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作者简介:

唐朝君,男,博士,副教授,主要从事矩阵理论与多智能体系统协调控制方面的研究.zhaojuntang@163.com

中图分类号:TP273

文献标识码:A

DOI:10.13878/j.cnki.jnuist.2022.02.013

参考文献 1
Olfati-Saber R,Murray R M.Consensus problems in networks of agents with switching topology and time-delays[J].IEEE Transactions on Automatic Control,2004,49(9):1520-1533
参考文献 2
唐朝君.基于自适应控制的非线性多智能体系统一致性[J].重庆理工大学学报(自然科学),2019,33(11):137-142;TANG Zhaojun.Adaptive consensus for nonlinear multi-agent systems[J].Journal of Chongqing University of Technology(Natural Science),2019,33(11):137-142
参考文献 3
陈军统,徐振华,项秉铜,等.具有控制器增益随机不确定性的多智能体一致性控制[J].南京信息工程大学学报(自然科学版),2018,10(2):173-177;CHEN Juntong,XU Zhenhua,XIANG Bingtong,et al.Leader-follower consensus of multi-agent systems with stochastic uncertainty of controller gain[J].Journal of Nanjing University of Information Science & Technology(Natural Science Edition),2018,10(2):173-177
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参考文献 9
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参考文献 10
Wang H,Yu W W,Wen G H,et al.Fixed-time consensus of nonlinear multi-agent systems with general directed topologies[J].IEEE Transactions on Circuits and Systems II:Express Briefs,2019,66(9):1587-1591
参考文献 11
Ji M,Ferrari-Trecate G,Egerstedt M,et al.Containment control in mobile networks[J].IEEE Transactions on Automatic Control,2008,53(8):1972-1975
参考文献 12
Cao Y C,Stuart D,Ren W,et al.Distributed containment control for multiple autonomous vehicles with double-integrator dynamics:algorithms and experiments[J].IEEE Transactions on Control Systems Technology,2011,19(4):929-938
参考文献 13
Wang X Y,Li S H,Shi P.Distributed finite-time containment control for double-integrator multiagent systems[J].IEEE Transactions on Cybernetics,2014,44(9):1518-1528
参考文献 14
贺清碧,唐朝君,黄大荣,等.有向切换网络拓扑下非线性多智能体系统的包含控制[J].科学技术与工程,2017,17(16):254-258;HE Qingbi,TANG Zhaojun,HUANG Darong,et al.Containment control of nonlinear multi-agent systems with directed switching topologies[J].Science Technology and Engineering,2017,17(16):254-258
参考文献 15
Godsil C,Royle G.Algebraic graph theory[M].New York,NY:Springer New York,2001
参考文献 16
Goldberg M.Equivalence constants for lp norms of matrices[J].Linear and Multilinear Algebra,1987,21(2):173-179
参考文献 17
Meng Z Y,Ren W,You Z.Distributed finite-time attitude containment control for multiple rigid bodies[J].Automatica,2010,46(12):2092-2099
参考文献 18
Wen G H,Hu G Q,Yu W W,et al.Consensus tracking for higher-order multi-agent systems with switching directed topologies and occasionally missing control inputs[J].Systems & Control Letters,2013,62(12):1151-1158
目录contents

    摘要

    针对具有本质非线性动态的多智能体系统,研究固定时间的包含控制问题.假设只有部分跟随智能体能够直接获取领导智能体的状态信息,而且跟随智能体的通信拓扑是有向的,设计分布式的控制协议来解决固定时间的包含控制问题.利用代数图论、矩阵理论和固定时间稳定性理论,得到了系统解决包含控制问题的拓扑条件.最后仿真实例验证了理论的正确性.

    Abstract

    The fixed-time containment control is investigated for multi-agent systems with inherent nonlinear dynamics.Assuming that not all followers can directly receive information from the leaders and the communication topology between the followers is directed,a distributed control law is designed to solve the fixed-time containment control problem.By using algebraic graph theory,matrix theory and fixed-time stability theory,the conditions on the communication topology are derived for realization of fixed-time containment control.Finally,a simulation example is given to verify the correctness of the theory.

