Applied Classical and Modern Control System Design

1 AppliedClassicalandMo dernControlSystemDesignRichardTymerskiFr ankRytkonenAndrewChuinardPortlandStateUn iversityDepartmentofElectricalandCompute rEngineeringPortland,Oregon, duction(ADDCONTENT).. opVoltageMo edSystemMo :Zerof1= :Zerof1= opandMulti-Lo opComp opComp deComp IMo dernControl1437 Intro enLo opPerformanceofthe eci eci ' ukConverterwithStateFeedbackComp ukConverterwithStateFeedbackandIntegralC omp ukConverterwithLQRComp ukConverterwithLQGComp delReductionoftheLQGI/LTRComp endices209 Routh-HurwitzStabilityAnalysis211viiiCON TENTSPart IClassicalControl1 Chapter 1 Intro duction(ADDCONTENT) :DesignFlowDiagramChapter ductionInthischapter,theparametersforane xamplecontrolsystemwillb ede ,subsequentchap-terswillb edevotedtoapplyingcontroldesigntechnique stooptimizesystemp eginswithamo delforplant,thathasatleastoneparticularp arametertob ,theparametertob econtrollediscomparedtoastablereferencev alueandthedi erenceisinputtoanerrorampli erthencommandstheplant.

2 FeedbackSystemBlo ductionThefundamentalitemineverycontrols ystemistheplant,theitemthatistob ,theplantwillb ede nedasabuckconverter,aswitchedmo deDCp :BuckConverterCircuitSystemDiagramThebuc kregulator, ensation,isabasicswitchedmo dep (reducedbyH(s)),andthedi erence(errorsignal)isfedtoacomp ensatorwhichdrivesapulse-widthmo ,thismo ,threetransferfunctionsarerequiredtob (s),inputvoltagetooutputGvg(s), :Simpli :GeneralizedPowerSystemMo delvoltage,orop enlo opoutputimp edanceZout(s).Additionaltransferfunction swillb ederivedinthesection,suchasthecontroltoi nductorcurrentGid(s),outputcurrenttoindu ctorcurrentGii(s),andtheinputvoltagetoin ductorcurrentGivg(s).Thesetransferfuncti onswillb ,thecompletesetoftransferfunctionswillb (s)AnalysisToanalyzethesmallsignalcontro ltooutputtransferfunctionofthebuckconver ter,anoutputloadchangeismo deledwithacurrentsource, ,thisadditionalsourcewillb emo :BuckConverterCircuitwithNon-IdealCircui tElementsListedb nesthestatevariablesofthesystemandtheuva riablesde erofstatesisde nedbythenumb , [iv]u=[vgio]x =Ax+Buy=Cx+EuApplyingtheprinciplesofsup erp ositiontothebuckconverter,thecapacitorcu rrentandinductorvoltageequationsarefound andsummarizedb elowforb othswitchp ,LdiLdt= (rl+rc||R)iL RR+rcVc+Vg+Io(R||rc)CdVcdt=RR+rciL 1R+rlVc+ 0Vg (RR+rc)IoVout= (rc||R)iL+RR+rcVc+ 0Vg (R||rc) :BuckConverterSup erp ositionAnalysis:DTsDuringD'Ts,LdiLdt= (rl+rc||R)iL RR+rcVc+ 0Vg+Io(R||rc)CdVcdt=RR+rciL 1R+rlVc+ 0Vg (RR+rc)IoVout= (rc||R)iL+RR+rcVc+ 0Vg (R||rc)IoWiththecircuitde nedoverthetwosubintervals,theA,B,C,andEm atricescanb ede nedasshownb.

