自动控制原理和其应用答案第二版黄坚课后答案市公开课一等奖百校联赛获奖课件.pptx

第二章习题课第二章习题课(2-1a)uoi2=R2CuiuoR1R22-1(a)试建立图所表示电路动态微分方试建立图所表示电路动态微分方程。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第1页第二章习题课第二章习题课(2-1b)2-1(b)试建立图所表示电路动态微分方试建立图所表示电路动态微分方程。程。uouiR1LR2Ci1=iL+icuL=LdiLdtuoiL=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)第2页习题课一习题课一 (2-2)求以下函数拉氏变换求以下函数拉氏变换。(1)f(t)=sin4t+cos4t解解:Lsinwt=ww2+s2sw2+s2Lsin4t+cos4t=4s2+16ss2+16 =s+4s2+16+Lcoswt=第3页(2)f(t)=t3+e4t3!解解:Lt3+e4t=+=+3!s3+11s-4s41s-4(3)f(t)=tneat解解:Ltneat=n!(s-a)n+1(4)f(t)=(t-1)2e2t解解:L(t-1)2e2t=e-(s-2)2(s-2)3第4页2-3-1 函数拉氏变换。函数拉氏变换。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-2t第5页F(s)=s(s+1)2(s+2)2-3-2 函数拉氏变换。函数拉氏变换。解解: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第6页F(s)=2s2-5s+1s(s2+1)2-3-3 函数拉氏变换。函数拉氏变换。解解:F(s)(s2+1)s=+j=A1s+A2s=+jA1=1,A2=-5A3=F(s)s =1s=0 f(t)=1+cost-5sintF(s)=+1ss2+1s-5s2+1第7页2-3-4 函数拉氏变换。函数拉氏变换。(4)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-t2311234t2第8页(2-4-1)求以下微分方程。求以下微分方程。d2y(t)dt2+5 +6y(t)=6，初始条件：初始条件：dy(t)dty(0)=y(0)=2。解解:s2Y(s)-sY(0)-Y(0)+5sY(s)-5Y(0)+6Y(s)=1sA1=sY(s)s=0 y(t)=1+5e-2t-4e-3tA2=(s+2)Y(s)s=-2A3=(s+3)Y(s)s=-3A1=1,A2=5,A3=-4 Y(s)=6+2s2+12ss(s2+5s+6)第9页 (2-4-2)求以下微分方程。求以下微分方程。d3y(t)dt3 +4 +29 =29,d2y(t)dt2dy(t)dt初始条件初始条件:y(0)=0 ,y(0)=17 ,y(0)=-122 解解:第10页2-5-a 试画题试画题2-1图所表示电路动态结构图图所表示电路动态结构图,并并求传递函数。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)第11页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)第12页2-5-b 试画出题试画出题2-1图所表示电路图所表示电路动态结构图动态结构图,并求并求传递函数。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)第13页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第14页解解:电路等效为电路等效为:2-6-a 用运算放大器组成有源电网络如图所用运算放大器组成有源电网络如图所表示表示,试采取复数阻抗法写出它们传递函数。试采取复数阻抗法写出它们传递函数。UO=R3SCR2R21UIR1UOR3SCR2R21SC1=第15页R1+R3+R2R3CS=R1(R2SC+1)R2R3=(+)R1(R2SC+1)R1R1R2=(+R3)(R2SC+1)1=R21R3R2SCR1C(S)=UO(S)UI(S)CR1R2R3uiuoCR1R2R3uiuo第16页CR1R2R3uiuoR4R52-6-b 用运算放大器组成有源电网络如力用运算放大器组成有源电网络如力所表示所表示,试采取复数阻抗法写出它们传递试采取复数阻抗法写出它们传递函数。