Containment analysis and design for high-order linear time-invariant singular swarm systems with time delays

Containment analysis and design problems for high-order linear time-invariant singular swarm systems on directed graphs with time delays are investigated. To eliminate impulse terms in singular swarm systems and ensure that the singular swarm systems can achieve containment, time-delayed protocols are presented for leaders and followers, respectively. By model transformation, containment problems of singular swarm systems are converted into stability problems of multiple low-dimensional time-delayed systems. In terms of linear matrix inequality, sufficient conditions are presented for time-delayed singular swarm systems to achieve containment, which are independent of the number of agents. By using the method of changing variables, an approach is provided to determine the gain matrices in the protocols. Numerical simulations are shown to demonstrate theoretical results.