Accounting for dynamics in self-optimizing control

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Journal of Process Control

jo u r n al h om ep ag e :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

Accounting for dynamics in self-optimizing control

Jonatan Ralf Axel Klemets

^∗

, Morten Hovd

DepartmentofEngineeringCybernetics,NorwegianUniversityofScienceandTechnology,Trondheim,Norway

a r t i c l e i n f o

Articlehistory:

Received13July2018

Receivedinrevisedform3January2019 Accepted13January2019

Availableonline19February2019

Keywords:

Self-optimizingcontrol LMI

Controlstructuredesign Staticoutputfeedbackcontrol

a b s t r a c t

Self-optimizingcontrolfocusesonminimizingthesteady-statelossforprocessesinthepresenceofdis- turbancesbyholdingselectedcontrolledvariablesatconstantset-points.Thelosscanfurtherbereduced bycontrollinglinearmeasurementcombinationsthathavebeenobtainedwiththepurposeofminimiz- ingeithertheworst-caselossortheaverageloss.Sinceself-optimizingcontrolmainlyfocusesonthe steady-stateoperation,littleemphasishasbeenputonthedynamicbehaviouroftheresultingclosed- loopsystem.Thegeneralapproachistofirstcomputetheoptimalcontrolledvariablesandthendesign theirrespectivecontrollers.However,theoptimalmeasurementcombinations,canoften(especiallyif manymeasurementsareused)resultinverydynamicallycomplexsystems,thatmakesdesigningthe feedbackcontrollersdifficult.Inthiswork,PIcontrollersandmeasurementcombinationsaresimultane- ouslyobtainedwiththeaimtofindanoptimaltrade-offbetweenminimizingthesteady-statelossand thetransientresponsefortheresultingclosed-loopsystem.Asolutioncanbefoundbysolvingabilinear matrixinequality(BMI),whichbecomesalinearmatrixinequality(LMI)byspecifyingastabilizingstate feedbackgain.Theoptimizationproblemcanalsobecombinedwiththesparsitypromotingweighted l1-norm,whichpenalizesthenumbermeasurementsusedandthus,attemptstofindanoptimalmea- surementsubset.TheproposedmethodrequiressolvingaBMI,forwhichaniterativeLMIapproachcan beusedtofindalocaloptimum,whichoftenseemstogivegoodresults,asillustratedontwocasestudies, consistingofabinaryandaKaibeldistillationcolumn.

©2019TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Theever-increasingcompetitivepressureintheglobalmarkets resultsintheneedforcontinuouslyimprovingtheperformance ofchemicalprocesses.Operatingtheprocessclosetoitseconomi- callyoptimaloperatingpointisthusessential.Thisleadstohigher demands on thecontrol system, which has to ensurethat the plantiskeptclosetothedesiredoperatingpoint.Iftherearelarge deviations fromtheoptimaloperation,caused by,e.g., external disturbances,itwould evidentlyresultinaneconomic lossand couldviolatesomeoftheoperatingconstraints.Therefore,boththe steady-state(economic)objectiveandthedynamicperformance oftheprocessshouldbeconsideredwhendesigningthecontrol system.

Chemicalprocessplantsaretypicallyoperatedwiththeaidof a multilayerhierarchicalcontrolstructure, consistingof several layersthataddressdifferenttimescales[1,2].Traditionally,theeco- nomicoptimizationandthedynamiccontrolofchemicalprocesses

∗Correspondingauthor.

E-mailaddresses:[email protected](J.R.A.Klemets), [email protected](M.Hovd).

areseparatedandoperateatdifferentlayers.Theeconomicopti- mizationisusuallylocatedinanupperlayerandusesreal-time optimization(RTO)[3]tocomputeandsendtheoptimalset-points tothelowerlayers.Theroleofthelowerlayeristodrivetheprocess tothedesiredset-pointusing,e.g.,modelpredictivecontrol(MPC) orotherlow-levelcontrollers(typicallyPIDControllers).Recently therehasbeenanincreasinginterestineconomicmodelpredic- tivecontrol(EMPC)[4],whichattemptstointegratetheeconomic optimizationandprocesscontrolperformancetogether.Despite therecentadvancements,therearestillchallengeswhenitcomes toimplementationinrealprocesses,mainlyduetothecomputa- tionalcomplexityandrequirementforaccuratedynamicmodelsof theprocess.

Anotherapproachistousesimplecontrolstructuresthatkeep specificcontrolled variables ata constant value,alsoknownas self-optimizing control [5]. The central idea of self-optimizing control is to select controlled variables such that in the pres- enceofdisturbances,thelossisminimizedbyholdingthechosen controlledvariables at constant set-points.Beyondusing single measurements,selectinglinearcombinationsofmeasurementsas controlledvariableswillfurtherimprovetheself-optimizingcon- trolperformance.Twomethodsthatachievethisaretheexactlocal

https://doi.org/10.1016/j.jprocont.2019.01.003

0959-1524/©2019TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).

method[6]andthenull-spacemethod[7].Mostresearchonself- optimizingcontrol(see, e.g.,thesurveypaper by[8])ismainly concernedwiththesteady-stateoperationwithoutconsideringthe dynamicbehaviouroftheresultingclosed-loopsystem.Thiscan leadtodynamicallycomplexsystemswithe.g.,righthand-plane zerosatlowfrequenciesthatposeslimitationsontheachievable controlperformance.Therefore,itwouldbepreferabletofinda combinationthatalsotakesthedynamicbehaviourintoaccount.

However,theresultingclosed-loopsystemisnotjustdependent onthemeasurementcombination,butalsoonthefeedbackcon- trollers.

Theproportionalintegral(PI)controllerisbyfarthemostcom- monlyusedcontrollerintheprocessindustriesduetoitssimplicity and robustperformance [9].With progressin numerical meth- ods,newconvexoptimizationmethodshavebeendevelopedfor designing controllers. However, for restricted-order controllers (e.g.,PI/PIDcontroller)theoptimizationproblemstendtobecome non-convexinthecontroller parameterspace.Theyareusually solvedbyemployingheuristicsorintelligentmethods[10,11].A loopshapingmethodwasproposedin[12],byspecifyingbounds onthephaseandgainmargins.

Thesemethodsoftenaimtominimizesomecommoncontrol performance criterion, e.g., the integrated absolute error (IAE).

