Type-1 and Type-2 effective Takagi-Sugeno fuzzy models for decentralized control of multi-input-multi-output processes

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ContentslistsavailableatScienceDirect

Journal

of

Process

Control

jo u rn al h om ep age :w w w . e l s e v i e r . c o m / l o c a t e / j p r o c o n t

Type-1

and

Type-2

effective

Takagi-Sugeno

fuzzy

models

for

decentralized

control

of

multi-input-multi-output

processes

Qian-Fang

Liao

a

_,

_Da

_Sun

b

_,

_Wen-Jian

_Cai

a,∗

_,

_Shao-Yuan

_Li

c

_,

_You-Yi

_Wang

a a_School_of_Electrical_and_Electronic_Engineering,_Nanyang_{Technological}_University,_639798,_Singapore

b_Department_of_Biomedical_Engineering,_National_University_of_Singapore,_118633,_Singapore c_Department_of_Automation,_Shanghai_Jiao_Tong_University,_Shanghai,_200240,_PR_China

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received5November2015

Receivedinrevisedform13January2017 Accepted16January2017

Availableonline16February2017 Keywords:

Interactions Looppairing

EffectiveTakagi-Sugeno(T-S)fuzzymodel Type-2fuzzysystem

Decentralizedcontrol

a

b

s

t

r

a

c

t

Effectivemodelisanoveltoolfordecentralizedcontrollerdesigntohandletheinterconnected inter-actionsin amulti-input-multi-output (MIMO)process. In this paper, Type-1 and Type-2effective Takagi-Sugenofuzzymodels(ETSM)areinvestigated.Bymeansofthelooppairingcriterion,simple cal-culationsaregiventobuildType-1/Type-2ETSMswhichareusedtodescribeagroupofnon-interacting equivalentsingle-input-single-output(SISO)systemstorepresentanMIMOprocess,consequentlythe decentralizedcontrollerdesigncanbeconvertedtomultipleindependentsingle-loopcontrollerdesigns, andenjoythewell-developedlinearcontrolalgorithms.Themaincontributionsofthispaperare:i) ComparedtotheexistingT-Sfuzzymodelbaseddecentralizedcontrolmethodsusingextratermsto characterizeinteractions,ETSMisasimplefeasiblealternative;ii)Comparedtotheexistingeffective modelmethodsusinglineartransferfunctions,ETSMcanbecarriedoutwithoutrequiringexact mathe-maticalprocessfunctions,andlaysabasistodeveloprobustcontrollerssincefuzzysystemispowerfulto handleuncertainties;iii)Type-1andType-2ETSMsarepresentedunderauniﬁedframeworktoprovide objectivecomparisons.AnonlinearMIMOprocessisusedtodemonstratetheETSMs’superiorityover theeffectivetransferfunction(ETF)counterpartsaswellastheevidentadvantageofType-2ETSMsin termsofrobustness.Amulti-evaporatorrefrigerationsystemisemployedtovalidatethepracticability oftheproposedmethods.

1. Introduction

Intheareaofmulti-input-multi-output(MIMO)processcontrol,theTakagi-Sugeno(T-S)fuzzymodelbaseddecentralizedcontrolisan attractivetopicbecauseofitsoutstandingmeritsincluding:i).itiseasytodesignandtunebecauseitusesthesimplestcontrolstructure whereeachmanipulatedvariable(processinput)isdeterminedbyonlyonecontrolledvariable(processoutput);ii)noexactmathematical processfunctionsarerequiredsincefuzzymodelscanbebuilttoahighdegreeofaccuracyfromdatasamplesandexpertexperience[1,2]; iii)itisrobusttodisturbancesincefuzzysystemexcelsinhandlinguncertainties[1–3];iv)linearcontrolalgorithmscanbeappliedto designcontrollersforanonlinearprocessviaparalleldistributedcompensation[4]sincetheT-Sfuzzymodeliscomposedofagroup oflinearlocalmodels[3,4].Anumberofacademicresultsconcerningthistopichavebeenproposed.Suchasthenetworkedandrobust decentralizedcontrolforlarge-scaleandinterconnectedMIMOprocessesin[5–8].Themaindifﬁcultyfordecentralizedcontrolistodeal withtheinteractionsamongthepairedinput-outputcontrol-loopsduetoitslimitedcontrolstructureﬂexibility.IntheexistingT-Sfuzzy modelbasedmethods,generally,foracertaincontrolpair,extratermsareaddedtoitsindividualopen-loopmodeltocharacterizethe interactingeffectsfromotherloops.Asimpleexampleisgivenasfollows:

Rule l:IF uj is Cl THEN yi=alij·uj+

n k=1,k/=jk(uk), l=1,···,M (1) ∗ Correspondingauthor.

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whereMisthenumberoffuzzyrules;yiistheithoutputandujisthejthinput(i,j=1,...,n),andyi−ujisoneofthecontrolpairsofan

n×nprocess;Cl_is_a_fuzzy_set;_y

i=al_ij·ujisthelthlocalmodeloftheT-Sfuzzymodelfortheindividualopen-loopyi−ujandal_ij isthe

coefﬁcient;k(uk)isanextratermstandingfortheinteractionscausedbyuk,and

n

k=1,k/=jk(uk)isthesumofextratermstodescribe

thetotalinteractingeffects.Eachlocalcontrollerofadecentralizedcontrolsystemisdevisedbasedonthemodelofacontrolpairbearing extratermsasshowninEq.(1)tocopewithinteractions.However,severalproblemsmayarise:

• Foralarge-scaleprocess,alargenumberofextratermsneedtobeidentiﬁed,whichwoulddrasticallyincreasethecostinprocess modeling;

• Foracomplexprocess,theinteractionsmaynotbedirectlymeasuredorevaluated,whichwouldformobstaclestoderivingtheextra terms;

• Foranonlinearprocess,differentworkingconditionsmayrequiredifferentcontrolpairconﬁgurationsandresultinchangedcoupling effects,whichwouldleadtochallengesinﬁndingsuitableextratermstodescribethevaryinginteractions;

• ThelocalmodelsofaT-Sfuzzymodelmaynotbelinearafteraddingtheextraterms,whichwouldincreasethecomplexityforcontroller design.

Giventheaboveproblems,amorepracticalmethodtoexpresstheinteractionsisrequired.Oneinterestingmannerdevelopedrecently istocreatetheeffectivemodels.Foreachcontrolpair,aneffectivemodelcanbebuiltbyrevisingthecoefﬁcientsofitsindividualopen-loop modeltoreﬂecttheinteractingeffects.UsingtheexampleinEq.(1),asimpleeffectiveT-Sfuzzymodel(ETSM)canbeexpressedas:

Rule l:IF u_j is Cl

THEN yi= ˆalij·uj, l=1,···,M

(2)

where ˆal

ijistherevisedcoefﬁcient.ComparedtoEq.(1),ETSMinEq.(2)usesadifferentmannertoexpressinteractionsthatcansolvethe

aforementionedproblemscausedbyusingextraterms,andgreatlysimplifydecentralizedcontrollerdesignbecause:i)theETSMmethod isusingagroupofnon-interactingsingle-inputsingle-output(SISO)systemstorepresentanMIMOprocesssuchthatthedecentralized controllerdesigncanbedecomposedintomultipleindependentsingle-loopcontrollerdesigns;ii)theETSMretainsthelinearityineach ofitslocalmodelswhichprovidesaplatformtoapplythematurelinearmethodstoregulateanonlinearprocess.Howtorevisethe coefficientstoachieveanETSMthatcancorrectlyreflecttheinteractingeffectsisakeyproblemtosolve.Currentlyseveralmethodstaking advantageoflooppairingcriteriatoconstructeffectivemodelsareavailable.Alooppairingcriterionisusedtopairinputsandoutputs todetermineaproperdecentralizedcontrolstructurewithminimumcouplingeffectsamongthepairedcontrol-loops,anditprovides quantifiedinterconnectedinteractionstocalculatetherevisedcoefficientsineffectivemodels.In[9],anapproachwaspresentedtoderive effectivetransferfunctions(ETF)todescribeagroupofequivalentopen-loopprocessesfordecentralizedcontrolintermsofdynamic relativegainarray(RGA)[10–13]basedcriterion,and[14]proposedamodelreductiontechniquetosimplifytheeffectiveopen-loop transferfunctionof[9].In[15],themethodtobuildETFsusingeffectiverelativegainarray[16]basedcriterionwasintroduced.In[17], analgorithmtomodifythecoefficientsforETFconstructionaccordingtorelativenormalizedgainarray(RNGA)[18]basedcriterionwas developed.Thesimulationsorexperimentsin[9,14,17,18]demonstratedthebetterperformancesofETFbasedcontrolmethodswhen comparedtoseveralotherpopularcontroltuningapproaches.Amongthese,RNGAbasedeffectivemodelhasprominentadvantages thatitprovidesacomprehensivedescriptionofdynamicinteractions,andworkswithsatisfactoryperformancesforbothlowandhigh dimensionalprocessesandwithoutrequiringthespecificsofcontrollers,andisabletoprovideauniqueresultwithlesscomputational complexity[17,18].WeinvestigatedRNGAbasedETSMfordecentralizedcontrolinaconferencearticle[19],whichis,tothebestofauthor’s knowledge,thefirstworkintheareaoflooppairingcriterionbasedeffectivefuzzymodel.ComparedtotheexistingETFmethods,ETSMis analternativetoprocesscontrollerdesignwhereexactmathematicalfunctionsareunavailable.Moreover,itlaysabasistodeveloprobust controllersincefuzzysystemisstrongincompensatingforuncertainties.

