隨機(jī)變時(shí)滯模糊神經(jīng)網(wǎng)絡(luò)的均方漸近穩(wěn)定性

1、第34卷/第4期/隨機(jī)變時(shí)滯、均方穩(wěn)定利用隨機(jī)過(guò)程、算子、模型可參考隨機(jī)變時(shí)滯模糊神經(jīng)網(wǎng)絡(luò)的均方漸近穩(wěn)定性趙 冬1, 陳立平2, 吳然超2(1.淮北職業(yè)技術(shù)學(xué)院基礎(chǔ)部,安徽淮北 235000;2.安徽大學(xué)數(shù)學(xué)科學(xué)學(xué)院,安徽合肥 230039)摘要:基于線性矩陣不等式、Lyapunov Krasovskill泛函和隨機(jī)分析方法,研究了一類具有隨機(jī)變時(shí)滯模糊神經(jīng)網(wǎng)絡(luò)的均方漸近穩(wěn)定性,得到了均方漸近穩(wěn)定性的充分條件.用數(shù)值例子說(shuō)明了結(jié)果的有效性.關(guān)鍵詞:均方漸近穩(wěn)定;隨機(jī);變時(shí)滯;模糊細(xì)胞神經(jīng)網(wǎng)絡(luò)中圖分類號(hào):O175.14 文獻(xiàn)標(biāo)識(shí)碼:A 文章編號(hào):1000 5854(2010)04 0389 05

2、細(xì)胞神經(jīng)網(wǎng)絡(luò)(CNNs)自從1988年由Chua與Yang提出以來(lái),在理論和應(yīng)用上得到了廣泛地研究.它在圖像處理、模式識(shí)別、信號(hào)處理、聯(lián)想記憶等方面有重要應(yīng)用1.1996年,Yang等2把模糊邏輯理論和傳統(tǒng)的細(xì)胞神經(jīng)網(wǎng)絡(luò)結(jié)合起來(lái),提出了另一類基本細(xì)胞神經(jīng)網(wǎng)絡(luò) 模糊細(xì)胞神經(jīng)網(wǎng)絡(luò)(FCNN).近年來(lái)對(duì)具有時(shí)滯的模糊神經(jīng)網(wǎng)絡(luò)穩(wěn)定性研究受到學(xué)者們的廣泛關(guān)注,并取得了一些好的成果,這其中包括常時(shí)滯、變時(shí)滯、分布時(shí)滯及擴(kuò)散時(shí)滯3 6.然而,這些結(jié)果都是針對(duì)確定性系統(tǒng)的.正如Haykin7指出,在真實(shí)的神經(jīng)網(wǎng)絡(luò)系統(tǒng)中,神經(jīng)信號(hào)傳輸是一個(gè)受隨機(jī)因素影響充滿噪音的過(guò)程.在這種噪音的影響下,系統(tǒng)的軌道將會(huì)變成一個(gè)

3、隨機(jī)過(guò)程.所以在人工神經(jīng)網(wǎng)絡(luò)的實(shí)現(xiàn)過(guò)程中,就必須考慮這種隨機(jī)過(guò)程.近年來(lái),學(xué)者們對(duì)隨機(jī)神經(jīng)網(wǎng)絡(luò)穩(wěn)定性做了大量研究,見文獻(xiàn)8及其所附文獻(xiàn).據(jù)了解,國(guó)內(nèi)外有關(guān)隨機(jī)模糊神經(jīng)網(wǎng)絡(luò)穩(wěn)定性研究的文獻(xiàn)不多.文獻(xiàn)9利用Lyapunov函數(shù)和半鞅收斂定理研究了常時(shí)滯隨機(jī)模糊神經(jīng)網(wǎng)絡(luò)的均方漸近穩(wěn)定性.然而,到目前為止,關(guān)于隨機(jī)變時(shí)滯模糊神經(jīng)網(wǎng)絡(luò)均方漸近穩(wěn)定性方面的研究很少.正因?yàn)槿绱?本文中,筆者利用線性矩陣不等式(LMI)、Lyapunov Krasovskill泛函和隨機(jī)分析的方法,研究了一類具有隨機(jī)變時(shí)滯模糊細(xì)胞神經(jīng)網(wǎng)絡(luò)的均方漸近穩(wěn)定性,得到了判斷這類神經(jīng)網(wǎng)絡(luò)均方漸近穩(wěn)定性的充分條件,并且通過(guò)一個(gè)數(shù)值例子說(shuō)

