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通讯作者:

余跃,E-mail:Yu.y@ntu.edu.cn

中图分类号:O31

文献标识码:A

文章编号:1672-6553-2022-20(2)-045-05

DOI:10.6052/1672-6553-2021-048

参考文献 1
Lee S H,Park M J,Kwon O M.Synchronization criteria for delayed Lur′e systems and randomly occurring sampled-data controller gain.Communications in Nonlinear Science and Numerical Simulation,2019,68:203~219
参考文献 2
Zhang R M,Zeng D Q,Liu X Z,et al.A new method for quantized sampled-data synchronization of delayed chaotic Lur′e systems.Applied Mathematical Modelling,2019,70:471~489
参考文献 3
Rakkiyappan R,Velmurugan G,George J N,et al.Exponential synchronization of Lur′e complex dynamical networks with uncertain inner coupling and pinning impulsive control.Applied Mathematics and Computation,2017,307:217~231
参考文献 4
Li T,Yuan R T,Fei S M,et al.Sampled-data synchronization of chaotic Lur′e systems via an adaptive event-triggered approach.Information Sciences,2018,462:40~54
参考文献 5
Zhang L,Stepan G.Exact stability chart of an elastic beam subjected to delayed feedback.Journal of Sound and Vibration,2016,367:219~232
参考文献 6
王贺元,尹霞.新超混沌系统的动力学行为及自适应控制与同步.动力学与控制学报,2017,15(4):335~341(Wang H Y,Yin X.Dynamical behaviors of a new hyperchaotic system and its adaptive control and synchronization.Journal of Dynamical and Control,2017,15(4):335~341(in Chinese))
参考文献 7
Liu Y J,Lee S M.Synchronization criteria of chaotic Lur′e systems with delayed feedback PD control.Neurocomputing,2016,189:66~71
参考文献 8
Huang H,Feng G,Cao J D.Exponential synchronization of chaotic Lur′e systems with delayed feedback control.Nonlinear Dynamics,2009,57(3):441~453
参考文献 9
Shi K B,Liu X Z,Zhu H,et al.Novel delay-dependent master-slave synchronization criteria of chaotic Lur′e systems with time-varying-delay feedback control.Applied Mathematics and Computation,2016,282:137~154
参考文献 10
Habib G,Rega G,Stepan G.Delayed digital position control of a single-DOF system and the nonlinear behavior of the act-and-wait controller.Journal of Vibration and Control,2016,22(2):481~495
参考文献 11
陈保颖.线性反馈实现Liu系统的混沌同步.动力学与控制学报,2006,4(1):1~4(Chen B Y.Linear feedback control for synchronization of Liu chaotic system.Journal of Dynamics and Control,2006,4(1):1~4(in Chinese))
参考文献 12
王琳,倪樵,黄玉盈.时滞反馈Liu系统的动力学行为.动力学与控制学报,2007,5(3):224~227(Wang L,Ni Q,Huang Y Y.Dynamical behaviors of Liu system time delayed feedbacks.Journal of Dynamics and Control,2007,5(3):224~227(in Chinese))
参考文献 13
Ge C,Hua C C,Guan X P.Master-slave synchronization criteria of Lur′e systems with time-delay feedback control.Applied Mathematics and Computation,2014,244:895~902
参考文献 14
Wang Y M,Xiong L L,Liu X Z,et al.New delay-dependent synchronization criteria for uncertain Lur′e systems via time-varying delayed feedback control.Journal of Nonlinear Science and Applications,2017,10(4):1927~1940
参考文献 15
Zhang H M,Cao J D,Xiong L G.Novel synchronization conditions for time-varying delayed Lur′e system with parametric uncertainty.Applied Mathematics and Computation,2019,350:224~236
参考文献 16
张旭,张丽.多自由度车削系统稳定性分析.动力学与控制学报,2019,17(6):508~513(Zhang X,Zhang L.Stability analysis of multi degree of freedom turning models.Journal of Dynamics and Control,2019,17(6):508~513(in Chinese))
参考文献 17
Wang Z H,Hu H Y,Xu Q,et al.Effect of delay combinations on stability and Hopf bifurcation of an oscillator with acceleration-derivative feedback.International Journal of Non-Linear Mechanics,2017,94:392~399
参考文献 18
刘喻,张思进,殷珊.高速切削过程中颤振现象的二自由度非光滑模型分析.动力学与控制学报,2018,16(4):350~356(Liu Y,Zhang S J,Yin S.Analysis on chatter vibration of a two-degree-of-freedom non-smooth system in high-speed cutting process.Journal of Dynamics and Control,2018,16(4):350~356(in Chinese))
参考文献 19
Zhang C K,He Y,Jiang L,et al.Notes on stability of time-delay systems:bounding inequalities and augmented Lyapunov-Krasovskii functionals.IEEE Transactions on Automatic Control,2017,62(10):5331~5336
参考文献 20
Park P,Lee W I,Lee S Y.Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems.Journal of the Franklin Institute,2015,352(4):1378~1396
目录contents