  • 0 引言

  • 近年来,多智能体系统一致性问题的研究吸引了来自计算机应用、自动控制、数学等领域研究学者的关注[1-4].其中系统中含有多个领导智能体的跟踪问题也被称为包含控制问题,包含控制在军事和民用领域都有着广泛的应用.在研究多智能体系统的有限时间一致性问题中,文献[5]构建了一个有效的分布式协议框架,在此协议下解决了有限时间的一致性问题,文献[6]通过二重积分器设计了用于实现在干扰下有限时间一致性问题的分布协议,文献[7]研究了在外部干扰下的非线性多智能体系统,通过利用滑模控制技术解决有限时间一致性的跟踪控制问题.然而上述文献的收敛时间是与初始状态息息相关的,因此,初始状态无法影响收敛时间的研究成为一个重要课题.文献[8]通过研究固定时间稳定性,从而提出了固定时间一致性问题,并且保证了收敛时间是与初始状态没有关系的.对于带有外部干扰的多智能体系统,文献[9]通过设计非线性协议解决了固定时间一致性问题.对于非线性的多智能体系统,并且该通信拓扑是有向的,其固定时间一致性问题在文献[10]中被解决.在多智能体系统包含控制问题中,文献[11]提出了固定无向网络拓扑下的包含控制问题,文献[12]研究了多智能体系统分别在静态和动态领导下的包含控制问题,文献[13]通过齐次控制技术设计了用于实现有限时间包含控制问题的控制协议,文献[14]针对有向切换网络拓扑研究其包含控制问题.但是在已有文献中,对固定时间包含控制的研究结果还比较少.特别是对于有向网络拓扑下固定时间包含控制的研究,相关的研究结果更少.

  • 基于上述原因,本文在文献[10]的基础上,通过设计更加一般的非线性系统和控制协议,研究多个领导的包含控制问题.本文与文献[10]的不同主要体现在两个方面:第一,文献[10]研究的是具有单个领导智能体的领导-跟随一致性问题,本文研究的是具有多个领导智能体的包含控制问题,文献[10]是本文的一个特例;第二,本文将文献[10]中的控制协议进行了推广,更具一般性.本文利用代数图论、矩阵理论和固定时间稳定性理论给出系统解决固定时间包含控制最弱的拓扑条件.

  • 注1 定义集合X的凸包为cos(X)=i=1n-m tjxjxjX,j=1n-m tj=1,tj[0,1].设xRn,x表示x的Euclidean范数.λmin(A)表示矩阵A的最小特征值.定义x[α]=sign(x)|x|α,其中α>0,xR,sign()是符号函数.如果x=x1,x2,,xnRn,则x[k]=x1[k],x2[k],,xn[k]T,k >0.对于矩阵A=aij,若aij≥0,则称矩阵A为非负矩阵.对于非负的方阵A,若每行元素的和均为1,则称矩阵A为随机矩阵.若矩阵A所有特征值具有负的实部,则称矩阵A稳定.

  • 1 预备知识

  • 1.1 图论

  • 有向图G=(V,E)由顶点集V={1,2,,n}和边集EV×V组成,智能体i能获取智能体j的信息可以用(j,i)E来表示.用A=aijRn×n表示由n个顶点构成的有向图的加权邻接矩阵.如果(j,i)E,aij>0,如果j,iE,aij=0,其中aij表示权重.用L=lijRn×n表示加权图的Laplacian矩阵,其中lij=-aij,ijj=1,jiN aij,i=j.更多图论相关知识可以参考文献[15].

