Performance of first and second order linear networked systems over digraphs

In this paper we investigate the performance of linear networked dynamical systems over digraphs with a globally reachable node. We consider first and second order systems subject to distributed disturbances and define an output that quantifies the performance through the input-output H
₂
norm of the system. We develop a generalized framework for computing the H
₂
norm for this class of systems, and apply this framework to evaluate two performance measures for systems whose underlying network graphs result in normal weighted graph Laplacian matrices. We find closed-form solutions for the measure that quantifies the total deviation of the states from the average, and bounds on the measure that quantifies the weighted squared difference between the states of neighboring nodes. Numerical examples indicate that a second order system connected over a cycle graph may have better performance when its underlying graph is directed due to complex eigenvalues of the Laplacian. The results also indicate that the H
₂
norm of a symmetric system is less than or equal to that of the corresponding perturbed non-symmetric system for either line or complete graphs when the network size is sufficiently large.