Second-order consensus for multi-agent systems with switching topology and communication delay
事件触发下随机非确定线性多智能体的指数同步
事件触发下随机非确定线性多智能体的指数同步邱丽;过榴晓【摘要】研究随机非确定线性多智能体系统在有向拓扑连接下的指数同步问题,为减少不必要的网络带宽资源的浪费,提出一种基于事件触发控制的协议.根据组合测量对系统中的所有节点设计相应的事件触发函数,使得节点之间的控制信号更新仅在事件触发时刻进行.基于Lyapunov稳定性理论和M矩阵理论,得到了多智能体系统指数同步结论,并给出了同步的收敛速度.同时,理论排除了事件触发控制过程中的芝诺(Zeno)现象.数值仿真结果进一步验证了理论分析的有效性.【期刊名称】《计算机工程与应用》【年(卷),期】2018(054)017【总页数】6页(P141-145,163)【关键词】事件触发控制;随机非确定;线性多智能体系统;指数同步;Zeno现象【作者】邱丽;过榴晓【作者单位】江南大学理学院,江苏无锡 214122;江南大学理学院,江苏无锡214122【正文语种】中文【中图分类】TP2731 引言多智能体系统是由多个能够相互作用、共同协作的个体组成的系统，其中每个个体具有自组织和通讯的能力，各个智能体能够通过彼此之间的信息交换来实现对整个系统的协调控制。
近年来，由于控制理论和应用的发展，多智能体系统已成为控制领域中一个重要的研究对象，其中多智能体系统的同步问题已取得不少成果[1-8]。
如：整体同步[1]，局部同步[2]，聚类同步[4]，指数同步[5-8]等。
指数同步因其在收敛速度方面的优势，成为学者们研究的热点问题之一。
在许多实际的多智能体系统中，智能体自身的能量和通信信道的带宽是有限的，为减少不必要的网络带宽资源的浪费，因此需要设计合适的通信控制方案，节省资源。
周期采样控制方法[9-11]是在等距离的离散时刻点上进行状态采样和信息通讯，有利于节约资源，但如果两个连续采样数据之间相差很小，继续周期采样控制，则明显浪费资源。
与周期采样控制相比，事件触发控制则执行较少的信息通讯，即当事先设定的触发条件不成立，控制器执行更新[12-13]。
非线性多智能体系统的间歇采样控制一致性研究
中文摘要中文摘要由于通信网络以及分布式控制的快速发展，多智能体系统的一致性研究成为系统与控制领域的研究热点，受到了国内外学者的广泛关注。
多智能体系统的一致性是指系统中每个智能体通过通信网络传递信息，使其在位置或者速度等状态量上趋于渐近相同，呈现出行为状态的一致，被广泛的应用于多无人机编队、多卫星角度校正、多传感器网络同步等。
由于现实世界中存在噪声、空气阻力等非线性因素，这些因素常常会给多智能体系统的一致性造成一定的影响，因此考虑带有非线性因素的多智能体系统的一致性具有重要意义。
在一致性控制策略的设计中，采样控制策略能降低控制器的更新频率，但控制器在每个控制时间段内依然要连续工作，而间歇控制策略可以减少控制器的工作时间，因此将采样控制和间歇控制策略相结合有利于统一考虑控制器的更新频率和工作时间。
本文研究基于间歇采样控制策略的非线性多智能体系统一致性问题，具体内容如下：首先，针对带有非线性因素的一阶多智能体系统，分别采用了周期间歇采样控制策略和非周期间歇采样控制策略，通过矩阵理论以及不等式的证明等得到了系统实现一致的充分性条件，从理论上分析证明了所设计的控制策略的可行性。
最后利用数值仿真验证了理论结果的有效性，并通过仿真结果进一步剖析得知，通信宽度和采样宽度对系统状态达到一致起着至关重要的作用。
其次，在以运动学为背景的物理世界中，研究带有非线性因素的二阶多智能体系统更符合实际情况。
并运用间歇采样控制策略，通过严格的理论证明，得到了二阶非线性多智能体系统达到一致的充分条件。
最后利用数值仿真验证了一致性理论的有效性，使得多智能体一致性算法具有更强的实用价值。
最后，为了进一步验证所研究的一致性算法的实用性，基于Anylogic软件仿真平台，搭建了多无人机系统一致性的虚拟原型环境，模拟多智能体之间信息交流，最后通过一致性耦合运算实现了无人机系统的一致性运动，从而验证了一致性理论的可行性。
关键词：多智能体系统；非线性；间歇采样控制；一致性；Anylogic仿真IABSTRACTABSTRACTDue to the rapid development of communication network and distributed control,the research on the consensus of multi-agent systems has become a hot topic in the field of systems and control,which has been widely concerned by scholars at home and abroad.A multi-agent system is a set of systems that work in a network environment and have multiple autonomous individuals.Consensus means that each intelligence in a system transmits information through a communication network to make it asymptotically identical in terms of position or velocity,showing a consensus behavior,and is widely used in multi-UA V formation,multi-satellite angle correction,multi-sensor network synchronization and so on.Because there are nonlinear factors such as noise and air resistance in the real world,these factors often affect the consensus of multi-agent systems, so it is important to consider the consensus of multi-agent systems with nonlinear factors.In the consensus analysis