TheETSMstudiedin[19]isbasedontraditional(Type-1)fuzzymodelswherethefuzzymembershipsarecrispnumbers.Whenlarge uncertaintiesappear,thecrispfuzzymembershipsmaystruggletodescribetheconditions.Inthiscase,Type-2fuzzymodel[20–22]with thefuzzymembershipsthatarethemselvesfuzzycanbeapplied.InaType-2fuzzyset,thefuzzymembershipofanelementincludes primaryandsecondarygradesthatcanbeconsideredasaType-1fuzzyset.AsshowninFig.1,Part(a)isageneralType-2fuzzysetwhere thesecondarygradesrangefrom0to1.Whenallsecondarygradesareeither0or1thatthefuzzymembershipforanelementisan interval,itbecomesanintervalType-2fuzzysetasshowninPart(b)whichismorewidelyusedbecauseofitsmanageablecalculations

[23].TheincreasedfuzzinessendowsaType-2fuzzysetadditionaldesigndegreesoffreedomthatmakeitpossibletodirectlydescribe theuncertainties[20–23].[24]gaveanintroductionofType-2T-Sfuzzymodels,andseveralresults[25–27]provedthatType-2T-Sfuzzy modeloutperformsitsType-1counterpartintermsofaccuracyandrobustnessinprocessmodelingandcontrol.

ThispaperinvestigatesbothType-1andType-2ETSMfordecentralizedcontrol.Firstly,theidentiﬁcationofType-1andType-2T-S fuzzymodelsforanMIMOprocessbasedondatasamplesisgiven.Afterwards,bymeansofRNGAbasedcriterion,theinput-outputpairing conﬁgurationisdeterminedandsimplecalculationsareintroducedtoconstructType-1andType-2ETSMs.AnumericalnonlinearMIMO processisusedtodemonstratethesuperioritiesofETSMsovertheirETFcounterparts,aswellastheevidentadvantageofType-2ETSMs withrespecttorobustness.Anexperimentalrefrigerationsystemisusedtovalidatethepracticabilityoftheproposedmethodsandcompare Type-1andType-2ETSMsinarealapplication.Themaincontributionsofthisworkare:

i) ComparedtotheexistingT-Sfuzzymodelbaseddecentralizedcontrolmethodsusingextratermstocharacterizeinteractions,ETSM methodexpressestheinteractingeffectsthroughrevisingthecoefﬁcientsoftheoriginalT-Sfuzzymodel,whichisasimplefeasible alternative;

ii)ComparedtotheexistingETFmethods,ETSMdoesnotrequireaccuratemathematicalprocessfunctions,andlaysabasistodevelop robustcontrollerssincefuzzysystemisapowerfultooltohandleuncertainties;

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Fig1. (a)GeneralType-2fuzzyset,secondarygradesarein[0,1](b)IntervalType-2fuzzyset,secondarygradesare0or1.

iii)Type-2ETSMisproposedtoenrichtheETSMstudyandoffersanimprovementintermsofrobustness.Also,Type-1andType-2ETSM arepresentedunderauniﬁedframeworktoallowobjectivecomparisons.

2. T-SfuzzymodelingforanMIMOprocess

Throughoutthispaper,itisassumedthattheMIMOprocessesconsideredareopen-loopstable,nonsingularatthesteady-state con-ditions,andsquareindimension(equalnumberofinputsandoutputs).ThefollowingT-Sfuzzymodelmatrixcanbeusedtodescribean MIMOprocesswithnoutputs(yi,i=1,...,n)andninputs(uj,j=1,...,n)[19,28]:

FTS=

fTS,ij

n×n=

⎡

⎢

⎣

fTS,11 fTS,12 ··· fTS,1n fTS,21 fTS,22 ··· fTS,2n . . . ... . .. ... fTS,n1 fTS,n2 ··· fTS,nn

⎤

⎥

⎦

(3)

wherefTS,ijistheindividualopen-loopT-Sfuzzymodelforyi−uj,whichisalwaysidentiﬁablethroughproperexcitations[29].WhenfTS,ij

isaType-1fuzzymodel,itsfuzzyrulescanbeexpressedas: Rule l: IF xij(k) is Cijl

THEN yl

i(k)=alij,0·uj(k−ij)+alij,1·uj(k−ij−1)+···+alij,p·uj(k−ij−p)

+bl

ij,1·yi(k−1)+···+b

l

ij,q·yi(k−q)

(4)

wherel=1, ...,Mij,Mij is thenumberof fuzzyrules infTS,ij.xij(k)∈Rn isa vectorconsisting ofpastinputs andoutputs as: xij(k)=

[ uj(k−ij) ··· uj(k−ij−p) yi(k−1) ··· yi(k−q) ] T

,pandqareintegers,ij=ij/T ,ij denotesthetimedelayinyi−uj,andT

isthesamplinginterval;yl

i(k)istheoutputoflthfuzzyrule;a l

ij,r(r=0,1,...,p)andb

l

ij,s(s=1,...,q)arethecoefﬁcients.TheoutputoffTS,ijis

aweightedsumoflocaloutputs:

yi(k)=

Mij

l=1

l

ij(xij(k))yli(k) (5)

l_ij(xij(k)) denotesthefuzzymembershipfunctionofxij(k)inthelthfuzzysetC_ijl.Astheweights,theysatisfy0≤l_ij(xij(k))≤1and

Mij

l=1lij(xij(k))=1.

WhenfTS,ijisanintervalType-2T-Sfuzzymodel,itsfuzzyrulescanbeexpressedas:

Rule l: IF xij(k) is C˜ijl THEN ˜yl i(k)= ã l ij,0·uj(k−ij)+ ã l ij,1·uj(k−ij−1)+···+ ã l ij,p·uj(k−ij−p) + ˜bl ij,1·yi(k−1)+···+ ˜blij,q·yi(k−q) (6)

l=1, ...,Mij,where ˜C_ijl is an interval Type-2 fuzzy set.The fuzzy membershipof xij(k) in ˜C_ijl is an interval denoted as ˜l_ij(xij(k))=

(4)

arealsointervalsas ˜al

ij,r=[ a

l

ij,lb,r, a

l

ij,rb,r](r=0,1,...,p)and ˜blij,s=[ b

l

ij,lb,s, b

l

ij,rb,s](s=1,...,q),and theoutputof lthfuzzyruleis

˜yl

i(k)=[ y

l

i,lb(k), yi,rbl (k) ]thatcanbeobtainedby[24]:

yl i,lb(k)=a l ij,lb,0·uj(k−ij)+···+a l ij,lb,p·uj(k−ij−p)+b l ij,lb,1·yi(k−1)+···+b l ij,lb,q·yi(k−q)

yl_i,rb(k)=al_ij,rb,0·uj(k−ij)+···+alij,rb,p·uj(k−ij−p)+blij,rb,1·yi(k−1)+···+blij,rb,q·yi(k−q)

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BasedonMijfuzzyrules,atype-reducedset[24],denotedby ˜yi(k)canbederived:

˜yi(k)=[ yi,lb(k), yi,rb(k) ] (8)

whereyi,lb(k)andyi,rb(k)canbecalculatedbyKarnik-Mendelmethod[24].However,Karnik-Mendelmethodrequiresiterativecalculations

thatmaybetimeconsuming.Inthispaper,thefollowingcalculations[25,27]isselectedforsimpliﬁcation:

⎧

⎨

⎩

yi,lb(k)=

Mij l=1lij,lb(xij(k))·yli,lb(k)/

Mij l=1lij,lb(xij(k)) yi,rb(k)=

Mij l=1lij,rb(xij(k))·yi,rbl (k)/

Mij l=1lij,rb(xij(k)) (9)

NotethatinanType-2fuzzyset,

Mij

l=1lij,lb(xij(k))and

Mij

l=1lij,rb(xij(k))maynotbeequalto1.Thecrispoutputcanbeobtainedby

defuzzifying ˜yi(k)as[25,27]:

yi(k)=

yi,lb(k)+yi,rb(k)

2 (10)

BothType-1andType-2T-Sfuzzymodelcanbeconstructedbasedontheinput-outputdatasamplesthatarebrieﬂyintroducedas follows[25]:

i)Foraninput-outputchannelyi−uj,collectNijdatasamplesaszij(k)=[xij(k)T yi(k) ] T

,k=1,...,Nij.Determinethenumberoffuzzy

rulesMij,whichimpliesMijfuzzysets/clusterswillbeusedtocharacterizethedata.

ii)UseGustafson-Kesselclusteringalgorithm[30]tolocateMijfuzzyclustercenterszl_c,ij=[ (xl_c,ij) T yl c,i] T (l=1,....Mij),wherexl_c,ij= [ ul_c,jij yl_c,i 1 yl_c,i2 ] T

isthelthcenterofinputvectors.Denotethedistancebetweenzij(k)andzlc,ijasDlij(zij(k))=(zij(k)−zlc,ij) T

·Aij·

(zij(k)−zlc,ij)(l=1,...Mij),whereAijisthenorm-inducingmatrixcalculatedbasedondatasamples.Dlij(zij(k))’s(l=1,...Mij)determinethe

Type-1fuzzymembershipsforzij(k)as:

l_ij(zij(k))=

⎧

⎪

⎨

⎪

⎩

0, if any Dr ij(zij(k))=0, r=1,···Mij, r /=l 1 Mij

r=1 Dl ij(zij(k)) Dr ij(zij(k)) , if Dr ij(zij(k)) /=0, r=1,···Mij 1, if Dl ij(zij(k))=0 (11)

iii)AssigneachdatumtotheclusterwhereithasthelargestType-1fuzzymembershiptodividethedataintoMijgroups.Foreach

group,utilizeleastsquaremethodtoidentifythecoefﬁcientsal

ij,r(r=0,1,...,p)andb

l

ij,s(s=1,...,q)foritsassociatedType-1fuzzyrule.

iv)Ineachgroup,evaluateavariantrangeforfuzzymembership,l_ij>0,toachieveanintervalType-2fuzzymembership˜l_ij(zij(k))=

[ l_ij,lb(zij(k)), lij,rb(zij(k)) ]foreachdatumzij(k)basedonitsType-1fuzzymembershipas:

l ij,lb(zij(k))=max

0, l ij(zij(k))−lij

l ij,rb(zij(k))=min

l ij(zij(k))+lij, 1

(12)

v)Ineachgroup,evaluateavariantrangeforoutput,yi>0,suchthattwodata,denotedaszij,lb(k)andzij,rb(k),canbederivedfrom

eachdatumzij(k)as

zij,lb(k)=[xij(k)T yi(k)−yi] T =[_x_ij(k)T yi,lb(k) ] T zij,rb(k)=[xij(k)T yi(k)+yi] T =[_x_ij(k)T yi,rb(k) ] T (13)

UseleastsquaremethodtoidentifythecoefﬁcientsoftwolinearpolynomialsasinEq.(7)basedonzij,lb(k)andzij,rb(k)respectivelytohave

theleftandrightboundsof ˜al

ij,r(r=0,1,...,p)and ˜b l

ij,s(s=1,...,q),foritsassociatedType-2fuzzyrule.

