Finite-time containment control of multi-agent systems with static or dynamic leaders☆
Introduction
In recent years, cooperative control of multi-agent systems have received considerable attention due to its extensive potential applications, such as group average in distributed computation, rendezvous of multiple vehicles, multiple space-craft alignment and formation control of multiple robots [1], [2], [3]. A fundamental but important issue that arises in such robotic applications is the consensus problem, where agents are required to reach an agreement on a state of interest [4], [5], [6]. In this line of research, significant efforts have been made on the development of various distributed consensus protocols for agents with different models. For examples, [4], [6], [7], [8] have studied the consensus problem for agents with first-order dynamics or second-order dynamics, consensus problem with time-delay, consensus of heterogeneous systems, and consensus of switched systems, respectively.
In the area of consensus control, convergence rate becomes an important performance index of the proposed consensus protocols. It has been shown that the second smallest eigenvalue of interaction graph Laplacian, called algebraic connectivity of graph, quantifies the speed of convergence of the consensus protocol [9]. In order to increase the algebraic connectivity, the problem of the weight design via semi-definite convex programming has been considered in [10]. Although maximizing the second smallest eigenvalue of interaction graph Laplacian allows for a better convergent rate of the linear protocols, the state consensus can never occur in finite time. In some practical situations, however, it may require that consensus has to be reached in finite time.
Then the idea of reaching consensus in finite time has been introduced to the consensus problem for multi-agent systems [11], [12], [13], [14], [15]. In [11], normalized and signed gradient dynamical systems associated with a differentiable function have been introduced and the conditions that guarantee finite-time convergence have been identified. In [12], finite-time consensus tracking of multi-agent systems has been reached on the terminal sliding-mode surface. [13] has investigated finite-time consensus problems for multi-agent systems and has presented a framework for constructing effective distributed protocols. In [14], weighted average consensus with respect to a monotonic function has been studied for a group of kinematic agents with time-varying topology. Some sufficient conditions of achieving finite-time consensus for second-order multi-agent systems have been obtained in [15].
Note that the above mentioned literatures mainly focus on consensus for a group of agents without leader. In fact, one or even multiple leaders might exist in both natural systems and artificial systems, for instance, the leading fish in a moving school and the navigation aircraft in a fight formation of unmanned aerial vehicles. When there is a single leader in the multi-agent system, the problem of controlling remaining agents, the so-called followers, to track the leader has been investigated in many studies [16], [17]. For the case of existing multiple leaders, the containment control problem arises, where the followers are required to be driven into the convex hull spanned by the leaders. Recently, containment control problem of multi-agent systems has been investigated by several researchers [18], [19], [20], [21], [22] due to its wide applications. For instance, a groups of mobile robots can arrive at their destination safely with low cost, when only a few of them (leaders) are equipped with proper sensors to detect the dangerous obstacles and then form a safety area while the rest robots (followers) just need remain in the moving safety area formed by the leaders. In [19], [20], some necessary and sufficient conditions for solving containment control problem of multi-agent systems have been obtained. [21] has investigated containment control for multi-agent systems with fixed undirected topology. Containment control for heterogeneous multi-agent systems has been addressed in [22]. However, the control protocols used in the above references can only achieve asymptotic convergence in infinite time. In contrast, finite-time control protocols offer many benefits including faster convergence, precise performance, and robustness to uncertainties and disturbances. Therefore, some researchers considered solve containment control in finite-time [23], [24], [25]. In [23], [24], the authors investigated finite-time attitude containment control for multiple rigid bodies. By using an observer-based approach, [25] dealt with the finite-time containment tracking problem for multi-agent systems.
Motivated by the above analysis, in this paper, we try to establish a novel framework on finite-time containment control for multi-agent systems. Different from the previous results, the aim of this paper is to ensure the system solving containment control at any preset convergence time. The main contribution of this paper is twofold. First, we propose a control protocol for multi-agent systems with static leaders, in which specific functions are designed as time-varying gains. We show that the system can reach containment at any preset time if the communication graph has a spanning forest. Then the results are extended to a general form. Second, we design a control protocol to solve containment control problem for multi-agent systems with dynamic leaders. We prove that the system can reach containment at any preset time if the communication graph has a spanning forest.
