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 -