外部法用系统输入输出之间关系来描述信号与chap8

——用n个状态变量8.1.1一个电路系统如下图所示，以u(t)和iC(t)为输变量uC(tiL1(tiL2(t)的d11du CCddCC8.1.1一个电路系统如下图所示，以u(t)和iC(t)为输变量uC(tiL1(tiL2(t)的d11du CCddCCd11d S1uuR L1uCdd11CSd1 1udiR2iL2uS2uCCSdLLd222d1C1C Cddi11 uuS1CdLLL111d1R1 du iuCSLLL222这是由三个内部d1C1C Cddi11 uuS1CdLLL111d1R1 du iuCSLLL222这是由三个内部变量uC(t)、iL1(t)和iL2(t)构成若初始值uC(t0)、iL1(t0)和iL2(t0)已知，则根据t≥t0时的给定激励uS1(t)和uS2(t)就可惟一地确定在t≥t0时。u(t)R2iL2(t)uS2(ti(t)(t)(tC状态变量是描述状态随时间t变化的一组变量，它们系统中任何响应均可表示成状态变量及输入的线性组合状态变量应线性独立(3)状态变量的选择并不是唯一。(4)对n阶动态系统需有n个独立的状态变量，通常用x1(t)x2(t)、…、xn(t)表示第一步是根第一步是根据系统的初始状态求出状态变量第二步是用这些状态变量来确定初始时刻以后的系统输组代数方程。通常将状态方程和输出方程总称为动态方程或系统方程8.1.2对于一般的n阶多输入多输，&1a11a12LLa1nb11e1b12e8.1.2对于一般的n阶多输入多输，&1a11a12LLa1nb11e1b12e2Lb1pep&aabeeLxxxe2 L 2 bn1e1bn2e2Lbnpepnan1x1an2Lanny1c11x1c12LLc1nd11e1d12e2d1pep xcdeedxxe2 p2 L 方yqcq1x1cq2Lcqndq1e1dq2e2dqpep……b1pe1ta1nx1t&1MLLLLttbbaa2n 2p 22b1pe1ta1nx1t&1MLLLLttbbaa2n 2p 22MMMMMMMMtn LeLaxtnnn&(t)Ax(t)Be(ty1(tc1nx1(t)d1pe1(t)LLLL(t(t(tccdd2n2p22MMMcqMMMdqMMMp(tq(tLLy(t)Cx(t)De(t&(t)Ax(t)Be(tb1pa1nLLMLLaabbB2pA2nMMMMMaLann&(t)Ax(t)Be(tb1pa1nLLMLLaabbB2pA2nMMMMMaLannbLby(t)Cx(t)De(tc1nd1pLLLLccddC2nD2pMMMMMMcLcdLdqnqqA、B、C、D都为常数矩阵：C为q×n矩阵，称为输出矩阵D为q×p矩阵，称为直达矩阵n阶多输入多输出线性离散系统，它有n阶多输入多输出线性离散系统，它有个输入，q个输出，n个状态变量，类x(k1)Ax(k)Be(ky(k)Cx(k)De(k。&&x(k1)Ax(k)Be(k)y(k)Cx(k)&(t)Ax(t)Be(ty(t)Cx(t)De(t1.2.3.4.1.2.3.4.B)建立状态方CdLd由于B)建立状态方CdLd由于CLdd对接对接写写电路如图，以电阻R1上的电压uR1和电阻R2上的电流x1(t)=x2(t)=L&1(t)＋R1x1(t)x2(t)us1(t电路如图，以电阻R1上的电压uR1和电阻R2上的电流x1(t)=x2(t)=L&1(t)＋R1x1(t)x2(t)us1(t&＋iiR2(t)，列右网孔KVL方程：R2iR2(tus2(tx2(t0x1(tus1(t&1(tLCL&(t)(t) (tR2CR2C 2s1 (t)[x(t)(t(t)Rx(t2sR 2uR(t)R1x1(t11 (t)[x(t)(t2uR(t)R1x1(t11 (t)[x(t)(t2sR2x(t00y(t(t1111x2(t (tsy2(tR2R22.