Research On Controllability Of Multi-agent Systems On Cartesian Network

Real life is full of various complex situations and tasks,such as traffic networks,social networks,predicting popularity thresholds,modeling and simulation of epidemic spread,and analysis and control of self-similar networks.Scholars have begun to study complex systems with multi-layer network topology,and the multilayer complex network constructed using the Cartesian product of graphs has emerged.The Cartesian network is a multi-layer network formed by the Cartesian product of graphs.Because there is a clear correspondence between nodes,the study of the controllability of multi-layer Cartesian networks from the perspective of nodes has become the research focus.This paper mainly studies the controllability of multiagent systems(MASs)on Cartesian networks,including the controllability of MASs on directed weighted Cartesian networks and signed Cartesian networks.The specific contents are introduced as follows:(1)For the controllability problem of MASs on directed weighted Cartesian networks,where the nodes that accept external control inputs are regarded as leaders of the entire network,and other nodes are regarded as followers.A new multi-layer multi-agent network model is constructed through the Cartesian product,and the multi-layer network model is transformed into a single-layer network model.Using the Kronecker product,the relationship between the Laplacian matrix L of the new multi-agent network and the Laplacian matrices L1,L2 of its two factor networks is derived,and the relationship between the eigenvalues and eigenvectors of the L matrix and the L1,L2 matrices is further discussed.The PBH controllability criterion for MASs on Cartesian networks is established,and the decomposition theorem of multi-layer Cartesian networks is finally given.(2)For the controllability MASs on signed Cartesian networks,where there are both positive and negative edges between network nodes.Firstly,the method and algorithm for obtaining the transformation matrix T of the structure balanced signed network are analyzed.Secondly,the relationship between the eigenvalues and eigenvectors of the signed Laplacian matrix L and the Laplacian matrix L of the corresponding unsigned network,as well as the transformation matrix T,is studied.The structural balance of signed Cartesian networks and the structural balance of their factor networks,as well as the relationship between the transformation matrix T and T1,T2,are derived.The process and algorithm for decomposing T into T1,T2 are given,and the PBH controllability criterion of structural balance signed Cartesian network systems is obtained.Finally,in order to verify the effectiveness of the results,corresponding simulations are provided.The research results of this paper lay a foundation for the subsequent study of the controllability of MASs on second-order and even higherorder Cartesian networks.