Algebraic criteria for consensus problems of continuous-time networked systems

Abstract

This article addresses algebraic criteria for consensus problems of continuous-time networked systems with fixed and switching topology. A special eigenvector ω of the Laplacian matrix is first constructed and correlated with the connectivity of digraph. And then, some necessary and/or sufficient algebraic conditions are presented by employing the vector ω, which can directly determine whether the consensus problem can be solved. More importantly, for the switching topology case, the obtained algebraic conditions for the average consensus problem are necessary and sufficient. Furthermore, the presented results clearly show that only the agents corresponding to the positive elements of ω contribute to the group decision value and decide the collective behaviour of the system. Particularly for the fixed topology case, the role of each agent is exactly measured by the value of the corresponding element of ω. Based on these results, some extended protocols are further proposed to solve the average consensus problem, in which the interaction digraphs are not needed to be balanced. Numerical examples are included to illustrate the obtained results.

Keywords:

• cooperative control
• continuous-time networked systems
• multi-agent systems
• consensus problem
• algebraic criteria
• digraph theory
• Laplacian matrix

Acknowledgements

This work was supported by the NSFC under Grant 60727002, 60774003, 60334030, the MOE under Grant 20030006003, the COSTIND under Grant A2120061303, and the national 973 program under Grant 2005CB321902.

Notes

Note

1. In Figure 2, all edges whose weights are not labelled take 1 as their weights.