Fast Consensus Of Multi-Agent Systems On Digraphs

In recent years,the research on distributed control of multi-agent systems has attracted the attention of many experts and scholars.In this paper,we extend the graph filtering method to the consensus of multi-agent systems under digraphs.By introducing the normalized Laplacian matrix of the digraph,a time-varying distributed control protocol is designed to make the system achieve fast concensus.The main research works are as follows:1)When the non-zero eigenvalues of normalized Laplacian matrix are real multiple roots,a graph filter based on the information of 1-HOP local neighborhood is designed.For the case of known digraphs,the conditions for the system to reach the finite-time consensus are obtained.If the digraph is a spanning tree,our graph filter design method can make the consensus time of the system equal to the depth of the tree.For the case of unknown digraphs,the priodic controller designed by Chebyshev interpolation method can make the system reach the worst-case fast consensus.Numerical simulations are given to illustrate the effectiveness of the results.2)When the non-zero eigenvalues of normalized Laplacian matrix have complex numbers,a graph filter based on the information of 2-HOP local neighborhood is designed.For the case of known digraphs,the graph filter designed with the characteristic that the complex eigenvalues exist in pairs in the form of conjugate,which can avoid the complex number of the state,and the system can reach the finite-time concensus.For the case of unknown digraphs,the polynomial optimization algorithm proposed in this paper can make the system reach fast consensus.Extensive examples are given to demonstrate the effectiveness of the proposed methods.