    摘要

    本文研究了混沌Lur′e系统的滞后同步问题.通过合理构造增广Lyapunov-Krasovskii泛函,使得增广向量不仅包括单积分项以及双积分项,而且加入了增广的三重积分项.在估计Lyapunov-Krasovskii泛函导数表达式的基础上,通过零等式引入自由矩阵,同时利用积分不等式,求得以线性矩阵不等式形式表达下的混沌Lur′e系统滞后同步的判据,即充分条件.最后,通过数值计算表明本文理论研究的有效性.

    Abstract

    The delayed synchronization of chaotic Lur′e system is studied. By constructing an appropriate augmented Lyapunov-Krasovskii functional, the augmented vector contains not only a single integral term and a double integral term, but also an augmented triple integral term. When estimating the derivative of the Lyapunov-Krasovskii functional, a free matrix is introduced through the zero equality. By using the integral inequality, the sufficient condition for the delayed synchronization of chaotic Lur′e systems is given in the form of linear matrix inequalities. Finally, numerical example demonstrates effectiveness of the proposed method.

  • 引言

  • 若系统的动态行为中存在混沌现象,则称该系统为混沌系统[1-4].对于混沌系统研究的一个重要方面是混沌同步.在过去的几十年,混沌同步广泛应用于物理、化学反应堆、生态系统、安全保密通讯等众多领域.目前对于实现混沌同步的方法已在大量文献中得到相关研究和发展[5-8].

  • 一些混沌系统如蔡氏电路以及超混沌吸引子是可以建立为Lur′e系统的模型.Lur′e系统的滞后反馈同步主要是利用主系统的滞后输出和从系统的滞后输出之差来作为反馈输入使得相应的误差闭环系统-时滞系统-渐近稳定[9-12].因此,关于时滞系统的若干研究成果可以应用到这个误差闭环系统.目前,关于混沌Lur′e系统的滞后反馈同步已有大量相关的研究成果[7-12].例如, 在Lyapunov-Krasovskii泛函(LKF)构造方面,Ge等[13]构造了时滞分解的LKF,Wang等[14]构造了增广的LKF,Zhang等[15]构造了含有多重积分的LKF.在估计LKF沿误差系统的导数方面,相关学者[16-18]则是利用了各种积分不等式,进而得到LKF导数更紧的上界.无论是构造更一般的LKF,还是利用先进的积分不等式,其目的都是为了得出时滞的最大上界.随着时滞系统分析方法的改进,可以期望在混沌Lur′e系统的滞后反馈同步问题上有更好的研究结果.

