Edge addition in directed consensus networks

We study the problem of performance enhancement in stochastically-forced directed consensus networks by adding edges to an existing topology. We formulate the problem as a feedback control design, and represent the links as the elements of the controller graph Laplacian matrix. The topology design of the controller network can be cast as an ℓ ₁ regularized version of the ℋ ₂ optimal control problem. The goal is to optimize the performance of the network by selecting a controller graph with low communication requirements. To deal with the structural constraints that arise from the absence of absolute measurements, we introduce a coordinate transformation to eliminate the average mode and assure convergence of all states to the average of the initial node values. By exploiting structure of the optimization problem, we develop a customized algorithm based on the alternating direction method of multipliers to design a sparse controller network that improves the performance of the closed-loop system.