A Complementarity Model for Closed-Loop

AComplementarityModelforClosed-Loop

PowerConverters

ValentinaSessa,Member,IEEE,LuigiIannelli,SeniorMember,IEEE,andFrancescoVasca,SeniorMember,IEEE

Abstract—Atacertainlevelofabstraction,powerconverterscanberepresentedaslinearcircuitsconnectedtodiodesandcontrolledelectronicswitches.Theevolutionsoftheelectricalvariablesaredeterminedbythestate-dependentswitchings,whichcomplicatethemathematicalmodelingofcontrolledpowerconverters.Dif-ferentlyfromthecomplementaritymodelspreviouslypresentedintheliterature,themodelproposedinthispaperallowstorepre-sentasalinearcomplementaritysystemalsoclosed-looppowerconverters,withoutrequiringtheaprioriknowledgeofthecon-vertermodes.Amodelconstructionprocedure,notdependentonthespecificconvertertopology,ispresented.Thediscretizationofthecontinuous-timemodelallowstoformulatemixedlinearcomplementarityproblemsforthecomputationofthecontrol-to-outputfrequencyresponseandtheevolutionsofbothtransientandsteady-statecurrentsandvoltages.Asillustrativeexamples,Z-source,boost,andbuckdc–dcpowerconvertersundervoltage-modecontrolandcurrent-modecontroloperatingbothincontin-uousanddiscontinuousconductionmodesareconsidered.IndexTerms—Circuitmodeling,closed-loopsystems,dc–dcpowerconversion,discretetimesystems,switchingcircuits.

I.INTRODUCTION

T

HEtypicalaveragedandswitchedmodelsofpowercon-vertersdependonthesequenceofmodesandoperatingconditions[1]–[4]whicharedeterminedbythespecificswitch-ingfunctions[5]andcontrolrequirements[6].Consequently,theuseoftheseclassicalmodelsfortheanalysisoftheclosed-looppowerconvertersrequirestheintroductionofsuitablesimplify-ingassumptions,[7],[8].

Complementaritymodels[9]representanattractivesolutiontoovercomesomeofthesedrawbacks.Recently,theyhavebeenproposedasanusefulframeworkformodelinglinearcircuitswithdiodesandidealswitches[10],[11].Indeed,thecomple-mentarityframeworkcanbeusedtomodelawideclassofnon-smoothdynamicalsystems,suchasmechanicalsystemswithCoulombfriction[12]andpiecewiselinearLur’esystems[13].Suchideahasbeenmorespecificallyappliedtopowerconvertersin[14].Linearcomplementarity(LC)mod-elshavebeenproposedformodelingspecificopen-looppowerconverterstopologies:single-phasediodebridges[15],

ManuscriptreceivedAugust1,2013;revisedNovember15,2013andJanuary15,2014;acceptedFebruary8,2014.DateofpublicationFebruary20,2014;dateofcurrentversionAugust13,2014.RecommendedforpublicationbyAssociateEditorC.K.Tse.

TheauthorsarewiththeDepartmentofEngineering,UniversityofSannio,PiazzaRoma21,82100Benevento,Italy(e-mail:valentina.sessa;luigi.iannelli,vasca@unisannio.it).

DigitalObjectIdentifier10.1109/TPEL.2014.2306975

three-phaserectifiers[16],switchedcapacitors[17],andres-onantconverters[18].Dealingwithmoregeneralclassesofpowerconverters,thecomplementaritymodelsproposedin[19]and[20]areconstructedbyconsideringthespecificswitchesstatesandbyassumingtheaprioriknowledgeofthepowerconverterconfigurationateachswitchingtimeinstant.Astronglimitationofthemodelsproposedin[10],[14],and[21]isthattheyrequireswitchingofthesets(cones)ofthecomplemen-tarityvariables.Unfortunately,thispropertydoesnotallowtoobtainthemodelsofcontrolledpowerconvertersinamanage-ableclosedform.Analternativeapproachwithfixedconesandcharacteristicsdependingonsomeexternalforcingvariableispresentedin[22]:theswitchmodelisasignumcharacteristichavingtheswitchcurrentastheargumentandwhoseamplitudeisapiecewiselinearfunctionofanexternalcontrolvoltage.In[23],transistorsaremodeledwithinthecomplementarityframework,buttheanalysisislimitedtostaticcircuits.In[15],theswitchesaremodeledbymeansofequivalentfiniteresistorsrepresentingtheON(conducting)andOFF(blocking)statesde-terminedbythesignumofanexternalvoltagesignal.Thatmodelcannotbeeasilygeneralizedtothecaseofzeroconductingandinfiniteblockingequivalentresistances.

ThemodelproposedinthispaperisabletorepresentinanLCexplicitformthelargesignaldynamicsofclosed-looppowerconverters.Thecomplementaritymodelissimpletobebuiltandcapturesallmodesoftheconverter,withoutenumeratingthem,norassumingtheaprioriknowledgeofthesequenceofmodesandoftheswitchingtimeinstants.SuchLCmodelisshowntobeusefulforthecomputationofthecontrol-to-outputfrequencyresponse,whichisacrucialissueforcontrolledpowerconverterssoasanalyzedin[18],[24],andthereferencestherein.Moreover,theproposedLCmodelofclosed-looppowerconvertersallowstoobtaindirectlytheclosed-loopsteady-statebehavior,whichisoftenanontrivialtasksoasinthecaseofLCCresonantconverters[25],[26]andmodularmultilevelconverters[27],[28].

Therestofthepaperisorganizedasfollows.InSectionII,thestaticLCmodelscorrespondingtotheidealdiodeandthecon-trolledswitchescharacteristicsarepresented.TheprocedurefortheconstructionofthedynamicLCmodelofapowerconverter,bothinopen-loopandclosed-loop,isdescribedinSectionIII.SectionIVshowshowthisrepresentationallowstocomputethesteady-stateperiodicoscillationasasolutionofamixedLCproblemderivedfromthediscretizedclosed-loopsystem.InSectionV,theapproachusedforthesteady-statesolutionisappliedforthecomputationofthecontrol-to-outputfrequencyresponseforaZ-sourceconverter[29],whereastheunique-nessofthesteady-statesolutionisguaranteedbythepassivity

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6822IEEETRANSACTIONSONPOWERELECTRONICS,VOL.29,NO.12,DECEMBER2014

hypothesis.InSectionVI,theusefulnessoftheproposedtech-niqueforthecomputationoftransientandsteady-statesolutionsofclosed-looppowerconvertersisnumericallydemonstratedbyconsideringcurrent-controlledandvoltage-controlledboostdc–dcpowerconverterswithstableandunstableperiodicsolu-tions[4],[30]andaclosed-loopbuckdc–dcconverterwhichexhibitsmultiplesolutions[31].SectionVIIconcludesthepaper.

II.COMPLEMENTARITYMODELSOFELECTRONICDEVICESTheidealizedvoltage–currentcharacteristicsofelectronicde-vicescanberepresentedbymeansofstaticLCmodels.Toshowthis,letusintroducethedefinitionofamixedLCproblem,[32].Givenavectorq∈Rr,amatrixM∈Rr×randlowerandupperboundsl,u∈Rr∪{−∞,+∞}r,amixedLCproblemconsistsoffindingz∈Rrsuchthat

w−v=Mz+q

(1a)0≤w⊥(z−l)≥0(1b)0≤v⊥(u−z)≥0

(1c)

where“⊥”representstheorthogonalitysymbol,i.e.,w⊥(z−l)standsforw󰀞(z−l)=0,andinequalitiesamongvec-torsaremeantcomponentwise.Inthecasel=u,theuniquesolutionof(1)isz=l=u.Forl=u,thenonnegativevari-ableswandvarecomplementarymeaningthatcomponentwiseatleastoneofthetwomustbezero.ThemixedLCproblemiswell-knowninthemathematicalprogrammingtheory,infact,itprovidesanaturalsettingfortheKarush–Kuhn–Tuckercondi-tionsofaquadraticprogramwithgeneralequalityandinequalityconstraints,[9].

Whenthelowerboundliszeroandtheupperbounduisinfinity,wegetv=0andthemixedLCproblembecomesaclassicalLCproblemwithwandzbeingtheusualcomple-mentarityvariables.Then,givenavectorq∈RrandamatrixM∈Rr×r,aLCproblemconsistsoffindingzsuchthat

w=Mz+q(2a)0≤w⊥z≥0.

