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Mastering Zero-Input and Zero-State Response in Systems

School
University of California, Los Angeles**We aren't endorsed by this school
Course
MAE 107
Subject
Mechanical Engineering
Date
Dec 11, 2024
Pages
7
Uploaded by ChefRose22801
TA : Shuo-Wu(Will) ShihEmail : swshih@g.ucla.eduOffice Hour : Thursday 10-11 amSolvingtheIVP->InitailValueProblem=ayce)+bult),110)=Yo,asSol:Y(t)=galt-tolyoStact-ibulusdiyLa-zero-inputzero-stateex1:Compute(t)whenefortoandY(t)=eaty-Stact-u(t)dYeatStatistbedT-enteT=e*yo+bet(ex-a(t))·et
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ex2:ComputeY(t)whenuct)=etforthoandC=aY(e)=2%+beatsendt,=n=eatyotbeatso1dy=eatyo+beeex3:ComputeY(t)whenult)=cost)Formula:eitheReImu(t)=coslve)=e(eirt)Y(t)=galt-tolyoStact-ibulusd=atStalt-bReLe=e%+ReSataTitT=e*+Yo+Re(beatnotLimad=e%o+Felder(east)
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=etyothe-alt-1)4=a+bj=0,estP=Jub2,0=tan")(2=c+dj=92ejtP2=J,E=tan"(2)(b+0)Ente↑A·Oor T=ey+Fellsere*)p:-tan"(Re=ey+xe)eb+we+Pljatkbl,=e%+(CS(Ob+we+P)-ecos(b++)aut yo,b=0=Yown(costvetys-ecosp),boEIIe**Yo(-cos(rt+1)+eatcosp),60Los(2+B)=cos&cosB-sinsing=>cos(+i)=CoS2·(1)-Sind0=-los
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=%+Fema(203 (we++)-eatcosa)zzero-inputzero-stateata co,texe->0=-Swe-transientresponsesteady-stateresponse
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ex4:-V-V+=v-=EI↓Wi-⑫?=⑫RY=4memy(t)=ay(+)+bu(t)Ia=-2,b=10=2y+10u()ComputYct)whenult)=etfor20,y(0):1Y(e)=eaty+Sta(t4)bu(i)di=hty+St-2(t-4).10.edi=e**Yo+10e+(e%)=e-+Yo+10e(et-1)It-t=+ezero-inputzero-state
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-2t(b)computey(t)whenact-eforT20y(0)=I-Y(e)=eaty+Sta(t4)bu(i)di=hty+St-2(t-4)·10.edi=ey+10 sed=e-Hy+10egdY=eyo-10+etYo=Y(0)=1=e-2+10+e()computey(t)whenUC)=sintfortho.Aft-Inlet,Y(e)=eaty+Sta(t4)bu(i)di=hty+St-2(t-4)·10.Im(est)di=e**Yo+Im(o10.ectgrind=e**Yo+[m(10etLehi)=(+St-1)S
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X=ety+Imet-1)=e%+Im=eA=(0-tan"(t))=-tan"(E)=e Y+Im=e*YotIn(-(sin(p++)-esind)unzero-stateYo=-1=e+(sin(p+t)-e* sind)=(-1-Esind)e*+sin(4++)-utrasientresponsesteady-stateresponse
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