Whengivenanewinputxij(k),itsType-1fuzzymembershipsl_ij(xij(k)),l=1,...,Mij,arecalculatedby:

l_ij(xij(k))=

⎧

⎪

⎨

⎪

⎩

0, if any Dr ij(xij(k))=0, r=1,···Mij, r /=l 1 Mij

r=1 Dl_ij(xij(k)) Dr ij(xij(k)) , if Dr ij(xij(k)) /=0, r=1,···Mij 1, if Dl ij(xij(k))=0 (14)

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Fig.2. Twotypicalunitstepresponses. whereDl ij(xij(k))=(xij(k)−x l c,ij) T

·(xij(k)−xl_c,ij).TheoutputfromtheType-1T-SfuzzymodeliscalculatedbyEq.(5).WhileitsType-2

fuzzymemberships˜l ij(xij(k))=[ lij,lb(xij(k)), l ij,rb(xij(k)) ]are:

⎧

⎨

⎩

l ij,lb(xij(k))=max

l ij(xij(k))−lij, 0

l ij,rb(xij(k))=min

l ij(xij(k))+lij, 1

(15)

TheoutputfromtheType-2T-SfuzzymodeliscalculatedbyEqs.(9)and(10).

3. Relativenormalizedgainarraybasedlooppairingcriterion

Looppairingdeﬁnesthedecentralizedcontrol-loopconﬁguration,i.e.,whichoftheavailableinputsshouldbechosentomanipulate eachoftheprocessoutputs.FromaT-Sfuzzymodel,twofactorscanbecalculatedforinteractionassessmentaccordingtoRNGAbased criterion[18,28]:steady-stategain,kTS,ij,whichindicatestheeffectofujonthegainofyiwhentheprocessreachesthesteady-state

condition,andnormalizedintegratederror,eTS,ij,whichreﬂectstheresponsespeedofyitouj.BothkTS,ijandeTS,ijaredeﬁnedfromthe

stepresponse.TwoexamplesaregiveninFig.2,wheretheshadedareaandkTS,ijdetermineeTS,ijas[28]:

eTS,ij=

∞ r=0 yi(∞)−yi(r·T ) kTS,ij · T (16)

whereTisthesamplinginterval,yi(∞)=yi(k)|k→∞isthesteady-stateoutputoffTS,ijinunitstepresponse.Itiseasytoknowthatyi(∞)=kTS,ij.

yi(r·T)istheoutputatrthsamplingtime.eTS,ijcanbeusedtorepresentthedynamicpropertysincesmaller/largereTS,ijmeansyigives

faster/slowerresponsetouj[18,28].

Becauseofthenonlinearnatureinafuzzymodel,anoperatingpointshouldbegiventocalculatekTS,ijandeTS,ijfromfTS,ijsincedifferent

operatingconditionsmayhavedifferentkTS,ijandeTS,ijandresultindifferentcontrolconﬁgurations[28].GivenanoperatingpointforfTS,ij

as:

x0,ij=[ u0,j(k0−ij) ··· u0,j(k0−ij−p) y0,i(k0−1) ··· y0,i(k0−q) ] T

(17) Inthevicinityofx0,ij,aT-Sfuzzymodelcanbeapproximatelyrepresentedbyalinearfunctionbylettingl_ij(xij(k))=l_ij(x0,ij)[28]:

yi(k)=fTS,ij(xij(k))

=aij,0·uj(k−ij)+···+aij,p·uj(k−ij−p)+bij,1·yi(k−1)+···+bij,q·yi(k−q)

(18)

WhenfTS,ijisaType-1T-SfuzzymodelasinEq.(4),thecoefﬁcientsofEq.(18)are

⎧

⎨

⎩

aij,r=

Mij l=1lij(x0,ij)aij,rl , r=0,1,···,p bij,s=

Mij l=1lij(x0,ij)blij,s, s=1,···,q (19)

WhenfTS,ijisaType-2T-SfuzzymodelasinEq.(6),thecoefﬁcientsofEq.(18)are

aij,r=(aij,lb,r+aij,rb,r)/2, r=0,1,···,p

bij,s=(bij,lb,s+bij,rb,s)/2, s=1,···,q

(6)

where aij,lb,r=

Mij l=1lij,lb(x0,ij)·a l ij,lb,r/

Mij

l=1lij,lb(x0,ij), aij,rb,r=

Mij

l=1lij,rb(x0,ij)·a

l

ij,rb,r/

Mij

l=1lij,rb(x0,ij), bij,lb,s=

Mij

l=1lij,lb(x0,ij)·

bl

ij,lb,s/

Mij

l=1lij,lb(x0,ij) and bij,rb,s=

Mij

l=1lij,rb(x0,ij)·b

l

ij,rb,s/

Mij

l=1lij,rb(x0,ij).Based on Eq. (18), kTS,ij and eTS,ij can be calculated by

followingequations[28]:

kTS,ij=

aij,0+aij,1+···+aij,p

1−(bij,1+bij,2+···+bij,q)

(21) eTS,ij=

p r=0raij,r−

p w=0

q s=1aij,w·bij,s·|w−s|·sgn(w−s)

(aij,0+aij,1+···+aij,p)(1−(bij,1+···+bij,q)) ·

T+ij·T (22)

Eqs.(21)and(22)couldbeverysimpleforrealapplicationssincepandqaregenerallynotlarge.Forexample,whenp=0andq=2,they become: kTS,ij= aij,0 1−(bij,1+bij,2) ,eTS,ij= bij,1+2bij,2 1−(bij,1+bij,2)· T+ij·T (23)

CollectingthecalculatedresultsofEqs.(21)and(22)ofeachelementinFTSformsasteady-stategainmatrixKTS=

kTS,ij

n×nanda

normalizedintegratederrormatrixETS=

eTS,ij

n×n.Next,weintroducetheimportantconceptsofRNGAlooppairingcriterionasfollows:

RGA:therelativegainofacontrolpairyi−uj,denotedbyTS,ij,isdeﬁnedas[10]:

TS,ij=

kTS,ij

ˆkTS,ij

(24)

where ˆkTS,ijisthesteady-stategainofyi−ujwhenallothercontrol-loopsareclosed.RGAisanarrayformedbyassemblingalltherelative

gainsasRGA=

TS,ij

n×n,whichcanbecalculatedonlyusingindividualopen-loopinformation[12]:

RGA=KTS⊗K−T_TS (25)

where⊗iselement-by-elementproduct,K−T

TS isthetransposeofinverseKTS.

RNGA:thenormalizedgainforcontrolpairyi−uj,denotedbykNTS,ij,reﬂectsthetotaleffectofujonyibyincludingbothkTS,ijandeTS,ij

as[18,28]:

kNTS,ij=

kTS,ij

eTS,ij

(26)

ExtendEq.(26)totheoverallprocesstoobtainthenormalizedgainmatrixKNTSas[18,28]:

KNTS=KTS ETS (27)

where iselement-by-elementdivision.Denotethenormalizedgainofloopyi−ujwhenallothercontrol-loopsareclosedas ˆkNTS,ij,where

ˆkNTS,ij= ˆkTS,ij/ˆeTS,ij, ˆeTS,ijisthenormalizedintegratederrorofyi−ujwhenotherloopsareclosed.Therelativenormalizedgain,denotedby

TS,ij,canbedeﬁnedas[18,28]:

TS,ij=

kNTS,ij

ˆkNTS,ij

(28)

RNGAisanarrayderivedbycollectingallthenormalizedgainsasRNGA=

TS,ij

n×n,whichcanbecalculatedonlyusingindividual

open-loopinformation[18,28]:

RNGA=KNTS⊗K−TNTS (29)

FromRGAandRNGA,thecontrolpairscanbeselectedaccordingtothefollowingrules[18,28]:

i)AllpairedRGAandRNGAelementsshouldbepositive;

ii)ThepairedRNGAelementsareclosestto1;

iii)LargeRNGAelementsshouldbeavoided;

Placethepairedelementsonthediagonalpositionsof_KTS throughcolumnswap,thevalueofNiederlinskiindex(NI)[31],canbe

calculatedas: NI= det [KTS]

˘n

i=1kTS,ii

(30)

wheredet [KTS] denotesdeterminantofKTSaftercolumnswap,˘_in₌₁kTS,iiistheproductofpairedelements.ApositiveNIisanecessary

conditionforpairedsystemtobestable[31].Therefore,anadditionalruleforpairingis

iv)NI>0

4. EffectiveT-Sfuzzymodel

TheETSMforacontrolpairyi−uj,denotedby ˆfTS,ij,istheopen-loopT-Sfuzzymodelforyi−ujwhenallothercontrol-loopsareclosed.

