Containment control of leader-following multi-agent systems with jointly-connected topologies and time-varying delays
Introduction
Distributed cooperative control of multi-agent systems (MASs) has attracted significant attention in recent years, due to its potential in the study of animal group behaviors such as swarms [1] and flocking [2], and its broad applications in both military and civilian sectors, such as the formation control of mobile robots, cooperative control of unmanned aerial vehicles, attitude adjustment and position of satellite, and so on [3], [4], [5]. The motivation of multi-agent distributed cooperative control is to guarantee a group of autonomous agents to complete some specific tasks via the distributed control strategy.
As one of the critical and fundamental research issues arising from distributed cooperative control of MASs, consensus problem, which means to reach an agreement on a common value for a group of agents by developing some distributed controllers based on the relative local information, has received increasing attention. The different investigation scenarios include leaderless consensus [6], [7], [8], [9], leader-following consensus [10], [11], [12], [13], and containment control [14], [15], [16], [17]. The leaderless consensus means to reach agreement state of each agent through interaction and coordination with the evolution of time, and the leader-following consensus means a leader is designated and all the followers track the leader using the consensus protocol. Containment control has been paid much attention as a special kind of consensus with multiple leaders, which aims to design appropriate control protocols to drive the followers to a target area (convex hull formed by the leaders) asymptotically.
In practical applications, communication delays and dynamically networked topologies always emerge in MASs. With this background, the consensus problem of MASs in the presence of jointly-connected topologies with diverse time-delays is investigated in [18], and sufficient conditions for consensus are derived by a contradiction approach. For high-order integral MASs, the consensus problem under switching directed topology is considered in [19], and two distributed protocols are proposed for consensus. For the uncertain nonlinear MASs, the consensus problem with probabilistic time-varying delay is investigated in [20], and several novel delay-dependent sufficient conditions are established by the Lyapunov–Krasovskii approach. For a class of complex dynamical networks, the issue of synchronization with uncertain inner coupling strength is discussed in [21], and two successive time-varying delays based on passivity theory are considered. For leader-following MASs with second-order dynamics, the consensus with communication delays is studied in [22], and some necessary and sufficient conditions are obtained by frequency domain analysis and control theory. For second-order MASs with jointly-connected topologies, the leader-following consensus problems with and without time-varying delays are respectively investigated in [23], and two distributed control algorithms for consensus are proposed.
In containment control problems, by regarding the collection of leaders as a virtual node in [24], containment control of first-order MASs with multiple leaders can be achieved when network topology is connected. For multiple rigid bodies with multiple stationary and dynamic leaders, two types of distributed control protocols are proposed for finite-time attitude containment control in [25]. For MASs with directed network topologies, the distributed containment control with multiple leaders is studied in [26], and both the case with fixed topologies and the case with switching topologies are discussed. For MASs with general linear dynamics, two classes of containment control problems with directed communication topologies are respectively investigated for continuous-time system and discrete-time system in [27]. Considering linear MASs, the distributed containment control algorithms are proposed in [28], and some necessary and sufficient containment conditions are presented by using spectral analysis and matrix theory. For singular swarm systems, high-order linear containment control problems with directed topologies and time delays are investigated in [29]. Considering communication delays, the containment control problems for second-order MASs with time-varying delays are studied in [30], and both the case with multiple dynamic leaders and the case with multiple stationary leaders are discussed. For MASs with time-varying communication topologies, the containment consensus problems with multiple leaders and jointly-connected topologies are studied in [31], and two types of distributed control protocols are proposed for containment with first-order dynamics and second-order dynamics, respectively. Considering uncertainties and communication delays, the containment control problem of first-order MASs with time-varying delays is considered in [32], and a distributed control algorithm is presented for containment for uncertain MASs with jointly-connected topologies.
Two important factors always emerge in some real situations. Firstly, the communication networks of MASs are usually time-varying or unconnected due to malfunctions, reconnection, and packet loss of network nodes or edges cannot be avoided in real systems. The second one is that time delays are usually inevitable because of the possible slow process of interactions among the agents in communication networks. Motivated by these two factors, lots of distributed consensus protocols have been developed by some researchers for MASs with unconnected topologies and delayed communications, so does the consensus tracking. However, differing from most of those current literatures, for the leader-following MASs with unconnected topologies and communication delays, we aim to analyze and investigate the distributed containment control problem with multiple leaders and jointly-connected topologies. For leader-following MASs with first-order dynamics and second-order dynamics, two types of containment control protocols are proposed for containment consensus with time-varying delays and dynamically switching topologies. The stability of MASs can be guaranteed with two proposed control algorithms, and some sufficient conditions for containment control are derived with the aid of Lyapunov–Krasovskii function. Several simulation results are presented to demonstrate the feasibility of the obtained theoretical results.
