Formation-containment analysis and design for high-order linear time-invariant swarm systems with time delays

Formation-containment analysis and design problems for high-order time-delayed linear time-invariant swarm systems with directed interaction topologies are studied respectively. Firstly, protocols with time delays are presented for leaders and followers respectively, where the formation for leaders can be time-varying. Then formation-containment problems for time-delayed swarm systems are transformed into asymptotic stability problems. Sufficient conditions for swarm systems to achieve formation-containment are proposed, which include eight linear matrix inequalities independent of the number of leaders and followers. Furthermore, an explicit expression of the formation reference function for leaders is derived, where the motion modes of formation reference can be specified. An approach to determine the gain matrices in the protocols is given using the method of changing variables. Finally, numerical simulations are provided to demonstrate theoretical results.