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Robust Macroscopic Density Control of Heterogeneous Multi-Agent Systems

Gian Carlo Maffettone, Davide Salzano, Mario di Bernardo

TL;DR

This work presents a robust macroscopic density control framework for large-scale, heterogeneous multi-agent systems by deriving upper and lower bounding advection–diffusion models under bounded unknown drifts, and designing a Lyapunov-based macroscopic feedback law that guarantees global exponential convergence of the density error in $L^2$. The approach follows a continuification paradigm, translating macro-level stability guarantees into distributed microscopic actuation through density-based discretization and velocity-field recovery. Theoretical results are supported by extensive numerical validation in one and two dimensions, including heterogeneous oscillators, traffic flow regulation on rings, and swarm robotics over partially unknown terrains. The framework offers scalable robustness margins before microscopic implementation and motivates future work on decentralization and experimental demonstrations for real-world large populations.

Abstract

Modern applications, such as orchestrating the collective behavior of robotic swarms or traffic flows, require the coordination of large groups of agents evolving in unstructured environments, where disturbances and unmodeled dynamics are unavoidable. In this work, we develop a scalable macroscopic density control framework in which a feedback law is designed directly at the level of an advection--diffusion partial differential equation. We formulate the control problem in the density space and prove global exponential convergence towards the desired behavior in $\mathcal{L}^2$ with guaranteed asymptotic rejection of bounded unknown drift terms, explicitly accounting for heterogeneous agent dynamics, unmodeled behaviors, and environmental perturbations. Our theoretical findings are corroborated by numerical experiments spanning heterogeneous oscillators, traffic systems, and swarm robotics in partially unknown environments.

Robust Macroscopic Density Control of Heterogeneous Multi-Agent Systems

TL;DR

This work presents a robust macroscopic density control framework for large-scale, heterogeneous multi-agent systems by deriving upper and lower bounding advection–diffusion models under bounded unknown drifts, and designing a Lyapunov-based macroscopic feedback law that guarantees global exponential convergence of the density error in . The approach follows a continuification paradigm, translating macro-level stability guarantees into distributed microscopic actuation through density-based discretization and velocity-field recovery. Theoretical results are supported by extensive numerical validation in one and two dimensions, including heterogeneous oscillators, traffic flow regulation on rings, and swarm robotics over partially unknown terrains. The framework offers scalable robustness margins before microscopic implementation and motivates future work on decentralization and experimental demonstrations for real-world large populations.

Abstract

Modern applications, such as orchestrating the collective behavior of robotic swarms or traffic flows, require the coordination of large groups of agents evolving in unstructured environments, where disturbances and unmodeled dynamics are unavoidable. In this work, we develop a scalable macroscopic density control framework in which a feedback law is designed directly at the level of an advection--diffusion partial differential equation. We formulate the control problem in the density space and prove global exponential convergence towards the desired behavior in with guaranteed asymptotic rejection of bounded unknown drift terms, explicitly accounting for heterogeneous agent dynamics, unmodeled behaviors, and environmental perturbations. Our theoretical findings are corroborated by numerical experiments spanning heterogeneous oscillators, traffic systems, and swarm robotics in partially unknown environments.
Paper Structure (30 sections, 4 theorems, 87 equations, 5 figures)

This paper contains 30 sections, 4 theorems, 87 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Work and Main Contributions
  3. Mathematical Modeling
  4. Problem Statement
  5. Control design
  6. Bounding system
  7. Macroscopic model
  8. Macroscopic control design
  9. Closed-loop analysis and extensions
  10. Well-posedness of the closed loop dynamics
  11. Extension to tracking problems
  12. Choice of alpha
  13. Periodic boundary conditions
  14. Reflective boundary conditions
  15. Discretization
  16. ...and 15 more sections

Key Result

Theorem 1

Choose where $k_p>0$, $\alpha(t)$ is any bounded function of time, and $k_s(t)$ satisfies with Then, the solution of eq:error_dynamics satisfies i.e., the density tracking error converges globally and exponentially to zero in $\mathcal{L}^2(\Omega)$.

Figures (5)

  • Figure 1: Validation of the controller in a 1D macroscopic setting. a. Evolution in time and space (bottom panel) of the density described by Equation \ref{['eq:upper_bounding']}. The upper panel shows the desired density using the same colormap used for the density portrayed in the bottom panel. b. Evolution in time and space of the velocity field $U(x,t)$ generated by the controller. c. $\mathcal{L}^2$ norm of the control error over time. d. $\mathcal{L}^2$ norm of the velocity field induced by the control action in time.
  • Figure 2: Control of an ensemble of heterogeneous stochastic oscillators. a. Evolution in time an space of all the agents in the ensemble. On the z-axis the estimated (solid) and desired (dashed) densities are displayed in four representative time instants. b. Evolution of $\Vert e\Vert_2$ in time. c. Residual steady state error $e(x,T)$ for increasing values of the oscillators heterogeneity $K_{dist}$. The red dashed vertical line represents the level of heterogeneity used to determine the control gains. $T_f$ is the final time instant of the simulation, set as in panels a and b as 0.15 time units. The inset shows the final density density (solid black line) against the reference density (dashed black line).
  • Figure 3: Control of a 2-dimensional Fokker--Planck equation.a. Plot of the desired density (shaded surface) and initial density $\rho(\mathbf{x},0)$ (solid surface). Darker colors correspond to lower density values, while lighter colors indicate higher densities. b. Plot of the desired density (shaded surface) and final density $\rho(\mathbf{x},T_f)$ (solid surface), where $T_f$ denotes the final simulation time. c. Evolution of $\lVert e \rVert_2$ over time. d. Evolution in time of the $\mathcal{L}^2$ norm of the first (blue solid line) and second (orange dashed line) component of the velocity field $\mathbf{U}(\mathbf{x},t)$.
  • Figure 4: Control of vehicles moving on a ring. a. Evolution in time an space of all the agents in the ensemble (x and y axes). On the z axis the estimated (solid) and desired (dashed) densities are displayed in four representative time instants. b. Evolution of the $\mathcal{L}^2$ norm of the control error in time. c. Evolution of the average velocity of the vehicles over time. This metric was computed as $\bar{u}(t) = 1/N \sum_{i=1}^{N}{|u_i(t)|}$, with $N$ being the number of vehicles in the group.
  • Figure 5: Control of an ensemble of Unmanned Ground Vehiclesa. position of the unmanned ground vehicles (represented as black dots) in the hilly landscape. The elevation of the ground is graphically represented as a surface with a color map where the color shifts from green to orange as the ground elevation increases. b. Agents positions at the final simulation step against the reference density. The reference density is portrayed as a greyscale shaded surface where the higher is the density the darker is the color. c. Evolution of $\mathcal{L}^2$ norm of the control error in time.

Theorems & Definitions (18)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1: Global exponential macroscopic convergence
  • proof
  • Remark 4
  • Remark 5: Recovery of the velocity field
  • Remark 6: Periodic boundary conditions
  • Theorem 2: Stability under a regularized control action
  • proof
  • ...and 8 more