The rest of this paper is organized as follows. In Section 2, some basic concepts in algebraic graph theory and some related lemmas are presented. In Section 3, main results for leader-following MASs with time-varying delays and jointly-connected topologies are obtained. Then two containment control protocols are presented for first-order MASs and second-order MASs, respectively. In Section 4, to validate our conclusion in this paper, some numerical simulations are given. Finally, conclusions are drawn in Section 5.
Notation., and represent the set of real numbers, n × n dimensional real vector space and n × n real matrix space, respectively. In denotes the identity matrix of dimension n, 0 represents the all-zeros matrix with appropriate dimension, and is a column vector. ΓT represents the transpose of matrix Γ, and γT represents the transpose of vector γ. λmax ( · ) (or (or λmin ( · ))) denotes the maximum (or minimum) eigenvalue of the corresponding matrix. C⊗D represents the Kronecker product of two matrices and .
Section snippets
Preliminaries
For an undirected network of n nodes, its interconnection topology can be modeled as a graph , where represents the set of nodes, and denotes the set of edges, and (wi, wj) ∈ ω⇔(wj, wi) ∈ ω, . Let denote the set of the neighbors of node vi, where (wj, wi) ∈ ω denotes the fact of the information exchange between wi and wj. A path from wi to wj is denoted by , where,
Main results
In this section, the main results are established.
Simulations
Consider a dynamic switching topology with 5 followers and 3 leaders as shown in Fig. 1. Suppose the topology graph of MASs is randomly switched in G1 to G3 at , , where .
From the union topology of graphs G1 ∼ G3, the system matrix Hσ can be obtained. The maximum eigenvalue of Hσ is 4.618 by calculating.
Conclusion
In this paper, distributed containment control for the first-order MASs and the second-order MASs with multiple leaders and unconnected topologies are studied. Two control algorithms with communication delays and jointly-connected topologies are proposed for containment control. By applying modern control theory and algebraic graph theory, convergences of MASs in the presence of dynamically switching topologies with time-varying delays are analyzed on Lyapunov–Krasovskii method. Finally, some
Acknowledgments
This research is supported by the National Natural Science Foundation of China (Grant Nos. 61573200, 61573199, 61673200), and the Tianjin Natural Science Foundation of China (Grant Nos. 14JCYBJC18700, 14JCZDJC39300).
Fuyong Wang received the B.S. degree in Electrical Engineering and Automation and M.S. degree in Computer Application Technology from the Ludong University, Yantai, China, in 2013 and 2016, respectively. He is now pursuing the Ph.D. degree in the College of Computer and Control Engineering, Nankai University, Tianjin, China. His research interest covers coordination of multi-agent systems.
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Fuyong Wang received the B.S. degree in Electrical Engineering and Automation and M.S. degree in Computer Application Technology from the Ludong University, Yantai, China, in 2013 and 2016, respectively. He is now pursuing the Ph.D. degree in the College of Computer and Control Engineering, Nankai University, Tianjin, China. His research interest covers coordination of multi-agent systems.
Hongyong Yang received his Ph.D. degree in Control Theory and Control Engineering from Southeast University in 2005. He is a professor in School of Information and Electrical Engineering, Ludong University. His research interest covers complex network, multi-agent systems, intelligence control.
Zhongxin Liu received the B.S. degree in Automation and Ph.D. degree in Control Theory and Control Engineering from the Nankai University, Tianjin, China, in 1997 and 2002, respectively. He has been at Nankai University, where he is currently a Professor in the Department of Automation. His main areas of research are in predictive control, complex networks and multi-agents system.
Zengqiang Chen received the B.S. degree in Mathematics, M.S. and Ph.D. degrees in Control Theory and Control Engineering from the Nankai University, Tianjin, China, in 1987, 1990 and 1997, respectively. He has been at Nankai University, where he is currently a Professor in the Department of Automation. His main areas of research are in neural network control, complex networks and multi-agents system.