Consensus in the network with uniform constant communication delay Xu Wang a , Ali Saberi a , Anton A. Stoorvogel b , Håvard Fjær Grip a , Tao Yang a a School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99164-2752, U.S.A. b Department of Electrical Engineering, Mathematics, and Computing Science, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. Abstract This paper studies the consensus among identical agents that are at most critically unstable and coupled through networks with uniform constant communication delay. An achievable upper bound of delay tolerance is obtained which explicitly depends on agent dynamics and network topology. The dependence on network topology disappears in the case of undirected networks. For any delay satisfying the proposed upper bounds, a controller design methodology without exact knowledge of the network topology is proposed so that the multi- agent consensus in a set of unknown networks can be achieved. Moreover, when the network topology is known, a larger delay tolerance is possible via a topology-dependent consensus controller. The results are illustrated by simulations. 1 Introduction The consensus problem in a network has received substantial attention in recent years, partly due to the wide applications in areas such as sensor networks and autonomous vehicle control. A relatively complete coverage of earlier work can be found in the survey paper of Olfati-Saber et al. (2007), the recent books by Wu (2007); Ren & Cao (2011) and references therein. Consensus in the network with time delay has been exten- sively studied in the literature. Most results consider the agent model as described by single-integrator dynamics (Bli- man & Ferrari-Trecate, 2008; Tian & Liu, 2008; Olfati-Saber & Murray, 2004), or double-integrator dynamics (Tian & Liu, 2009; Lin et al., 2007; Yu et al., 2010). Specifically, it is shown by Olfati-Saber & Murray (2004) that a net- work of single-integrator agents subject to uniform constant communication delay can achieve consensus with a partic- ular linear local control protocol if and only if the delay is bounded by a maximum that is inversely proportional to the ? This paper was not presented at any IFAC meeting. The work of Xu Wang, Ali Saberi and Tao Yang is partially supported by NAVY grants ONR KKK777SB001 and ONR KKK760SB0012. ?? Corresponding author Xu Wang Email addresses: xwang@eecs.wsu.edu (Xu Wang), saberi@eecs.wsu.edu (Ali Saberi), A.A.Stoorvogel@utwente.nl (Anton A. Stoorvogel), grip@ieee.org (Håvard Fjær Grip), tyang1@eecs.wsu.edu (Tao Yang). largest eigenvalue of the graph Laplacian associated with the network. This result was later on generalized in Bliman & Ferrari-Trecate (2008) to non-uniform constant or time- varying delays. Sufficient conditions for consensus among agents with first order dynamics were also obtained in Tian & Liu (2008). The results in Olfati-Saber & Murray (2004) were extended in Lin et al. (2007); Yu et al. (2010) to double integrator dynamics. An upper bound on the maximum net- work delay tolerance for second-order consensus of multi- agent systems with any given linear control protocol was obtained. In this paper, we study the multi-agent consensus problem with uniform constant communication delay. The agents are assumed to be at most critically unstable, i.e. each agent has all its eigenvalues in the closed left half plane. The contri- bution of this paper with respect to Olfati-Saber & Murray (2004); Bliman & Ferrari-Trecate (2008); Lin et al. (2007); Yu et al. (2010) is twofold: first, we find a sufficient condi- tion on the tolerable communication delay for agents with high-order dynamics, which has an explicit dependence on the agent dynamics and network topology. For undirected network, this upper bound can be independent of network topology provided that the network is connected. Moreover, in a special case where the agents only have zero eigenvalues, such as single- and double-integrator dynamics, arbitrarily large but bounded delay can be tolerated. Another layer of contribution is that for delay satisfying the proposed upper bound, we present a controller design methodology without exact knowledge of network topology so that the multi-agent consensus in a set of unknown networks can be achieved. Preprint submitted to Automatica 4 April 2012