Sensing and Control in Symmetric Networks

TL;DR

This paper investigates how symmetries in network structures affect controllability and observability, proposing algorithms to improve control design in symmetric networked systems using representation theory.

Contribution

It analyzes the impact of graph symmetries on controllability and observability and introduces an algorithm for designing sparse input/output matrices based on isotypic projections.

Findings

Symmetries can reduce controllability in networked systems.

Representation theory helps analyze controllability related to symmetries.

Proposed algorithms facilitate the design of sparse control matrices.

Abstract

In engineering applications, one of the major challenges today is to develop reliable and robust control algorithms for complex networked systems. Controllability and observability of such systems play a crucial role in the design process. The underlying network structure may contain symmetries -- caused for example by the coupling of identical building blocks -- and these symmetries lead to repeated eigenvalues in a generic way. This complicates the design of controllers since repeated eigenvalues might decrease the controllability of the system. In this paper, we will analyze the relationship between the controllability and observability of complex networked systems and graph symmetries using results from representation theory. Furthermore, we will propose an algorithm to compute sparse input and output matrices based on projections onto the isotypic components. We will illustrate our results with the aid of two guiding examples, a network with $ D_4 $ symmetry and the Petersen graph.