Research On The Controllability Of Several Basic Topologies

Controllability of topology structure is a hot topic in the field of control. In this paper, we mainly study the controllability of graph topology by exploring the information of complex network protocol and the second-order neighbors protocol. The main contents and contributions of this paper are list as follows:In this paper, the basic knowledge of graph theory, graph related matrices, basic graph topology, theorem and others are also sorted out, and system models of the research in this paper(Laplacian system model, complex network model and the second-order neighbors model) were introduced and defined in detail.The controllability under Laplacian system model was analyzed in this paper. The controllability of path graph under complex network model was analyzed in detail. When the controlled nodes are any single or multiple nodes, necessary and sufficient conditions of path graph are given and conditions are sorted out by the simple numerical theory for the conclusion more clear. Discrimination methods of the controllability of star graph and complete graph under complex network system were given.Through decomposing the system matrix and analyzing the relationship between the eigenvalues of submatrices, the controllability of path graph was studied by exploring the information of second-order neighbors. Some differences between the protocol of second-order neighbors and first-order neighbors for path graph are presented. A necessary and sufficient condition is proposed for the controllability of star graph by taking advantage of the protocol of second-order neighbors. Finally, the results are verified by examples.