However,in self-optimizing control(SOC) minimizing IAE may not beideal.Typically, the SOCvariables are controlled bythe remainingdegreesoffreedom,oncetheplantfirsthasbeensta- bilizedandalltheactiveconstraintsarebeingcontrolled.Although itis economicallyoptimaltooperatetheplantas closeaspos- sibletoitsactive constraints,it is usuallynecessary toemploy some “back off” to avoid dynamic and steady-state problems.

“Back off”is the differencebetween theoptimal set-point and theactualset-point,andisestimatedbasedontheinformationof thedisturbances andtheexpectedcontrolperformance[13,14].

Therefore,the SOCvariables shouldpreferably,when subjected todisturbances,drivetheprocesstothenewoptimaloperating pointwhileminimizingdeviations intheactiveconstraints(i.e., reducingthe“backoff”)orinothervariableswithlargeeconomic impact.

Instead,ofminimizingIAE,itmightbebettertofindtheSOC variablesbyrecastingitasanoptimizationproblemforfinding,e.g., theH2 ortheH_∞optimalstaticoutputfeedback(SOF)controller thatminimizesthedeviationsinaspecifiedperformanceoutput.

Contrarytofullstate-feedbackorfull-ordercontrollers,whichcan besolvedusingLinearMatrixInequalities(LMI),structuredstatic outputfeedbackgenerally resultsinBilinear MatrixInequalities (BMI)andremainsanopenproblem[15,16].Theyareoftensolved toalocaloptimumbyiterativelyfixingsomevariablesandsolving theresultingLMI.

The optimal measurement combination has been shown to benon-unique, and ifmultiplied withanynon-singular matrix it results in the same steady-state loss. Based on [17,18], an iterative LMI algorithm was proposed in [19] for simultane- ously determining this non-singular matrix and PI controller parameterstoimprovethedynamicperformance. However,the optimal self-optimizing control variable is only non-unique if thedegreesoffreedom availablearegreater thanone.Further- more,while[19]maintainstheoptimalsteady-statesolution,it maybebeneficialtochooseameasurementcombinationwitha largersteady-statelossifitwouldprovideasignificantimprove- ment in the closed-loop performance. That is, if the dynamic improvementsresulted in betterdisturbancerejection,it could allow for a reduction in the “back off” applied to the active constraintsandasaconsequencefurtherincreasetheprofitabil- ity.

Therefore,inthiswork,aniterativeLMIalgorithmisproposed thatsolvesaParetooptimizationproblemthatgivesatrade-off

betweenminimizingthesteady-statelossandthedynamicperfor- mance.Theproposedmethodcanthenbeexpandedonasin[20],to includeanadditionalpenaltyfunctioninthemulti-objectiveopti- mizationproblemthatpromotessparsitybypenalizingthenumber ofmeasurementsused.Thesparsitypromotingfunctionisknown astheweightedl₁-norm[21],andhasbeenusedinseveralpapers forpromotingsparsityincontrollerdesign,see,e.g.,[22–24].The proposedmethodisvalidatedbyapplicationstomodelsofabinary andaKaibeldistillationcolumn.

Thepaperisorganizedasfollows.Section2introducesthenota- tionusedinthispaperwhiletheconceptofself-optimizingcontrol isdescribedinSection3.ThemaincontributionispresentedinSec- tion4,wheretheoptimizationproblemoffindingameasurement combinationtogetherwithPIcontrollersisformulatedasaBMI.

Theoptimizationproblemissolvedtoalocaloptimumusingan iterativeLMIalgorithm.InSection5,resultsfromsimulationsare presented,wheretheproposedmethodhasbeenusedtodesign thecontrolstructuresfortwodifferentdistillationcolumnmodels.

Finally,aconclusionisgiveninSection6.

2. Preliminaries

LetR^n×mdenotethesetofn×mrealmatrices.ForamatrixA, itstransposeisdenotedA^T,andA⁻¹denotesitsinverse.Theiden- tityandthenullmatrixofsuitabledimensionisgivenbyIand0.

ThenotationA≺0,A0meansthematrixispositiveandnegative definiterespectively.·1,·2,·∞,and·Frepresentsthel1, H₂,H_∞,andFrobeniusnorms,respectively.

3. Self-optimizingcontrol

Self-optimizingcontrolisachievedwhenanacceptablelossis obtainedwithconstantset-pointswithouttheneedtoreoptimize when(changesin)disturbancesoccur[5].Moreprecisely,theaim istoselectcontrolledvariablesratherthandeterminingoptimal set-points.Here,thelossLisdefinedasthedifferencebetweenthe actualvalueofagivencostfunctionandthetrulyoptimalvalue (accountingforthecorrectvalueofthedisturbance),i.e.,

L(u,d)=J(u,d)−Jopt(d), (1) wheretrulyoptimaloperationisachievedwhenL=0.However, ingeneral,L≥0andthusasmallervalueforthelossfunction,L impliesthattheplantisoperatingclosertoitsoptimum.Byusing theavailabledegreesoffreedom(u),thegoalistominimizethe constrainedcostfunction(J),inordertofindtheoptimaloperating pointfortheprocess.Typically,Jdefinestheeconomiccostofthe processandcanoftenbeexpressedas

J=feedcost+utilitiescost−productvalue.

However,otherobjectivessuchasenergyefficiencyandindirect control[25]arealsopossible.

Forspecifieddisturbances(d),theoptimizationproblemcanbe formulatedas,

minx,u J(x,u,d) (2)

s.t.f(x,u,d)=0 (3)

g(x,u,d)≤0 (4)

y=fy(x,u,d) (5)

wherex ∈R^nx,u ∈R^nu,andd∈R^nd arethestates,inputs, and disturbancesrespectively.Theequalityconstraintsarerepresented byf(·)andcontainthesteady-statemodelequations;theinequal- ityconstraintsing(·)definetheconstraintsontheoperation,and theavailablemeasurementsaregiven byy. Thesolutiontothe

Fig.1. Blockdiagramoftheself-optimizingcontrolstructure.

optimizationproblemusuallyresultsinsomeoftheconstraints beingactive, i.e.,g_i(x,u, d)=0. Toachieve optimaloperationat steady-state,thevariablesrelatedtotheactiveconstraintsshould becontrolledandkeptascloseaspossibletotheiroptimalset- points.Stabilizingtheplantandcontrollingtheactiveconstraints, therefore,requiresacorrespondingnumberofdegreesoffreedom.

