含生成森林有向图的零特征值及在编队控制中的应用  

Zero eigenvalue of directed graphs with spanning forests with application to formation control

作　　者：李修贤 李莉[1] 谢立华[3] LI Xiu-xian;LI Li;XIE Li-hua(College of Electronics and Information Engineering,Department of Control Science and Engineering,Shanghai Research Institute for Intelligent Autonomous Systems,and Shanghai Institute of Intelligent Science and Technology,Tongji University,Shanghai 200240,China;Institute for Advanced Study,Tongji University,Shanghai 220240,China;School of Electrical and Electronic Engineering,Nanyang Technological University,50 Nanyang Avenue 999002,Singapore)

机构地区：[1]同济大学电子与信息工程学院控制科学与工程系上海自主智能无人系统科学中心上海智能科学与技术中心,上海200240 [2]同济大学高等研究院,上海220240 [3]南洋理工大学电气与电子工程学院,新加坡999002

出　　处：《控制理论与应用》2022年第10期1799-1806,共8页Control Theory & Applications

基　　金：Supported by the National Natural Science Foundation of China(62003243,72171172);the Basic Science Centre Program by National Natural Science Foundation of China(62088101);the Fundamental Research Funds for the Central Universities(22120210099);the Shanghai Municipal Commission of Science and Technology(19511132101);the Shanghai Municipal Science and Technology Major Project(2021SHZDZX0100)。

摘　　要：本文研究了含有m-生成森林有向图拉普拉斯矩阵的零特征值重数,其中m1是一个整数.对于这个问题,这个图一般不含有生成树.即使初始时具有生成树,受到隐秘的攻击或经过障碍物造成的智能体之间的通信阻挡(如在分布式控制、分布式(在线)优化、多智能体算子等问题中)等因素后,这个图也可能不再含有生成树了.另外,作为一个研究方向,它本身亦是个有趣的科学问题.为了解决这个问题,本文证明了拉普拉斯矩阵的零特征值重数等于这个图中的生成森林个数,这个结论可以看作是在带有生成树的有向图情形(即m=1时)的一个推广.再者,结合分布式优化方法,所得结论被应用于单积分器多智能体系统下的编队控制,表明了达到的编队队形处在通信图拉普拉斯矩阵的核空间中.最后给出了一个例子用以展示在编队控制中的应用.This paper investigates the multiplicity of zero eigenvalue of the Laplacian matrix for a directed graph,which has a spanning m-forest,where m 1 is an integer.For this problem,the graph usually does not contain a spanning tree,and this scenario may occur due to insidious attacks or communication blocking by obstacles between two agents in distributed control,(online)optimization,multi-agent operators,and so on,even though it indeed has a spanning tree at the beginning.In addition,this problem is of interest as a research direction in its own right.To deal with this problem,it is shown that the multiplicity of the Laplacian’s zero eigenvalue amounts to the number of spanning forests in the studied graph,which can be seen as an extension of the directed graph case with a spanning tree,in which case it has m=1.Moreover,the obtained result is applied to formation control for single-integrator multi-agent systems along with distributed optimization methods,indicating that the achieved formation shape lies in the kernel space of the Laplacian matrix associated with the communication graph.Finally,an example is provided to demonstrate the applicability to formation control.

关 键 词：拉普拉斯矩阵 多智能体网络 有向图 生成森林 编队控制 

分 类 号：O157.5[理学—数学]