  • 1.2 相关引理

  • 引理1[16]x1,x2,,xn0,0<p<1<q,

  • k=1n xkpk=1n xkp,k=1n xkqn1-qk=1n xkq
    (1)
  • 引理2[8] 考虑如下系统:

  • x˙(t)=f(x(t),t),x(0)=x0,
    (2)
  • 其中:x(t)Rn表示信息状态;f:Rn×R+Rn是连续非线性函数.假设原点是平衡点,如果存在一个连续的径向无界函数V:RnR和一些常数a,b >0,0<μ<1<ν使得:

  • V˙(t)-aVμ-bVν
    (3)
  • 则原点是全局固定时间稳定,且满足:

  • Tx0Tmax:=1a(1-μ)+1b(ν-1).
    (4)
  • 2 模型描述与结果

  • 设所研究的多智能体系统包含m个跟随智能体和n-m个领导智能体,分别用F={1,2,,m}S={m+1,m+2,,n}表示跟随智能体和领导智能体的集合.该系统的动力学模型为

  • xi˙(t)=ui(t)+fxi(t),t,iF,xi˙(t)=fxi(t),t,iS,
    (5)
  • 其中: xi(t)表示第i个智能体在时刻t的信息状态; ui(t)R是第i个跟随智能体的控制输入;fxi(t), t) :R×RR是第i个智能体的非线性动态.

  • 定义1 如果存在控制协议ui,i=1,,m,且对于一个不依赖于初始值的正实数T和有界正常数T max,满足TT max,使得:

  • limtT xi(t)-co(X)=0,iF,
    (6)
  • 则称该协议解决固定时间包含控制问题.

  • 假设1 对于每个跟随智能体,都存在来源于领导智能体的有向路径.

  • 为研究方便,将Laplacian矩阵L写成如下分块矩阵的形式:

  • L=L1L200
    (7)
  • 其中L1Rm×m,L2Rm×(n-m).

  • 引理3[17] 在满足假设1条件下的矩阵-L1是稳定且可逆的,并且矩阵-L1-1L2是随机矩阵.

  • 定义2[18] 若对于一个非奇异的实方阵A,其非对角线元素是非正的,并且每个特征值具有正实部,则称矩阵AM-矩阵.

  • 引理4[18]L1M-矩阵,则存在对角阵W=diagw1,w2,,wm>0,使得WL1+L1TW是正定矩阵.

  • 假设2 已知l为正常数,对于常数k1,k2,kn-m,满足i=1n-m ki=1,且ki0,i=1,2,,n-m.非线性函数满足如下条件:

  • fxi(t),t-i=1n-m kifyi(t),tlxi(t)-i=1n-m kiyi(t).
    (8)
  • 由假设2明显看出当m=n-1时,即系统只有一个领导智能体时,该条件为Lipschitz条件.

  • 考虑如下控制协议:

  • ui(t)=-αj=1n aijxi(t)-xj(t)[μ]-βj=1n aijxi(t)-xj(t)[ν],
    (9)
  • 其中βα>0,μ>1>ν>0.

  • xF(t)=x1(t),x2(t),,xm(t)T,xS(t)=xm+1(t),xm+2(t),,xn(t)T,FxF(t)=fx1(t),t,,fxm(t),tT,FxS(t)=fxm+1(t),t,,fxn(t),tT,

  • 则有:

  • x˙F(t)=-αL1xF(t)+L2xS(t)[μ]-βL1xF(t)+L2xS(t)[ν]+FxF(t)
    (10)
  • 令加权跟踪误差δ(t)=L1xF(t)+L2xS(t),可以得到:

  • δ˙(t)=L1x˙F(t)+L2x˙S(t)=L1-αδ(t)[μ]-βδ(t)[ν]+L1FxF(t)+L2FxS(t).
    (11)
  • 在下文中,为了表示方便,在不引起混淆的情况下,我们有时去掉时间变量t.下面给出本文的主要结论.

  • 定理1 在假设1和假设2的条件下,如果控制增益参数α,β满足以下条件:

  • α2λminWL1+L1TW>βL1TW2+βl2L1-12
    (12)
  • 则协议(9)解决固定时间包含控制问题.

