Research Article Predefined-Time Consensus of Nonlinear First-Order Systems Using a Time Base Generator J. Armando Colunga , 1 Carlos R. Vázquez , 2 Héctor M. Becerra , 1 and David Gómez-Gutiérrez 2,3 1 Centro de Investigaci´ on en Matem´ aticas (CIMAT), A.C., Computer Science Department, GTO, Mexico 2 Tecnologico de Monterrey, Escuela de Ingenier´ ıa y Ciencias, Zapopan, Mexico 3 Intel Tecnolog´ ıa de M´ exico, Multi-Agent Autonomous Systems Lab, Intel Labs, Zapopan, Mexico Correspondence should be addressed to H´ ector M. Becerra; hector.becerra@cimat.mx Received 16 June 2018; Revised 11 September 2018; Accepted 17 September 2018; Published 9 October 2018 Academic Editor: Ju H. Park Copyright © 2018 J. Armando Colunga et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Tis paper proposes a couple of consensus algorithms for multiagent systems in which agents have frst-order nonlinear dynamics, reaching the consensus state at a predefned time independently of the initial conditions. Te proposed consensus protocols are based on the so-called time base generators (TBGs), which are time-dependent functions used to build time-varying control laws. Predefned-time convergence to the consensus is proved for connected undirected communication topologies and directed topologies having a spanning tree. Furthermore, one of the proposed protocols is based on the super-twisting controller, providing robustness against disturbances while maintaining the predefned-time convergence property. Te performance of the proposed methods is illustrated in simulations, and it is compared with fnite-time, fxed-time, and predefned-time consensus protocols. It is shown that the proposed TBG protocols represent an advantage not only in the possibility to defne a settling time but also in providing smoother and smaller control actions than existing fnite-time, fxed-time, and predefned-time consensus. 1. Introduction Recent years have seen an increasing interest in algorithms allowing a group of systems to reach a common value for its internal state through local interaction. Tis problem has been addressed, from diferent viewpoints, in the consensus of multiagent systems (MASs) [1, 2] and in the synchroniza- tion of complex dynamical networks [3–5]. On the one hand, consensus protocols have been applied to focking [6], forma- tion control [7, 8], and distributed resource allocation [9, 10]. On the other hand, the results on synchronization of complex dynamical networks have been applied to neuroscience [3], power-grids [3], and the chaotic synchronization for secure communication in swarms [11]. Several works have been published proposing consensus and synchronization algorithms for diferent types of systems, considering static [12–14] and dynamic networks [4, 5]. Regarding frst-order agents, the standard protocol (the input of an agent is a linear combination of the errors between the agent’s state and those of its neighbors) achieves consensus if the graph topology is strongly connected [15, 16]. Tis algorithm achieves consensus also for dynamic topologies, switching among connected graphs [15, 17]. For directed graphs topologies, a common requirement is that the graph contains a spanning-tree (e.g., [18]). All these algorithms are based on linear protocols and achieve consensus in an asymptotic fashion, where convergence rate is a function of the algebraic connectivity of the graph. With the aim of developing consensus protocols satisfy- ing real-time constraints, fnite-time consensus has received a great deal of attention. Te main focus has been in defning nonlinear protocols evaluated either on each of the neighbors’ errors or on the sum of the neighbors’ errors. For fnite- time convergence, binary protocols based on the sign(∙) function have been proposed to achieve consensus to the average value [19, 20], the average-min-max value [21], the Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 1957070, 11 pages https://doi.org/10.1155/2018/1957070