Zero eigenvalue of directed graphs with spanning forests with application to formation control
LI Xiu-xian
LI Li
XIE Li-hua
Abstract：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.
Keywords：Laplacian matrixmulti-agent networksdirected graphsspanning forestformation control
Publication Date：2022-10-28
Online Publishing Date：2025-08-15(First online date of this platform, not the publication date of the document)
Pages：8( 1799-1806 )