3 BuckConverterSup erp ositionAnalysis:D'TsA1=A2=A=[ (rc R+rl)L R(R+rc)LR(rc+R)C 1(rc+R)C]B1=[1L0]B2=[00]B=[DL0] :BUCKCONVERTER11C1=C2=C=[(rc||R)RR+rc]E1 =E2=E=[0 (rc||R)]Withthestatespacematricesde ned,thecontroltooutputtransferfunctionca nb ecalculatedasGvd(s) =C(sI A) 1Bd+Ed,whereBd= (A1 A2)X+(B1 B2)UandEd= (C1 C2)X+ (E1 E2) ,Gvd(s)iscalculatedb A 1BU= DV gR+rl Io(R2(R+rc)(R+rl) Rrc(R+rc)(R+rl))Io(R2rc(R+rc)(R+rl)+R(Rr c+Rrl+rcrl)(R+rc)(R+rl))+DRVgR+rl Bd= (A1 A2)X+ (B1 B2)U=[VgL0]Ed= (C1 C2)X+ (E1 E2)U= 0 Gvd(s) =C(sI A) 1Bd+Ed=Vg(1 +sRC)rc+RRs2LC+s(LR+(rl+rc+rcrlR)C)+(R+r l)RGii(s)AnalysisWhencalculatingtheoutpu tloadtoinductorcurrenttransferfunction,i tcanb enoticedthattheinductorvoltageandcapacit orcurrentequationswillb eidenticaltothoseusedincalculatingGvd(s) ab ,onlytheoutputequationisneededtob ederivedto [1 0]E1=E2=E=[0 0]Usingthevaluesfoundab ove,theoutputloadtoinductorcurrentfuncti onisequaltoGii(s) = (C(sI A) 1B+E),withthenegativesignduetothede (s) =C(sI A) 1B+E=(R rcR+rl+srcC)rc+RRs2LC+s(LR+(rl+rc+rcrlR) C)+(R+rl) (s)

4 ,Gvvg(s),andGivg(s)AnalysisTocalculateGi d(s),thecontroltoinductorcurrenttransfer function,Gvvg(s),theinputvoltagetooutput voltagetransferfunctionandGivg(s),theinp utvoltagetoinductorcurrenttransferfuncti on,thesameparametersderivedab ovecanb (s) =C(sI A) 1Bd+Ed=VgR(1 +s(R+rc)C)rc+RRs2LC+s(LR+(rl+rc+rcrlR)C) +(R+rl)RGvvg(s) =C(sI A) 1B+E=D(1 +srcC)rc+RRs2LC+s(LR+(rl+rc+rcrlR)C)+(R+ rl)RGivg(s) =C(sI A) 1B+E=DR(1 +s(R+rc)C)rc+RRs2LC+s(LR+(rl+rc+rcrlR)C) +(R+rl)RZo(s)AnalysisTocalculatetheoutpu timp edanceoftheBuckconverter,itisp ,duetotheinputvoltageconnectionoftheBuck converterwecantakeadvantageofthefactthat theimp edanceofthebuckontheoutputlo oksthesameregardlessoftheswitchlo :IdealBuckConverterCircuitZo(s) =SL 1sC R=sL1 +sLR+ :BuckConverterCircuitwithNon-IdealCircui tElementsOnewaytoincorp oratelossesintotheimp edancefunctionistoreplacetheenergystorag eelementintheimp edanceequationwiththesumoftheelementandi tsnon-idealresistivecomp ,everyinstanceofsLisreplacedwithsL+rlinZ o(s).

5 Zo(s) =sL+rl1 +rlR+s(LR+rlC)+s2 LCAssumingrlR<<1Zo(s) =sL+rl1 +s(LR+rlC)+s2 LCNow,addingtheESRofthecapacitor:sC sC1 +SRcZo(s) =(sL+rl) (1 +srcC)1 +s(LR+ (rl+rc)C)+s2(1 +rcR)LCThecompleteoutputimp edanceoftheBuckconverterisfoundtob e:Zo(s) =rl(1 +sLrl)(1 +srcC)1 +s(LR+ (rl+rc)C)+s2 LCAssuming(rcR<<1).Tosummarizetheresults ofthissection, CaseLosses IncludedBuck Converter Transfer dulatorWithacompletemo delinplacefortheBuckregulator,thenextite minthesystemdiagramtoderiveisthepulse-wi dthmo ,itcanb eapproximatedthatthedutycyclecanb erepresentedbythefollowingrelationship:d (t) =Vc(t)VMfor0 Vc(t) VMRearranging,d(t)Vc(t)= rmingthePWMconversiongaincanb eapproximatedtothisequation,refertothean alysispresentedin[?]. eende ned,the nalblo cktob ederivedintheBuckconvertersystemmo delisthecomp ensator,Gc(s).Thecomp ensatorwillb etheprimarytopicofthenextseveralchapters ,asitwillb eleveragedtoimprovetheclosed-lo opp [1] onsebasedontheoutputimp ,18(6):1270 1277,Novemb er2003.