  • 本文主要是考虑了混沌Lur′e系统滞后反馈的同步问题.同时,受Zhang等[19]学者的启发,构造了一个适当的增广LKF,在估计LKF沿误差系统的导数时,通过零等式引入自由矩阵,利用积分不等式得出该系统存在滞后反馈控制器的充分条件,并以LMI的形式表示.最后数值算例表明本文方法的有效性和优越性.

  • 1 系统描述及引理

  • 考虑如下的主从混沌Lur′e系统

  • 1.1 统描述及引理

  • M:x˙(t)=Ax(t)+Bf(Cx(t))p(t)=Hx(t)
    (1)
  • S:y˙(t)=Ay(t)+Bf(Cy(t))+u(t)q(t)=Hy(t)
    (2)
  • U:u(t)=K(p(t-d)-q(t-d))
    (3)
  • 其中,x(t),y(t)分别是主系统和从系统的状态,A,B,C,H则是具有适当维数的常数矩阵,K是待定的控制器的增益矩阵.f(·)满足扇形条件,即

  • 0fi(u)-fi(v)u-vli,u,v,i=1,2,,nh
    (4)
  • e(t)=x(t)-y(t),则误差系统可表示为

  • e˙(t)=Ae(t)+Bg(Ce(t))-KHe(t-d)
    (5)
  • 其中g(Ce(t))=f(Cx(t))-f(Cy(t)).本文的主要目的是设计形如系统(3)的控制器,使得系统(5)渐近稳定.在给出本文的主要结果之前,我们首先给出相关引理.

  • 引理[20]ω:(a,b)Rn是向量函数,R,Z是对称正定矩阵,使得如下的积分有意义,则

  • ab ω˙T(s)Rω˙(s)ds1b-aΓ1ξ(ω,a,b)TRΓ1ξ(ω,a,b),ab ωT(s)Rω(s)ds1(b-a)Γ2ξ(ω,a,b)TR^Γ2ξ(ω,a,b),ab θb ω˙T(s)Zω˙(s)dsdθΓ3ξ(ω,a,b)TZΓ3ξ(ω,a,b),ab θb ωT(s)Zω(s)dsdθ2(b-a)2ab θb ω(s)dsdθT×Zab θb ω(s)dsdθ,ξ(ω,a,b)=colω(b)ω(a)ω1(a,b,s)ω2(a,b,s),

  • ω1(a,b,s)=1b-aab ω(s)ds,ω2(a,b,s)=1(b-a)2ab θb ω(s)dsdθ,Γ1=I-I00II-2I0I-I-6I12I,Γ2=00I000I-2I,Γ3=I0-I0I02I-6I,R¯=diagR^5R),Z-=diag2Z 4Z,R^=diagR 3R.

  • 2 主要结果

  • 为表述简单,定义如下符号

  • η(t)=col(e(t)e˙(t)e(t-d)v1(t)v2(t)e˙(t-d)g(Ce(t)),v1(t)=1dt-dt e(s)ds,v2(t)=1d2t-dt θt e(s)dsdθ,L=diagl1l2lnh,

  • ei=0n,(i-1)nIn0n,(6-i)n+nh,i=1,2,,6,0nh,i-1nInh, i=7,

  • 定理1 对于给定的d,a,误差系统(5)渐近稳定,若存在对称矩阵P,T,以及对称正定矩阵Q, Zi(i=1,2,3,4),对角正定矩阵Λ1,Λ2,具有适当维数的矩阵G,Y,使得如下矩阵不等式成立,

  • Ω=Ω1+Ω2+Ω3+Ω4+Ω5<0
    (6)
  • P+Γ1TQ~Γ1>0,Z-2=Z2+0 TT 0>0
    (7)
  • 其中,

  • Q~=diag(Q 3Q 5Q),Ω1=2E1TPE2+de2TQe2-e6TQe6Ω2=d2e2TZ1e2+E3TZ2E3-Γ1E1TZ~1Γ1E1-E4TZ~2E4-3E5TZ2E5+de1TTe1-e3TTe3,Ω3=d22e2TZ3e2+E3TZ4E3-Γ3E1TZ~3Γ3E1-2E6TZ4E6,Ω4=2LCe1-e7TΛ1Ce2+2e7TΛ2LCe1-e7,Ω5=2e1+ae2TG-e2+Ae1+Be7T-2e1+ae2TYHe3,

