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High-dimensional continuification control of large-scale multi-agent systems under limited sensing and perturbations

Gian Carlo Maffettone, Mario di Bernardo, Maurizio Porfiri

TL;DR

The paper addresses robust control of large-scale multi-agent swarms by developing a high-dimensional continuification framework that begins from a microscopic ODE model, derives a macroscopic density PDE on $\Omega=[-\\pi,\\pi]^d$, and discretizes the macroscopic control back to microscopic inputs via a Poisson potential. A Lyapunov-based error control $q = K_p e - \nabla\cdot (e\mathbf{V}^d) - \nabla\cdot(\rho\mathbf{V}^e)$ ensures global asymptotic convergence of the density error in the ideal case, and the authors rigorously prove semi-global stability under limited sensing and bounded perturbations by bounding nonlocal terms and perturbations. They introduce a limited-sensing variant using a truncated interaction kernel and establish bounds showing the error remains bounded and can be made arbitrarily small with sufficient gain $K_p$, plus a perturbation analysis that yields a explicit bound $\lim_{t\to\infty}\sup \|e\|_2 \le H/(2K_p-\hat{W})$. Numerical validation with 100 agents in 2D confirms the theoretical results, demonstrating bounded, small errors under both limited sensing and perturbations, and highlighting the practical viability of the approach for large swarms.

Abstract

This paper investigates the robustness of a novel high-dimensional continuification control method for complex multi-agent systems. We begin by formulating a partial differential equation describing the spatio-temporal density dynamics of swarming agents. A stable control action for the density is then derived and validated under nominal conditions. Subsequently, we discretize this macroscopic strategy into actionable velocity inputs for the system's agents. Our analysis demonstrates the robustness of the approach beyond idealized assumptions of unlimited sensing and absence of perturbations.

High-dimensional continuification control of large-scale multi-agent systems under limited sensing and perturbations

TL;DR

The paper addresses robust control of large-scale multi-agent swarms by developing a high-dimensional continuification framework that begins from a microscopic ODE model, derives a macroscopic density PDE on , and discretizes the macroscopic control back to microscopic inputs via a Poisson potential. A Lyapunov-based error control ensures global asymptotic convergence of the density error in the ideal case, and the authors rigorously prove semi-global stability under limited sensing and bounded perturbations by bounding nonlocal terms and perturbations. They introduce a limited-sensing variant using a truncated interaction kernel and establish bounds showing the error remains bounded and can be made arbitrarily small with sufficient gain , plus a perturbation analysis that yields a explicit bound . Numerical validation with 100 agents in 2D confirms the theoretical results, demonstrating bounded, small errors under both limited sensing and perturbations, and highlighting the practical viability of the approach for large swarms.

Abstract

This paper investigates the robustness of a novel high-dimensional continuification control method for complex multi-agent systems. We begin by formulating a partial differential equation describing the spatio-temporal density dynamics of swarming agents. A stable control action for the density is then derived and validated under nominal conditions. Subsequently, we discretize this macroscopic strategy into actionable velocity inputs for the system's agents. Our analysis demonstrates the robustness of the approach beyond idealized assumptions of unlimited sensing and absence of perturbations.
Paper Structure (14 sections, 11 theorems, 55 equations, 3 figures)

This paper contains 14 sections, 11 theorems, 55 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Model and Problem Statement
  4. Problem statement
  5. High-dimensional Continuification Control
  6. Continuification
  7. Macroscopic Control Design
  8. Discretization and Microscopic control
  9. Robustness to limited sensing
  10. Structural Perturbations
  11. Numerical validation
  12. Robustness to limited sensing
  13. Robustness to perturbations
  14. Conclusions

Key Result

Lemma 1

Given $n$$L^p$ functions, $f_i$, with $i=1, 2, \dots n$, we have For instance, if $n=2$, $\Vert f_1f_2\Vert_1\leq \Vert f_1\Vert_2 \Vert f_2\Vert_2$ or $\Vert f_1f_2\Vert_1\leq \Vert f_1\Vert_1 \Vert f_2\Vert_\infty$.

Figures (3)

  • Figure 1: Continuification control scheme (inspired by nikitin2021continuation).
  • Figure 2: Robustness to limited sensing ($K_\mathrm{p} = 100$, (a, b) $\Delta = 0.1\pi$, (c, d) $\Delta = \pi$). (a, c) Final displacement of the system on top of the desired density; (b, d) time evolution of the percentage error for the discrete (blue) and continuous case (orange).
  • Figure 3: Robustness to perturbations ($K_\mathrm{p} = 100$): percentage error in time for (a) a continuous trial, (b) a discrete trial.

Theorems & Definitions (18)

  • Definition 1: $L^p$-norm on $\Omega$ Axler2020
  • Lemma 1: Holder's inequality Axler2020
  • Lemma 2: Minkowsky inequality Axler2020
  • Lemma 3: Young's convolution inequality Axler2020
  • Lemma 4: Comparison lemma khalil2002nonlinear
  • Lemma 5: Chapter 1.2 of griffiths2023introduction
  • Lemma 6
  • Theorem 1: Macroscopic convergence
  • proof
  • Remark
  • ...and 8 more