Thusitssteady-stategainandnormalizedintegratederrorare ˆkTS,ijand ˆeTS,ij.Sincetheopen-loopmodelforacertaincontrolpairwhen

(7)

tokeeppartofthecoefﬁcientsof ˆfTS,ijsametothatoffTS,ij.InspiredbytheETFconstructionproposedin[15],wechoosetheType-1ETSM

consistingoffollowingfuzzyrules: Rule l: IF xij(k) is C_ijl THEN yl i(k)= â l ij,0·uj(k− îj)+ â l ij,1·uj(k− îj−1)+···+ â l ij,p·uj(k− îj−p) +bl ij,1·yi(k−1)+···+blij,q·yi(k−q) (31) where âl

ij,r(r=0,1,...,p)and ˆijarethecoefﬁcientsrevisedfroma l

ij,r(r=0,1,...,p)andijoftheindividualopen-loopType-1T-Sfuzzy

modelasinEq.(4).Similarly,wechoosetheType-2ETSMconsistingoffollowingfuzzyrules: Rule l: IF xij(k) is C˜ijl THEN ˜yl i(k)= ˆã l ij,0·uj(k− îj)+ ˆã l ij,1·uj(k− îj−1)+···+ ˆã l ij,p·uj(k− îj−p) + ˜bl ij,1·yi(k−1)+···+ ˜blij,q·yi(k−q) (32)

where ˆãlij,r=[ âlij,lb,r, âij,rb,rl ]and îjarerevisedfrom ãlij,r=[ a l

ij,lb,r, alij,rb,r]andijoftheindividualopen-loopType-2T-Sfuzzymodel

asinEq.(6).

Thequantiﬁedinteractingeffectsonsteady-stategainofyi−ujcanbederivedfromrelativegainTS,ij=kTS,ij/ˆkTS,ij,whilethequantiﬁed

interactingeffectsondynamicpropertycanbederivedfrombothrelativegainTS,ijandrelativenormalizedgainTS,ijby[17]:

TS,ij

TS,ij =

ˆeTS,ij

eTS,ij ≡

TS,ij (33)

whereTS,ijistherelativenormalizedintegratederror[17].Forthewholeprocesswehave:

TS=

TS,ij

n×n=RNGA RGA (34)

BasedonTS,ijandTS,ij,therevisedcoefﬁcientsofType-1andtheType-2ETSMcanbecalculatedasfollows:

AccordingtoEq.(21),thesteady-stategain ˆkTS,ijofanETSM ˆfTS,ijbasedonagivenoperatingpointx0,ijcanbecalculatedas:

ˆkTS,ij=

â_ij,0_{+ â}_ij,1₊_·_·_·_{+ â}_ij,p 1−(bij,1+bij,2+···+bij,q)

(35)

ForaType-1ETSM,thecoefﬁcients ˆaij,r(r=0,1,...,p)inEq.(35)arecalculatedby

ˆa_ij,r₌

Mij

l=1

l

ij(x0,ij)ˆalij,r (36)

SubmittingEqs.(21)and(24)intoEq.(35)tohavethefollowingequationtodetermine ˆal

ij,r: ˆal_ij,r= a l ij,r TS,ij (37)

ForaType-2ETSM,thecoefﬁcients ˆaij,r(r=0,1,...,p)inEq.(35)aredeterminedby

ˆaij,r=

ˆaij,lb,r+ ˆaij,rb,r

2 (38)

where ˆaij,lb,r=

Mij

l=1lij,lb(x0,ij)· ˆalij,lb,r/

Mij

l=1lij,lb(x0,ij)and ˆaij,rb,r=

Mij

l=1lij,rb(x0,ij)· ˆalij,rb,r/

Mij

l=1lij,rb(x0,ij).SubmittingEqs.(20),(21),

(24)and(38)intoEq.(35),thefollowingequationstoderive ˆãlij,r=[ âlij,lb,r âlij,rb,r]canberevealed:

ˆal ij,lb,r= al ij,lb,r TS,ij , ˆal ij,rb,r= al ij,rb,r TS,ij (39)

AccordingtoEq.(22),thenormalizedintegratederror ˆeTS,ijofaType-1orType-2ETSM ˆfTS,ijbasedonthegivenoperatingpointx0,ijis

computedby: ˆeTS,ij=

p r=0r ˆaij,r−

p w=0

q

s=1ˆaij,w·bij,s·|w−s|·sgn(w−s)

(âij,0+ âij,1+···+ âij,p)(1−(bij,1+···+bij,q))

·T+ ˆij·T (40)

ForaType-1ETSM, ˆaij,risdeterminedbyEq.(36)andbij,sisdeterminedbyEq.(19).WhileforaType-2ETSM, ˆaij,risdeterminedbyEq.

(38)andbij,sisdeterminedbyEq.(20).SubmittingEqs.(22),(33),(37)/(39)into(40),afterarrangement,givesthefollowingequationto

calculate ˆij: ˆ_ij₌

p r=0raij,r−

p w=0

q s=1aij,w·bij,s·|w−s|·sgn(w−s)

(aij,0+aij,1+···+aij,p)(1−(bij,1+···+bij,q)) ·

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Fig.3. TheworkingprocedureforETSMbaseddecentralizedcontrollerdesign.

SeveralexperimentalresultsdemonstratethatforwellpairedMIMOprocesses,thevaluesofTS,ij’sofpairedcontrol-loopsareclosed

to1.ThusEq.(41)canbesimpliﬁedas:

ˆij≈ij·TS,ij (42)

Eqs.(37),(39)and(42)providesimplecalculationstorevisethecoefficientstodescribeinteractingeffects.However,animportant andnecessaryfactwhichcannotbeignoredisthatacontrolsystemshouldpossessintegrityproperty[15,17],whichmeans,thesystem shouldremainstablewhetherotherloopsareputinortakenout.Moreover,theintegrityrequiresthatwhencontrollingacertainloop afterallotherloopsremove,theperformanceofthecontrollerdesignedbasedontheETSMshouldbenomoreaggressivethanthatof thecontrollerdesignedbasedontheindividualopen-loopmodel[17].Notethatlargerabsolutevalueofsteady-stategainandlargertime delayimplymorechallengesforastablecontrolsystemdesign.Inabidtomaintaintheintegrityproperty,anETSMshouldchoosethe coefficientsbetweenoriginalandrevisedonesthatcanreflect“worsecondition”forcontrollerdesign.Therefore,wehavethefollowing criteriontodetermine âij,r, ˆãij,rand îjforType-1/Type-2ETSMs:

⎧

⎪

⎨

⎪

⎩

ˆal ij,r=

max{al ij,r, a l

ij,r/TS,ij}, kTS,ij>0

min{al

ij,r, a

l

ij,r/TS,ij}, kTS,ij<0

ˆal ij,lb,r=

max{al ij,lb,r, a l

ij,lb,r/TS,ij}, kTS,ij>0

min{al

ij,lb,r, a

l

ij,lb,r/TS,ij}, kTS,ij<0

, ˆal ij,rb,r=

max{al ij,rb,r, a l

ij,rb,r/TS,ij}, kTS,ij>0

min{al

ij,rb,r, a

l

ij,rb,r/TS,ij}, kTS,ij<0

ˆ_ij₌max

ij, ij·TS,ij

(43)

BasedonETSMs,linearSISOcontrolalgorithmscanbedirectlyappliedtodesigndecentralizedcontrollersfornonlinearMIMOprocesses. ThestepstodevisetheETSMbaseddecentralizedcontrollersaresummarizedasfollowswithaﬂowchartgiveninFig.3.

i)Forann_×nprocess,collectdatasamplesfromeachinput-outputchanneltobuildanindividualType-1orType-2open-loopT-Sfuzzy modeltoformann×nType-1orType-2fuzzymodelmatrixFTS.

ii)Atacertainworkingcondition,calculatedsteady-stategainkTS,ijandnormalizedintegratederroreTS,ijforeachindividualelementin

FTStoobtainKTSandETS.

iii)UseRNGAbasedcriteriontopairinputsandoutputstodetermineadecentralizedcontrolconﬁguration.

iv)Foreachcontrolpair,revisethecoefﬁcientsofitsindividualType-1orType-2open-loopT-SfuzzymodelaccordingtoEq.(43)toobtain aType-1oraType-2ETSM.Afterwards,designalocalcontrollerbasedoneachETSMtoachieveadecentralizedcontrolsystem. v)Iftheworkingconditionchanges,repeatstepii)–iv).