  • 证明 首先构造如下李雅普诺夫函数:

  • V(t)=αμ+1i=1m wiδiμ+1+βν+1i=1m wiδiν+1,
    (13)
  • 其中W=diagw1,w2,,wm是引理4中定义的对角阵.

  • 对其求导得到:

  • V˙(t)=αi=1m wiδi[μ]δ˙i+βi=1m wiδi[ν]δ˙i=αδ[μ]TWδ˙+βδ[ν]TWδ˙=αδ[μ]+βδ[ν]TWδ˙=-αδ[μ]+βδ[ν]TWL1αδ[μ]+βδ[ν]+αδ[μ]+βδ[ν]TWL1FxF+L1-1L2FxS=-12αδ[μ]+βδ[ν]TWL1+L1TWαδ[μ]+βδ[ν]+αδ[μ]+βδ[ν]TWL1FxF+L1-1L2FxS-12λmin WL1+L1TWα2δ[μ]+βαδ[ν]2+βαβδ[μ]+δ[ν]TWL1FxF+L1-1L2FxS.
    (14)
  • 又由于βα>0,所以

  • δ[μ]+βαδ[ν]2=i=1m δi2μ+β2α2δi2ν+2βαδiμ+ν

  • i=1m δi2μ+δi2ν+2δiμ+ν=δ[μ]+δ[ν]2
    (15)
  • 由引理3可知-L1-1L2是随机矩阵.所以L1-1L2=-kijm×(n-m),其中kij ≥0,且j=1n-m kij=1,则有

  • FxF+L1-1L2FxS=fx1,t-j=1n-m k1jfxm+jfxm,t-j=1n-m kmjfxm+j
    (16)
  • 再由假设2可以得到:

  • FxF+L1-1L2FxS2l2xF+L1-1L2xS2=l2L1-1δ2l2L1-12||δ||2.
    (17)
  • 又由于||δ||2||δ[μ]+δ[ν]2,所以再由不等式xTy12||x2+||y||2可以得到:

  • V˙(t)-12λmin WL1+L1TWα2δ[μ]+δ[ν]2+12βL1TWαβδ[μ]+δ[ν]2+12βl2FxF+L1-1L2FxS2-12λmin WL1+L1TWα2δ[μ]+δ[ν]2+12βL1TW2αβδ[μ]+δ[ν]2+12βl2L1-12δ[μ]+δ[ν]2.
    (18)
  • 又根据βα>0,所以

  • αβδ[μ]+δ[ν]2=i=1m δi2ν+α2β2δi2μ+2αβδiμ+νi=1m δi2μ+δi2ν+2δiμ+ν=δ[μ]+δ[ν]2
    (19)
  • γ=12α2λminWL1+L1TW-12βL1TW2-12βl2L1-12,由(12)可知γ>0,则

  • V˙(t)-12α2λminWL1+L1TWδ[μ]+δ[ν]2-βL1TW2+βl2L1-12δ[μ]+δ[ν]2=-γδ[μ]+δ[ν]2
    (20)
  • 又由于

  • V=αμ+1i=1m wiδiμ+1+βν+1i=1m wiδiν+1βwmaxν+1i=1m δiμ+1+i=1m δiν+1
    (21)
  • 其中wmax=maxw1,w2,,wm,由于0<μ+νμ+1<1,所以由引理1可知:

  • i=1m δiμ+1+i=1m δiν+1μ+νμ+1i=1m δiμ+ν+δiν+1μ+1(μ+ν).
    (22)
  • 又由于δiv+1μ+1(μ+ν)δi2μ+δi2v,则有

  • i=1m δiμ+1+i=1m δiν+1μ+νμ+1i=1m 2δiμ+ν+δi2μ+δi2ν
    (23)
  • ν+1βwmaxVμ+νμ+1i=1m δiμ+1+i=1m δiν+1μ+νμ+1i=1m 2δiμ+ν+δi2μ+δi2ν.
    (24)
  • 又由2μμ+1>1可知:

