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BuildingServ.Eng.Res.Technol.31,1(2023)pp.39–55

RobustMPCfortemperaturecontrolofair-conditioningsystemsconcerningonconstraintsandmultitypeuncertainties

XinhuaXua,PhDShengweiWangaPhDMASHRAECEngandGongshengHuangbPhDa

DepartmentofBuildingServicesEngineering,theHongKongPolytechnicUniversity,Kowloon,HongKongb

DivisionofBuildingScienceTechnology,CityUniversityofHongKong,Kowloon,HongKong

Thispaperpresentsarobustmodel-basedpredictivecontrol(MPC)strategyfortemperaturecontrolofanair-conditioningsystem,whichconsistsofmultiplelocal-loopprocessesandeachprocesssuffersfromdifferentdynamicsuncertaintiesorvariations.Whenanappropriatesamplingperiodischosentodiscretisethesystemforcomputercontrol,astate-spacediscretemodelwithanuncertaintypolytopeisdevelopedtodescribethemixinguncertaintiesofthesetypeofsystems.Themainbenefitoftheproposeddescriptionisthatrobustmodelpredictivecontrolcanbeeasilyusedtodesignarobustcontrollerforsuchasystemwhiletakingaccountofconstraintsassociatedwiththissystem.Alinearmatrixinequality-basedMPCalgorithmisemployedforcontroldesign.Casestudywasconductedonadynamicsimulationplatformofanair-conditioningsystem,whichevaluatedthedevelopedstrategyinvarioussimulationtestsbycomparingwiththeconventionalPIDcontrol.Resultsdemonstratedthatthedevelopedstrategyisabletodealwithconstraintsandallowsstableandrobustcontrolwhilemaintainingacceptablethermalcomfort.Althoughthestrategyisillustratedandvalidatedusingaconstantairvolumeair-conditioningsystem,itcanbeappliedtootherconstraintHVACprocessessufferingfromsimilaruncertainties.

Practicalapplications:Themainbenefitofthedevelopedstrategy(includingtheproposeddescriptionandtheadoptedrobustcontrolalgorithm)forpracticalapplicationisthatuncertaintiesandconstraintscanbedealtwithsimultaneouslyinoneframework.ConstraintsinHVACsystemsexistduetotheapplicationofactuators,forexample,theratelimitconsideredinthepaper.Takingaccountofconstraintsisusefultopreventunnecessarydamagestoequipmentsandmaintainthesystemoperatinginasafemode.Mostimportantly,thecontrolobjectivescanbeachievedinaconstraintmanner.Theconsiderationofuncertainties,probablyduetothechangesofoperatingenvironment,isusefultoreleasetheworkofaccuratemodellingofHVACprocessesandonlinetuningofcontrollers.

1Introduction

Currentaddress:DepartmentofBuildingEnvironmentServicesEngineering,SchoolofEnvironmentalScienceEngineering,HuazhongUniversityofScienceTechnology,Wuhan,China.

Addressforcorrespondence:GongshengHuang,DivisionofBuildingScienceTechnology,CityUniversityofHongKong,Kowloon,HongKong.

E-mail:gongsheng.huang@.hk

Moderncommercialandofficebuildingsarealwaysequippedwithair-conditioningsys-temsprovidingcomfortableenvironmentforoccupants.Variable-air-volume(VAV)andconstant-air-volume(CAV)air-conditioningsystemarecurrentlythetwotypicalprocesseswidelyequippedincommercialandofficebuildings.WhetherinVAVorCAVsystems,

10.1177/0143624409352420

TheCharteredInstitutionofBuildingServicesEngineers2023

XXuetal.

(LMI)-basedMPC16wasemployedtodesignarobustcontroller.

