Cycles and sparse design of consensus networks

This work considers the role that cycles play in consensus networks. We show how the presence of cycles improve the H2 performance of the consensus network. In particular, we provide an explicit combinatorial characterization relating the length of cycles to the improvement in the performance of the network. This analysis points to a general trade-off between the length of the cycle and how many edges the cycle shares with other cycles. These analytic results are then used to motivate a design procedure for consensus networks based on an ℓ
₁
relaxation. This relaxation method leads to sparse and {0, 1}-solutions for the design of consensus graphs. A feature of the ℓ
₁
relaxation is the ability to include weighting terms in the objective. The choice of weighting functions are related to the combinatorial properties of the graph. The applicability of this scheme is then shown via a set of numerical examples.