  • (2m)1-2μμ+1i=1m δiμ+1+i=1m δiν+12μμ+1i=1m 2δi2μ+δiν+1μ+12μ.
    (25)
  • 又由于2ν<ν+1μ+12μ<μ+νδiv+1μ+12μi=1m δiμ+ν+δi2ν,则有

  • (2m)1-2μμ+1i=1m δiμ+1+i=1m δiν+12μμ+1i=1m 2δiμ+ν+δi2μ+δi2ν.
    (26)
  • (2m)1-μμ+1ν+1βwmaxV2μμ+ν(2m)1-μμ+1i=1m δiμ+1+i=1m δiν+12μμ+1i=1m 2δiμ+ν+δi2μ+δi2ν
    (27)
  • 最后结合不等式(20)、(24)和(27),可以得到:

  • V˙(t)-γi=1m δi2μ+δi2ν+2δiμ+ν-γ2ν+1βwmaxVμ+νμ+1-γ2(2m)1-μμ+1ν+1βwmaxV2μμ+1=-γ2ν+1βwmaxμ+νμ+1(V)μ+νμ+1-γ2(2m)1-μμ+1ν+1βwmax2μμ+1(V)2μμ+1.
    (28)
  • 根据引理2得到:V(t)=0,即可推出δ在固定时间T 0达到0,也就是说xF(t)=-L1-1L2xS(t),tT0,其中

  • T02(μ+1)βwmaxμ+νμ+1γ(ν+1)μ+νμ+1(1-ν)+2(μ+1)βwmaxμ+ν+1γ(2m)1-μμ+1(ν+1)2μμ+1(μ-1).
    (29)
  • 注意到-L1-1L2是随机矩阵,即-L1-1L2xS(t)cosxS,因此协议(9)解决固定时间包含控制问题.

  • 3 数值仿真

  • 多智能体系统的通信拓扑如图1所示,分别由2个领导智能体(用顶点5、6表示)和4个跟随智能体(用顶点1、2、3、4表示)组成.设智能体的非线性动态为fxi,t=xisint.当l=1时,满足条件假设2.经过计算,取对角矩阵W=diag(2,0.8,1,0.5),选择参数μ=1.5,ν=0.5,α=30,β=35,取跟随智能体的初始状态xF(0)=(5,-2,3,-7)T,领导智能体的初始状态xS(0)=(4,-9)T,可以得到初始加权跟踪误差的状态δ(0)=(1,-12,12,2)T,加权跟踪误差的状态随时间的变化曲线如图2所示.从图2可以看出,加权跟踪误差大约在0.01s内收敛到0,比T max=0.89s要小得多,这意味着系统能在固定时间内解决包含控制问题,验证了理论结果的正确性.

  • 图1 系统的通信拓扑

  • Fig.1 System communication topology

  • 图2 加权跟踪误差的状态随时间的变化曲线

  • Fig.2 State curve of the weighted tracking error over time

  • 4 结论

  • 本文研究了非线性多智能体系统在有向网络拓扑下的固定时间包含控制问题.假设所有智能体之间的通信拓扑是有向的,而且每个跟随智能体都至少有一个领导智能体能够直接或间接地与其通信,在这最弱拓扑条件下,所给出的控制协议能够解决固定时间的包含控制问题.仿真实例验证了所提理论结果的正确性.

  • 参考文献

    • [1] Olfati-Saber R,Murray R M.Consensus problems in networks of agents with switching topology and time-delays[J].IEEE Transactions on Automatic Control,2004,49(9):1520-1533

    • [2] 唐朝君.基于自适应控制的非线性多智能体系统一致性[J].重庆理工大学学报(自然科学),2019,33(11):137-142;TANG Zhaojun.Adaptive consensus for nonlinear multi-agent systems[J].Journal of Chongqing University of Technology(Natural Science),2019,33(11):137-142

