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Exponential Consensus through Z-Control in High-Order Multi-Agent Systems

Angela Monti, Fasma Diele

TL;DR

This work addresses achieving consensus in $k$-th order multi-agent systems when direct actuation of the top-order state is not always feasible. It develops a hierarchical Z-control framework that enforces exponential convergence to consensus in the highest-order state while preserving the average under weight-balanced interactions, with both direct and indirect actuation pathways. The paper generalizes Z-control to arbitrary order, provides explicit control laws, derives a unified least-squares solver for indirect control, and demonstrates effectiveness on first-order opinion dynamics and second- and third-order Cucker-Smale flocking models. It also analyzes scalability to higher dimensions, highlights numerical challenges, and suggests model-order reduction as a compelling direction for real-time, large-scale applications. Overall, Z-control offers a scalable, analytically tractable approach for coordinating complex multi-agent systems across diverse domains, with tunable convergence through the parameter $\lambda$ and practical pathways for indirect actuation.

Abstract

In this work, we introduce a Z-control strategy for multi-agent systems of arbitrary order, aimed at driving the agents toward consensus in the highest-order observable state. The proposed framework supports both direct and indirect control schemes, making it applicable in scenarios where high-order derivatives such as acceleration cannot be directly manipulated. Theoretical analysis ensures exponential convergence while preserving the average dynamics, and a hierarchy of control laws is derived accordingly. Numerical experiments up to third-order models, including opinion dynamics and Cucker-Smale flocking systems, demonstrate the robustness and flexibility of Z-control under varying interaction regimes and control intensities.

Exponential Consensus through Z-Control in High-Order Multi-Agent Systems

TL;DR

This work addresses achieving consensus in -th order multi-agent systems when direct actuation of the top-order state is not always feasible. It develops a hierarchical Z-control framework that enforces exponential convergence to consensus in the highest-order state while preserving the average under weight-balanced interactions, with both direct and indirect actuation pathways. The paper generalizes Z-control to arbitrary order, provides explicit control laws, derives a unified least-squares solver for indirect control, and demonstrates effectiveness on first-order opinion dynamics and second- and third-order Cucker-Smale flocking models. It also analyzes scalability to higher dimensions, highlights numerical challenges, and suggests model-order reduction as a compelling direction for real-time, large-scale applications. Overall, Z-control offers a scalable, analytically tractable approach for coordinating complex multi-agent systems across diverse domains, with tunable convergence through the parameter and practical pathways for indirect actuation.

Abstract

In this work, we introduce a Z-control strategy for multi-agent systems of arbitrary order, aimed at driving the agents toward consensus in the highest-order observable state. The proposed framework supports both direct and indirect control schemes, making it applicable in scenarios where high-order derivatives such as acceleration cannot be directly manipulated. Theoretical analysis ensures exponential convergence while preserving the average dynamics, and a hierarchy of control laws is derived accordingly. Numerical experiments up to third-order models, including opinion dynamics and Cucker-Smale flocking systems, demonstrate the robustness and flexibility of Z-control under varying interaction regimes and control intensities.

Paper Structure

This paper contains 23 sections, 6 theorems, 98 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. General $k$-th Order Consensus Models
  3. Consensus and invariance of average
  4. Z-Controlled $k$-th Order Multi-Agent Models
  5. Direct and Indirect Z-Control
  6. Direct Z-Control.
  7. Indirect Z-control.
  8. Applications to Opinion Dynamics and Flocking Models
  9. First-Order Opinion Dynamics Model
  10. Numerical example.
  11. Second-Order Cucker-Smale Flocking Model
  12. Direct control on velocities
  13. Indirect control of velocities through positions
  14. Third-Order generalized Cucker-Smale Flocking Model
  15. Direct control on accelerations
  16. ...and 8 more sections

Key Result

Lemma 1

Let $x_i^{(k)}(t)\in\mathbb{R}^d$ evolve according to k-order_system where the interaction matrix $A=(a_{ij})$ is weight-balanced. Then the population average is preserved.

Figures (10)

  • Figure 1: Opinion dynamics for four different values of the smoothness parameter $\alpha$. Left panels: uncontrolled dynamics. As $\alpha$ increases, the interaction function becomes more localized, transitioning from global consensus to the persistence of multiple opinion clusters. Center panels: Z-controlled dynamics converging to consensus. The controlled dynamics are similar for different values of $\alpha$: they converge to consensus with an error that decreases at the same rate, governed by $\lambda = 1$. Right panels: control signals required to steer the system toward consensus. For $\alpha = 0.1$ the controls remain around $\pm0.2$, while for $\alpha = 300$, significantly larger control values are needed to force the dynamics towards consensus.
  • Figure 2: Consensus parameter of the controlled opinion dynamics for $\alpha = 1.6$ and increasing values of the parameter $\lambda$. Larger values of $\lambda$ induce a faster convergence. Note that achieving consensus requires setting the product $\lambda \, T$ to approximately $10$.
  • Figure 3: Cucker-Smale flocking model: uncontrolled trajectories (left) and velocities (right) for $\beta = 0.1$ are shown. Consensus is always achieved for values of the smoothness parameter $\beta \leq 1/2$.
  • Figure 4: Cucker-Smale flocking model. Uncontrolled and direct Z-controlled dynamics for $\beta = 1$ and $\lambda=1$. Top panels: uncontrolled (left) and controlled (right) agent mean value trajectories. Central panels: uncontrolled (left) and controlled (right) mean velocity evolution. Bottom panels: the control inputs (left) and the consensus parameter (right) over time. Initial conditions are selected such that consensus is not naturally achieved ($\beta > 1/2$).
  • Figure 5: Cucker-Smale flocking model. Indirect Z-controlled system dynamics for $\beta = 1$ and $\lambda=1$. The first row shows agent mean value trajectories and mean velocity evolution, and the second row shows the control inputs and the consensus parameter over time. Initial conditions are selected such that consensus is not naturally achieved ($\beta > 1/2$).
  • ...and 5 more figures

Theorems & Definitions (15)

  • Lemma 1: Invariance of the average under weight-balanced interactions
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1: Conservation of the average of $x^{(k)}$
  • Remark 2
  • Remark 3
  • Theorem 3: Indirect velocity control via positions
  • ...and 5 more