Joint leader and link weight selection for fast convergence in multi-agent systems

Multi-agent system protocols, such as consensus and formation control, require agents to converge rapidly to their desired states. Smooth convergence is achieved through selection of leader nodes (in leader-follower systems) or choosing optimal link weights (in leader-less systems). In this paper, we introduce a framework for minimizing convergence time through optimization of joint leader and link weight selection for a class of systems that include leader-follower and leader-less systems as special cases. We present a semidefinite programming approach for two problems: first, the problem of selecting a fixed number of leaders and the corresponding link weights in order to minimize the convergence time, and second, the problem of selecting the minimum number of leaders and corresponding link weights to achieve a bound on the convergence time. We prove that our approach is equivalent to classical results in discrete location theory, with provable optimality gaps, under special cases. Our results are illustrated through numerical simulations.