Algebraic criteria for consensus problem of networked systems with continuous-time dynamics

This paper addresses algebraic criteria for consensus problem of continuous-time networked systems, in which both fixed and switching topology cases are considered. A special eigenvector omega of Laplacian matrix is first constructed and correlated with the connectivity of digraph. And then, based on this tool, some necessary and/or sufficient algebraic conditions are proposed, which can directly determine whether the consensus problem can be solved or not. Furthermore, it is clearly shown that only the agents corresponding to the positive elements of omega contribute to the group decision value and decide the collective behavior of all agents. Particularly for the fixed topology case, not only the role of each agent is exactly measured by the value of the corresponding element of omega but also the group decision value can be calculated by such a vector and the initial states of all agents.