Consensus Of Linear Multi-agent Systems With Communication Delay

This thesis investigates the consensus problem for linear multi-agent systems with communication delay. Under the assumption that the network topology is undi-rected, two issues are considered. Firstly, we study the consensus of critically un-stable high-order multi-agent systems that are subjected to time-varying commu-nication delay; Secondly, the consensus problem of unstable agents with constant communication delay is concerned.The main contribution are as:For the high-order continuous multi-agent sys-tems that are critically unstable, consensus conditions are provided for time-varying communication delay, which improves the existing results essentially; when the agent dynamic is unstable (at least critically unstable), agent’s own historical in-put information is used in the protocol design, and the results weaken the effect of delay on consensus.In the order of chapters, the main contents and innovations are listed as follows:1. Suppose that the systems are critically unstable, we study the consensus problem of high-order continuous multi-agent systems with time-varying communi-cation delay. The consensus gain is designed with the unique positive-definite so-lution of a parametric algebra Riccati equation. Then, by applying the Razumikhin stability Theorem, consensus conditions are provided respectively based on the de-layed relative information and the delayed neighbors’ informationThe main innovation is that consensus conditions are obtained for high-order multi-agent systems with time-varying and unknown communication delay, which extends the existing researches.2. For the first-order unstable continuous-time multi-agent systems, the con-sensus problem are addressed in case of constant communication delay. When the delayed relative information is available, agent’s own historical input information is used in the protocol to recede the effect of delay on consensus, and obtain the following results:● If the protocol contains agent’s own historical input information, consensus conditions related to communication delay, network topology and agent dy-namics are acquired. For any fixed agent dynamics, the better synchronizabil-ity corresponds to larger delay bound for consensus. On the other hand, if the communication delay is fixed, more unstable agent dynamic is allowed for consensus if the synchronizability of the network is better. Especially, when the undirected graph is completed, any bounded communication delay and un-stable agents are tolerant for consensus.●If the relative historical input information among agent and neighbors are used in the protocol, any bounded delay is allowed for consensus just as the undi-rected graph is connected, which is similar to the delay-free case.If the delay only influences the neighbors’ information, consensus condition related to network topology is provided for any bounded delay, and the delay in this case is allowed to be time-varying and unknown.The main innovations are that we study the unstable (the poles of the system matrix are on the closed right-half plane) agent dynamics, which are different from the results in the literature that are mainly on single-integrator or the systems that are critically unstable (the poles of the system matrix are on the closed left-half plane). Furthermore, the effect of delay on consensus is significantly weakened by introducing agent’s historical input information in the protocol design. In addition, we extend the constant delay to time-varying case on condition that the delay only affects the neighbors’information.3. The consensus problem for unstable first-order discrete multi-agent systems that are subjected to constant communication delay is studied. To the best of our knowledge, the relevant results are rare. Similarly as case 2, we respectively study the consensus problem on account of the delayed relative information and the de- layed neighbors’information, and the corresponding consensus conditions are fol-lows. Especially, when the protocol includes agent’s own historical input informa-tion, necessary and sufficient condition is obtained in case of one step delay.The main innovations are extending the existing results which are aimed at single-integrator or critically unstable(the poles of the system matrix are inside or on the closed unit circle) systems to the unstable(the poles of the system matrix are on or outside the unit circle) case. The historical input information in protocol definitely weakens the influences of delay on consensus. Furthermore, when the delay only affects neighbors’information, we also improve the constant delay to the time-varying case.4. In case of constant communication delay, the consensus problem of unstable high-order continuous multi-agent systems is discussed. We use the delay informa-tion and the unique positive-definite solution of a parametric algebra Riccati equation to design the control gain, and the consensus conditions are given by means of the variation of constants formula and Razumikhin stability Theorem.● If the protocol only contains the relative information, an allowable delay bound related to the maximum of a concave function is obtained, and the biggest delay bound is achieved in case of complete graph.● If the protocol includes the relative information and agent’s own historical input information, the allowable delay bound is also related to the maximum of a concave function. Especially, any bounded delay and high-order unstable agent dynamics are tolerant for consensus if the graph is complete.Comparing with the existing work, the system investigated are of high-order and at least critically unstable. The adoption of agent’s own historical input infor-mation in the protocol weakens the constraint of communication delay on consensus.