线性控制系统分析与设计期末考试(大学).docx

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1、一.(a)求位置y(t)与力f(t)有关的微分方程；(b)画出机械网络图；(C)确定传递函数G(D)=y/f。ri(a)nodexa(K2MyD2)xa-K2xh=fnodeXb(/C2+K3+BD+M2D2)xh-K2xa=O(c)G(D)=-4v7D4+BD3+KD2+BKa+KaKh-KwhereKa=KI+K?,Kb=K2+KyiK=Ka+Kb二、So1vethefo11owingdifferentia1equations.Assumezeroinitia1conditions.Sketchtheso1utions.D2x+6x=(1),尸1,k=0,HUO.q=k-w=0Theste

2、adystateoutputistherefore:xss=b0GXSSz0.Insertingtheseva1uesintopreviousequation(1):16XSS=16f1XSS=b匠(2)16Thehomogeneousequationisformedby1ettingtherightsideofthedifferentia1equationequa1zero:D2xz+16xz=0(3)thetransientresponseistheso1utionofthehomogeneousequation,isobtainedbyassumingaso1utionoftheform

3、(4)xt=Atwheremisaconstantyettobedeterminedthecharacteristicequationofsystem:/+i6m=/+16=0加尸4上忻一4jva1uesofmarecomp1ex,byusingtheEu1eridentityeij,t,=cosdtjsndtandthencombiningterms,transientso1utionsare(6)x1=A1e4j,+A2e4j,=B1cos4r+B2sin4z=Asin(4r+)x=xt+XSS=ASin(4，+。)+一16Assumezeroinitia1conditions,i.e.,

4、t=0,x(O)=Q,DX(O)=Q、insertingtheseva1uesintopreviousequation(7):X(O)=ASin+=0,Dx(O)=4ACOSo=O16gjA=-1216x=xt+xss=-sin(4r+)+16216三、Writethe1ap1acetransformsofthefo11owingequationsandso1vefoex);theinitia1conditionsaregiventotheright.万户2.8ZZv+4产10X(O)=2,以(0)=3The1ap1acetransformsoftheequations(s)-sx(0)-(0

5、)+2.8(SX(S)-X(O)+4X(S)=IOSTX(s)(s2+2.8s+4)-(2s+8.6)=IOs1X(s)(s2+2.8s4)=2a+8,65+10XG)=s2j-8.65102s+8.6s+10S(S2+2.85+4)S($+1.4)2+(04)2)=A+As+B522.8+4Theinverse1ap1acetransformsoftheequation(p637,appendxA36)x(t)=2.51.69e,4zsin(2X)4r-17.25)rvr1252+8.65+10A=sX(S)1=SZ8s+4=2.55=0XX=21+8.6s+102.5-0.55+1.65(

6、52+2.85+4)s+2.8S4山、252+8.65+102.5-0.55+1.62.515-3.2X(S)=1=1rS(S+2.8S+4)ss2.8s+4s2(5+4)?+(j2.04jTheinverse1ap1acetransformsoftheequation(p637,appendxA26)x(t)=2.5-0.54-3.2sin(04r+tan,)V2.04-3.2-1.4x(t)=2.5-1.69e,4/sin(Mr-17.25)四、Forthefo11owingsystem,C(S)(a)Drawanequiva1entsinga1f1owgraph,(b)Derivetra

7、nsferfunctionsforE(s)R(s),X(s)R(s),B(s)R(s),C(s)R(s),andY(s)R(s).H(1)(2)(3)(4)(5)(6)(a) 11=-HG1.1ZA0=1-Z1+Z2-z3+.=1+HG1Ti-GiT2-G21=12=1+HG1T二C(S)二TA,G+G2(1+HG)一丽-1+7G1五、Foreachofthefo11owingcases,determinetherangeofva1uesofKforwhichtheresponsec(t)isstab1e,wherethedrivingfunctionisastepfunction.Deter

8、minetherootsontheimaginaryaxisthatyiedsustainedosci1ations.C(S)=ss(s+2)(s2+4s+20)+KSo1ution:B)=%O)=1,R(S)=1r(t)=1u,i(t)=sK、C(s)由G+2)(s?+4s+20)+KKG(S)=H=：=AR(S)2s(s+2)(/+4s+20)+KThecharacteristicequationofthesystemis:Q(S)=S(S+2)(/+4s+20)+K=/+6T+28s2+40s+KTheRouthianarray:s128Ks36405264/3Ksx-6C+2560/3

9、s。KBasedRouth,sstabi1itycriterion,forstab1eoperationofthesystem,therangeofK:K0-6C+2560/30therangeofva1uesofK。一六、Aunity-feedbackcontro1systemhas(1)G(S)=20Ks(s+I)(S+5)+20wherer(t)=21.(a)IfK=1.5,determinee(8ss；(b)Itisdesiredthatforarampinpute(t)51.5,whatminimumva1ueKhaveforthisconditiontobesatisfied?(1

10、) So1ution:(a)determiningetssr(0=2r,R(S)=1卜)=E(S)=RG)二21+H(s)G(s)-5212(s+1)(s+5)+4020K7SKS+1)(s+5)+20+20Ks(s+1)(s+5)+20e(t)ss=e()=IimSE(S)=IimSJ2(-+5)+40_SSvoSToSs(s+1)(s+5)+20+20K_Iim2(S+1)(s+5)+40_2(0+1)(0+5)+40_5_5-I。5(5+1)(5+5)+20+20K-20K2K3(Z?)findingtheminimumva1ueKxIfe(。&s15isdesired,then1.5,so2KK-=K.=K.=-3,m,n3七、ForeachofthetransferfunctionsG(S)=24(1+0.5)_52(1+0.25)(1+0.055)(a)drawthe1ogmagnitude(exactandasymptotic)andphasediarams;(b)drawthepo1arp1ot