of nonlinear multi-agent systems,the sampled-data control strategy can reduce the update frequency of the controller, but the controller still has to work continuously in each control time period,and the intermittent control strategy can reduce the working time of the controller, Therefore,the combination of sampled-data control and intermittent control strategy is beneficial to consider the update frequency and working time of the controller consensus.This paper studies the consensus problem of nonlinear multi-agent system based on intermittent sampled-data control strategy,the details of the paper are as follows:Firstly,for the first-order multi-agent system with nonlinear factors,the control strategy of periodic intermittent sampled-data and aperiodic intermittent sampled-data are adopted respectively.By means of matrix theory and the proof of inequality,we get the conditions for the system to achieve consensus adequacy.the feasibility of the designed control strategy is proved by theoretical analysis.Finally,numerical simulation is used to verify the validity of theIII非线性多智能体系统的间歇采样控制一致性研究theoretical results,and further analysis of the simulation results shows that the communication width and sampled-data width play a vital role in the system state to achieve consensus.Secondly,in the physical world with kinematics as the background,it is more realistic to study the second-order multi-agent system with nonlinear factors.By using the intermittent sampled-data control strategy,it is proved by strict theory that the consensus condition of the second-order nonlinear multi-agent system.Finally,the validity of the consensus theory is verified by numerical simulation,which makes the multi-agent consensus algorithm more practical.Finally,in order to further verify the practicability of the studied consensus algorithm,based on the above theoretical results,based on the Anylogic software simulation platform,a virtual prototype environment of multi-UA V system consensus is built,and the information exchange between multi-agent is simulated.Finally,the consensus motion of UAV system is realized by consensus coupling operation,which verifies the feasibility of consensus theory. Key words:Multi-agent systems;Nonlinear;Intermittent sampled-data control; Consensus;Anylogic simulationIV目录目录第一章绪论 (1)1.1课题背景及研究意义 (1)1.2多智能体系统一致性简介 (2)1.3一致性问题研究现状及分析 (3)1.4基于间歇采样控制的一致性研究概况 (6)1.4.1基于采样控制的一致性 (6)1.4.2基于间歇控制的一致性 (6)1.5本文研究内容及结构安排 (7)第二章预备知识 (9)2.1基本符号 (9)2.2代数图论 (10)2.3一致性相关理论 (13)2.4本章小结 (14)第三章一阶非线性多智能体系统间歇采样控制的一致性 (15)3.1引言 (15)3.2系统模型的建立及预备知识 (15)3.3周期间歇采样控制策略 (18)3.4非周期间歇采样控制策略 (20)3.5数值仿真 (23)3.6本章小结 (28)第四章二阶非线性多智能体系统间歇采样控制的一致性 (29)4.1引言 (29)4.2系统模型的建立及预备知识 (29)4.3理论分析与证明 (32)4.4数值仿真 (34)4.5本章小结 (39)第五章基于Anylogic的多智能体系统一致性仿真 (41)5.1引言 (41)5.2无人机系统仿真平台创建 (42)V非线性多智能体系统的间歇采样控制一致性研究5.3无人机系统仿真前端设计 (45)5.4无人机系统仿真实验结果 (49)5.5本章小结 (52)第六章总结与展望 (53)6.1全文总结 (53)6.2工作展望 (53)参考文献 (55)致谢 (61)攻读学位期间发表的学术论文目录 (63)VI第一章绪论1第一章绪论1.1课题背景及研究意义洞察自然界，随处可见许多奇妙有趣的现象。
