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This paper proposes a double-layered framework (or form of network) tointegrate two mechanisms, termed consensus and conservation, achievingdistributed solution of a linear equation. The multi-agent framework consideredin the paper is composed of clusters (which serve as a form of aggregatingagent) and each cluster consists of a sub-network of agents. By achievingconsensus and conservation through agent-agent communications in the samecluster and cluster-cluster communications, distributed algorithms are devisedfor agents to cooperatively achieve a solution to the overall linear equation.These algorithms outperform existing consensus-based algorithms, including butnot limited to the following aspects: first, each agent does not have to knowas much as a complete row or column of the overall equation; second, each agentonly needs to control as few as two scalar states when the number of clustersand the number of agents are sufficiently large; third, the dimensions ofagents' states in the proposed algorithms do not have to be the same (while incontrast, algorithms based on the idea of standard consensus inherently requireall agents' states to be of the same dimension). Both analytical proof andsimulation results are provided to validate exponential convergence of theproposed distributed algorithms in solving linear equations.

A Double-Layered Framework for Distributed Coordination in Solving Linear Equations