函数。第17页=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第18页c(t)t0TK(t)(t)2-8 设有一个初始条件为零系统，系统设有一个初始条件为零系统，系统输入、输出曲线如图，求输入、输出曲线如图，求G(s)。c(t)t0TK(t)(t)c(t)=KTt-(t-T)KT C(s)=K(1-e )Ts2-TSC(s)=G(S)第二章习题课第二章习题课(2-8)解解:第19页2-9 若系统在单位阶跃输入作用时，已若系统在单位阶跃输入作用时，已知初始条件为零条件下系统输出响应，知初始条件为零条件下系统输出响应，求系统传递函数和脉冲响应。求系统传递函数和脉冲响应。r(t)=I(t)c(t)=1-e +e-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-t第二章习题课第二章习题课(2-)第20页2-10 已知系统微分方程组拉氏变换式，已知系统微分方程组拉氏变换式，试画出系统动态结构图并求传递函数。试画出系统动态结构图并求传递函数。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)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)=第二章习题课第二章习题课(2-10)第21页解解:2-11(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)求系统传求系统传递函数递函数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)第二章习题课第二章习题课(2-11a)第22页2-11(a)G1(s)G2(s)G3(s)H1(s)_+R(s)C(s)H2(s)求系统传求系统传递函数递函数解解:L1L1=-G2H1L2L2=-G1G2H1P1=G1G2P2=G3G21=12=1R(s)C(s)=nk=1Pkk=1+G2H1+G1G2H21+G2H1+G1G2H2G2G1+G2G3=第二章习题课第二章习题课(2-11a)第23页解解:2-11(b)G1(s)G2(s)G3(s)G4(s)_+R(s)C(s)H(s)求系统传求系统传递函数递函数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+G1G2H+G1G4HG1G2+G2G3+G1G2G3G4 H第二章习题课第二章习题课(2-11b)第24页解解:2-11(b)G1(s)G2(s)G3(s)G4(s)_+R(s)C(s)H(s)求系统传求系统传递函数递函数R(s)C(s)=1+G1G2H+G1G4HG1G2+G2G3+G1G2G3G4 HL1L1=-G1G2HL1=-G1G4HL2P1=G1G21=1P2=G3G2=1+G4G2H+G1G2H2=1+G1G4H第二章习题课第二章习题课(2-11b)第25页H1_+G1+C(s)R(s)G3G22-11c 求系统闭环传递函数求系统闭环传递函数。解解:H1_+G1+C(s)R(s)G3G2H1R(s)C(s)1+G1G2+G1H1G3H1G1G2(1 G3H1)=_G1C(s)R(s)G2H1+G21-G3H11第二章习题课第二章习题课(2-11c)第26页H_G1+C(s)R(s)G22-11d 求系统闭环传递函数求系统闭环传递函数。解解:(1)_G1+C(s)R(s)G2HG21+G2H1(G1+G2)R(s)C(s)=(2)L1L1=-G2HP1=G11=1P2=G22=1第二章习题课第二章习题课(2-11d)第27页-_G1+C(s)R(s)G2G3G42-11e 求系统闭环传递函数求系统闭环传递函数。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)第二章习题课第二章习题课(2-11e)L1L2L3L4L2=G1G4L3=-G2G3L4=G2G4(2)L1=-G1G3P1=G11=1P2=G22=11+G1G3+G2G3G1G4-G2G4=(G1+G2)C(s)R(s)第28页_G1+C(s)R(s)G22-11f 求系统闭环传递函数求系统闭环传递函数。