Brief PapersContainment for linear multi-agent systems with exogenous disturbances
Introduction
In recent years, distributed cooperative control of multi-agent systems has attracted much attention for its broad potential applications in sensor networks, combat intelligence, surveillance, etc. [1]. Lots of works have been reported including formation control [2], [3], flocking [4], [5], [6], leaderless consensus [7], [8], [9], leader-following consensus [10], [11], [12], containment [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], and so forth.
Consensus is one of the most fundamental problems in distributed cooperative control, which means that the state of the agents reaches an agreement on a common physical quantity of interest by implementing an appropriate consensus protocol based on the information from local neighbors. According to the number of leaders, consensus problem and its extensions can be roughly classified into leaderless consensus (there is no leader in the network), consensus tracking or leader-following consensus (there is only one leader in the network), containment control (multiple leaders case) [13].
Containment control for multi-agent systems with multiple leaders has been intensively investigated for the past several years. Under fixed topology, for guaranteeing that all the follower agents asymptotically converge into the convex hull formed by multiple stationary/moving leaders, a hybrid control protocol based on the stop-and-go strategy was proposed [14]. Distributed containment control for multi-agent systems with single/double-integrator dynamics was investigated in the presence of multiple stationary or dynamic leaders under fixed and switching directed topologies, respectively [15], [16]. Distributed finite-time attitude containment control was studied for multiple rigid bodies [17]. Necessary and sufficient conditions were established for containment of multi-agent systems with stationary or dynamic leaders via both continuous and sampled control protocols [18]. Distributed containment control with multiple dynamic leaders was studied for double-integrator dynamics under the constraints that the velocities and the accelerations of the agents are not available [19]. Based on sample-data control protocol, necessary and sufficient containment conditions were obtained for second-order multi-agent systems without velocity measurement [20]. Impulsive containment for second-order multi-agent systems under a directed network topology was investigated [21]. And the containment was investigated for multi-agent systems with heterogeneous dynamics [22]. Distributed containment was studied for second-order multi-agent systems with inherent nonlinear dynamics [23], in which an adaptive containment protocol was proposed. Considering that linear system can model more natural dynamics in reality, the containment control for general linear multi-agent systems (LMAS) began to be discussed. A state feedback protocol was proposed by solving an algebraic Riccati equation [24]. Both output feedback and state feedback protocols were proposed for containment of linear multi-agent systems [25]. The semi-global containment control for general linear multi-agent systems with input saturation was studied [26].
However, external disturbances widely exist in real processes, which is a main source of instability and poor performance. Thereby, this paper focuses the containment for linear multi-agent systems (LMAS) with exogenous disturbances. The consensus of multi-agent systems with disturbances has been studied by many researchers [27], [28], [29]. The adaptive consensus problem of multi-agent systems with partly unknown parameters and bounded external disturbances is studied by adopting the model reference adaptive control method and an adaptive disturbance compensator in [27]. By proposing a distributed protocol using the neighbors׳ measured outputs, output consensus problem of directed networks of multiple high-order agents with external disturbances is studied [28]. The consensus of second-order multi-agent dynamical systems with exogenous disturbances is studied by designing disturbance-observer-based consensus protocol [29]. Motivated by [29], [30], disturbance-observer-based containment protocol is used to compensate for the influence of the exogenous disturbances. When the state can be used, a state feedback control protocol is designed based on a state feedback disturbance-observer. While the state cannot be used, an output feedback control protocol is designed based on an output feedback disturbance-observer. The main contribution of this paper is as follows: (1) The plant in this paper is general linear dynamic, which can be used model more real plants. (2) Output feedback containment protocol based disturbance-observer is proposed in this paper, which is more difficult and more practical than the state feedback case. (3) Disturbance-observer is used for disturbance attenuation.
The rest of the paper is organized as follows. In Section 2, we state the model considered in the paper and give some basic definitions, lemmas and assumptions. In Section 3, both state feedback and output feedback containment protocol based disturbance-observer are proposed, and sufficient containment conditions are obtained. Numerical examples are given in Section 4. Finally, we conclude the paper in Section 5.
Section snippets
Preliminaries and model description
In this section, some notations and preliminaries are introduced. The following notations are used throughout this paper. In denotes the identity matrix. For a matrix A (or a vector x), AT (or xT) represents the transpose of A (or x).
Let be a directed graph with a nonempty set of nodes , a set of edges , and a weighted adjacent matrix . An edge is denoted by in a directed graph which means that vertex j can obtain information from vertex i, but
Main results
In this section, the distributed containment control problem with exogenous disturbances will be solved in virtue of disturbance observers, i.e., to design distributed containment protocols to ensure that all the followers can be driven into the convex hull spanned by the leaders asymptotically.
