1、应用数学MATHEMATICA APPLICATA2023,36(4):868-876Lure主从系统的二次反馈型脉冲同步控制王万林,黄振坤,赵玲(集美大学理学院,福建 厦门 361021)摘要：本文研究Lure主从系统的脉冲同步控制问题.考虑脉冲控制器带有二次反馈的情况,利用多面体凸组合、线性矩阵不等式(LMI)和Lyapunov稳定性理论,设计两种在二次输出反馈型脉冲控制下主从系统同步的新策略.最后,通过数值模拟验证了所得结果的可行性与有效性.关键词：脉冲控制;二次反馈;同步;Lure主从系统中图分类号：O193AMS(2010)主题分类：93C10;93D15文献标识码：A文章编号：10
2、01-9847(2023)04-0868-091.引言混沌系统是一类非常重要的系统,近年来时滞反馈控制1、滑模控制23、脉冲控制4等多种控制方法应用于混沌系统的同步控制,特别是脉冲控制在耦合混沌系统中应用引起了人们的广泛关注58,并在通信保密应用中取得了丰富的成果912.对于脉冲控制在耦合混沌系统同步问题上的研究,大部分工作都是通过线性输出反馈脉冲来进行的,比如文13-14对Lure主从系统采用了线性静态测量反馈脉冲控制,得到了一系列使得Lure主从系统达到同步的稳定性结果.然而带有二次输出反馈的脉冲控制作用在Lure混沌主从系统上甚至其它耦合混沌系统上都尚无相关结果.由于其脉冲控制器上带有二
3、次输出反馈项,处理起来十分麻烦且无现有方法可直接套用.Amato等人在文15-16中利用多面体凸组合方法研究非线性二次系统这个工作非常具有启发性,此方法加以运用可以解决脉冲控制器上二次输出反馈项所带来的困难.基于以上观点与启发,本文研究了带有二次输出反馈的脉冲控制在Lure混沌主从系统上的同步问题.利用了多面体凸组合与线性矩阵不等式(LMI)17并结合Lyapunov稳定性理论,得到主从系统同步的两个新准则.最后针对蔡氏电路与二维Lure型动态网络系统进行数值模拟.2.主从Lure系统模型与预备工作考虑以下带有二次反馈型脉冲的主从Lure系统M（主系统）:x(t)=Ax(t)+B(Dx),p(
4、t)=Cx(t),(2.1)S（从系统）:z(t)=Az(t)+B(Dz),q(t)=Cz(t),t=tk,(2.2)收稿日期：2022-08-30基金项目：国家自然科学基金(61573005);福建省自然科学基金(2019J01330)作者简介：王万林,男,汉族,福建人,研究方向：复杂网络分析与控制.第 4 期王万林等：Lure主从系统的二次反馈型脉冲同步控制869C（控制器）:z|t=tk=K(p(tk)q(tk)eT(tk)H1e(tk)eT(tk)H2e(tk).eT(tk)Hne(tk),k=1,2,3,(2.3)其中x(t)=(x1(t),x2(t),xn(t)T Rn,z(t)=
5、(z1(t),z2(t),zn(t)T Rn是系统的状态向量,A Rnn,B Rnnh,D Rnhn,e(tk)=x(tk)z(tk).这里假设():Rnh7Rnh是一个非线性函数,并且i()都满足扇区条件0,k,i=1,2,nh.p,q Rl分别是主系统M与从系统S的输出向量,其中C Rln且l n.设脉冲时刻tk给定并且满足0=t0 t1 t2 tk 0,1,使得下面不等式成立eT(i)Pce(i)1,i=1,2,p,aTk(Pc)1ak 1,i=1,2,q,870应用数学2023那么P P,其中为椭球体e Rn:eTPe c.由多面体P的凸性可得引理2.2若存在一个正实数,使得(H1e(
6、i),H2e(i),Hne(i),i=1,2,q,则对于任意的e P,有(H1e,H2e,Hne)0,0 1,使得下面不等式成立eT(i)Pce(i)1,(3.1)aTk(Pc)1ak 1,(3.2)ATP+PA PPB+kDTBTP+kD2 0,(3.3)cond(P)(I KC 2+HT1(e(i)HT2(e(i)HTn(e(i)2)2,(3.4)对于初始条件e(t0)满足eT(t0)Pe(t0)eT c,(3.5)T+ln 0,(3.6)则主系统M与从系统S达到同步,其中T=supkN(tk+1 tk),cond(P)=max(P)min(P),max(P)和min(P)分别是P的最大特
