Graph Neural Networks for Distributed Linear-Quadratic Control

TL;DR

This paper proposes using graph neural networks to design distributed controllers for linear-quadratic problems in network systems, enabling scalable, stable, and efficient control solutions learned through self-supervised methods.

Contribution

It introduces a novel GNN-based approach for distributed control in linear-quadratic systems, with stability conditions and extensive simulation validation.

Findings

GNN controllers are computationally efficient and scalable.

The method achieves stable closed-loop systems under certain conditions.

GNNs demonstrate transferability across different network sizes.

Abstract

The linear-quadratic controller is one of the fundamental problems in control theory. The optimal solution is a linear controller that requires access to the state of the entire system at any given time. When considering a network system, this renders the optimal controller a centralized one. The interconnected nature of a network system often demands a distributed controller, where different components of the system are controlled based only on local information. Unlike the classical centralized case, obtaining the optimal distributed controller is usually an intractable problem. Thus, we adopt a graph neural network (GNN) as a parametrization of distributed controllers. GNNs are naturally local and have distributed architectures, making them well suited for learning nonlinear distributed controllers. By casting the linear-quadratic problem as a self-supervised learning problem, we are able to find the best GNN-based distributed controller. We also derive sufficient conditions for the resulting closed-loop system to be stable. We run extensive simulations to study the performance of GNN-based distributed controllers and showcase that they are a computationally efficient parametrization with scalability and transferability capabilities.