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Distributed optimisation and control of graph Laplacian eigenvalues for robust consensus via an adaptive multilayer strategy
Author(s) -
Kempton L. C.,
Herrmann G.,
di Bernardo M.
Publication year - 2017
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.3808
Subject(s) - algebraic connectivity , laplacian matrix , spectral radius , eigenvalues and eigenvectors , connectivity , consensus , mathematical optimization , strongly connected component , robustness (evolution) , spectral graph theory , mathematics , nonlinear system , computer science , algebraic graph theory , algebraic number , graph , theoretical computer science , discrete mathematics , multi agent system , line graph , mathematical analysis , biochemistry , physics , chemistry , quantum mechanics , artificial intelligence , graph power , gene
Summary Functions of eigenvalues of the graph Laplacian matrix L , especially the extremal non‐trivial eigenvalues, the algebraic connectivity λ 2 and the spectral radius λ n , have been shown to be important in determining the performance in a host of consensus and synchronisation applications. In this paper, we focus on formulating an entirely distributed control law for the control of edge weights in an undirected graph to solve a constrained optimisation problem involving these extremal eigenvalues. As an objective for the distributed control law, edge weights must be found that minimise the spectral radius of the graph Laplacian, thereby maximising the robustness of the network to time delays under a simple linear consensus protocol. To constrain the problem, we use both local weight constraints that weights must be non‐negative, and a global connectivity constraint, maintaining a designated minimum algebraic connectivity. This ensures that the network remains sufficiently well connected. The distributed control law is formulated as a multilayer strategy, using three layers of successive distributed estimation. Adequate timescale separation between the layers is of paramount importance for the proper functioning of the system, and we derive conditions under which the distributed system converges as we would expect for the centralised control or optimisation system to converge. © 2017 The Authors International Journal of Robust and Nonlinear Control published by John Wiley & Sons Ltd.