Topology design for stochastically-forced consensus networks

We study an optimal control problem aimed at adding a certain number of edges to an undirected network, with a known graph Laplacian, in order to optimally enhance closed-loop performance. The performance is quantified by the steady-state variance amplification of the network with additive stochastic disturbances. To promote controller sparsity, we introduce `1-regularization into the optimal H2 formulation and cast the design problem as a semidefinite program. We derive a Lagrange dual, provide interpretation of dual variables, and exploit structure of the optimality conditions for undirected networks to develop customized proximal gradient and Newton algorithms that are well-suited for large problems. We illustrate that our algorithms can solve the problems with more than million edges in the controller graph in a few minutes, on a PC. We also exploit structure of connected resistive networks to demonstrate how additional edges can be systematically added in order to minimize the H2 norm of the closed-loop system.