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Consensus approximation and impulsive control for a class of uncertain multi-agent dynamics

Zoltan Nagy, Irinel-Constantin Morarescu, Lucian Busoniu

TL;DR

This work addresses consensus formation under time-varying, uncertain interaction gains by deriving practical bounds on the consensus value $\alpha$ via left eigenvectors of $\mathrm{diag}(\gamma)L$, and by formulating an LP-based method to compute these bounds efficiently. Building on these bounds, it develops an impulsive, budget-bounded control framework that jointly optimizes initial opinion shifts and uncertainty to bring the consensus near a target, casting the problem as tractable linear programs through a Charnes–Cooper transformation. Empirical results on scale-free networks show the proposed bounds are tight relative to naive extrema and that the LP-based budget allocation consistently outperforms linear baselines in both small- and large-scale networks. The approach offers scalable, theory-grounded tools for controlling consensus in uncertain multi-agent systems with applications in robotics and social influence campaigns.

Abstract

This paper studies a class of consensus dynamics where the interactions between agents are affected by a time-varying unknown scaling factor. This situation is encountered in the control of robotic fleets over a wireless network or in opinion dynamics where the confidence given to the peers varies in time. Firstly, we establish conditions under which practical upper and lower bounds on the consensus value can be determined. Secondly, we propose control strategies for allocating a given control budget to shift agent states towards a desired consensus value despite the uncertainty. We provide computationally efficient linear programming-based approaches for both problems and validate the obtained results in numerical simulations.

Consensus approximation and impulsive control for a class of uncertain multi-agent dynamics

TL;DR

This work addresses consensus formation under time-varying, uncertain interaction gains by deriving practical bounds on the consensus value via left eigenvectors of , and by formulating an LP-based method to compute these bounds efficiently. Building on these bounds, it develops an impulsive, budget-bounded control framework that jointly optimizes initial opinion shifts and uncertainty to bring the consensus near a target, casting the problem as tractable linear programs through a Charnes–Cooper transformation. Empirical results on scale-free networks show the proposed bounds are tight relative to naive extrema and that the LP-based budget allocation consistently outperforms linear baselines in both small- and large-scale networks. The approach offers scalable, theory-grounded tools for controlling consensus in uncertain multi-agent systems with applications in robotics and social influence campaigns.

Abstract

This paper studies a class of consensus dynamics where the interactions between agents are affected by a time-varying unknown scaling factor. This situation is encountered in the control of robotic fleets over a wireless network or in opinion dynamics where the confidence given to the peers varies in time. Firstly, we establish conditions under which practical upper and lower bounds on the consensus value can be determined. Secondly, we propose control strategies for allocating a given control budget to shift agent states towards a desired consensus value despite the uncertainty. We provide computationally efficient linear programming-based approaches for both problems and validate the obtained results in numerical simulations.

Paper Structure

This paper contains 14 sections, 3 theorems, 52 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Consensus bounds
  4. Analytical results
  5. Computing the bounds efficiently
  6. Experimental results
  7. Consensus Control
  8. Analytical results
  9. Experimental setup and baselines
  10. Experimental results
  11. Consensus control in small-scale networks
  12. Consensus control in large-scale networks
  13. Conclusions
  14. Acknowledgement

Key Result

Lemma 1

The nonlinear opinion dynamics eq:coca_model_matrix_form asymptotically converge to a consensus value, which we denote by $\alpha$.

Figures (6)

  • Figure 1: Evolution of $\underline{\theta}(k)$ and $\overline{\theta}(k)$
  • Figure 2: Distributions of $\alpha_{\min}-\alpha$, $\alpha_{\max}-\alpha$ for biased $x(0)$.
  • Figure 3: Histogram of $\alpha_{\max}-\alpha_{\min}$.
  • Figure 4: Network topology
  • Figure 5: Examples of $\beta$-distributions
  • ...and 1 more figures

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • proof
  • Corollary 1
  • proof