Inthispaper,atypicalbutmoresophisti-catedconstraintsystemisconsidered,whichconsistsofmorethanoneconstraintprocess.Eachprocesscanbedescribedusingafirst-orderplustimedelaymodel,butsuffersfromdifferentdynamicsuncertaintiesorvariations.Whenanappropriatesamplingperiodischosentodiscretisethesystemforcomputercontrol,anuncertaintypolytopeisdevelopedtodescribethemixinguncertaintiesofthistypeofsystems.ThemainbenefitoftheproposeddescriptionisthattherobustcontrolschemedevelopedinHuangetal.15canbeeasilyusedtodesignarobustcontrollerforsuchasystem.Asanexample,aCAVair-conditioningsystemisusedtoillustratetheprocedureofusingoneuncertaintypoly-topetodescribedifferentuncertaintiesasso-ciatedwithdifferentprocesses.ThisisbecauseatypicalCAVair-conditioningsystemconsistsofnotonlytheAHUprocessbutalsotheconditionedroomtemperatureprocess,asshowninFigure1.TheAHUprocessmainlysuffersfromprocess-gainandtime-delayvari-ation(typeIuncertainties),whiletheroomtemperatureprocessmainlysuffersfrom

41

process-gainandtime-constantvariations(typeIIuncertainties).Casestudiesshowthattheproposedstrategy,includingtheuncertaintydescriptionandthecontrollerdesign,canachieveagoodrobustnesscom-paredwithaconventionalPIcontrolmethod.Thisismainlybecauseconstraintsanddiffer-enttypesofuncertaintiesaredealtwithinastraightforwardmanner.2

DynamicmodellingoftheCAVair-conditioningsystem

ThemaincomponentsofaCAVair-conditioningsystemarethecoolingcoilandtheconditionedroomincludingtheairduct.TheglobaldynamicsofthesystemcanbedescribedasshowninFigure2andbeformulatedby:

y2s

G1sG2sus

1

wherey2isthereturnairtemperature(usuallyrepresentingtheroomairtemperature),uisthecontrolsignaldenotingtheopennessofthewatervalve,Gistransferfunctionandthesubscript1and2denotesthetransfer

Figure1CAVsystemanditscontrolschematicdiagram

42RobustMPCfortemperaturecontrol

Figure2SchematicofthecontrolprocessofaCAVair-conditioningsystem

functiondescribingthedynamicsofthecool-ingcoilsubsystemandtheconditioningroomsubsystem,respectively.2.1

DynamicmodellingoftheCAVcoolingcoilprocess

Thecoolingcoildynamicscanbeapproxi-matedbyafirst-orderplustimedelaymodel11

y1sK1

G1se1sus1T1s

2

DynamicmodellingoftheCAV-conditionedroomtemperatureprocess

TheconditionedroomisoftenmodelledasEquation(4)basedonasimpleenergybalancefortheroomair.ItcanberewrittenasEquation(5)forthecontrolpurpose.ByperformingLaplacetransformonEquation(5)withtheroomairtemperatureandsupplyairtemperatureasvariables,theroomtrans-ferfunctionGroomcanbeachievedas(Equations(6)and(7)):Vracpa

dy2

cpamsy1y2dtX

fqgainAUiy2

2.2

wherey1isthesupplyairtemperature,K1,T1

and1aretheprocessgain,timeconstantandtimedelay,respectively.Steptestsareusedtocalculatetheprocessparameters.11Itisfoundthattheprocessdynamicsvarieswithrespecttotheoperatingenvironment,17suchastheinletairtemperature,thechilledwaterflowrateinthecoil,etc.Inthisstudy,thedynamicsuncertaintiesofthecoolingprocessaredescribedbyconsideringtheprocessgainandthetimedelayinacertainrange,whilethetimeconstantkeepsconstant,asinUnderwood.11Thedynamicsvariationsarerepresentedas:

LUU

and3K12KL,K21,1111wherethesuperscriptsLandUindicatethe

upperlimitandthelowerlimit,respectively.TheuncertaintiesdescribedbyEquation(3)aretitledastypeIuncertaintiesinthispaperinordertodistinguishwithtypeIIuncertainties.

