图 1 n=11的路图 Fig. 1 The path graph with n=11

 图 2 n=14的路图 Fig. 2 The path graph with n=14

3 结束语

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DOI: 10.3969/j.issn.1673-4785.201411031

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#### 文章信息

CHAO Yongcui, JI Zhijian, WANG Yaowei, DONG Jie

Necessary and sufficient conditions for the controllability of complex networks with path topology

CAAI Transactions on Intelligent Systems, 2015, 10(04): 577-582.
DOI: 10.3969/j.issn.1673-4785.201411031