7、征值与最小特征值.证考虑Lyapunov函数V(e(t)=eT(t)Pe(t)和椭球体=e Rn:eTPe c.根据引理2.1,由(3.1)式和(3.2)式可得P P.设椭球体=e Rn:eTPe V(t0)eT,由(3.5)式可得 P.根据脉冲时间序列构造椭球体序列k与+k(k N):0=+0=e Rn:eTPe V(t0),k=e Rn:eTPe V(tk),+k=e Rn:eTPe V(t+k).当t (t0,t1,V(t)沿误差系统(2.4)求导可得V(t)=eT(t)Pe(t)+eT(t)P e(t)=(Ae(t)+B)TPe(t)+eT(t)P(Ae(t)+B)(Ae(t)+B)T
8、Pe(t)+eT(t)P(Ae(t)+B)2nhi=1ii(ikdTie)=(Ae(t)+B)TPe(t)+eT(t)P(Ae(t)+B)2T(kDe(t)=TS1+V(t).第 4 期王万林等：Lure主从系统的二次反馈型脉冲同步控制871由(3.3)式可得V(t)V(t),其中=e,S1=ATP+PA PPB+kDTBTP+kD2,从而在t (t0,t1上,可知V(t)V(t0)eT,V(t1)V(t0)eT,即0 P,1 P,从而e(t0)P,e(t1)P.由误差系统(2.4)可得V(t+1)=eT(t1)(I KC)TP(I KC)e(t1)+eT(t1)(I KC)TPeT(t1)H
9、1eT(t1)H2.eT(t1)Hne(t1)+eT(t1)HT1e(t1)HT2e(t1)HTne(t1)P(I KC)e(t1)+eT(t1)HT1e(t1)HT2e(t1)HTne(t1)PeT(t1)H1eT(t1)H2.eT(t1)Hne(t1)eT(t1)Pe(t1)+V(t1)eT(t1)(2HT1e(t1)HT2e(t1)HTne(t1)PeT(t1)H1eT(t1)H2.eT(t1)Hn+2(I KC)TP(I KC)e(t1)eT(t1)Pe(t1)+V(t1)2max(P)(I KC 2+HT1e(t1)HT2e(t1)HTne(t1)2)eT(t1)e(t1)min(P
10、)eT(t1)e(t1)+V(t1),由e(t1)P、引理2.2和(3.4)式可得V(t+1)V(t1),即+1 1 P,从而e(t+1)P.同理,当t (t1,t2,可得V(t2)V(t+1)eT V(t1)eT(eT)V(t0)eT,由(3.6)式可得eT 1,从而V(t2)V(t0)eT,即2 P,从而e(t2)P.与V(t+1)同样处理过程可得V(t+2)V(t2),即+2 2 P,从而e(t+2)P.重复相同步骤,可得当t (tk1,t+k,有V(tk)0,c 0,1,使得下面不等式成立eT(i)Pce(i)1,(3.7)aTk(Pc)1ak 1,(3.8)ATP+PA+PBBTP+
11、k2DTD P,(3.9)cond(P)(I KC 2+HT1(e(i)HT2(e(i)HTn(e(i)2)eT12,(3.10)其中T=infkN(tk+1 tk),cond(P)=max(P)min(P).对于任意初始状态e(t0)P,误差系统(2.4)渐近稳定,即主系统M与从系统S达到同步.证考虑Lyapunov函数V(e(t)=eT(t)Pe(t)和椭球体=e Rn:eTPe c.根据引理2.1,由(3.7)式和(3.8)式可得P P.根据脉冲时间序列构造椭球体序列k与+k(k N):0=+0=e Rn:eTPe V(t0),k=e Rn:eTPe V(tk),+k=e Rn:eTPe
12、 V(t+k).当t (t0,t1,V(t)沿误差系统(2.4)求导可得V(t)=eT(t)Pe(t)+eT(t)P e(t)=(Ae(t)+B)TPe(t)+eT(t)P(Ae(t)+B)eT(t)ATPe(t)+TBTPe(t)+eT(t)PAe(t)+eT(t)PB nhi=12i(ikdTie)=eT(t)ATPe(t)+TBTPe(t)+eT(t)PAe(t)+eT(t)PB 2T(kDe(t)eT(t)(ATP+PA)e(t)+T+eT(t)PBBTPe(t)2T+T(kDe(t)+(keT(t)DT)eT(t)(ATP+PA)e(t)+eT(t)PBBTPe(t)+k2eT(t)
