Brian. D. O. Anderson

Jingqiu Zhou, Wang Xuan, Shaoshuai Mou et al.

This paper proposes distributed algorithms for multi-agent networks to achieve a solution in finite time to a linear equation $Ax=b$ where $A$ has full row rank, and with the minimum $l_1$-norm in the underdetermined case (where $A$ has more columns than rows). The underlying network is assumed to be undirected and fixed, and an analytical proof is provided for the proposed algorithm to drive all agents' individual states to converge to a common value, viz a solution of $Ax=b$, which is the minimum $l_1$-norm solution in the underdetermined case. Numerical simulations are also provided as validation of the proposed algorithms.

Xuan Wang, Shaoshuai Mou, Brian. D. O. Anderson

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.