Elsevier

Automatica

Volume 136, February 2022, 110049
Automatica

Brief paper
Output feedback based adaptive consensus tracking for uncertain heterogeneous multi-agent systems with event-triggered communication

https://doi.org/10.1016/j.automatica.2021.110049Get rights and content

Abstract

In this paper, a backstepping based distributed adaptive output feedback control scheme is proposed, to achieve output consensus tracking for uncertain heterogenous linear multi-agent systems with event-triggered communication and directed communication topology. To deal with the unknown system parameters while addressing the non-differentiability problem of the virtual control inputs, auxiliary systems are elaborately constructed for the agents with relative degree larger than one. Different from most of the existing results, both the system orders and the relative degree of all the agents are allowed to be nonidentical. To decrease the amount of communication, a decentralized triggering condition with time-varying triggering threshold is well designed such that continuous monitoring of neighboring states can be avoided. Besides, only the output of each agent needs be shared among neighbors, which contributes to further reduction of communication burden. It is shown that with the proposed control scheme, all the closed-loop signals are uniformly bounded and desired output consensus tracking can be achieved without Zeno behavior. Moreover, the tracking performance in the mean square sense can be improved by appropriately adjusting design parameters. Simulation results are shown to validate the effectiveness of the proposed control scheme.

Introduction

Distributed consensus/synchronization of multi-agent systems has been widely investigated in the past two decades. Some seminal references can be found in Lu and Chen, 2007a, Lu and Chen, 2007b and Qin, Ma, Shi, and Wang (2017). Event-triggered consensus/synchronization is a hot research topic in this field since event-triggered communication mechanism can save communication resources effectively (Dimarogonas et al., 2012, Lu et al., 2015). In recent years, a great number of results for simple integrator-type multi-agent systems have been reported. Interested readers may refer to Dimarogonas et al., 2012, Kia et al., 2015, Nowzari et al., 2019, Seyboth et al., 2013 and Yi, Lu, and Chen (2015) and the references therein.

For more general linear multi-agent systems, the event-triggered consensus problem is also extensively investigated. In Garcia, Cao, and Casbeer (2014), a decentralized event-triggered leaderless consensus control scheme is proposed. In Li, Tang, and Karimi (2020) and Xing, Wen, Guo, Liu, and Su (2017), the distributed adaptive event-triggered leader-following consensus problem is investigated, in which external disturbances are also considered. In He, Xu, Han, and Qian (2020), a dynamic triggering condition is designed, where an auxiliary parameter is introduced into the triggering threshold. Compared with the traditional static condition, a dynamic one results in less triggering time instants. Note that the event-triggered consensus algorithms in Garcia et al., 2014, He et al., 2020, Li et al., 2020 and Xing et al. (2017) are all developed by using the full state feedback. For the output feedback case, two novel observer-based event-triggered control schemes are proposed in Zhang, Feng, Yan, and Chen (2014). In Hu, Liu, and Feng (2017), the event-triggered output consensus problem for heterogeneous linear multi-agent systems is investigated. To rule out Zeno behavior, fixed timers are introduced into the designed triggering conditions. In Qian, Liu, and Feng (2020), event- and self-triggered adaptive output feedback based consensus control strategies are proposed. It is worth mentioning that all the currently available results are obtained based on the assumption that the system matrices or parameters of each agent are precisely known. However, such an assumption may be restrictive when system parameters cannot be precisely measured. Therefore, exploring how to solve the distributed consensus problem for linear multi-agent systems with unknown system parameters and event-triggered communication is of great significance and importance.

In this paper, an output feedback based distributed adaptive output consensus tracking control scheme will be presented for heterogenous linear multi-agent systems with unknown system parameters, event-triggered communication and directed communication topology. Compared with currently available results, the main features of this paper are summarized as follows.

(i) It is known that adaptive backstepping control (Krstic, Kanellakopoulos, & Kokotovic, 1995) is widely used to deal with linear systems with totally unknown parameters. There are also some backstepping based adaptive consensus results on uncertain linear multi-agent systems (Ding, 2017, Wang et al., 2016). However, these results are all obtained based on continuous communication mechanisms. As observed in Ding (2017) and Wang et al. (2016), the neighboring states are usually involved to design virtual control inputs in each step. However, for event-triggered communication case, only discrete-time neighbors’ states are available in the design of virtual control inputs, which makes the virtual controllers be non-differentiable. Hence, the standard backstepping design procedure cannot be applied. To overcome this difficulty, an auxiliary system with the triggered local consensus error being its input is introduced for the agents with relative degree greater than one. Then the introduction of triggered neighbors’ states can be postponed until the last step. Besides, due to the introduced auxiliary systems, the original system orders and relative degree of all the agents are allowed to be arbitrary. This is different from all the aforementioned references.