有两个连续系统，描述它们的微分方程(1)y(t)3y(t)2y(t)52.有两个连续系统，描述它们的微分方程(1)y(t)3y(t)2y(t)5y(t)e(t(2)y(t)3y(t)2y(t)5y(t)6e(t)3e(t)4e(tx1y(t)：解：(1) y(t)2 y(t3&1y(t)&2y(t)01011&1x0223x1&3y(t)5y(t)2y(t)3y(t)e(t523exy 1232y1x3(2)y(t)3y(t)(2)y(t)3y(t)2y(t)5y(t)6e(t)3e(t)4e(t引入一个辅助函数q(t)q(t)3q(t)2q(t)5q(t)e(tx1q(t) q(t)2 q(t3Qy(t)6q(t)3q(t)4q(t输出方y4x13x26&q(t)12&q(t)23&q(t35q(t)2q(t)3q(t)e(t5x12x23x3输出方程为：y4x13x26&1输出方程为：y4x13x26&10x110&1x0223x1xy32(1)y(t)3y(t)2y(t)5y(t)e(t(2)y(t)3y(t)2y(t)5y(t)6e(t)3e(t)4e(tny(n1)(t)Lay(t)ny(n1)(t)Lay(t)ay(ty(n)(t)10bne(n)(t)bn1e(n1)(t)Lb1e(t)b0e(t为更具一般这里设y(t)和e(t)的最高阶次相同引入一个辅助函数q(t)q(n)(t)an1q(n1)(t)La1q(t)a0q(t)e(t&1q(t)x1q(t) q(t)&q(t)232 q(t&q(n)(t3n，a0q(t)a1q(t)Lan1q(n1)(t)e(ta0x1a1x2an1ey(t)bnq(n)(t)y(t)bnq(n)(t)bn1q(n1)(t)Lb1q(t)b0q(t将q(nte(taq(taq(tLq(n1)(t)代入01y(t)be(t)abq(t)abq(t)Lbq(n1)(t n bn1q(n1)(t)Lb1q(t)b0q(tbne(t)(b0a0bn)q(t)(b1a1bn)q(tLa)q(n1)(tb (b0a0bn)x1(b1a1bn)x2L(bn1an1bn)&001001LL00x22&001001LL00x22MMMLLn1n1&x1xybab(baba2Lb 0 1 nMxn3.3.选积分器的输出(或微分器的输入)作为状态变量围绕加法器列写状态方程或者输出方程图所示，从右到左分别取x1图所示，从右到左分别取x1x2、x3&1y436&1232&35x12x23x3例例设状态变量x1(t)&1&22x13x2设状态变量x1(t)&1&22x13x2y(t)=8x1+201x011e3 2 2xy12x 2x1xx1&&设状态变量x(t)x12&4x1xx1&&设状态变量x(t)x12&423x1y132ey(t)=2x0x11y1211e2 2 2 2x1设状态变量x(t)x12x&1x1e&22x2e系统输出端，有x1设状态变量x(t)x12x&1x1e&22x2e系统输出端，有x2y(t)=0x1x1y111e2 2 2 22(sH(s)63sH(s)ss2(sH(s)63sH(s)ss2(s 3ss2(s2H(s)gs1s3s4.方4.