Introducing network Gramians to undirected network systems for structure-preserving model reduction

In this paper, we propose the notion of controllability Gramian for linear network systems. In contrast to the conventional Gramians defined for asymptotically stable systems, the new Gramian is generalized to semi-stable systems and can be computed for network systems with imaginary axis poles. We also extend the Lyapunov equations to solve the network controllability Gramian. Based on this Gramian, we propose an efficient method to evaluate the H2-norms of network systems. The result is then applied to design a model reduction procedure for network systems using the clustering-based projection