(ii) Similar to Wang, Wen, Huang, and Zhou (2020) and Xing et al. (2017), a decentralized triggering condition is designed to transmit signals among neighbors such that continuous monitoring of neighboring states can be avoided. However, unlike the triggering conditions in Wang et al. (2020) and Xing et al. (2017) with fixed constant triggering thresholds, the triggering threshold in this paper is time-varying and closely related to the changing rate of the signal to be transmitted. Such a triggering mechanism contributes to improving the accuracy of transmitted signal when its changing rate is small.

(iii) In most of the existing output feedback based consensus results, multiple signals need be transmitted among the connected agents to implement the designed consensus algorithms, such as multi-dimensional states of the dynamic compensators in Hu et al. (2017) and Qian et al. (2020), multi-dimensional states of the observers in Zhang et al. (2014), the local control inputs and system outputs in Wang et al. (2016). Different from these results, each agent needs only transmit its one-dimensional output to its out-neighbors by using the consensus tracking control scheme presented in this paper. This is beneficial to reduce the communication burden.

The rest of this paper is organized as follows. In Section 2, the system model, graph theory and control objective are introduced. In Section 3, the local state estimation filters and distributed adaptive controllers are designed. The stability analysis of the entire closed-loop system is provided in Section 4 followed by an illustrative example in Section 5. Finally, the conclusion is drawn in Section 6.

Section snippets

System model

Similar to Ding (2017) and Wang et al. (2016), we consider a group of N heterogeneous linear multi-agent systems of the following form. ẋi=Aixiyiϕi+biuiyi=CiTxi,fori=1,,N, where Ai=0(ni1)×1Ini1001×(ni1),bi=0(ρi1)×1b̄i and Ci=[1,0,,0]Tni. xi=[xi,1,,xi,ni]Tni, yi and ui are the state vector, output and control input of agent i, respectively. 0p×q and Ip are p×q zero matrix and p×p identity matrix, respectively. ϕi=[ϕi,ni1,,ϕi,1,ϕi,0]Tni and b̄i=[bi,mi,,bi,1,bi,0]Tmi+1 are

Design of local state estimation filters

For each agent i, the following local filters are designed to estimate its unmeasurable states (Krstic et al., 1995 Chapter 10) η̇i=Ai0ηi+eni,niyiλ̇i=Ai0λi+eni,niui where ηi=[ηi,1,,ηi,ni]Tni and λi=[λi,1,, λi,ni]Tni are the state vectors of local state estimation filters. In this paper, ep,q is used to denote the qth coordinate vector in p. Ai0 is a Hurwitz matrix of the form Ai0=Aikieni,1T, where ki=[ki,1,ki,2,,ki,ni]T is a constant vector chosen to render Ai0 Hurwitz. Since Ai0 is

Stability and consensus analysis

In this section, the main results of this paper are formally stated as follows.

Theorem 1

Consider a group of uncertain heterogeneous linear multi-agent systems (1) consisting of the decentralized event-triggering conditions (6), auxiliary systems (24), distributed adaptive control inputs (13), (37) and a set of parameter estimators (15)(19), (29)(31), (38). Suppose that Assumption 1, Assumption 2, Assumption 3 can be satisfied. For any initial conditions satisfying V(0)p, where V(t) is a Lyapunov

An illustrative example

We consider a group of 4 single-link manipulators of the form (Spong & Vidyasagar, 2008). Jiϑ̈i+Biϑ̇i+Nisin(ϑi)=τifori=1,2,3,4.where ϑi denotes the angle of the link. τi denotes the generalized force, which is the control input of manipulators 1 and 3. Ji=1kg/m2 is the mechanical inertia. Bi=0.1Nms/rad is the coefficient of viscous friction at the joint. Ni=5 is a positive constant related to the mass of the load and the coefficient of gravity. For manipulators 2 and 4, the motor dynamics

Conclusion

In this paper, a novel distributed adaptive control scheme is proposed for heterogenous linear multi-agent systems with unknown system parameters, event-triggered communication and directed communication topology via output feedback control. It is shown that with the proposed control scheme, all the closed-loop signals are uniformly bounded and the desired output consensus tracking can be achieved. Besides, Zeno behavior in each agent is ruled out. Finally, the effectiveness of the proposed

Jiang Long received the B.Eng. degree in school of aviation automation from the Civil Aviation University of China, Tianjin, China in 2015. He is currently working toward the Ph.D. degree in control theory and control engineering at Beihang University, Beijing, China. His research interests include adaptive control of uncertain systems, distributed cooperative control of multi-agent systems, event-triggered control of multi-agent systems and attitude synchronization of multiple spacecraft.

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  • Cited by (39)

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    Jiang Long received the B.Eng. degree in school of aviation automation from the Civil Aviation University of China, Tianjin, China in 2015. He is currently working toward the Ph.D. degree in control theory and control engineering at Beihang University, Beijing, China. His research interests include adaptive control of uncertain systems, distributed cooperative control of multi-agent systems, event-triggered control of multi-agent systems and attitude synchronization of multiple spacecraft.