  • E1=cole1e3e4e5,E2=cole2 e6 1de1-e3 1de1-e4,E3=cole1 e2,E4=colde4e1-e3,E5=colde4-2e52e4-e1-e3,E6=colde5 e1-e4,Z~1=diagZ1 3Z1 5Z1,Z¯3=diag2Z34Z3Λ1=diagλ11λ12λ1nh,Λ2=diagλ21λ22λ2nh,

  • 相应的控制器增益矩阵为K=G-1Y

  • 证明 构造如下的LKF

  • V(t)=V1(t)+V2(t)+V3(t)+V4(t),
    (8)
  • 其中

  • V1(t)=ξ1T(t)Pξ1(t)+dt-dt e˙T(s)Qe˙(s)ds,V2(t)=d-d0 t+θt e˙T(s)Z1e˙(s)dsdθ+d-d0 t+θt ξ2TsZ2ξ2sdsdθ,V3(t)=-d0 θ0 t+δt e˙T(s)Z3e˙(s)dsdδdθ+-d0 θ0 t+δt ξ2T(s)Z4ξ2(s)dsdδdθV4(t)=2i=1nh 0c1Te(t) λ1ilis-ggi(s)dsξ1(t)=colet et-d v1t v2(t)ξ2(t)=col(et e˙(t)),

  • 由引理可知

  • dt-dt e˙T(s)Qe˙(s)dsΓ1ξ(e,t-d,t)TQ~Γ1ξ(e,t-d,t)

  • 由(7)可知,V(t)≥V1(t)>0,即V(t)正定.

  • Vi(t)(i=1,2,3,4)沿误差系统求导得,

  • V˙1(t)=2ξ1T(t)Pξ˙1(t)+de˙T(t)Qe˙(t)-e˙T(t-d)Qe˙(t-d)=ηT(t)Ω1η(t),V˙2(t)=d2e˙T(t)Z1e˙(t)+ξ2T(t)Z2ξ2(t)-dt-dt e˙T(s)Z1e˙(s)ds+t-dt ξ2T(s)Z2ξ2(s)ds,V˙3(t)=0.5d2e˙T(t)Z3e˙(t)+ξ2T(t)Z4ξ2(t)-t-dt θt eT(s)Z3e˙(s)dsdθ-t-dt θt ξ2T(s)Z4ξ2(s)dsdθ,V˙4(t)=2LCe(t)-g(Ce(t))TΛ1Ce˙(t).

  • 由引理可以得到

  • -dt-dt e˙T(s)Z1e˙(s)ds-ηTtΓ1E1TZ~1Γ1E1ηt,

  • 对于任意的对称矩阵T,如下等式恒成立

  • deT(t)Te(t)-eT(t-d)Te(t-d)-2t-dt eT(s)Te˙(s)ds=0
    (9)
  • 因此

  • -dt-dt ξ2T(s)Z2ξ2(s)ds+deT(t)Te(t)-eT(t-d)Te(t-d)-2t-dt eT(s)Te˙(s)ds-ηT(t)E4TZ~2E4+3E5TZ~2E5-de1TTe1-e3TT3η(t),-t-dt θt e˙T(s)Z3e˙(s)dsdθ-ηT(t)Γ3E1TZ~3Γ3E1η(t)-t-dt θt ξ2T(s)Z4ξ2(s)dsdθ-2ηT(t)E6TZ4E6η(t)

  • 注意到g(Ce(t))的定义,由式(4)可得对任意的正对角矩阵Λ2,有如下的不等式成立

  • 2gT(Ce(t))Λ2(LCe(t)-g(Ce(t)))>0

  • 2ηT(t)e7TΛ2LCe1-e7η(t)>0.

