第二章自控习题课

1、uoi2=R2CuiuoR1R2试建立图所示电路的动态微分方程。试建立图所示电路的动态微分方程。解：解：输入量为输入量为ui，输出量为，输出量为uo。ui=u1+uou1=i1R1ic=Cducdt=dtd(ui-uo)i1=i2-icu1= R1+uouo-Cd(ui-uo)dtR2R2ui=uoR1-C R1R2+C R1R2+uoR2duidtdtduouoR1+C R1R2+uoR2=R2ui+C R1R2duoduidtdtCucR1R2uii1i2uoicC试建立图所示电路的动态微分方程。试建立图所示电路的动态微分方程。uouiR1LR2Ci1=iL+icuL=LdiLdtuoiL

2、=i2=R2uL=LR2duodtic= = +CducdtCLR2d2uodt2duodt + uoR2CLR2d2uodt2duodti1= +Cuoi2=R2输入量为输入量为ui，输出量为，输出量为uo。ui=u1+uou1=i1R1ic=Cducdt=dtd(ui-uo) 函数的拉氏变换。函数的拉氏变换。F(s)=s+1(s+1)(s+3)解解：A1=(s+2)s+1(s+1)(s+3)s=-2= -1(s+1)(s+3)A2=(s+3)s+1s=-3= 2F(s)= - 2s+31s+2f(t)=2e-3t-e-2tF(s)=s(s+1)2(s+2)函数的拉氏变换。函数的拉氏变换。解

3、解:f(t)= est +lim ests(s+1)2s=-2ddsss+2s -1 =-2e-2t+lim( est+ est)s -1 sts+22(s+2)2=-2e-2t-te-t+2e-t=(2-t)e-t-2e-2t函数的拉氏变换。函数的拉氏变换。F(s)=s+2s(s+1)2(s+3)解解:f(t)= est + ests+2(s+1)2(s+3)s=0s+2s(s+1)2s=-3+ lim s -1d est s+2s(s+3)ds= + e-3t+lim + 23112s -1(-s2-4s-6)est(s2+3)2(s+2)tests2+3s= + e-3t- e-t- e

4、-t2311234t2试图所示电路的动态结构图试图所示电路的动态结构图,并求传递函数。并求传递函数。CucR1R2uii1i2uoicC解解:ui=R1i1+uo ，i2=ic+i1UI(s)=R1I1(s)+UO(s)ducic=CdtI2(s)=IC(s)+I1(s)IC(s)=CsUC(s)即即: =I1(s)UI(s)-UO(s)R1UI(s)-UO(s)Cs=IC(s)UO(s)UI(s)=1R1( sC)R21+1R1( sC)R2=R2+R1R2sCR1+R2+R1R2sC1R1sCR2UI(s)-UO(s)IC(s)I1(s)I2(s)1R1sC R2( )UI(s)-UO(s

5、)试画图所示的电试画图所示的电路的动态结构图路的动态结构图,并求传递函数。并求传递函数。 uouiR1LR2C解：解：ui=R1I1+ucuc=uo+uLuL=LdiLdtiL=uoR2i1=iL+icic=CducdtUi(s)=R1I1(s)+UC(s)UC(s)=UO(s)+UL(s)UL(s)=sLIL(s)I1(s)=IL(s)+IC(s)1R1CssLR2I1UOUiIC-UC=UO+ULILULI2(s)=UO(s)R2IC(s)=CsUC(s)I1(s)=UO(s)R2I1(s)=UI(s)+UC(S)R1即：即：IL(s)=I1(s)-IC(s)IC(s)=UC(s)Cs解解

