TY - JOUR

T1 - The Smallest Eigenvalue of the Generalized Laplacian Matrix, with Application to Network-Decentralized Estimation for Homogeneous Systems

AU - Giordano, Giulia

AU - Blanchini, Franco

AU - Franco, Elisa

AU - Mardanlou, Vahid

AU - Montessoro, Pier Luca

PY - 2016/10/1

Y1 - 2016/10/1

N2 - 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.

AB - 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.

KW - generalized Laplacian matrix

KW - Graph theory

KW - network problems

KW - network-decentralized estimation

KW - network-decentralized observer

UR - http://www.scopus.com/inward/record.url?scp=85012869448&partnerID=8YFLogxK

U2 - 10.1109/TNSE.2016.2600026

DO - 10.1109/TNSE.2016.2600026

M3 - Article

AN - SCOPUS:85012869448

VL - 3

SP - 312

EP - 324

JO - IEEE Transactions on Network Science and Engineering

JF - IEEE Transactions on Network Science and Engineering

SN - 2327-4697

IS - 4

M1 - 7542498

ER -