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

Formation analysis and feasibility problems for high-order linear time-invariant swarm systems with time-delayed protocols are dealt with. Firstly, formation problems are transformed into consensus problems. Then necessary and sufficient conditions for swarm systems to achieve a given time-varying formation are presented. An explicit expression of the time-varying formation center function is given. It is shown that the moving mode of the formation center can be specified. Furthermore, necessary and sufficient conditions for formation feasibility are proposed. The approach to expand the feasible formation set and an algorithm to design the protocol for swarm systems to achieve formation are provided respectively. Finally, numerical simulations are presented to demonstrate theoretical results.