## Abstract

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.

Original language | English |
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Pages (from-to) | 353-360 |

Number of pages | 8 |

Journal | International Journal of Applied Mathematics and Computer Science |

Volume | 25 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2015 Jun 1 |

## Keywords

- adjacency weights
- cooperative control
- distributed consensus algorithm
- generalized graph Laplacian
- graph Laplacian

## ASJC Scopus subject areas

- Computer Science (miscellaneous)
- Engineering (miscellaneous)
- Applied Mathematics