Algorithms for leader selection in stochastically forced consensus networks

We examine the leader selection problem in stochastically forced consensus networks. A node is a leader if, in addition to relative information from its neighbors, it also has an access to its own state. This problem arises in several applications including control of vehicular formations and localization in sensor networks. We are interested in selecting an a priori specified number of leaders such that the steady-state variance of the deviation from consensus is minimized. Even though we establish convexity of the objective function, combinatorial nature of constraints makes determination of the global minimum difficult for large networks. We introduce a convex relaxation of constraints to obtain a lower bound on the global optimal value. We also use a simple but efficient greedy algorithm and the alternating direction method of multipliers to compute upper bounds. Furthermore, for networks with noise-free leaders that perfectly follow their desired trajectories, a sequence of convex relaxations is used to identify the leaders. Several examples ranging from regular lattices to random graphs are provided to illustrate the effectiveness of the developed algorithms.