Controllability and Fraction of Leaders in Infinite Network

TL;DR

This paper investigates how the controllability of large linear networks depends on network topology and leader node placement, providing theoretical bounds and insights for designing controllable large-scale systems.

Contribution

It offers theoretical analysis of controllability in infinite networks, highlighting the impact of node degree and leader fraction, with explicit bounds for common network topologies.

Findings

High degree nodes challenge controllability as network size grows.

Bounded maximum degree networks maintain well-conditioned controllability Gramian.

A non-zero fraction of leader nodes can ensure controllability in path and cycle networks.

Abstract

In this paper, we study controllability of a network of linear single-integrator agents when the network size goes to infinity. We first investigate the effect of increasing size by injecting an input at every node and requiring that network controllability Gramian remain well-conditioned with the increasing dimension. We provide theoretical justification to the intuition that high degree nodes pose a challenge to network controllability. In particular, the controllability Gramian for the networks with bounded maximum degrees is shown to remain well-conditioned even as the network size goes to infinity. In the canonical cases of star, chain and ring networks, we also provide closed-form expressions which bound the condition number of the controllability Gramian in terms of the network size. We next consider the effect of the choice and number of leader nodes by actuating only a subset of nodes and considering the least eigenvalue of the Gramian as the network size increases. Accordingly, while a directed star topology can never be made controllable for all sizes by injecting an input just at a fraction $f<1$ of nodes; for path or cycle networks, the designer can actuate a non-zero fraction of nodes and spread them throughout the network in such way that the least eigenvalue of the Gramians remain bounded away from zero with the increasing size. The results offer interesting insights on the challenges of control in large networks and with high-degree nodes.