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The Smallest Eigenvalue of the Generalized Laplacian Matrix, with Application to Network-Decentralized Estimation for Homogeneous Systems

Giordano, Giulia LU ; Blanchini, F.; Franco, Elisa; Mardanlou, V. and Montessoro, P. L. (2016) In IEEE Transactions on Network Science and Engineering 3(4). p.312-324
Abstract
The problem of synthesizing network-decentralized observers arises when several agents, corresponding to the nodes of a network, exchange information about local measurements to asymptotically estimate their own state. The network topology is unknown to the nodes, which can rely on information about their neighboring nodes only. For homogeneous systems, composed of identical agents, we show that a network-decentralized observer can be designed by starting from local observers (typically, optimal filters) and then adapting the gain to ensure overall stability. The smallest eigenvalue of the so-called generalized Laplacian matrix is crucial: stability is guaranteed if the gain is greater than the inverse of this eigenvalue, which is strictly... (More)
The problem of synthesizing network-decentralized observers arises when several agents, corresponding to the nodes of a network, exchange information about local measurements to asymptotically estimate their own state. The network topology is unknown to the nodes, which can rely on information about their neighboring nodes only. For homogeneous systems, composed of identical agents, we show that a network-decentralized observer can be designed by starting from local observers (typically, optimal filters) and then adapting the gain to ensure overall stability. The smallest eigenvalue of the so-called generalized Laplacian matrix is crucial: stability is guaranteed if the gain is greater than the inverse of this eigenvalue, which is strictly positive if the graph is externally connected. To deal with uncertain topologies, we characterize the worst-case smallest eigenvalue of the generalized Laplacian matrix for externally connected graphs, and we prove that the worst-case graph is a chain. This general result provides a bound for the observer gain that ensures robustness of the network-decentralized observer even under arbitrary, possibly switching, configurations, and in the presence of noise. (Less)
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author
organization
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Contribution to journal
publication status
published
subject
keywords
Eigenvalues and eigenfunctions, Laplace equations, Network topology, Observers, Robustness, Topology, Graph Theory, Network problems, generalized Laplacian matrix, network-decentralized estimation, network-decentralized observer
in
IEEE Transactions on Network Science and Engineering
volume
3
issue
4
pages
312 - 324
publisher
IEEE--Institute of Electrical and Electronics Engineers Inc.
external identifiers
  • scopus:85012869448
ISSN
2327-4697
DOI
10.1109/TNSE.2016.2600026
language
English
LU publication?
yes
id
e7247903-7020-43d5-a06e-dd1872637709
date added to LUP
2016-11-26 14:53:24
date last changed
2017-10-29 04:54:35
@article{e7247903-7020-43d5-a06e-dd1872637709,
  abstract     = {The problem of synthesizing network-decentralized observers arises when several agents, corresponding to the nodes of a network, exchange information about local measurements to asymptotically estimate their own state. The network topology is unknown to the nodes, which can rely on information about their neighboring nodes only. For homogeneous systems, composed of identical agents, we show that a network-decentralized observer can be designed by starting from local observers (typically, optimal filters) and then adapting the gain to ensure overall stability. The smallest eigenvalue of the so-called generalized Laplacian matrix is crucial: stability is guaranteed if the gain is greater than the inverse of this eigenvalue, which is strictly positive if the graph is externally connected. To deal with uncertain topologies, we characterize the worst-case smallest eigenvalue of the generalized Laplacian matrix for externally connected graphs, and we prove that the worst-case graph is a chain. This general result provides a bound for the observer gain that ensures robustness of the network-decentralized observer even under arbitrary, possibly switching, configurations, and in the presence of noise.},
  author       = {Giordano, Giulia and Blanchini, F. and Franco, Elisa and Mardanlou, V. and Montessoro, P. L.},
  issn         = {2327-4697},
  keyword      = {Eigenvalues and eigenfunctions,Laplace equations,Network topology,Observers,Robustness,Topology,Graph Theory,Network problems,generalized Laplacian matrix,network-decentralized estimation,network-decentralized observer},
  language     = {eng},
  number       = {4},
  pages        = {312--324},
  publisher    = {IEEE--Institute of Electrical and Electronics Engineers Inc.},
  series       = {IEEE Transactions on Network Science and Engineering},
  title        = {The Smallest Eigenvalue of the Generalized Laplacian Matrix, with Application to Network-Decentralized Estimation for Homogeneous Systems},
  url          = {http://dx.doi.org/10.1109/TNSE.2016.2600026},
  volume       = {3},
  year         = {2016},
}