多智能体系统一致性综述
多智能体系统一致性综述一 引言多智能体系统在20世纪80年代后期成为分布式人工智能研究中的主要研究对象。
研究多智能体系统的主要目的就是期望功能相对简单的智能体系统之间进行分布式合作协调控制，最终完成复杂任务。
多智能体系统由于其强健、可靠、高效、可扩展等特性，在科学计算、计算机网络、机器人、制造业、电力系统、交通控制、社会仿真、虚拟现实、计算机游戏、军事等方面广泛应用。
多智能体的分布式协调合作能力是多智能体系统的基础，是发挥多智能体系统优势的关键，也是整个系统智能性的体现。
在多智能体分布式协调合作控制问题中，一致性问题作为智能体之间合作协调控制的基础，具有重要的现实意义和理论价值。
所谓一致性是指随着时间的演化，一个多智能体系统中所有智能体的某一个状态趋于一致。
一致性协议是智能体之间相互作用、传递信息的规则，它描述了每个智能体和其相邻的智能体的信息交互过程。
当一组智能体要合作共同去完成一项任务，合作控制策略的有效性表现在多智能体必须能够应对各种不可预知的形式和突然变化的环境，必须对任务达成一致意见，这就要求智能体系统随着环境的变化能够达到一致。
因此，智能体之间协调合作控制的一个首要条件是多智能体达到一致。
近年来，一致性问题的研究发展迅速，包括生物科学、物理科学、系统与控制科学、计算机科学等各个领域都对一致性问题从不同层面进行了深入分析，研究进展主要集中在群体集、蜂涌、聚集、传感器网络估计等问题。
目前，许多学科的研究人员都开展了多智能体系统的一致性问题的研究，比如多智能体分布式一致性协议、多智能体协作、蜂涌问题、聚集问题等等。
下面，主要对现有文献中多智能体一致性协议进行了总结，并对相关应用进行简单的介绍。
1.1 图论基础多智能体系统是指由多个具有独立自主能力的智能体通过一定的信息传递方式相互作用形成的系统；如果把系统中的每一个智能体看成是一个节点，任意两个节点传递的智能体之间用有向边来连接的话，智能体的拓扑结构就可以用相应的有向图来表示。
异质多智能体系统二分一致性的充要条件
DOI : 10.11992/tis.201901008异质多智能体系统二分一致性的充要条件王晓宇1，刘开恩1，纪志坚2，梁静娴1（1. 青岛大学 数学与统计学院，山东 青岛 266071; 2. 青岛大学 自动化工程学院，山东 青岛 266071）摘 要：针对由一阶智能体和二阶智能体组成的异质多智能体系统的二分一致性问题,对连续和离散系统情形分别设计了二分一致性协议。
基于结构平衡的拓扑,通过规范变换实现了从具有敌对关系的系统到具有非负连接权重系统的转化，将二分一致性问题转变为一般一致性问题。
进一步,运用代数图论和矩阵理论分析闭环控制系统的动态特性,得到了异质多智能体系统渐近实现二分一致性的充要条件。
最后通过数值模拟验证了所得结果的有效性。
关键词：异质多智能体系统；二分一致性；规范变换；结构平衡；连续系统；离散系统；代数图论；矩阵理论中图分类号：TP18 文献标志码：A 文章编号：1673−4785(2020)04−0679−08中文引用格式：王晓宇, 刘开恩, 纪志坚, 等. 异质多智能体系统二分一致性的充要条件[J]. 智能系统学报, 2020, 15(4):679–686.英文引用格式：WANG Xiaoyu, LIU Kaien, JI Zhijian, et al. Necessary and sufficient conditions for bipartite consensus of hetero-geneous multi-agent systems[J]. CAAI transactions on intelligent systems, 2020, 15(4): 679–686.Necessary and sufficient conditions for bipartite consensus ofheterogeneous multi-agent systemsWANG Xiaoyu 1，LIU Kaien 1，JI Zhijian 2，LIANG Jingxian 1(1. School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China; 2. School of Automation Engineering, Qing-dao University, Qingdao 266071, China)Abstract : To investigate the bipartite consensus problem of heterogeneous multi-agent systems composed of first- and second-order agents, in this study, we designed bipartite consensus protocols for continuous and discrete systems. Based on a structurally balanced topology, we employ gauge transformation to transform a system with antagonistic interac-tions into one with non-negative connection weights. Accordingly, the bipartite consensus problem is transformed into a general consensus problem. We use algebraic graph theory and matrix theory to analyze the dynamic characteristics of the closed-loop control system and obtain the necessary and sufficient conditions to guarantee that heterogeneous multi-agent systems reach bipartite consensus asymptotically. Finally, we present numerical simulations to illustrate the effect-iveness of the obtained theoretical results.Keywords : heterogeneous multi-agent systems; bipartite consensus; gauge transformation; structural balance; continu-ous systems; discrete systems; algebraic graph theory; matrix theory多智能体系统的一致性问题作为多智能体系统协作控制的基本问题，因具有重要的理论和现实意义受到国内外研究人员的广泛关注[1-9]。