(2b)

Itiswell-knownthatanLCproblemintheform(2)hasauniquesolutionforallqifandonlyifMisaP-matrix,[9].AmatrixMiscalledaP-matrixifallitsprincipalminorsarepositive.Accordingtothedefinition,everypositivedefinitematrixisaP-matrix,buttheconverseisnottrue.IfthematrixMisonlypositivesemidefinite,thentheuniquenessoftheLCproblemsolutioncannotbeproved,butitispossibletoprovethatamongallsolutionsoftheLCproblemtheleast-normoneisunique,[33].

TheLCframeworkcanbeusedtorepresentidealizedvoltage–currentcharacteristicsofelectronicdevices.Tothisaim,letusassociateto(2)an‘output’equationintheform

ϕ=Nz

(3)

whereNisarealmatrixofsuitabledimensions.Theexpressions(2)–(3)canrepresentanLCmodelfora(ϕ,λ)electronicdevicecharacteristic:thevariableλcanbeinterpretedastheinputof

Fig.1.Idealdiode:(a)symbol,(b)idealizedvoltage–currentcharacteristic,(c)idealizedcurrent–voltagecharacteristic,(d)characteristic(ϕandoutputd,λd)withinputλd=−vDandoutputϕd=iDorwithinputλd=iDϕ−vd=D.

thecharacteristicandentersintotheLCproblemifthevectorqdependslinearlyonit,andϕcanbeseenasthecorrespondingoutput.Asanexample,letusconsidertheidealizedpiecewiselineardiodecharacteristicshowninFigs.1(b),1(c)andassumethattheoppositeofthediodevoltage,sayλd=−vD,isthegiveninput.Then,theoutputvariableofthediodemodelcanbechosenasthediodecurrentϕd=iD.ByconsideringtheresultingcharacteristicinFig.1(d),thediodecharacteristiccanberepresentedwiththefollowingLCmodel

ϕd=zd(4a)wd=λd

(4b)0≤wd⊥zd≥0.

(4c)

Theequation(4a)isintheform(3)withN=1and(4b)–(4c)areintheform(2a)–(2b)withM=0andq=λd.SinceMiszero,thecurrentzdwhichsatisfies(4)foragivenvoltageλdisnotnecessarilyunique,whiletheleast-normsolutionmustbeunique.Indeed,ifλd=0,thenanyzd≥0satisfies(4),buttheleast-normsolution(zd=0)isunique.Obviously,(4)representstheidealcharacteristicofadiodealsoifλd=ivDandϕd=−D,i.e.,whentheproblemconsistsoffindingadiodevoltageϕdgiventhediodecurrentλd,seeFigs.1(b)and1(c).

AnothertypicalpiecewiselinearrepresentationforthediodecharacteristicisthatshowninFig.2.Thevoltage–currentchar-acteristicinFig.2(c)canberepresentedbythefollowingLCmodel:

ϕ=zd

(5a)wd=λ+rDzd+VF(5b)0≤wd⊥zd≥0

(5c)

SESSAetal.:COMPLEMENTARITYMODELFORCLOSED-LOOPPOWERCONVERTERS6823

Fig.2.Piecewiselinearcharacteristicsofadiode:(a)voltage–currentchar-acteristic,(b)current–voltagecharacteristic,(c)(ϕ,λ)characteristicwithinputλoutput=−vϕand=−outputv.

ϕ=i,and(d)(ϕ,λ)characteristicwithinputλ=iandwhiletheLCmodelofthecurrent–voltagecharacteristicinFig.2(d)isgivenby

ϕ=zd−VF−rDλ(6a)wd=λ

(6b)0≤wd⊥zd≥0.

(6c)

ItisnowpossibletoconstructanLCmodelforanelectronicswitchcharacteristic,seeFig.3.Byconsideringabipolarjunc-tiontransistor,theswitchvariableshavethefollowingmeanings:iSisthecollectorcurrent,vSisthecollector-emittervoltage,imaxisthemaximumcollectorcurrentthatthetransistorcanprovidewithoutdamage,σ∈[0,1]isthenormalizedswitchcontrolsignalproportionaltothebasecurrent,-analogousin-terpretationsofthephysicalmeaningsforiS,vS,imax,andσarepossibleforotherelectronicswitches.Then,forσ=0(OFF),theswitchingisblockingandtheswitchcurrentiSisidenti-callyzeroforanyswitchvoltagevS.Forσ=1,theswitchisconducting(ON).Themodelalsocapturestheswitchoperatingconditionintheso-calledsaturationregionwherethedevicecanprovideanycurrentbetween0andimaxσ.

Letusassumethattheoppositeofthediodevoltage,sayλc=−vSandthecontrolsignalσaregiven.Then,theoutputvariableoftheswitchmodelischosenastheswitchcurrentϕc=iS.Thesubscript“c”isusedtoindicatethatthevariableϕinthiscaseisacurrent.Then,thevoltage–currentcharacteristicinFig.3(d)canberepresentedwiththefollowingLCmodel:

ϕc=zc1

(7a)wc1=zc2+λc(7b)wc2=−zc1+imaxσ(7c)0≤wc⊥zc≥0

(7d)

wherewc=col(wc1,wc2),zc=col(zc1,zc2),andcol(·)indi-catesavectorobtainedbystackinginauniquecolumnthe

Fig.3.Electronicswitch:(a)symbol,(b)idealizedvoltage–currentchar-acteristic,(c)idealizedcurrent–voltagecharacteristic,(d)characteristicwithλc=−vWithoutlossSandϕc=iofgenerality,S,and(e)characteristicwithλv=iitisassumedthattheswitchisSandϕv=−vabletoblockalsoS.negativevoltages.

columnvectorsinitsargument.Themodel(7)isintheform(2)–(3)withN=[10],q=col(λ󰁴c,i󰁵maxσ)and

M=01

−10.(8)

Expressions(7)canbeexplainedbylookingatthepossiblevaluesassumedbyλc.Thefollowingcasesarepossible,seeFig.3(d):

λc>0⇒wc1>0⇒zc1=0⇒ϕc=0

(9a)λc<0⇒zc2>0⇒wc2=0⇒ϕc=zc1=imaxσ

(9b)

λc=0⇒{wc1≥0,zc2≥0}⇒ϕc=zc1∈[0,imaxσ].

(9c)

Forσ∈[0,1],themodel(7)describesthesetofpossiblevoltage–currentcharacteristicsobtainedbyvaryingσbetween0and1.Thematrix(8)ispositivesemidefinite(andnotP-matrix)andforλc=0thesolutionzcof(7)isnotunique,see(9c).Ontheotherhand,amongallpossiblesolutionsof(7),thefollowingleast-normsolution:

λc>0⇒zc=col(0,0)(10a)λc<0⇒zc=col(imaxσ,−λc)

(10b)

6824IEEETRANSACTIONSONPOWERELECTRONICS,VOL.29,NO.12,DECEMBER2014

λc=0⇒zc=col(0,0)

(10c)

isunique.

AnanalysissimilartothatpresentedabovecanbedoneiftheswitchcurrentiSandtheswitchcommandσaregivenandthecorrespondingswitchvoltagemustbefound.Indeed,onecanconsiderthecurrent–voltagecharacteristicshowninFig.3(e),whereϕv=−vSistheswitchvoltageandλv=iSistheswitchcurrent.ApossibleLCmodelcanbewrittenas

ϕv=zv1−zv2(11a)wv1=λv

(11b)wv2=−λv+imaxσ(11c)0≤wv⊥zv≥0

(11d)

wherewv=col(wv1,wv2)andzv=col(zv1,zv2).Themodel(11)isintheform(2)–(3)withN=[1−1],q=col(λv,−λv+imaxσ)andMbeingthezeromatrix.Theex-pressions(11)canbeexplainedbyconsideringthedifferentrangesforλv.Thefollowingcasesarepossible:

λv=0⇒{wv1=0,wv2=imaxσ}

󰁺

⇒

0<σ≤1⇒zv2=0⇒ϕv=zv1≥0

σ=0⎧⇒ϕ−z(12a)v=zv1v2∈R

⎪⎨0<σ≤1⇒wv1>0⇒zv1=0

λv=imaxσ⇒⎪⎩

⇒ϕv=−zv2≤0

(12b)σ=0⇒see(12a)0<λv0,wv2>0}⇒ϕv=0.(12c)Consideralsotheantiparallelconnectionofanelectronic

switchandadiode.Assumethatλistheoppositeofthevoltageacrosstheparallel,thatisequaltothevoltageacrosstheswitchandiisthetotalcurrentthroughtheparallel,thatisthedifferencebetweentheswitchandthediodecurrents.Bycombiningthevoltage–currentcharacteristic(ϕc,λc)oftheelectronicswitchinFig.3(d)withthevoltage–currentcharacteristic(ϕd,λd)ofthediodeinFig.1(d),onegetsthevoltage–currentpiecewiselinearcharacteristicoftheantiparallelconnectioninFig.4(b).TheresultingcharacteristichasonebreakingpointanditcanbemodelledbytheLCmodel

ϕ=−z+imaxσ(13a)w=−λ(13b)0≤w⊥z≥0.