4

Xdy2

VracpaAUiy2cpams

dtX

fqgaincpamsy1AUi

y2sKr

y1s1Trs

5

Grooms

6

Kr

cpams

,

cpamsAUi

7

Vracpa

Tr

cpamsAUi

44RobustMPCfortemperaturecontrol

1d1h

~b1,0~1h.Theparametersa1,b1,d1and1,d11arecomputedby:

a1eh=T1,b1,deh

~1=T1

1K11andb1,dK1eh~1=T

eh=T11Notethatthesumofb1,d1,b1,d11satisfies15

b1,d1b1,d11K11a1

14

Whenthetimedelay1variesintherange

d12L1,U

1,d1hasthevalueintherange

12dL1,dU1.dL1anddU

1aredefinedas:

dL1flL1=handdU

1clU1=h,wherefl(x)isafunctiontoroundxtothenearestintegertowardnegative,andcl(x)isafunctiontoroundxtobenearestintegerpositive.Therefore,whentowardLUEquation(13)canbewrittenin12ageneral1,1,formasfollows:

y1,k1a1y1,kb1,dL1ukdL

1

b1,dUukdU11

15

Defineastatevectorx0...,u1,ky1,k,ukdU1

,k1,thenEquation(15)canberefor-mulatedas:

x1,k1A1x2,kB1u1,k

yx16

1,kC12,kwherethecoefficientmatricesA1,Bappropriateformcorresponding1,Cto1areinanthedefinitionofx(14),itcanbeshown1,k.BasedontherelationshipthatwhenLUand12KL,12K1,K11U1

,A1,B1lieinanuncertaintypolytopeX(

1definedby:

1:A1,B1

XL1l1,iA1,i,B1,i,i1

0l1,XL1)

i1,

l1,i1

17

i1

whereL12dU1dL

11.ThedefinitionofA1,i,B1,iisgiveninAppendixA.

Second,usingthesamplingintervalh,themodel(9)issampledinto:e2,k11a2e2,ka2e2,k1

b2,d2y1,kd2b2,d21y1,kd21

18

wheree2,ky2,ky(yfor2,risthetrackingerror2,risthesetpointysatisfying2),d2isthetimedelay

2d2h~2,0~2hand

a2,b2,d2,b2,d21are:

a2eh=T2,b2,d

~2=T2

2K21andb2,d1K2eheh~2=T2

eh=T22Definex0y2,ke2,k,e2,k1,y1,kd21,...,

1,k1,then(18)becomes:

x2,k1A2x2,kB2y1,ke19

2,kC2x2,k

wherethecoefficientmatricesA2,B2,C2are

inanappropriateformcorrespondingdefinitionx2,k.WhenK22KLUtotheTLof2,K2and22T2,TU

coefficient2,theuncertaintiesassociatedwiththematrices(A2,B2)canalsobedescribedusinganuncertaintypoly-topeanddefinedas:

(

2:A2,BXL22

l2,jA2,j,B2,j,

j1

XL2)

0l2,j1,

l2,j1

20

j1

ThedefinitionofA2,j,B2,jisgivenin

AppendixB.

x0

Finally,defineaglobalstatevectoraskx01,k,x02,k,then(16)and(19)arecom-binedtogetherandreformulatedintotheformof(11),wherethecoefficientmatricesA,Bare:

A

A10

!

B1!

B2C1

A2

,B

021

46RobustMPCfortemperaturecontrol

Inthisstudy,an16offlinemodelpredictivecontrolalgorithmusingLMItechniqueisemployedtodesignthecontrollerasinHuangetal.15Thecontrollerisintheformofastatefeedbackcontrolanddescribedby:

ukFxkxk

28

whereF(xisstatek)indicatesthegainofthecontrollawdependentandF2{Fj,j1,...,Np}.Npisauser-definedparameter,seeAppendixC.Thedesignofthefeedbacklawsandthesearchofanappropriatefeed-backgainforagivenxCforthekarebrieflyintroducedinAppendixsakeofcompleteness.Itshouldbenotedthatthesamplingratiohhasasignificanteffectonthecontrolperfor-mance,andgenerally,thesampling18ratiohshouldsatisfyhmin(T1,T2)/10.Althoughthecontrolalgorithmdevelopmentseemsquitecomplex,itsonlinecomputationisactuallyassimpleassearchingalookuptableandallthecomplexcomputationisdoneoffline.Sincey1(thesupplyairtemper-ature)andy2(thereturnairtemperature)couldbemeasured,nostateobserverisneededforthefeedbackcontrollerintheformofEquation(28).5

