Solution Manual For Modern Control Engineering, 5th Edition

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B-2-1..:.RG)—[Zo’0<):-B.-B.paa8GFG2.cs)G+6:RE)/t(G+G2)(Gs~Ge):CS—B-2-2.:e6)re62)6)[¢.].a7—RE[-a6)-+6,———en:14Go(Hy—He)C6)__(/+G06G-.RS)1+GulH=e)-ISE——————————.B-2-3..}’Co2%)<=![622=&=x):[]EE:Ld1StudyXY©2010PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThispublicationisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpeslission(s),writeto:RightsandPermissionsDepartment,-PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.-

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DownloadedfromStudyXY.com®+StudyXYSdYe.o>\|iF’prE\3SStudyAnythingThisContentHasbeenPostedOnStudyXY.comassupplementarylearningmaterial.StudyXYdoesnotendroseanyuniversity,collegeorpublisher.Allmaterialspostedareundertheliabilityofthecontributors.wv8)www.studyxy.com

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“.g©:-£Hat—=—)nsrrrmeKimg4N.C.6:93(GatH)w«@).’I4G)(Gt)Bs3CiGGG+GGzHyBNBCeB —2£y:RO1+Gell¥G61+GyHH:+G6,G,+G,G3H,B-2-4.Inthefollowingdiagrams,(a)denotestheunit-stepresponseand(b)~~correspondstotheunit-rampresponse.R.:-.al)“e)‘©:P3®’:..Rd.)-H2).|6/28+%oe723#¢©’YlKL=k-rIbaz.gt)ue.@.:é.#[44®].R*'®+*--.BsoPy.2--.6723F£F.oa3§¢.Loening)=trnd) fhet(rnd=tlread’+StudyXxy©2010PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThispublicationisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic,niechanical,photocopying,recording,orlikewise.Forinformationregardingpermi@sion(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.

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wo.:“)|®oo’oo.®fR£):0rz2gp&=[3.»=k(ts+s)=#(rth+ars)B-2-5.WhenD(s)iszero,theclosed-looptransferfunctionCr(s)/R(s)isiGl)__Ge€)GE)Rs)-AtGels)Gpé):WhenR(s)=0,theclosed-looptransferfunctionCp(s)/D(s)is-:C6)_/-.pe)1+Gel®Gb)Whenboththereferenceinputanddisturbanceinputarepresent,theoutputC(s)aisthesumofCg(s)andCp(s).Hence-Lo/-:CE)=Cefo)+C,p=—L—|9GolRE)+PO)|CL%p65)PEP)59Gl?B-2-6.WhenonlythereferenceinputR(s)ispresent,theoutputCg(s)isgivenRs)1G05)626)ForthereferenceinputR(s),thedesiredoutputisR(s)fortheunity-feedback.systemsuchasthepresentsystem.Thus,theerrorEp(s)isthedifference‘betweenR(s)andtheactualoutputCp(s).TheerrorEp(s)isgivenby.:)Cels)BE=RO-G@)=RO[/-=|—reG,6)G26)]—/yy’.#&)[&146,6)62091G,6)5:094.Assumingthesystemtobestable,thesteady-stateerroreggp(t)canbegivenby2AR)Css(8)=Li,Cp)=LiSEH=.ITY)©2010PearsonEducation,Inc.,UpperSaddFRAN.AlrightsreservedThspublicationisprotectZdByopyrightaidFeissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermjssion(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458..

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WhenonlythedisturbanceinputD(s)ispresent,theoutputCp(s)isgivenby’.’Spb)_G5.26)1+G(s)G5)sincethedesiredoutputtothedisturbanceinputD(s)iszero,theerrorEp(s)-canbegivenby.E6)=0—Cpls)=—C6).Hence-Gels?R-.Ep)=—Cp6)=—————————Df)°id146,096.5)Forthestablesystem,thesteady-stateerroreggp(t)isgivenby’&;&)=Lom5)=Da.==-5G2(5)DE).wr)=LzCott)=Ly560)=Looe5500Thesteady-stateerrorwhenboththereferenceinputR(s)anddisturbanceinputD(s)arepresentisthesumOfeggp(t)andeggp(t)andisgivenby:Cssft)=Crp(@)+Essp(BD:=/;[sRE)_SG(5)PE)]BEEdd136,676,9/1G,6)G2(3)~=La.=[#8).=—————r—6,6)PB):.ssofIt6,6)6:(9%I.B-2-7..whenD(s)=0,theblockdiagramofthesystemcanbesimplifiedas-.follows::ER2Wvs|saG®>e/+G,6.H,3Theclosed-looptransferfunctionCg(s)/R(s)canbegivenby.-GeGrGzGsSoCris)=I+G)Ga#7_GeG1G2Gy:RE)ypSGGsll16,6.H+Go,BYStudyxy|1+6,6,Hy©2010PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThispublicationisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpersassion(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458..~

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WhenR(s)=0,theblockdiagramofthesystemshowninFigure2-34canbeifiedasfollows:.modi.DE).Cpl:$CLarasef—4,~B—%1%.Dé&)G6)|&|7Hence%6)_%6,ooG28s—_——=26)(+G:G3G,(6th+4)IEG.GGMtGGl,EO,B-2-8.Thereareinfinitelymanystate-spacerepresentationsforthissystem.Weshallgivetwoofthepossiblestate-spacerepresentations.State-spacerepresentation1:FromFigure2-35,weobtain+.YoSssez=BE—.TE)43LLSppstys+E:StP52.whichisequivalentto~ase-..RNFrpytyr2y=d+zu:Comparingthisequationwiththestandardequation.fodooopoe.Adray+ayrary=4+hi+hutbha.weobtain©g|~v;.:UY=p,a:=/,az=2,by=0,b=0,be=1,1StudyXY©2010PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThispublicationisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpéignission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.