    • [3] 陈军统,徐振华,项秉铜,等.具有控制器增益随机不确定性的多智能体一致性控制[J].南京信息工程大学学报(自然科学版),2018,10(2):173-177;CHEN Juntong,XU Zhenhua,XIANG Bingtong,et al.Leader-follower consensus of multi-agent systems with stochastic uncertainty of controller gain[J].Journal of Nanjing University of Information Science & Technology(Natural Science Edition),2018,10(2):173-177

    • [4] 邵劭,胡元发,刘小洋,等.多智能体系统的有限时间与固定时间一致性[J].南京信息工程大学学报(自然科学版),2019,11(4):409-413;SHAO Shao,HU Yuanfa,LIU Xiaoyang,et al.Finite-time/fixed-time consensus of multi-agent systems[J].Journal of Nanjing University of Information Science & Technology(Natural Science Edition),2019,11(4):409-413

    • [5] Wang L,Xiao F.Finite-time consensus problems for networks of dynamic agents[J].IEEE Transactions on Automatic Control,2010,55(4):950-955

    • [6] Li S H,Du H B,Lin X Z.Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics[J].Automatica,2011,47(8):1706-1712

    • [7] Zhang Y J,Yang Y,Zhao Y,et al.Distributed finite-time tracking control for nonlinear multi-agent systems subject to external disturbances[J].International Journal of Control,2013,86(1):29-40

    • [8] Polyakov A.Nonlinear feedback design for fixed-time stabilization of linear control systems[J].IEEE Transactions on Automatic Control,2012,57(8):2106-2110

    • [9] Hong H F,Yu W W,Wen G H,et al.Distributed robust fixed-time consensus for nonlinear and disturbed multiagent systems[J].IEEE Transactions on Systems,Man,and Cybernetics:Systems,2017,47(7):1464-1473

    • [10] Wang H,Yu W W,Wen G H,et al.Fixed-time consensus of nonlinear multi-agent systems with general directed topologies[J].IEEE Transactions on Circuits and Systems II:Express Briefs,2019,66(9):1587-1591

    • [11] Ji M,Ferrari-Trecate G,Egerstedt M,et al.Containment control in mobile networks[J].IEEE Transactions on Automatic Control,2008,53(8):1972-1975

    • [12] Cao Y C,Stuart D,Ren W,et al.Distributed containment control for multiple autonomous vehicles with double-integrator dynamics:algorithms and experiments[J].IEEE Transactions on Control Systems Technology,2011,19(4):929-938

    • [13] Wang X Y,Li S H,Shi P.Distributed finite-time containment control for double-integrator multiagent systems[J].IEEE Transactions on Cybernetics,2014,44(9):1518-1528

    • [14] 贺清碧,唐朝君,黄大荣,等.有向切换网络拓扑下非线性多智能体系统的包含控制[J].科学技术与工程,2017,17(16):254-258;HE Qingbi,TANG Zhaojun,HUANG Darong,et al.Containment control of nonlinear multi-agent systems with directed switching topologies[J].Science Technology and Engineering,2017,17(16):254-258

    • [15] Godsil C,Royle G.Algebraic graph theory[M].New York,NY:Springer New York,2001

    • [16] Goldberg M.Equivalence constants for lp norms of matrices[J].Linear and Multilinear Algebra,1987,21(2):173-179

    • [17] Meng Z Y,Ren W,You Z.Distributed finite-time attitude containment control for multiple rigid bodies[J].Automatica,2010,46(12):2092-2099

    • [18] Wen G H,Hu G Q,Yu W W,et al.Consensus tracking for higher-order multi-agent systems with switching directed topologies and occasionally missing control inputs[J].Systems & Control Letters,2013,62(12):1151-1158

  • 参考文献

    • [1] Olfati-Saber R,Murray R M.Consensus problems in networks of agents with switching topology and time-delays[J].IEEE Transactions on Automatic Control,2004,49(9):1520-1533

