Adaptive Containment Control for Fractional-Order Nonlinear Multi-Agent Systems With Time-Varying Parameters Yang Liu, Huaguang Zhang, Fellow, IEEE, Yingchun Wang, Member, IEEE, and Hongjing Liang, Member, IEEE Abstract—This paper investigates adaptive containment control for a class of fractional-order multi-agent systems (FOMASs) with time-varying parameters and disturbances. By using the bounded estimation method, the difficulty generated by the time- varying parameters and disturbances is overcome. The command filter is introduced to solve the complexity problem inherent in adaptive backstepping control. Meanwhile, in order to eliminate the effect of filter errors, a novel distributed error compensating scheme is constructed, in which only the local information from the neighbor agents is utilized. Then, a distributed adaptive containment control scheme for FOMASs is developed based on backstepping to guarantee that the outputs of all the followers are steered to the convex hull spanned by the leaders. Based on the extension of Barbalat’s lemma to fractional-order integrals, it can be proven that the containment errors and the compensating signals have asymptotic convergence. Finally, three simulation examples are given to show the feasibility and effectiveness of the proposed control method. Index Terms—Adaptive backstepping control, command filter, fractional-order multi-agent system, time-varying parameters.    I. Introduction D URING the past few decades, there has been a great deal of research in regards of the cooperative control of the nonlinear multi-agent systems (MASs), such as [1]–[6]. Early works focused on MASs with integer-order dynamics. However, some phenomena can be explained by the cooperative group behavior of agents with fractional-order dynamics rather than classical integer-order dynamics, such as robot formation [7] and high-speed aircraft formation. Then, the cooperative control for nonlinear fractional-order multi- agent systems (FOMASs) has been developed. In [8], an adaptive feedback control was constructed to guarantee consensus control for FOMASs under direct topology. Further, the authors designed fault-tolerant consensus control for the FOMASs in [9]. Note that the existing results for the fractional-order nonlinear systems were limited to time- invariant systems (the uncertain parameters were constants). However, the time-varying parameters and disturbances exist in the practical operating environment, which affect the control performance of the systems. The adaptive backstepping method, as a classical control method, has been widely used for the control design of the nonlinear systems in the lower triangle structure [10]–[13]. Especially with the development of the intelligent algorithms, such as fuzzy systems [14]–[17] and neural networks [18]–[23], the adaptive control methods for uncertain nonlinear systems have been greatly promoted. For nonlinear systems with uncertain time-varying parameters, the adaptive control method is still valid, and some pioneering works have been proposed. In [24], the authors proposed the so-called congelation of variables method, and a structured adaptive approach was provided for the strict-feedback nonlinear system. For time-varying MASs, a cooperative tracking control was designed in [25], in which tracking errors were steered to converge into the neighborhood of origin. In [26], a smooth nonlinear function was utilized to compensate for time-varying parameters and disturbances, and asymptotic tracking was achieved. In [27], two identification algorithms were developed to identify the time-varying parameters in finite-time and fixed-time. In [28], three stable adaptive tracking schemes for multi-input-multi-output nonlinear systems with time-varying control coefficients were designed. However, all the results are concerned with integer-order nonlinear systems. To the best of our knowledge, there is no result for the fraction-order nonlinear system with time- varying parameters. Due to the specific characteristics of the fractional-order calculation, the obtained results on the integer-order dynamic system can not be directly extended to the fractional-order case. As a result, it is important to develop the control method for the fractional-order nonlinear system with time-varying parameters. It is noteworthy that the problem of “complexity explosion” Manuscript received August 4, 2021; revised November 2, 2021; accepted January 24, 2022. This work was supported by National Key R&D Program of China (2018YFA0702200), and National Natural Science Foundation of China (61627809, 62173080), and Liaoning Revitalization Talents Program (XLYC1801005). Recommended by Associate Editor Tengfei Liu. (Corresponding author: Huaguang Zhang.) Citation: Y. Liu, H. G. Zhang, Y. C. Wang, and H. J. Liang, “Adaptive containment control for fractional-order nonlinear multi-agent systems with time-varying parameters,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 9, pp. 1627–1638, Sept. 2022. Y. Liu and Y. C. Wang are with the College of Information Science and Engineering, Northeastern University, Shenyang 110004, China (e-mail: myliuyang@yeah.net; drwangyc@gmail.com). H. G. Zhang is with the State Key Laboratory of Synthetical Automation for Process Industries, and also with the School of Information Science and Engineering, Northeastern University, Shenyang 110004, China (e-mail: hgzhang@ieee.org). H. J. Liang is with the School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China (e- mail: lianghongjing99@163.com). 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.2022.105545 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 9, NO. 9, SEPTEMBER 2022 1627