Brief paperTime-varying formation control for general linear multi-agent systems with switching directed topologies☆
Introduction
Cooperative control of multi-agent systems has received significant attention from both scientific and engineering communities in recent years. This research field includes consensus control (Ren and Beard, 2005, Zhou and Lin, 2014), rendezvous control (Dong and Huang, 2014, Zavlanos et al., 2009), containment control (Ji et al., 2008, Notarstefano et al., 2011) and formation control (Navaravong et al., 2012, Wang et al., 2014), etc. As one of the most important research topics, formation control of multi-agent systems has broad range of applications in various areas, such as unmanned aerial vehicles (Dong et al., 2015, Karimoddini et al., 2013), mobile robots (Yoo and Kim, 2015, Zheng et al., 2015), autonomous underwater vehicles (Leonard et al., 2010, Wang et al., 2012). As a matter of fact, formation control problems have been studied a lot in robotics community during the past decades, and three formation control approaches, namely, leader–follower based approach (Das, Fierro, Kumar, & Ostrowski, 2002), behavior based approach (Balch & Arkin, 1998) and virtual structure based approach (Lewis & Tan, 1997), have been proposed.
One of the main challenges in formation control of multi-agent systems lies in the fact that each agent usually cannot rely on centralized coordination and has to use local information to achieve the desired formation (see the latest survey paper (Oh, Park, & Ahn, 2015) for more details).Ren (2007) proposed a consensus based formation control approaches for second-order multi-agent systems and proved that leader–follower, behavior and virtual structure based approaches can be unified in the framework of consensus based approaches. Consensus or graph based formation control problems for first-order and second-order multi-agent systems were studied in Antonelli, Arrichiello, Caccavale, and Marino (2014), Du, Li, and Lin (2013), Guo, Zavlanos, and Dimarogonas (2014), Guzey, Dierks, and Jagannathan (2015), Hawwary (2015), Liu and Jiang (2013), Lin, Wang, Han, and Fu (2015), Mylvaganam and Astolfi (2015), Tian and Wang (2013), Xiao, Wang, Chen, and Gao (2009) and Wang, Xie, and Cao (2014). In some practical applications, the dynamics of each agent can only be described by high-order model. Results on time-invariant formation control of high-order linear multi-agent systems can be found in Fax and Murray (2004), Lafferriere, Williams, Caughman, and Veerman (2005), Ma and Zhang (2012) and Porfiri, Roberson, and Stilwell (2007). Time-varying formation control problems for high-order linear multi-agent systems with fixed and switching undirected interaction topologies were addressed in Dong, Shi, Lu, and Zhong (2014) and Dong, Xi, Lu, and Zhong (2014), respectively. In practice, undirected interaction topologies mean that the communication among agents is bidirected, which may consume twice the communication and energy resources used in the directed links. Directed interaction among agents is more practical. Therefore, it is meaningful to study time-varying formation control problems for general linear multi-agent systems with switching directed interaction topologies. Moreover, the Laplacian matrix for undirected interaction topology is symmetric while the Laplacian matrix for directed interaction topology does not have the symmetric structure, and the eigenvalues of the Laplacian matrix can be complex, which makes the analysis and design much complicated.
Motivated by the facts and challenges stated above, in this paper, time-varying formation analysis and design problems for multi-agent systems with general linear dynamics and switching directed interaction topologies are investigated. Compared with previous results on formation control, the contributions of the current paper are threefold. Firstly, the formation can be time-varying and each agent has general linear dynamics. In Du et al. (2013), Fax and Murray (2004), Lafferriere et al. (2005), Liu and Jiang (2013)Ma and Zhang (2012), Porfiri et al. (2007), Ren (2007), Tian and Wang (2013) and Xiao et al. (2009), the formation is assumed to be time-invariant. Because the time-varying formation will bring the derivative of the formation information to both the analysis and design, the results for time-invariant formations cannot be directly applied to time-varying formations. In Antonelli et al. (2014), Hawwary (2015) and Wang et al. (2014), the formation can be time-varying, but the dynamics of each agent is first-order. Secondly, the interaction topology can be switching and directed, and each possible topology only needs to have a spanning tree. However, the topologies in Dong et al. (2014) is fixed. Although the topologies in Dong et al. (2014) can be switching, each topology is required to be undirected and connected. The common Lyapunov functional approach used in Dong et al. (2014) cannot be applied to solve the switching directed topology problems in the current paper. Thirdly, a description of the feasible time-varying formation set and an explicit expression of the time-varying formation reference function are derived. It is revealed that switching topologies, dynamics of each agent, initial states of all the agents and the time-varying formation have effects on the macroscopic movement of the whole formation.
