Controllability scores for selecting control nodes of large-scale network systems

To appropriately select control nodes of a large-scale network system, we propose two control centralities called volumetric and average energy controllability scores. The scores are the unique solutions to convex optimization problems formulated using the controllability Gramian. The uniqueness is proven for stable cases and for unstable cases that include multi-agent systems. We show that the scores can be efficiently calculated by using a proposed algorithm based on the projected gradient method onto the standard simplex. Numerical experiments demonstrate that the proposed algorithm is more efficient than an existing interior point method, and the proposed scores can correctly capture the importance of each state node on controllability, outperforming existing control centralities.