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X=Y-Pe«@€-Tamj-pid—fu=*"fi%.%=dfwwhereooby=be=0.|p=bape=o.b=be=afa:fo=/]:Also,defineAa=by—a;fs—4afr—p=Z—pThen,state-spaceequationscanbegivenby.-:p4eo1oflxl|4:H=1°e/+plu:[=1—rIps]Lad.%Yalr0x|+fk::Xsor-%eo+oll|2’.%|=lo¢o/+}7ju:5)|2-1rls][=r:y=[/¢lx©|x:Siz_st+p+2-p1+EL.)_.S+ps+pP+Studyxywecanredravtheblockdiagramasshownbelow.:©2010PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.Thispublicationis-protectedbyCopyrightandwrittenpermissionshouldbe~~obtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorby.anymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.-6-

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vayvxXoR-%272ht)..®Q+4+hooFromthisblockdisgramwegetthefollowingequations:.V=v-X+XzRN:T=X, s+p’—_—-T-xi+Xss::XLXsfromwhichweobtaing./+py=(2~-—pa+PR=(2p)a=(ep)x,Hw—x+Atu::°-.N=:Rewriting,wehave.=[JAAETT....A=u-—(z-p)x,—pXs+(2-plu.Lo7=%Coor’.-A£147dfx°Zl=1-/0/|le|+]1|e.|VvEApzeo—=plixs2-1StudyXY§©2010PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThispublicationisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermissions).writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.eeB

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EFi7=C/eo.2]EA):.xeeB-2-9..Ld..’Lt3y+2y=«Define.m=#LdGg‘Then.o[+30+2X=uoo.;EA=;R=A,.or:.CLxe(oflx[JLl=|eoolxt+|oleAE)le—2=3)m](4-Z¥=[7oo]|%:.Xz-B-2-10.R-¢-/an=RhB.4[5~|Br&=U°1ThetransferfimctionG(s)ofthesystemisgivenbyCo:s+¢771StudyXY.-/1udyXY.)=C{sT—-Ar=[7o::G6)gCI4)EL1-3S+¢/©2010PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThispublicationisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.-8--R

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[ro]/s+/Aik:Gro+)+2|3selfs|Logs,)stp&s+7s+7:—so.S24Ss+73eeB-2-11,:S—5-/2,=-==z©oa=[T0][Fera|ThetransferfunctionG(s)ofthesystemisgivenby:oo~s+7172)Ge)=CC:I~27B=2]|;5]|s./s+-/|l2}|.=[7s—del[r=]+s)(s+)+3|3sts1s./I2]25-3=/25+5.FstgstsFoTd|ssazss*+gs+EE’AMATIABsolutiontothisproblemisgivenbelow.-:EEE:A=[513I;.:B=[25];.c=02%-D=0;:.*|frumde}=ss2ABCD)|:i.0125-.dean168.)EEB-2-12).-o/oV7c=/0A={o27s|+B=1001sS%1s15:—-2-¢—§|/0-)©2010PearsonEducation,Inc.,Upper$addleRiver,NJ.Allrightsreserved.ThisPublicatioTsprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermissions),writeto:RightsandPermissionsDepartment,~~_PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.I=-

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Thetransfermatrixofthesystemcanbegivenby:co-1s=/[4ao.-LoClips—o)=0(sI-AB=]kao=gr-oe=[s0ole=lol Tsteseeste[eo'-[1°d/-zsess{lo7]aS345rgs+2’—25—es-2SEO,)//St€=——2,:SSrssi+astz|SSHES:/S+&SpF5544s5tT2.Streepgst==.S%gs)Sitesgs+2s¥+fs+gst+2|-.AMATLABsolutiontothisproblemisgivenbelow.-.A=[0100.01;246}.Co:B=[000L1ofiC=[100010};::D=[00;00[oum,den]=ss2{(A,B,C.D,1)’om=::-00.00000.00001.0000.-000000100000.0000:)1.00006.00004.00002.0000i[num,den]=ss2{A.B,C.D,2)ico)-num=)}’-00.00001.00006.0000.01.00006.00000.0000:}):)+StudyXxyden=©2010PearsonEducation,Inc.,UpperSaddleRiverJNJ.NOs6.0000714.9000.0090.tedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedrefkoductioavsiorage-a-a-retreve~ortramehissioninanyformorbyanymeans,electronic,©mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.~10-

