On Stability and Inverse Optimality for a Class of Multi-agent Linear Consensus Protocols

This paper studies stability and inverse optimality for a class of linear consensus protocols applied to the identical linear time-invariant multi-agent systems, where communications among the agents are described by a fixed digraph, containing a spanning tree, whose scaled Laplacian is diagonalizable. The concept of the scaled Laplacian, normal Laplacian multiplied by a positive diagonal matrix, allows a more general graph topology to be handled. First, we show that partial stability and, even more, inverse optimality hold if either the scaling factors in each protocol are sufficiently large depending on graph properties, or the system matrix A satisfies the Lyapunov inequality. Then, duality principles between the agents’ and the consensus error dynamics are presented, which provide additional properties regarding consensus-related stability and inverse optimality. And the results are characterized in terms of the related symmetric Laplacian and its algebraic connectivity when the given graph is scaled undirected. Finally, through formation control simulation of a multi-agent mobile robot system for Г-scaled undirected and Г-scaled directed graphs, we have verified the theory of this paper on the partial stability and the inverse optimality conditions.