4、明了所得結(jié)果的有效性.1 模型及預(yù)備知識(shí)考慮下述具有變時(shí)滯隨機(jī)模糊細(xì)胞神經(jīng)網(wǎng)絡(luò):dxi(t)=-cixi(t)+nj=1nj=1 aijfj(xj(t)+nnj=1! ijfj(xj(t- j(t)+!Tijuj(t)+j=1nn ijfj(xj(t(t- j(t)+Hijuj(t)+Iidt+j=1j=1 !ij(t,xj(t),xj(t- j(t)dj(t),(1) xi(t)=#i(t),- #t#0,i1,2,%,n.其中:ci>0為自反饋?lái)?xiàng), 模糊向后最大模板、模糊向前最小模ij, ij,Tij和Hij分別代表模糊向后最小模板、板、模糊向前最大模板;!和代表模糊&and

5、和&or算子;xi(t),ui(t)和Ii分別代表第i個(gè)神經(jīng)元狀態(tài)變量、輸入和偏倚.變時(shí)滯 j(t)滿足0# j(t)# .fj()表示神經(jīng)元激勵(lì)函數(shù),!()=(!ij()n)n為擴(kuò)散系數(shù)矩陣.(t)=1(t),2(t),%,n(t)為定義在具有自然濾波Ftt>0且完備的概率空間(,Ft,P)上的n維Brownian運(yùn)動(dòng).系統(tǒng)(1)的初值函數(shù)#LF0(- ,0,R),這里L(fēng)F0(- ,0,R)表示所有F0可測(cè)且滿足2n2nTF|0- #(s)|ds<+的隨機(jī)過(guò)程#(s),- #s#0.E(表示相應(yīng)于概率測(cè)度P數(shù)學(xué)期收稿日期:2009 08 18;修回日期:2009 12 1

6、5基金項(xiàng)目:安徽省自然科學(xué)基金(070416225);安徽省高校省級(jí)自然科學(xué)基金(KJ2007A003): ),(390(望.系統(tǒng)(1)有唯一解記為x(t)=(x1(t),x2(t),%,xn(t)T,t>0.定義C=C(- ,0,Rn)為所有從- ,0上到Rn連續(xù)函數(shù)組成的全體,對(duì) x(t)=(x1(t),x2(t),%,xn(t)TRn,定義范數(shù),x(t),=(i=1 n|xi(t)|2)1/2.設(shè)C2,1(- ,+)Rn,R+)表示在- ,+)Rn上關(guān)于x2次可微,關(guān)于t1次可微的所有非負(fù)函數(shù)V(t,x)的全體,對(duì)任意V(t,x)C2,1(- ,+)Rn,R+)定義如下算子:LV(

7、t,x)=Vt(t,x)+Vx(t,x)-cixi(t)+ ijfj(xj(t- j(t)+j=1其中Vt(t,x)=Vxx=nj=1 aijfj(xj(t)+nj=1! ijfj(xj(t- j(t)+(2)n,Vx(t,x)=2,!=!(t,x(t),x(t- (t).i yjn)ntrace!TVxx!,2,%, x1 x2 xn在本文中,假設(shè)下列條件成立:H1)激勵(lì)函數(shù)fj()滿足Lipschitz條件,即 x1,x2Rn有|fj(x1)-fj(x2)|#Lj|x1-x2|,Lj>0,jN.(t,x(t),x(t- (t)R+)R)RRTnnn)n(3)H2)!(t,x(t),x

8、(t- (t):R+)Rn)RnRn)n滿足局部Lipschitz條件和線性增長(zhǎng)條件,且對(duì),存在正對(duì)角陣 P=diag(p1,p2,%,pn),D=diang(d1,d2,%,dn),使得 trace!(t,x(t),x(t- (t)(P+d0L)!(t,x(t),x(t- (t)#xT(t)Ex(t)+xT(t- (t)Fx(t- (t).其中 L=diag(Li),d0=%H3)=*i=1(4) di.PA00<0,(5)n2DA+D(| |+| |)+R*&其中%=-2PC+P(| |+| |)+E-2LDC,&=(D+P)(| |+| |)+L-1FL-1-(1-