Undertheassumptionthattheactiveconstraintsremainthesame duringoperation,thenitresultsinthereducedspaceoptimization problem:

minu J^∗(u,d). (6)

Here,themodelequations andactiveconstraints,areimplicitly included in J^*. Whatremains is to determinewhich of uncon- strained controlled variables (y and u) that should be kept at constantset-pointbyusingtheremainingdegreesoffreedom(u), inordertominimizeloss.Toquantifythelossresultingfromkeep- ingtheselectedcontrolledvariablesatconstantvalues,methodsfor calculatingtheworstcaseandaveragelosswerederivedin[6,26]

respectively.

3.1. Optimalmeasurementcombination

Rather than selecting single measurements for the uncon- strainedoptimizationproblemin(6),afurtherreductioninloss canbeobtainedbyselectingthecontrolvariablesasoptimallinear measurementcombinationsc=Hy,resultinginthecontrolstruc- tureseeninFig.1.ThematrixH ∈R^nu×nydefinesthemeasurement combinations,and y∈R^ny isasubsetoftheavailablemeasure- ments.

3.1.1. Theexactlocalmethod

WiththeexpectationoperatordenotedE[·],andassumingthe disturbancesdandmeasurementnoisenareindependentanduni- formlydistributedinthesetsd∈D,andn ∈N.Then,theworst caseandaveragelosswerederivedin[6,26]respectivelyandare givenby

Lworst=max_d∈D,n∈NL= 1

2J_uu^1/2(HG^y)⁻¹HY²2, (7) Lavg= E

d∈D,n∈N[L]=1

2J_uu^1/2(HG^y)⁻¹HY²_F. (8) Here,Y:=

FW_d Wn

,withW_dandWnrepresentingtheexpected magnitudesofthedisturbancesandimplementationerrorsrespec- tively.F=^∂y_∂^opt_d isthesensitivitymatrixfortheoptimaldeviations inthemeasurements(∂y^opt)withrespecttochangesinthedis- turbances(∂d);Juu=_∂^∂_u²2^J denotesthesecondderivativeofthecost function(6),andG^y=^∂_∂^y_u,representsthegainfromtheinputstothe

availablemeasurements.Theauthorsof[26]provedthatobtain- ingtheHthatminimizestheaveragelossin(8)issuper-optimal andhence,thesameHalsominimizestheworstcaselossin(7).

However,theoppositeisnotnecessarilytrue.Therefore,onlythe minimization of theFrobenius norm willbe consideredin this paper,wheretheoptimizationproblemcanbeformulatedas min

H

1

2J_uu^1/2(HG^y)⁻¹HY²F. (9) Atfirstglance,thisseemslikeanon-linearoptimizationproblem.

However,animportantobservationwasdiscoveredin[27],which foundthat(9)canberecastasaconvexoptimizationproblem.

Theorem1. IfHisafullmatrix (withnostructuralconstraints) thentheproblemin(9)canbeformulatedasaconvexconstrained optimizationproblem[27]:

min

H

1

2HY²_F (10)

s.t.HG^y=J^1/2_uu (11)

Proof. Fromtheoriginalproblemin (9),it canbeshownthat theoptimalsolutionforHisnon-uniqueandforanynon-singular matrixQ ∈R^nu×nu,

Hˆ =Q⁻¹H (12)

resultsinthesameloss.Thiscanbeshownby[8]:

Lavg =1

2J_uu^1/2( ˆHG^y)⁻¹HYˆ ²F

=1

2J_uu^1/2(Q⁻¹HG^y)⁻¹Q⁻¹HY²_F

=1

2J_uu^1/2(HG^y)⁻¹QQ⁻¹HY²F

=1

2J_uu^1/2(HG^y)⁻¹HY²_F.

(13)

Thenon-uniquenessofHcanbeusedtoaddtheconstraintin(11), whichguaranteesthatthefirstpartin(9)becomesJ^1/2_uu (HG^y)⁻¹=I.

Hence,thenonlinearoptimizationin(9)canberecastastheconvex optimizationproblemin(10)and(11).

Remark1. Someadditionalinsightwasgivenin[28],whereit wasnotedthatJuuisnotneededforfindingtheoptimalHin(10) and(11).

ThismeansJuucanbereplacedwithanynon-singularmatrixQ andstillgivetheoptimalH.Thismaysimplifythecalculations,as Juucanbedifficulttoobtainnumerically.However,Juuwouldstill berequiredtofindthecorrectnumericalvalueoftheloss.

Remark2. ForameasurementcombinationHwithnu≥2,anon- singularmatrixQcanbechosenasin(12),togetameasurement combinationwithbetterdynamicpropertieswhilestillmaintain- ingthesamesteady-stateloss.

Theoptimalsolutionto(10)providesthemeasurementcom- bination that gives the locally best steady-state performance.

However,theoptimalsolutiondoesnotconsidertheresultingtran- sientresponseandcangiverisetocomplexdynamicbehaviour.

Ifnu≥2,then anon-singularmatrix Qcan beselectedtogeta measurementcombinationwithbetterdynamicbehaviourwith- out affectingsteady-state performance. However,ifnu=1,then Qbecomesascalarandcanonlychangethesteady-stategainof Hˆ withnoeffectonthedynamicbehaviour.Furthermore,evenif nu≥2itmaybebeneficialtosacrificesomesteady-statelossifit

significantlyimprovestheclosed-loopperformance.Therefore,a methodisproposedinSection4thattriestoobtaintheoptimal trade-offbetweenthesteady-stateanddynamicperformance.

3.2. Selectingameasurementsubset

Theloweststeady-statelosscanbeachievedwhenthemeasure- mentcombinationHiscomputedusingallavailablemeasurements.

However,formostpracticalcasesthisisnotdesirableasitleadsto overlycomplexcontrolstructuresandincreasesthelikelihoodof gettingsensorfailures.Besides,often,thereexistsasubsetofthe availablemeasurementsthatcanbeusedwithoutanysignificant reductioninthesteady-stateperformance.

Findingthebestmeasurementsubsetisacombinatorialopti- mizationproblem,andthelosshastobeevaluatedateverypossible measurementcombination.Tosolvethisproblem,[29]developed atailor-madebranchandboundalgorithm.Anotherapproachwas presentedin[28],wherethecombinatorialproblemwasformu- latedusingmixedintegerquadraticprogramming(MIQP)thatcan besolvedusingstandardMIQPsolvers.

Analternativeapproach,forfindingtheoptimalmeasurement subsetistosolveamulti-objectiveoptimizationproblem,thatgives theoptimaltrade-offbetweensteady-statelossandthenumberof measurementsused.Ifacolumn-wisesparsitypromotingfunction isincludedin(10),theoptimizationproblemcanbeformulatedas, J=min

H

1

2HY²F+card(H) (14)

subjectto(11).Byspecifyingascalarvaluefor,therewillbea trade-offbetweenthesteady-statelossandthecardinalityofH, wherethecardinalityofthemeasurementmatrixHisdefined:

card(H):=thenumberofnon−zerocolumnsofH.

