第15卷第4期 智能系统学报 Vol.15 No.4 2020年7月 CAAI Transactions on Intelligent Systems Jul.2020 D0:10.11992tis.201908027 含时延约束的多智能体系统二分一致性 冀秀坤，谢广明2，文家燕'，罗文广 (1.广西科技大学电气与信息工程学院，广西柳州5450062.北京大学工学院，北京100871) 摘要：针对多智能体系统中信息交互存在通信时延这一约束，在无向符号图拓扑结构下分别研究了含固定时 延和时变时延的一阶多智能体系统二分一致性问题。通过设计相应的控制协议，使得该系统收敛到两个模值 相同但符号不同的状态。在稳定性分析中，利用广义Nyquist准则的方法，得到含固定时延多智能体系统实现 二分一致性的充分条件；对含时变时延系统构造包含三重积分项的Lyapunov函数，利用积分不等式和线性矩 阵不等式理论，并结合自由矩阵的方法得到含时变时延多智能体系统实现二分一致性的充分条件。最后，数值 仿真验证了所得结论的有效性和正确性。 关键词：多智能体系统：无向图；二分一致性：固定时延：时变时延；yquist准则：线性矩阵不等式；自由矩阵 中图分类号：TP273文献标志码：A文章编号：1673-4785(2020)04-0780-07 中文引用格式：冀秀坤，谢广明，文家燕，等.含时延约束的多智能体系统二分一致性{J引.智能系统学报，2020,15(4)： 780-786. 英文引用格式：JIXiukun,XIE Guangming,WEN Jiayan,.ctal.Bipartite consensus for multi-agent systems subject to time delays J CAAI transactions on intelligent systems,2020,15(4):780-786. Bipartite consensus for multi-agent systems subject to time delays JI Xiukun',XIE Guangming,WEN Jiayan',LUO Wenguang (1.School of Electrical and Information Engineering,Guangxi University of Science and Technology,Liuzhou 545006,China; 2.College of Engineering,Peking University,Beijing 100871,China) Abstract:The aim of this paper is to consider the practical constraint of interaction network associated with multi-agent systems (MASs)subject to communication delay,the bipartite consensus seeking is investigated for the first-order MASs with fixed time-delays and time-varying delays over an undirected signed graph,respectively.To solve the con- cerned problems,the corresponding algorithms oriented for fixed time-delays or time-varying counterpart are proposed. In other words,the consensus is achieved within each subgroup during evolution,in which the state of each subgroup will converge to the same modulus but different symbol.Sufficient condition of bipartite consensus for the interaction network of MASs suffered from fixed time-delays can be obtained by the use of the generalized Nyquist criterion;Fur- thermore,based on the tools of integral inequality and linear matrix inequality,the sufficient condition for the case of bi- partite consensus for the interaction network of MASs with time-varying delays also can be derived through construct- ing a reasonable Lyapunov function with triple integral term.Numerical simulations are provided that demonstrate the effectiveness of our theoretical results. Keywords:multi-agent systems;undirected graph;bipartite consensus,fixed-time delays,time-varying delays,Nyquist criterion;linear matrix inequality;free matrix 收稿日期：2019-08-23. 自然界里存在着许多生物聚集的现象，比如 基金项目：国家自然科学基金项目(61963006,61973007 61633002,61563006):广西省自然科学基金项目 候鸟迁徙、鱼群逆流、蚁群迁徙等，通过对自然界 (2018 GXNSFAA050029.2018 GXNSFAA294085):广酉 自动检测技术与仪器重点实验室基金项目 各种群集现象的研究以及对现有的网络模型结构 (YQ20208). 通信作者：文家燕.E-mail:wenjiayan20I5@pku.edu.cn. 的理解，学者们提出了多智能体系统(multi-agentDOI: 10.11992/tis.201908027 含时延约束的多智能体系统二分一致性 冀秀坤1 ，谢广明1,2，文家燕1 ，罗文广1 （1. 广西科技大学 电气与信息工程学院，广西 柳州 545006; 2. 北京大学 工学院，北京 100871） 摘 要：针对多智能体系统中信息交互存在通信时延这一约束，在无向符号图拓扑结构下分别研究了含固定时 延和时变时延的一阶多智能体系统二分一致性问题。通过设计相应的控制协议，使得该系统收敛到两个模值 相同但符号不同的状态。在稳定性分析中，利用广义 Nyquist 准则的方法，得到含固定时延多智能体系统实现 二分一致性的充分条件；对含时变时延系统构造包含三重积分项的 Lyapunov 函数，利用积分不等式和线性矩 阵不等式理论，并结合自由矩阵的方法得到含时变时延多智能体系统实现二分一致性的充分条件。最后，数值 仿真验证了所得结论的有效性和正确性。 关键词：多智能体系统；无向图；二分一致性；固定时延；时变时延；Nyquist 准则；线性矩阵不等式；自由矩阵 中图分类号：TP273 文献标志码：A 文章编号：1673−4785(2020)04−0780−07 中文引用格式：冀秀坤, 谢广明, 文家燕, 等. 含时延约束的多智能体系统二分一致性 [J]. 智能系统学报, 2020, 15(4): 780–786. 英文引用格式：JI Xiukun, XIE Guangming, WEN Jiayan, et al. Bipartite consensus for multi-agent systems subject to time delays[J]. CAAI transactions on intelligent systems, 2020, 15(4): 780–786. Bipartite consensus for multi-agent systems subject to time delays JI Xiukun1 ，XIE Guangming1,2 ，WEN Jiayan1 ，LUO Wenguang1 (1. School of Electrical and Information Engineering, Guangxi University of Science and Technology, Liuzhou 545006, China; 2. College of Engineering, Peking University, Beijing 100871, China) Abstract: The aim of this paper is to consider the practical constraint of interaction network associated with multi-agent systems (MASs) subject to communication delay, the bipartite consensus seeking is investigated for the first-order MASs with fixed time-delays and time-varying delays over an undirected signed graph, respectively. To solve the con￾cerned problems, the corresponding algorithms oriented for fixed time-delays or time-varying counterpart are proposed. In other words, the consensus is achieved within each subgroup during evolution, in which the state of each subgroup will converge to the same modulus but different symbol. Sufficient condition of bipartite consensus for the interaction network of MASs suffered from fixed time-delays can be obtained by the use of the generalized Nyquist criterion; Fur￾thermore, based on the tools of integral inequality and linear matrix inequality, the sufficient condition for the case of bi￾partite consensus for the interaction network of MASs with time-varying delays also can be derived through construct￾ing a reasonable Lyapunov function with triple integral term. Numerical simulations are provided that demonstrate the effectiveness of our theoretical results. Keywords: multi-agent systems; undirected graph; bipartite consensus; fixed-time delays; time-varying delays; Nyquist criterion; linear matrix inequality; free matrix 自然界里存在着许多生物聚集的现象，比如 候鸟迁徙、鱼群逆流、蚁群迁徙等，通过对自然界 各种群集现象的研究以及对现有的网络模型结构 的理解，学者们提出了多智能体系统 (multi-agent 收稿日期：2019−08−23. 基金项目：国家自然科学基金项 目 (61963006， 61973007， 61633002， 61563006);广西省自然科学基金项目 (2018GXNSFAA050029，2018GXNSFAA294085);广西 自动检测技术与仪器重点实验室基金项 目 (YQ20208). 通信作者：文家燕. E-mail：wenjiayan2015@pku.edu.cn. 第 15 卷第 4 期 智 能 系 统 学 报 Vol.15 No.4 2020 年 7 月 CAAI Transactions on Intelligent Systems Jul. 2020