13、DTDe(t)=eT(t)(ATP+PA+PBBTP+k2DTD)e(t),由(3.9)式可知V(t)V(t),从而在t (t0,t1上可知V(t)V(t0)eT,V(t1)V(t0)eT并且V(t1)V(t0),即1 0,从而e(t1)0.假设V(t0)c,即0 P,并且1 0,从而e(t0),e(t1)P.由误差系统(2.4)可得V(t+1)=eT(t1)(I KC)TP(I KC)e(t1)+eT(t1)(I KC)TPeT(t1)H1eT(t1)H2.eT(t1)Hne(t1)+eT(t1)HT1e(t1)HT2e(t1)HTne(t1)P(I KC)e(t1)+eT(t1)HT1e(
14、t1)HT2e(t1)HTne(t1)PeT(t1)H1eT(t1)H2.eT(t1)Hne(t1)第 4 期王万林等：Lure主从系统的二次反馈型脉冲同步控制873eT(t1)(2HT1e(t1)HT2e(t1)HTne(t1)PeT(t1)H1eT(t1)H2.eT(t1)Hn+2(I KC)TP(I KC)e(t1)2max(P)(I KC 2+HT1e(t1)HT2e(t1)HTne(t1)2)eT(t1)e(t1),由于e(t1)P、引理2.2和(3.10)式可得V(t+1)min(P)eTeT(t1)e(t1)eTV(t1)V(t0),即+1+0 P,从而e(t+1)P.同理当t
15、(t1,t2可得V(t2)V(t+1)eT V(t1),即2 1 0 P,从而e(t2)P;与V(t+1)同样处理过程可得V(t+2)eTV(t2)V(t+1),即+2+1+0 P,从而e(t+2)P;重复相同步骤可得当t (tk1,t+k,有k k1 0 P,+k+k1 +0 P,从而e(tk)P,e(t+k)P.以上已证得对k N,e(tk)P都成立,再结合(3.10)式保证对k N,V(t+k+1)V(t+k)都成立,从而存在k(0,1),使得V(t+k+1)=kV(t+k).当t (tk,tk+1时,由(3.9)式可得V(t)V(t+k)eT=k1V(t+k1)eT=k1i=0iV(t
16、0)eT 0,k +.因此当任意初始状态e(t0)时,误差系统(2.4)渐近稳定.由于P ,从而当任意初始状态e(t0)P也可使得误差系统(2.4)渐近稳定,即主系统M与从系统S达到同步.注3.1定理3.2条件下,多面体P是该系统一吸引域估计.注3.2定理3.1与定理3.2的区别在于:1)定理3.2中的状态矩阵A必须要求为Hurwitz矩阵,定理3.1中的A并无要求;2)定理3.1是在给定初值状态e(t0)下,其对合适的多面体P条件更苛刻,并且初值状态e(t0)不一定在符合条件的多面体P上;3)定理3.1与定理3.2中的T含义不同.综上可知定理3.1与定理3.2是相互独立的.4.数值模拟例1考
17、虑蔡氏电路主从系统,主系统M如下 x1=x2 f(x1),x2=x1 x2+x3,x3=x2,(4.1)其中f(x1)=m1x1+12(m0 m1)(|x1+c|x1 c|)，,m0,m1,c为常数,这里取=9,=14.286,m0=17,m1=27,c=1,相应的从系统S与主系统M参数一致.图4.1为主系统M的混沌状态图.主系统M可写成 x=Ax+B(Dx),其中x=x1x2x3,A=18790111014.2860,B=9700,D=100,(Dx)=12(|x1(t)+1|x1(t)1|)属于扇区0,1.现在假设输出向量p=x1(t)x2(t)x3(t)T,q=z1(t)z2(t)z3(
18、t)T.假设初始状态x(0)=422T,z(0)=313.2T,即e(0)=874应用数学2023图4.1主系统M状态图111.2T.取多面体P=2,22,21,1,利用LMI工具箱求得可行解P与,c,T,如下P=1.41750.72060.03060.72063.87611.16500.03061.16501.9084,=20,c=34,=4.8,=10.令T=0.05,脉冲增益矩阵如下K=100010001,H1=000.02000000,H2=000000.02000,H3=000000000.025.假设脉冲步长一致,即T=T.图4.2为主从系统的状态分量图,图4.3为相应的误差系统状