_C(s)R(s)G11-G2G2C(s)=R(s)1+1-G2G1G1G21+G1G2G2G1(1 G2)=第二章习题课第二章习题课(2-11f)解解:(1)(2)L1L1=-G1G2L2L2=G2P1=G11=1-G2=1+G1G2-G2C(s)R(s)1+G1G2G2G1(1 G2)=第29页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)=01-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)=第二章习题课第二章习题课(2-12a)第30页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+G1G2HG1G2 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系统传递函数系统传递函数:)C(s)=D(s)1+1(1+1+G1G2HG1G2 1+G1G2HGnG2 1+G1G2+G1G2H=1+GnG2+G1G2H第二章习题课第二章习题课(2-12b)第31页2-13(a)求求:R(s)E(s)R(s)C(s)C(s)E(s)G1G2G3_+R(s)解解:L1L1=-G2L2L2=-G1G2G3P1=G2G3P2=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系统传递函数系统传递函数:)E(s)=R(s)1+1(1-G1G2G31+G2 1+G2G2G31+G2+G1G2G3=1+G2-G2G3第二章习题课第二章习题课(2-13a)第32页R(s)G4(s)+C(s)E(s)G1(s)G2(s)-+G3(s)第二章习题课第二章习题课(2-14)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)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)第33页D(s)第二章习题课第二章习题课(2-14)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)2-14解解:D(s)=0结构图变换为结构图变换为 G3(G1+G2)第34页D(s)C(s)D(s)R(s)C(s)G4(s)+E(s)G1(s)G2(s)-+G3(s)X(s)求求:2-14解解:R(s)=0D(s)C(s)=1E(s)X(s)=G2(s)E(s)X(s)第二章习题课第二章习题课(2-14)第35页C1(s)R1(s)第二章习题课第二章习题课(2-15)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)=第36页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)R2(s)-H1H2G2G1-第二章习题课第二章习题课(2-15)1+G4G4G5G1H1H21-G1G21+G4+G1G4G5H1H2-G1G2-G1G2G4G4G5G6(1-G1G2)=1+1+G4G6G4G5G1H1H21-G2G21+G4G4G5C2(s)R2(s)第37页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)第二章习题课第二章习题课(2-15)C2(s)R1(s)1+G4H2G4G5G11-G1G21+G4+G1G4G5H1H2-G1G2-G1G2G4G1G4G5G6H2=1+G6H1C2(s)R1(s)G11-G1G21+G4H2G4G5G11-G1G21+G4H2G4G5第38页C2(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 