Closed-form solution for optimal convergence speed of multi-agent systems with discrete-time double-integrator dynamics for fixed weight ratios

This paper considers the convergence speed of multi-agent systems with discrete-time double-integrator dynamics. The communication topology is assumed to be fixed and undirected. First the speed of convergence of the associated average consensus protocol is analyzed, and the problem of maximizing the convergence speed over the free parameters in the consensus protocol is considered under the assumption that the weights of communication links are fixed. A closed-form solution to this problem is proposed. Next it is shown that when the weights of the communication links are considered as decision variables, a global optimum of the convergence speed can be obtained by solving an LMI problem. Simulation results are provided that demonstrate the effectiveness of the proposed approach.