title: Lower Bounds on the Proportion of Leaders Needed for Expected Consensus of 3-D Flocks.
abstract: This paper considers the consensus behavior of a spatially distributed 3-D dynamical network composed of heterogeneous agents: leaders and followers, in which the leaders have the preferred information about the destination, while the followers do not have. All followers move in a 3-D Euclidean space with a given speed and with their headings updated according to the average velocity of the corresponding neighbors. Compared with the 2-D model, a key point lies in how to analyze the dynamical behavior of a "linear" nonhomogeneous equation where the nonhomogeneous term strongly nonlinearly depends on the states of all agents. Using the network structure and the estimation of some characteristics for the initial states, we present a proper decaying rate for the nonhomogeneous term and then establish lower bounds on the ratio of the number of leaders to the number of followers that is needed for the expected consensus by considering two cases: 1) fixed speed and neighborhood radius and 2) variable speed and neighborhood radius with respect to the population size. Some simulation examples are given to justify the theoretical results.This paper considers the consensus behavior of a spatially distributed 3-D dynamical network composed of heterogeneous agents: leaders and followers, in which the leaders have the preferred information about the destination, while the followers do not have. All followers move in a 3-D Euclidean space with a given speed and with their headings updated according to the average velocity of the corresponding neighbors. Compared with the 2-D model, a key point lies in how to analyze the dynamical behavior of a "linear" nonhomogeneous equation where the nonhomogeneous term strongly nonlinearly depends on the states of all agents. Using the network structure and the estimation of some characteristics for the initial states, we present a proper decaying rate for the nonhomogeneous term and then establish lower bounds on the ratio of the number of leaders to the number of followers that is needed for the expected consensus by considering two cases: 1) fixed speed and neighborhood radius and 2) variable speed and neighborhood radius with respect to the population size. Some simulation examples are given to justify the theoretical results.
authors.affiliation: College of Applied Sciences, Beijing University of Technology, Beijing, China.
authors.forename: Xuejing
authors.initials: X
authors.lastname: Li
authors.affiliation: Department of Automation, Shanghai Jiao Tong University, Shanghai, China.
authors.forename: Lin
authors.initials: L
authors.lastname: Wang
authors.affiliation: Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China.
authors.forename: Zhixin
authors.initials: Z
authors.lastname: Liu
authors.affiliation: School of Engineering and Information Technology, University of New South Wales, Canberra, ACT, Australia.
authors.forename: Daoyi
authors.initials: D
authors.lastname: Dong
keywords: Birds; Estimation; Manganese; Mathematical model; Nickel; Sociology; Solid modeling