(13c)

Thesamereasoningcanberepeatedalsowhenthevoltageoftheantiparallelconnectionisconsideredastheoutputandthecurrentastheinput,seeFig.4(c).ThecorrespondingLCmodelcanbewrittenas

ϕ=−z(14a)w=−λ+imaxσ(14b)0≤w⊥z≥0.

(14c)

Remark1:TheLCmodelsofdiodesandcontrolledswitchesarethebasicelementsfromwhichmorecomplexpiecewise

Fig.4.Antiparallelconnectionofanelectronicswitchandadiode:(a)symbol,(b)idealizedvoltage-currentcharacteristic,and(c)idealizedcurrent–voltagecharacteristic.

linearcharacteristicsofelectronicdevicescanberepresentedinthecomplementarityform.Forthesakeofsimplicityinthesequel,onlytheLCmodels(4),(7),and(11)willbeconsidered,thoughthemodelingapproachproposedinthispapercanbeextendedtothecaseofmoreinvolvedpiecewiselinearcharac-teristicsofelectronicdevices.

TheLCmodelsoftheelectronicdevicescanbesuitablycol-lectedintoacompactcomplementarityrepresentationwhichisusefulfortheformulationoftheentirepowerconvertercom-plementaritymodel.LetusassumethatthepowerconverterunderinvestigationhasNddiodes,NcswitcheswithcurrentsasoutputvariablesandNvswitcheswithvoltagesasoutputvari-ables.Withsomeabuseofnotation,letusredefinethefollowingvariables:ϕdandλdarethecolumnvectorswhosecomponentsarethevoltagesandcurrentsthroughtheNddiodes,ϕcisthecolumnvectoroftheNcswitchescurrentsandλcarethecorre-spondingvoltages,ϕvisthecolumnvectoroftheNvswitchesvoltagesandλvarethecorrespondingcurrents.Bydefiningthecolumnvectors

ϕ=col(ϕd,ϕc,ϕv)(15a)λ=col(λd,λc,λv)(15b)wo=col(wd,wc,wv)(15c)zo=col(zd,zc,zv)(15d)σ=col(σc,σv)

(15e)

SESSAetal.:COMPLEMENTARITYMODELFORCLOSED-LOOPPOWERCONVERTERS6825

Fig.5.BlockdiagramoftheLCmodelforaclosed-looppowerconverter.

theLCmodelobtainedbycollectingalldevicescharacteristicscanbewrittenas

ϕ=Bϕzo

(16a)wo=Cϕλ+Dϕzo+Gϕσ(16b)0≤wo⊥zo≥0,

(16c)

whereBϕ,Cϕ,Dϕ,andGϕcanbeeasilyobtained.

Themodel(16)isintheform(2)–(3)withN=Bλ+Gϕ,q=CϕϕσandM=Dϕ.Then,theidealizedcharacteristicofelectronicdevicescommonlypresentinagenericpowercon-vertercanberepresentedinthecompactLCform(16).

III.COMPLEMENTARITYDYNAMICMODEL

Inthissection,itisshownthatthemodelofawideclassofclosed-looppowerconverterscanbewritteninthefollowingdynamicLCform:

x˙=Ax+Bz+Ee(17a)w=Cx+Dz+Fe(17b)0≤w⊥z≥0

(17c)

wherexisthestatevector,wandzarecomplementarityvari-ables,eisthevectoroftheexogenousinputs,andA,B,C,D,

E,Fareconstantmatricesofsuitabledimensions.

Theconstructionofthecomplementaritymodelismodularinthesensethatitdirectlyfollowsbyassociatingtheconverterdynamicmodel—derivedbystandardmethodsanddependingonlyonitstopology—withtheelectronicdevicescharacteris-tics.Ablockschemerepresentation,correspondingtothemodel(17)decomposedintoitsmajorsubsystems,isshowninFig.5.Themodelconstructionprocedureisappliedstep-by-steptoaboostdc–dcconverterwithvoltage-modepulse-widthmodula-tion(PWM)control,seeFigs.6and7.A.Open-LoopLCModel

Asapreliminarystep,itisshownthatadynamicmodelofopen-looppowerconverterscanbewrittenintheform(17).Atypicalapproachforderivingapowerconverterdynamical

Fig.6.Boostdc–dcconverter.

Fig.7.Blockdiagramofaproportional–integralcontroller:eσfortheoutputvariabley;misthemodulationsignal;1istherefer-encesignaleσthecontrollerstatevariable(theintegraloftheerror);2isthecarriersignal;xσiskpandkiaretheproportionalandintegralgains,respectively;σistheswitchingsignal.

modelconsistsofextractingthediodesandswitchestotheports

andapplyingtheKirchhofflawstothebranches(andnodes)ofapropertreeofthegraphassociatedtothecircuit,[34].Inparticular,supposethatthecapacitorsdonotformaloop(withorwithoutvoltagesources)andinductorsdonotformacutset(withorwithoutcurrentsources).Then,byapplyingtheKirchhofflaws,astate-spacemodelinthefollowingformcanbeobtained:

x˙o=Aoxo+Boϕ+Eoeo(18a)λ=Coxo+Doϕ+Foeo

(18b)

wherexo∈RNoisthestatevectoroftheopen-loopsystem,

eodenotesthevectoroftheexternalsourcesfortheopen-loopsystem,and(ϕ,λ)arethevoltage–currentorcurrent–voltagepairsoftheelectronicdevices.

Byusing(16)with(18),themodelofanopen-looppowerconvertercanbewritteninthefollowingLCformx˙o=Aoxo+BoBϕzo+Eoeo(19a)wo=CϕCoxo+(CϕDoBϕ+Dϕ)zo+CϕFoeo+Gϕσ(19b)

0≤wo⊥zo≥0.

(19c)

Themodel(19)correspondstothedashedboxedblockinFig.5,withwoandzobeing‘internal’complementarityvari-ables,xothestatevector,eotheexogenousinputandσthecontrolinput,i.e.,thevectorofthecommandstotheelectronicswitches.Foropen-loopcontrolledconverters,thevectorσcanbeincludedintotheexogenousvectoreo,then,themodel(19)isintheform(17).

Asanillustrativeexample,considertheboostdc–dcconverterdepictedinFig.6.Denoteeoastheinputvoltage,xo1astheinductorcurrent,xo2asthecapacitorvoltage,(ϕc,λc)asthecurrent–voltagepairoftheelectronicswitch,and(ϕd,λd)asthevoltage–currentpairofthediode.ByapplyingtheKirchhoff

6826IEEETRANSACTIONSONPOWERELECTRONICS,VOL.29,NO.12,DECEMBER2014

laws,thecircuitmodelcanbewrittenasfollows:

x˙Ro11=−Lx111

o1−Lxo2+ϕd+e(20a)11L1L1ox˙o2

=1Cx11o1−xo2−ϕc

(20b)1R2C1C1

λd=xo1−ϕc(20c)λc=−xo2+ϕd

(20d)

whichisintheform(18)withxo=col(xo1,xo2),ϕ=col(ϕd,ϕc)andλ=col(λd,λc).