Thecardinalityfunctionisnon-convexandnon-smooth,thatstill makestheoptimizationformulationin(14)acombinatorialprob- lem,whichisdifficulttosolve.

Toaddressthisissue,severalconvexrelaxationslikethel1-norm andtheweightedl₁-normhavebeenproposed[21].Byusingthe weightedl₁-norm,thecardinalityfunctioncanbereplacedwith:

card(H)=

i,j

W_i,jH_i,j1 (15)

Theauthorsof[21] notedthatiftheweights W_i,jarechosento beinversely proportionaltothel1-norm,thenthereis anexact correspondencebetweenthel₁-normandthecardinalityfunction.

However,thisrequires aprioriknowledge oftheHmatrix, and therefore,are-weightedschemeneedstobeimplemented,where theweightsareupdatedaftereveryiteration(k)as,

W_i,j^(k+1)= 1

H^(k)_i,j1+ (16)

where1>0ensurestheupdateiswell-defined.

Theweightedl₁-normin(15)promoteselement-wisesparsity.

However,itcaneasilybemodifiedtopromotecolumn(orrow) sparsityas,e.g.,shownin[30]byrevisingitas,

card(H)=

i,j

W_jH_i,j1 (17)

withtheupdaterule:

W_j^(k+1)= 1

iH_i,j^(k)1+ (18)

Foragivenvalueof,theiterativeprocedure,describedinAlgo- rithm1canbeusedtofindasubsetoftheavailablemeasurements.

Algorithm1:

Initialize:obtainH,bysolving(10)s.t.(11)andcomputeW^(k)using(18).

1: FortheobtainedW^(k)solve:

Jk=min

H

1 2HY²_F+

i,j

W_j^(k)H_i,j1

s.t.(11).

2: IfH^(k−1)−H^(k)2<gotostep3,elseupdateW^(k+1)using(18) andrepeatstep1and2.

3: Removethemeasurementsthatcorrespondtothezero columnsinH.

Whiletheconvergencepropertiesforthere-weightedl1-norm arestillnotclearlyunderstood,numericalexperimentshaveshown ittobeaveryefficientmethodforpromotingsparsity,whichalso isdemonstratedinSection5.1.1.

4. Staticoutputfeedbackcontrol

Inself-optimizingcontrol,thefocusmainlyliesontheeconomic steady-statebehaviour.However,intheresultingclosed-loopsys- temshowninFig.1,itcanbeseenthatthedynamicbehaviouris heavilydependentonboththemeasurementcombinationHand thePIcontroller.Thus,whendeterminingH,itwouldbeadvanta- geoustoconsiderboththesteady-stateandthedynamicbehaviour oftheresultingclosed-loopsystem.

In this section, themethodfor simultaneously selectingthe measurementcombinationanddesigningthefeedbackcontrollers is presented. The method is based on the two-step procedure forstatic outputfeedbackcontroller design,which issimilarto [17,18,31–33].Thesemethodsformulatetheoptimizationproblem asa bilinearmatrixinequality(BMI)thatbecomeslinearmatrix inequalities(LMI),ifinitialized withastabilizingstatefeedback controller.

4.1. ProcessmodelandPIcontrollers

Consider a system described by the discrete linear time- invariantstate-spacemodel,

x_k+1=Axx_k+Buu_k+Bww_k (19)

y_k=Cyxx_k+Dyww_k (20)

wherex_k ∈R^nx,y_k ∈R^ny,u_k ∈R^nu,w_k ∈R^nwarethestates,mea- surements,inputs,anddisturbancesrespectively.Theaimistofind ameasurementcombinationmatrix ˆHanddesignafeedbackcon- trollerthatgivesthedesiredtrade-offbetweenthesteady-state anddynamicperformance.Duetotheirpopularityintheindustry, decentralizedPIcontrollerswillbeusedasthefeedbackcontrollers.

Intheiridealform,theyarerepresentedby

u_k=Kp(e_k+K_i

k−1 n=0

en), (21)

whereu_k ∈R^nuande_k ∈R^nurepresentthecontrolvariableandthe errorvalueofthePIcontroller.TheparametersKp ∈R^nu×nu and K_i∈R^nu×nu arediagonalmatrices,representingtheproportional andintegralgainsrespectively.Here,thesamplingtimeisassumed tobeincludedinK_i,i.e.,K_ivariesdependingonthesamplingtime.

Whenameasurementcombination ˆHisused,thentheerrorvalue isdefinede_k:=r_k−Hyˆ _k,wherer_kisthereferenceinput.Setting thereferencevaluetor_k=0(usingdeviationvariables),theerror valuebecomese_k=−Hyˆ _k,andthus,thePIcontrollerin(21)canbe expressedas:

u_k=−Kp( ˆHy_k+K_i

k−1 n=0

Hyˆ n) (22)

Fromtheaboveformulation,itcanbeseenthatthisdescriptionis overparameterized,whereKpcanbeconsideredasasimplescaling valuefor ˆH.Therefore,Kpcanbeselectedtobeanynon-zerovalue, e.g.,settingKp=Igives:

u_k=−( ˆHy_k+K_i

k−1 n=0

Hyˆ n) (23)

Defininganauxiliarystatevectorq_k ∈R^nufortheintegralterm (thesummationterm

k−1

n=0Hyˆ n),thenthestatespacerepresenta- tionofthePIcontrollerin(23)becomes:

u_k q_k

=

−( ˆHy_k+K_iq_k−1)

q_k−1+Hyˆ _k

. (24)

Byintroducingtheaugmentedstatevector ¯x_k ∈R^(nx+nu)

¯ x_k:=

x_k q_k−1

, (25)

theaugmentedmeasurementvector ¯y_k ∈R^(ny+nu)

¯ y_k:=

y_k q_k−1

, (26)

andtheaugmentedcontrolinputvector ¯u_k ∈R^2nu,givenbythe controllaw:

¯ u_k:=−

Hyˆ _k

K_iq_k−1

. (27)

Thentheclosedloopsystemfortheprocessmodelin(19)and(20) withthePIcontrollersin(24)canbegivenbytheaugmentedmodel

¯

x_k+1=A¯xx¯_k+B¯uu¯_k+B¯ww_k (28)

¯

y_k=C¯yxx¯_k+D¯yww_k (29) wheretheaugmentedsystemmatricesare:

A¯x =[Ax 0

0 I], B¯u=[Bu Bu

I 0 ], B¯w=[Bw

0 ], C¯yx =[Cyx 0

0 I

], D¯yw=[Dyw

0 ].