    • [2] 唐朝君.基于自适应控制的非线性多智能体系统一致性[J].重庆理工大学学报(自然科学),2019,33(11):137-142;TANG Zhaojun.Adaptive consensus for nonlinear multi-agent systems[J].Journal of Chongqing University of Technology(Natural Science),2019,33(11):137-142

    • [3] 陈军统,徐振华,项秉铜,等.具有控制器增益随机不确定性的多智能体一致性控制[J].南京信息工程大学学报(自然科学版),2018,10(2):173-177;CHEN Juntong,XU Zhenhua,XIANG Bingtong,et al.Leader-follower consensus of multi-agent systems with stochastic uncertainty of controller gain[J].Journal of Nanjing University of Information Science & Technology(Natural Science Edition),2018,10(2):173-177

    • [4] 邵劭,胡元发,刘小洋,等.多智能体系统的有限时间与固定时间一致性[J].南京信息工程大学学报(自然科学版),2019,11(4):409-413;SHAO Shao,HU Yuanfa,LIU Xiaoyang,et al.Finite-time/fixed-time consensus of multi-agent systems[J].Journal of Nanjing University of Information Science & Technology(Natural Science Edition),2019,11(4):409-413

    • [5] Wang L,Xiao F.Finite-time consensus problems for networks of dynamic agents[J].IEEE Transactions on Automatic Control,2010,55(4):950-955

    • [6] Li S H,Du H B,Lin X Z.Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics[J].Automatica,2011,47(8):1706-1712

    • [7] Zhang Y J,Yang Y,Zhao Y,et al.Distributed finite-time tracking control for nonlinear multi-agent systems subject to external disturbances[J].International Journal of Control,2013,86(1):29-40

    • [8] Polyakov A.Nonlinear feedback design for fixed-time stabilization of linear control systems[J].IEEE Transactions on Automatic Control,2012,57(8):2106-2110

    • [9] Hong H F,Yu W W,Wen G H,et al.Distributed robust fixed-time consensus for nonlinear and disturbed multiagent systems[J].IEEE Transactions on Systems,Man,and Cybernetics:Systems,2017,47(7):1464-1473

    • [10] Wang H,Yu W W,Wen G H,et al.Fixed-time consensus of nonlinear multi-agent systems with general directed topologies[J].IEEE Transactions on Circuits and Systems II:Express Briefs,2019,66(9):1587-1591

    • [11] Ji M,Ferrari-Trecate G,Egerstedt M,et al.Containment control in mobile networks[J].IEEE Transactions on Automatic Control,2008,53(8):1972-1975

    • [12] Cao Y C,Stuart D,Ren W,et al.Distributed containment control for multiple autonomous vehicles with double-integrator dynamics:algorithms and experiments[J].IEEE Transactions on Control Systems Technology,2011,19(4):929-938

    • [13] Wang X Y,Li S H,Shi P.Distributed finite-time containment control for double-integrator multiagent systems[J].IEEE Transactions on Cybernetics,2014,44(9):1518-1528

    • [14] 贺清碧,唐朝君,黄大荣,等.有向切换网络拓扑下非线性多智能体系统的包含控制[J].科学技术与工程,2017,17(16):254-258;HE Qingbi,TANG Zhaojun,HUANG Darong,et al.Containment control of nonlinear multi-agent systems with directed switching topologies[J].Science Technology and Engineering,2017,17(16):254-258

    • [15] Godsil C,Royle G.Algebraic graph theory[M].New York,NY:Springer New York,2001

    • [16] Goldberg M.Equivalence constants for lp norms of matrices[J].Linear and Multilinear Algebra,1987,21(2):173-179

    • [17] Meng Z Y,Ren W,You Z.Distributed finite-time attitude containment control for multiple rigid bodies[J].Automatica,2010,46(12):2092-2099

    • [18] Wen G H,Hu G Q,Yu W W,et al.Consensus tracking for higher-order multi-agent systems with switching directed topologies and occasionally missing control inputs[J].Systems & Control Letters,2013,62(12):1151-1158

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