异构非线性多智能体系统的一致性
Advances in Applied Mathematics 应用数学进展, 2023, 12(9), 3872-3885 Published Online September 2023 in Hans. https:///journal/aam https:///10.12677/aam.2023.129381异构非线性多智能体系统的一致性谢浩浩，李超越，贺 鑫长安大学理学院，陕西 西安收稿日期：2023年8月9日；录用日期：2023年9月3日；发布日期：2023年9月8日摘要针对一阶智能体和二阶智能体组成的异构多智能体系统，在无向通讯拓扑下研究了具有输入饱和与非输入饱和的异构非线性多智能体系统的一致性问题。
首先，分别提出了基于牵制控制和事件触发控制的一致性控制协议，其次，通过对每个智能体设计事件触发条件，当满足事件触发条件时，智能体才向周围的邻居传递自身的状态信息和更新控制器，且每个智能体只在自己的触发时刻进行传递和更新。
然后利用图论、Lyapunov 稳定性理论和LaSalle 不变集理论，证明了在满足某些条件下，该系统不仅达到了期望的一致性状态，而且减少了控制器的更新次数，有效地节省了通讯资源。
最后，通过数值模拟验证了理论的正确性。
关键词异构多智能体系统，牵制控制，事件触发控制，一致性，饱和输入，非线性Consensus of Heterogeneous Nonlinear Multi-Agent SystemsHaohao Xie, Chaoyue Li, Xin HeSchool of Sciences, Chang’an University, Xi’an ShaanxiReceived: Aug. 9th , 2023; accepted: Sep. 3rd , 2023; published: Sep. 8th, 2023AbstractThe consensus problem of heterogeneous nonlinear multi-agent systems with and without input saturation is investigated under the undirected communication topology for heterogeneous mul-ti-agent systems composed of first-order agents and second-order agents. First, consensus control protocols based on pinning control and event-triggered control are proposed respectively, and second, by designing event-triggered conditions for each agent, the agent transmits its own state information and updates its controller to its surrounding neighbors only when the event-triggered谢浩浩等conditions are satisfied, and each agent transmits and updates only at its own triggering moments. Then using graph theory, Lyapunov stability theory and LaSalle invariance principle, it is proved that the systems not only achieve the desired consensus state, but also reduce the number of con-troller updates and effectively save the communication resources under the fulfillment of certain conditions. Finally, the correctness of the theory is verified by numerical simulation. KeywordsHeterogeneous Multi-Agent Systems, Pinning Control, Event-Triggered Control, Consensus, Saturated Inputs, NonlinearThis work is licensed under the Creative Commons Attribution International License (CC BY 4.0)./licenses/by/4.0/1. 引言近年来，多智能体系统的一致性问题引起了学者们的广泛关注，并且在传感器网络[1]、编队控制[2]、群居昆虫的集群[3]、机器人[4]等具有广泛的实际应用价值。
广义多自主体系统的一致性
