Performance of leader-follower multi-agent systems in directed networks

TL;DR

This paper analyzes how the steady-state deviation of followers from a leader in multi-agent systems scales with network dimension, revealing unbounded growth in 1D and 2D but bounded variance in 3D directed lattices.

Contribution

It provides a detailed asymptotic analysis of variance scaling in directed lattice networks across different dimensions, highlighting the impact of network topology.

Findings

Variance diverges in 1D and 2D lattices as distance from the leader increases.

Variance remains bounded in 3D lattices.

Variance scales as a square-root in 1D and logarithmically in 2D.

Abstract

We consider leader-follower multi-agent systems in which the leader executes the desired trajectory and the followers implement the consensus algorithm subject to stochastic disturbances. The performance of the leader-follower systems is quantified by using the steady-state variance of the deviation of the followers from the leader. We study the asymptotic scaling of the variance in directed lattices in one, two, and three dimensions. We show that in 1D and 2D the variance of the followers' deviation increases to infinity as one moves away from the leader, while in 3D it remains bounded. We prove that the variance scales as a square-root function in 1D and a logarithmic function in 2D lattices.