On the Role of the Necessary Conditions of Optimality in Structuring Dynamic Real-Time Optimization Schemes

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Computers

and

Chemical

Engineering

jo u r n al h om ep a ge :ww w . e l s e v i e r . c o m / l o c a t e / c o m p c h e m e n g

On

the

role

of

the

necessary

conditions

of

optimality

in

structuring

dynamic

real-time

optimization

schemes

Dominique

Bonvin

a,∗

_,

_Bala

_Srinivasan

b

a_{Laboratoired’Automatique,EcolePolytechniqueFédéraledeLausanne,CH-1015Lausanne,Switzerland} b_{DepartmentofChemicalEngineering,EcolePolytechniqueMontreal,Montreal,Canada}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received28March2012

Receivedinrevisedform9July2012 Accepted12July2012

Available online 2 August 2012 Keywords:

Dynamicreal-timeoptimization Plant-modelmismatch Modelingforoptimization Inputparameterization Plantmodel

Solutionmodel

a

b

s

t

r

a

c

t

Indynamicoptimizationproblems,theoptimalinputprofilesaretypicallyobtainedusingmodelsthat predictthesystembehavior.Inpractice,however,processmodelsareofteninaccurate,andon-linemodel adaptationisrequiredforappropriatepredictionandre-optimization.Inmostdynamicreal-time opti-mizationschemes,theavailablemeasurementsareusedtoupdatetheplantmodel,withuncertainty beinglumpedintoselecteduncertainplantparameters;furthermore,apiecewise-constant parameteri-zationisusedfortheinputprofiles.Thispaperarguesthattheknowledgeofthenecessaryconditionsof optimality(NCO)canhelpdevisemoreefficientandmorerobustreal-timeoptimizationschemes.Ideally, thestructuringdecisionsinvolvetheNCOasfollows:(i)onemeasuresorestimatestheplantNCO,(ii) aNCO-basedinputparameterizationisused,and(iii)modeladaptationisperformedtomeettheplant NCO.ThebenefitofusingtheNCOindynamicreal-timeoptimizationisillustratedinsimulationthrough thecomparisonofvariousschemesforsolvingafinal-timeoptimalcontrolprobleminthepresenceof uncertainty.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Optimizationisimportantinscienceandengineeringasaway offindingthebestsolutions,designsoroperatingconditions. Opti-mizationis typicallyperformed onthebasis ofa mathematical modeloftheobjectofattention.Forexample,engineersmightbe interestedintheoptimaloperationofprocessesthateitheroperate atsteadystateorundergotransientchanges.Theobjectof atten-tion,orreality,iscalledthe“plant”,whereasthe“model”isasetof algebraic,differentialordifferential-algebraicequations.

In practice, optimization is complicated by the presence of uncertaintyintheformofplant-modelmismatchand unknown disturbances. Withoutuncertainty, one could usethemodel at hand,optimizeitnumericallyoff-lineandimplementtheoptimal inputsinanopen-loopfashion.However,becauseofuncertainty, additional informationsuchasuncertainty description or plant measurementsmustbeincluded.Intheformercase,robust opti-mizationcomputesasetofinputsthatguaranteesfeasibilityeither for all possiblerealizations or with a desired probability level, however at theexpense of a conservative solution(Srinivasan, Bonvin, Visser, & Palanki, 2003; Terwiesch, Agarwal, & Rippin, 1994). In the latter case, the inputs are updated in real-time

∗Correspondingauthor.Tel.:+41216933843;fax:+41216932574.

E-mailaddress:dominique.bonvin@epfl.ch(D.Bonvin).