第二章习题课第二章习题课(2-15)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-G2G2G2G3第39页G3C1(s)R1(s)+G1G2+-H2H1G4G5-G6C2(s)R2(s)C1(s)R1(s)求求2-15第二章习题课第二章习题课(2-15)解解:L1=G1G2L3=-G4L2=-G1G4G5H1H2P1=G1G2G3=1-G1G2+G1G4G35H1H2+G4-G1G2G41=1+G41+G4+G1G4G5H1H2-G1G2-G1G2G4G1G2G3(1+G4)=C1(s)R1(s)第40页G3C1(s)R1(s)+G1G2+-H2H1G4G5-G6C2(s)R2(s)C2(s)R2(s)求求2-15解解:L1=G1G2L3=-G4L2=-G1G4G5H1H2P1=G4G5G6=1-G1G2+G1G4G5H1H2+G4-G1G2G41=1-G1G21+G4+G1G4G5H1H2-G1G2-G1G2G4G4G5G6(1-G1G2)=C2(s)R2(s)第二章习题课第二章习题课(2-15)第41页G3C1(s)R1(s)+G1G2+-H2H1G4G5-G6C2(s)R2(s)C1(s)R2(s)求求2-15解解:L1=G1G2L3=-G4L2=-G1G4G5H1H2=1-G1G2+G1G4G5H1H2+G4-G1G2G41=1P1=-G1G2G3G4G5H11+G4+G1G4G5H1H2-G1G2-G1G2G4-G1G2G3G4G5H1=C1(s)R2(s)第二章习题课第二章习题课(2-15)第42页G3C1(s)R1(s)+G1G2+-H2H1G4G5-G6C2(s)R2(s)C2(s)R1(s)求求2-15解解:L1=G1G2L3=-G4L2=-G1G4G5H1H2=1-G1G2+G1G4G5H1H2+G4-G1G2G41=1P1=G1G4G5G6H21+G4+G1G4G5H1H2-G1G2-G1G2G4G1G4G5G6H2=C2(s)R1(s)第二章习题课第二章习题课(2-15)第43页3-1 设温度计需要在一分钟内指示出响设温度计需要在一分钟内指示出响应值应值98%，而且假设温度计为一阶系统，而且假设温度计为一阶系统，求时间常数求时间常数T。假如将温度计放在澡盆内，。假如将温度计放在澡盆内，澡盆温度以澡盆温度以10oC/min速度线性改变，求速度线性改变，求温度计误差。温度计误差。第三章习题课第三章习题课 (3-1)解解:c(t)=c()98%t=4T=1 min r(t)=10t e(t)=r(t)-c(t)c(t)=10(t-T+e)-t/T=10(T-e)-t/Tess=lim t e(t)=10T=2.5T=0.25第44页3-2 电路如图，设系统初始状态为领电路如图，设系统初始状态为领第三章习题课第三章习题课 (3-2)解解:(1)求系统单位阶跃响应求系统单位阶跃响应,及及uc(t1)=8 时时t1值值R1Cs+1R1/R0G(s)=R0=20 k R1=200 k C=2.5F uc(t)=K(1 e tT-)KTs+1=T=R1C=0.5 K=R1/R0=10 =10(1 e -2t)8=10(1 e -2t)0.8=1 e -2te -2t=0.2 t=0.8-+R1R0Curuc第45页g(t)=e-t/TTK t1=0.8 =4(2)求系统单位脉冲响应，单位斜坡求系统单位脉冲响应，单位斜坡 响应响应,及单位抛物响应在及单位抛物响应在t1时刻值时刻值第三章习题课第三章习题课 (3-2)解解:uc(t)=K(t-T+Te-t/T)=4 R(s)=1s2R(s)=1R(s)=1s3T2=K(ss+1/T+Ts2-1s3-T2)=1.2 1s3KTs+1Uc(s)=-0.5t+0.25-0.25e-2t)12t2uc(t)=10(第46页3-3 已知单位负反馈系统开环传递函数，已知单位负反馈系统开环传递函数，求系统单位阶跃响应。求系统单位阶跃响应。4s(s+5)G(s)=第三章习题课第三章习题课 (3-3)解解:=s2+5s+4 C(s)R(s)4s(s+1)(s+4)C(s)=4R(s)=s11/3s+41s+=4/3s+1-13c(t)=1+-4t-t43-e e 第47页3-4 已知单位负反馈系统开环传递函数，已知单位负反馈系统开环传递函数，求系统上升时间求系统上升时间tr、峰值时间、峰值时间tp、超调量、超调量%和调整时间和调整时间ts。