第2章 系统辨识数学模型及常用输入信号

1、 11111111( )( )( )( )( )( )( )aaaaaabbbbnnnnnnnnnabndy tdy tdy taaa y tdtdtdtdu tdu tbbb u tnndtdt111111( )( )( )( )( )bbbaaaannnnnnnbsbsbY sB sG sU ssa sasaA s11( )(1)()(1)()abnanby ka y ka y knbu kb u knz121211( )( )( )1bbaannnnb zb zb zz y kG zz u ka za z11() ( )() ( )A zy kB zu k11212()1aannA za

2、 za za z 11212()bbnnB zb zb zb z111() ( )() ( )() ( )A zy kB zu kD zk11212()1ddnnD zd zd zd z 11() ( )() ( )( )A zy kB zu kk111() ( )() ( )() ( )A zy kB zu kD zk 零均值白噪声)(k11() ( )() ( )A zy kD zk1() ( )( )A zy kk1( )() ( )y kD zk1111()()( )( )( )()()B zC zy ku kkF zD z1212112121121211212()()1()1()1b

3、bffccddnnnnnnnnB zbb zb zF zf zf zf zC zc zc zc zD zd zd zd z ,bn,fn,cndn( )g tG( )u t( )y t0)()()(dtugty( )( )utt)()()()(0tgdtgty()G j( )( )G sL g t1( )( )g tLG s( )( )u tt( )g t( )G ss()G j()( )j tG jg t edt( )g k0,1,2,k 0( )() ( )kiy kg ki u i( )g k1()G z10()( )kkG zg k z)(tu)(ty)(tg)(tg)(1zG )(t

4、u)(ty)(tg)(tg)(1zGG( )u t( )y t)(tu)(ty)(tgTTk) 1( kT)(tu0)()()(dtugty)(tu)(tyT),(tu)(ty)(tgT, 2 , 1 , 0)()() 1()()()()(kkTgtgTktkTkTytykTutu1032203020200)()()()0()2()()()2()0()3()()3()()3()()3()()3()0()()()0()2()()2()()2()()2()0()0()()()(NiTTTTTTTTTTTTiTNTuiTgTNTyuTgTuTgTugTdTugdTugdTugdTugTyuTgTug

5、TdTugdTugdTugTygTudTugTy令)0()()()()2()0()(00)0()()()0()()2()(uTuTNTuTuTuuTuuUTNTgTggGNTyTyTyYTUGY YUTG11)(tg)(1zG12111111)()2() 1 (1)(kknnnnzkgzgzgzazazbzbzGnnniinnniinnniinnnzigangzigangzigangzgagzgzbzb2122)1(111121111)()2()() 1()()() 1 ()2() 1 ()()2() 1 (10010001121121ngggaaaabbbnnn)22()() 1()()2(

6、) 1() 1() 1()(),(lkglkglkglkgkgkglkgkgkgklH)2()2() 1() 12() 1()() 1() 3()2()()2() 1 (11ngngngaaangngngngggngggnn) 1 ,(nHibia( )g t1231123123123()1b zb zb zG za za za z)2()2() 1() 12() 1()() 1() 3()2()()2() 1 (11ngngngaaangngngngggngggnn1239.4910778.5638895.9305069.4910778.5638895.9305062.8459728.563

7、8895.9305062.8459720.1447.157 310691aaa 1232.232575,1.764088,0.496585aaa 563889. 8491077. 9157039. 712326. 27641. 1012326. 2001321bbb)()2() 1 (10010001121121ngggaaaabbbnnn0 . 0,4875. 6,1570. 7321bbb32121111114966. 07641. 12326. 214875. 61570. 71)(zzzzzzazazbzbzGnnnn( )v t( )g t( )u t( )y t( )v t( )z

8、 t+)(tg0)()()(dtugtz)(ty)(tv)()()()()()(0tvdtugtvtzty)(tu)()()()()()()(0tutvdtutugtutydttutvTddttutuTgdttutyTTTTTT0000)()(1lim)()(1lim)()()(lim)(),(tutv0)()(1lim0dttutvTTT)(tu)(uyR)(tydttutyTRTTuy0)()(1lim)()(tu)(uuRdttutuTRTTuu0)()(1lim)()(uyRdRguu)()(00dRgRuuuy)()()(0Wiener-Hopf方程)(tg)(uuR)(tu)(ty

9、)(uyR)(tudttutvTddttutuTgdttutyTTTTTT0000)()(1lim)()(1lim)()()(lim( )u t( )y t( )v t( )z t)(tg)(uuR)(uyR)()(2uuR)()()()()()(2200gdgdRgRuuuy)(1)(2uyRg)(g2、伪随机信号及其产生方法伪随机信号及其产生方法1()JM|log( | )log( | )Typ yp yME 1det()M0wE(w)()(2wR2)(wS2( )( )wR 2( )wS ( )x k2( )( ) ()( ),0, 1, 2,xR nE x k x knnn ( )n1

10、0( )00nnn2( )( )jnxxnSR n e111( ,)iiif )(modMAxxii1 ixMAx,0Mxii iMA,0 x0 x010203040506070809010000.10.20.30.40.50.60.70.80.91kv(0,+1)均 匀 分 布 的 白 噪 声l 混合同余法混合同余法混合同余法产生伪随机数的递推同余式为：Mxii icMAx,0)(mod(1McAxxii34212nAnc0 xn7,235nM,2)6/35 . 0(35c( )uuR2aNTN02a NT2/ ) 1(N2/ ) 1(N( )x tt01234567aa1x2xkxnx11