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5. Casestudies

5.1. Simulations

Considerathree-input-three-outputnonlinearprocess[19]:

⎧

⎪

⎨

⎪

⎩

˙x1=x2+5x₁2x2+6x2₂ ˙x2=−4x1−5x2+8x1x2+u1 ˙x3=x4 ˙x4=−6x3−5x4+3x₃3+10x3x4x5+u2 ˙x5=x6+4x72 ˙x6=x7+5x5x26x7 ˙x7=−14x5−23x6−10x7+7x5x6x7+u3 y1=5x1+5x2+6x3+2x4+14x5+9x6+x7 y2=8x1+2x2+3x3+4x5+6x6+2x7 y3=x1+x2+4x3+2x4+1.4x5+0.2x6 (44)

wherexr’s(r=1,...,7)arestatevariables.Thetimedelaysinthisprocessarei1=i2=2 (sec)andi3=1 (sec)fori=1,2,3.Choose

thesamplingintervalasT=0.1sec,supposetherearedisturbancesrandombutboundedin[-0.2,0.2]ontheinputsofthesampleddata pairs,constructaType-1andaType-2fuzzymodelwithp=0andq=2foreachinput-outputchannel(theresultsareshowninAppendix A).Giventheoperatingpointsasx0,ij=

u0,j(k0−ij) y0,i(k0−1) y0,i(k0−2)

=

0 0 0

⎦

AccordingtoRNGAbasedcriterion,thedecentralizedcontrolconﬁgurationcanbedeterminedy1−u3/y2−u1/y3−u2,NI=0.9598>0,

andthenormalizedintegratederrormatrixis:

TS=

⎡

⎣

4.35451.3138 −0.48611.9981 1.15892.9520 −0.6065 0.8125 0.3405

⎤

⎤

⎦

Thegainandphasemarginsbasedcontrolalgorithmemployedin[15,17]isselectedtodesigncontrollersbasedonType-1andType-2 ETSMs(ThedetailsaregiveninAppendixA),andtherequiredgainandphasemarginsfortheETSMbasedcontrolsystemaresetas3and /3.Forcomparison,welinearizethefunctionsofEq.(44)atthegivenoperatingpointstoobtainatransferfunctionmatrixtoapplythe RNGAbasedETFmethod[17]usingthesamecontrolalgorithmwiththesamerequiredgainandphasemargins(thedetailsarealsogiven inAppendixA).Letthereferencevaluesberv1=1.5,rv2=1andrv3=0,thecontrolperformancesareshowninFig.4.

AscanbeseeninFig.4,whengiventhesamegainandphasemarginrequirements,thecontrollersbasedonfuzzymodelsbuiltfromdata withinexactnesscanachievesmallerovershootscomparedtothatbasedontransferfunctionslinearizedfromexactmathematicalmodel.

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0 5 10 15 20 25 0 0.5 1 1.5 y1 0 5 10 15 20 25 0 0.5 1 y2 0 5 10 15 20 25 0 0.1 0.2 y3 Time(sec) 4 6 1.451.5 1.551.6 1.65

ETF based control Type-1 ETSM based control Type-2 ETSM based control

4 6 0.75 0.8 0.85 4 5 6 0.16 0.18

Fig.4.ThecomparisonsofETFandETSMbasedcontrolforEq.(44).

Table1

PerformanceindexesofType-1andType-2ETSMbasedcontrolforEq.(44).

IAE ISE ITAE ITSE

y1 y2 y3 y1 y2 y3 y1 y2 y3 y1 y2 y3

Type-1 4.83 4.99 0.90 5.49 2.99 0.10 10.81 21.81 6.08 7.19 5.97 0.54

Type-2 4.77 4.81 0.89 5.42 2.91 0.10 10.56 19.89 5.83 6.97 5.59 0.54

Table2

PerformanceindexesofType-1andType-2ETSMbasedcontrolforCase-IandCase-II.

IAE ISE ITAE ITSE

y1 y2 y3 y1 y2 y3 y1 y2 y3 y1 y2 y3

Case-I Type-1 8.71 4.84 2.59 7.63 2.71 0.78 57.18 27.05 20.90 21.30 5.74 4.21

Type-2 8.15 4.60 2.42 6.73 2.67 0.73 48.90 23.33 18.55 18.44 5.21 3.92

Case-II Type-1 33.2 17.8 11.2 21.4 6.6 2.8 1077.5 562.3 372.4 328.4 89.4 41.7

Type-2 24.7 13.5 8.33 16.2 5.2 2.1 586.5 307.3 204.7 182.3 50.2 24.5

TheperformancesofType-1andType-2ETSMbasedcontrolarecamparableinthiscase.Inabidtoexplicitlydemonstratetheirdifferences,

thecomparisionsoffourperformanceindexes,IAE=

∞k=0|r

v

i−yi(k)|·T ,ISE=

∞

k=0(r

v

i−yi(k))2·T ,ITAE=

∞

k=0k·|r

v

i−yi(k)|·Tand

ITSE=

∞k=0k·(r

v

i−yi(k))2·T ,betweenType-1andType-2ETSMbasedcontrolareemployedandshowninTable1,whichproveType-2

controlcanachievesmallerintegratederrors.

Supposethethirdstateequation ˙x3=x4inEq.(44)ischangedtothefollowingtwocasesduetotheuncertainties:

Case-I: ˙x3=x4+1.5u1;Case-II: ˙x3=x4+2.31u1

Applythecontrollerstothechangedprocesses,thecomparisonsareshowninFig.5,andthefourindexesofETSMbasedcontrolare giveninTable2.

AscanbeseeninFig.5andTable2,ETSMbasedcontrolcanprovidemuchbetterresultsthanETFbasedcontrol,andType-2ETSM

basedcontrolcanachievesmallerovershoots,lesssettlingtimeandsmallerintegratederrorscomparedtoitsType-1counterpart.When thechangedcoefﬁcientbecomeslargerduetotheincreaseddegreeofuncertaintyas:

Case-III: ˙x3=x4+2.5u1

TheprocessundertheETFbasedcontrolbecomesinstablewhileitsoutputsundertheETSMbasedcontrolcanstillreachthereference valuesasshowninFig.6.ThefourindexesofETSMbasedcontrolinTable3provethatType-2ETSMbasedcontrolismorerobust.

ConcludedfromFig.4–6andTable1–3,ETSMbasedcontrolcangivebetterperformanceswhencomparedtotheirETFbasedcounterparts ineverycase.Asthedegreeofuncertaintybecomelager,Type-2ETSMbasedcontrolcanprovidemoresatisfactoryresultsthanType-1 ETSMbasedcontrolintermsofrobustness.

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0 10 20 30 40 50 0 1 2 3 Time(sec) y1 0 10 20 30 40 50 0 0.5 1 1.5 Time(sec) y2 Case-I 0 10 20 30 40 50 -0.2 0 0.2 0.4 0.6 0.8 Time(sec) y3 0 50 100 -1 0 1 2 3 4 Time(sec) y1 0 50 100 -0.5 0 0.5 1 1.5 2 Time(sec) y2 Case-II 0 50 100 -0.5 0 0.5 1 Time(sec) y3

ETF based control

Type-1 ETSM based control

2 4 6 2 2.2 2 4 6 81012 1.2 1.4 6 8 10121416 -0.2 -0.1 0 90 95 100 1.2 1.4 1.6 90 95 100 0.8 1 90 95 100 -0.1 0 0.1

ETF based control

Fig.5.ComparisonsofETFandETSMbasedcontrolforCase-IandCase-II.

0 10 20 30 40 50 -4 -2 0 2 4 6 8 10 12 Time (sec) Outputs

ETF based control

0 50 100 150 200 0 1 2 3 y1

ETSM based control

0 50 100 150 200 0 1 2 y2 0 50 100 150 200 -0.5 0 0.5 Time(sec) y3 y1 y2 y3

Type-1 ETSM based control Type-2 ETSM based control

Fig.6. ComparisonsofETFandETSMbasedcontrolforCase-III.

Table3

PerformanceindexesofType-1andType-2ETSMbasedcontrolforCase-III.

IAE ISE ITAE ITSE

y1 y2 y3 y1 y2 y3 y1 y2 y3 y1 y2 y3

Type-1 116 62.4 40.4 67.5 19.8 8.51 1457.1 7758 5079 4051 1145 495.4

Type-2 53.1 28.9 18.4 32.4 9.96 4.24 296.1 1582 1038 832.4 237.0 105.0

5.2. Applicationinamulti-evaporatorrefrigerationsystem

Anexperimentalmulti-evaporatorrefrigerationsystemwiththreeevaporators(EVAP1,EVAP2andEVAP3)isshowninFig.7,andits

schematicdiagramandpressure(P)-enthalpy(h)chartareshowninFig.8.Inthissystem,R134aisusedastherefrigerant.ForEVAP1,water isusedasheattransferﬂuidtoconveythecoolingtomeettheair-conditioningrequirements.WhileforEVAP2usedforperishablefood

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Fig.7.Theexperimentalmulti-evaporatorrefrigerationsystem.

Fig.8.Theschematicdiagramandpressure(P)-enthalpy(h)chartofthemulti-evaporatorrefrigerationsystem.

storageandEVAP3usedforfreezingwhereevaporatingtemperaturesarelowthatwatermaybefrozen,ethyleneglycolsolutionisused instead.TheworkingprocessasshowninFig.8is:therefrigerantasasaturatedvaporwithalowpressure(state1)entersthecompressor andiscompressedisentropicallytoasuperheatedvaporwithahighpressure(state2).Thenitentersintothecondenserwhereitiscooled andcondensedintoliquidphase(state3)byrejectingheattotheexternalenvironment.Afterwardsitisdividedintothreeﬂows(states

4,5and6),whichgointoEVAP1(state7),EVAP2(state8)andEVAP3(state9)aftertheirpressuresarereducedthroughthreeexpansion valves(EV1,EV2,andEV3)respectively.Byabsorbingheatsfromtheambientenvironmentsoftheevaporators,thethreeflowsevaporate atspecifiedtemperaturestobecomesaturatedvapor(states10,11and14).Inthepressureregulationdevice,theflowsatstate10and11

arethrottledtostate12and13respectivelysuchthattheirpressuresareequaltothatoftheﬂowatstate14fromEVAP3whichhasthe lowestevaporatingpressureandtemperature.Finally,thethreeﬂowsmixupintooneatstate1andreturntocompressortocompletethe refrigerationcycle.

