Leader selection in directed networks

Abstract

We study the problem of leader selection in directed consensus networks. In this problem, certain ‘leader’ nodes in a consensus network are equipped with absolute information about their state. This corresponds to diagonally strengthening a dynamical generator given by the negative of a directed graph Laplacian. We provide a necessary and sufficient condition for the stabilization of directed consensus networks via leader selection and form regularized H 2 and H ∞ optimal problem leader selection problems. We draw on recent results that establish the convexity of the H 2 and H ∞ norms for structured decentralized control of positive systems and identify sparse sets of leaders by imposing an l 1 penalty on the vector of leader weights. This allows us to develop a method that simultaneously assigns leader weights and selects a limited number of leaders. We use proximal gradient and subgradient method to solve the optimization problems and provide examples to illustrate our developments.