Leader-follower coherence in noisy ring-trees networks
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ORIGINAL PAPER
Leader-follower coherence in noisy ring-trees networks Weigang Sun Kai Fan
· Meidu Hong · Suyu Liu ·
Received: 1 February 2020 / Accepted: 9 October 2020 © Springer Nature B.V. 2020
Abstract This paper investigates leader-follower network coherence in a noisy ring-trees network model with preassigned leaders at the initial state. Different from existing works on designing consensus algorithms in the multi-agent systems, the leader-follower coherence characterized by the eigenvalues of a principal submatrix obtained from the Laplacian matrix is a measure of deviation from the state of the leaders in an H2 norm. The recursive properties of ring-trees networks allow analytical calculations of this network coherence. Based on the relationship of the eigenvalues of the submatrix in two successive steps, an analytical expression for the leader-follower coherence is determined depending on the number of leaders and network parameters. This network model shows better consensus with the increasing number of leaders in the ring network and the ring-trees topology has a profound impact on the coherence. Keywords Consensus · Leader-follower coherence · Ring-trees network W. Sun (B) · S. Liu School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China M. Hong Information Engineering School, Hangzhou Dianzi University, Hangzhou 311305, China K. Fan College of Automation, Hangzhou Dianzi University, Hangzhou 310018, China
1 Introduction The consensus problems have widely been studied in the multi-agent systems, where consensus means that each agent reaches agreement on position, velocity and spacing arrangement by local communication [1–4]. However, the agents do not converge to the consensus in the presence of stochastic external disturbances. Patterson and Bamieh proposed a concept of network coherence to capture the variance of the deviation from consensus of the system and showed that it is determined by the Laplacian spectrum in terms of an H2 norm [5]. Different from designing robust consensus algorithms of multi-agent systems, this coherence is helpful to investigate the interplay between network coherence and network attributes. For example, the fractal dimension has a profound impact on the scalings of network coherence in undirected Vicsek fractal graphs [5], while in some directed graphs, they display better speed and robustness than those of undirected graphs [6]. Recently, increasing attention has been addressed to the scalings of network coherence in deterministic networks [7,8] because exact scalings are well determined. It has been shown that there is a prominent difference in the network coherence with diverse network topology. In [9], Qi et al. showed that the structure difference of two self-similar graphs with same numbers of nodes and edges displays a distinct performance of the studied consensus algorithms. To investigate the role of
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weighted factors on the coherence, Dai et al. proposed a family of weighted recursive trees and showed that the weight plays
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