based onmeasurements. Thisis thefield of real-time optimiza-tion,whichislabeledRTOforstaticoptimizationproblems(Marlin &Hrymak,1997)andDRTO fordynamicoptimizationproblems (Biegler,2009).Thispaperdealswithtwomajorimplementation issuesinDRTO,namely,modelqualityandcomputationalaspects. Theissueofmodelqualityraisesanimportantquestion:Does goodperformancerequireagoodmodel?Thisisnotnecessarily thecaseforcontrol,sinceerrorsresultingfromapoormodelcan beoffsetbytheactionoffeedback.Inoptimization,without feed-backtomakeupformodelingerrors,themodelneedstorepresent therealityaccurately,inparticulartheoptimalityconditionsofthe plant.Thesituationisslightlydifferentinreal-timeoptimization sincethemeasurementsavailableon-linerepresentsomeformof feedback.However,thisfeedbackisonlypartialasitistypically limitedtooutputinformation.Furthermore,itisimportanttoadapt themodelappropriately,thatis,therewhereitmattersmostfor thepurposeofoptimization.Theseissues ofmeasurement loca-tionandinputupdateinthecontextofimperfectmodelarecrucial forreachingoptimality.Itisarguedinthispaperthatthe neces-saryconditionsofoptimality(NCO)predictedbythemodelneed tomatchthoseoftheplantforplantoptimality.Wewilldiscussed howtheNCOmeasurementscanbeincorporatedinamodelsoas tobemostusefulforoptimization.

Thecomputationalaspectsarealsocrucialforimplementation. Avery reliableoptimizationschemeismodelpredictivecontrol (MPC), whichincorporatesstatefeedback,usesa receding hori-zonandcarriesouttheoptimizationrepeatedlyateachsampling 0098-1354/$–seefrontmatter© 2012 Elsevier Ltd. All rights reserved.

time(Rawlings&Mayne,2009).MPCwasinitiallydevelopedto trackareferencetrajectorybyminimizationofaquadraticerror term.Ithasrecentlybeenextendedto“economicMPC”thatuses anon-quadraticcost function(Heidarinejad, Liu,&Christofides, 2012;Rawlings&Amrit,2009).Furthermore,therehasbeen con-siderableeffortsinrecentyearstospeedupthecomputationsby formulatingconvexoptimizationproblemsand alsousing algo-rithmsthatexploitthestructure oftheproblem(Diehl,Ferreau, &Haverbeke,2009;Richter,Morari,&Jones,2011;Wang&Boyd, 2010).Ontheotherhand,recent trendsinDRTOhaveincluded attemptstomove theheavycomputations off-line,where time andcomputationalpoweraremoreavailable,andlimittheon-line operationstoquickdecisionsandeasycomputations.Forexample, multi-parametricprogramminggeneratesoff-linealookuptableof controllaws,whicharethenusedon-linebasedontheestimated statesoftheplant(Bemporad,Morari,Dua,&Pistikopoulos,2002; Pistikopoulos,Georgiadis,&Dua,2007;Zeilinger,Jones,&Morari, 2011).Also,“advancedstepNMPC”strategieshavebeenproposed, whichsolvethedetailedoptimizationprobleminbackgroundand applysensitivity-basedupdateon-line(D’Amato,Kumar, Lopez-Negrete,&Biegler,2012;Zavala&Biegler,2009).Anotherapproach isthenonlinearreal-timeiterationscheme,whichusesa continu-ationNewton-typeframeworkandsolvesoneQPateachiteration (Diehl, Bock, &Schlöder, 2005; Diehl et al., 2002).This allows formultiple activeset changesand thus ensures that the non-linearMPC algorithmcannotperform worsethan a linearMPC controller.YetadifferentapproachisNCOtracking,whichuses aNCO-based parameterizationof theinputprofiles todesigna multivariable feedback schemethat tracksthe first-order opti-malityconditions,therebypushingthesystemtowardoptimality (Srinivasan&Bonvin,2007).

Thispaperdealswiththemodel-qualityissueinDRTO.Inthe presenceofsignificantplant-modelmismatch,theuseofafixed nominalmodelistypicallyinsufficienttodrivetheplantto opti-mality. With MPC for example, the estimated states are often inaccurate,andone would needtoupdatethe model,which is difficulttodoinclosed-loopoperationduetotheso-calleddual controlproblem(Aström&Wittenmark,1995).Thisworkadopts theviewpointthat,inreal-timeoptimization,themodelisa vehi-cletoprocessplantmeasurementsandcomputetheoptimalinputs. Thisstepinvolvestwomajordecisions,namely,thechoiceofthe measuredquantitiesandthechoiceofafinitenumberofdecision variablesviainputparameterization.Notethatthesechoicescan benefitfromknowledgeoftheNCOsincetheNCOareintimately linkedtoplantoptimality.Thestructure oftheoptimalsolution andthecorrespondingNCOcanbedeterminedoff-lineby numeri-caloptimization.Thesemeasurementandinput-parameterization issuesarebrieflyaddressednext.

