Almost equitable partitions and controllability of leader–follower multi-agent systems

In this paper, the controllability of leader–follower multi-agent systems is studied from the perspective of graph theory. It is known that, for a given graph, the number of cells of an almost equitable partition (AEP) provides an upper bound on the controllability index. In this context, the main contribution here is an efficient algorithm to construct the maximal AEP for a given graph with a set of leaders. Specifically, we first establish the relationship between an AEP and the distance partition (DP) for the single-leader setting. It is shown that the cells of an AEP are subsets of cells of the DP. We extend the results to the multiple-leader case and develop a convergent algorithm to efficiently compute the maximal AEP. Finally, numerical examples are provided to show the effectiveness of the proposed method.