9、 )R,C=diag(ci),A=(aij)n)n,| |=(| ij|)n)n,| |=(| ij|)n)n.在假設(shè)H1)成立下,則系統(tǒng)(1)對(duì)應(yīng)的確定性系統(tǒng)dxi=-cixi(t)+nj=1j=1 aijfj(xj(t)+nnj=1! ijfj(xj(t- j(t)+!Tijuj(t)+j=1nn ijfj(xj(t(t- j(t)+Hijuj(t)+Iidt,j=1(6) xi(t)=#i(t),- #t#0,i1,2,%,n,*T有唯一的平衡點(diǎn)x*=(x*1,x2,%,xn).證明與文獻(xiàn)10類似,在此省略.*H4)!ij(t,x,x)=0.引理12 設(shè)x(t)=(x1(t),x2(t)

10、,%,xn(t)T和y(t)=(y1(t),y2(t),%,yn(t)T是系統(tǒng)(1)的2個(gè)狀態(tài)變量,則有|j! ijfj(xj)-! ijfj(yj)|#=1j=1|j ijfj(xj)-j ijfj(yj)|#=1=12 主要結(jié)論下面給出一個(gè)使系統(tǒng)(1)均方漸近穩(wěn)定的充分條件.14nnnnj=1 nn| ij,fj(xj)-fj(yj)|,| ij,fj(xj)-fj(yj)|.j=1(391(證 作變換yi=xi-x*,則系統(tǒng)(1)變?yōu)閐yi(t)=-ciyi(t)+nj=1 naijgj(yj(t)+j=! ijgj(yj(t- j(t)+1j=1nj=1 ijgj(yj(t- j(t)

11、dt+ !ij(t,yj(t),yj(t-nj(t)dj(t).(7)*其中y(t)=(y1(t),y2(t),%,yn(t)T,gj(yj(t)=fj(yj(t)+x*j)-fj(xj)且gj(0)=0.因此,研究系統(tǒng)(1)平衡點(diǎn)的穩(wěn)定性只要研究系統(tǒng)(7)零點(diǎn)的穩(wěn)定性.定義如下Lyapunov泛函V=V1+V2+V3,其中V1=i=1 npiy2i(t),V2=2 dii=1nN0yi(t)gi(s)ds,V3=ng(y(s)Rg(y(s)ds.tT- 結(jié)合H1),H2),H3)和引理1有如下列不等式:LV1(y(t)=2 piyi(t)-ciyi(t)+i=1nj=1j=1n aijgj(

12、yj(t)+! ijgj(yj(t- j(t)+ ijgj(yj(t- j(t)+j=1ntrace!T(t,y(t),y(t- (t)P!(t,y(t),y(t- (t)#2 pi-i=1nciy2i(t)+j=1 aijyj(t)gj(yj(t)+nj=1 (|n ij|+| ij|)|yi(t)|gj(yj(t- j(t)|+nntrace!T(t,y(t),y(t- (t)P!(t,y(t),y(t- (t)#i=1n pi- (|2ciy2i(t)+j=1 2aijyi(t)gj(yj(t)+ (|j=12 ij|+| ij|)|yi(t)+j=1 ij|+| ij|)|gj(yj(

13、t- j(t)+2trace!T(t,y(t),y(t- (t)P!(t,y(t),y(t- (t)=yT(t)(-2PC+P(| |+| |)y(t)+yT(t)2PAg(y(t)+gT(y(t- (t)P(| |+| |)g(y(t- (t)+trace!(t,y(t),y(t- (t)P!(t,y(t),y(t- (t),LV2(y(t)=2 digi(yi(t)-ciyi(t)+i=1nnnj=1T(8) aijgj(yj(t)+n! ijgj(yj(t- j(t)+ ijgj(yj(t- j(t)+j=1j=1trace!T(t,y(t),y(t- (t)P!(t,y(t),y(t-