Measurements anduncertaintydescription.Themeasurements aretypically the plantoutputs yplant.The uncertainty, which is observedasthedifferencebetweentheplantmeasurementsand thecorrespondingmodelpredictions,canberepresentedas para-metricvariations of theplant model. Alternatively, if the NCO elementscanbemeasured,sayyNCO,themodeluncertaintycanbe expressedasthedifferencebetweenthemeasuredand the pre-dictedyNCO values. It is interesting tonotice the closerelation betweenthetype ofmeasurements(the plantoutputsyplant vs. theNCOelementsyNCO)andtheuncertaintydescription(theplant parametersplantvs.theNCOdeviationsNCO).

Parameterizationandupdateoftheinputs.Thetraditionalway of parameterizing infinite-dimensional inputs is control vector parameterization(CVP),wherebytheinputsareapproximatedas piecewise-constantprofiles.Themain advantageisuniversality, thatis,anysolutioncanbecloselyapproximatedbyintroducinga sufficientnumberofpieces(barringcertainnumericalissuessuch asringingarounddiscontinuities).However,CVPtypicallycontains

Fig. 1.Measurement and input-parameterization features of various DRTO schemes.Themeasuredplantoutputsarelabeledyplant,themeasuredNCOelements

yNCO;theinputsareparameterizedviacontrolvectorparameterization,CVP,orvia

theelementsoftheNCO,NCO.Plant-modelmismatchcanbeabsorbedintheplant

modelparametersplantorviaanadditivedisturbancetotheNCOvalues,NCO.

“Ident”meanstheuseofparameteridentification,“Diff”thecomputationofa dif-ference,“Opt”theuseofnumericaloptimization,and“Control”theuseoffeedback control.

alargenumberofpiecewise-constantinputvalues,denotedhere

CVP.Incontrast,aparsimoniousinputparameterization,NCO,can beobtainedfrom theknowledge oftheNCO, that is, theinput elementscorrespondtoswitchingtimesbetweenarcsandinput valuesassociatedwithcertainarcs.Thewaytheinputsare param-eterizedimpactsonthewaythereareupdated.WithCVP,theonly efficientway tocomputetheinputsis throughnumerical opti-mization.WithNCO,thefewinputparameterscanbeadjustedvia feedbackcontroltoregulatethedeviationNCOtozero.

VariousDRTOschemesarepossiblebasedonthechoiceofthe measurementandinput-parameterizationoptions,someofwhich areillustratedinFig.1anddiscussednext.

•Inthe“two-stepapproach”ofrepeatedparameteridentification andperformance optimization,themeasurementsareusedto adaptthemodelparametersandestimatethecurrentstates.The estimatedstatesserveasinitialconditionsfortheoptimization thatisrepeatedon-linewiththeupdatedmodel(Chen&Joseph, 1987;Eaton&Rawlings,1990).Theinputparameterizationisof theCVPtype.

•Inthemodifier-adaptationapproach,modifiertermsareadded tothecostandconstraintfunctions.UponmeasurementofyNCO, themodifiersareupdatedinorderforthemodelandtheplant tohave matching first-order optimalityconditions (Chachuat, Srinivasan,&Bonvin,2009;Marchetti,Chachuat,&Bonvin,2007). TheseschemesusethemeasurementsyNCOandtheinputsCVP foroptimization.

•Itisalsopossibletoperformnumericaloptimizationusinga NCO-basedinputparameterization.Suchaschemehasbeendeveloped by Schlegel and Marquardt(2006a) and applied toan indus-trialpolymerizationprocessinSchlegelandMarquardt(2006b). ThecorrespondingDRTOschemeconsistsinmeasuringyplantand updatingthemodelparametersaccordingly,followedby numer-icaloptimizationusingtheNCOparameterization.