The rest of the paper is organized as follows. Section 2 gives the preliminary knowledge about graph theory. Section 3 presents finite-time containment control protocol of multi-agent systems with static leaders. In Section 4 we propose finite-time containment control protocol of multi-agent systems with dynamic leaders. Simulation results are given in Section 5. Finally, Section 6 concludes this paper.
Notations: let be the set of real numbers. is the set of n×m real matrices. Denote (or ) as the column vector with all entries equal to one (or all zeros). For a matrix , is the eigenvalue with the greatest absolute value. is the spectral radius of A. Notation represents the diagonal matrix with , on its diagonal.
Section snippets
Preliminaries on algebraic graph theory
In this section, we present some definitions and properties about algebraic graph theory that will be used in this paper. For more details, we refer to [26].
Graph will be used to describe the communication topology among agents. Let be a weighted directed graph with the set of vertices , the set of edges , and a weighted adjacency matrix with nonnegative adjacency elements aij. An edge of is denoted by , where vj is
Finite-time containment control with static leaders
In this section, we consider a multi-agent system of n identical agents with single-integrator dynamics:where and are the state and the control input of the agent i, respectively. Definition 4 Given any finite time , if system (1) satisfies that for any initial state , all the followers move into the convex hull spanned by the leaders, as , i.e.,then we say that the control protocol ui(t) solves the finite-time
Finite-time containment control with dynamic leaders
In this section, we consider a multi-agent system which consists of multiple dynamic leaders:where , , and are the state, the velocity, and the control input of the agent i, respectively. Definition 5 For any finite time , we say that the control protocol ui(t) solves the finite-time containment control problem at time Tf for system (6), if system (6) satisfies that for any initial state , all the followers move into the convex hull spanned
Simulations
In this section, we present the simulation results to demonstrate our theoretical results. Considering a multi-agent system with . The communication graph is given in Fig. 1, where is a directed graph containing a directed spanning forest. Agents 1, 2, 3 are the leaders, while the others are the followers.
The Laplacian matrix of is given by
In the following two examples, we will show that systems (1), (6) solve containment control at a
Conclusion
Finite-time containment control problems for multi-agent systems with static and dynamic leaders have been investigated in this paper. Firstly, for multi-agent systems with static leaders, a time-varying nonlinear feedback control protocol has been proposed under which the followers move into the convex hull spanned by the leaders at any preset time. The result has been further extended to a general form. Secondly, for multi-agent systems with dynamic leaders, we have proposed a protocol to
Huaizhu Wang received his Bachelor and Master degrees in computer engineering from Ningxia University in 1999 and 2008, respectively. Since 1999, he has been at School of Mathematics and Statistics, Ningxia University, where he is now an associate professor. His current research interests are multi-agent systems, robotic fish and computer networks.
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Huaizhu Wang received his Bachelor and Master degrees in computer engineering from Ningxia University in 1999 and 2008, respectively. Since 1999, he has been at School of Mathematics and Statistics, Ningxia University, where he is now an associate professor. His current research interests are multi-agent systems, robotic fish and computer networks.
Chen Wang received the Bachelor's degree in electronic and information engineering from Xi'an Jiaotong University, Xi'an, China, in 2007. After finishing the Bachelor's degree, in the same year, she was enrolled in a joint Ph.D. training program, for which she received the first Ph.D. degree in general mechanics and foundation of mechanics from Peking University, Beijing, China, in 2013 and the second Ph.D. degree in systems and control from the University of Groningen, Groningen, The Netherlands, in 2014.
She is currently a Postdoctoral Researcher with the College of Engineering, Peking University. Her research interests include multi-agent systems, biomimetic robotics, and game theory.
Guangming Xie received B.S. degrees in applied mathematics and electronic and computer technology, the M.E. degree in control theory and control engineering, and the Ph.D. degree in control theory and control engineering from Tsinghua University, Beijing, China, in 1996, 1998, and 2001, respectively.
He is currently a Professor of dynamics and control with the College of Engineering, Peking University, Beijing. His research interests include smart swarm theory, multi-agent systems, biomimetic robots, switched and hybrid systems, and networked control systems.
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This work was supported in part by grants from the National Natural Science Foundation of China (NSFC, No. 51575005, 61503008, 61633002, 61562069, 61563043), the China Postdoctoral Science Foundation (No. 2015M570013, 2016T90016), and the Natural Science Foundation of Ningxia (No. NZ14022).