  • 又对于任意的矩阵G和常数a,存在下式恒成立

  • (e(t)+ae˙(t))G×(-e˙(t)+Ae(t)+Bg(Ce(t))-KH(t-d))=0

  • Y=GK,上式可表示为

  • 2ηT(t)e1+ae2TG-e2+Ae1+Be7-e1+ae2TYHe3η(t)=0.

  • 综上可得

  • V˙i(t)ηT(t)Ωiη(t)

  • 从而V˙(t)ηT(t)Ωη(t).由定理条件可知,若Ω<0,则误差闭环系统渐近稳定.证毕.

  • 注1 定理构造了适当的增广LKF, 不要求所有的矩阵正定,在增广向量中,不仅含有单积分项,还含有双积分项,不仅如此,在V3(t)中,还含有增广的三重积分项,充分利用了时滞及相关误差状态的信息.

  • 注2 在计算LKF沿误差系统的导数时,通过零等式(9)引入了自由的对称矩阵T,在估计V˙i(t)(i=2,3)中的负定积分项时,利用了更紧的不等式,可以期望该定理的保守性较小.

  • 3 数值算例

  • 考虑混沌Lur′e系统,其系统参数矩阵如下

  • A=-187901-110-14.280,

  • B=27700,C=100T,

  • fx1(t)=0.5x1(t)+1-x1(t)-1,易知L=1.取a=4,应用定理所得的最大时滞值为d=0.358,和已有文献的结果比较如表1所示.

  • 表1 不同方法下的结果比较

  • Table1 Comparison of results under different methods

  • 取主从系统的初始状态分别为

  • x(0)=-0.2-0.33-0.2Ty(0)=0.30.4-0.8T

  • d=0.358,K=2.8469 0.0894 -2.4639时,误差系统的状态如图1所示,表明利用本文的设计方法得出的控制器,能使误差系统快速达到稳定状态,即实现了混沌系统的主从同步.

  • 图1 误差系统数值仿真

  • Fig.1 Numerical simulation of error system

  • 4 结论

  • 对于混沌Lur′e系统,研究了基于滞后反馈的主从同步问题.通过构造含有较多时滞状态信息的LKF泛函,由此得出了系统实现主从同步的充分条件.最后,通过数值算例表明了本文的方法保守性较小.

  • 参考文献

    • [1] Lee S H,Park M J,Kwon O M.Synchronization criteria for delayed Lur′e systems and randomly occurring sampled-data controller gain.Communications in Nonlinear Science and Numerical Simulation,2019,68:203~219

    • [2] Zhang R M,Zeng D Q,Liu X Z,et al.A new method for quantized sampled-data synchronization of delayed chaotic Lur′e systems.Applied Mathematical Modelling,2019,70:471~489

    • [3] Rakkiyappan R,Velmurugan G,George J N,et al.Exponential synchronization of Lur′e complex dynamical networks with uncertain inner coupling and pinning impulsive control.Applied Mathematics and Computation,2017,307:217~231

    • [4] Li T,Yuan R T,Fei S M,et al.Sampled-data synchronization of chaotic Lur′e systems via an adaptive event-triggered approach.Information Sciences,2018,462:40~54

    • [5] Zhang L,Stepan G.Exact stability chart of an elastic beam subjected to delayed feedback.Journal of Sound and Vibration,2016,367:219~232

    • [6] 王贺元,尹霞.新超混沌系统的动力学行为及自适应控制与同步.动力学与控制学报,2017,15(4):335~341(Wang H Y,Yin X.Dynamical behaviors of a new hyperchaotic system and its adaptive control and synchronization.Journal of Dynamical and Control,2017,15(4):335~341(in Chinese))

    • [7] Liu Y J,Lee S M.Synchronization criteria of chaotic Lur′e systems with delayed feedback PD control.Neurocomputing,2016,189:66~71