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B-2-13,DefineZF=z*tPry+35=Fy)CL25xX<4,rosy</zLetuschooseX=3and¥=11.ThenF=T™+535+39=prade+365=65¢-WeshallobtainalinearizedequationforthenonlinearequationnearthepointX=3,¥=11.ExpandingthenonlinearequationintoaTaylorseriesaboutpointx=X,y=¥andneglectingthehigher-orderterms,weobtainE-F=k(x=Z)+t=(=F)-where5=ee=2%re=6+88=2£;[£23l=y=FTeyI#=2£=PTET=24+48=90ke=#...gazTETBE.g=u“HencethelinearizedequationisF—g36=FF(x=3)+70(y—1/)orR=F7¢x+70y—634«B-2-14.DefineCoY=o2=fx),E=2—2=¥=do=f+xx).Sincethehigher-ordertermsinthisequationaresmall,neglectingthoseterms,weobtain.4-f&)=o.2(x-2)or.Y-0.2x2%=0.4X2°(X-2)Thus,linearapproximationofthegivennonlinearequationneartheoperatingpointis+StudyXY:4=2.4X~-3.2i©2010PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThispublicationisprotectedbyCopyrightandwrittenpermissionshouldbe.obtainedfremmte-pubtSHerprortoamyProttotetreprottetom-Storagefraretevat-SySteTTorTassoITEnyTorOrBY3nymeans,slectonic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,-PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.~11-

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B-3-1.sincethesameforcetransmitstheshaft,wehavefeb(3-2)=b,(§-8)+bs(7-2)Co(&3)wheredisplacementzisdefinedinthefigurebelow.5R=1by0|GByB2|.x¥zIntermsoftheequivalentviscousfrictioncoefficient,theforcefisgivenby$=.»(P_*)(2)FromEquation(1)wehave.....byZrbyE+bz=pZtbag+bs7”Lhhth]sz=by+(batbs)Yy(3©b,tbatbs'iBysubstitutingEquation(3)intoEquation(1),wehave’..:4—Z.sFybdThx(batb3)T]ff=4(2x)ThLh-by+bs2=p—(g-X(4)bgapesCF)Hence,bycomparingEquations(2)and(4),weobtainber.batbs/=Bb)————=——bitb.tbts!>¢by+bs&;)eee—————————rrrTTTB32,’(2)i.wx+X=:|~,XO_J’TE)ns*+%©2010PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThispublicationisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.-12-

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(b)Definethedisplacementofapointbetweeningskjandkyasy.Thentheequationsofmotionforthisaystembecomha1z7..WXFy(X-y)=u%Y=Xa(x2)Fromthesecondequation,wehave..#,=k.oryy+kH2y=_%_i8Foytka:mE+_biksdH=uori}ths:X&/TemsePuke..7+#sy-8-8-3..)hdLdTemg,thGof)hy=a,te.*mgth(Gg~3th,=u,x=F)ZL=:Ce[’Ee7jpAWw=#23*=%Then...MX,+b(a=)+%&Xx=,.WyXp+(t=)rby=4,...iXG=X:.B-==[b-z)tbx]+55,4=xSomiLnCromeyrba]+©2010PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThispublicationisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermigs®a(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.E

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.oratlCMwlas)-[C/A21£L4wmTw©wm1%]|9[4]’,=+%o¢[HlleOflu:3be_ke4AXs4mewzwm||%#emyx&_/eoooooxX,Co:Ya0¢(ollx;Xf————B34...Je=T.:.whereT=—24a*—~mglawrtJ=ml*:Forsmall6,:wd26=~2ka*6—myleor.428a|2\Fee2)m0misvq)"B-3-5.Notethat:Gxtdato,Yo=Aens’Forxdirection:’edLd-’MAX+2Zs=uSoarozeoz=(x+=uMX+mpr(X+fan7)i.||©2010PEducation,Inc.,UpperS:£2bets.Bi@sresoffe&:cBDbdwrittionshouldbyearsonEducation,Inc.,Upper:ain)orfandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanypo2reproduction,storageiharetrievalsystem,ortransmissionanyformorbyanyans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.-14-h.

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©wehave.*’g2pa(Mim)x—md(anf)Etml(en8)d=ccForsmall6andsmall62,wehavese[Ad(Mim)X+md6=(1)Forrotationalmotion:RTo=mpdaint—mxLengwhere’.tween,=mg~Thus,o.(Tm?)=mplais—milcn8Forsmall6,wehavete.-(T+mL?)6=mgdo-mlx=@FromEquation(2),..I+.YL}.X=9Q8—-——___§¢7mlco.SubstitutingthislastequationintoEquation(1),weobtain:Tr+ml®oo15.m)(38~25)+mlb=u_(mm)(38=~———EIT(Mm)36—(pm)T+pmd®meThus,sosnd(Mim)mdPR{CiBpm="(Mm)z+pmdL(Msi)IT+mdAlso,fromEquation(1)vehave-oemgdls—mlxrm)%rmEEEmex_(+m)’I4+mLe.or.BNa”B[MTomzee]+7428=4(T+rion.fromwhichweget’©2010PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThispublicationisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermissions),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.==

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