Byusing(4)and(7),orequivalentlybyconsidering(16)withNd=Nc=1andNv=0,thedevicescharacteristicscanberepresentedasfollows:

ϕd=zd(21a)ϕc=zc1(21b)wd=λd(21c)wc1=λc+zc2

(21d)wc2=−zc1+imaxσ(21e)0≤wo⊥zo≥0

(21f)

whichareintheform(16)withϕ=col(ϕλ=col(λd,ϕc),

d,λc),

zo=col(zd,zc1,zc2),andwo=col(wd,wc1,wc2).Then,bycombining(20)and(21),theopen-loopLCmodeltakestheform(17)withe=col(eo,σ).B.Closed-LoopLCModel

Theclosed-loopLCmodelofpowerconverterscanbeob-tainedbyspecifyingthedependencesoftheNs=Nc+Nvswitchingsignalsσonthestateandontheexogenousinputs.Asanexample,considerthattheswitchesarecontrolledbymeansofaPWMtechnique.Then,defineλσasthedifferencebetweenthemodulationsignalmandthecarriersignaleσ2,andσasastepfunctionwhoseargumentisλσ,seeFig.7.AscalarswitchingsignalσcanberepresentedbymeansofthefollowingLCmodel:

σ=zσ1

(22a)wσ1=zσ2−λσ(22b)wσ2=−zσ1+1(22c)0≤wσ⊥zσ≥0

(22d)

wherewσ=col(wσ1,wσ2),zσ=col(zσ1,zσ2).Themodel(22)canbeexplainedbyconsideringthesignofλσ.Ifλσ<0,sincezσ2isnonnegative,from(22b),itwillbewσ1strictlypos-itiveandbytheusingcomplementarityconstraint(22d),from(22a),wegetσ=zσ1=0,i.e.,theswitchisOFF.Ifλσ>0from(22b),itmustbezσ2strictlypositiveandthen,byusingthecomplementarityconstraint(22d),itwillbewσ2=0,and(22c)with(22a)leadstoσ=zσ1=1,i.e.,theswitchisON.Fi-nally,ifλσ=0,the(22b)doesnotimposeanyconstraintonzσ2andthen(22b)withthenonnegativeconstraintonwσ1impliesthatσ=zσ1cantakeanyvaluewithintheinterval[0,1].

Byusing(22),theNsswitchingsignalscanberepresented

alltogetherinthefollowingcompactcomplementarityform:

σ=Bσzσ

(23a)wσ=Cσλσ+Dσzσ+γσ(23b)0≤wσ⊥zσ≥0

(23c)

wherezσ∈R2Nsisthevectorofallcomplementarityvariablesrequiredtorepresenttheswitchingsignals,andthematricescanbesimplyobtained.

Inordertocompletetheclosed-loopLCmodelofapowerconverter,amodelofthecontrollermustbeconstructed,seeFig.5.Inparticular,forawideclassofpracticalcontrollers,thecontrolsignalcanberepresentedastheoutputofalineardynamicmodelthatcomputesitscontrolactionbyusinginfor-mationonthepowerconverterstatexoand,possibly,onotherexogenoussignals

x˙σ=Aσxσ+Aσoxo+Eσoeo+Eσeeσ(24a)λσ=Cmσxσ+Cmoxo+Fmeeσ

(24b)

withxσ∈RNσbeingthestateofthecontrollerandeσhavingthereferenceinputsandothercontrolinputsascomponents.Remark2:Thecomplementarityframeworkcanbeusedtorepresentamoregeneralclassofcontrollers,e.g.,thoseconsist-ingoflinearpartscombinedwithpiecewiselinear,quantized,orhystereticcharacteristics,ormoreingeneralallrelationshipsrepresentablewithinthecomplementarityframework.Inthesecases,LCmodelsofpiecewiselinearcharacteristicsandalge-braicmanipulations,similartothosepresentedinthispaper,allowtowritetheclosed-looppowerconverterintheform(17).InSectionVI-C,thiswillbedemonstratedforcurrent-modecontrolledpowerconverters.

ThecomplementaritymodelcorrespondingtotheblockschemeinFig.5isthencomplete.Inparticular,(24)isthecontrollermodel,(23)isthemodulatormodel,and(19)isthemodelofthecircuitincludingtheelectronicdevicescharacter-istics(thedashboxedblock).Bycollecting(19)–(24)andbydefining

x=col(xo,xσ)(25a)e=col(eo,eσ,1Ns)(25b)z=col(zo,zσ)(25c)w=col(wo,wσ)

(25d)

thecompletemodelofthecontrolledpowerconvertercanbewrittenintheform(17)󰀂with

A=A0

󰀃oA(26a)

󰀂

σoAσ

B=

B󰀃oBϕ0

00(26b)󰀂C=C󰀃ϕCo0

CC(26c)

σmoCσCmσ

SESSAetal.:COMPLEMENTARITYMODELFORCLOSED-LOOPPOWERCONVERTERS6827

󰀂D=

CϕDoBϕ+Dϕ

G󰀃

ϕBσ0D(26d)

󰀂σ

E=

Eo00󰀃

EE(26e)

󰀂σoσe

0F=

CϕFo00󰀃0

CσFme

γ.

(26f)

σ

Itisimportanttohighlightthatthemodel(17)containsallthepossibleoperatingmodesoftheclosed-looppowerconverterandthatthematricesA,B,C,D,E,Fareconstant.

Goingbacktotheboostillustrativeexample,consideraproportional–integralcontrollerwhoseinputistheerrorbe-tweentheoutputvoltagey=xo2anditsreferencevalue,sayeσ1,seeFig.7.Byindicatingwithxσthestatevariablecor-respondingtotheoutputoftheintegrator,witheσ2thecarriersignal,withkpandkiastheproportionalandintegralgainsofthecontroller,respectively,thecontrollerdynamicscanbewrittenas

x˙σ=−xo2+eσ1

(27a)λσ=kixσ−kpxo2+kpeσ1−eσ2

(27b)

whichareintheform(24)witheσ=col(eσ1,eσ2).

Theswitchingsignalσcanbeexpressedbyusingthemodel(22).Letusdefinex=col(xo,xσ),e=col(eo,eσ).Byusing(26),orequivalentlybysubstituting(22a)in(21e)and(27b)in(22b),theclosed-loopmodelcanbewritteninthedynamicLCform(17)withthefollowingmatrices:

⎡R11

⎤⎢−−L0

1⎥A=⎢

L⎢1

⎢⎣11⎥⎥(28a)

C1−R0⎥2C1

⎦0−10⎡1⎤

⎢L0000

1⎥B=⎢⎢⎢⎣0−1⎥

(28b)

C000⎥⎥1

⎦⎡000⎤00⎢100⎢0−10⎥⎥C=⎢⎢⎢⎢00

0⎥⎥

(28c)

⎢⎥⎣0k⎥p−ki⎥⎦

⎡000

⎢0

−1000⎤

⎢10100⎥⎥D=⎢⎢⎢⎢0

−10i0⎥⎢max

⎥⎥(28d)

⎣

00001⎥⎥⎦

0

0

0−1

0

⎡100

0

⎤

E=⎢⎢L1

⎣⎥

0

000⎥⎦(28e)

⎡0100⎢

0

0

00⎤

⎢00⎥

⎥F=⎢00⎢

⎢⎢

0000⎥⎥⎢⎥(28f)

⎣

0−k

p10⎥.⎥⎦0

0

0

1

TheLCmodel(17)withthematrices(28)representsthe

dynamicevolutionsoftheclosed-loopdc–dcboostconverterforallpossibleoperatingmodes,initialconditionsandexogenousinputs.

IV.TIME-STEPPINGANDSTEADY-STATESOLUTIONSThemodel(17)canbeusedtocomputethetransienttime-steppingevolutionandthesteady-stateoscillationexhibitedbyacontrolledconverter.Tothisaim,letusdiscretizethesystem(17)withasamplingperiodh.Byconsidering,forinstance,theTustinmethodoneobtainsthefollowingdiscrete-timesystem:

xk−Azxk−1=Bz(zk−1+zk)+Ez(ek−1+ek)(29a)wk=Cxk+Dzk+Fek(29b)0≤wk⊥zk≥0

(29c)

withk=1,2,...,Nh,xk=x(kh)andanalogouslyforthe

othervariables,and󰁲

thematricesaregivenby

AIh󰁳−1󰁲h

󰁳z=Nx−2AINx+2

A

(30a)Bh󰁲

h󰁳−1z=2INx−2

A

B(30b)󰁳−1Eh󰁲

h

z=2INx−2A

E(30c)whereINxistheNx×NxidentitymatrixandNx=No+Nσ

isthenumberofthestatevariablesoftheclosed-looppowerconverter.