    Wei Wang received her B.Eng. degree in Electrical Engineering and Automation from Beihang University (China) in 2005, M.Sc. degree in Radio Frequency Communication Systems with Distinction from University of Southampton (UK) in 2006 and Ph.D degree from Nanyang Technological University (Singapore) in 2011. From January 2012 to June 2015, she was a Lecturer with the Department of Automation at Tsinghua University, China. Since July 2015, she has been with the School of Automation Science and Electrical Engineering, Beihang University, China, where she is currently a Full Professor. Her research interests include adaptive control of uncertain systems, distributed cooperative control of cyber–physical systems, fault tolerant control, and robotic control systems. Prof. Wang received Zhang Si-Ying Outstanding Youth Paper Award in the 25th Chinese Control and Decision Conference (2013) and the First Prize of Science and Technology Progress Award by Chinese Institute of Command and Control (CICC) in 2018. She is the Principle Investigator for a number of research projects including the Distinguished Young Scholars Fund of NSFC (2021–2023). She has been serving as Associate Editor for IEEE Transactions on Industrial Electronics, Journal of Control and Decision, IEEE Open Journal of Circuits and Systems.

    Changyun Wen received the B.Eng. degree from Xi’an Jiaotong University, China, in 1983 and the Ph.D. degree from the University of Newcastle, Australia in 1990. From August 1989 to August 1991, he was a Postdoctoral Fellow at University of Adelaide, Australia. Since August 1991, he has been with Nanyang Technological University, Singapore, where he is currently a Full Professor. His main research activities are in the areas of control systems and applications, cyber–physical systems, smart grids, complex systems and networks. Some of his publications in these areas are available in https://publons.com/researcher/2816818/changyun-wen/publications/.

    Prof Wen is a Fellow of IEEE, was a member of IEEE Fellow Committee from January 2011 to December 2013 and a Distinguished Lecturer of IEEE Control Systems Society from 2010 to 2013. He is currently the co-Editor-in-Chief of IEEE Transactions on Industrial Electronics, Associate Editor of Automatica (from Feb 2006) and Executive Editor-in-Chief of Journal of Control and Decision. He also served as an Associate Editor of IEEE Transactions on Automatic Control from 2000 to 2002, IEEE Transactions on Industrial Electronics from 2013 to 2020 and IEEE Control Systems Magazine from 2009 to 2019. He has been actively involved in organizing international conferences playing the roles of General Chair (including the General Chair of IECON 2020 and IECON 2023), TPC Chair (e.g. The TPC Chair of Chinese Control and Decision Conference since 2008) etc.

    He was the recipient of a number of awards, including the Prestigious Engineering Achievement Award from the Institution of Engineers, Singapore in 2005, and the Best Paper Award of IEEE Transactions on Industrial Electronics in 2017.

    Jiangshuai Huang received his B.Eng. and M.Sc. degree in School of Automation from Huazhong University of Science & Technology, Wuhan, China in July 2007 and August 2009 respectively, and Ph.D. from Nanyang Technological University in 2015. He was a Research Fellow in the Department of Electricity and Computer Engineering, National University of Singapore from August 2014 to September 2016. He is currently with the School of Automation, Chongqing University, Chongqing, China. His research interests include adaptive control, nonlinear systems control, underactuated mechanical system control and multi-agent system control.

    Jinhu Lü received the Ph.D. degree in applied mathematics from the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China, in 2002.

    He was a Professor with RMIT University, Melbourne, VIC, Australia, and a Visiting Fellow with Princeton University, Princeton, NJ, USA. Currently, he is the Dean with the School of Automation Science and Electrical Engineering, Beihang University, Beijing, China. He is also a Professor with the Academy of Mathematics and Systems Science, Chinese Academy of Sciences. He is a Chief Scientist of the National Key Research and Development Program of China and a Leading Scientist of Innovative Research Groups of the National Natural Science Foundation of China. His current research interests include complex networks, industrial Internet, network dynamics and cooperaton control.

    Dr. Lü was a recipient of the prestigious Ho Leung Ho Lee Foundation Award in 2015, the National Innovation Competition Award in 2020, the State Natural Science Award three times from the Chinese Government in 2008, 2012, and 2016, respectively, the Australian Research Council Future Fellowships Award in 2009, the National Natural Science Fund for Distinguished Young Scholars Award, and the Highly Cited Researcher Award in engineering from 2014 to 2020. He is/was an Editor in various ranks for 15 SCI journals, including the Co-Editor-in-Chief of IEEE TII. He served as a member in the Fellows Evaluating Committee of the IEEE CASS, the IEEE CIS, and the IEEE IES. He was the General Co-Chair of IECON 2017. He is the Fellow of IEEE and CAA.

    This work was supported by National Natural Science Foundation of China under grants 62022008 and 61973017. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Gang Tao under the direction of Editor Miroslav Krstic.

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