Controlperformancetests

5.1Testfacility

AplatformfordynamicallyemulatingtheoperationofaCAVair-conditioningsystemwasdevelopedtoevaluatethecontrolperfor-manceoftheproposedrobustcontrolstrat-egy.TheplatformwasconstructedusingaTransientSimulationProgramTRNSYS.19Thebuildingsystemisanopenplanofficeabout1200m2,whichisonehalfofafloorofahigh-risecommercialbuilding.ACAVair-conditioningsystemservesthisbuildingsystembymaintainingtheroomtemperaturetoauser-predefinedsetpoint.Theindoorthermalcomfortisrealisedbydeliveringthesupplyair,whichiscooleddownafterthe

coolingcoil,toindoorspacethroughdiffu-sers.Returnairisdrawnbackthroughtheceilingplenum.Thedesignairflowrateis7.2kg/s.Thedesignsupplychilledwaterflowrateandtemperatureare7.0kg/sand78C,respectively.ThissystemasshowninFigure1includesasupplyairfanandareturnairfanwithconstantspeed,acoolingcoilandinterlockeddampersforintroducingconstantfreshairflowrateforensuringindoorairquality,etc.

Inthissimulation,thecomponentsofconcernarethebuildingmodel,thecoolingcoilmodeloftheAHU,andtheairductmodelandtheauxiliarymodelssuchassensorandactuatormodels,etc.Thebuildingmodelisasimplifiedmodelsimulatingthedynamicbalanceofenergyandmoistureofthebuild-ingsystem,20developedbasedonIEAAnnex17.21Themodelrepresentstheopenofficespaceusingathermalnetworkofthermalresistance,thermalcapacitanceandairvolume.Thisofficeisdividedintodifferentsubspacesfordiffuserconfiguration,whileeachsubspaceismodelledasanodeofwell-mixedairvolume.Theexternalwallofeachsubspaceisrepresentedbyanodeofthermalcapacitanceandresistancelinkingthespacewithoutside.Theinternalstructureandfurnitureineachsubspacearerepresentedbyanodeofthermalcapacitanceconnectedtothespacethroughathermalresistance,respectively.

Thecoolingcoilmodelwasdevelopedonthebasisofthemathematical22modelproposedbyLebrunetal.inIEAAnnex17.Thismodelwasusedindevelopingthesimulationplatform.20Afirst-orderdifferentialequationisusedtorepresentthedynamicsofacoilwithlumpedthermalmassbasedonenergybalance.Theoutletairandwatertempera-turesarecomputedusingthesteady-stateapproachbasedontheheatbalancesattheairsideandwaterside,respectively,separatedbythecoil.Classicalnumberoftransferunitmethodandheattransfereffectiveness

50RobustMPCfortemperaturecontrol

wereillustratedinFigure5,whichresultedintheoscillationsinthewaterflowrateaswellasinthesupplyairtemperatureasshowninFigures6and7,respectively.Althoughthe

parameterstunedinthesummerdaywerenotsuitableanymoreforthePIcontrolinthewinterdayduetothechangedoutdoorconditions.Theoscillationsincontrolinputs

Waterflowrate(kg/s)

Time(h)

Figure6WaterflowrateusingPIDcontrolandRMPCcontrolinthesunnysummertestdayandthesunnywintertestday

Outletairtemperature(C)

891011121314Time(h)

151617181920

Figure7OutletairtemperatureofthecoolingcoilusingPIDcontrolandRMPCcontrolinthesunnysummertestdayandthesunnywintertestday

XXuetal.

oscillationmagnitudeissmallandmightnotcausethermaluncomfortable,itcannotbeacceptablebecausethefrequentoscillationsareharmfulfortheequipments.4

Predictedpercentageofdissatisfied(PPD)isanindexthatestablishesaquantitativepredictionofthepercentageofthermallydissatisfiedpeopleforevaluatingthethermalcomfortofanoccupiedspace.Inthisstudy,PPDvaluesoftheair-conditioningspaceusingtraditionalPIDcontrolandtheadvancedRMPCcontrolwerealsoevaluatedtoassesstheimpactsofbothcontrolsonthethermalcomfort.Figure8presentsthePPDprofilesusingbothcontrolsinthesunnysummertestdayandthesunnywintertestday.ThePPDprofilesusingRMPCandPIcontrolarealmostthesame.ThemaximumPPDvalueofabout12.5occurredbetween8:00amand9:00amduetothesignificantincreaseofinternalheatloadresultinginhighindoorairtemperature.Inthesunnywintertestday,bothPPDprofilesalmostoverlap,andthePPDvalueisabout5.ThesePPDvaluesinthesummertestdayandthewintertestdayareacceptablesinceASHRAE

