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Consensus-Based Stability Analysis of Multi-Agent Networks

Ingyu Jang, Ethan J. LoCicero, Leila Bridgeman

TL;DR

This work addresses stability verification for large-scale multi-agent networks under sparse interconnections while preserving agent privacy. It introduces two distributed algorithms based on Vidyasagar's Network Dissipativity Theorem and ADMM: a privacy-preserving method that avoids sharing dynamics, and a chordal-decomposition variant that reduces computational burden by breaking large LMIs into smaller ones. Key contributions include (1) a distributed ADMM framework for QSR-dissipativity–based stability without disclosing dynamics, (2) a chordal decomposition technique that scales more favorably with network size, and (3) an extension to nonlinear/uncertain agents. A 2D UAV swarm example demonstrates convergence and reveals significant speedups for larger networks, underscoring the practical potential for privacy-preserving, distributed stability analysis and paving the way for distributed sparse controller synthesis.

Abstract

The emergence of large-scale multi-agent systems has led to controller synthesis methods for sparse communication between agents. However, most sparse controller synthesis algorithms remain centralized, requiring information exchange and high computational costs. This underscores the need for distributed algorithms that design controllers using only local dynamics information from each agent. This paper presents a consensus-based distributed stability analysis. The proposed stability analysis algorithms leverage Vidyasagar's Network Dissipativity Theorem and the alternating direction methods of multipliers to perform general stability analysis. Numerical examples involving a 2D swarm of unmanned aerial vehicles demonstrate the convergence of the proposed algorithms.

Consensus-Based Stability Analysis of Multi-Agent Networks

TL;DR

This work addresses stability verification for large-scale multi-agent networks under sparse interconnections while preserving agent privacy. It introduces two distributed algorithms based on Vidyasagar's Network Dissipativity Theorem and ADMM: a privacy-preserving method that avoids sharing dynamics, and a chordal-decomposition variant that reduces computational burden by breaking large LMIs into smaller ones. Key contributions include (1) a distributed ADMM framework for QSR-dissipativity–based stability without disclosing dynamics, (2) a chordal decomposition technique that scales more favorably with network size, and (3) an extension to nonlinear/uncertain agents. A 2D UAV swarm example demonstrates convergence and reveals significant speedups for larger networks, underscoring the practical potential for privacy-preserving, distributed stability analysis and paving the way for distributed sparse controller synthesis.

Abstract

The emergence of large-scale multi-agent systems has led to controller synthesis methods for sparse communication between agents. However, most sparse controller synthesis algorithms remain centralized, requiring information exchange and high computational costs. This underscores the need for distributed algorithms that design controllers using only local dynamics information from each agent. This paper presents a consensus-based distributed stability analysis. The proposed stability analysis algorithms leverage Vidyasagar's Network Dissipativity Theorem and the alternating direction methods of multipliers to perform general stability analysis. Numerical examples involving a 2D swarm of unmanned aerial vehicles demonstrate the convergence of the proposed algorithms.

Paper Structure

This paper contains 17 sections, 4 theorems, 23 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Notation
  4. Graph Structure and Chordal Decomposition
  5. ADMM
  6. QSR-Dissipativity of Large-Scale Multi-Agent Systems
  7. PROBLEM STATEMENT
  8. MAIN RESULTS
  9. Information-Secure Network Stability Analysis
  10. Chordal Decomposition of VNDT
  11. Distributed Network Stability Analysis
  12. Primal X, Clones Z, W Update
  13. Clone J Update
  14. Dual Update
  15. Extension to Nonlinear Systems
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\mathcal{Z}$ be a chordal graph with maximal cliques $\{C_i\}_{i=1}^M$. Then $\mathbf{Z}\in\mathbb{S}_{-}^N(\mathcal{E}(\mathcal{Z}),0)$ (block-wise) if and only if there exist $\mathbf{Z}_p\in\mathbb{S}_-^{|C_p|}$ (block-wise) for $p\in\mathbb{N}_M$ such that where $\mathbf{E}_{C_p}\in\mathbb{R}^{|C_p|\times N}$ (block-wise) is defined as $(\mathbf{E}_{C_p})_{ij}=\mathbf{I}$ if $C_p(i)=j$ a

Figures (4)

  • Figure 1: Example of chordal decomposition of graph $\overline{\mathcal{Q}}$: $\overline{\mathcal{Q}}_o$ represents the overlapped graph from \ref{['thm:Chordal Decomposition']}.
  • Figure 2: Mapping among all block diagonal components of variables in \ref{['tab:ADMM Variables']}: The gray rectangles represent the block component indices defining each variable. The QSR parameters are grouped by agent, with blocks for each dissipativity matrix, while the chordal parameters are grouped by clique with blocks for each and a matrix equation
  • Figure 3: Graph network of .
  • Figure 4: Computation time of \ref{['alg:01', 'alg:02']}.

Theorems & Definitions (7)

  • Theorem 1: Chordal Block-Decomposition zheng2021chordal
  • Definition 1: QSR-Dissipativity vidyasagar1981input
  • Lemma 2: gupta1996robust
  • Definition 2: $\mathcal{L}_2$-stability vidyasagar1981input
  • Theorem 3: vidyasagar1981input
  • Theorem 4
  • proof