Automatica 50 (2014) 1026–1036 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica H ∞ almost output synchronization for heterogeneous networks of introspective agents under external disturbances ✩ Ehsan Peymani a,b,1 , Håvard Fjær Grip b , Ali Saberi c , Xu Wang c , Thor I. Fossen b a Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, Trondheim, Norway b Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway c School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA, USA article info Article history: Received 30 June 2012 Received in revised form 23 June 2013 Accepted 25 November 2013 Available online 6 March 2014 Keywords: Synchronization Regulation Formation Interconnection networks Multi-agent systems abstract This paper brings forward the notion of ‘‘H ∞ almost output synchronization’’ for heterogeneous multi- agent systems under directed interconnection structures. Agents are assumed to be linear, right-invertible and introspective (i.e. agents have partial knowledge of their own states). The objective is to suppress the impact of disturbances on the synchronization error dynamics in terms of the H ∞ norm of the corresponding closed-loop transfer function. In addition, the problem of regulating the consensus trajectories to a reference signal is addressed. The application of the proposed method to the problem of formation is furthermore elucidated. Simulation results are provided to illustrate the method. © 2014 Elsevier Ltd. All rights reserved. 1. Introduction Consensus in networked dynamical systems has received con- siderable attention in the past decade. Its widespread application can be seen in satellite formation, sensor networks, flocking, co- ordinated decision making, etc. (see Olfati-Saber, Fax, & Murray, 2007). The objective is to make a function of the state of agents converge to a common trajectory. The seminal works of Wu and Chua (1995a,b) have substantially contributed in analysis and design of multi-agent systems by in- troducing the application of the graph theory and the Kronecker product. The theoretical framework for solving consensus prob- lems has been strengthened by the works of Jadbabaie, Lin, and Morse (2003); Olfati-Saber and Murray (2004) and Ren and Beard (2005). ✩ The work of H.F. Grip was supported by the Research Council of Norway. The material in this paper was partially presented at the American Control Conference (ACC13), June 17–19, 2013, Washington, DC, USA and 12th Biannual European Control Conference (ECC), July 17–19, 2013, Zurich, Switzerland. This paper was recommended for publication in revised form by Associate Editor Tamas Keviczky under the direction of Editor Frank Allgöwer. E-mail addresses: ehsan@ntnu.no (E. Peymani), grip@ieee.org (H.F. Grip), saberi@eecs.wsu.edu (A. Saberi), xw665@ieee.org (X. Wang), fossen@ieee.org (T.I. Fossen). 1 Tel.: +47 94224458; fax: +47 51648988. Synchronization of networks of general linear agents is ad- dressed in Fax and Murray (2004); Li, Duan, Chen, and Huang (2010); Seo, Shim, and Back (2009); Yang, Stoorvogel, and Saberi (2011); Zhang, Lewis, and Das (2011) where partial-state informa- tion is given to each agent via the network and dynamic proto- cols are introduced. Seo et al. (2009) propose a low-gain approach by filtering the information that each agent receives whereas Fax and Murray (2004) consider self-feedback for all agents. A signifi- cant breakthrough in the design of dynamic protocols is presented by Li et al. (2010) where conventional observers are expanded to distributed observers by allowing agents to exchange information about the state of their protocols over the network. The result is ex- tended to LQR-based optimal design by Zhang, Lewis et al. (2011). Predictive pinning control is proposed by Zhang, Chen, and Stan (2011) for accelerated consensus. Synchronization of heterogeneous networks (where agents are non-identical) is an active research. The common assumption is that agents are introspective; that is, agents possess some knowl- edge about their own states. For networks of nonlinear agents, Zhao, Hill, and Liu (2011) present criteria for state consensus. Output consensus for weakly minimum-phase systems of relative degree one is studied in Chopra and Spong (2008). Embedding ad- ditional models within agents, Kim, Shim, and Seo (2011) propose a controller for SISO minimum-phase systems. Using the principle of internal model, Wieland, Sepulchre, and Allgöwer (2011) pro- pose a dynamic controller for general linear systems. Yang, Saberi, Stoorvogel, and Grip (2011) develop a method to represent a 0005-1098/$ – see front matter © 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.automatica.2013.12.021