In order to describe the interconnection among agents with multi-dimensional states, we generalize the notion of a graph Laplacian by extending the adjacency weights (or weighted interconnection coefficients) from scalars to matrices. More precisely, we use positive definite matrices to denote full multi-dimensional interconnections, while using nonnegative definite matrices to denote partial multi-dimensional interconnections. We prove that the generalized graph Laplacian inherits the spectral properties of the graph Laplacian. As an application, we use the generalized graph Laplacian to establish a distributed consensus algorithm for agents described by multi-dimensional integrators.
@article{bwmeta1.element.bwnjournal-article-amcv25i2p353bwm,
author = {Guisheng Zhai},
title = {A generalization of the graph Laplacian with application to a distributed consensus algorithm},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {25},
year = {2015},
pages = {353-360},
zbl = {1322.93008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv25i2p353bwm}
}
Guisheng Zhai. A generalization of the graph Laplacian with application to a distributed consensus algorithm. International Journal of Applied Mathematics and Computer Science, Tome 25 (2015) pp. 353-360. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv25i2p353bwm/
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