# On the Coherence of Large-Scale Networks with Distributed PI and PD Control

## TL;DR

This paper introduces distributed PI and PD controllers for large-scale double-integrator networks, significantly improving coherence by bounding state variance in the presence of stochastic disturbances, especially when agents have access to absolute measurements.

## Contribution

It proposes novel distributed PI and PD control schemes that overcome known limitations of standard consensus protocols, ensuring bounded network coherence with limited sensing capabilities.

## Key findings

- Controllers achieve bounded variance in network states
- Optimal tuning improves network coherence
- Simulation results validate theoretical improvements

## Abstract

We consider distributed control of double-integrator networks, where agents are subject to stochastic disturbances. We study performance of such networks in terms of coherence, defined through an H2 norm metric that represents the variance of nodal state fluctuations. Specifically, we address known performance limitations of the standard consensus protocol, which cause this variance to scale unboundedly with network size for a large class of networks. We propose distributed proportional integral (PI) and proportional derivative (PD) controllers that relax these limitations and achieve bounded variance, in cases where agents can access an absolute measurement of one of their states. This case applies to, for example, frequency control of power networks and vehicular formation control with limited sensing. We discuss optimal tuning of the controllers with respect to network coherence and demonstrate our results in simulations.