Controllability Gramian Spectra of Random Networks

TL;DR

This paper develops a theoretical framework using random matrix theory to analyze the eigenvalue spectra of the controllability Gramian in systems with random network structures, aiding understanding of control energy requirements.

Contribution

It introduces a novel approach to characterize the eigenvalue distribution of the controllability Gramian for systems with random matrices, including closed-form expressions for certain ensembles.

Findings

Eigenvalue spectra can be derived for various random graph ensembles.

The distribution informs about the energy needed for control.

Framework applies to Wigner, GOE, and regular graphs.

Abstract

We propose a theoretical framework to study the eigenvalue spectra of the controllability Gramian of systems with random state matrices, such as networked systems with a random graph structure. Using random matrix theory, we provide expressions for the moments of the eigenvalue distribution of the controllability Gramian. These moments can then be used to derive useful properties of the eigenvalue distribution of the Gramian (in some cases, even closed-form expressions for the distribution). We illustrate this framework by considering system matrices derived from common random graph and matrix ensembles, such as the Wigner ensemble, the Gaussian Orthogonal Ensemble (GOE), and random regular graphs. Subsequently, we illustrate how the eigenvalue distribution of the Gramian can be used to draw conclusions about the energy required to control random system.