(30)

Usingtheproposedcontrollawfor ¯u_kin(27),whichisequivalent to

¯ u_k=−

Hˆ 0

0 K_i C¯yxx¯_k−

Hˆ 0

0 K_i D¯yww_k, (31)

=−

HCˆ yxx_k+HDˆ yww_k

K_iq_k−1

. (32)

Thenitcaneasilybeshownthattheaugmentedprocessmodelin (28)and(29)becomes

x_k+1 q_k

=

Axx_k−Bu( ˆHy_k+K_iq_k−1)+Bww_k

q_k−1+Hyˆ _k

,

whichcorrespondstoaclosed-loopsystemfor(19)and(20)with thePIcontrollersin(24).

Theproposedaugmentedsystemin(28)and(29)differsfrom howstatespacemodelstypicallyareaugmentedwithPIcontrollers [34,35].E.g.,in(27)thelengthoftheaugmentedcontrolinputvec- tor ¯u_khavebeendoubledbyseparating ˆHy_k,andK_i

_k−1

n=0Hyˆ nfrom eachotherin(23).Thebenefitofthisapproachisthat ˆHandK_iare

Fig.2.Controlconfigurationforstaticoutputfeedbackcontrol.

decoupledinthecontrollaw(31),whichwillbebeneficialwhen computingtheirvaluesinSection4.2.

Itispossiblethatbetterperformancecanbeachievedifmore advancedcontrollerformulationsareusedinsteadofdecentralized PIcontrollers.Themaindifferencewouldbehowthecontroller descriptionisincludedintheformulationoftheaugmentedsys- tem matrices in (30). One important requirement is that the controllersincorporatesomeintegralactiontoensurethattheself- optimizingcontrolcriterionismetatsteady-state.However,the simulationsinSection5.2,andtheworkin[20]seemstoindicate that well-tuned decentralizedPI controllers can givecompara- bleresultstomoreadvancedcontrolstructuresifthecontrolled variables(measurementcombinations)arechosenproperly.Also, usingadecentralizedPIcontrolstructureallowsforrelativelyeasy retuning of the controllers, when comparing tothe case using advancedmultivariablecontrollers,shouldfuturechangesinoper- atingconditionsmakeretuningnecessary.Thus,thesmallpotential improvementsinperformanceformoreadvancedcontrolstrate- giesmaynotbeworththeiradditionalcomplexity.

4.2. H_∞staticoutputfeedbackcontrol

Considerthefollowinggeneralizedextensionoftheaugmented systemin(28)and(29):

P:

⎧ ⎪

⎨

⎪ ⎩

¯

x_k+1 =A¯xx¯_k+B¯uu¯_k+B¯ww_k

¯

y_k =C¯yxx¯_k+D¯yww_k z_k =C¯zxx¯_k+D¯zuu¯_k+D¯zww_k

(33)

wherez_k ∈R^nz istheperformanceoutputvector.Theaugmented plantPmapstheexogenousdisturbanceinputsw_kandthecontrol inputs ¯u_ktotheperformanceoutputz_kandthemeasuredoutputs y¯_kasshowninFig.2.

TheH_∞ optimalcontrolproblemthenconsistsofminimizing theH_∞-normoftheclosed-loopsystemfromtheexogenousdis- turbancesignalsw_ktotheperformanceoutputsignalsz_k[36,37].

Definingtheclosedloopmatrices,

A_cl=A¯x+B¯uKC¯yx (34) B_cl=B¯_d+B¯uKD¯yw (35) C_cl=C¯zx+D¯zuKC¯yx (36) D_cl=D¯zw+D¯zuKD¯yw (37) where Kis the staticoutput feedbackcontroller. When a state feedbackcontrolleris used,thenKC¯yx,KD¯yw,KC¯yx,andKD¯yw in (34)–(37)arereplacedwiththestatefeedbackgainKSF.Iftheresult- ingclosed-loopsystemisdefinedas

(38) thentheobjectiveistofindthestaticoutputfeedbackcontrollerK suchthatTw,z∞isminimized.TheH_∞-normhasseveralinterpre-

tationsregardingperformance.Oneisthatitminimizesthepeak ofthesingularvalueofTw,z(jω).Alternatively,fromatimedomain interpretation,itcanbeconsideredastheworst-case2-norm[37]:

Tw,z∞=max_w(t)=/0z(t)2

w(t)2

. (39)

Next,letsrecallthewell-knownBoundedRealLemma.

Lemma1. (BoundedRealLemma[36]),Tw,zisasymptoticallystable andT_w,z∞<iffthereexistsasymmetricmatrixP0suchthat thefollowinginequalityholds:

⎡

⎢ ⎣

A^T_clPA_cl−P A^T_clPB_cl C_cl^T B^T_clPA_cl B^T_clPB_cl−²I D^T_cl

C_cl D_cl −I

⎤

⎥ ⎦

^{≺0 (40)}

Lemma2. ProjectionLemma[38]

GivenasymmetricmatrixandtwomatricesUandV,thereexists amatrix thatsatisfies

U^T V+V^T U+≺0 (41)

iffthefollowingprojectioninequalitiesaresatisfied:

N_U^TN_U≺0 (42)

N_V^TNV≺0 (43)

whereNUandNVarearbitrarymatriceswhosecolumnsformabasisof thenullspacesofUandV,respectively.BasedonLemmas1and2,an H_∞optimalsolutionfor ˆH,andK_ithatensuresastableclosed-loop andaminimumupperboundforTw,z∞canbeobtainedfrom thefollowingtheorem.