6、:电路等效为电路等效为:用运算放大器组成的有源电网络如图所示用运算放大器组成的有源电网络如图所示,试试采用复数阻抗法写出它们的传递函数。采用复数阻抗法写出它们的传递函数。UO= R3SCR2R21UIR1UOR3SCR2R21SC1= R1+R3+R2R3CS=R1(R2SC+1)R2R3=( + )R1(R2SC+1)R1R1R2= ( +R3 )(R2SC+1)1=R21R3R2SCR1C(S)=UO(S)UI(S)CR1R2R3uiuoCR1R2R3uiuoCR1R2R3uiuoR4R5用运算放大器组成的有源电网络如力所用运算放大器组成的有源电网络如力所示示,试采用复数阻抗法写出它们的传

7、递函试采用复数阻抗法写出它们的传递函数。数。=R5R4+ R5UO(R3SC+1)R2R3SC+R2+R3UOUI=(R2R3SC+R2+R3)(R4+R5)R1(R3SC+1)R5=(R4+R5)(R2+R3)( SC+1) R2R3R2+R3R1R5(R3SC+1)UIR1=R5R4+ R5UOR2R3SCSCR3SC1R5R4+ R5UOR2R3R3SC 1若系统在单位阶跃输入作用时，已知初若系统在单位阶跃输入作用时，已知初始条件为零的条件下系统的输出响应，始条件为零的条件下系统的输出响应，求系统的传递函数和脉冲响应。求系统的传递函数和脉冲响应。r(t)=I(t)c(t)=1-e +e-

8、2t-t解解:R(s)=1sG(S)=C(s)/R(s)1s+21s-C(s)=1s+1+=s(s+1)(s+2)(s2+4s+2) =(s+1)(s+2)(s2+4s+2) C(s)=(s+1)(s+2)(s2+4s+2) 脉冲响应脉冲响应:2s+2=1+1s+1-c(t)=( (t t) )+2e +e-2t-t2-10 已知系统的微分方程组的拉氏变换已知系统的微分方程组的拉氏变换式，试画出系统的动态结构图并求传递式，试画出系统的动态结构图并求传递函数。函数。解解:X1(s)=R(s)G1(s)-G1(s)G7(s)-G8(s)C(s)X2(s)=G2(s)X1(s)-G6(s)X3(s)

9、X3(s)=G3(s)X2(s)-C(s)G5(s)C(s)=G4(s)X3(s)G1G2G3G5-C(s)-R(s)G4G6G8G7X1(s)=R(s)-C(s)G7(s)-G8(s)G1(s)C(s)G7(s)-G8(s)G6(s)X3(s)X1(s)X2(s)C(s)G5(s)X3(s)G1G2G3G5-C(s)-R(s)G4G2G6G8G7G1G2G5-C(s)-R(s)G7-G81+G3G2G6G3G4-C(s)R(s)G7-G81+G3G2G6 +G3G4G5G1G2G3G41+G3G2G6 +G3G4G5+G1G2G3G4(G7 -G8)G1G2G3G4R(s)C(s)=解解:2

10、-11(a)G1(s)G2(s)G3(s)H1(s)_+R(s)C(s)H2(s)G1(s)G2(s)H1(s)_+R(s)C(s)H2(s)G3(s)求系统的求系统的传递函数传递函数1+G2H1G2 G1+G31+G1H21+G2H1G2 1+G2H1G2 =1+G2H1+G1G2H2G2 R(s)C(s)=1+G2H1+G1G2H2G2G1+G2G3G1(s)G2(s)G3(s)H1(s)_+R(s)C(s)G 1(s)H2(s)G1(s)G2(s)G3(s)H1(s)_+R(s)C(s)H2(s)求系统的求系统的传递函数传递函数解解:L1L1=-G2H1L2L2=-G1G2H1P1=G1

11、G2P2=G3G21 =12 =1R(s)C(s)=nk=1Pkk=1+G2H1+G1G2H21+G2H1+G1G2H2G2G1+G2G3 =解解:2-11(b)G1(s)G2(s)G3(s)G4(s)_+R(s)C(s)H(s)求系统的求系统的传递函数传递函数G1(s)G2(s)G3(s)G4H_+R(s)C(s)H(s)1+G4G1HG1 G2(s)G3(s)_+R(s)C(s)H(s)1+G4HG1G1 G2G3_+R(s)C(s)1+G4HG1G1 HG1 1+G4HG1G1+G3 (1+HG1G4)1+G4HG1G2 (1+HG1G4)1+G4G1H+G1G2HR(s)C(s)=1+

