On leader selection for performance and controllability in multi-agent systems

In a leader-follower multi-agent system (MAS), a set of leader agents act as external control inputs and are used to influence the dynamics of the remaining follower agents. Current approaches to selecting leaders are based on either achieving controllability of the follower agents or optimizing performance criteria such as robustness to noise, but not both. In this paper, we present a framework for selecting leaders based on joint consideration of controllability and performance. We first show that for the case where the number of nodes that can act as leaders is sufficient to guarantee controllability, the leader selection problem can be posed within a matroid optimization framework. For the case where the number of nodes that can serve as leaders is fixed and may not be sufficient for controllability, we introduce a new metric, the graph controllability index (GCI), defined as the fraction of network nodes that are controllable using the leader set. We prove that the GCI is a submodular function of the set of leader agents, leading to a submodular relaxation to the problem of achieving controllability. Our results are demonstrated using simulation study and compared to other leader selection algorithms, including random, average degree and descending order of degree based leader selection.