| In complex network systems,the research of topological structures is a hot issue,and the change of topological structure is also an important factor affecting the controllability of the system.In this thesis,matrix and graph theory are used as the main research tools to study the strong structural controllability of complex network systems under topological edge addition,and the influence of dynamic changes of topological structure on the strong structural controllability of systems and relevant conclusions are obtained.The main research contents are as follows:Firstly,a criterion for determining the strong structural controllability of network systems is obtained,that is,the full rank of the structural matrix [A,B] is only a necessary condition for the strong structural controllability of network systems,which provides a new algebraic method for judging the strong structural controllability of complex network systems.Secondly,based on the stem-bud topology with strong structure controllability,the influence of different edging methods on the strong structure controllability of the network system is analyzed.The following four cases are mainly considered:(1)Adding the directed edge from the stem to the bud on the stem-bud topology;(2)Adding the directed edge from the bud to the stem on the stem-bud topology;(3)Adding a reverse edge or a forward edge to the stem or bud in the stem-bud topology;(4)Adding both the reverse edge and forward edge to the stem or bud in the stem-bud topology.In addition,the sufficient conditions for the system to achieve strong structural controllability under these four edging methods are obtained.The edging operation makes the single stem-bud topology more complex and diversified,and maintains the strong structural controllability of the network system while enhancing the network connectivity.Thirdly,by analyzing the structure of the bipartite graph corresponding to the network system which has achieved structural controllability but not achieved strong structural controllability,it is obtained that a kind of cross bipartite graph in bipartite graphs will affect the strong structural controllability of the system,and a kind of cross topology graph which is not strong structural controllability is obtained correspondingly.This kind of cross topology graph is also a basic topology of the research content of this thesis.Finally,the original topology is based on the single cross graph and the connected cross graph(multiple cross graphs)respectively.By analyzing the relationship between the number of leaf nodes and the number of cross graphs in the generated graph,the generated graph of a single cross graph containing one leaf node or multiple leaf nodes and the generated graph of connected cross graphs containing one leaf node or multiple leaf nodes are obtained.Moreover,these four types of generated graphs are not strongly structurally controllable.Then,the minimum number of addition edges for these four kinds of generated graphs to achieve strong structure controllability is given.In addition,for the above four types of generated graphs,the corresponding edging algorithm satisfying the minimum number of edging is designed respectively to get the specific edging method to make the network system achieve strong structure controllability. |
