Leaderless Consensus of Linear Multi-agent Systems: Matrix Decomposition Approach

This paper considers the leaderless consensus problem of linear multi-agent systems with static and dynamic consensus controllers. The communication topology is modeled by a directed graph which contains a spanning tree. A special type of matrix decomposition is performed on the graph Laplacian matrix which can be factored into the product of two specific matrices. Base on this property of graph Laplacian matrix, a novel analysis approach for leaderless consensus problem is introduced in which the consensus problem can be converted into a stabilization problem of a system with lower dimensions by performing a proper variable transformation. Sufficient conditions are obtained based on Lyapunov stability analyses and algebraic graph theory. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.