  This paper proposes a double-layered framework (or form of network) to
integrate two mechanisms, termed consensus and conservation, achieving
distributed solution of a linear equation. The multi-agent framework considered
in the paper is composed of clusters (which serve as a form of aggregating
agent) and each cluster consists of a sub-network of agents. By achieving
consensus and conservation through agent-agent communications in the same
cluster and cluster-cluster communications, distributed algorithms are devised
for agents to cooperatively achieve a solution to the overall linear equation.
These algorithms outperform existing consensus-based algorithms, including but
not limited to the following aspects: first, each agent does not have to know
as much as a complete row or column of the overall equation; second, each agent
only needs to control as few as two scalar states when the number of clusters
and the number of agents are sufficiently large; third, the dimensions of
agents' states in the proposed algorithms do not have to be the same (while in
contrast, algorithms based on the idea of standard consensus inherently require
all agents' states to be of the same dimension). Both analytical proof and
simulation results are provided to validate exponential convergence of the
proposed distributed algorithms in solving linear equations.
