Published in IET Control Theory and Applications Received on 15th January 2011 Revised on 25th April 2011 doi: 10.1049/iet-cta.2011.0033 ISSN 1751-8644 Consensus of heterogeneous multi-agent systems Y. Zheng 1 Y. Zhu 1 L. Wang 2 1 Center for Complex Systems, School of Mechano-electronic Engineering, Xidian University, Xi’an 710071, People’s Republic of China 2 Center for Systems and Control, College of Engineering and Key Laboratory of Machine Perception (Ministry of Education), Peking University, Beijing 100871, People’s Republic of China E-mail: zhengyuanshi2005@163.com Abstract: In this study, the consensus problem of heterogeneous multi-agent system is considered. First, the heterogeneous multi-agent system is proposed which is composed of first-order and second-order integrator agents in two aspects. Then, the consensus problem of heterogeneous multi-agent system is discussed with the linear consensus protocol and the saturated consensus protocol, respectively. By applying the graph theory and Lyapunov direct method, some sufficient conditions for consensus are established when the communication topologies are undirected connected graphs and leader-following networks. Finally, some examples are presented to illustrate the theoretical results. 1 Introduction In recent years, distributed coordination of multi-agent systems has attracted more and more attention in a wide range including system control theory, applied mathematics, statistical physics, biology, communication, computer science etc. Consensus problem, which is fundamental to distributed coordination, has been studied as an active research area in many fields. Consensus means that a group of agents reaches an agreement on a common value by negotiating with their neighbours asymptotically or in a finite time. Roughly speaking, the main objective of the consensus problem is to design an appropriate control input such that a group of agents converges to a consistent quantity of interest. The control input is usually called consensus protocol, and the consistent quantity that depends on the initial state is usually called consensus state. Up to now, by using the matrix theory, the graph theory, the frequency-domain analysis method, the Lyapunov direct method etc., consensus problem of multi-agent systems has been studied in detail, and the consensus criterions have been obtained under first-order, second-order or high-order multi-agent systems [1]. The consensus problem of first-order multi-agent systems is primarily studied. Vicsek et al. [2] proposed a discrete- time model of n agents all moving in the plane with the same speed and demonstrated by simulation that all agents move to one direction asymptotically. Jabdabaie et al. [3] provided a theoretical explanation of the consensus behaviour in the Vecsek model, and analysed the alignment of a network of agents with switching topologies that are periodically connected. Olfati-Saber and Murray [4] discussed the consensus problem for networks of dynamic agents with switching topologies and time delays in a continuous-time model by defining a disagreement function, and obtained some useful results for solving the average- consensus problem. Ren and Beard [5] presented some more relaxable conditions for consensus of states under dynamically changing interaction topologies. With the development of issue, a lot of new consensus results were given out with different models and protocols by a single integrator. Xiao and Wang studied the consensus problem of discrete-time multi-gent systems with time-delays [6, 7], asynchronous consensus of multi-agent systems with switching topologies and time-varying delays [8]. The finite-time consensus problem of multi-agent systems for both the bidirectional and unidirectional interaction case was also considered by Wang and Xiao [9]. Sun et al. [10] discussed the average-consensus problem in undirected networks of multi-agent systems with fixed and switching topologies as well as multiple time-varying communication delays. Hui and Haddad [11] investigated the consensus problem for non-linear multi-agent systems with fixed and switching topologies, and Liu et al. [12] considered the consensus problem in directed networks via non-linear protocols. Li and Zhang [13] gave the necessary and sufficient condition of mean square average-consensus for multi-agent systems with noises. Hatano and Mesbahi [14] considered the consensus problem for multi-agent systems with random topology for the first time and Tahbaz-Salehi and Jadbabaie [15] gave a necessary and sufficient condition for consensus with random topology. Unlike the first-order case, a (directed) spanning tree is a necessary rather than a sufficient condition for consensus seeking with second-order dynamics. Therefore the extension of consensus algorithms from first order to second order is non-trivial [16]. Xie and Wang [17] investigated the consensus problem of second-order multi-agent systems with fixed and switching topologies. Ren [18] considered the consensus problem of multi-agent systems with IET Control Theory Appl., 2011, Vol. 5, Iss. 16, pp. 1881–1888 1881 doi: 10.1049/iet-cta.2011.0033 & The Institution of Engineering and Technology 2011 www.ietdl.org