Intheexperiment,thecompressorpowerandthespeedsoffansarefixed.Theflowratesofrefrigerantinthreeevaporatorscanbe adjustedtosatisfydifferentcoolingloadsthroughregulatingtheopeningdegreesofthreeEVs.TheopeningchangeinanyoneoftheEVs willhaveimpactsonthreerefrigerantflowratesofthreeevaporatorssubsequentlyaffectthetemperaturesofheattransferfluidsT1,T2

andT3asshowninFig.8.Therefore,aninterconnectednonlinearthree-input-three-output(3×3)processcanbeformedwherethethree

EVopeningdegreesareusedtoregulatetheheattransferﬂuidtemperaturesofthreeevaporators.Sincethedesignedworkingcondition forthismulti-evaporatorrefrigerationsystemisT1,d=17oC,T2,d=3oCandT3,d=−8oCwiththecorrespondingEVopeningdegreesas

85%,43%and16%respectively,lettheoutputsofthis3×3processbeyi=Ti−Ti,d(i=1,2,3),andtheopeningrangesofEV1,EV2andEV3be

[70%,100%],[31%,55%]and[12%,20%]whichareuniformlyscaledto[−3,3]tobethevariationrangesofinputsuj(j=1,2,3)forconstructing

fuzzymodels.Thestepresponsesforthis3×3processareshowninFig.9.

Thetimedelaycanbemeasuredas:11=1 (min),12=1.6 (min),13=1.5 (min),21=1.4 (min),22=1.2 (min),23=

1.4 (min),31=1.4 (min),32=1.5 (min)and33=1.2 (min),andthesamplingintervalischosenasT=0.5(min).Basedonthedata

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Fig.9.Stepresponsesforthe3×3process.

outintheareaofdesignedworkingconditionaroundtheoperatingpointsx0,ij=

u0,j(k0−ij) y0,i(k0−1) y0,i(k0−2)

=

0 0 0

fori,j=1,2,3.FromType-1fuzzymodels,thefollowingresultscanbeobtained:

KTS=

−1.7958 0.6011 0.2011 0.7983 −0.6962 0.0996 0.2005 0.0993 −0.2961

, ETS=

1.0409 1.8491 1.7683 1.5136 1.3125 1.6252 1.5769 1.6966 1.4837

RGA=

2.2836 −0.9984 −0.2852 −1.0239 2.2168 −0.1929 −0.2597 −0.2184 1.4780

, RNGA=

1.3730 −0.2860 −0.0870 −0.2936 1.3614 −0.0679 −0.0794 −0.0755 1.1549

TS=

0.6012 0.2864 0.3051 0.2867 0.6141 0.3519 0.3058 0.3456 0.7813

Thedecentralizedcontrolstructureisy1−u1/y2−u2/y3−u3,whereNI=0.4169>0.FromType-2fuzzymodels,theresultsare:

KTS=

−1.7995 0.6000 0.2009 0.7968 −0.7012 0.0994 0.1997 0.0990 −0.2970

, ETS=

1.0427 1.8483 1.7692 1.5118 1.3163 1.6258 1.5762 1.6959 1.4844

RGA=

2.2432 _−0.9669 _−0.2763 −0.9916 2.1777 −0.1861 −0.2516 −0.2108 1.4624

, RNGA=

1.3688 _−0.2829 _−0.0859 −0.2905 1.3573 −0.0668 −0.0784 −0.0743 1.1527

TS=

0.6102 0.2926 0.3109 0.2929 0.6233 0.3589 0.3115 0.3526 0.7882

Thedecentralizedcontrolstructureisy1−u1/y2−u2/y3−u3,sameasthatobtainedfromType-1fuzzymodels,NI=0.4247>0.Usingthe

gainandphasemarginsbasedcontrolmethodtodevisethelocalcontroller,giventherequiredgainandphasemarginsare4and3 /8 respectively,theperformancesofType-1andType-2ETSMbaseddecentralizedcontrolforthismulti-evaporatorrefrigerationsystemare showninFig.10.

ItcanbeseenfromFig.10thattheoutputsunderbothType-1andType-2ETSMbasedcontrolcantracktheirreferencevaluesinspiteof disturbances.Theirperformanceindexesintegratedk=6tok=60/TarecomparedinTable4,whichdemonstratethatType-2ETSMbased controlcanachievesmallerintegratederrorsinrealapplications.

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0 10 20 30 40 50 60 14 16 18 20 22 T1 (° C) 0 10 20 30 40 50 60 -2 0 2 4 T2 (° C) 0 10 20 30 40 50 60 -8.4 -8.2 -8 -7.8 -7.6 T3 (° C) Time(min) Reference

Type-1 ETSM based control Type-2 ETSM based control

Fig.10.Type-1andType-2ETSMbaseddecentralizedcontrolfortherefrigerationsystem.

Table4

PerformanceindexesofType-1andType-2ETSMbasedcontrolforthemulti-evaporatorrefrigerationsystem.

IAE ISE ITAE ITSE

y1 y2 y3 y1 y2 y3 y1 y2 y3 y1 y2 y3

Type-1 38.3 25.6 9.4 62.3 34.7 1.96 973.7 609.4 280.3 1272.2 690.6 58.2

Type-2 34.6 25.0 6.4 52.3 34.5 0.96 850.8 575.6 184.2 1036.7 629.2 26.7

6. Conclusions

ThispaperpresentedType-1andType-2ETSMmethodstomeasureanddescribeinteractionstofacilitatedecentralizedcontroller

design.ForeachcontrolpairofanMIMOprocess,aType-1/Type-2ETSMcanbebuiltbymergingthesteadyandthedynamicinteracting

effects,quantiﬁedbyRNGAbasedcriterion,intoitsindividualopen-loopType-1/Type-2 T-Sfuzzymodelthroughsimplyscalingthe

coefﬁcients.BasedoneachETSM,alocalcontrollercanbeindependentlydevisedbyusinglinearSISOcontrolalgorithms,andthena

decentralizedcontrolsystemcanbeformedbyassemblingalltheselocalcontrollerstomanipulateanonlinearMIMOprocess.Compared

totheexistingdecentralizedfuzzycontrolmethodsinsertingextratermsintheindividualopen-loopmodelstoexpresstheinteracting

effects,ETSMmethodcangreatlyreducethecostandcomplexityinmodelingandcontrollerdesigns.WhilecomparedtotheexistingRNGA

basedETFmethod,ETSMmethodcanbeimplementedwithoutrequiringexactmathematicalprocessfunctionsandisabletoprovidemore

satisfactorycontrolresults.Type-2ETSMswithadditionaldegreesoffuzzinesscanachievemorerobustperformancesthantheType-1

counterpartsundertheinﬂuenceofuncertainties,whichhavebeenprovedbysimulationandexperimentalresults.SinceanETSMcan

expresstheinteractingeffects,moreinterestingtopics,suchasblockdecentralizedcontrolandsparsecontrol,canbeinvestigated.These

topicswillbereportedlater.

Acknowledgements

TheworkwasfundedbyNationalResearchFoundationofSingapore:NRF2011NRF-CRP001-090.SchoolofElectrical&Electronic

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AppendixA.

TheparametersofType-1andType-2T-SfuzzymodelsfortheprocessinEq.(44)aregiveninTableA1andA2,whereRl_denotes_Rule

TableA1

ThecentersoffuzzyclustersfortheprocessinEq.(44).

CentersofCl

ij’s

inloopyi−uj

No.offuzzyclusters

R1_(l₌₁₎ _R2_(l₌₂₎ _R3_(l₌₃₎ _R4_(l₌₄₎ _R5_(l₌₅₎ _R6_(l₌₆₎ fTS ,11 (l 11=0.05) ul c,111 0.5622 1.3647 0.8080 1.2917 1.3446 0.6003 yl c,11 1.3286 0.9915 1.2043 1.1904 1.5125 0.9530 yl c,12 1.1450 1.1523 1.3147 1.0916 1.3482 1.1075 yl c,1 1.1055 1.2062 1.0991 1.3174 1.6284 0.8485 fTS ,12 (l 12=0.05) ul c,212 1.2909 0.8570 1.3501 0.5789 0.6350 1.2364 yl c,11 1.0727 0.9838 0.9385 1.0686 0.9503 1.0267 yl c,12 1.0145 1.0085 0.9997 1.0040 1.0149 0.9979 yl c,1 1.1222 0.9727 1.0153 0.9847 0.8888 1.0575 fTS ,13 (l 13=0.05) ul c,313 0.6910 0.6976 1.2953 1.2328 1.3997 0.6252 yl c,11 1.1078 1.0586 1.0414 1.0761 1.0703 1.0272 yl c,12 1.0911 1.0471 1.0593 1.0804 1.0460 1.0584 yl c,1 1.0822 1.0196 1.0782 1.0774 1.1201 1.0064 fTS ,21 (l 21=0.05) ul c,121 1.1255 0.9491 0.5658 1.4285 0.5358 1.3894 yl c,21 2.0363 2.1622 2.1112 2.0109 1.9872 2.1456 yl c,22 1.9934 2.2249 2.0555 2.0733 2.0470 2.0585 yl c,2 2.0275 2.1363 2.0055 2.1058 1.9364 2.2626 fTS ,22 (l 22=0.05) ul c,222 1.1127 0.8016 0.5747 1.4044 1.2765 0.7716 yl c,21 0.5326 0.4769 0.5144 0.5023 0.4934 0.5007 yl c,22 0.5303 0.4824 0.5160 0.4970 0.5015 0.4931 yl c,2 0.5327 0.4761 0.5152 0.5022 0.5039 0.4902 fTS ,23 (l 23=0.05) ul c,323 1.2047 0.7098 1.3109 0.6307 0.8037 1.2922 yl c,21 0.3152 0.2388 0.2396 0.3261 0.2656 0.3328 yl c,22 0.2842 0.2888 0.2839 0.2673 0.2936 0.2966 yl c,2 0.3275 0.2264 0.3076 0.2541 0.2461 0.3571 fTS ,31 (l 31=0.05) ul c,131 1.3618 0.6121 0.7111 1.2980 1.3706 0.6179 yl c,31 0.2010 0.2782 0.2186 0.2335 0.2980 0.2037 yl c,32 0.2405 0.2419 0.2171 0.2216 0.2627 0.2448 yl c,3 0.2425 0.2354 0.1998 0.2574 0.3258 0.1773 fTS ,32 (l 32=0.05) ul c,2 32 1.3431 0.6205 1.2799 0.6305 1.2034 0.8962 yl c,31 0.6055 0.7323 0.7242 0.6044 0.7122 0.6536 yl c,32 0.6633 0.6627 0.6880 0.6667 0.6706 0.6782 yl c,3 0.6858 0.6514 0.7517 0.5572 0.7462 0.6429 fTS ,33 (l 33=0.03) ul c,333 0.8772 1.2857 0.6684 1.2590 0.5560 1.3119 yl c,31 0.1077 0.1117 0.1109 0.1092 0.1096 0.1102 yl c,32 0.1080 0.1111 0.1090 0.1113 0.1105 0.1093 yl c,3 0.1082 0.1130 0.1086 0.1100 0.1109 0.1086

l,Mij’s(i,j=1,2,3)arechosenas6.