51

Standard55specifiesconditionsorcomfortzones,where80%ofsedentaryorslightlyactivepersonsfindtheenvironmentthermallyacceptable.24Therefore,bothcontrolmeth-odsofPIDcontrolandRMPCcontrolalmostdonotaffecttheindoorthermalcomfort,althoughtheymayresultindifferentindoortemperatureresponse.However,itisworthytopointoutthatinappropriatepara-metersofPIDcontrolmayresultindamagetovalvesand/oractuatorsduetounstablecontrol,althoughsuchcontrolcouldsatisfythethermalcomfortrequirementofoccu-pants.AsforRMPCcontrol,theuncertain-tiesoftheair-conditioningsystemandthebuildingsystemareconsideredseriouslywhendesigningthecontroller.Thiscontrolcanobtainnotonlyacceptablethermalcomfortbutalsostablecontrolwhichmayprolongthelifecycleofdevicesandequipmentsofair-conditioningsystems.6

Conclusions

Thisarticledevelopsarobustcontrolstrategyforimprovingthetemperaturecontrolofan

AveragePPD(–)

891011121314Time(h)

151617181920

Figure8AveragePPDusingPIDcontrolandRMPCcontrolinthesunnysummertestdayandthesunnywintertestday

52RobustMPCfortemperaturecontrol

air-conditioningsystemmainlyincludingthe

coolingprocesswhenairpassesthroughthecoolingcoilandtheroomcoolingprocesswhencoolingairisdeliveredintothecondi-tionedspace.Thecoolingcoilsubsystemandtheair-conditioningroomsubsystemaremod-elledasafirst-orderform,anddifferentuncertaintiesassociatedwiththeprocessparametersofbothmodelsareinvestigatedanddescribedusinganuncertaintypolytope.Ithasbeenshownthatbasedonthisdescrip-tion,RMPCcouldbeemployedtodesignarobustcontrollerforthissystemanddealwiththeuncertaintiesandconstraintsinastraight-forwardway.

CasestudyshowsthattheapplicationofRMPCintheCAVair-conditioningsystemcanachieveabettersetpointtrackingoftheroomtemperatureandabetterdisturbancerejectionofdisturbanceswhencomparedwithconventionallyusedPIDcontrol.RMPCmayachievestabletemperaturecontrolwhilemain-tainingacceptablethermalcomfort,althoughitmayresponseslightlyslowlythanthefinetunedPIDcontrol.InappropriateparametersofPIDcontrolmayresultinunstablecontrolcausingdamagetocontroldevices,whilethispointmaybeignoredsincesuchunstablecontroldoesnothaveobviouseffectonthermalcomfort.TheproposedmethodcanalsobeappliedtootherHVAClocalprocessesthatsufferheavilyfromdynamicsvariationsoruncertainties.Appendix

A:DefinitionLof(A1,i,B1,i)Whend10,

2

1a1a101,dU

3A61

1,166101,dU1

17407dU11,3Id11,dU1

175,

U01,dU1

2

A1,2A1,1

261a1a

01,dU1bML

131

A01,dU71,3

661611740dU

11,30dU711,dU1

175,2

01,dU1261a1a01,dUMU31

1b11,dU7A66101,4

611740dU

0dU711,311,dU1

17051,dU122

6

1a1abML1

01,dU13

1

A1,2dU6101,dU7117116640dU711,3IdU11,dU1

175,2

01,dU1

26

1a1abMU1

01,dU311A01,71,2dU116616dU11740dU11,3IdU11,dU1

1772051,dU1

2ML32MUB1,16b40dU,175,B1,26b340dU71

1

,15,

1

1B01,3BdU!