12、G1G2H+G1G4HG1G2+G2G3+G1G2G3G4 H解解:G1(s)G2(s)G3(s)G4(s)_+R(s)C(s)H(s)求系统的求系统的传递函数传递函数R(s)C(s)=1+G1G2H+G1G4HG1G2+G2G3+G1G2G3G4 HL1L1=-G1G2HL1=-G1G4HL2P1=G1G21 =1P2=G3G2=1+G4G2H+G1G2H2=1+G1G4HH1_+G1+C(s)R(s)G3G2求系统的闭环传递函数求系统的闭环传递函数 。解解: H1_+G1+C(s)R(s)G3G2H1R(s)C(s)1+G1G2+G1H1G3H1G1G2 (1 G3H1)=_G1C(s)R

13、(s)G2H1+G21-G3H11H_G1+C(s)R(s)G2求系统的闭环传递函数求系统的闭环传递函数 。解解: (1)_G1+C(s)R(s)G2HG21+G2H1(G1+G2 )R(s)C(s)=(2)L1L1=-G2HP1=G11 =1P2=G22 =1-_G1+C(s)R(s)G2G3G4求系统的闭环传递函数求系统的闭环传递函数 。解解: (1) _C(s)R(s)G1+G2G3-G4C(s)=R(s) 1+(G1+G2)(G3-G4)(G1+G2)1+G1G3+G2G3G1G4-G2G4=(G1+G2)L1L2L3L4L2=G1G4L3=-G2G3L4=G2G4(2)L1=-G1G

14、3P1=G11 =1P2=G22 =11+G1G3+G2G3G1G4-G2G4=(G1+G2)C(s)R(s)_G1+C(s)R(s)G2求系统的闭环传递函数求系统的闭环传递函数 。_C(s)R(s)G11-G2G2C(s)=R(s)1+1-G2G1G1G21+G1G2G2G1 (1 G2)=解解: (1) (2)L1L1=-G1G2L2L2=G2P1=G11 =1-G2=1+G1G2-G2C(s)R(s) 1+G1G2G2G1 (1 G2)=2-12(a)R(s)G1(s)G2(s)H2(s)_+C(s)H3(s)H1(s)_+D(s)解解:求求:D(s)C(s)R(s)C(s)D(s)=0

15、1-G2H2G2 G(s)=1-G2H2G1G2 C(s)=R(s)1+1-G2H2G1G2H31-G2H2G1G2 1-G2H2+G1G2H3G2G1=R(s)=0结构图变结构图变 换成：换成：G2(s)H2(s)_+C(s) G1H3G1H1_D(s)1-G2H2G2 1-G1H1C(s)=D(s)1+1-G2H2G21-G2H2G2 G1H3(1-G1H1 )1-G2H2+G1G2H3G2(1-G1H1 )=2-12(b)求求:D(s)C(s)R(s)C(s)R(s)Gn+D(s)解解: D(s)=0G(s)=1+G1G2HG1G2 G1G2H_C(s)C(s)=R(s)1+1+G1G2

16、HG1G2 1+G1G2HG1G2 1+G1G2H+G1G2G1G2=R(s)=0Gn+D(s)结构图变结构图变 换成：换成：G1G2H-C(s)Gn+D(s)G1G2H-C(s)Gn/G1+D(s)1+G1G2HG1G2 -C(s)+D(s)1+G1G2HG2Gn 1+G1G2HG2G1系统的传递函数系统的传递函数:)C(s)=D(s)1+1(1+1+G1G2HG1G2 1+G1G2HGnG2 1+G1G2+G1G2H=1+GnG2+G1G2H2-13(a)求求:R(s)E(s)R(s)C(s)C(s)E(s)G1G2G3_+R(s)解解:L1L1=-G2L2L2=-G1G2G3P1=G2G