广义多自主体系统的一致性魏菊梅;支慧敏【摘要】考虑了一类切换拓扑下的广义多自主体系统,利用代数图论和广义系统理论,分两种情形(无领导者和领导者跟随)来研究其一致性.通过研究慢子系统的一致性从而得到了广义多自主体系统的两个一致性.【期刊名称】《郑州大学学报（理学版）》【年(卷),期】2019(051)001【总页数】5页(P29-33)【关键词】广义多自主体系统;切换拓扑;一致性【作者】魏菊梅;支慧敏【作者单位】郑州大学数学与统计学院河南郑州450001;郑州大学数学与统计学院河南郑州450001【正文语种】中文【中图分类】O2310 引言共识问题即一致性问题，是多自主体网络的一个最基本的分布式协调控制问题.在过去几十年里，多自主体的共识问题在许多领域都引起了极大关注[1-5].其中，出现了一类线性切换多自主体系统一致性问题，包括无领导者一致性问题[6]和领导者跟随一致性问题[7-8].广义系统有动态系统的自然表示，比一般的线性系统有更广泛的背景[9-10].文献[11]和[12]分别利用状态反馈和输出反馈来设计控制协议，给出了广义多自主体系统达到一致的充分必要条件.但都只考虑了固定拓扑情形下的一致性问题，对于更一般的动态拓扑(切换拓扑)没有研究，给出的条件虽然是充分必要条件，但是证明过程却是从系统达到一致性的条件出发.本文将文献[8]中的一般线性系统推广到广义系统，并讨论在切换拓扑下的一类广义多自主体系统的一致性问题.与文献[12]相比，这里讨论的拓扑图是动态图, 而且分两种情形(无领导者和领导者跟随)来研究其一致性.通过代数图论[13]和广义系统理论[14]得到结论：要解决广义多自主体系统的两个一致性问题，只需要相应的慢子系统达到一致性.1 预备知识对于给定的向量或矩阵X，‖X‖表示X的欧几里得模.向量1N表示所有元素都是1的列向量，span{X}表示由X的列向量张成的线性子空间，a表示不超过实数a的最大的整数，A1/2表示正定矩阵A的二次方根.⊗代表Kronecker积，满足如下性质：(A⊗B)T=AT⊗BT,(A⊗B)(C⊗D)=(AC)⊗(BD),(A+B)⊗C=A⊗C+B⊗C,A⊗(B+C)=A⊗B+A⊗C.通常，一个多自主体系统中每个自主体之间的信息交换可以通过有向图或无向图来描述[13].2 主要结果本文将考虑式如(1)广义多自主体系统的稳定性，其中：xi∈Rn，ui∈Rm分别是第i个自主体的状态和输入；E, A∈Rn×n，B∈Rn×m是常数矩阵；(E,A)是正则无脉冲的，且rankE=r≤n.对于系统(1)，定义动态图Gσ(t)=(V,εσ(t))，其中V={1,…，N}，且(j,i)∈εσ(t)，当且仅当控制ui在时刻t利用(xj-xi)作为反馈.令Aσ(t)=[aij(t)]N×N是动态图Gσ(t)的邻接权矩阵，则可定义状态反馈拓扑，i=1,…,N，(2)这里K∈Rm×n是增益矩阵.定义1[8] 无领导者一致性问题. 给定系统(1)和一个动态图Gσ(t)，找到一个状态反馈拓扑(2)的反馈增益矩阵K，使得对于i,j=1,…,N，当t→∞时，xi(t)-xj(t)→0.对于以上描述的无领导者的一致性问题，每一个子系统解的稳态行为是无足轻重的.还有一个一致性问题称为领导者跟随一致性问题，而这个问题就要求每一个子系统的解都要渐近趋近于信号x0(t).假设信号x0(t)由线性系统(3)产生，其中x0∈Rn.下面分别称系统(1)和系统(3)为跟随者系统和领导者系统.联合系统(1)和系统(3)，定义另外一个动态图,,这里{0,1,…,N}. 显然G是的子图，因为可以从图移去中的结点0，和在t时刻中所有属于结点0的边得到.令Δσ(t)是一个N×N的非负对角矩阵，其中第i个对角元是ai0(t)，这里如果(0,i)∈，则ai0(t)>0，否则ai0(t)=0.考虑状态反馈拓扑，i=1,…,N.(4)定义2[8] 领导者跟随一致性问题.给定领导者系统(3)、跟随者系统(1)和一个动态图，找到一个状态反馈拓扑(4)的反馈增益矩阵K，使得对于i=1,…,N，当t→∞时，xi(t)-x0(t)→0.为了解决以上两个一致性问题,我们将系统(1)进行正则性分解.由于(E,A)是正则无脉冲的，由文献[14]可知，存在可逆矩阵P,Q∈Rn×n，使得，，，(5)进行坐标变换,，i=1,…,N，(6)其中：∈Rr,∈Rn-r.由式(5)得到无领导者系统(1)等价于，(7)，i=1,…,N，(8)而领导者系统(3)等价于，(9).(10)这种分解通常称为快、慢子系统分解，式(7)和(9)为慢子系统，式(8)和(10)为快子系统.通过这种分解，证明无领导者系统和领导者跟随系统的一致性可分别由相应的慢子系统的一致性来得到.定理1 如果无领导者慢子系统(7)达到一致性,则无领导者系统(1)也达到一致性.证明假设存在一个增益矩阵K1∈Rm×r，使得，i=1,…,N，(11)解决无领导者系统(7)的一致性问题，由定义1，我们有.则.由式(8)得到，i=1,…,N.由变换(6)，.由式(2)可得，系统(1)的一致性问题也得以解决.定理2 如果无领导者慢子系统(7)和领导者慢子系统(9)的领导者跟随一致性问题得以解决，则系统(3)和系统(1)的领导者跟随一致性问题也得以解决.证明假设存在一个增益矩阵K1∈Rm×r，使得，i=1,…,N，解决无领导者慢子系统(7)和领导者慢子系统(9)的领导者跟随一致性问题，由定义2，有.则.由式(8)得到，i=0,1,…,N.由变换(6)得.取，所以系统(3)和(1)的领导者跟随一致性问题也得以解决.要解决无领导者系统(1)的一致性问题，只需讨论无领导者慢子系统(7)的一致性问题，而系统(7)是一个一般的线性多自主体系统，研究其一致性问题就简单得多.这里考虑关于慢子系统(7)的线性切换系统⊗A1-Fσ(t)⊗，σ(t)∈P，(12)其中：X是正定矩阵；∈Rr;A1、B1 如式(5)中定义一样；IN∈RN×N是单位矩阵；σ(t):[0,+∞)→P={1,2,…,ρ},ρ≥1，是右连续的分段常值的切换信号，切换瞬时{ti:i=0,1,…}满足对任意i≥1和正常数τ，都有ti-ti-1≥τ，且对所有的t≥0；Fσ(t)∈RN×N是半正定的矩阵.假设1 (A1,B1)能控, 令X是一个正定矩阵，满足不等式.(13)假设2 动态图Gσ(t)是无向图，∀t≥0.假设3 存在{i:i=0,1,…}的子序列{ik}，tik+1-tik<v,v>0，使得连接图G([tik,tik+1))是连通的.在假设2下，图Gσ(t)的拉普拉斯矩阵Lσ(t)是对称半正定的，∀t≥0.若一个动态图满足假设3，就称图在[0,∞)上是一致连通的，或者称在[tik,tik+1)上是共连通的. 注意到是一个行和为零的Metzler矩阵，关于的图是连接图,tik+1)).因此，在假设2下，矩阵是半正定的.在假设3下，矩阵恰好有一个零特征值，且零空间是span{1N}.引理1[8] 考虑系统(12)，在假设1下，X是满足式(13)的正定矩阵.σ(t)是驻留时间为τ的分段常值切换信号，对任意的t≥0，Fσ(t)是对称半正定的矩阵，则1) 如果系统(12)的解满足性质：如果存在{i:i=0,1,…}的一个子序列ik，tik+1-tik<v,v>0，使得与矩阵⊗In的零空间正交，则.2) 如果存在{i:i=0,1,…}的一个子序列ik，tik+1-tik<v,v>0，使得矩阵是非奇异的，则系统(12)的原点是渐近稳定的.