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Sufficient Condition on Bipartite Consensus of Weakly Connected Matrix-weighted Networks

Chongzhi Wang, Haibin Shao, Ying Tan, Dewei Li

Abstract

Recent advances in bipartite consensus on matrix-weighted networks, where agents are divided into two disjoint sets with those in the same set agreeing on a certain value and those in different sets converging to opposite values, have highlighted its potential applications across various fields. Traditional approaches often depend on the existence of a positive-negative spanning tree in matrix-weighted networks to achieve bipartite consensus, which greatly restricts the use of these approaches in engineering applications. This study relaxes that assumption by allowing weak connectivity within the network, where paths can be weighted by semidefinite matrices. By analyzing the algebraic constraints imposed by positive-negative trees and semidefinite paths, we derive new sufficient conditions for achieving bipartite consensus. Our findings are validated by numerical simulations.

Sufficient Condition on Bipartite Consensus of Weakly Connected Matrix-weighted Networks

Abstract

Recent advances in bipartite consensus on matrix-weighted networks, where agents are divided into two disjoint sets with those in the same set agreeing on a certain value and those in different sets converging to opposite values, have highlighted its potential applications across various fields. Traditional approaches often depend on the existence of a positive-negative spanning tree in matrix-weighted networks to achieve bipartite consensus, which greatly restricts the use of these approaches in engineering applications. This study relaxes that assumption by allowing weak connectivity within the network, where paths can be weighted by semidefinite matrices. By analyzing the algebraic constraints imposed by positive-negative trees and semidefinite paths, we derive new sufficient conditions for achieving bipartite consensus. Our findings are validated by numerical simulations.

Paper Structure

This paper contains 11 sections, 12 theorems, 36 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Notations
  4. Graph Theory
  5. Problem Statement
  6. Network Properties
  7. Main Results
  8. Continents
  9. Continents Connected with Semidefinite Paths
  10. Simulation Example
  11. Concluding Remarks

Key Result

Lemma 2.6

For matrix-weighted network $\mathcal{G}$ in eq:protocol whose Laplacian has ${\bf null}(L)={\bf span}\{\eta_{1},\eta_{2},...,\eta_{m}\}$, where $\eta_{1},\eta_{2},...,\eta_{m}$ are the orthonormal eigenvectors, $\underset{t\rightarrow\infty}{\lim}x(t)$ exists and $\underset{t\rightarrow\infty}{\lim

Figures (5)

  • Figure 1: Network $\mathcal{G}_{1}$ and $\mathcal{G}_{3}$. Red solid (resp., dashed) lines represent edges weighted by positive definite (resp., semidefinite) matrices; blue solid (resp., dashed) lines represent edges weighted by negative definite (resp., semidefinite) matrices. Network $\mathcal{G}_{3}$ modifies $\mathcal{G}_{1}$ by letting $\mathcal{W}((7,8))=A_{4}$, so that only Condition (3) does not hold.
  • Figure 2: Network $\mathcal{G}_{2}$. Network $\mathcal{G}_{2}$ connects the two continents with a single semidefinite path whose weight matrices do not satisfy eqn. \ref{['eq:bipartite_continent_suff']}, thereby violating Condition (1).
  • Figure 3: Network $\mathcal{G}_{4}$. Network $\mathcal{G}_{4}$ modifies $\mathcal{G}_{1}$ by letting $(1,7)\in\mathcal{E}^{nb}$, while the bases for the null spaces of $\mathcal{W}((7,8)),\mathcal{W}((8,4)),\mathcal{W}((4,6))$ are linearly dependent, so that only Condition (4) does not hold.
  • Figure 4: Metric $e_{b}(t)$ on $\mathcal{G}_{1}$, $\mathcal{G}_{2}$, $\mathcal{G}_{3}$, and $\mathcal{G}_{4}$ (left to right). Bipartite consensus is achieved on $\mathcal{G}_{1}$ but not on $\mathcal{G}_{2},\mathcal{G}_{3}$, and $\mathcal{G}_{4}$.
  • Figure 5: State trajectories of the nine agents of $\mathcal{G}_{1}$. Agents of $\mathcal{V}_{1}=\{1,3,6,7,8,9\}$ and $\mathcal{V}_{2}=\{2,4,5\}$ are marked with solid lines and dashed lines respectively.

Theorems & Definitions (25)

  • Definition 2.1: Weakly Connected
  • Definition 2.2: Structural Balance
  • Definition 2.3: Nontrivial Balancing Set
  • Remark 2.4
  • Definition 2.5: Bipartite Consensus
  • Lemma 2.6
  • Proposition 2.7
  • Remark 2.8
  • Proposition 2.9
  • Proposition 2.10
  • ...and 15 more