17、3P2=G1G2G3R(s)C(s)=1+G2+G1G2G3G2G3+G1G2G31 =12 =1E(s)结构图变结构图变 换成：换成：G1G2+-E(s)G3-R(s)G1+-E(s)R(s)1+G2G3G2 -E(s)-R(s)1+G2G2G3G1G2G31+G2系统的传递函数系统的传递函数:)E(s)=R(s)1+1(1-G1G2G31+G2 1+G2G2G31+G2+G1G2G3=1+G2-G2G3R(s)G4(s)+C(s)E(s)G1(s)G2(s)-+G3(s)D(s)X(s)2-14 求求:R(s)C(s)解解: D(s)=0结构图结构图变换为变换为 R(s)G4(s)+C(s

18、)G1(s)G2(s)-+G3(s)G3(s)(G1+G2)(G3+G4)1+(G1+G2)G3G1+G2C(s)R(s)=1+G3(G1+G2)(G1+G2)(G3+G4)D(s)R(s)+-E(s)G3G2G1E(s)R(s)=1+G3(G1+G2)1R(s)G4(s)+C(s)E(s)G1(s)G2(s)-+G3(s)X(s)求求:R(s)E(s)解解: D(s)=0结构图变换为结构图变换为 G3(G1+G2)D(s)C(s)D(s)R(s)C(s)G4(s)+E(s)G1(s)G2(s)-+G3(s)X(s)求求:解解:R(s)=0D(s)C(s)=1E(s)X(s)=G2(s)E(s

19、)X(s)C1(s)R1(s)求求:2-15+G1G2G3C1(s)R1(s)+-H2H1G4G5-G6C2(s)R2(s)解解:结构图结构图变换为变换为 +G1G2G3C1(s)R1(s)-H2H1G4G5-1+G4G4G5H1H21+G4G4G5H1H21+G1G1G21+G4+G1G4G5H1H2-G1G2-G1G2G4G1G2G3(1+G4 )=C1(s)R1(s)1+G4+G1G4G5H1H2G1G2(1+G4 )=C2(s)R2(s)G3C1(s)R1(s)+G1G2-H2H1G4G5G6-C2(s)R2(s)求求:2-14解解: 结构图结构图变换为变换为 +G4G5G6C2(s)

20、R2(s)-H1H2G2G1-1+G4G4G5G1H1H21-G1G21+G4+G1G4G5H1H2-G1G2-G1G2G4G4G5G6(1-G1G2)=1+1+G4G6G4G5G1H1H21-G2G21+G4G4G5C2(s)R2(s)C2(s)G4G5G6-+G1G2-H2H1R1(s)G3C1(s)R2(s)求求:2-14解解:结构图变换为结构图变换为 G4G5G6-+-G1G2H2H1R1(s)C2(s)C2(s)R1(s)1+G4H2G4G5G11-G1G21+G4+G1G4G5H1H2-G1G2-G1G2G4G1G4G5G6H2=1+G6H1C2(s)R1(s)G11-G1G21+G4H2G4G5G11-G1G21+G4H2G4G5C2(s)G6R1(s)G4G5-+G1G2-H2H1G3C1(s)R2(s)求求:2-14解解:结构图变换为结构图变换为 G1G2G3+G5G4-H1R2(s)C1(s)H2-C1(s)R2(s)G1G2G3+G5G4-H1R2(s)C1(s)H2-G21+G4G4G51-G1G2-G1H11+G4+G1G4G5H1H2-G1G2-G1G2G4-G1G2G3G4G5H1=1+1+G4G4G5G1H11-G2G21+G4G4G5C1(s)R2(s)H2-G1H11-G2G2G2G3G3C1(s)R1(s) +G1G2+-H2H