方法一：将H(s)转化为微分方程，再建立状态方程2(s例5已知系统函数H(s)3syt3yt2(s例5已知系统函数H(s)3syt3yt2yt2et8etx01011e3 2 2xy12x 22(sH(s)3s设状态变量x1(t)&1&222(sH(s)3s设状态变量x1(t)&1&22x13x2y(t)=8x1+2xx010y111e2x3 2 2 2已知一个二输入已知一个二输入、二输出系统由下列微分方程态变量，分别记为x1(t)x2(t、&12x2e1&2x3&3&1态变量，分别记为x1(t)x2(t、&12x2e1&2x3&3&1x33x1y1(t)y2(t)x1010x132xeey02312312 202&10x1 000110e2232 3例解:s1ss&x1121y2e例解:s1ss&x1121y2es&1&22&23&33x3&14x13x32ey(t)=12&3x23y2(t)=-x3+&(t)1x(&(t)1x(t)1e(t例80x(ty(t)y(t)=x1(t)=–4x1(t)+x2(t)+y(t)=–4x1(t)+x2(t)+e=–4[–4x1(t)+x2(t)+e(t)]+[–3x1(t)+e(t)]+e=13x1(t)–4x2(t)–3e(t)+ey+ay+by=(13–4a+b)x1+(–4+a)x2+e(t)+(a–3)ey+4y+3y=e(t)+通过适当选取状态变量把描述通过适当选取状态变量把描述离散系统的输入输出关系的n阶差分方:有两个(1)y(k)3y(k1)2y(k2)5y(k3)e(k(2)y(k)3y(k1)2y(k2)5y(k3)6e(k1)3e(k(1)y(k)3y(k1)(1)y(k)3y(k1)2y(k2)5y(k3)e(k解：选取状态变量如下 x1(k)y(kx(k)y(k2)2x(k)y(k3x1(k1)y(k2)x2(kx(k1)y(k1)x(k2x3(k1)y(k35y(k3)2y(k2)3y(k1)e(k5x1(k)2x2(k)3x3(k)e(ky(k)x3(k5x1(k)2x2(k)3x3(k)e(kx1(k0x1(kx1(k0x1(k0010(k1)1x(k)0e(k22x1(kx(k)e(ky(k)2(2)y(k)3y(k(2)y(k)3y(k1)2y(k2)5y(k3)6e(k1)3e(k引入辅助函数q(k)q(k)3q(k1)2q(k2)5q(k3)e(kx1(k)q(kx(k)q(k2)2x(k)q(k3x1(k1)q(k2)x2(kx(k1)q(k1)x(k2x3(k1)q(k35q(k3)2q(k2)3q(k1)e(k5x1(k)2x2(k)3x3(k)e(k(2)y(k)3y(k1)2y(k2)(2)y(k)3y(k1)2y(k2)5y(k3)6e(k1)3e(ky(k)6q(k1)3q(k3x2(k)6x3(k输出方程x1(k0x1(k0010x(k1)1x(k)0e(k22x1(kx(ky(k)32(1)选取延(1)选取延时器(即z-)的输出作为状态变(2)围绕加法器列写状态方程和输出方程例10已知一个二输入二输出的离散系统方框图，由左端加法器列为x1例10已知一个二输入二输出的离散系统方框图，由左端加法器列为x1(k1)a1x1(k)a2x2(k)e1(kx2(k1)x1(kx3(k1)a3x3(k)e2(k由右端加法器列为y1(k)x2(k)x3(ky2(k)x3(k)e1(kx1(k1)a1x1(k)a2x2(k)e1(kx2(k1)x1(kxx1(k1)a1x1(k)a2x2(k)e1(kx2(k1)x1(kx3(k1)a3x3(k)e2(kx1(k0x1(k0e(k000(k1)(k)1(k22(k(ka3 3y1(k)x2(k)x3(ky2(k)x3(k)e1(k1x1(ky(k0e(k10(k)11(k10(k2 (k31．把1．把H(z)转换为差分方程，由差分方程建立状态方2．