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
IS - 4
M1 - 7542498
ER -