Event-triggered Bipartite Consensus For Multi-agent Systems

Under the undirected signed graph topology,we consider event-triggered bipartite consensus problem of multi-agent systems for first-order continuous-time system.Based on the properties of the connected signed graph,a first-order event-triggered interaction rule is proposed for the multi-agent systems with coexistence of cooperative and competitive interactions,where all the agents reach agreement with identical magnitude but opposite sign.We analyze the bipartite consensus and obtain the sufficient conditions for bipartite consensus of the first-order multi-agent systems based on the algebraic graph and matrix theories.We then obtain the bipartite consensus equilibriums of the subgroups.In continuous-time systems,it is usually assumed that agents can communicate continuously to ensure control.This is obviously unrealistic,so periodic sampling control is considered,but it will also cause unnecessary consumption of resources and occupied bandwidth.In order to solve the problem of energy waste caused by sampling control,event-trigger control is introduced.The event trigger function is constructed.When the state measurement error of the agent meets the event-trigger function,the event is triggered and the system updates the control input.Moreover,although the designed event-triggered control protocol of multi-agent systems can achieve convergence,it often converges in asymptotic time,and the convergence speed is slow,which is not suitable for engineering applications.Therefore,it is considered to introduce a finite time consensus protocol to enable the multi-agent systems to achieve convergence within a finite time.In the paper,a centralized event-triggered condition is designed firstly for an undirected signed graph for first-order multi-agent systems whose network topology is structurally balanced.When an agent's state measurement error meets the triggering function,all agents trigger events and update the control input.No continuous communication between agents is required so as to reduce the communication utilization.The control protocol after the gauge transformation proves that the system can reach bipartite consensus convergence based on the algebraic graph and matrix theories and designed Lyapunov function.When the multi-agent systems achieve bipartite consensus,it also proves that the Zeno behavior can be avoided while ensuring the stability of the systems.Numerical simulation results are shown to illustrate the efficiency of the proposed theoretical results.Secondly,considering the consumption of bandwidth and communication resources in the centralized event-triggered conditions,a distributed event triggering condition is designed for the first-order multi-agent systems,and the communication topology is designed as an undirected signed graph and a directed strong connected signed graph.Using the Lyapunov function shows that the systems can achieve bipartite consensus.It can also avoid the Zeno behavior while ensuring the stability of the systems.Numerical simulation results prove the effectiveness of the control protocol.Thirdly,considering the slow convergence rate and engineering inapplicability of asymptotic convergence,finite-time convergence is introduced,and a control protocol with better robustness and faster convergence rate is designed.For a first-order multiagent systems with a network topology of structurally balanced undirected signed graphs,considering distributed event-triggered control and finite-time convergence,the use of Lyapunov function shows that the system can achieve finite-time bipartite consensus.And,while ensuring the stability of the system,it can avoid the Zeno behavior.Numerical simulation results prove the effectiveness of the control protocol.Finally,we have summarized all the results in this paper.