Interaction topology optimization by adjustment of edge weights to improve the consensus convergence and prolong the sampling period for a multi-agent system

The second smallest eigenvalue and the largest eigenvalue of the Laplacian matrix of a simple undirected connected graph G are called the algebraic connectivity λ2(G) and the Laplacian spectral radius λn(G), respectively. For a first-order periodically sampled consensus protocol multi-agent system (MAS), whose interaction topology can be modeled as a graph G, a larger λ2(G) results in a faster consensus convergence rate, while a smaller λn(G) contributes to a longer sampling period of the system. Adjusting the weights of the edges is an efficient approach to optimize the interaction topology of a MAS, which improves the consensus convergence rate and prolongs the sampling period. If λ2(G) increases, then the weight of one edge {vs,vt} increases, i.e., the increment δst>0, and the entries of its eigenvector with respect to vs and vt are not equal. If λn(G) decreases, then the weight of one edge {vs,vt} decreases, i.e., the increment δst<0, and the entries of its eigenvector with respect to vs and vt are not equal. Moreover, when considering adjusting the weights of edges, some necessary conditions for increasing λ2(G) and decreasing λn(G) are also given respectively, both of which are determined by the entries of their eigenvectors with respect to the vertices of edges and the increment of edge weights. A number of numerical exemplifications are presented to support the theoretical findings.