Givenzk−1,ek−1,ekandxk−1,onecansolve(29a)forxkandsubstitutethisvaluein(29b)that,togetherwith(29c),willbeusedforfindingzk.Therefore,thetime-steppingevolutioncanbeobtainedbyiteratingthesolutionofLCproblemsintheform

wk=Mzk+qk

(31a)0≤wk⊥zk≥0

(31b)

with

M=CBz+D

(32)

and

qk=CAzxk−1+(CEz+F)ek+CBzzk−1+CEzek−1

(33)

fork=1,2,...,Nh.EachLCproblem(31)canbesolvedbyusingstandard(andefficient)algorithmswidelystudiedinthecomplementarityliterature[9].

6828IEEETRANSACTIONSONPOWERELECTRONICS,VOL.29,NO.12,DECEMBER2014

ThedynamicLCmodel(29)canbeusedalsotocom-putesteady-stateoscillationsexhibitedbyclosed-looppowerconverters.Letusassumethatthesystem(17)hasaperi-odicabsolutelycontinuoussolutionx(t)=x(t+T)foreverycontinuous-timeinstanttwithTbeingtheknownperiodofthesolution.Weassumethattheconvergenceproperty,alsocalledconsistency,holds,i.e.,thecontinuouspiecewiselinearinter-polationofthesamplessequencexkgivenby(29)convergestothecontinuous-timesolutionx(t)of(17)ash=T/Nhgoestozero.Inparticular,weassumethat(29)hasaperiodicsolu-tionofperiodNh,approximatingtheperiodiccontinuous-timesolutionofperiodTofthesystem(17).Foralargeclassofconstrainedcomplementaritysystems,theconsistencypropertyholds[33].ThenumericalresultsinSectionsVandVIwillcon-firmthevalidityofthispropertyalsoforthepowerconvertersconsidered.Bydefining

x¯=col(x1,x2,...,xNh)

(34a)e¯=col(e1,e2,...,eNh)(34b)z¯=col(z1,z2,...,zNh)

(34c)

w¯=col(w1,w2,...,wwithx∈RN,z∈R(4NNandh)(34d)

x·Nhs+Nd)·Nh

byusingtheperi-odicityconditionx0=xNh,wecanwritesimultaneouslytheequations(29)alongtheperiodNhwhichleadsto

0=Ax+Bz+Ee(35a)

w=Cx+Dz+Fe

(35b)

with

⎡

−INx00···

0A⎢z⎤⎢⎢Az−INx0···00⎥0···0⎥⎥A=⎢⎢Az−INx0⎢⎢...⎢..

.......

....⎥

.⎥.⎥⎣⎥(36a)0···0Az

−INx0⎥⎦⎡00···0ABz00···0⎤z−INx

B⎢z⎢

⎢BzBz0···00⎥B=⎢⎢0BzBz0···0⎥⎥⎢⎢...⎥⎢

..

..........

..⎥.⎥⎣⎥(36b)

0···0BzBz

0⎥⎦

00···0BzBz

C=INh⊗C(36c)D=I⎡Nh⊗DE0

···0E⎢z0z⎤(36d)

⎢

⎢EzEz0···00⎥0⎥⎥E=⎢⎢Ez

Ez0···0⎢⎥⎢.

..⎥⎢

..

.............⎥⎣⎥(36e)

0···0EzEz0⎥⎦

00···

0

Ez

Ez

F=INh⊗F

(36f)

where⊗denotestheKroneckerproduct.

Remark3:Bydiscretizing(17)withthebackwardEulerorthe

Zero–Order–Holdtechniquesandbyusingsimilararguments,onecanobtaincorrespondingrepresentationsintheform(35)withsuitableA,B,C,D,E,F.From(36a),oneobtains

det(A)=(−1)Nhdet(INzh

x−AN)

(37)

thenA¯issingularifANhhassomeeigenvaluein1,whichhappensforinstancewhenz

Ahaseigenvaluesintheorigin,see(30a).

IfthematrixAisinvertible,theperiodicsolutionof(29)canbeobtainedbyformulatingasuitableLCproblem.Indeedbysolving(35a)forthevectorx,weobtain

x=−A−1

(Bz+Ee).(38)

Then,bysubstituting(38)in(35b),weobtain

w=Mz+q(39a)

0≤w⊥z≥0

(39b)withM=−CA−1

B+D

(40)and

q=(−CA−1

E+F)e

(41)

whichisaclassicalLCproblemintheform(2).Thesolutionzcanbeusedin(38)toobtaintheperiodicsteady-statesolutionx.

Remark4:IfthematrixAisnotinvertible(whichisthecasefortheclosed-loopboostconverterconsideredpreviously,see(28a),(30a),and(37)),onecandefineamixedLCproblem(1)whichallowstoavoidtheinversionofAforthecomputationoftheperiodicsolution.Indeed,thesystem(35)canbewrittenintheform(1)bydefining

z=col󰀂

(x,z)

(42a)M=

AB

󰀃CD(42b)󰁴󰁵q=EFe

(42c)l=col(−∞Nx·Nh,0)

(42d)u=col(+∞Nx·Nh,+∞(4Ns+Nd)·Nh)

(42e)

where∞NistheNthdimensionalvectorof∞.Thechoiceofinfinitelowerandupperboundsimpliesthatthevariablewandthevariablevassociatedtoxarezero.Thisallowstoformulate(35)asamixedLCproblem.

V.OPEN-LOOPPOWERCONVERTERSANALYSIS

Theapproachforthecomputationofthesteady-statesolu-tioncanbeusedtodeterminethecontrol-to-outputfrequencyresponseofpowerconverters,thatisusuallycalculatedbycon-sideringopen-loopmodels.Then,inthefollowing,werefertotheapplicationofthetechniquepresentedintheprevioussectionbyconsideringthematrices

A=Ao,B=BoBϕ,C=CϕCo,D=CϕDoBϕ+Dϕ

(43)

SESSAetal.:COMPLEMENTARITYMODELFORCLOSED-LOOPPOWERCONVERTERS6829

Fig.8.Z-sourcedc–dcpowerconverter.

see(17)and(19).Ausefulpropertyforsuchanalysisistheuniquenessofthesteady-statesolutionxwhichcorrespondstotheuniqueleast-normsolutionzoftheLCproblem(39)–(41),orifAisnotinvertibletothesolutionofthemixedLCproblem(1)with(42).Conditionsforsuchuniquenesscanbeobtainedbyusingthepassivityconcept.Inparticular,iftheopen-loopmodel(18)ofapowerconverterispassivewithrespecttotheinputϕandtheoutputλ(withϕandλbeingtheinput–outputofthedis-cussedidealizedelectronicdevices),thenthecomplementarityproblem(39)hasauniqueleast-normsolution.

Asanexample,letusconsidertheZ-sourcedc–dcconverterinFig.8,whereeoistheinputvoltage,xo1,xo3,xo5arethecurrentsoftheinductors,xo2,xo4,xo6arethevoltagesofthecapacitors,(ϕc,λc)isthecurrent–voltagepairoftheelectronicswitch,and(ϕd,λd)isthevoltage–currentpairofthediode.ByapplyingtheKirchhofflawstothecircuit,oneobtainsx˙(Ro1+R2)1R1=−

LxoL+2x1

1−x1o2L1o5+Lϕ11

d

+

R2Lϕ1

1c+Le(44a)1ox˙o2

=1Cx11(44b)

1o1−Cx1o5−Cϕ1

c

x˙(Ro1+R2)3=−

Lx1R21

o3−xo4+xo5+Lϕ1L1L11

d

+

R2Lϕ1c+Le(44c)11ox˙o4

=1Cx11o3−Cxo5−ϕ(44d)

11C1

c

x˙o5=

R21RLx2oL2o21R2

1+x2+Lx2o3+Lx2o4−2Lx2o5−

1Lx1ϕR21

(44e)2o6−L2d−2Lϕ2c−Le2ox˙o6

=1Cx1o5−Rx(44f)23C2o

6

λd=xo1+xo3−xo5−ϕc

(44g)

λc=−R2xo1−xo2−R2xo3−xo4+2R2xo5+ϕd

+2R2ϕc+eo

(44h)

whichisthemodelintheform(18)withxo=

col(xo1,xo2,xo3,xo4,xo5,xo6),ϕ=col(ϕd,ϕc),andλ=col(λd,λc).Itiseasytoshowthatthemodelispassivewithrespecttotheinputϕandtheoutputλ.

Moreover,bysubstitutingtheswitchandthediodecom-plementaritymodels(7)and(4),respectively,theresultingZ-sourceconvertermodelintheform(19)ispassivewithrespect

totheinputzoandtheoutputwo.