Theorem2. There exist decentralized PI controllers anda mea- surement combinationHˆ ∈R^nu×ny,that gives astable closed-loop systemandminimizeswhileachievingTw,z∞≤,ifthereexists astabilizing state feedbackgain K_SF ∈R^2nu×(nx+nu),a matrix H∈ R^nu×ny, a non-singular matrix Q ∈R^(nu×nu), a matrix P=P^T ∈ R^{(nx+nu)×(nx+nu)}, diagonalmatrices X₁,X₂ ∈R^2nu×2nu, where X₁ hastobeinvertible,andmatricesZ₁,Z₂ ∈R^{(nx+nu)×(nx+nu)}thatsolves thefollowingnon-convexoptimizationproblem:

J_k= min

KSF,Z1,Z2,P,X1,X2,Q,H² (44)

s.t.P0 (45)

M+N≺0 (46)

X₁=diag(x₁₁···x_1nu) (47) X₂=diag(x₂₁···x_2nu) (48) whereMisdefined

M:=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎣

₁₁ ∗ ∗ ∗ ∗

_{21 22} ∗ ∗ ∗ B¯^T_uZ₂^T B¯^T_uZ₁^T 0 ∗ ∗ ₄₁ 0 D¯zu −I ∗ B¯^T_wZ₂^T B¯^T_wZ^T₁ 0 D¯^T_zw −²I

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎦

(49)

with

₁₁ =−P+( ¯Ax+B¯uK_SF)^TZ₂^T+Z₂( ¯Ax+B¯uK_SF) 21 =−Z₂^T+Z1( ¯Ax+B¯uKSF)

₂₂ =P−Z₂−Z^T₂, ₄₁ =C¯_zx+D¯_zuK_SF

andNisgivenby

N:=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎣

0 ∗ ∗ ∗ ∗

0 0 ∗ ∗ ∗

₂C¯yx−₁K_SF 0 −₁−^T₁ ∗ ∗

0 0 0 0 ∗

0 0 D¯^T_yw^T₂ 0 0

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎦

(50)

with 1=

Q 0

0 X₁ , 2=

H 0 0 X₂

Proof. Similartotheproofin[31],theexpressionin(46)canbe rewrittenas

U₁V^T+V^T₁U^T+M≺0, (51) whereVandUaredefinedas

V:=

⁻¹₁ 2C¯yx−KSF 0 −I 0 ⁻¹₁ 2D¯yw

T

, (52)

U:=

0 0 I 0 0

T

. (53)

Choosingthematrices,

NV=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎣

I 0 0 0

0 I 0 0

⁻₁¹₂D¯yw 0 0 ⁻₁¹₂C¯yx−K_SF

0 0 I 0

I 0 0 I

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎦

(54)

N_U=

⎡

⎢ ⎢

⎢ ⎢

⎣

I 0 0 0 0 I 0 0 0 0 0 0 0 0 I 0 0 0 0 I

⎤

⎥ ⎥

⎥ ⎥

⎦

⁽⁵⁵⁾

whosecolumnsformthenullspacesofV,andU,respectively.Then, accordingtotheProjectionlemma,theexpressionin(51)isequiv- alentto

⎡

⎢ ⎢

⎢ ⎢

⎣

A^T_clZ₁^T+Z1A_cl−P ∗ ∗ ∗

−Z₁^T+Z₂A_cl P−Z₂−Z₂^T ∗ ∗

C_cl 0 −I ∗

B^T_clZ₁^T B^T_clZ₂^T D^T_cl −²I

⎤

⎥ ⎥

⎥ ⎥

⎦

^{≺0 (56)}

whichisamultiplicationofMin(51)byN^T_VontheleftandN_Von theright.ReplacingKin(34)–(37),with⁻₁¹₂,guaranteesthe stabilityoftheclosed-loopsystemwhenusingthestaticoutput feedbackcontroller.MultiplyingMin(51)byN^T_UontheleftandN_U ontherightgivesthesecondconditionintheProjectionLemma andensuresstabilityforthesystemwhenusingthestatefeedback controllerK_SF.Theresultingexpressionisthesameasin(56),but withK_SFreplacingthestaticoutputfeedbackcontrollersforA_cl,B_cl, C_cl,andD_cl.Finally,multiplying(56)withˇ^Tontheleftandˇon theright,with

ˇ=

⎡

⎢ ⎢

⎣

I 0 0

A_cl B_cl 0

0 0 I

0 I 0

⎤

⎥ ⎥

⎦

⁽⁵⁷⁾

resultsintheboundedreallemma.

Theoptimizationproblemin(44)–(48)requiressolvingabilin- earmatrixinequality(BMI),whichisNP-hardandthus,difficult

tosolve.However,byspecifyingastablestatefeedbackgainK_SF, itbecomesalinearmatrixinequality(LMI),andaniterativealgo- rithmcanbeusedtofindalocaloptimum,followingtheprocedure described in Algorithm 2. The controller parameters and mea- surementcombinationcanbeobtainedfromK_i=X₁⁻¹X2 and ˆH= Q⁻¹H.

Algorithm2:

Initialize:chooseastabilizingstatefeedbackgainKSF. 1: ForfixedKSF,solvetheLMI:

Jk,1= min

Z1,Z2,P,X1,X2,Q,H

² s.t.(45),(46),(47),and(48)

2: FixZ1,Z2,X1,andQatthevaluesobtainedinstep1andsolve theLMI:

Jk,2= min

K_SF,P,X2,H

² s.t.(45),(46),and(48).

3: IfJk,1−Jk,2<stop,elseupdateKSFandrepeatstep1to3.

IfAlgorithm2isinitializedwithastabilizingfeedbackgainK_SF suchthatstep 1 givesa feasiblesolution forthefirst iteration, thenitwillgenerateasequenceofnon-increasingsolutionssuch that:

J_k+1,1≤J_k,2≤J_k,1, ∀^k

Remark3. IfanoptimalmeasurementcombinationHopthasbeen obtainedaprioriusinge.g.,(10)and(11),thenAlgorithm2canbe modifiedbyreplacingtheaugmentedmeasurementmatrices ¯Cyx

and ¯Dywin(29)with

C¯yx=

HoptCyx 0

0 I

, D¯yw=

HoptDyw

0

.

ThisresultsinthecontrollerK_i=X₁⁻¹X₂andanewmeasurement combination ˆH=Q⁻¹HHoptthatgivesthesamesteady-stateloss, butshouldimprovethedynamicbehaviouroftheclosed-loopsys- tem.

4.3. Staticoutputfeedbackandself-optimizingcontrol

Intheprevioussection,theproblemoffindingPIcontrollers andameasurementcombination, ˆHthatminimizestheH_∞norm oftheclosed-loopsystemwasinvestigated,whileinSection3.2 theoptimaltrade-offbetweensteady-statelossandthenumber ofmeasurementsusedwasconsidered.Thesetwoproblemscan easilybecombined,resultinginthefollowingoptimizationprob- lem:

J_k= min K_SF,Z₁,Z₂,P

X1,X2,Q,H

²+˛1

2HY²_F+

i,j

W_jH_i,j1 (58)

s.t.(11),(45),(46),(47),and(48)

Theiterativeprocedure,describedinAlgorithm3canbeused to finda local optimum, where the controller parameters and measurementcombinationcanbeobtainedfromK_i=X₁⁻¹X₂ and Hˆ =Q⁻¹Hrespectively.

