发表机构
School of Automation and Intelligent Sensing, Shanghai Jiao Tong University（上海交通大学生物医学工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究矩阵加权多智能体网络的子空间一致性问题，从代数和拓扑角度推导相关充要、充分及必要条件，包括基于边权重零空间、\(\mathbb{V}\)连通性等，还给出树网络条件，揭示其基本能力并建立分析框架。
AI 中文摘要
本文研究矩阵加权多智能体网络的子空间一致性问题，其中每个智能体在\(\mathbb{R}^{d}\)中有向量值状态，相邻智能体间交互由矩阵值边权重表征。除了所有维度达到全状态一致性，许多实际应用要求智能体仅在某些维度上达成一致，其余维度保持期望相对配置。为此引入子空间一致性概念，若智能体状态差在子空间\(\mathbb{V}\subseteq\mathbb{R}^{d}\)上的投影渐近收敛到零，则称矩阵加权网络在\(\mathbb{V}\)上达到子空间一致性，传统一致性是其\(\mathbb{V}=\mathbb{R}^{d}\)时的特殊情况。从代数和拓扑角度分别推导了子空间一致性的充要条件、充分条件和必要条件，还给出了树网络的精细化充要条件。该工作揭示了矩阵加权网络的基本能力并建立了分析规定子空间上一致性行为的系统框架。
英文摘要
This paper examines the subspace consensus problem for matrix-weighted multi-agent networks. In this setting, each agent holds a high-dimensional state vector in $\mathbb{R}^{d}$, and matrix-valued edge weights characterize inter-agent interactions across different dimensions. Unlike the classical scalar-weighted consensus problem, where all dimensions of agents' states must converge to respective common values, this paper examines scenarios in which agents must agree along certain dimensions while preserving prescribed relative configurations in the remaining ones. To address this need, we introduce the concept of subspace consensus for a matrix-weighted network, which is said to achieve consensus on a subspace $\mathbb{V}\subseteq\mathbb{R}^{d}$ if the projections of all pairwise state differences onto $\mathbb{V}$ asymptotically vanish. This formulation subsumes the conventional consensus problem as the special case $\mathbb{V}=\mathbb{R}^{d}$. We first derive a purely algebraic necessary and sufficient condition for subspace consensus by characterizing the interplay between the null spaces of the matrix-valued Laplacian and those of the individual matrix-valued edge weights. Then, we propose the limiting constraint network as a means to analyze the steady state of matrix-weighted networks. We establish the relationship between the steady state of a matrix-weighted network and the null space of the matrix-valued Laplacian of the associated limiting constraint network. Under mild assumptions, we further show that a matrix-weighted network achieves subspace consensus on $\mathbb{V}$ if and only if the associated limiting constraint network contains a $\mathbb{V}$-spanning tree. We further show that a matrix-weighted network achieves consensus if and only if the associated limiting constraint network contains a positive spanning tree, as a special case with $\mathbb{V}=\mathbb{R}^{d}$.
