Disturbance Attenuation in a Consensus Network of Identical Linear Systems: An $ {\cal H}_{\infty }$ Approach

We consider two problems related to disturbance attenuation in undirected consensus networks of identical linear systems subject to exogenous disturbances: 1) network interconnection design and 2) design of distributed and decentralized controllers. We use the H∞ norm of the transfer function from the disturbance vector to the disagreement vector of the network as the performance metric for disturbance attenuation. We show that the disturbance attenuation performance is enhanced by maximizing the second smallest eigenvalue of the graph Laplacian under a certain condition, which can be checked using a linear matrix inequality. For the case of a consensus network with fixed interconnection weights, e.g., as the result of physical constraints, we provide algorithms for the design of both decentralized and distributed controllers that ensure a prescribed disturbance attenuation performance.