11、0,000101,011 1A2A3A4ATime stepOutput of modulo-2 adder00111110011120001131000140100050010061001071100180110091011010010111110101Output of Output of Output of 1A2A3AOutput of 4A12 nN12 nN2/) 1(N2/) 1(N234576a-aTN1,1rqxD xD xrqN:10001001 10101 1 1: 1 1 1 10001001 1010rxD x: 01 1 1 10001001 101qD x12N

12、0011( )( ) ()( ) ()TNxxRx t x tdtx t x tdtTN02220011(0)( )( )TNxxRx t dtx t dtaTN2211( )/(1)xxNaNRaNN ii) 2( )x t2a( )x tt01234567at( ) ()x t x t2at012345672a22aN()x tt01234567a2222(1)( )21(1)xxaNRaNNaN 2( )/xxRaN 1,2,1N10011( )( ) ()() ()NNxxkRx t x tdtx kx kNN 2() ()x kx ka 2() ()x kx ka 2( )()xxa

13、RN() ()x kx k ()()x kx k 2a2a2a2a2a()x k()x k ()()x kx k ()x k ()x k11,22NN2211( )()22xxaNNaRNN22.5() 0,2N( ) (2 )x t x t 2at012345672a( ) (2.5 )x t x t 2at012345672a( )x tt01234567a(2.5 )x t t01234567a2.520.57a20.57a20.57a20.57a2( )xxaRN02(0)xxRa 21( )(1)xxNRaN1,2,1N2( )xxaRN2( )xxaRN2211( )/(1)xxN

14、aNRaNN ) 1()(xxR222002200sin(/ 2)22(1)( )( )()/ 2xxnnnaaNnNNn 022TN2aNTN02a)(xxR( )xxR(1)(2)( )( )( )xxxxxxRRR2(1)(2)21(1)(1)( )0(1)( )/xxxxaRNNRaN 2aNTN02a)(xxR2aNTN021NaN)(xxR(1)(2)( )( )( )xxxxxx(1)(1)( )( )jxxxxRed0(1)( )jnxxnnRc e (1)( )xxR0/2(1)/21( )TjnnxxTcRedT 0()n 0011()22jnjnede 000()(1)(

15、)( )jnjnjxxnnnnjnnnc eedc edced 0jne )(20)(00ndedeenjjjn(1)0( )2()xxnncn 0n/2/2(1)(1)/20/2202222212( )( )cos21(1)(1)cos0cos2111(1)sin(cos1sin)21(1 cos)1 sin(/ 2)()/ 2TTjnxxxxTNcRedRdTTaddNNaNNaNNNaNNN 22200sin(/ 2)1()/ 2naNNNn0sinlim1xxx22(1)000222002200sin(/ 2)21( )()/ 2sin(/ 2)2(1)2(1)( )()/ 2xxnn

16、nna NnNNnnaNaNnNNn 2(2)2( )( )xxaN 2( ) (1)(2)222002200( )( )( )sin(/ 2)22(1)( )()/ 2xxxxxxnnnaaNnNNn (0)24()xx222aN2234()xx2N3db0.707 (0)2 /32sin(/2)0.707/2( )xx02N23,2 , NNn ( )0 xx02( )( )wR 2( )wS ( )uuR2aNTN0(0)24()xx222aN0.707 (0)2234()xx2Namax23 (1.2 1.5)/sNTmin2 /N 0( )( )()xyxxRgRd0( )( )()

17、TxyxxRgRd( )xxR2aNTN021NaN221( )( )xxNaRaNN ( )xxR2211( )/xxNaNRaNN 2(1)(2)21(1)(1)( )0(1)( )/xxxxaRNNRaN 2202201( )()( )1( )( )TxyTNaRagdNNNaaggdNN 20( )TagdN21NSaN( )( )/xygRCS(0)2(0)/xygRCS01( )( ) ()TxyRx t y tdT101( )() ()NxyiRx iy iNa10( )( ()()NxyiaRsign x iy iN10( )( ()()rNxyiaRsign x iy irN

18、)(),(tytx( )xyRtteetg10/101)(ms15k100)(tx)(tyF100.10.20.30.40.50.60.70.80.91-2.5-2-1.5-1-0.500.511.522.521NSaN()()/xyg iRiCS (0)2()/xygRiCS 10( )( ()()NxyiaRsign x iy iNTotaldelanCR,clc; close all; clear all;R = 100e3; % 100k ohmC = 1e-6; % 1uftc = R*C; % Time Constant% generate M-sequencen=5;a=2; %

19、 Level of the PRBS 1 - -a 0 - +adel = 15e-3; % clock pulse periodN=2n-1; % Period of M sequenceTotal=2*N; % Generate m-sequence using the genPRBS functionOut = genPRBS(n,a,del,total);% Generate response y(t) of the systems = tf(s); G = 1/(tc*s+1)tf = total*del;tim = 0:del:tf-del;y = lsim(G, Out, tim

20、);%plot input and output of the systemfigurestairs(tim,Out);axis(0 1.0 -2.5 2.5);hold onplot(tim,y,r);hold off% Compute Rxy(i*del)sum = 0.0;Rxy = ;iDel_vec=;for i=1:N tau=i-1; iDel_vec=iDel_vec;tau*del; for j=1:N sum=sum+sign(Out(j)*y(j+tau); endRxy_i = (a/N)*sum;sum=0.0;Rxy = Rxy; tau Rxy_i;end% Compute ghat & gind = length(Rxy);C = -Rxy(ind, 2);S = (N+1)*a2*del/N;Rxy_iDel = Rxy(:,2);ghat=(Rxy_iDel+ C )/S;ghat(1)=2*ghat(1);g = 10*exp(-10.*iDel_vec);