定理3 如果假设1～3成立，则系统(1)在控制协议(2)下达到一致性.证明设式(11)的增益矩阵为，系统(7)的第i个自主体的闭环系统为.(14)令xc(t)=(x1(t)+x2(t)+…+xN(t))/N，xc(t)称在t时刻所有自主体的中心.图是无向图，得//N)=A1xc(t)，(15)将分解为ωi(t)，i=1,…,N，(16)则⊗xc(t)+ω(t)，(17)其中，且ω(t)=[ω1(t)T,ω2(t)T,…,ωN(t)T]T.由式(14)、(15)和(17)得向量ω(t)满足⊗A1-Lσ(t)⊗ω(t)，具有与系统(12)同样的形式，只不过这里Fσ(t)=Lσ(t).因为ω，对任意的t≥0，ω(t)与span{1N⊗In}正交.由定理2可知，的零空间是span{1N}，所以⊗In的零空间是span{1N⊗In}.因此，ω(tik)与⊗In的零空间正交.那么由引理1得ω(t)=0.从(16)式可知所有的状态都渐近趋近于xc(t)，则有.即系统(7)在控制协议(11)下达到一致性，根据定理1，取，即系统(1)在控制协议(2)下也达到一致性.3 仿真结果例1 考虑无领导者广义多自主体系统(1)，N=4，系统矩阵为，，.按(5)式将系统矩阵进行正则性分解，得到，，，其中：切换动态图Gσ(t)，由分段常值的切换信号定义σ其中：s=0,1,2,…；切换拓扑图Gi=(Vi,εi,Ai)，i=1,2,3,4，ε1={(1,2),(2,1)}，ε2={(2,3),(3,2)}，ε3={(3,4),(4,3)}，ε4={(1,4),(4,1)}.没有一个图Gi是连通的，但是对于任意的s=0,，Gi是连通的.因此假设2和假设3成立，可以证明假设1也成立.解不等式，得到正定解×10-11，考虑慢子系统的线性切换(12)，由定理1，控制增益为×10-12，因此×10-12.最后得到广义系统的状态与中心的误差趋近于零，即系统最终达到了一致性.4 结论本文讨论了一类具有切换拓扑的广义多自主体系统，利用代数拓扑和广义系统理论知识，将广义多自主体系统进行快慢子系统分解，通过研究其慢子系统的性质，即设计了状态反馈控制协议，得到慢子系统的一致性，从而得到广义多自主体系统的一致性问题，包括无领导者一致性问题和领导者跟随一致性问题.参考文献：【相关文献】[1] LIU Q, JIANG D Q, SHI N Z, et al. 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多智能体系统自适应跟踪控制
多智能体系统自适应跟踪控制赵蕊;朱美玲;徐勇【摘要】The leader-follower tracking problems of second-order multi-agent systems with intrinsic nonlinear dynamics are studied. It assumes that each following agent can access the relative position and velocity information with its neigh-bors, the position and velocity information of the leader is only accessed by a subset of the following agents, and the leader's non-zero reference input cannot be available by any following agents. To track the active leader, a distributed adaptive consensus protocol is proposed for each following agent in the case that the interaction relationship among agents is undi-rected connected graph. The protocol effectively avoids the uncertainty of global information. The consensus tracking problem can be transformed into the stability problem of error system. Based on the theory of Lyapunov stability and matrix theory, it gets the sufficient conditions which guarantee the system to reach a leader-follower tracking consensus. Finally, a simulation example is given to verify the effectiveness of the obtained.%基于带有非线性动态的二阶多智能体系统,研究了在有动态领导者条件下的跟踪一致性问题.假设跟随者只能获取邻居智能体的相对状态信息,只有一部分跟随者可以获得领导者的位置和速度信息,领导者的控制输入非零且不被任何一个跟随者可知.在通信拓扑为无向连通图的条件下,为了避免全局信息的不确定性,设计了分布式自适应控制协议.将系统的一致性问题转化为误差系统的一致性问题,通过Lyapunov稳定性理论和矩阵理论分析得到了该协议使系统达到一致的充分条件.最后用仿真例子证明了设计方法的有效性.【期刊名称】《计算机工程与应用》【年(卷),期】2017(053)018【总页数】5页(P39-43)【关键词】多智能体系统;一致性;分布式控制;自适应控制;领导者【作者】赵蕊;朱美玲;徐勇【作者单位】河北工业大学理学院,天津 300401;河北工业大学理学院,天津300401;河北工业大学理学院,天津 300401【正文语种】中文【中图分类】TP13一致性，是智能体组成的网络系统的一类集体行为，近年来由于它广泛应用在生物系统、传感器网络、无人机编队控制等领域，引起了许多学者的关注，得到了大量研究成果[1-5]。
非参数不确定多智能体系统一致性误差跟踪学习控制