When ˆfTS,ijofcontrolpairyi−ujisaType-1ETSMwithp=0andq=2,itis:

Rule l: IF xij(k) is C_ijl

THEN yl_i(k)= ˆal

ij,0·uj(k− ˆij)+blij,1·yi(k−1)+blij,2·yi(k−2)

(A.1)

Applyinggainandphasemarginsbasedcontrolalgorithm[15,17]onthelinearpolynomialoflthfuzzyrulecancalculateacontrol

variableul j(k)by[19]: ul_j(k)=uj(k−1)+ (r

v

i(k)−yi(k))− blij,1(r

v

i(k−1)−yi(k−1))− blij,2(r

v

i(k−2)−yi(k−2)) 2ˆal ij,0Am,ijˆij (A.2)

whereAm,ijisthegainmarginforthecontrolsystem.Accordingtotherequirement,Am,ij=3,whichassociatedwithaphasemarginof /3

[15,17].Accordingtoparalleldistributedcompensation[4],thetotalcontrolvariableuj(k)isaweightedsumofulj(k)(l=1,...,Mij)andshare

thesamefuzzymembershipswithEq.(A.1):

uj(k)=

Mij l=1 l ij

xij(k)

ul_j(k)

(16)

TableA2

TheparametersofType-1andType-2T-SfuzzymodelsfortheprocessinEq.(44).

No.offuzzyrules Type–1fuzzymodel Type–2fuzzymodel

al

ij,0 blij,1 blij,2 alij,lb,0 bij,lb,1l blij,lb,2 alij,rb,0 blij,rb,1 blij,rb,2

fTS ,11 R1_(l₌₁₎ _0.4168 _0.6234 _0.0503 _0.4362 _0.6849 _0.0205 _0.3974 _0.5620 _0.0801 R2_(l₌₂₎ _0.3746 _0.8545 _−0.1103 _0.4063 _0.8622 _−0.1055 _0.3429 _0.8468 _−0.1150 R3_(l₌₃₎ _0.3863 _0.8449 _−0.1765 _0.4012 _0.8446 _−0.1448 _0.3714 _0.8452 _−0.2082 R4_(l₌₄₎ _0.3621 _0.7834 _−0.1002 _0.3859 _0.8059 _−0.0979 _0.3383 _0.7608 _−0.1025 R5_(l₌₅₎ _0.4159 _0.7726 _−0.0442 _0.4317 _0.8055 _−0.0558 _0.4001 _0.7396 _−0.0327 R6_(l₌₆₎ _0.3931 _0.7623 _−0.0971 _0.4217 _0.7919 _−0.0774 _0.3646 _0.7327 _−0.1169 fTS ,12 R1_(l₌₁₎ _0.1853 _0.7450 _0.0959 _0.1948 _0.7958 _0.0751 _0.1758 _0.6943 _0.1168 R2_(l₌₂₎ _0.1745 _0.6941 _0.1611 _0.1825 _0.7393 _0.1618 _0.1665 _0.6489 _0.1604 R3_(l₌₃₎ _0.1845 _0.7492 _0.0625 _0.1997 _0.7782 _0.0625 _0.1693 _0.7202 _0.0626 R4_(l₌₄₎ _0.1831 _0.7475 _0.0791 _0.1858 _0.8249 _0.0418 _0.1805 _0.6702 _0.1164 R5_(l₌₅₎ _0.1519 _0.6249 _0.1768 _0.1594 _0.6537 _0.1946 _0.1443 _0.5960 _0.1590 R6_(l₌₆₎ _0.1822 _0.6979 _0.0894 _0.1916 _0.7502 _0.0701 _0.1727 _0.6456 _0.1086 fTS ,13 R1_(l₌₁₎ _0.0980 _0.7854 _0.1420 _0.0991 _0.8415 _0.1272 _0.0969 _0.7293 _0.1569 R2_(l₌₂₎ _0.0931 _0.8095 _0.0692 _0.0971 _0.8475 _0.0773 _0.0891 _0.7716 _0.0610 R3_(l₌₃₎ _0.0946 _0.8083 _0.1186 _0.1047 _0.8468 _0.1152 _0.0845 _0.7699 _0.1219 R4_(l₌₄₎ _0.0919 _0.7504 _0.1254 _0.0987 _0.7647 _0.1474 _0.0851 _0.7361 _0.1034 R5_(l₌₅₎ _0.1062 _0.6681 _0.2587 _0.1147 _0.7141 _0.2457 _0.0978 _0.6222 _0.2717 R6_(l₌₆₎ _0.0956 _0.8711 _0.0580 _0.1010 _0.9326 _0.0447 _0.0902 _0.8096 _0.0712 fTS ,21 R1_(l₌₁₎ _0.1881 _1.0126 _−0.1523 _0.2011 _1.0316 _−0.1280 _0.1751 _0.9935 _−0.1766 R2_(l₌₂₎ _0.2028 _0.8467 _0.0421 _0.2154 _0.8228 _0.0992 _0.1903 _0.8705 _−0.0151 R3_(l₌₃₎ _0.2350 _1.0616 _−0.1896 _0.2459 _1.0841 _−0.1675 _0.2242 _1.0391 _−0.2116 R4_(l₌₄₎ _0.2127 _0.8061 _0.0828 _0.2404 _0.8094 _0.1061 _0.1851 _0.8028 _0.0595 R5_(l₌₅₎ _0.2605 _1.0580 _−0.1312 _0.2805 _1.0763 _−0.1038 _0.2406 _1.0397 _−0.1586 R6_(l₌₆₎ _0.1848 _0.9333 _0.0234 _0.2087 _0.9642 _0.0203 _0.1608 _0.9024 _0.0265 fTS ,22 R1_(l₌₁₎ _0.0254 _1.1676 _−0.2225 _0.0262 _1.2035 _−0.2183 _0.0245 _1.1316 _−0.2267 R2_(l₌₂₎ _0.0159 _1.2794 _−0.2973 _0.0166 _1.3471 _−0.3081 _0.0151 _1.2116 _−0.2864 R3_(l₌₃₎ _0.0326 _1.1339 _−0.1470 _0.0336 _1.1633 _−0.1308 _0.0316 _1.1045 _−0.1632 R4_(l₌₄₎ _0.0242 _1.3626 _−0.4637 _0.0299 _1.4041 _−0.4739 _0.0185 _1.3212 _−0.4534 R5_(l₌₅₎ _0.0173 _1.4796 _−0.4639 _0.0218 _1.5131 _−0.4600 _0.0128 _1.4460 _−0.4678 R6_(l₌₆₎ _0.0140 _1.4328 _−0.5218 _0.0169 _1.4874 _−0.5301 _0.0112 _1.3782 _−0.5134 fTS ,23 R1_(l₌₁₎ _0.1431 _0.4784 _0.0008 _0.1470 _0.5093 _−0.0062 _0.1393 _0.4475 _0.0078 R2_(l₌₂₎ _0.1472 _0.5180 _0.0102 _0.1520 _0.5509 _0.0226 _0.1424 _0.4850 _−0.0023 R3_(l₌₃₎ _0.1460 _0.4818 _−0.0028 _0.1525 _0.4978 _−0.0001 _0.1395 _0.4657 _−0.0055 R4_(l₌₄₎ _0.1448 _0.4781 _0.0295 _0.1475 _0.5224 _0.0189 _0.1422 _0.4337 _0.0400 R5_(l₌₅₎ _0.1413 _0.4808 _−0.0026 _0.1466 _0.5059 _0.0115 _0.1360 _0.4557 _−0.0167 R6_(l₌₆₎ _0.1495 _0.5094 _−0.0118 _0.1539 _0.5300 _−0.0112 _0.1451 _0.4889 _−0.0124 fTS ,31 R1_(l₌₁₎ _0.0788 _0.8838 _−0.1595 _0.0849 _0.8820 _−0.1458 _0.0727 _0.8857 _−0.1731 R2_(l₌₂₎ _0.0852 _0.6130 _0.0631 _0.0864 _0.6812 _0.0276 _0.0839 _0.5447 _0.0985 R3_(l₌₃₎ _0.0823 _0.6856 _−0.0409 _0.0855 _0.7638 _−0.0673 _0.0792 _0.6073 _−0.0145 R4_(l₌₄₎ _0.0747 _0.7549 _−0.0992 _0.0796 _0.7771 _−0.0971 _0.0698 _0.7328 _−0.1014 R5_(l₌₅₎ _0.0802 _0.7467 _−0.0008 _0.0841 _0.7766 _−0.0119 _0.0763 _0.7168 _0.0103 R6_(l₌₆₎ _0.0818 _0.8178 _−0.1564 _0.0864 _0.7850 _−0.0949 _0.0773 _0.8506 _−0.2180 fTS ,32 R1_(l₌₁₎ _0.1778 _0.6767 _0.0550 _0.1896 _0.7009 _0.0570 _0.1661 _0.6526 _0.0530 R2_(l₌₂₎ _0.1789 _0.6433 _0.1093 _0.1830 _0.7056 _0.0848 _0.1748 _0.5811 _0.1338 R3_(l₌₃₎ _0.1711 _0.6716 _0.0518 _0.1784 _0.7101 _0.0411 _0.1639 _0.6331 _0.0624 R4_(l₌₄₎ _0.1670 _0.6305 _0.0966 _0.1762 _0.6755 _0.1000 _0.1579 _0.5854 _0.0932 R5_(l₌₅₎ _0.1775 _0.7149 _0.0543 _0.1857 _0.7500 _0.0474 _0.1693 _0.6797 _0.0613 R6_(l₌₆₎ _0.1799 _0.6413 _0.0997 _0.1944 _0.6931 _0.0796 _0.1654 _0.5895 _0.1198 fTS ,33 R1_(l₌₁₎ _0.0024 _1.0770 _−0.0830 _0.0029 _1.1021 _−0.0621 _0.0019 _1.0520 _−0.1040 R2_(l₌₂₎ _0.0038 _0.6604 _0.3255 _0.0045 _0.6959 _0.3242 _0.0030 _0.6248 _0.3268 R3_(l₌₃₎ _0.0005 _1.0561 _−0.1007 _0.0007 _1.0956 _−0.0965 _0.0003 _1.0166 _−0.1050 R4_(l₌₄₎ _0.0035 _0.6166 _0.3367 _0.0041 _0.5863 _0.4023 _0.0029 _0.6469 _0.2711 R5_(l₌₅₎ _0.0027 _0.9986 _0.0155 _0.0029 _1.0034 _0.0547 _0.0025 _0.9938 _−0.0236 R6_(l₌₆₎ _0.0029 _0.8095 _0.1177 _0.0035 _0.8531 _0.1108 _0.0023 _0.7660 _0.1246