Simulations
In this section, several simulation results are presented to illustrate the previous theoretical results. The network includes six agents and three leaders. The topology can be described in Fig. 1, in which agents 1–6 are followers and 7–9 are leaders. In the figures, red lines denote the trajectories of the leaders. Example 1 In this example, we consider the state feedback containment for system (1). Choosing matrices A, B, E, F as and
Conclusions
]In this paper, containment for linear multi-agent systems with exogenous disturbances under fixed topologies is investigated. Both state feedback and output feedback control protocols are proposed, under which all the followers will asymptotically converge to the convex hull spanned by the leaders without any global information except a connection assumption. By using tools from matrix, graph and Lyapunov stability theories, sufficient conditions for containment of linear multi-agent systems
Acknowledgments
This work was partially supported by the National Natural Science Foundation of China under Grant nos. 61473129, 61374139, 61104140, 61034006, the Program for New Century Excellent Talents in University from Chinese Ministry of Education under Grant NCET-12-0215, Natural Science Foundation of Ministry of Education in Hunan Province(12C0077).
Chengjie Xu is a doctor candidate in the School of Automation, Huazhong University of Science and Technology. Since July 2009, he has been with the School of Science, Hunan University of Technology, Zhuzhou, China, where he is currently a lecturer of automatic control engineering. His research interests are multi-agent systems, process control.
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Cited by (31)
Containment control of linear discrete-time fractional-order multi-agent systems with time-delays
2020, NeurocomputingCitation Excerpt :In [20], containment control of MAS with random graphs has been verified. Containment control of linear MAS under exogenous disturbances [21] and finite-time consensus of MAS [22] have been reported. Finite-time containment control of MAS without velocity and acceleration measurements has been studied in [23].
Large-scale algebraic Riccati equations with high-rank constant terms
2019, Journal of Computational and Applied MathematicsSemi-global containment control of discrete-time linear systems with actuator position and rate saturation
2019, NeurocomputingCitation Excerpt :A containment control law based on high-frequency feedback robust control is constructed for each follower agent such that all the followers will converge to the dynamic convex hull formed by dynamic leaders. Ref. [13–18] deal with containment control for high-order linear systems. In [14], the authors discuss adaptive containment control for multi-agent systems with multiple leaders containing parametric uncertainties.
Chengjie Xu is a doctor candidate in the School of Automation, Huazhong University of Science and Technology. Since July 2009, he has been with the School of Science, Hunan University of Technology, Zhuzhou, China, where he is currently a lecturer of automatic control engineering. His research interests are multi-agent systems, process control.
Ying Zheng received her B.S. degree in industrial electric automation engineering in 1997 from Huazhong University of Science and Technology, China; M.S. and Ph.D. degrees in control theory and engineering in 2000 and 2003 from Huazhong University of Science and Technology, China, respectively. She has been a postdoctor in the Chemical Engineering Department, National Tsing-Hua University, Hsinchu, Taiwan, from 2004 to 2005, and a faculty member in the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China (associate professor 2005, professor 2010). She has been involved in many projects granted by National Science Foundation Committee of China for recent years. Her research interests include process control, data-driven method, fault diagnosis, networked control system.
Housheng Su received the B.S. degree in automatic control and the M.S. degree in control theory and control engineering from Wuhan University of Technology, Wuhan, China, in 2002 and 2005, respectively, and the Ph.D. degree in control theory and control engineering from Shanghai Jiao Tong University, Shanghai, China, in 2008. From December 2008 to January 2010, he was a postdoctoral researcher with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong. Since January 2010, he has been an associate professor with the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China. His research interests lie in the areas of multi-agent coordination control theory and its applications to autonomous robotics and mobile sensor networks.
Hong-bin Zeng received the B.S. degree in electrical engineering from Tianjin University of Technology and Education, Tianjin, China, in 2003, M.S. degree in computer science from Central South University of Forestry, Changsha, China, in 2006, and Ph.D. degree in control science and engineering from Central South University, Changsha, China, in 2012. Since July 2003, he has been with the Department of Electrical and Information Engineering, Hunan University of Technology, Zhuzhou, China, where he is currently an associate professor of automatic control engineering. Now, he is working as a postdoctoral research associate in the Department of Electrical Engineering, Yeungnam University, Kyongsan, Korea. His current research interests are time-delay systems, neural networks and networked control systems.