•Finally,NCOtrackingusesthemeasurementsofyNCOtoupdate

NCOusingfeedbackcontroltomeettheplantNCO(Srinivasan& Bonvin,2007).

Thispaperconsiderstheimplementationofoptimalcontrolin thepresenceofsignificantuncertaintyintheformofplant-model mismatch,whichrequiressomeformofadaptationbasedonplant measurements.Three possibleadaptationstrategiesare consid-ered,namelyadaptationoftheprocessmodel,adaptationofthe

costandconstraintfunctions,anddirectadaptationoftheinputs.A similarinvestigationhasalreadybeenproposedforthestatic opti-mizationcase(Chachuatetal.,2009).Notealsothattheemphasis inthepresentstudyisnotoncomputationalaspectsasthishas beenthecaseinnumerousrecentinvestigations involving non-linearMPC (Diehl et al.,2009),butrather onalternativesways tocompensateforplant-modelmismatchanddrivetheplantto optimality.

Thepaperisorganizedasfollows.Section2formulatesthe opti-mizationproblemandpresentsthecorrespondingNCO.Section3

addressesnumericaloptimizationusingaplantmodelandtouches uponthetopicsof“modeladequacy”and“modelingfor optimiza-tion”.Optimizingcontrolusingasolutionmodelispresentedin

Section4.Section5discussesthechoiceofaDRTOschemewith regardtobothmeasurementsandinputparameterization.The var-iousideasdevelopedinthispaperareillustratedthroughasimple dynamicexampleinSection6,andSection7concludesthepaper.

2. Dynamicoptimizationproblem

2.1. Problemformulation

Considertheconstraineddynamicoptimizationproblemswith finiteoperationaltime,where theobjectiveistodeterminethe inputprofilesthatoptimizeafinal-timecostfunction.Inadditionto constraintscorrespondingtothedynamicsystemequations,there mightbe pathconstraints(involving inputs and states)as well

asterminalconstraints.Inputboundsaredictatedbyactuator lim-itations,whilestate-dependentconstraintstypicallyresult from safetyandoperabilityconsiderations.Terminalconstraintsusually arisefromqualityorperformanceconsiderations.

Sincetheinputsareinfinitedimensional,numericalsolutionof theoptimizationproblemrequiresinputparameterization.Here theinputsareparameterizedusingasetoffiniteparametersand numericaloptimizationthencomputestheoptimalvaluesof.

Themainchallengeaddressedin thispaperisthat of uncer-tainty,thatis,theplantdoesnotquitebehaveaspredictedbythe model.Onewaytoincorporatetheuncertaintyinthemodelisvia thesetofuncertainparameters.Thedynamicoptimization prob-lemcanbeformulatedmathematicallyasfollows(Bryson&Ho, 1975;Kirk,1970): min (x(tf),,) (1) s.t. x˙(t)=F(x(t),u(t),,), x(0)=x0(,) (2) S(x(t),u(t),,)≤0 (3) T(x(tf),,)≤0 (4) [u(t),]=U(t,x,,), (5)

whereisthefinal-timecostfunctionaltobeminimized,xthe stateswiththeknowninitialconditionsx0,utheinputs,the

time-invariantdecisionvariables,Sthepathconstraints,Ttheterminal constraints,Fthesystemdynamics,theuncertainplantmodel parameters,Utheinputparameterization,theinputparameters, andtfthefinaltimethatisfinitebutcanbeeitherfixedorfree.If

tfisfree,itispartof.Notethattheinitialconditionscanalsobe consideredasdecisionvariables.

ThesolutionofProblems(1)–(5)istypicallydiscontinuousand consistsofseveralintervalsorarcs(Srinivasan,Palanki,&Bonvin, 2003).Yet,theinputprofilesarecontinuousanddifferentiablein each interval.Eachinterval ischaracterizedbyadifferentsetof activepathconstraints,thatis,thissetchangesbetweentwo suc-cessiveintervals.