    • [8] Huang H,Feng G,Cao J D.Exponential synchronization of chaotic Lur′e systems with delayed feedback control.Nonlinear Dynamics,2009,57(3):441~453

    • [9] Shi K B,Liu X Z,Zhu H,et al.Novel delay-dependent master-slave synchronization criteria of chaotic Lur′e systems with time-varying-delay feedback control.Applied Mathematics and Computation,2016,282:137~154

    • [10] Habib G,Rega G,Stepan G.Delayed digital position control of a single-DOF system and the nonlinear behavior of the act-and-wait controller.Journal of Vibration and Control,2016,22(2):481~495

    • [11] 陈保颖.线性反馈实现Liu系统的混沌同步.动力学与控制学报,2006,4(1):1~4(Chen B Y.Linear feedback control for synchronization of Liu chaotic system.Journal of Dynamics and Control,2006,4(1):1~4(in Chinese))

    • [12] 王琳,倪樵,黄玉盈.时滞反馈Liu系统的动力学行为.动力学与控制学报,2007,5(3):224~227(Wang L,Ni Q,Huang Y Y.Dynamical behaviors of Liu system time delayed feedbacks.Journal of Dynamics and Control,2007,5(3):224~227(in Chinese))

    • [13] Ge C,Hua C C,Guan X P.Master-slave synchronization criteria of Lur′e systems with time-delay feedback control.Applied Mathematics and Computation,2014,244:895~902

    • [14] Wang Y M,Xiong L L,Liu X Z,et al.New delay-dependent synchronization criteria for uncertain Lur′e systems via time-varying delayed feedback control.Journal of Nonlinear Science and Applications,2017,10(4):1927~1940

    • [15] Zhang H M,Cao J D,Xiong L G.Novel synchronization conditions for time-varying delayed Lur′e system with parametric uncertainty.Applied Mathematics and Computation,2019,350:224~236

    • [16] 张旭,张丽.多自由度车削系统稳定性分析.动力学与控制学报,2019,17(6):508~513(Zhang X,Zhang L.Stability analysis of multi degree of freedom turning models.Journal of Dynamics and Control,2019,17(6):508~513(in Chinese))

    • [17] Wang Z H,Hu H Y,Xu Q,et al.Effect of delay combinations on stability and Hopf bifurcation of an oscillator with acceleration-derivative feedback.International Journal of Non-Linear Mechanics,2017,94:392~399

    • [18] 刘喻,张思进,殷珊.高速切削过程中颤振现象的二自由度非光滑模型分析.动力学与控制学报,2018,16(4):350~356(Liu Y,Zhang S J,Yin S.Analysis on chatter vibration of a two-degree-of-freedom non-smooth system in high-speed cutting process.Journal of Dynamics and Control,2018,16(4):350~356(in Chinese))

    • [19] Zhang C K,He Y,Jiang L,et al.Notes on stability of time-delay systems:bounding inequalities and augmented Lyapunov-Krasovskii functionals.IEEE Transactions on Automatic Control,2017,62(10):5331~5336

    • [20] Park P,Lee W I,Lee S Y.Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems.Journal of the Franklin Institute,2015,352(4):1378~1396

  • 参考文献

    • [1] Lee S H,Park M J,Kwon O M.Synchronization criteria for delayed Lur′e systems and randomly occurring sampled-data controller gain.Communications in Nonlinear Science and Numerical Simulation,2019,68:203~219

    • [2] Zhang R M,Zeng D Q,Liu X Z,et al.A new method for quantized sampled-data synchronization of delayed chaotic Lur′e systems.Applied Mathematical Modelling,2019,70:471~489

    • [3] Rakkiyappan R,Velmurugan G,George J N,et al.Exponential synchronization of Lur′e complex dynamical networks with uncertain inner coupling and pinning impulsive control.Applied Mathematics and Computation,2017,307:217~231

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