Consensus control of homogeneous networks of general second order discrete-time systems

We consider the problem of obtaining linear state feedback controllers to achieve consensus in a homogeneous network of general second-order discrete-time systems whose graph is directed. Consensus is achieved if certain polynomials associated with the closed-loop network are Schur stable. The coefficients of these polynomials depend on the eigenvalues of the network Laplacian matrix and can therefore be complex. This paper provides a new and simple proof and condition for a second-order polynomial with complex coefficients to be Schur. This condition was applied to obtain controllers which achieve consensus with a guaranteed rate of convergence. We show that consensus can always be achieved for marginally stable systems and discretized systems. Simple conditions for consensus achieving controllers are obtained when the Laplacian eigenvalues are all real. We also demonstrate how these results can be applied to some robust control problems.