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Robust multi-scale leader-follower control of large multi-agent systems

Davide Salzano, Gian Carlo Maffettone, Mario di Bernardo

Abstract

In many multi-agent systems of practical interest, such as traffic networks or crowd evacuation, control actions cannot be exerted on all agents. Instead, controllable leaders must indirectly steer uncontrolled followers through local interactions. Existing results address either leader-follower density control of simple, unperturbed multi-agent systems or robust density control of a single directly actuated population, but not their combination. We bridge this gap by deriving a coupled continuum description for leaders and followers subject to unknown bounded perturbations, and designing a macroscopic feedback law that guarantees global asymptotic convergence of the followers' density to a desired distribution. The coupled stability of the leader-follower system is analyzed via singular perturbation theory, and an explicit lower bound on the leader-to-follower mass ratio required for feasibility is derived. Numerical simulations on heterogeneous biased random walkers validate our theoretical findings.

Robust multi-scale leader-follower control of large multi-agent systems

Abstract

In many multi-agent systems of practical interest, such as traffic networks or crowd evacuation, control actions cannot be exerted on all agents. Instead, controllable leaders must indirectly steer uncontrolled followers through local interactions. Existing results address either leader-follower density control of simple, unperturbed multi-agent systems or robust density control of a single directly actuated population, but not their combination. We bridge this gap by deriving a coupled continuum description for leaders and followers subject to unknown bounded perturbations, and designing a macroscopic feedback law that guarantees global asymptotic convergence of the followers' density to a desired distribution. The coupled stability of the leader-follower system is analyzed via singular perturbation theory, and an explicit lower bound on the leader-to-follower mass ratio required for feasibility is derived. Numerical simulations on heterogeneous biased random walkers validate our theoretical findings.
Paper Structure (14 sections, 2 theorems, 41 equations, 4 figures)

This paper contains 14 sections, 2 theorems, 41 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Mathematical modeling
  3. Microscopic model
  4. Macroscopic model
  5. Problem statement
  6. Control design
  7. Followers control design
  8. Deconvolution and choice of $\alpha(t)$
  9. Minimum leader mass for feasibility
  10. Leaders' control
  11. Coupled stability analysis
  12. Numerical Validation
  13. Finite population effects
  14. Conclusions

Key Result

Theorem 1

We choose $v^{FL}$ in eq:control_action, with $v^{FB}$ such that Here where $\beta$ is a bounded function of time. If $\Vert g_1 \Vert_\infty<2$, $k_p^F > 0$, then $e^F$ globally asymptotically converges to 0 in $\mathcal{L}^2(\Omega)$, $\forall \alpha\in(0,1]$.

Figures (4)

  • Figure 1: Block diagram of the multi-scale leader-follower control architecture. An outer loop regulates the followers' density by comparing the reference $\bar{\rho}^F$ with the estimated density $\rho^F$, generating a reference density $\bar{\rho}^L$ for the leaders. An inner loop tracks this reference, producing the macroscopic control field $u(x,t)$. The macro-to-micro bridge maps $u(x,t)$ to individual leader inputs $u_i$ via spatial sampling. The multi-agent dynamics produce leader and follower positions $\mathbf{x}_i$, which the micro-to-macro bridge converts back to estimated densities $\rho^L$ and $\rho^F$ via density estimation, closing both feedback loops.
  • Figure 2: Control of biased random walkers moving on a ring. a. Evolution in time and space of all the followers 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 in time and space of all the leaders 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. c. Evolution in time and space of the velocity field $u(x,t)$ generated by the controller and used to steer the leaders towards their desired density. d. Evolution of the $\mathcal{L}^2(\Omega)$ norm of the followers control error in time.
  • Figure 3: Robustness to heterogeneity. a. $\mathcal{L}_2(\Omega)$ norm of the followers control error for increasing heterogeneity in the follower population. The red shaded area represents the conditions where \ref{['eq:minimum_mass']} is not satisfied. $T=1$ is the terminal instant of the simulation. b. Evolution in time and space of all the followers in the ensemble when $b_i\sim\mathcal{U}([-20,20])$. On the z axis the estimated (solid) and desired (dashed) densities are displayed in four representative time instants.
  • Figure 4: Effects of finite population size. $\mathcal{L}_2(\Omega)$ norm of the followers control error when (a) varying the number of leaders $N^L\in[10,5000]$ when $N^F=1000$, and (b) varying the number of followers $N^F\in[10,5000]$ when $N^L=1000$ (vertical dashed lines denote the threshold above which the error norm is below $10^{-2}$). $T=1.5$ is the terminal instant of the simulation. $N^L$($N^F$) were sampled in the interval with $[10,5000]$ using 30 samples equally spaced in logarithmic scale.

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • proof
  • Remark 4