由H(z)画出系统模拟方框图或信号流图z1H(z)z1H(z)12z11zz状态方程x1(k+1)=x2x2(k+1)=x1(k)–2x2(k)+输出方程y(k)=–x1(k)+某离散系统有两个输入e1(k)、e2(k)和两个输出y1(k)某离散系统有两个输入e1(k)、e2(k)和两个输出y1(k)y2(k)，其信号流图如图示，列写该系统的状态方程和输出方p1(k2x1(kp2(k)=3p1(k)-x3(k)1z=6x(k)+5x(k)+e132x(kx(k2z1x1(k1)3x1(k)x2(kx2(k1)x1(k)2x2(k)p2(k)e1(kx2(k1)7x1(k)2x2(k)5x3(k)e1(k)e2(kp2(k)(k3zx3(k1)p2(k)2x3(k)6x1(k)7x3(k)e2(k zx1(k0x1(k0e(k1(k1)x1(k0x1(k0e(k1(k1)5x(k)1(k7221(k(k 30x1(ky(k00(k1(k22 (k31zp1(k)=2x1(k)1z2.根据矩阵函数积分的概念，一个n维状态2.根据矩阵函数积分的概念，一个n维状态矢量x(t)[x(t)],,ℒ[x2(t)]ℒℒ[x1(tℒ[xn(t[X(s)ℒ[x(t它是n维矢量ℒ[e(tℒ[y(tE(s)Y(s)pq&(t)&(t)Ax(t)Be(ty(t)Cx(t)De(t(8-sX(s)-x(0-)=AX(s)+(sI-A)X(s)=x(0-)X(s)=(sI-A)-1x(0-)+(sI-A)-=Φ(s)x(0-)式中Φ(s)=(sI-A)-1常称为。Y(s)=CX(s)=CΦ(s)x(0-)+[CΦ(s)B+D]Y(s)=CΦ(s)x(0-)+[CΦ(s)BYzs(s)YziY(s)=CΦ(s)x(0-)+[CΦ(s)BYzs(s)Yzi(s)Yzi(s)=CΦ(s)x(0-Yzs(s)=[CΦ(s)B+D]E(s)=H(s)H(s)=CΦ(s)Bh11(s)h1p(s)LL (s)(s)h(s)H(s)2MMM (s)(s)h(s)Lq转移函数矩阵中第i行第j列的元素Hij(s)ijYzs(sCΦ(sB+Yzs(sCΦ(sB+DE(sH(s(s)]L1[H(s)E(s)]L1[H1E(s)]yzs(t)1[H(s)]e(t)h(t)e(t即其中h(t1[H(s)]为冲激响应H(s)=H(s)=CΦ(s)BH(s)C(sIA)-1BDCadj(sIA)BDdet(sIdet(sIΦ(s)的极点就是H(s)的极点，即|sI-A|=0具体地说，如果系统矩阵A的特征值全部位于s平面如果系统函数矩如果系统函数矩阵H(s)在j轴上收（亦即H(s)的所有元素在j轴上收敛jIsjBH(j)H(s)例描述LTI因果系统的状态方程和输出方程&1(t)2x1(t)0e(ty(t)例描述LTI因果系统的状态方程和输出方程&1(t)2x1(t)0e(ty(t) 1x1(t)[1]e(t&(t4x(t(t 初始状态x1(0-)=3，x2(0-)=2，输入 =(t)02s(sIA)s s1 sadj(sI21(s)(sI(s2)(s3)sdet(sIX(s)=Φ(s)[x(0-)3s21(s2)(s3)1s1 3(s 12 (s2)(s3)ss3X(s) 3(s 12 (s2)(s3)ss3X(s)96(s2)(s3)s212e2t9(tx(t)3t6e299x(t121y(t) 1(t)(t119e3t6e2x2(t6(t)(tH(s)的极点就是|sI-A|=0的根|sI-由于H(s)的极点均在左半平面，故该因果系统稳定2.一个n维状态矢量x(k)的z变换,x(k)2.一个n维状态矢量x(k)的z变换,x(k)x(k)ZTxZ,Z21nX(z)Zx(k它是n维矢量同样地，输入、输出矢量的z变换简记e(k)E(z)Y(z)Zy(k)pqx(k1)Axx(k1)Ax(k)Be(k)y(k)Cx(k)De(k(8-对式(8–90)取zzX(z)-zx(0)=X(z)=(zI-A)-1zx(0)+(zI-A)-设Φ(z)