| Regarding the research on the performance analysis of multi-agent systems,controllability,as the basic property of modern control theory,has received more and more attention and research in recent years.Based on the leader-follower model framework,with the help of matrix theory,graph division theory,and other theoretical knowledge,this paper studies and analyzes the controllability of multi-agent systems by using the characteristics of equitable partition and automorphism,and gives the defect that how to break through equitable partition can only obtain necessary conditions under specific conditions.The thesis focuses on the following four parts:Firstly,we clarify the background and significance of multi-agent systems research,discuss the research status at home and abroad,and describe the basic theoretical knowledge of graph theory,matrix theory and equitable partition.Secondly,the controllability of the first-order integrator system with different connection modes is analyzed by using the characteristics of equitable partition and automorphism.For the case that cells are incompletely connected outside but completely connected inside,a necessary condition for controllability is given from the perspective of the rank of connection matrix.For the case of multiple cells being completely connected outside and incompletely connected inside,in terms of the eigenvalues and eigenvectors of the Laplacian matrix of the original topological graph and the quotient graph,several sufficient and necessary conditions for controllability are presented.For the case that cells are incompletely connected outside and inside,a necessary condition for controllable is given by using the characteristics of automorphism.Then,the difference between the first-order integrator systems and the general linear systems model is analyzed,the controllability of equitable partition under the general linear systems model is discussed,and the controllability determination method for the incompletely connected case of cells is obtained.Finally,the dimension of controllable subspace is defined by analyzing the specific form of eigenvector under equitable partition and using the method of PBH rank criterion.And by dividing the Laplacian matrix of the quotient graph into blocks,we analyze the possibility that the right eigenvector of the corresponding nontrivial cell element may have zero elements,and reveal the gap between necessary conditions and sufficient conditions for controllability under equitable partition. |
