Matrix-weighted Bidirectional Convergence Of Second-order Multi-agent System

The bipartite consensus of multi-agent systems is a hot topic in the research of cooperative control of multi-agent systems.Most of the existing studies on bipartite consensus are carried out under the scalar-weighted signed graph.There are few researches on the bipartite consensus problem of multi-agent systems under the matrix-weighted signed graph.In view of the fact that velocity information is often difficult to obtain under certain circumstances,this paper will design a matrix-weighted control protocol without velocity information for second-order multi-agent systems,and study the bipartite consensus problem of the systems for the communication process with time delay and without time delay.The main results are as follows:1.The bipartite consensus problem of second-order multi-agent systems under the matrix-weighted undirected signed graph is studied.By introducing an auxiliary variable,a control protocol without velocity information is designed for each agent.The error system is constructed by using the structural balance of the communication topology between agents.The performance of the error system is analyzed with the help of polynomial theory,matrix analysis and algebraic graph theory,and a sufficient condition for the second-order multi-agent systems to achieve bipartite consensus under the designed control protocol is obtained,where the matrix-weighted undirected signed graph is structurally balanced,and the null space of the Laplacian matrix satisfies certain algebraic condition,which are the requirements for the topology of the communication relationship between agents.In particular,when the second-order multi-agent system degenerates into the second-order integrator-type multi-agent system,it not only gives the limit form of the bipartite consensus state of the agents but also proves that the algebraic condition satisfied by the null space of the Laplacian matrix is also the necessary condition for the system to realize bipartite consensus.2.The influence of time delay on the bipartite consensus of second-order multi-agent systems under the matrix-weighted undirected signed graph is studied.Considering the universality of the existence of time delay and the difficulty of obtaining velocity under certain circumstances,we design a velocity-free information control protocol with time delay for each agent.Compared with the case without time delay,the characteristic equation of the closed-loop system with time delay usually has transcendental functions,which makes it difficult to solve the characteristic roots of the closed-loop system,and thus brings difficulties to the state analysis of the closed-loop system.For this reason,the error system is decomposed into (N-1)d subsystems with time delay by using reversible transformation.Then,using the tools of linear matrix inequality and Lyapunov stability theory,the sufficient condition for the second-order multi-agent systems to achieve bipartite consensus is obtained,that is,when the time delay is within a certain range,if the linear matrix inequalities related to the dynamics of agents and the topology of the communication relationship between agents exist positive definite solutions,then the second-order multi-agent system can achieve bipartite consensus.