Time-varying group formation control for general linear multi-agent systems with directed topologies

Time-varying group formation control problems for general linear multi-agent systems with directed topologies are studied, where the agents in the multi-agent system are classified into subgroups and each subgroup is required to form a specified time-varying sub-formation. Different from the traditional complete formation, where only one formation is realized by the multi-agent system, in the group formation, there could be multiple sub-formations. Firstly, a time-varying group formation protocol is constructed by local relative information of each agent and its neighbors. Then nonsingular transformations are applied to the closed-loop multi-agent systems. Sufficient conditions for the multi-agent systems to achieve time-varying group formation are further presented together with the time-varying group formation feasibility constraints. Explicit expressions of the subgroup formation reference functions are derived to describe the macroscopic movement of the time-varying subgroup formations. It is shown that the subgroup formation reference functions are jointly determined by the dynamics, the initial states and formation information of each agent in the same subgroup, and also by the interaction topologies among agents from the same or different subgroups. Moreover, an approach to design the time-varying group formation protocol is proposed by solving an algebraic Riccati equation. Finally, a numerical example with three subgroups is provided to demonstrate the effectiveness of the obtained theoretical results.