Distributed optimisation and control of graph Laplacian eigenvalues for robust consensus via an adaptive multi-layer strategy

Louis Kempton, Guido Herrmann, Mario Di Bernardo

Research output: Contribution to journalArticle (Academic Journal)

5 Citations (Scopus)
366 Downloads (Pure)

Abstract

Functions of eigenvalues of the graph Laplacian matrix L, especially the extremal non-trivial eigenvalues, the algebraic connectivity λ2 and the spectral radius λn, 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 in the simple linear consensus protocol [1]. 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 multi-layer strategy, using three layers of successive distributed estimation. Adequate time-scale 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.
Original languageEnglish
Pages (from-to)1499-1525
Number of pages27
JournalInternational Journal of Robust and Nonlinear Control
Volume27
Issue number9
Early online date28 Mar 2017
DOIs
Publication statusPublished - 1 Jun 2017

Bibliographical note

Special Issue: Consensus‐based Applications in Networked Systems

Keywords

  • Distributed control and optimisation
  • Robust consensus
  • Multi-layer networks
  • Singular perturbation theory
  • Graph Laplacian eigenvalues

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