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Leader-Follower Density Control of Spatial Dynamics in Large-Scale Multi-Agent Systems

Gian Carlo Maffettone, Alain Boldini, Maurizio Porfiri, Mario di Bernardo

TL;DR

This work develops a macroscopic density-control framework for large-scale leader–follower multi-agent systems on periodic domains, formulating coupled convection–diffusion PDEs with a nonlocal interaction kernel. It derives feasibility conditions linking leader mass, diffusion, and kernel scale to ensure a desired follower density is attainable, and presents two globally stable control architectures: a feed-forward scheme tracking a reference leader density, and a reference–governor scheme that adapts targets based on both populations. The methods are extended from 1D to higher dimensions, including a continuification path to connect macroscopic controllers with finite-agent implementations, and are validated numerically in 1D and 2D with robustness analyses under disturbances and model perturbations. The framework offers closed-form feedback laws, explicit convergence guarantees, and practical pathways to bridge continuum models with discrete swarm behavior, enabling applications in traffic control and swarm robotics. Future work includes relaxing kinematic assumptions, incorporating inter-follower interactions, and developing localized, distributed experiments.

Abstract

We address the problem of controlling the density of a large ensemble of follower agents by acting on a group of leader agents that interact with them. Using coupled partial integro-differential equations to describe leader and follower density dynamics, we establish feasibility conditions and develop two control architectures ensuring global stability. The first employs feed-forward control on the followers' and a feedback on the leaders' density. The second implements a dual feedback loop through a reference-governor that adapts the leaders' density based on both populations' measurements. Our methods, initially developed in a one-dimensional setting, are extended to multi-dimensional cases, and validated through numerical simulations for representative control applications, both for groups of infinite and finite size.

Leader-Follower Density Control of Spatial Dynamics in Large-Scale Multi-Agent Systems

TL;DR

This work develops a macroscopic density-control framework for large-scale leader–follower multi-agent systems on periodic domains, formulating coupled convection–diffusion PDEs with a nonlocal interaction kernel. It derives feasibility conditions linking leader mass, diffusion, and kernel scale to ensure a desired follower density is attainable, and presents two globally stable control architectures: a feed-forward scheme tracking a reference leader density, and a reference–governor scheme that adapts targets based on both populations. The methods are extended from 1D to higher dimensions, including a continuification path to connect macroscopic controllers with finite-agent implementations, and are validated numerically in 1D and 2D with robustness analyses under disturbances and model perturbations. The framework offers closed-form feedback laws, explicit convergence guarantees, and practical pathways to bridge continuum models with discrete swarm behavior, enabling applications in traffic control and swarm robotics. Future work includes relaxing kinematic assumptions, incorporating inter-follower interactions, and developing localized, distributed experiments.

Abstract

We address the problem of controlling the density of a large ensemble of follower agents by acting on a group of leader agents that interact with them. Using coupled partial integro-differential equations to describe leader and follower density dynamics, we establish feasibility conditions and develop two control architectures ensuring global stability. The first employs feed-forward control on the followers' and a feedback on the leaders' density. The second implements a dual feedback loop through a reference-governor that adapts the leaders' density based on both populations' measurements. Our methods, initially developed in a one-dimensional setting, are extended to multi-dimensional cases, and validated through numerical simulations for representative control applications, both for groups of infinite and finite size.
Paper Structure (26 sections, 11 theorems, 133 equations, 8 figures)

This paper contains 26 sections, 11 theorems, 133 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Related work and main contributions
  3. Mathematical preliminaries
  4. Model
  5. Problem statement
  6. Feasibility analysis
  7. Example
  8. Followers' Feed-Forward and Leaders' Feedback Control
  9. Leaders' control design
  10. Followers' stability analysis
  11. Reference-governor control
  12. Governor design
  13. Numerical validation
  14. Robustness analysis
  15. Perturbations
  16. ...and 11 more sections

Key Result

Lemma 1

Given $f_1, \dots, f_n \in \mathcal{L}^p(\mathcal{S})$, we have For instance, if $n=2$, we have $\Vert f_1f_2\Vert_1\leq \Vert f_1\Vert_2 \Vert f_2\Vert_2$, as well as $\Vert f_1f_2\Vert_1\leq \Vert f_1\Vert_1 \Vert f_2\Vert_\infty$.

Figures (8)

  • Figure 1: Feasibility plots: minimum amount of leaders' mass, $\widehat{M}^L$ for (a) fixed $L$ and varying $\kappa$ and $D$, (b) fixed $D$ and varying $\kappa$ and $L$, and (c) fixed $\kappa$ and varying $D$ and $L$. In red we show the curve indicating when $\widehat{M}^L$ becomes greater than 1. $\widehat{M}^L$ has been saturated to 1 for visualization purposes.
  • Figure 2: (a) Feed-forward control scheme. (b) Detail of the leaders' reference generator block (where the feasibility analysis is performed).
  • Figure 3: (a) Reference-governor control scheme. (b) Detail of the governor block.
  • Figure 4: Monomodal trial: (a) initial and final densities; (b) time evolution of the percentage error and KL divergences using the feed-forward control scheme; and (c) time evolution of the percentage error, KL divergences, and $\alpha$ using the reference-governor scheme.
  • Figure 5: Robustness to external disturbance. Percentage error (top panel) and evolution of $\alpha$ (bottom panel) in time for the feedback control schemes (orange line for feed forwards, yellow line for reference-governor and purple line for reference-governor with an improved choice of $\alpha$).
  • ...and 3 more figures

Theorems & Definitions (33)

  • Definition 1: Unit circle
  • Definition 2: $\mathcal{L}^p$ norms on $\mathcal{S}$ axler2020
  • Lemma 1: H$\ddot{\mathrm{o}}$lder's inequality axler2020
  • Lemma 2: Poincaré-Wirtinger inequality for $\mathcal{S}$ heinonen2001lectures
  • Remark 1
  • Remark 2
  • Lemma 3: Comparison lemma khalil2002nonlinear
  • Lemma 4
  • proof
  • Remark 3
  • ...and 23 more