Atypicalapproachinordertoobtainthecontrol-to-outputfrequencyresponseofpowerconvertersconsistsofconsid-eringaPWMwithsinusoidalmodulationsignaleσ1=V0+V1sin(2πt/T)andaperiodiccarriersignal,sayeσ2.Withoutlossofgenerality,itisusefultoconsiderthemodulationsig-nalperiodTproportionaltothecarriersignalperiodTc,i.e.,T=NpTcwhereNpisapositiveinteger.Thesamplingperiodhcanbechosenash=T/Nh=TcNp/Nh.ByconsideringFig.7withoutthefeedback,i.e.,y=0,kp=1andki=0,onecanwrite

λσ=eσ1−eσ2

(45)

whichisintheform(24b).Themodelofthepowercon-vertercanbewrittenintheform(17)byusing(44)with(4)forthepair(ϕd,λd),(7)forthepair(ϕc,λc),(22)fortheswitchingsignalσ,(45)forthevariableλσandbydefiningx=xo,e=col(eo,eσ1,eσ2,1),z=col(zd,zc1,zc2,zσ1,zσ2),andw=col(wd,wc1,wc2,wσ1,wσ2).

TheproposedLCapproachforthecomputationofthesteady-statesolutioncanbeappliedtotheresultingmodel.

ConsiderthefollowingcircuitparametersL1=175μH,C1=220μF,R1=0.42Ω,R2=0.22Ω,R3=50Ω,L2=330μH,C2=470μF,andimax=5A,andeo=12V.Themodulationvoltageeσ1isasinusoidalsignalwithV0=0.3VandV1=0.03Vandthecarriersignaleσ2isasawtoothwithunitaryamplitudeandfrequency1/Tc=100kHz.Fig.9showstheamplitudeandthephaseofthefirstharmonicofthesteady-stateoutputvoltagexo6obtainedbyvaryingNp=2,3,...,10withNh=4000,comparedwiththeresultsobtainedbyus-ingtheaveragedmodelandtheanalyticalsolutionobtainedbyconsideringtheaprioriknowledgeofthecommutationtimeinstantsandthesequenceofthemodes.Thesedataarenotrequiredforthecomputationofthecomplementaritysolutionwhichprovidestheseinformationasresults.

Fig.9confirmsthegoodaccuracyofthecomplementarityapproachwhoseresultsareveryclosetotheanalyticalsolution.Instead,at50kHz,correspondingtoNp=2,wecannotetheaccuracyreductionoftheaveragedmodelwhenthefrequenciesofthemodulationandcarriersignalsbecomecloser.

TheLCmodelcapturestheswitchedbehavioroftheconverterindependentlyofthemodulationfrequency,providedthatasuf-ficientlysmallsamplingperiodhisselected.

Onceagain,itisimportanttohighlightthatthesameLCmodelisvalidforalloperatingconditionsanddoesnotrequiretheaprioriknowledgeofthemodessequence.

Inthefollowing,thecontrol-to-outputresponseisobtainedbyconsideringaconstantcontrolvoltage,asin[35],whereaZ-sourceconverterwiththeparasiticresistancesisconsidered.ThemodeloftheZ-sourceconverterisbuiltbyconsideringthecomplementaritymodelofthediodeandtheelectronicswitchwithparasitics,seeSectionII.Inordertoprovetheefficiencyoftheproposedapproach,wehavecomparedourresultswiththeexactsteady-statesolution,computedwithanalyticalmap,see

6830IEEETRANSACTIONSONPOWERELECTRONICS,VOL.29,NO.12,DECEMBER2014

Fig.9.FirstharmonicsoftheoutputvoltageoftheZ-sourceconverterobtainedwiththeLCsteady-stateapproach(*),theaveragedmodel(o),andtheanalyticalsolution(󰁣)fordifferentvaluesofNp.

Fig.10,andthetheoreticalandexperimentalresultspresentedin[35],seeTableI.

Inparticular,Fig.10showstheevolutionoftheerrordefinedasthedifferencebetweenthevalueoftheexactsteady-statesolutionattheswitchingtimeinstantandtheonecomputedbyusingthecomplementarityprocedure.Thecomplementaritysolutioniscomputedbyvaryingthenumberofthesamplesperperiod.Asonecouldexpect,theerrordecreaseswhenthenumberofsamplesincreases.Thenonregularityofthecomputederrorismainlyduetothechosendiscretizationtechnique.Thisphenomenonisgenerallyobservedwhenahigh-ordermethodisusedforasolutionwithlimitedsmoothness,see[12,Section9.1].

TableIsummarizesthevaluesofsomekeyparameterscorre-spondingtothesteady-stateanalysis.Thevalidityofthecom-plementarityapproachisconfirmed.

VI.SOMEEXAMPLESOFCLOSED-LOOPPOWER

CONVERTERSANALYSIS

TheproposedLCmodelingapproachcanbeusedforthecomputationofthetransientandsteady-statesolutionsof

Fig.10.ErrorinthetrajectorycomputationobtainedbyvaryingN[50,500]fortheopen-loopZ-sourcedc–dcconverter.Thehorizontalaxish∈is

log10(1/Nh)andtheverticalaxisisthelog10|xˆ∗oi−ˆxoi|,wherexˆ∗oiandˆx

aretheanalyticalandthecomplementaritysolutions,respectively,evaluatedoiattheswitchingtimeinstant,with(a)i=1(o)andi=5(*),and(b)i=2(o)andi=6(*).Thedashedlinesarethelinearleast-squaresinterpolations.

TABLE1

THEORETICAL,EXPERIMENTAL,ANDSIMULATIONRESULTS

SESSAetal.:COMPLEMENTARITYMODELFORCLOSED-LOOPPOWERCONVERTERS6831

closed-looppowerconverters.Beforeanalyzingsomeexam-ples,itisinterestingtomakeapreliminaryconsiderationonthepassivityofclosed-looppowerconverters.Wewerenotabletofindapowerconverterwithaclosed-loopmodelbeingpassivewithrespecttothecomplementarityvariables:forallcasesanalyzed,thepassivitypropertyislostwhengoingfromopen-looptoclosed-loopmodels.Iftheclosed-loopcomple-mentaritymodelisnotpassive,itisnotpossibletoconcludetheuniquenessofthesolutionof(39).However,theinterest-ingthingisthatclosed-loopconverterscanbestillanalyzedbyusingthecomplementaritymodelingapproachandthesteady-statecomputationprocedurepresentedpreviously.Indeed,fordeterminingthemultiplesolutions,onecanenlargethecomple-mentarityproblemwithopportuneconstraints.Forinstance,byfollowingtheprocedureshownin[36],itispossibletofindalldifferentsolutionsofaLCproblem.Briefly,sayzˆasolutionoftheLCproblem(39):onecanconstructanewLCproblemthathasallsolutionsof(39)exceptforzˆ,i.e.,toexcludezˆfromthesolutionsof(39).Thiscanberepeatedforallsolutions.See[36]andthereferencesthereinformoredetails.AnotherapproachcouldbetoinitializetheLCproblemsolverwithdifferentini-tialguesses.AllLCproblemsaresolvedbyusingthePATHtool[32].

AsinSectionV,thenumericalresultspresentedinthissectionareverifiedbyconsideringtheexactsteady-statesolutioncom-putedbyconstructingthenonlinearclosed-loopmapswhichcanbeanalyticallyobtainedoncethesequenceofmodes(butnottheswitchingtimeinstant)isfixed.

