TY - JOUR

T1 - Decentralized gradient algorithm for solution of a linear equation

AU - Anderson, Brian D.O.

AU - Mou, Shaoshuai

AU - Morse, A. Stephen

AU - Helmke, Uwe

N1 - Publisher Copyright:
© 2016 American Institute of Mathematical Sciences. All rights reserved.

PY - 2016/9

Y1 - 2016/9

N2 - The paper develops a technique for solving a linear equation Ax = b with a square and nonsingular matrix A, using a decentralized gradient algorithm. In the language of control theory, there are n agents, each storing at time t an n-vector, call it xi(t), and a graphical structure associating with each agent a vertex of a fixed, undirected and connected but otherwise arbitrary graph G with vertex set and edge set V and E respectively. We provide differential equation update laws for the xi with the property that each xi converges to the solution of the linear equation exponentially fast. The equation for xi includes additive terms weighting those xj for which vertices in G corresponding to the i-th and j-th agents are adjacent. The results are extended to the case where A is not square but has full row rank, and bounds are given on the convergence rate.

AB - The paper develops a technique for solving a linear equation Ax = b with a square and nonsingular matrix A, using a decentralized gradient algorithm. In the language of control theory, there are n agents, each storing at time t an n-vector, call it xi(t), and a graphical structure associating with each agent a vertex of a fixed, undirected and connected but otherwise arbitrary graph G with vertex set and edge set V and E respectively. We provide differential equation update laws for the xi with the property that each xi converges to the solution of the linear equation exponentially fast. The equation for xi includes additive terms weighting those xj for which vertices in G corresponding to the i-th and j-th agents are adjacent. The results are extended to the case where A is not square but has full row rank, and bounds are given on the convergence rate.

KW - Autonomous systems

KW - Distributed algorithms

KW - Linear equations

UR - http://www.scopus.com/inward/record.url?scp=84989352556&partnerID=8YFLogxK

U2 - 10.3934/naco.2016014

DO - 10.3934/naco.2016014

M3 - Article

SN - 2155-3289

VL - 6

SP - 319

EP - 328

JO - Numerical Algebra, Control and Optimization

JF - Numerical Algebra, Control and Optimization

IS - 3

ER -