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

Abstract

The second smallest eigenvalue and the largest eigenvalue of the Laplacian matrix of a simple undirected connected graph G are called the algebraic connectivity ${\lambda }_2\left(G\right)$ and the Laplacian spectral radius ${\lambda }_n\left(G\right)$, 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 ${\lambda }_2\left(G\right)$ results in a faster consensus convergence rate, while a smaller ${\lambda }_n\left(G\right)$ 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 ${\lambda }_2\left(G\right)$ increases, then the weight of one edge $\{v_s,v_t\}$ increases, i.e., the increment ${\delta }_{st}>0$, and the entries of its eigenvector with respect to $v_s$ and $v_t$ are not equal. If ${\lambda }_n\left(G\right)$ decreases, then the weight of one edge $\{v_s,v_t\}$ decreases, i.e., the increment ${\delta }_{st}<0$, and the entries of its eigenvector with respect to $v_s$ and $v_t$ are not equal. Moreover, when considering adjusting the weights of edges, some necessary conditions for increasing ${\lambda }_2\left(G\right)$ and decreasing ${\lambda }_n\left(G\right)$ 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.