Speeding up finite-time consensus via minimal polynomial of a weighted graph - a numerical approach

Reaching consensus among states of a multi-agent system is a key requirement for many distributed control/optimization problems. Such a consensus is often achieved using the standard Laplacian matrix (for continuous system) or Perron matrix (for discrete-time system). Recent interest in speeding up consensus sees the development of finite-time consensus algorithms. This work proposes an approach to speed up finite-time consensus algorithm using the weights of a weighted Laplacian matrix. The approach is an iterative procedure that finds a low-order minimal polynomial that is consistent with the topology of the underlying graph. In general, the lowest-order minimal polynomial achievable for a network system is an open research problem. This work proposes a numerical approach that searches for the lowest order minimal polynomial via a rank minimization problem using a two-step approach: the first being an optimization problem involving the nuclear norm and the second a correction step. Several examples are provided to illustrate the effectiveness of the approach.