Consensus on Nonlinear Spaces *

Abstract

Consensus problems have attracted significant attention in the control community over the last decade. They act as a rich source of new mathematical problems pertaining to the growing field of cooperative and distributed control. This paper is an introduction to consensus problems whose underlying state-space is not a linear space, but instead a highly symmetric nonlinear space such as the circle and other relevant generalizations. A geometric approach is shown to highlight the connection between several fundamental models of consensus, synchronization, and coordination, to raise significant global convergence issues not present in linear models, and to be relevant for a number of engineering applications, including the design of planar or spatial coordinated motions.

Synchronization in complex networks and its application – A survey of recent advances and challenges

“…, do not suppose relative angular coordinate formulations accompanied by a uniform damping, and they do not assume the transfer conductances to be ''sufficiently small''. In addition, based on nonlinear consensus protocols (Saber et al, 2007;Sepulchre, 2012) and synchronization theory, the conditions are presented, which can be interpreted as ''the network connectivity has to dominate the network's nonuniformity and the network's losses'' .…”

“…Synchronization is a widely studied topic in physics, while the consensus problem of multi-agent systems is an important research problem in engineering (Bakule, 2014;Lovisari & Zampieri, 2012;Sepulchre, 2012). Mathematically, the definitions for synchronization and consensus are quite similar (Cao, Yu, Ren, & Chen, 2013;Wielanda et al, 2011).…”

Synchronization of diffusively coupled systems on compact Riemannian manifolds in the presence of drift

“…Lately, significant effort has been dedicated to the study of synchronizing systems that live on manifolds [1][2][3][4][5]. Scardovi et al have studied synchronizing systems on tori [1] and Sarlette et al have focused on synchronizing systems on the orthogonal group [2].…”

“…projective) point of view. A very general review paper regarding the study of synchronization on manifolds was published by Sepulchre [4]. Tron et al [5] studied synchronization from an intrinsic point of view whilst characterizing the region of attraction of the synchronization manifold precisely, therein treating systems that evolve in discrete time.…”

An overview of research in distributed attitude coordination control

“…Some special consensus problems are introduced in [4], whose underlying state-space is not a linear space, but instead a highly symmetric nonlinear space such as the circle and other relevant generalizations. In [5], some fundamental problems are discussed, such as consensus, rendezvous, flocking and formation, and based upon the characteristics of system dynamics and communication topology, the latest research achievements on coordination control of networked Euler-Lagrange systems are presented intensively.…”