1,2dU11

1,11

1

WhendL140,

2

61a1a101,dU1

dL

1bML101,dL131A01,dU7

1,1661611740dU

11,3IdU1,dU7111705,1,dU2

121a1a101,dU61

dL

1bMU101,dL311A66101,dU71,261

1

740dU

11,3I7dU11,dU1

1705

1,dU1

2

2

61a1a1bML1

1,dU31

160U7

A1,d1

171,2dU71dL6

11

166640dU11,3I7dU711,dU1

17,051,dU22

161a1a1bMU101,dU3116071,dU1

1

7A1,2dU6

171dL1

1666I740dUdU711,311,dU11705

1,dU12

B0dU1,1B1

1,1#

1,2dU1

dL1

1

1

Inthesedefinitions,0d,disaddmatrixwithallitemsbeingzero;Imatrix;andbMLd,disaddidentity

1,bMU

1arebML1KL11abMUK1,1U

11a1.

B:Definitionof(A2,j,B2,j)

WhenhTlinearisedwith2)/10,brespect2,tod2andba2,d21canbe2

b2,dK21a2e

~2=T22%K211a212b2,d21K21a2b2,d2%K221a222,wherenWhenK211LnU11andn1nandT2212values22Kof2,Kthea2

22TL2,TU2.,2,b2,d2,b2,d21liesinanuncertaintypolytopewiththecornersdefinedby:C12:aL2,KL211aL212,KL221aL

222C22

:aL2,KU211aL212,KU221aL

222

C32:aU2,KL211aU212,KL221aU

222C42:aU2,KU211aU212,KU2

21aU

222,whereaLh=TL

U

2e2andaUh=T2affineto2e

.Since(A2,B2)area2,b2,d2,b2,d21

XXuetal.53

accordingtodefinitionofxas:

2,kinEquation

(20),(A2,j,B2,j)aredefined21ajj2aj2bj2,d21b2,d2

01,d21

3

6

A2,j

666101,d272767

40d2,3d7,I2,d275

1,d23B

0d22,1!1

,

whereajj2,bi2,d2,b2,d21obtainitsvaluefromthecornerCj2

,j1,...,4.

C:OfflineoptimisationofthefeedbacklawInordertooptimisethegainofthefeedbackcontrollawFxispartitionedintoj,thespaceofthestate0asequencesubspaces,denotedasxS1

ofnested

Njx1(j1,...,p),whereNEachpisthenumberofthenestedsubspaces.subspaceisassignedtoastatefeedbackruleFj.Ifthesmallestspacewherethecurrentstateliesisthej-thsub-space,thenthecontrollawisusubspacecanbecomputedaccordingkFjxk.Thetothesizeofthetrackingerror.Whendenotingthelargestpossibletrackingerrorwithemandsettingx1(em,em,0),asequenceofstatesforcomputingthenestedsubspacescanbedefinedby:

xjx1

Npj1

N,j1,...,Np

p

Givenxj,thematrixSminimisation.jandFjcanbe16computedusingaLMI-basedThefor-mulationoftheLMI-basedminimisationisgivenby:SjargSjminj,Sj,Yjjsubjectto

j0!

C:1x

2IYj!1xjS!0;C2

:!0;j

Y0jQC3:Sj1Sj!0

and

54

RobustMPCfortemperaturecontrol

SSB

j

jA0i,lY0jB0i,lSjQ1=2Y0jR1=21C:BBBAlSj

BlYjSj00CC4BC@

Q1=2Sj0jI0C

CAR1=2Yj0

jI

!0,

i1,...,dUdL1l1,2

,wherejandYjaretemporaryvariables.Theconstraint(C1)definesthesubspaceStheconstraints(Cj,(25),theconstraint2)comesfromtheconstraint(CS3)ensuresthatSnestedintojisj1and(Cat4)indicatesthattheoptimiserisobtainedthecornersoftheuncertaintypolytope.ThefeedbacklawFgivenbyFofjis

jYjSj1

.ThecomputationFandSj1,...,N)areperformedoffline,jandtheyj(arestoredpinalookuptable.

Foronlineapplication,itisnecessarytofindthesmallestsubspaceinwhichthecurrent

statexx0klies,i.e.xksatisfiesx0kSm1

xk1andkSm11xk1,wherem2[1,2,...,Np].Whenm5Np,FkisalinearcombinationofFFcomputedby:

mandm1FkkFm1kFm1,

wherethecoefficientakispositivevalue,satisfying

x011kkSm

1