Control- Theoretic Distributed Algorithm for Solving Linear Equations: An Observer Perspective

This paper addresses the distributed computation of the linear algebraic equations (LAEs) from a control perspective. The fundamental methodology involves interpreting the unknown variables of the LAE as the states of a related dynamic system and harnessing the full potential of distributed state observers, within the context of multi-agent systems. The proposed distributed algorithm leverages the combination of observer-based design and consensus-based design principles. It is shown that regardless of whether the LAE of interest has a unique solution or multiple solutions, the presented distributed algorithm can converge exponentially fast to the solution, independent of the choice of initial values. This control-oriented approach demonstrates the idea of control design by effectively leveraging control tools, specifically the distributed state observer, to address the mathematical problem of equation-solving. Finally, simulation examples are presented to demonstrate the efficacy of the proposed methods.