Minimal Leader Selection in General Linear Multi-Agent Systems With Switching Topologies: Leveraging Submodularity Ratio

In multi-agent systems with leader-follower dynamics, choosing a subset of agents as leaders is a critical step in achieving the desired coordination performance. In this study, by considering consensus tracking for general linear multi-agent systems under switching topologies, we address the problem of selecting a minimum-size set of leaders by leveraging the submodularity ratio. First, using the dwell time technique, a criterion is derived to ensure that the states of all agents can converge to a reference trajectory that is directly tracked by each leader. Second, exploiting the derived consensus tracking criterion, the metrics with a structure of the Euclidean distance between specific vectors and the space spanned by an iteratively updated matrix are established to identify a set of leaders, and then the corresponding bound of the submodularity ratio is proposed. Third, combining the derived criterion and the constructed metrics, a leader selection scheme is presented together with three polynomial-time algorithms, and the related provable optimality bound of each algorithm can be obtained by leveraging the proposed bound of the submodularity ratio. Finally, illustrative examples are provided to verify the effectiveness of the proposed leader selection scheme.