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 <inline-formula><tex-math notation="LaTeX">$\ell _1$</tex-math></inline-formula>-regularization into the optimal <inline-formula><tex-math notation="LaTeX">${\mathcal H}_2$</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">${\mathcal H}_2$</tex-math></inline-formula> norm of the closed-loop system.