A Distributed Algorithm for Solving Linear Equations in Clustered Multi-Agent Systems

A new approach to solving the linear algebraic equation
$Ax=b$
is presented by introducing a leaderless clustered multi-agent system. Each agent is associated with a certain submatrix of
$A$
and a vector obtained from
$b$
by a certain decomposition process. A distributed algorithm has each agent processing information solely with its own information about
$A$
and
$b$
, but sharing a time-varying estimate of part of the solution of
$Ax=b$
with its neighbors, as defined by a graphical structure overlaying the agents. This graphical structure divides the agents into clusters, with agents in one cluster being associated with one block column of
$A$
, and different rows of that block column; with each intra-cluster graph being connected. The update law uses consensus within individual clusters, utilizing their estimated states together with a process of inter-cluster conservation to ensure that the concatenated sub-solutions reached by the clusters solve the overall system of linear equations. Unlike previous literature that uses clustered multi-agent systems or double-layered networks, this framework does not require the presence of an aggregator or central node in the network’s clusters. The algorithm demonstrates exponential convergence, evidenced by both theoretical proof and numerical simulations.