A.BoostConverterinDiscontinuousConductionModeConsidertheboostDC–DCconvertershowninFig.6withavoltage-modePWMandaproportional–integralcon-troller.Thecircuitandcontrollerparametersare:R1=0.1Ω,L1=100μH,C1=200μF,R2=20Ω,kp=0.1,ki=400,imax=5A,eo=10V,eσ1=15V,andeσ2isasawtoothwithunitaryamplitudeandfrequency1/Tc=5kHz.Theclosed-looppowerconvertercanbemodeledintheform(17)withthematricesgivenby(28).Suchmodelisvalidforalloperatingcon-ditions(continuousanddiscontinuousconductionmodes)ofthepowerconverter.AperiodicsolutionofperiodTcisexpected.BydiscretizingthesystemwithNh=130samplesperpe-riod,whichcorrespondstoh=1.53μs,wecanconstructthemixedLCproblem(1)withthematricesdefinedin(30),(36),and(42).Fig.11showsthesolutionsobtainedthroughtheLCprocedure.ThecomputationtimerequiredtosolvethemixedLCproblemis1.06s.Theresultsshowthattheproposedalgorithmisabletodetectthesteady-statebehavioroftheclosed-loopconverteroperatingindiscontinuousconductionmode,whichisasituationdifficulttobeaprioripredictedwithoutintroduc-ingsomesimplifyingassumptions.Letusdefinetheerrorasthedifferencebetweenthevalueoftheswitchingtimeinstantinwhichthecurrentbecomeszerocomputedbyconsideringtheexactsteady-statesolutionandtheoneobtainedbyusingthecomplementarityapproachwithdifferentvaluesofNh.Fig.12showsthatsucherrordecreaseswhenNhincreases,thuscon-firmingtheeffectivenessoftheproposedapproach.

Fig.11.Steady-stateinductorcurrentandoutputvoltagecomputedbytheLCprocedurefortheclosed-loopboostdc–dcconverterindiscontinuousconductionmode.

Fig.12.ErrorintheswitchingtimeinstantinwhichthecurrentbecomeszeroobtainedbyvaryingNindiscontinuousconductionh∈[42,mode.2]Theforthehorizontalclosed-loopaxisboostislogdc–dcconverter10(1/Nh)and

theverticalaxisisthelog10|t∗sw−tsw|wheret∗swandtswaretheswitching

timeinstantsinwhichthecurrentbecomeszeroobtainedfortheanalyticalandthecomplementaritysolutions,respectively.Thedashedlinesarethelinearleast-squaresinterpolations.

B.BoostConverterWithanUnstableSolution

Considertheboostdc–dcconvertershowninFig.6withaPWMandaproportionalcontrollerwhoseschemeisinFig.7,wherey=k1xo1+k2xo2,eσ1isthereferenceoutputvoltageandtheintegraltermiszero,i.e.,ki=0.ThisexampleshowsthattheLCapproachisabletocomputealsounstablesteady-statesolutions.Thematricesoftheform(17)canbesimplyobtainedbytheprocedureinSectionIII.Thefollowingparam-etersareused:R1=0Ω,L1=5.24μH,C1=0.2μF,R2=16Ω,k1=−0.1,k2=0.01,kp=1,imax=5A,eo=4V,eσ1=0.48V,andeσ2isasawtoothwithunitaryamplitudeandfrequency1/Tc=500kHz.

InFig.13,itisdepictedthetimeevolutionofthestablesolutionobtainedwiththetime-steppingnumericalintegrationoftheLCmodelandbychoosingasinitialconditionsxo1=0.1Aandxo2=8V.Theresultsconfirmthosereportedin[30].

6832IEEETRANSACTIONSONPOWERELECTRONICS,VOL.29,NO.12,DECEMBER2014

Fig.13.Timeevolutionobtainedwiththetime-steppingLCprocedurefortheclosed-loopboostdc–dcconverter.

Fig.14.Phaseplanecorrespondingtothestable(continuousline)andtheunstable(dashedline)solutionsobtainedwiththeLCtechniqueappliedtotheclosed-loopboostdc–dcconverter.

In[30],itisshownthattheclosed-looppowerconverterex-hibitstwoperiodicsolutionswithperiodTc:onestableandanotherunstable.Fig.14showsthestable(continuousline)andtheunstable(dashedline)solutionsinthecurrent–voltageplaneobtainedwiththeLCapproachandNh=400,whichcorre-spondsto0.005μs.Theresultsarealsocoherentwiththosepresentedin[37].ThetwosolutionsareobtainedbyusingthePATHalgorithm[32]withzero(nonzero)initialguessforthestable(unstable)solution.Thecomputationtimeis2.04sforthestablesolutionand2.06sfortheunstableone.InFig.15,itisshownthecomparisonofthecomplementaritysolutionwiththeexactsteady-statesolutionofthenonlinearclosed-loopmapobtainedbyassumingthesequenceofmodes.ThequalitativecommentsontheerroraccuracyreportedforFig.10arecon-firmedfortheboostconverter.

C.BoostConverterWithCurrent-ModeControl

Considertheboostdc–dcconverterinFig.6undercurrent-modecontrolwithslopecompensation.Becauseofthepresence

Fig.15.ErrorinthetrajectorycomputationobtainedbyvaryingN[200,600]fortheclosed-loopboostdc–dcconverter.Thehorizontalaxish∈islog10(1/Nh)andtheverticalaxisisthelogtheanalyticalandthecomplementaritysolutions,10|xˆ∗oi−respectively,ˆxoi|,wherexˆ∗evaluatedoiandˆxareoiattheswitchingtimeinstant,with(a)i=1(o)andi=2(*)forthestablesolutionand(b)i=1(o)andi=2(*)fortheunstablesolution.Thedashedlinesarethelinearleast-squaresinterpolations.

Fig.16.Blockdiagramofthecurrent-modecontroller:eσthecompensation1isthereferencesignalfortheoutputvariabley,thatisacurrent;eσ2isramp;eσ3istheclock;σistheswitchingsignal.

ofamemoryelement,thatistheflip-flop,itisopportunetoformalizethecurrent-modecontrolprobleminthediscrete-timedomain.AcorrespondingblockdiagramisshowninFig.16,whereeσ1isthereferencecurrentvalue,eσ2istheexternalcompensationramp,eσ3istheclocksignalthatis1foronesampleatthebeginningoftheperiod,i.e.,k=1anditiszerofork=2,3,...,Nh.

ThebehavioroftheS-Rflip–flopcanbemodeledbyusingthefollowingfunction:

σk=1−(1−eσ3,k)sat[0,1](1−σk−1+step(λσ,k))(46a)

SESSAetal.:COMPLEMENTARITYMODELFORCLOSED-LOOPPOWERCONVERTERS6833

Fig.17.ErrorinthetrajectorycomputationobtainedbyvaryingN[50,900]fortheclosed-loopboostdc–dcconverterwithcurrent-modecontrol.

h∈

Thehorizontalaxisislogwherexˆ∗andxˆarethe10(1/Nanalyticalh)andtheverticalaxisisthelogandthecomplementaritysolutions,10|ˆx∗oirespec-−ˆx

oi|tively,evaluatedoi

oiattheswitchingtimeinstant,withi=1(o)andi=2(*).Thedashedlinesarethelinearleast-squaresinterpolations.

λσ,k=xo1,k−eσ1,k+eσ2,k

(46b)

wheresat[0,1]isthesaturationfunctionbetweenzeroandoneandstepistheunitaryamplitudestepfunction,thatwehavealreadyanalyzedinSectionIII-B,seeFig.3(b).Fork=1,theclockvalueeσ3,k,issetto1and,from(46a),thecontrolsignalσissetto1independentlyonthevalueofthecurrent(switchON).TheswitchstaysON,i.e.,σkeepsthevalue1,untilthecurrentxo1,k=ykreachesthecompensationrampeσ1,k−eσ2,kforsomevalueofk.Atthisvalueofk,λσ,kbecomesnonnegativeandtheoutputofthestepfunctionchangesfrom0to1.Afterthat,thecontrolvariableσiszero,i.e.,theswitchisOFFuntilthebeginningofthenextperiodalsointhecaseλσ,kchangesagainitssign.Byreplacingin(46a)thesaturationandthestepfunctionwiththecorrespondingcomplementarityrepresentation,weobtainthecomplementaritymodelofthecontrolsignalinthecaseofcurrent-modecontrol

σk=1−(1−eσ3,k)zσ2,k(47a)wσ1,k=−zσ2,k+1

(47b)wσ2,k=zσ1,k+zσ2,k−zσ3,k−1+σk−1(47c)wσ3,k=zσ4,k−λσ,k(47d)wσ4,k=−zσ3,k+1

(47e)0≤wσ,k⊥zσ,k≥0

(47f)

whereλσ,kisgivenin(46b).Bydiscretizingtheopen-loopLCmodelofthepowerconverterin(19)andbyconsidering(47)andtheperiodicitycondition,theclosed-loopdiscretizedLCmodelintheformof(35)beeasilyobtained.In[4],thefollowingparametersarechosenR1=0Ω,R2=22Ω,C1=880μF,L1=69μH,imax=10A,eo=16.9V,eσ1=12.58A,andeσ2isthecompensationrampwithslopeequalto8.4As−1andfrequency1/Tc=100kHz.

