Distributed Optimal Control of Graph Symmetric Systems via Graph Filters

TL;DR

This paper introduces a novel approach to distributed optimal control for graph symmetric systems using graph filters, enabling efficient communication and stability guarantees, bridging control theory and graph signal processing.

Contribution

It proposes a new class of graph filters for optimal control in GSSs, with methods for distributed implementation and stability analysis, connecting control and graph signal processing.

Findings

Distributed controllers can be designed using graph filters.

Tradeoff between communication cost and control performance is demonstrated.

Stability and suboptimality guarantees are provided for the proposed controllers.

Abstract

Designing distributed optimal controllers subject to communication constraints is a difficult problem unless structural assumptions are imposed on the underlying dynamics and information exchange structure, e.g., sparsity, delay, or spatial invariance. In this paper, we borrow ideas from graph signal processing and define and analyze a class of Graph Symmetric Systems (GSSs), which are systems that are symmetric with respect to an underlying graph topology. We show that for linear quadratic problems subject to dynamics defined by a GSS, the optimal centralized controller is given by a novel class of graph filters with transfer function valued filter taps and can be implemented via distributed message passing. We then propose several methods for approximating the optimal centralized graph filter by a distributed controller only requiring communication with a small subset of neighboring subsystems. We further provide stability and suboptimality guarantees for the resulting distributed controllers. Finally, we empirically demonstrate that our approach allows for a principled tradeoff between communication cost and performance while guaranteeing stability. Our results can be viewed as a first step towards bridging the fields of distributed optimal control and graph signal processing.