自动控制原理及其应用(第2版)黄坚第二章习题课

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1、第二章习题课(2-1)2-1试建立图所示电路的动态微分方程解：输入ui输出uou1=ui-uoi2=Cdu1dt)-R2(ui-uo)=R1u0-CR1R2(duidtdtduoC＋-+-uiuoR1R2i1ii2u1i1=i-i2uoi=R2u1i1=R1=ui-uoR1dtd(ui-uo)=CCd(ui-uo)dtuo-R2=ui-uoR1CR1R2duodtduidt+R1uo+R2u0=CR1R2+R2ui(a)第二章习题课(2-1)i=i1+i2i2=Cdu1dtuoi1=R2u1-uo=LR2duodtR1i=(u

2、i-u1)(b)C＋-+-uiuoR1R2i1ii2Lu1=R1ui-u1uo+CR2du1dtu1=uo+LR2duodtduodtR1R2Lduodt+CLR2d2uodt2=--uiR1uoR1uoR2+C)uoR1R2Lduodt)CLR2d2uodt2=++(uiR11R11R2+(C+解：2-2求下列函数的拉氏变换。(1)f(t)=sin4t+cos4tL[sinωt]=ωω2+s2=s+4s2+16L[sin4t+cos4t]=4s2+16ss2+16+sω2+s2L[cosωt]=第二章习题课(2-2)解：(2

3、)f(t)=t3+e4t解：L[t3+e4t]=3!s41s-4+6s+24+s4s4(s+4)=(3)f(t)=tneatL[tneat]=n!(s-a)n+1(4)f(t)=1(t-τ)e2tL[1(t-τ)e2t]=e-τs1s-2第二章习题课(2-2)解：解：2-3求下列函数的拉氏反变换。A1=(s+2)s+1(s+2)(s+3)s=-2=-1=2f(t)=2e-3t-e-2t(1)F(s)=s+1(s+2)(s+3)第二章习题课(2-3)解：A2=(s+3)s+1(s+2)(s+3)s=-3F(s)=2s+31s+2

4、-=A1s+2s+3+A2(2)F(s)=s(s+1)2(s+2)f(t)=-2e-2t-te-t+2e-t第二章习题课(2-3)解：=A2s+1s+2+A3+A1(s+1)2A1=(s+1)2s(s+1)2(s+2)s=-1A3=(s+2)s(s+1)2(s+2)s=-2ddsss+2][A2=s=-1=-1=2=-2(3)F(s)=2s2-5s+1s(s2+1)F(s)(s2+1)s=+j=A1s+A2s=+jA2=-5A3=F(s)ss=0f(t)=1+cost-5sint第二章习题课(2-3)解：=s+A3s2+1A1

5、s+A2=1s2s2-5s+1=A1s+A2s=js=jj-2-5j+1=jA1+A2-5j-1=-A1+jA2A1=1F(s)=1ss2+1s-5s2+1++(4)F(s)=s+2s(s+1)2(s+3)第二章习题课(2-3)解：=+s+1A1s+3A2(s+1)2+sA3+A4-12A1=23A3=112A4=A2=d[s=-1ds](s+2)s(s+3)-34=-34A2=+-43+f(t)=e-t32e-3t2-te-t121=s=-1[s(s+3)]2[s(s+3)-(s+2)(2s+3)]2-4求解下列微分方程。y

6、(0)=y(0)=2·A1=1y(t)=1+5e-2t-4e-3tA2=5A3=-4Y(s)=6+2s2+12ss(s2+5s+6)第二章习题课(2-4)+6y(t)=6+5d2y(t)dt2dy(t)dt(1)解：s2Y(s)-sy(0)-y'(0)+5sY(s)-5y(0)+6Y(s)=6ss2Y(s)-2s-2+5sY(s)-10+6Y(s)=6s=A1s+2s+3+A3s+A2+y(t)=t2dy(t)dt(2)y(0)=0第二章习题课(2-4)2-5试画题图所示电路的动态结构图，第二章习题课(2-5)并求传递函数。(

7、1)ii2+-uruc+-R2R1ci1解：I2(s)I1(s)+Uc(s)Ur(s)_Cs1R1+R2Uc(s)I(s)Ur(s)Uc(s)=1R1(1+(+sC)R21R1+sC)R2=R2+R1R2sCR1+R2+R1R2sC第二章习题课(2-5)(2)C＋-+-urucR1R2Lu1i2i1iI(s)Ur(s)_1R1U1(s)解：I1(s)-I2(s)L1=-R2/LsL2=-/LCs2L3=-1/sCR1L3Δ1=1L1L3=R2/LCR1s21CsU1(s)Uc(s)-1LsR2I1(s)Uc(s)L1L2P1=

8、R2/LCR1s2=R1CLs2+(R1R2C+L)s+R1+R2Ur(s)Uc(s)R2第二章习题课(2-6)2-6试采用复阻抗写出传递函数。R1ucCR2ur-Δ∞++R3第二章习题课(2-7)2-7试证明两系统为相似系统。i＋-+-uiuoR1R2i1i2c1c2xix