1s(s+1)G(s)=第三章习题课第三章习题课 (3-4)解解:=s2+s+1 C(s)R(s)12=1n 2n=1=0.5=1n=0.866d=n2 1-=60o-1=tg 21-tr=d -=3.14-3.14/30.866=2.42 tp=d3.140.866=3.63%=100%e-1-2=16%-1.8ets=3n=6 ts=4n=8第48页3-6 已知系统单位阶跃响应已知系统单位阶跃响应:c(t)=1+0.2e -60t-10t-1.2e(1)求求系统闭环传递函数。系统闭环传递函数。(2)求求系统阻尼比和无阻尼振荡频率。系统阻尼比和无阻尼振荡频率。第三章习题课第三章习题课 (3-6)解解:0.2s+601s+C(s)=1.2s+10-s(s+60)(s+10)=600=s2+70s+600 C(s)R(s)600R(s)=s12=600n 2n=70=1.43=24.5n 第49页3-7 设二阶系统单位阶跃响应曲线如图，系设二阶系统单位阶跃响应曲线如图，系统为单位反馈，求系统传递函数。统为单位反馈，求系统传递函数。tc(t)010.11.3解解:第三章习题课第三章习题课 (3-7)tp=0.121-n=0.3e-1-2e1-2=3.3n2 1-3.140.1=31.421-/=ln3.3=1.1921-)2/(=1.429.862=1.42-1.422=0.35=33.4n nn s(s+2)G(s)=21115.6s(s+22.7)=第50页3-8 已知单位负反馈系统开环传递函数，已知单位负反馈系统开环传递函数，求系统求系统K、T值以满足动态指标：值以满足动态指标：%30%30%，ts0.3(5%)0.3(5%)。Ks(Ts+1)G(s)=第三章习题课第三章习题课 (3-8)解解:=Ts2+s+K C(s)R(s)Ks2+=T1s+TKTKT12n=ts=3n 0.3n 10T0.050.3e-1-20.3528.6n 2=n TK=817.96K40.9第51页3-11 已知闭环系统特征方程式，试用劳已知闭环系统特征方程式，试用劳斯判据判断系统稳定性。斯判据判断系统稳定性。第三章习题课第三章习题课 (3-11)(1)s3+20s2+9s+100=0解：解：劳斯表以下：劳斯表以下：s1 s0 s3 s2 1 9 20 100 4 100系统稳定。系统稳定。(3)s4+8s3+18s2+16s+5=0 1 18 5 s4 s3 8 16 劳斯表以下：劳斯表以下：s2 16 5 s1 216 16 s0 5系统稳定。系统稳定。第52页3-12 已知单位负反馈系统开环传递函数，已知单位负反馈系统开环传递函数，试确定系统稳定时试确定系统稳定时K值范围。值范围。第三章习题课第三章习题课 (3-12)G(s)=s(s+1)(0.5s2+s+1)K(0.5s+1)解解:0.5s4+1.5s3+2s2+s+0.5Ks+K=0 0.5 2 K s4 s3 1.5 1+0.5K s2 b31 b31=1.5*2-0,5(1+0.5K)1.5 3-0,5-0.25K0 0.25K00第54页3-14 已知系统结构如图，试确定系统稳已知系统结构如图，试确定系统稳定时定时值范围。值范围。第三章习题课第三章习题课 (3-14)R(s)R(s)-s+1s10s(s+1)C(s)C(s)解解:G(s)=s2(s+1)10(s+1)(s)=s3+s2+10 s+1010(s+1)s3 s2 1 10 1 10 s1 b31 s0 10 b31=10 -10 1 01第55页r(t)=I(t)+2t+t23-16 已知单位反馈系统开环传递函数，试已知单位反馈系统开环传递函数，试求求K p、Kv和和Ka 并求稳态误差并求稳态误差ess第三章习题课第三章习题课 (3-16)解：解：(0.1s+1)(0.2s+)(1)G(s)=20Kp=20=0ess1=R01+Ks2R(s)=1s2+s32+=121K =0 ess2=Ka=0 ess3=ess=1K =10 Ka=0 ess3=ess=s(s+2)(s+10)(2)G(s)=200s(0.5s+1)(0.1s+1)=10Kp=ess1=0ess2=K 2=210s2(s2+4s+10)(3)G(s)=10(2s+1)=2Kp=ess1=0s2(0.1s2+0.4s+1)=(2s+1)K =ess2=0Ka=1 ess3=2 ess=2第56页3-17 已知系统结构如图。