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Time-varying formation control problems for high-order linear time-invariant swarm systems with switching interaction topologies are investigated. A general formation control protocol is proposed firstly. Then using a consensus based approach, necessary and sufficient conditions for swarm systems with switching interaction topologies to achieve a given time-varying formation are presented. An explicit expression of the time-varying formation reference function is given. It is revealed that the switching interaction topologies have no effect on the formation reference function and the motion modes of the formation reference can be specified. Furthermore, necessary and sufficient conditions for formation feasibility are presented. An approach to expand the feasible formation set is given and an algorithm to design the protocol for swarm systems with switching interaction topologies to achieve time-varying formations is provided. Finally, numerical simulations are presented to demonstrate theoretical results.

Stable-protocol (SP) output consensus analysis and design problems for high-order linear time-invariant swarm systems with time-varying delays are investigated. First, a dynamic output feedback consensus protocol is proposed on the basis of the observable decomposition, and a necessary and sufficient condition for SP output consensus is given, which transforms the SP output consensus problem into asymptotical stability problems of multiple lower-dimensional subsystems. Then, in terms of linear matrix inequalities (LMIs), a sufficient condition for SP output consensualisation is presented, which can guarantee the scalability of swarm systems since it includes only five LMI constrains independent of the number of agents. Furthermore, an explicit expression of the output consensus function, which is independent of the time-varying delay, is shown and the impacts of initial states of agents and consensus protocols on the output consensus function are determined, respectively. Finally, a numerical example is given to demonstrate theoretical results.