Algorithm3:

Initialize:chooseastabilizingstatefeedbackgainKSFand obtainHbysolving(10)subjectto(11)forallavailable measurements.

1: ComputeW^(k)using(18).

2: ForthefixedKSF,andW^(k)solvetheLMI:

J_k,1= min Z1,Z2,P X1,X2,Q,H

²+˛1

2HY²_F+

i,j

W_j^(k)H_i,j1

Subjectto:(11),(45),(46),(47),and(48).

3: UpdateW^(k)using(18)withtheHobtainedfromstep2.

4: FixQ,X1,Z1,andZ2atvaluesobtainedinstep2andsolve theLMI:

Jk,2= min KSF,P,X2,H

²+˛1 2HY²_F+

i,j

W_j^(k)H_i,j1

Subjectto:(11),(45),(46),and(48).

5: IfH^(k−1)−H^(k)2<1,andJk,1−Jk,2<2gotostep6,else updateKSFandrepeatstep1to5.

6: Removethemeasurementsthatcorrespondtothezero columnsinH.

TheconvergenceofAlgorithm3followstheconvergenceprop- ertiesofAlgorithms1and2.When=0thentheconvergenceis monotonicallydecreasing:

J_k+1,1≤J_k,2≤J_k,1, ∀^k

Ifthere-weightedl₁-normisincorporated,thentheconvergenceis unknown.However,numericalexperiments(e.g.,[23,30])indicates thatittendstoconvergetoalocalminimum.

5. Simulations

Inthissection,theeffectivenessoftheproposedalgorithmsis validatedbyapplicationtotwodifferentdistillationcolumnmod- els.TheoptimizationproblemswereformulatedbytheYALMIP toolbox[39]andsolvedusingthesolverMOSEK[40].

5.1. Binarydistillationcolumn

Inthefirstexample,theproposedalgorithmsareappliedtothe

“columnA”distillationcolumnmodel[41],whereabinarymixture isseparatedthathasarelativevolatilityof1.5.Thedistillationcol- umnhas41stages,whichincludesthereboilerandthecondenser.

Thestagesarecountedfromthebottomwiththereboilerasstage1 andwiththefeedatstage21.Forthedistillationcolumn,thefeedis assumedtobegiven.Thus,ithasfourdegreesoffreedom;bottoms flowrate(B),distillateflowrate(D),refluxflowrate(L_R)andvapor boilup(V_B).Thedistillateboilupandbottomflowrateareusedto stabilizethetwoliquidlevelsinthecondenserandthereboiler.

ThisresultsintheLVconfigurationshowninFig.3wherethetwo remainingdegreesoffreedomare:

u=

L_R V_B

T

. (59)

Theobjectiveistogetatopproductwith99%lightcomponent(1%

heavy)andabottomproductwith1%lightcomponent,i.e.,thecost functionis

J=

x^top_H −x^top,s_H x^top,s_H

2

+

x_L^btm−x_L^btm,s x_L^btm,s

²

, (60)

where the specifications for the top and bottom products are denotedwiththesuperscript,s.

Ascompositionoftenisdifficulttomeasure,theywillbecon- trolledindirectlyusingthetemperaturesinsidethecolumnasin [25].ItisassumedthatthetemperaturesTi(^◦C)oneachstageican becalculatedusingthelinearfunction[41],

T_i=0x_L,i+10x_H,i (61)

Fig.3. AtypicaldistillationcolumnwithLVconfigurations.

Table1

Controlledvariablesforsteady-stateloss,wherecopt[28]iscomparedtocalg,which hasbeenobtainedusingAlgorithm1.

No.meas. Controlledvariables Loss¹₂HY²_F

2 copt=

T12

T30

0.548 calg=

T12

T29

0.553

3 copt=

T12+0.0446T31

T30+1.0216T31

0.443 calg=

T12+1.3030T13

T29−0.0705T13

0.463

4 copt=

1.0316T11+T12+0.0993T31

0.0891T11+T30+1.0263T31

0.344 calg=

1.0811T12+T13+0.2075T30

0.1205T12+T29+1.0859T30

0.358

withanaccuracyof±0.5^◦C.

Themaindisturbancesconsideredarechangesinfeedflowrate (F),feedcomposition(z_F)andfeedliquidfraction(q_F),whereFand z_Fcanvarybetween1±0.2,and0.5±0.1respectively.Thenominal valueforqFis1.0withalowerboundof0.9.

5.1.1. Measurementselectionforsteady-stateloss

Findingtheoptimalsubsetofmeasurementsthatminimizesthe steady-statelossforthebinarydistillationcolumnexamplehas previouslybeenstudiedin[29,28].Theproblemwassolvedin[29]

withabranchandboundmethod,whilein[28]aMIQPformula- tionwasused.Theoptimalcontrolledvariablesandtheirrespective steady-statelosswhenusing2,3,and4measurementsobtained from[28]canbeseeninTable1.

Theoptimalcontrolledvariables arecompared tocontrolled variablescomputedusingAlgorithm1.Thetrade-offbetweenthe steady-statelossandthenumberofmeasurementusedcanbeseen inFig.4whenusingdifferentvalues for(obtainedusingtrial anderror).ThemeasurementcombinationHforsetsof2,3,and 4measurements,canbeseeninTable1.

Algorithm1promotessparsity,usingtheweightedl1-normas aconvexrelaxationforthecardinalityoftheHmatrix.Asacon- sequence, it cannot guarantee that it converges totheoptimal measurementsubsetandtherefore,itgivessubsetswithaslightly largersteady-statelosscomparedtotheonesobtainedin[28].

5.1.2. Dynamicandsteady-stateperformance

Fortheoptimalcontrolvariablewhenusing3measurements (inTable1),theauthorof[42]implementedtwoPIcontrollers,that

Fig.4. Steady-statelossvsno.ofmeasurementsforthebinarydistillationcolumn.

weretunedusingtheSIMCmethod[43]andresultedinc_A,3shown inTable2.However,byusingAlgorithm2asdescribedinRemark3, itshouldbepossibletofindadifferentmeasurementcombination (cB,3)togetherwithPIcontrollersthatfurtherimprovesthetran- sientresponse,withoutaffectingthesteady-stateloss.Similarly,a controlledvariablewhenusingthe4optimalmeasurementsgiven inTable1isalsoobtained,usingthesameprocedure,resultingin cA,4.ForcomparisonAlgorithm3isusedtofindacB,4,whichshould giveanimprovementinthedynamicbehaviourattheexpenseof thesteady-stateloss.Finally,thecontrolledvariablesc_A,7,andcB,7

arecomputedwhenusing7measurements.Thecontrolledvariable c_A,7putsmoreemphasisonthesteady-stateloss(large˛),while c_B,7prioritiesthedynamicperformanceoftheresultingclosed-loop system(small˛).Allthecontrolledvariablesc_A,n,andcB,n(ndefines thenumberofmeasurements)canbeseeninTable2,together,with theirrespectiveH_∞-normandsteady-stateloss.

