Formation-containment tracking for high-order linear multi-agent systems on directed graphs

Formation-containment tracking problems for high-order linear multi-agent system on directed graphs are studied, where the multi-agent system is composed of a virtual leader, several real leaders and followers. The real leaders need to not only accomplish a predefined time-varying formation but also track the expected trajectory generated by the virtual leader, while the followers are required to converge to the convex hull formed by the real leaders. Firstly, a formation-containment tracking protocol is presented by using the neighboring relative information. Then an algorithm to design the control parameters is given, where the formation-containment tracking feasible constraint is proposed. Furthermore, based on the Lyapunov theory, it is proved that the multi-agent system can achieve the given formation-containment tracking under the designed protocol. Finally, the effectiveness of the theoretical results is verified by a simulation example.