Automatica 48 (2012) 2262–2270 Contents lists available at SciVerse ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief paper Dynamic average consensus via nonlinear protocols ✩ Shahram Nosrati 1 , Masoud Shafiee, Mohammad Bagher Menhaj Electrical Engineering Department, Amirkabir University of Technology, 424 Hafez Ave, Tehran, Iran article info Article history: Received 4 October 2010 Received in revised form 7 March 2012 Accepted 29 March 2012 Available online 4 July 2012 Keywords: Consensus algorithms Multi-agent systems Cooperative control abstract This paper addresses the dynamic average consensus problem under nonlinear protocols for networks of dynamic agents. In this problem, each agent aims to track the average of time-varying reference inputs of all the agents in the network by local communication with neighbors. We propose a class of continuous-time nonlinear protocols for this problem, and theoretical analyses for two cases are provided: (1) undirected networks with switching topologies and (2) balanced directed networks with switching topologies. Based upon the analysis results, design procedures for nonlinear protocols are presented. Simulations are also provided to demonstrate the effectiveness of the proposed design procedures. © 2012 Elsevier Ltd. All rights reserved. 1. Introduction In many applications involving multi-agent systems, groups of agents are required to reach a consensus on the values of certain quantities of interest. These quantities are referred to as coordination variables (Kingston, Ren, & Beard, 2005; McLain & Beard, 2005). In the dynamic average consensus problem, the coordination variable is the average of individually measured time-varying signals. The dynamic average consensus problem plays a crucial role in a very broad spectrum of applications, including distributed sensor fusion (Olfati-Saber & Murray, 2004; Olfati-Saber & Shamma, 2005; Spanos, Olfati-Saber, & Murray, 2005), distributed formation control (Freeman, Yang, & Lynch, 2006; Sepulchre, Paley, & Leonard, 2007; Yang, Freeman, & Lynch, 2008), distributed synchronization (Scardovi & Sepulchre, 2006), decentralized environmental modeling (Lynch, Schwartz, Yang, & Freeman, 2008), and distributed constrained optimization (Nedic, Ozdaglar, & Parrilo, 2010), to cite but a few examples. These applications require that all agents agree on the average of time- varying signals. Furthermore, in some applications involving the consensus problem, for example, where a finite-time consensus is required (Shang, 2009; Wang & Xiao, 2007) or just a nonlinear function of the agent’s state of interest is observable (Jiang & Wang, 2009), ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Hideaki Ishii under the direction of Editor Ian R. Petersen. E-mail addresses: sh_nosrati@aut.ac.ir (S. Nosrati), mshafiee@aut.ac.ir (M. Shafiee), tmenhaj@ieee.org (M.B. Menhaj). 1 Tel.: +98 912 3980190; fax: +98 251 3342428. one should consider the nonlinear protocols. Although nonlinear protocols for the static consensus problem (i.e. the consensus on a static value) have better performance and robustness than linear ones (Qu, Wang, & Chunyu, 2007), a rigorous demonstration of the merits of nonlinear consensus protocols versus linear ones requires more research. Moreover, in many practical applications involving multi-agent systems, the information flow between the individual agents may be unidirectional. Therefore, in this work, we investigate both directed and undirected networks. Literature review. The dynamic average consensus problem under linear protocols is studied in Freeman et al. (2006), Nosrati, Shafiee, and Menhaj (2009), Olfati-Saber and Shamma (2005), Ren (2007), Spanos et al. (2005) and Zhu and Martínez (2010). Although many nonlinear protocols for the static consensus problem have been developed over the last several years (see, e.g., Chen, Chen, Xiang, Liu, & Yuan, 2009; Hui & Haddad, 2008; Hui, Haddad, & Bhat, 2008; Liu & Chen, 2008; Liu, Chen, & Lu, 2009, and the references therein), the development of nonlinear protocols for the dynamic average consensus problem has not been considered so far. Furthermore, the input-to-state stability property of the existing nonlinear protocols for the static consensus problem in the presence of external disturbances has been largely ignored. In the context of robustness against model uncertainty, Hui et al. (2008) proposed homogeneous nonlinear protocol functions for the static average consensus problem and showed that the consensus property of the perturbed protocol with additive model uncertainty of a specified structure is retained. However, the robustness property of nonlinear consensus protocols needs more investigation, and the analysis of their input-to-state stability property, which is dealt with in this paper, is a fundamental step toward this goal. 0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.06.031