General linear second-order consensus control with guaranteed rate of convergence

This paper presents necessary and sufficient conditions to achieve consensus for a homogeneous network of linear second-order systems. Simple controllers to achieve consensus with a guaranteed rate of convergence are proposed. The only requirement is that the network graph contain a spanning tree. The eigenvalues of the graph Laplacian can be complex and the consensus behavior of the systems will not be altered. The paper concludes with an example where the proposed controllers were used to synchronize a network of damped oscillators.