Subspace Consensus

Abstract

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 $\mathbb{V}={\mathbb{R}}^d$.