非参数不确定多智能体系统一致性误差跟踪学习控制严求真;孙明轩;李鹤【摘要】This paper presents a consensus-error-tracking iterative learning control method to tackle the consensus problem for a class of leader-following non-parametric uncertain multi-agent systems, which perform a given repetitive task over a finite interval with arbitrary initial error. The iterative learning controllers are designed by applying Lyapunov synthesis. As the iteration increases, each following multi-agent’s consensus-error can track its desired consensus-error trajectory, and the all following multi-agents’ states perfectly track the leader’s state on the specified interval. The robust learning technique is applied to deal with the nonparametric uncertainties, and the hyperbolic tangent function is used to design feedback terms, in order to compensate the cycle-varying but bounded uncertainty. Numerical results demonstrate the effectiveness of the learning control scheme.%针对一类在有限时间区间上执行重复任务的主−从型非参数不确定多智能体系统，提出一致性误差跟踪学习控制方法，用于解决在任意初始误差情形下的一致性问题。
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Contents lists available at ScienceDirect
Systems & Control Letters
journal homepage: /locate/sysconle
1. Introduction During the past years, distributed coordination for networks of dynamical agents has been an active area of research due to the broad applications of multi-agent systems in such areas as sensor networks, rendezvous tasks, swarming and flocking models, consensus problem, and congestion control in communication networks. One of the critical problems is how to design a network ‘‘consensus algorithm’’ (or protocol) such that the group of agents can reach consensus on the shared information in the presence of limited or unreliable information exchanges and dynamically changing network topologies. The consensus problem for agents with first-order dynamics has recently been investigated from various perspectives [1–13]. Taking into account the fact that second-order dynamics can be used to model more complicated processes in reality, cooperative control for second-order agents have been studied in
✩ This work was partially supported by National Natural Science Foundation of China (60825303, 60834003, 61028008), the 973 Project of China (2009CB320600), the Foundation for the Author of National Excellent Doctoral Dissertation of China (2007B4), the Key Laboratory of Integrated Automation for the Process Industry Northeastern University, Ministry of Education, China, and by a research grant from the Australian Research Council. ∗ Corresponding author. Fax: +61 2 4736 0867. E-mail addresses: qinjiahu301@ (J. Qin), hjgao@ (H. Gao), w.zheng@.au (W.X. Zheng).
Article history: Received 28 July 2010 Received in revised form 25 January 2011 Accepted 17 March 2011 Available online 17 April 2011 Keywords: Consensus Second-order agents Graphic method Switching topology Time delay