When ˆfTS,ijisaType-2T-Sfuzzymodelwithp=0andq=2,itis:

Rule l: IF xij(k) is C˜l_ij

THEN ˜yl

i(k)= ˆ˜a

l

ij,0·uj(k− ˆij)+ ˜bl_ij,1·yi(k−1)+···+ ˜bl_ij,2·yi(k−2)

(A.3) where ˜yl i(k)=[ y l i,lb(k), y l i,rb(k) ]that {y l

i,lb(k)= ˆalij,lb,0·uj(k− ˆij)+bij,lb,1l yi(k−1)+blij,lb,2·yi(k−2)

yl

i,rb(k)= ˆalij,rb,0·uj(k− ˆij)+bij,rb,1l yi(k−1)+blij,rb,2·yi(k−2)

(A.4)

BasedonthetwolinearpolynomialsinEq.(A.4),twocontrolvariables,denotedbyul

j,lb(k)andulj.rb(k),canbecalculatedusingthegain

(17)

˜uj(k)=[ uj,lb(k), uj,rb(k) ]=

Mij l=1lij,lb

xij(k)

ul j,lb(k)

Mij l=1lij,lb

xij(k)

,

Mij l=1lij,rb

xij(k)

ul j,rb(k)

Mii l=1lij,rb

xij(k)

(A.5)

Thetotalcontrolvariableuj(k)isderivedbydefuzzifying ˜uj(k)[25]as

uj(k)=

uj,lb(k)+uj,rb(k)

2 (A.6)

LinearizetheprocessinEq.(44)atthegivenoperatingpointstohavethefollowingtransferfunctionmatrix:

G(s)=

gij(s)

n×n=

⎡

⎢

⎣

1.25 0.25s+1e −2s 1 0.5s+1e −2s 1 s+1e −s 2 s+1e −2s 0.5 0.1667s2₊_0.8333s₊₁e−2s 0.2857 0.1429s+1e −s 0.25 0.25s+1e−2s 0.6667 0.3333s+1e−2s 0.1 0.5s2₊_1.5s₊₁e−s

⎤

⎥

⎦

LooppairingstructureselectedusingRNGAbasedcriterionisy1−u3/y2−u1/y3−u2.TheETFs ˆgij(s)’sandthecontrollersGc,i(s)(i=1,2,

3)designedwiththerequiredgainandphasemarginswhichare3and /3are:

Loopy1−u3: ˆg13(s)=_1.2756s+11 e−1.2756s,andGc,1(s)=0.5236s+0.4105s .

Loopy2−u1: ˆg21(s)=1.3462s2 +1e−2.6924s,andGc,2(s)=0.1309s+0.0972s .

Loopy3−u2: ˆg32(s)=_0.3333s+10.6667 e−2,andGc,3(s)= 0.1309ss+0.3927.

AppendixB.

TheparametersofType-1andType-2T-Sfuzzymodelsforthemulti-evaporatorrefrigerationsystemaregiveninTableB1andB2, whereRl_denotes_Rule_l,_M

ij=6,i,j=1,2,3).

TableB1

Thecentersoffuzzyclustersforthemulti-evaporatorrefrigerationsystem.

CentersofCl

ij’sinloopyi−uj No.offuzzyclusters

R1_(l₌₁₎ _R2_(l₌₂₎ _R3_(l₌₃₎ _R4_(l₌₄₎ _R5_(l₌₅₎ _R6_(l₌₆₎ fTS ,11 (l 11=0.05) ul c,111 −0.0895 −0.0753 −0.0958 −0.0379 −0.0614 −0.1178 yl c,11 0.1123 0.1214 0.0602 0.2745 0.2665 0.0027 yl c,12 0.0469 0.2542 0.1135 0.1654 0.0066 0.2075 yl c,1 0.1584 0.1340 0.1632 0.0865 0.1211 0.1965 fTS ,12 (l 12=0.05) ul c,212 −1.2258 0.4658 −0.6004 0.7367 −0.6717 0.8497 yl c,11 −0.5747 0.2136 0.2547 −0.1123 0.3494 −0.4642 yl c,12 −0.0492 −0.3563 0.1002 0.2441 0.1283 −0.3066 yl c,1 −0.6855 0.2591 −0.1491 0.2547 −0.1602 0.1935 fTS ,13 (l 13=0.03) ul c,313 −0.7445 0.8525 −1.0112 0.3896 0.8216 −0.8990 yl c,11 0.1154 −0.1251 −0.2065 0.0716 0.0239 0.0259 yl c,12 0.0342 0.0011 −0.0611 −0.1126 0.0145 0.0301 yl c,1 −0.0589 0.0698 −0.2045 0.0768 0.1141 −0.1077 fTS ,21 (l 21=0.05) ul c,121 −0.1187 −0.0572 −0.1087 −0.0453 −0.0830 −0.0632 yl c,21 −0.0307 −0.1017 −0.0422 −0.0838 −0.0611 −0.0490 yl c,22 −0.0365 −0.0673 −0.0680 −0.0661 −0.1000 −0.0122 yl c,2 −0.0823 ‘−0.0565 −0.0786 −0.0449 −0.0653 −0.0502 fTS ,22 (l 22=0.05) ul c,222 −0.0990 −0.0686 −0.1687 0.0210 −0.0938 −0.0671 yl c,21 0.0301 0.1051 0.0699 0.0324 0.0323 0.0510 yl c,22 0.0287 0.0783 0.0851 −0.0026 0.0421 0.0729 yl c,2 0.0615 0.0588 0.1090 −0.0065 0.0589 0.0477 fTS ,23 (l 23=0.03) ul c,323 0.0016 −0.1283 −0.1004 −0.0738 −0.1648 −0.0101 yl c,21 −0.0108 −0.0053 −0.0030 −0.0115 −0.0073 −0.0075 yl c,22 −0.0086 −0.0086 −0.0094 −0.0052 −0.0033 −0.0083 yl c,2 −0.0034 −0.0103 −0.0078 −0.0087 −0.0136 −0.0030 fTS ,31 (l 31=0.03) ul c,131 −0.0210 −0.1501 −0.1141 −0.0507 −0.0651 −0.0758 yl c,31 −0.0058 −0.0259 −0.0092 −0.0208 −0.0268 −0.0039 yl c,32 −0.0156 −0.0110 −0.0052 −0.0257 −0.0141 −0.0163 yl c,3 −0.0046 −0.0291 −0.0193 −0.0130 −0.0167 −0.0123 fTS ,32 (l 32=0.05) ul c,232 −0.0602 −0.1107 −0.0780 −0.1081 −0.0713 −0.0484 yl c,31 −0.0037 −0.0127 −0.0134 −0.0020 −0.0092 −0.0052 yl c,32 −0.0050 −0.0064 −0.0096 −0.0064 −0.0082 −0.0083 yl c,3 −0.0053 −0.0116 −0.0094 −0.0083 −0.0078 −0.0050 fTS ,33 (l 33=0.05) ul c,333 0.4512 0.5290 0.1056 −1.2862 −0.9391 0.8002 yl c,31 −0.1045 0.1306 −0.1123 0.1961 −0.1869 0.2330 yl c,32 −0.0183 −0.1162 0.0914 0.0075 −0.0369 0.2063 yl c,3 −0.1251 −0.0543 −0.0580 0.3157 0.1087 −0.0659