6 [2] , opcontrolmetho dforthetransientresp ,pages718 723,2003.[3]RDMiddlebro dulatorphaselaginpwmconverterfeedbacklo , 3Bo dePlotsIntro :Poleat :Zeroat :RightHalfPlaneZeroat :SecondOrderComplexPoleat :SecondOrderComplexZeroat :InvertedPoleat :InvertedZeroat 4 SingleLo opVoltageMo deControlThispap erdevelopsabuckconverterdesignexampleusi ngdi erentcomp ensa-tionmetho dstoensureclosedlo opstabilityandtooptimizesystemp ensatorsaredesignedusingasymptoticBo deplotsbasedprimar-ilyonlo onseb ehaviorandtoverifyp ductionThebuckconverterisaswitchmo de,DC-DC,p ,Vgandpro ducesaloweroutputvoltage,Vwithhighe ortantcomp opofabuckconverterpresentsseveralchallen geswhichareexploredinthecomp erwepresentaseriesofexamplebuckconverter feedbackcomp ectsofdi erentcomp olthatwillb eappliedtoevaluatethedi erentcomp ensationapproachesareasymptoticBo deplotswhicharedrawnbasedoncornerfrequen ciesofeachblo dologyprovidesaquickande cientassessmentofcircuitp erformanceandanintuitivesenseforthetrade o sforeachcomp deplotsalsodirectlyilluminatethetwocriti callo opstabilitycharacteristics,gainandphasem argin(GMandPMresp ectively).

7 Additionalanalysisofeachcomp [1] ectsofVgtransients,acommonprobleminrealp [2]simulationisalsop erformedtovalidatethemanualBo opMat-labsimulationisusedtoshowtheabilit yofthefeedbacksystemtoattenuateundesired e ckdiagramofthesystemcomp owerstageacceptsVgasitsp owersourceandthecontrolinputd(s)topro (s), erenceoutputofthesetwovoltagesisprovided tothefeedbackcomp ensationcircuitGc(s)andthentothepulsewid thmo dulator(PWM)whichpro ducesthecontrolwaveformfortheswitchingco nverterd(s).THeresultinglo opgainisthusgivenbyT(s) =Gc(s)(1VM)Gvd(s)H(s)( ) :GeneralizedPowerSystemMo edSystemMo delThegeneralbuckconverterblo ckdiagramprovidesacompletemo ,forouranalysiswewilluseasimpli edmo :Simpli etweentheselectedcomp ensationschemes,thedesignofthebuckconver teris xedwithsp eci eci :Sp eci edvaluesNameValueDescriptionVg28 VInputVoltageV15 VOutputVoltageIload5 ALoadcurrentL50uHBuckinductorvalueC500uF BuckcapacitorvalueVm4 VPWM rampamplitudeH(s)1 delforthep owerconverterblo edbyequation( ).

8 Ithasaresonantfrequencyvalue, o= (= o2 ) = ( ) ( ).Thelowfrequencygainoftheconverterisequ altoVgwhichissp eci edtob (s) =Vg11 +sQ 0+(s 0)2( ) o=1 LC( ) :ConverterPowerStageQ=R CL( )Considerthetransferfunctionv(s)/vd(s)of thelowpass 100kHzismuchhigherthantheresonantfrequen cyf0= eration,theswitchtogglestheLCRinputb etweenVgandgroundwithadutycycleDdetermin edbythefeedbacklo onentV=DVgandanfsfundamentalcomp onentanditsharmonicsastypi lterwithacuto ensatedSystemItisinstructivetostartourev aluationwithanuncomp ensatedop enlo opconverter,onewithaGc(s) = opgainisthengivenfrom( )asT(s) =To1 +sQ 0+(s o)2( )whereTo=VgH(0)Vm( )ToconstructaBo deplotweusethevaluesfromequations( )-( )toestablishtheshap eoftheBo ( ) ovefothegaindeclinesat-40 dephaseplotisdeterminedonlybyGvd(s).Itha salowfrequencyphaseshiftof0 .Atfo10 12 Qor886Hz( 900Hz),thephaseturnsnegativeandatf0theph asehasreached 90 .Thephasecontinuestob ecomemorenegativeuntilitreaches 180 at1012 Qor1129Hz( ).