已知系统结构如图。第三章习题课第三章习题课 (3-17)解解:-R(s)R(s)-K1ss2C(s)C(s)(1)单位阶跃输入单位阶跃输入:G(s)=s2+K1sK1(s)=s2+K1s+K1K1确定确定K1 和和值值。%=20%=20%ts=1.8(5%)2n=K12=K1n=0.2e-1-2ts=3n=1.8=0.45n31.8*0.45=3.72 n K1=13.7=0.24第57页(2)求系统稳态误差：求系统稳态误差：r(t)=I(t),t,12t2 解解:G(s)=s2+K1sK1=s(s+1)K111=1Kp=ess1=0R(s)=1sR(s)=s21K =K ess2=0.24R(s)=s31Ka=0 ess3=第三章习题课第三章习题课 (3-17)第58页3-18 已知系统结构如图。为使已知系统结构如图。为使=0.7时时 单位斜坡输入稳态误差单位斜坡输入稳态误差ess=0.25第三章习题课第三章习题课 (3-18)解解:确定确定 K 和和值值。-s-Ks(s+2)R(s)R(s)C(s)C(s)G(s)=s2+2s+K sK=s(s+1)2+K12+KK(s)=s2+(2+K)s+KK2n=2+K2=Kn =2*0.7 K ess=2+K K =0.25=0.25K-2 K K K=31.6=0.186第59页+R(s)G(s)F(s)C(s)+-D1(s)D2(s)E(s)3-19 系统结构如图。系统结构如图。第三章习题课第三章习题课 (3-19)解解:r(t)=d1(t)=d2(t)=I(t)(1)求求r(t)作下稳态误差作下稳态误差essr=lim s 1+G(s)F(s)s0s1=1+G(0)F(0)1(2)求求d1(t)和和d2(t)同时作用下稳态误差同时作用下稳态误差essd=lim s -F(s)1+G(s)F(s)s0-11+G(s)F(s)+s1Ed(s)=-G2(s)H(s)1+G1(s)G2(s)H(s)D(s)=1+G(0)F(0)-1+F(s)第60页(3)求求d1(t)作用下稳态误差作用下稳态误差G(s)=Kp+KsJs1F(s)=essd=lim s -F(s)1+G(s)F(s)s0s1-s0s1=lim s 1+(Kp+KsJs1)Js1=0返回返回第61页4-1 已知系统零、极点分布如图，大致已知系统零、极点分布如图，大致绘制出系统根轨迹。绘制出系统根轨迹。第四章习题课第四章习题课 (4-1)解解:j0(1)(2)j0(3)j0(4)j0600900600第62页第四章习题课第四章习题课 (4-1)(5)j0(6)j0j0(7)(8)j060045013503601080第63页4-2 已知开环传递函数，试用解析法绘制已知开环传递函数，试用解析法绘制出系统根轨迹，并判断点出系统根轨迹，并判断点(-2+j0),(0+j1),(-3+j2)是否在根轨迹上。是否在根轨迹上。第四章习题课第四章习题课 (4-2)解解:Kr(s+1)G(s)=Kr(s)=s+1+KrKr=0s=-1-Krj0系统根轨迹系统根轨迹 s=-1-1Kr=s=-s=-2+j0-2s=0+j10+j1-3+j2s=-3+j2第64页j04-3 已知系统开环传递函数，试绘制出根已知系统开环传递函数，试绘制出根轨迹图。轨迹图。第四章习题课第四章习题课 (4-3)解解:s(s+1)(s+5)(1)G(s)=Kr(s+1.5)(s+5.5)1）开环零、极点）开环零、极点 p1 p1=0p2p2=-1p3 p3=-52）实轴上根轨迹段）实轴上根轨迹段 p1p2z1 z1=-1.5z2 z2=-5.5z1p3z2-3）根轨迹渐近线）根轨迹渐近线 n-m=1=+180o4）分离点和会合点）分离点和会合点A(s)B(s)=A(s)B(s)A(s)=s3+6s2+5sB(s)=s2+7s+8.25A(s)=3s2+12s+5B(s)=2s+7解得解得 s1=-0.63s2=-2.5s3=-3.6s4=-7.28第65页第四章习题课第四章习题课 (4-3)s(s+1)(s+4)(2)G(s)=Kr(s+1.5)1）开环零、极点）开环零、极点 p1 p1=0p2p2=-1p3 p3=-4j02）实轴上根轨迹段）实轴上根轨迹段 p1p2z1 z1=-1.5p3z1 