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 with time delays on directed graphs are dealt with. 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 the asymptotically stable 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 applying changing variable methods, an approach is provided to determine the gain matrix in the protocols. Numerical simulations are shown to demonstrate theoretical results.