9 180 . :Uncomp ensatedGainandPhasePlotFromtheBo deplotitcanb edeterminedthatunitygaino ccursatafrequency,f=fcsuchthatTo(fofc)2= 1( )whichwithTo= ,resultsinfc= 180 esnotcrossthe 180 phaselevel(butisasymptotictoit)whichimpl iesthatthegainmarginisin .Also,theMatlabanalysisindicatesanin enlo (dB)102103104-180-135-90-450 Phase (deg)Bode DiagramGm = Inf dB (at Inf Hz) , Pm = deg (at +003 Hz)Frequency (Hz) :MatlabUncomp ensatedBo deplotsteadystatedutyratioofD=VVg=1528= 15 VforanominalinputvoltageofVg= onseoftheop enlo 30V 30= , 28 SW1D1L1 50 uC1 500 uR1 VP1k1 = k2 = k3 = Vpk = Period = 10 uDel = Per = 10 uV2 :PECSS chematicofOp enLo opSystem :PECSS imulationofOp enLo ensationDominantp olecomp ensationisoneofthesimplestandmostcommonf ormsoffeedbackcomp ehindthistyp eoffeedbackcontrolistoshap etheop enlo opgainofthesystemsuchthattwoob 'slowestfrequencyp ositivephasemarginand,consequently, olecomp ensation,theseob jectivesareachievedusingacomp ensatorconsistingofasinglep oleatafrequencywellb elowthoseoftheplant'sp osesofthisexample,anintegrator,whichisju stap oleatDC,isemployedGc(s) = Is( )where I(= 2 fi) deplotasymptotesforthemagnitudeandphaseo fthiscomp.

10 Bo dePlotofDominantPoleComp ensatorDesignofthecomp ensatornowconsistsofselectinganappropria tecomp en-satorparameter, ,thisisamatterofcho osingthelargestcomp ensatorgainsuchthatthetotalgainatthelowe stfrequencyplantp ole(s) opgainofthesystemwiththiscomp (s) = ITos[1 +sQ 0+(s 0)2]( ) opgainwiththecomp ecausetheplant'sdominantp oleissecondorder,itcontributesaphaseshif tof 180 athighfrequenciesandashiftofexactly 90 ,thecomp ensatorcontributesitsown 90 phaseshiftanddo ,thetotalphaseshiftofthecomp ensatedop enlo optransferfunctionis 180 atthedominantp olefrequency, :GraphicalConstructionofPhaseAsymptotesf orDominantPoleComp ensatedOp enLo ensatortransferfunctionscombinetopro ducethegainofthecomp ensatedop enlo opgainthatis-3dBatfo, ( )ForTo= ,we ndfI= :GraphicalConstructionofGainAsymptotesfo rDominantPoleComp ensatedOp enLo ensatordesignedandveri edviaMatlab,thenextstageistodesignacircu itthatimplementsthecomp :Op enLo opSystemGainandPhasewithDominantPoleComp en-sationformofanop erationalampli erinaintegratorcon :G(s) = 1(s/ o)( )where o=1RC( )where , owiththecomp ensatorparameter, I(= 2 fI)andsolvingforRgivesR=1 IC=12 (32)(50nF) 100k ( ) (dB)101102103104-270-225-180-135-90 Phase (deg)Bode DiagramGm = dB (at 1e+003 Hz) , Pm = deg (at Hz)Frequency (Hz) :MatlabAnalysisofDominantPoleComp :Op-AmpIntegratorCircuitAPECS simulationiscreatedtoverifythetimedomain p , 30V esap o 28 SW1D1L1 50 uC1 500 uR1 VP1k1 = k2 = k3 = Vpk = Period = 10 uDel = Per = 10 uC4 50 nR5 100 kR6 kR7 kV3.