Second-order consensus for multi-agent systems with switching topology and communication delay✩
Jiahu Qin a,b , Huijun Gao a , Wei Xing Zheng b,∗
a b
Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001, China School of Computing and Mathematics, University of Western Sydney, Penrith NSW 2751, Australia
article
info
abstract
In this paper, two kinds of consensus problems for second-order agents under directed and arbitrarily switching topologies are investigated, that is, the cases without and with communication delay. For the former, by constructing a new kind of digraph and employing a new graphic method, we can specify the least convergence rate for all the agents to reach consensus. For the latter, in virtue of a matrix inequality method, a sufficient condition in the form of feasible matrix inequalities is presented for all the agents to reach consensus. This, on the other hand, shows that consensus can be reached if the delay is small enough. Finally, two numerical examples are given to demonstrate the effectiveness and advantages of the proposed results. © 2011 Elsevier B.V. All rights reserved.
[14–23]. In particular, extensions of first-orde second-order case with undirected network topology are discussed in [21], while [19] proposes and analyzes a different consensus algorithm for second-order agents under directed and fixed network topology under the assumption that the velocity information is measurable by the neighboring agents. A generalized consensus algorithm of those considered in [19,21] is investigated in [24] under directed and fixed topology. [18] considers various consensus algorithms for second-order dynamics by using different methods also under fixed topology. It is worth mentioning that the case with directed network topology is much more challenging than that of the undirected one due to the fact that the Laplacian matrix of a digraph is usually nonsymmetric. On the other hand, it is well known that communication delay and switching of the network topology are the key factors which may destabilize the multi-agent systems. So far, little attention has been paid to the consensus as well as the related convergence rate analysis for second-order agents under directed and switching topology and with communication delay. With this background, this paper investigates two kinds of second-order consensus problems for agents under directed and switching topology. For the case without communication delay, we perform the consensus analysis by employing a new graph theoretical method. This is different from most of the existing literature which heavily relies on analyzing the eigenvalues of the transformed system matrix. Moreover, the least convergence rate for all the agents to reach consensus can also be specified, which, to the best of our knowledge, has not been investigated