Optimal convergence speed of consensus under constrained damping for multi-agent systems with discrete-time double-integrator dynamics

This paper considers the optimization of the convergence speed of consensus under given damping constraints for multi-agent systems with discrete-time double-integrator dynamics with fixed interconnection topology. This work summarizes and details existing results in the case of undirected topologies and extends them to directed ones. The interconnection topology is assumed to be connected or to contain a rooted-out branching, respectively. Depending on the minimum required damping, for undirected interconnection topologies in most cases analytic solutions are provided. The structure of these solutions is independent of the size of the network and only depends on the largest and second smallest eigenvalue of the corresponding Laplacian. For the remaining cases without analytic solutions provided, a combined bisection grid search is presented that solves the constrained optimization problem efficiently. This algorithm can also be applied to directed interconnection topologies and, as for the undirected case, converges to the single optimum. Simulation results are provided that demonstrate the effectiveness of the proposed approach.