Topology design for stochastically-forced consensus networks

TL;DR

This paper presents an efficient method for designing sparse, optimally enhanced consensus networks by adding edges to improve performance while maintaining communication efficiency, using semidefinite programming and customized algorithms.

Contribution

It introduces a novel semidefinite programming approach with $$-regularization for sparse network design, along with tailored algorithms for large-scale problems.

Findings

Algorithms solve networks with over a million edges in minutes.

Structured approach effectively minimizes the $$ norm of the closed-loop system.

Systematic edge addition improves network coherence and performance.

Abstract

We study an optimal control problem aimed at achieving a desired tradeoff between the network coherence and communication requirements in the distributed controller. Our objective is to add a certain number of edges to an undirected network, with a known graph Laplacian, in order to optimally enhance closed-loop performance. To promote controller sparsity, we introduce $\ell_1$-regularization into the optimal ${\cal H}_2$ 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 ${\cal H}_2$ norm of the closed-loop system.