Section snippets
Preliminaries and problem description
In this section, firstly, basic notations, definitions and useful results on graph theory are introduced. Then the problem description is presented.
Time-varying formation analysis
In this section, firstly, necessary and sufficient conditions for multi-agent system (3) with switching directed interaction topologies to achieve time-varying formation specified by are presented. Then an explicit expression of the formation reference function is given to describe the macroscopic movement of the whole formation.
Let and . Then multi-agent system (3) with switching directed interaction topologies can be rewritten as
Time-varying formation protocol design
In this section, firstly an algorithm to design the time-varying formation protocol (2) is proposed. Then it is proven that using the algorithm, time-varying formation can be achieved by multi-agent system (3) with switching directed topologies if the formation feasibility condition is satisfied and the dwell time is larger than a positive threshold.
Since the interaction topology has a spanning tree, from Lemma 1 and the structure of , one knows that the real parts of all the eigenvalues
Numerical simulations
In this section, a numerical example is given to illustrate the effectiveness of theoretical results obtained in the previous sections.
Consider a third-order multi-agent system with six agents, where the dynamics of each agent is described by (1) with and .Suppose that there are three different 0–1 weighted directed topologies, namely, , and as shown in Fig. 1. These six agents are required to keep a periodic time-varying
Conclusions
Time-varying formation control problems for general linear multi-agent systems with switching directed interaction topologies were studied. Necessary and sufficient conditions for general linear multi-agent systems with switching directed topologies to achieve time-varying formations were presented. A description of the feasible time-varying formation set and an explicit expression of the time-varying formation reference function were proposed. Approaches to expand the feasible formation set
Xiwang Dong received his B.E. degree in Automation from Chongqing University, Chongqing, China, in 2009, and Ph.D. degree in Control Science and Engineering from Tsinghua University, Beijing, China, in 2014. From December 2014 to December 2015, he was a Research Fellow in the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He is now a Lecturer in the School of Automation Science and Electrical Engineering, Beihang University, Beijing, China. His
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Xiwang Dong received his B.E. degree in Automation from Chongqing University, Chongqing, China, in 2009, and Ph.D. degree in Control Science and Engineering from Tsinghua University, Beijing, China, in 2014. From December 2014 to December 2015, he was a Research Fellow in the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He is now a Lecturer in the School of Automation Science and Electrical Engineering, Beihang University, Beijing, China. His research interests include consensus control, formation control and containment control of multi-agent systems. Dr. Dong is the recipient of the Academic Rookie Award of the Department of Automation, Tsinghua University in 2014, Outstanding Doctoral Dissertation Award of the Tsinghua University in 2014 and Springer Theses Award in 2015.
Guoqiang Hu received the B.Eng. degree in Automation from the University of Science and Technology of China, Hefei, China, in 2002, the M.Phil. degree in Automation and Computer-Aided Engineering from the Chinese University of Hong Kong, Hong Kong, in 2004, and the Ph.D. degree in Mechanical Engineering from the University of Florida, Gainesville, FL, USA, in 2007. He is currently with the School of Electrical and Electronic Engineering at Nanyang Technological University, Singapore. Prior to his current position, he was a Post-doctoral Research Associate at University of Florida, Gainesville, FL, USA, in 2008, and an Assistant Professor at Kansas State University, Manhattan KS, USA, from 2008 to 2011. His current research interests include analysis, control and design of distributed intelligent systems.
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This work was supported by Singapore MOE AcRF Tier 1 Grant RG60/12(2012-T1-001-158). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Dimos V. Dimarogonas under the direction of Editor Christos G. Cassandras.