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A Novel Edge Laplacian-based Approach for Adaptive Formation Control of Uncertain Multi-agent Systems with Unified Relative Error Performance

Kun Li, Kai Zhao, Yongduan Song, Lihua Xie

TL;DR

The paper tackles enforcing prescribed performance on relative inter-agent errors $\tilde{e}$ for uncertain high-order MASs under directed graphs. It introduces a unified performance function $\mathcal{P}$, nonlinear mappings, and an edge-Laplacian-based adaptive backstepping controller that transforms the relative-error constraints into a strict-feedback-like system with a formal stability proof. Key contributions include plug-and-play tuning to realize different performance forms without redesign, elimination of initial-constraint verification under global performance, and a fully distributed design applicable to high-order nonlinear MASs on general graph topologies. Simulations across directed spanning-tree, directed-cycle, and undirected graphs validate convergence of edge errors within prescribed bounds and illustrate robustness and practicality of the approach.

Abstract

For most existing prescribed performance formation control methods, performance requirements are not directly imposed on the relative states between agents but on the consensus error, which lacks a clear physical interpretation of their solution. In this paper, we propose a novel adaptive prescribed performance formation control strategy, capable of guaranteeing prescribed performance on the relative errors, for uncertain high-order multi-agent systems under a class of directed graphs. Due to the consideration of performance constraints for relative errors, a coupled nonlinear interaction term that contains global graphic information among agents is involved in the error dynamics, leading to a fully distributed control design more difficult and challenging. Here by proposing a series of nonlinear mappings and utilizing the edge Laplacian along with Lyapunov stability theory, the presented formation control scheme exhibits the following appealing features when compared to existing results: 1) different performance requirements can be guaranteed in a unified way by solely tuning the design parameters a priori, without the need for control redesign and stability reanalysis under the proposed fixed control protocol, making the design more user-friendly and the implementation less demanding; 2) the complex and burdensome verification process for the initial constraint, often encountered in existing prescribed performance controls, is completely obviated if the performance requirements are global; and 3) nonlinear interaction is completely decoupled and the asymptotic stability of the formation manifold is ensured via using the adaptive parameter estimate technique. Finally, simulations of various performance behaviors are performed to show the efficiency of the theoretical results.

A Novel Edge Laplacian-based Approach for Adaptive Formation Control of Uncertain Multi-agent Systems with Unified Relative Error Performance

TL;DR

The paper tackles enforcing prescribed performance on relative inter-agent errors for uncertain high-order MASs under directed graphs. It introduces a unified performance function , nonlinear mappings, and an edge-Laplacian-based adaptive backstepping controller that transforms the relative-error constraints into a strict-feedback-like system with a formal stability proof. Key contributions include plug-and-play tuning to realize different performance forms without redesign, elimination of initial-constraint verification under global performance, and a fully distributed design applicable to high-order nonlinear MASs on general graph topologies. Simulations across directed spanning-tree, directed-cycle, and undirected graphs validate convergence of edge errors within prescribed bounds and illustrate robustness and practicality of the approach.

Abstract

For most existing prescribed performance formation control methods, performance requirements are not directly imposed on the relative states between agents but on the consensus error, which lacks a clear physical interpretation of their solution. In this paper, we propose a novel adaptive prescribed performance formation control strategy, capable of guaranteeing prescribed performance on the relative errors, for uncertain high-order multi-agent systems under a class of directed graphs. Due to the consideration of performance constraints for relative errors, a coupled nonlinear interaction term that contains global graphic information among agents is involved in the error dynamics, leading to a fully distributed control design more difficult and challenging. Here by proposing a series of nonlinear mappings and utilizing the edge Laplacian along with Lyapunov stability theory, the presented formation control scheme exhibits the following appealing features when compared to existing results: 1) different performance requirements can be guaranteed in a unified way by solely tuning the design parameters a priori, without the need for control redesign and stability reanalysis under the proposed fixed control protocol, making the design more user-friendly and the implementation less demanding; 2) the complex and burdensome verification process for the initial constraint, often encountered in existing prescribed performance controls, is completely obviated if the performance requirements are global; and 3) nonlinear interaction is completely decoupled and the asymptotic stability of the formation manifold is ensured via using the adaptive parameter estimate technique. Finally, simulations of various performance behaviors are performed to show the efficiency of the theoretical results.
Paper Structure (14 sections, 3 theorems, 44 equations, 3 figures)

This paper contains 14 sections, 3 theorems, 44 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Graph & Edge Laplacian
  4. Problem Statement
  5. System and Performance Transformation
  6. Unified Performance Function
  7. Transformed Error Dynamic Model
  8. Controller Design and Stability Analysis
  9. Controller Design
  10. Stability Analysis
  11. SIMULATION RESULTS
  12. Asymmetric Performance Behavior
  13. Global Performance Behavior
  14. Conclusion

Key Result

Lemma 1

restrepo2021edge For a digraph that is a spanning tree, it holds that $L^s_e=\frac{1}{2}\left(E^\top E_{\odot}+E^\top_{\odot}E\right)$ is positive definite; For a digraph that is a directed cycle, it holds that $E^\top_tE_t$ is positive definite.

Figures (3)

  • Figure 1: The sensing graph for a group of 5 mobile robots: (a) directed spanning tree; (b) directed cycle; and (c) undirected graph.
  • Figure 2: Asymmetric performance behavior. (a) directed spanning tree. (b) directed cycle, and (c) undirected graph cases.
  • Figure 2: Global performance behavior. (a) directed spanning tree. (b) directed cycle, and (c) undirected graph cases.

Theorems & Definitions (8)

  • Lemma 1
  • Definition 1
  • Lemma 2
  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4