19、态分量图,很明显可看出主系统M与从系统S产生同步现象.00.511.5t-15-10-5051015x(t)x1(t)z1(t)x2(t)z2(t)x3(t)x3(t)图4.2主从系统状态分量x(t),z(t)00.050.10.150.20.25t-2-1.5-1-0.500.511.5e(t)e1(t)e2(t)e3(t)图4.3误差系统状态分量e1(t),e2(t),e3(t)例2考虑二维Lure型动态网络主从系统,主系统M参数选取如下x=x1x2,A=2012,B=0.20.3,D=10,(Dx)=12(|x1(t)+1|x1(t)1|)属于扇区0,1.相应地,从系统S与主系统M参数一
20、致.令C=1001,即输出向量p=x1(t)x2(t)T,q=z1(t)z2(t)T.为了验证定理3.2的有效性,取多面体P=4,4 2,2,利用LMI 工具箱求得可行解P,c,如下P=0.98100.26580.26580.8413,=0.6,c=24,=2.8.第 4 期王万林等：Lure主从系统的二次反馈型脉冲同步控制875选取满足定理3.2条件(3.10)中的脉冲增益矩阵K,Hi,i=1,2与T,如下K=1001,H1=0.040.0200,H2=000.040.02,T=1.5.取一组误差系统初始状态ei(0)P(i=0,1,2,3),在脉冲步长一致即T=T下进行数值模拟,其中e0=
21、42,e1=32,e2=42,e3=32.图4.4为相应的误差系统状态分量图,很明显只要误差系统初始状态在多面体P所包含的范围内,主系统M与从系统S产生同步现象.00.511.522.533.5t-4-3-2-101234e(t)e01(t)e02(t)e11(t)e12(t)e21(t)e22(t)e31(t)e32(t)图4.4误差系统状态分量ei(t),i=0,1,2,35.总结本文研究了二次输出反馈脉冲控制下Lure混沌主从系统的同步问题.基于Lyapunov稳定性理论,结合多面体凸组合与线性矩阵不等式(LMI)方法得到了在二次输出反馈脉冲控制下Lure混沌主从系统同步的充分条件.最后
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31、AM Press,1994.Quadradtic Feedback Type Impulsive SynchronizationControl of Lure Master-Slave SystemWANG Wanling,HUANG Zhenkun,ZHAO Ling(School of Sciences,Jimei University,Xiamen 361021,China)Abstract:In this paper,the impulsive synchronization control of Lure master-slave system isstudied.For the c
32、ase that the impulsive controller has quadratic feedback,two new synchronizationstrategies of master-slave system under quadratic output feedback type impulsive control are designed byusing polytope convex combination,linear matrix inequality(LMI)and Lyapunov stability theory.Finally,the feasibility and effectiveness of the results are verified by numerical simulations.Key words:Impulsive control;The quadratic feedback;Synchronization;Lure master-slave system