A Distributed Algorithm for Solving A Time-Varying Linear Equation

This paper studies the problem of cooperatively solving a time-varying linear equation of the form $\mathrm{A}(t)x(t)= b(t)$, which always has a unique solution. Each agent has access to only some rows of the time-varying augmented matrix $[\mathrm{A}(t)\ b(t)]$. We propose a distributed algorithm for solving the time-varying linear equation. The proposed distributed algorithm enforces local solutions to track local time-varying manifolds corresponding to local linear sub-equations while simultaneously reaching a consensus. This enables all local solutions to converge to the solution of the original time-varying linear equation. Finally, the effectiveness of the proposed algorithm is demonstrated through its application in a cooperative monitoring task.