14、 (t)#2 digi(yi(t)-ciyi(t)+i=1nnj=1 aijgj(yj(t)+ni=1j=1 2di(|Tnn ij|+| ij|)|gi(yi(t)|gj(yj(t- j(t)|)+ntrace!(t,y(t),y(t- (t)d0L!(t,y(t),y(t- (t)#2 digi(yi(t)+i=1ni=1j=1j=1 aijgj(yj(t)+g2i(yi(t)+i=1j=1 di(|n ij|+| ij|)| di(|nn2 ij|+| ij|)|gi(yi(t- j(t)+trace!T(t,y(t),y(t- (t)d0L!(t,y(t),y(t- (t)=gT(y(

15、t)(-2DC)y(t)+gT(y(t)(2DA+D(| |+| |)g(y(t)+Tt)|(392(trace!T(t,y(t),y(t- (t)d0L!(t,y(t),y(t- (t).LV3(y(t)=gT(y(t)RG(y(t)-(1- .(t)gT(y(t- (t)Rg(y(t- (t).由(8)-(10),再根據(jù)H2)有LV(y(t)#yT(t)(-2PC+P(| |+| |)+E-2LDC)y(t)+yT(t)2PAg(y(t)+gT(y(t)(2DA+D(| |+| |)+R)g(y(t)+gT(y(t- (t)(D+P)(| |+| |)+LFL-1-(1- )Rg(y(t-

16、 (t).LV(y(t)#(#0.(11)(12)(9)(10)即其中=yT(t),gT(y(t),gT(y(t- (t).這意味著系統(tǒng)(1)的平衡點(diǎn)是均方漸近穩(wěn)定的.如果令定理1中L=I,則有下面的推論.推論1 若H1),H2.):!(t,x(t),x(t- (t):R+)Rn)RnRn)n滿足局部Lipschitz條件和線性增長(zhǎng)條件,且對(duì) (t,x(t),x(t- (t)R+)Rn)RnRn)n,存在對(duì)角陣P=diag(p1,p2,%,pn),D=diag(d1,d2,%,dn),E,F>0,使得trace!T(t,x(t),x(t- i(t)(P+d0I)!(t,x(t),x(t-

17、 (t)#yT(t)Ex(t)+yT(t- (t)Fy(t- (t).其中 L=I,d0=i=1(13) di.(14)n%PA0 H3.)=*2DA+D(| |+| |)+R0<0.*&其中%=-2PC+P(| |+| |)+E-2DC,&=(D+P)(| |+| |)+F-(1- )R.C=diag(ci),A=(aij)n)n,| |=(| ij|)n)n,| |=(| ij|)n)n,則系統(tǒng)(1)的平衡點(diǎn)是均方漸近穩(wěn)定的.3 數(shù)值例子考慮下述二維的變時(shí)滯隨機(jī)模糊細(xì)胞神經(jīng)網(wǎng)絡(luò):dxi=-cixi(t)+2j=12j=1 aijfj(xj(t)+22j=1! ijfj

18、(xj(t- j(t)+!Tijuj(t)+j=122 ijfj(xj(t- j(t)+Hijuj(t)+Iidt+j=1j=1 !ij(t,xj(t),xj(t-,A=0.4 0.30.7 0. j(t)dj(t),i=1,2. -44 4 4(15)這里 C=, =,-4 44 4g1(x)=g2(x)=0.5(|x+1|-|x-1|),!11(x1(t),x1(t- 1(t)=0.1x1(t).0 !12(x2(t),x2(t- 2(t)=0.1x2(t- 2(t),!21(x1(t),x1(t- 1(t)=0.3x1(t),!22(x1(t),x2(t- 2(t)=0.3x2(t- 2

19、(t), 1(t)= 2(t)=0.3cost,5 0, =顯然,Li=1, =0.3,設(shè)E=F=I.H3),可以得到通過(guò)簡(jiǎn)單的計(jì)算可知P,D,E,F滿足H2).由定理1知,(15)的平衡點(diǎn)是均方漸近穩(wěn)定的.4 結(jié) 論 P=本文中,筆者利用線性矩陣不等式(LMI),Lyapunov Krasovskill泛函和It 微分公式,研究了均方意義下的一類具有變時(shí)滯隨機(jī)模糊細(xì)胞神經(jīng)網(wǎng)絡(luò)的穩(wěn)定性,給出了均方漸近穩(wěn)定性的判斷準(zhǔn)則,理論和數(shù)值例子,(393(參考文獻(xiàn):1 CHUALO,YANGL.CellularNeuralNetwork:TheoryandApplicationsJ.IEEETransCi

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