292 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL.7, NO.1, JANUARY 2020 Distributed Adaptive Cooperative Tracking of Uncertain Nonlinear Fractional-order Multi-agent Systems Zhitao Li, Lixin Gao, Wenhai Chen, and Yu Xu Abstract—In this paper, the leader-following tracking prob- lem of fractional-order multi-agent systems is addressed. The dynamics of each agent may be heterogeneous and has unknown nonlinearities. By assumptions that the interaction topology is undirected and connected and the unknown nonlinear uncertain dynamics can be parameterized by a neural network, an adaptive learning law is proposed to deal with unknown nonlinear dy- namics, based on which a kind of cooperative tracking protocols are constructed. The feedback gain matrix is obtained to solve an algebraic Riccati equation. To construct the fully distributed cooperative tracking protocols, the adaptive law is also adopted to adjust the coupling weight. With the developed control laws, we can prove that all signals in the closed-loop systems are guaranteed to be uniformly ultimately bounded. Finally, a simple simulation example is provided to illustrate the established result. Index Terms—Adaptive control, consensus, distributed control, fractional-order systems, multi-agent system. I. I NTRODUCTION I N recent years, coordination problem of multi-agent sys- tems has received a great deal of attention and become a heated topic due to the reason of its extensive applications in many areas, which include formation flight of UAV, col- laborative rescue, multi-robot cooperative actions, distributed sensor networks and so on. Consensus problem is well-known as one of the foremost and basic issues in the area of coordination control for multi-agent systems, whose purpose is to develop distributed control protocols which make a group of agents reach an agreement on some quantities. There are many absorbing issues of coordination control linked with consensus such as synchronization, swarm, flock, formation, rendezvous, containment [1]. Till now, numerous constructive results have been obtained for the consensus problems with different agent dynamics including single integration system, Manuscript updated December 20, 2019. This work was supported by the National Natural Science Foundation of China (61303211) and Zhejiang Provincial Natural Science Foundation of China (LY17F030003, LY15F030009). Recommended by Associate Editor Yanjun Liu. (Correspond- ing author: Lixin Gao.) Citation: Z. T. Li, L. X. Gao, W. H. Chen, and Y. Xu, “Distributed adaptive cooperative tracking of uncertain nonlinear fractional-order multi- agent systems,” IEEE/CAA J. Autom. Sinica, vol. 7, no. 1, pp. 292-300, Jan. 2020. Z. T. Li, L. X. Gao, and W. H. Chen are with the Institute of Intelligent Systems and Decision, Wenzhou University, Zhejiang 325035, China (e-mail: 1451649267@qq.com; lxgao@wzu.edu.cn; whchen@wzu.edu.cn). Y. Xu is with the College of Physics and Electronic Information Engineer- ing, Wenzhou University, Wenzhou 325035, China (e-mail: yxu@wzu.edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JAS.2019.1911858 double integration system, general linear system, discrete-time system, time-delay system, descriptor system, fractional-order system, nonlinear system [2]−[9]. As is known to all, neural networks (NNs) and fuzzy logic systems (FLSs) have been extensively used to model and design the control for uncertain interconnected (largescale) nonlinear systems. In [10]−[12], fuzzy decentralized control schemes have been developed for some classes of uncertain interconnected nonlinear systems. By assumption that the uncertainty can be linearly parameterised by a neural network, distributed consensus protocols were developed to solve multi- agent problems in [13]−[18]. In comparison with integer-order systems, fractional-order systems are more suitable to model some practical application systems such as viscoelastic systems, dielectric polarization, electromagnetic waves and so on. It is easy to see that the traditional integer-order systems can be viewed as a special case of the fractional-order systems. In [19]−[21], the authors investigated the stability problem of fractional-order systems, which is more complex than that of integer-order systems. Recently, some researchers have focused on the coordination problem of fractional-order systems. The first-order consensus problem was generalized to the case of networked fractional- order systems in [8]. The consensus problem of fractional- order multi-agent systems with input delay and communication delay was studied by [22]. In [23], the authors probed the con- sensus problem of fractional-order with uncertainty dynamics via output feedback protocol. The synchronization problem for a general fractional-order dynamical network model was addressed in [24]. In [25], the relative state error feedback laws were used to solve the leader-following fractional-order consensus problem with Lipschitz nonlinear dynamics. The leader-following consensus problem of fractional-order multi- agent systems was addressed via adaptive pinning control by [26]. The multi-consensus problem of fractional-order uncertain linear multi-agent systems was investigated by [27], and the related containment problem was addressed by [28]. Generally, the above established consensus conditions are related to the interaction topology. The well-known consensus condition for the coupling parameters is determined by the smallest real part of the non-zero Laplacian eigenvalues of the interaction topology, which plays a key role in the consensus stability analysis. For a large-scale interaction topology, it is very hard to estimate its eigenvalues, which limits the appli- cations of the obtained results [29]. Because the eigenvalues