Towards event-triggered extended state observer for multi-agent systems☆
Introduction
Over the past two decades, cooperative control of multi-agent systems has been studied with great attention due to its potential applications in smart grid, sensor networks, and so forth [1], [2], [3]. In the literature mentioned above, a fundamental theme is consensus control, meaning that a group of agents reach an agreement by interacting with their local neighbors. According to whether there is a leader in communication topology, consensus can be divided into two categories, leader-follower consensus, and leaderless consensus. For leaderless consensus issues, readers can refer to these articles [4], [5], [6]. The leader-follower consensus problem is also called distributed tracking problem [7], [8], [9], [10], [11], [12], which is the main research topic in this paper. The authors in [13], [14] assumed the leader has no control input and then designed a distributed estimator to track the status of the leader. The case in which the control input of the leader is non-zero but available by followers was considered in [15]. A discontinuous sliding mode controller was proposed in [16] by using upper bound information of the leader’s control input. In many practical situations, the leader has control input, e.g., to avoid hazardous obstacles, and it is restrictive to assume the follower is able to receive the control input of the leader. Besides, there exists a tremble phenomenon by sliding mode control in [16].
It is well known that unnecessary communications in controller actuation will cause a waste of energy. Event-triggered was proposed to remove the requirement for continuous communication among controllers. Dimos et al. [17] considered the problem of event-triggered multi-agent with first-order integrators. The general linear dynamics form of this problem has been investigated in [18]. Yang and Liu proposed a new protocol to solve the event-triggered problem over directed graphs [19]. It is noteworthy that the event-triggered protocol will be useless if Zeno behavior exists (i.e., the interval time between adjacent two trigger behaviors is zero). Some of the latest results can be found in [20], [21], [22]. It is a severe problem to exclude Zeno behavior of event-triggered consensus problem, especially for the general linear dynamics of multi-agent.
Moreover, it is worth noting that the unknown external disturbance is widespread, including measurement noise, environmental impacts, and changes in physical processes. Therefore, it is significant and interesting to solve distributed event-triggered consensus problem with unknown external disturbances. In existing event-triggered results, anti-disturbance methods can be mainly divided into three categories: (1) H∞ control. By assuming the energy of disturbance is bounded, adjustable control parameters are introduced to reduce the effects of disturbances [23], [24]; (2) Sliding control. The upper bound of disturbance is needed to be available in advance to construct controller [25], [26]; (3) Disturbance observer. The structure of the disturbance needs to be acquired beforehand (e.g., disturbance for each agent is sinusoidal signal), and output information is used to estimate the disturbance [27]. For method (1) and (3), only bounded consensus can be achieved, i.e., consensus error always exists. Furthermore, tremble phenomena cannot be avoided in the control process of method (2).
Motivated by the results mentioned above, we aim to address the event-triggered tracking consensus problem for general linear multi-agent systems with unknown disturbances. The main contributions of this paper are as follows:
- 1)
Above all, we develop the event-triggered extended state observer to the multi-agent system, which can save energy and do not need to use the state information of agents.
- 2)
Next, we consider unmodeled disturbances. Besides, the leader’s input can be nonzero, and not all followers need to know the leader’s input information.
- 3)
Finally, a novel event-triggered leader-follower consensus protocol is designed. The tracking error asymptotically approaches zero with fewer controller updates. Furthermore, Zeno behavior is excluded.
The remainder of this paper is organized as follows. In Section 2, the preliminaries are proposed. The main control results are shown in Section 3. Simulation results are provided in Section 4, and Section 5 concludes the paper.
Section snippets
Notations and graph theory
Let and be the set of m × n real and complex matrices respectively. λi( · ) denotes the ith eigenvalue of a matrix. Let ‖ · ‖ denote the Euclidean norm for vectors and the induced 2-norm for matrices. In addition, dim( · ) describes the dimension of a square matrix. For a complex number, Re( · ) denotes its real part.
The adjacency matrix associated with the directed graph is defined as and otherwise. The Laplacian matrix is
Main result
In this section, by using relative output information, we propose a novel class of extended state observers to estimate the information of the leader’s control input, leader tracking error, and disturbance of each agent, simultaneously.
Leader tracking error is defined as follows:
Then, from (1) to (4), we have
By adding an extended state variable ηi(t) coupled with u0(t) and ωi(t), we can rewrite system (5) to the extended system form
Numerical simulations
In this section, a controller simulation example is offered to verify the theory. Consider the multi-agent systems consisting of one leader and five agents, whose communication topology is shown as 1. Then, matrix can be computed as
The system dynamics can be discribed asthe leader’s control input and disturbances of each agents are
Conclusion
This paper has studied the distributed event-triggered leader-follower consensus problem for multi-agent systems with general linear dynamics over directed graphs. A novel class of extended state observers based on relative output information has been developed to estimate leader tracking error and coupled state related to disturbance and leader’s control input. We proposed a distributed event-triggered protocol by using estimated value for each follower, to achieve leader-follower consensus.
CRediT authorship contribution statement
Xiang Wu: Conceptualization, Visualization, Writing - original draft, Writing - review & editing. Kexin Liu: Writing - review & editing. Yuqi Bai: Writing - original draft, Writing - review & editing. Jinzhi Wang: Conceptualization, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing for financial interests or personal relationships that could haveappeared to influence the work reported in this paper. The authors also confirmed that the work reported in this paper has not been published elsewhere. In addition, the authors have confirmed that their respective contributions are the same as in the authorship contribution statement.
Xiang Wu received the B.S. degree from University of Electronic Science and Technology of China in 2017. He is currently a postgraduate in the State Key Laboratory for Turbulence and Complex System, Department of Mechanics and Engineering Science, Peking University, Beijing, China. His research interests include multi-agent systems, IoT and complex networks.
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Xiang Wu received the B.S. degree from University of Electronic Science and Technology of China in 2017. He is currently a postgraduate in the State Key Laboratory for Turbulence and Complex System, Department of Mechanics and Engineering Science, Peking University, Beijing, China. His research interests include multi-agent systems, IoT and complex networks.
Kexin Liu received the M.Sc. degree in control science and engineering from Shandong University, Jinan, China in 2013, and Ph. D degree in System Theory from Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China in 2016, respectively. From 2016 to 2018, he was a postdoctoral fellow in Peking University, Beijing, China. Currently, he is an associated professor with the School of Automation Science and Electrical Engineering, Beihang University, Beijing, China. His research interests include multi-agent systems and complex networks.
Yuqi Bai received the B.S. degree from Northwestern Polytechnical University in 2017. She is currently a Ph.D. Candidate in the State Key Laboratory for Turbulence and Complex System, Department of Mechanics and Engineering Science, Peking University, Beijing, China. Her research interests include multi-agent systems, fault detection and isolation, and resilient control.
Jinzhi Wang received the M.S. degree in mathematics from Northeast Normal University, China, in 1988 and Ph.D. degree in control theory from Peking University in 1998. From July 1998 to February 2000 she was a post-doctor at Institute of Systems Science, the Chinese Academy of Sciences. From March 2000 to August 2000 she was a research associate in the University of Hong Kong. She is currently a professor at the Department of Mechanics and Engineering Science, Peking University. Her research interests include robust control and control of nonlinear dynamical systems.