3）根轨迹渐近线）根轨迹渐近线 n-m=2=+90o2=-1-4+1.5=-1.75-1.754）分离点和会合点）分离点和会合点A(s)=s3+5s2+4sB(s)=s+1.5A(s)=3s2+10s+4B(s)=1解得解得 s=-0.625）系统根轨迹）系统根轨迹第66页第四章习题课第四章习题课 (4-3)s(s+1)2(3)G(s)=Kr 1）开环零、极点）开环零、极点 p1=0p2=-1p3 p3=-12）实轴上根轨迹段）实轴上根轨迹段 p1p2p3-3）根轨迹渐近线）根轨迹渐近线 n-m=3j0p1 p23=-1-1=-0.67-0.67 4）根轨迹与虚轴交点）根轨迹与虚轴交点=+180o+60o,闭环特征方程为闭环特征方程为 s3+2s2+s+Kr=0Kr=0 Kr=2 2,3=11=01-15）分离点和会合点）分离点和会合点A(s)=s3+2s2+sB(s)=1A(s)=3s2+4s+1B(s)=0解得解得 s=-0.336）系统根轨迹）系统根轨迹第67页第四章习题课第四章习题课 (4-3)1）开环零、极点）开环零、极点 p1=0p2=-3 p3=-7j02）实轴上根轨迹段）实轴上根轨迹段 p1p2p4 p4=-15z1 z1=-8p3z1p4-3）根轨迹渐近线）根轨迹渐近线 n-m=3s(s+3)(s+7)(s+15)(4)G(s)=Kr(s+8)p1 p2p3 3=-3-7-15+8=-5.67=+180o+60o,-5.67 4）根轨迹与虚轴交点）根轨迹与虚轴交点闭环特征方程为闭环特征方程为 s4+25s3+171s2+323s+8Kr=0Kr=0 1=0Kr=638 2,3=6.26.2-6.25）分离点和会合点）分离点和会合点A(s)=s4+25s3+171s2+315sB(s)=s+8A(s)=4s3+75s2+342s+315B(s)=2s+7解得解得 s=-1.46）系统根轨迹）系统根轨迹第68页第四章习题课第四章习题课 (4-4)第69页4-5 已知系统开环传递函数。已知系统开环传递函数。(1)试绘制出试绘制出根轨迹图。根轨迹图。(2)增益增益Kr为何值时，复数特征为何值时，复数特征根实部为根实部为-2。第四章习题课第四章习题课 (4-5)解解:，s(s+1)Kr(s+2)G(s)=j0p1=0p1 p2=-1p2 z1=-2z1 p1p2z1-分离点和会合点分离点和会合点s2+4s+2=0s1=-3.41s2=-0.59s2+s+Krs+2Kr=0闭环特征方程式闭环特征方程式系统根轨迹系统根轨迹s=-2+j(-2+j)2+(-2+j)(1+Kr)+2Kr=0-4+(1+Kr)=04-2-2(1+Kr)+2Kr=0Kr=3=1.41第70页4-6 已知系统开环传递函数，试确定闭环已知系统开环传递函数，试确定闭环极点极点=0.5时时Kr值。值。第四章习题课第四章习题课 (4-6)解解:p1=0 p2=-1 p3=-3p1p2p3-8+60o=+180o,3=-1-3=-1.3根轨迹分离点：根轨迹分离点：A(s)B(s)=A(s)B(s)3s2+8s+3=0s1=-0.45s2=-2.2s2没有位于根轨迹上，舍去。没有位于根轨迹上，舍去。js(s+1)(s+3)KrG(s)H(s)=与虚轴交点与虚轴交点 s3+4s2+3s+Kr=02=0Kr-43+3=0-Kr=0 Kr=12 2,3=1.71=01.7-1.7p3-30p1 p2-1系统根轨迹系统根轨迹s1=0.5得得 s1=-0.37+j0.8Kr=|s3|s3+1|s3+3|s3=-4+0.372=-3.26=3.262.260.26=1.9s3第71页4-7 已知系统开环传递函数，已知系统开环传递函数，(1)试绘制出试绘制出根轨迹图。根轨迹图。第四章习题课第四章习题课 (4-7)解解:p1=0 p2=-2 p3=-4p1p2p3-8+60o=+180o,3=-2-4=-2根轨迹分离点：根轨迹分离点：A(s)B(s)=A(s)B(s)3s2+12s+8=0s1=-0.85s2=-3.15s2没有位于根轨迹上，舍去。没有位于根轨迹上，舍去。j(3)与虚轴交点与虚轴交点 s3+6s2+8s+Kr=02=0Kr-63+8=0-Kr=0 Kr=48 2,3=2.81=02.8-2.8p3-40p1 p2-2系统根轨迹系统根轨迹s1s3s(s+2)