InFig.17,itisshownthattheerrorbetweentheexactsteady-statesolutionandtheonecomputedbyusingtheapproach

Fig.18.Buckdc–dcconverter.

proposedinthispaperdecreaseswhenthetimestepchosenforthediscretizationdecreases.

D.BuckConverterWithPeriodDoublingBifurcationConsiderthebuckdc–dcconvertershowninFig.18withavoltage-modePWMandaproportionalcontrollerwhoseschemeisinFig.7,wherey=k1xo1+k2xo2,eσ1isthereferenceoutputvoltageandtheintegraltermiszero,i.e.,ki=0.In[31],itisshownthatthispowerconverterpresentsdifferentkindsofbifurcations,underthevariationoftheinputvoltageeoandthefrequencyofthecarriersignaleσ2.WhenthefollowingparametersarechosenR1=22Ω,C1=47μF,L1=5.8mH,kp=1,k1=0.02,k2=2,imax=5A,eo=20V,eσ1=8V,andeσ2isasawtoothsignalwithafrequency1/Tc=1.013kHzandwhoseamplitudevariesbetween2.4and8V,thebuckdc–dcconverterhasaperioddoublingbifurcation.ByusingtheproceduredescribedinSectionIII,i.e.,byapplyingtheKirchhofflawstothecircuitinFig.18andbyusing(4),(7),and(22),theLCmodeloftheclosed-looppowerconvertercanbewrittenasfollows:

x˙11o1=−Lx1o1−

Lx1

1o2+Lzd(48a)x˙11o2=Cx1o1−Cx1R1o2

(48b)wd=xo1−zc1(48c)wc1=zd+zc2−eo(48d)wc2=−zc1+imaxzσ1

(48e)wσ1=kpxo2+zσ2−kpeσ1+eσ2(48f)wσ2=−zσ1+1(48g)0≤w⊥z≥0

(48h)

whichisintheform(17)withx=col(xo1,xo2),z=

col(zd,zc1,zc2,zσ1,zσ2),

w=col(wd,wc1,wc2,wσ1,wσ2)ande=col(eo,eσ1,eσ2,1).Theconverterhasanunstablesolutionwiththesameperiodofthesawtoothandastablesteady-statesolutionwithdouble-period,thatisT=2Tc.

BydiscretizingthesystemwithNh=350samples(h=5.6μsfortheunstablesolutionandh=2.8μsforthestableone)andbyconstructingthemixedLCproblem(1)withthema-tricesdefinedin(30),(36),and(42),thePATHalgorithmwithzeroandnonzeroinitialguessesprovidesthesolutionsshowninFigs.19and20.

Fig.21showsthecomparisonbetweenthenumericalre-sultsobtainedwiththecomplementarityprocedureandtheexperimentalresultspresentedin[31]andconfirmsthe

6834IEEETRANSACTIONSONPOWERELECTRONICS,VOL.29,NO.12,DECEMBER2014

Fig.19.Carriersignaleσ2(dottedline)andmodulationsignalsforstabledouble-periodsolution(continuousline)andsingle-periodicunstablesolution(dashedline)fortheclosed-loopbuckdc–dcconverter.

Fig.20.Phaseplanecorrespondingtothedifferentsolutionsoftheclosed-loopbuckdc–dcconverterinFig.19:unstable(dashedline)andstable(continuousline).Notethepresenceofmultiplediscontinuousconductionmodephasesduringthedouble-periodicsolution.

Fig.21.Comparisonwithexperimentalresults(dots)fortheclosed-loopbuckdc–dcconverter:carriersignaleσtinuousline).Numericalsolutionwith2(dottedline)andsimulationsolution(con-slidingmotion(dot-dashedline)obtainedwithdifferentcontrolparameters.

effectivenessoftheproposedapproach.Inthesamefigure,withthedash–dottedline,wepresentalsothesolutioncharacterizedbyslidingmotionobtainedbysettingthevalueofthereferencevoltageeσ1=12Vandthecontrolparametersk1=0.01andk2=1.05.

VII.CONCLUSIONS

LCsystemsareaninterestingclassofmodelssuitablefortherepresentationofthedynamicevolutionsofpowerconverters.OneofthemajordrawbacksoftheLCmodelspreviouslypro-posedintheliteratureforpowerconverterswasthepresenceofswitchedsets(cones)requiredfortherepresentationoftheswitchescharacteristics.Thatmadeverycomplicatedthecon-structionofcomplementaritymodelsforcontrolledconverters.TheLCmodelproposedinthispaperforthevoltage–currentpiecewiselinearcharacteristicsofswitchesallowstorepresentpowerconvertersasnonswitchedLCsystems.TheLCmodelstructurewithconstantmatricesisvalidforbothopen-loopandclosed-looppowerconverters,anditcanbeobtainedbycom-biningthedynamicalequationsdescribingthecircuitwiththeLCrepresentationsoftheelectronicdevices,modulator,andcontroller.TheLCmodelcanbeusedforthetime-steppingevolutionandforthesteady-stateoscillationcomputations.Inparticular,thesteady-statesolutionofthediscretizedclosed-loopsystemhasbeenformulatedasthesolutionofastaticmixedLCproblem.Theproposedtechniquehasbeenshowntobeeffectiveforthecomputationofthetransientandthesteady-stateperiodicoscillationsexhibitedbyclosed-loopZ-source,boost,andbuckdc–dcpowerconvertersoperatingincontinu-ousordiscontinuousconductionmodeandalsointhepresenceofmultiplesolutions.

Futureworkwillbededicatedtoimprovethenumericalef-fectivenessoftheLCmodelintegration,totheordermodelreductionissuewithinthecomplementarityframeworkandtotheapplicationoftheproposedtechniquefortheanalysisofmorecomplexnonlinearphenomenainswitchedcircuits.

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ValentinaSessa(S’10–M’13)wasborninAvellino,Italy,in1983.Shereceivedthemaster’sdegreeinau-tomaticcontrolengineeringandthePh.D.degreeininformationengineeringfromUniversityofSannio,Benevento,Italy,in2010and2013,respectively.In2012,shewasaVisitingStudentatINRIARhˆone-Alpes,France.SheiscurrentlyaPostdoctoralFellowattheDepartmentofEngineering,UniversityofSannio.Hercurrentresearchinterestsincludetheanalysisofnonsmoothdynamicalsystems,inpartic-ular,piecewiselinearandcomplementaritysystems,

andthemodelingandcontrolofpowerelectronicconverters.

LuigiIannelli(S’00–M’02–SM’12)wasborninBenevento,Italy,in1975.Hereceivedthemaster’sdegree(Laurea)incomputerengineeringfromtheUniversityofSannio,Benevento,Italy,in1999,andthePh.D.degreeininformationengineeringfromtheUniversityofNapoliFedericoII,Naples,Italy,in2003.

HewasaGuestResearcherattheDepartmentofSignals,Sensors,andSystems,RoyalInstituteofTechnology,Stockholm,Sweden,andthenasaRe-searchAssistantattheDepartmentofComputerand

SystemsEngineering,UniversityofNapoliFedericoII.Currently,heisanAssistantProfessorofautomaticcontrolwiththeDepartmentofEngineering,UniversityofSannio.Hiscurrentresearchinterestsincludeanalysisandcontrolofswitchedandnonsmoothsystems,stabilityanalysisofpiecewise-linearsys-tems,smartgridcontrolandapplicationsofcontroltheorytopowerelectronics.HeisaMemberofIEEEControlSystemsSociety,theIEEECircuitsandSys-temsSociety,andtheSocietyforIndustrialandAppliedMathematics.

FrancescoVasca(M’94–SM’12)wasbornin1967inGiugliano,Napoli,Italy.In1995,hereceivedthePh.D.degreeinautomaticcontrolfromtheUniver-sityofNapoliFedericoII,Naples,Italy.

Since2000,hehasbeenanAssociateProfes-sorofautomaticcontrolattheUniversityofSannio,Benevento,Italy.Hisresearchinterestsincludetheanalysisandcontrolofswitchedsystems(averaging,complementarity,dithering,real-timehardwareintheloop)withapplicationstopowerelectronics,railwaycontrol,andautomotivecontrol.

Since2008,hehasbeenanAssociateEditorfortheIEEETRANSACTIONSONCONTROLSYSTEMSTECHNOLOGY.