OutofthethreedisturbancesF,zF,andqF,changesinfeedflow Fhavenosteady-stateeffectonthecost.Therefore,thedynamic performanceforchangesinFisonlyconsidered,asitmakesiteas- iertocomparethetransientresponses.Thesimulationforastep changeinFcanbeseeninFig.5,andshowsasignificantimprove- mentintheresponseforc_B,3obtainedusingAlgorithm2compared toc_A,3thatwasproposedin[42].Asexpected,abetterresponseis alsoachievedforcB,4comparedtocA,4,whenatrade-offbetween minimizingtheH_∞-normandsteady-statelossiscomputedusing Algorithm3.Itisinterestingtonotethatusing7measurements (c_A,7andcB,7),givesthebestandnearlytheworstdynamicresponse dependingonwhetherthefocusliesonminimizingtheH_∞-norm oftheclosed-loopsystemor thesteady-stateperformance.This would suggest that thedynamic considerationswhen selecting measurementcombinationsbecomemorecrucial,themoremea- surementsareused.Furthermore,c_B,7alsohasthesecondlowest steady-state lossand thus,it canbe seenthat a smallsacrifice insteady-statelossmightgiveasignificantimprovementinthe controlperformance.

5.2. Kaibeldistillationcolumn

AKaibeldistillation column[44] isa thermally coupleddis- tillation column that canseparate fourproductsusing a single condenserandasinglereboiler(seeFig.6).Toachievethesame fourproductstreamsfromasinglefeedstreamwhenusingcon- ventionalbinarycolumns,itwouldrequireasetupconsistingof threedifferentbinarycolumns.

AKaibeldistillationcolumnisanextensiontothePetlyukdistil- lationcolumn[45]andhastogetherwithotherdivingwallcolumns, toagreatextentbeenstudiedintheliterature[46].Incomparison

Table2

Controlledvariables,PIparameters(Kp=I),andtheirdynamicandsteady-stateperformanceforthebinarydistillationcolumn.

Controlledvariables Ki Tw,z∞ 1

2HY²_F cA,3=

−0.022T12+0.383T30+0.390T31

−0.906T12+0.149T30+0.111T31

0.125 0.125

0.343 0.443

cB,3=

−2.289T12+0.451T30+0.358T31

−3.715T12−0.818T30−1.001T31

0.175 0.145

0.074 0.443

cA,4=

−0.970T11−0.923T12−0.200T30−0.297T31

−1.093T11−1.040T12−0.227T30−0.336T31

0.164 0.173

0.144 0.344

cB,4=

−0.479T11−0.479T12+1.110T32+0.852T33

−0.782T11−0.801T12−1.265T32−1.147T33

0.601 0.736

0.070 0.383

cA,7=

−0.118T11−0.128T12−0.123T13−0.015T21+0.125T28+0.139T29+0.126T30

−0.204T11−0.220T12−0.209T13+0.090T21+0.062T28+0.054T29+0.030T30

0.170 0.206

0.341 0.222

cB,7=

−0.272T11−0.276T12−0.267T13+0.251T25+0.645T32+0.530T33+0.416T34

−0.536T11−0.528T12−0.505T13+0.244T25−1.055T32−0.971T33−0.784T34

0.624 0.731

0.066 0.271

Fig.5.Distillateandbottomcompositionschangesinthebinarydistillationcolumn forastepdisturbanceof−10%inF.

tothetraditionalconfigurations,theKaibelcolumnhasthepoten- tialtogiveupto40%reductioninenergyconsumption,aswellas significantlyreducingthecapitalinvestmentcostandthephysical spacerequiredintheprocessplant.However,theenergysavings areonlyachievedifthedistillationcolumnoperatesclosetoits optimalvalue,whichremainsachallengeasitisahighlyinteractive multivariablesystemthatisdifficulttocontrol.

TheKaibelcolumnisdivedintosevensections,andeachsection consistsofseveralstages.Forthesimulatedmodel,thereareatotal of64stagesasshowninFig.6,wherethestagesarenumbered withtheprefractionatorsectionfirst.Theternaryfeedislocated betweenstage12and13andconsistsofthreecomponentswith themolefractionsz_D,z_S1,andz_S2.Fourproductstreamsaredrawn of,wherethelightcomponentzDdominatesthedistillatestream (D),componentz_S1andz_S2 dominateintheside-streams(S₁,and S₂)andtheremainingheavycomponentsz_Bdominatesthebottom stream(B).Foramoredetaileddescriptionoftheprocessmodel anditsnominalvalues,thereaderisreferredto[47].

Thedistillateboilup(D)andbottomflowrate(B)areusedto stabilizethelevelsinthecondenserandthereboiler,respectively.

Furthermore,vaporboilupVBandthevaporsplitRVwillbekept constantasithasbeenshowntobedifficulttocontrolinprac- tice.Instead,theywillbetreatedasdisturbances.Therefore,the remainingdegreesoffreedomuanddisturbancesdare:

u=

RL LR S1 S2

T

(62) d=

V_B R_V F z_D z_S1 z_S2 q_F

T

. (63)

Fig.6.Kaibeldistillationcolumn.

FortheKaibelcolumn,theobjectiveistooptimizetheprod- uctdistributionforgivenfeedrateFandboiluprateVB.Assuming equalvalueoftheproductsandthatonlythemaincomponentsin eachproductstreamareofvalue,thentheobjectiveisequivalent tominimizingthesumoftheimpurityflows.ThecostfunctionJ canthenbewrittenas[48]:

J=D(1−x_D)+S₁(1−x_S1)+S₂(1−x_S2)+B(1−x_B). (64)

5.2.1. Controlstructuredesign

ThecontrolstructuredesignfortheKaibelcolumnmodelhas previouslybeenstudiedin[47–49].Mostoftheworkhasmainly beenfocusedonthesteady-stateoperationofthecolumn;how- ever,differentcontrolstructureswaspresentedin[49].