Study On Consensus Control Of Linear Multi-agent System With Time Delay

In recent years,with the continuous development of control theory,computer technology,network signal processing and communication technology,the consen-sus control problem of network multi-agent systems has been paid more and more attention by scholars at home and abroad,and many valuable research results have appeared.On the basis of the existing work,this paper studies the consensus prob-lem with constant communication delay and with constraints of linear multi-agent systems.And we obtain some results as follows:1.We consider a consensus control problem of general high-order linear time-invariant multi-agent systems in undirected communication network.The generalityof state matrix and input matrix and the existence of delays make it difficult to studythe consensusability problem.By introducing an appropriate linear transformation,we turn the consensusability problem into an equivalent stability problem of time-delayed subsystems associated with the characteristic roots of the Laplacian matrix.That established a relationship between consensusability problem and stability prob-lem.We use the linear matrix inequality method and Lyapunov stability theory to obtain the synchronous stability conditions of subsystems.For first-order integrator system,we directly use the frequency domain analy-sis method to obtain explicit conditions guaranteeing the consensusability.We also find a relationship between delay margin and parameter,and so regardless of the size of delay,the multi-agent systems can always reaching consensus as long as the characteristic roots of the Laplacian matrices satisfy some relationship.2.In this chapter,we consider a consensus problem for a class of general form of second-order linear dynamic model,also involves the communication delay.We want to find the maximum allowable bound of time delay to guarantee the second-order consensus of the this multi-agent system.Firstly,we consider the delay-free control protocol,and by introducing an appropriate linear transformation,the con-sensusability is transformed into an equivalent stability problem.By analysing of the stability of the characteristic polynomial of the closed-loop system,we obtain we obtain the sufficient and necessary conditions guaranteeing consensus for the delay-free multi-agent system.In particular,the corresponding consensusability conditions can be obtained for the second-order integrator system.Then,based on theses con-ditions,we consider the control protocol with constant communication delay,and by analyzing of the relations between the roots of the closed-loop system and the time-delay parameter,we obtain an explicit expressions for the bound of time delay guaranteeing consensus.This makes us aware that there is a compromise between the convergence properties of consensus and the value of time delay.We also give a more detailed analysis of the special system,the second-order integrator system,and a simpler explicit expression for the bound of time-delay is presented.3.In the undirected connected network,we consider the consensus problem of a class of general linear dynamic model.The significance of studying the con-sensus problem of third-order system is that the third-order system can simulate a class of physical systems in real life,such as the the longitudinal dynamics model of the vehicle,and that consensus problems can represent some kinds of group be-havior of multiple physical systems.In this chapter,one of the contributions is that the necessary and sufficient conditions for a class of general third-order consensus are obtained by analyzing the distribution of the roots of the closed-loop system.The second contribution is that for the third-order integrator system,we provides a method to calculate the allowable communication delay by assuming that the condi-tions guaranteeing consensus for the case of delay free are hold.4.The consensus problem of discrete-time multi-agent systems with time delay and state constraint is considered.Firstly,a sufficient and necessary condition for unconstrained consensus is given.Then for the case of delay-free but with state con-straint,the closed-loop system is decomposed into two parts,and we will find that the requirement of consensus is higher than that of delay-free and state unconstrained.Next,based on the consensus conditions of delay-free and state unconstrained,we consider the consensus problem of time delay but without state constraint,and the necessary and sufficient condition for this case is obtained.Here the condition has the form of exponential equation,also we only give an explicit form for the case of the delay of 1 or 2 in the undirected graph.Finally,we consider the consensus problem with time delay and state constraint,and we find that the condition for the consensus with delay-free and state constraint can still guarantee the consensus for the problem with time delay and state constraint.