2.2. Necessaryconditionsofoptimality

Letusdefinethefollowingfunctions:

H(t)=T(t)F(x(t),u(t),,)+T(t)S(x(t),u(t),,) (6) ˚(tf)=(x(tf),,)+ TT(x(tf),,) (7) (t_f)₌˚(t_f)₊

tf 0 H(t)dt (8) ˙ T_(t)=−

∂

H

∂

x(t), T_(t f)=

∂

˚(tf)

∂

x(tf) , (9)

whereH(t)istheHamiltonianfunction,˚(tf)theaugmented ter-minal cost, (tf) the total terminal cost, (t)=/ 0 the adjoint variables(Lagrangemultipliersforthesystemequations),(t)≥0 theLagrange multipliersfor thepath constraints,and ≥0the Lagrangemultipliersfortheterminalconstraints.

The NCOfor optimization problem (1)–(4), that is, without parameterizationoftheinput,canbewrittenasfollows(Bryson &Ho,1975;Srinivasan,Palanki,etal.,2003):

l

a

n

i

m

r

e

T

h

t

a

P

Constr

aints

μ

T

₍

_t

₎

_S

_x

₍

_t

₎

_,

_u

₍

_t

₎

_,

_ρ,

_θ

₌

₀

_,

_μ

₍

_t

₎

_≥

₀

_ν

T

_T

_x

₍

_tf

₎

_,

_ρ,

_θ

₌

₀

_,

_ν

_≥

₀

Sensiti

vities

∂H

_∂u

(

t

)

=

0

∂

Ψ(

_∂ρ

tf

)

=

0

(10)

3. Numericaloptimizationusingaplantmodel

Thebestrepresentativeofthisclassofmethodsforthecaseof plant-modelmismatchisthetwo-stepapproach,which(i)usesa plantmodelandoutputmeasurementstoidentifyuncertainmodel parametersand (ii)optimizetheupdated modelusingCVP.The plantmodelisoftheform:

˙

x(t)=F(x(t),u(t),,), ymodel(t)=G(x(t),u(t),,), (11)

withthestatevectorxandtheoutputvariablespredictedbythe modelymodel.Afterabriefdescriptionofthemodel-building pro-cedure,thetwoconceptsof“modeladequacy”and“modelingfor optimization”willbedeveloped.Specialattentionwillbepaidto thechoiceandadaptationoftodealwithuncertainty.

3.1. Modelbuilding

Toconstructamathematicalmodel,themodelertypicallyuses bothpriorknowledgeandmeasurementsfromtheplant.The mod-elergoesthroughseveralstepsthatinclude(i)abstractionfrom therealitytodefinethe“system”,(ii)simplificationtoarriveat amathematicalmodelofmanageablecomplexity,(iii)parameter identificationtofitthemodeltotheplant,and(iv)modelvalidation toensurethatthemodelwillbeusefulforitsintendedgoal.Since thelaterstepsinfluencetheearlyones,thisprocedureistypically iterative.

Modelidentificationandvalidationaretypicallydoneby com-paring the model prediction ymodel(t) with the observed plant outputs yplant(t).However,parameteridentificationisa difficult taskinthepresenceofstructuralplant-modelmismatch,thatis,

whentheplantdoesnotbelongtothemodelset(Ljung,1999). Inthiscase,parameteridentificationrequiresappropriate exper-imentaldesign(Montgomery,2005)andpersistencyofexcitation (Walter&Pronzato,1997).

Themodelvalidationstepisveryimportant.Howcanoneensure thatthemodelwillbeadequateforsolvingtheoptimization prob-lemathand?ThecriterionJid_=y

model(t)−yplant(t)isconvenient becausetheoutputsare,bydefinition,available.Furthermore,itis fullyjustifiedifthemainpurposeofthemodelistopredictthe out-puts,forexampleinasimulationstudy.Butisitstilljustifiedifthe modelisusedforoptimization?Thisquestionisaddressednext.

3.2. Modeladequacy

It is well known that the two-step approach of repeated parameteridentificationandperformanceoptimizationworkswell provided that (i) there is no structural plant-model mismatch, that is, the plant lies in the model set and (ii) the operating conditionsyieldsufficientexcitationforalltheuncertainmodel parameters to be estimated (Chen & Joseph, 1987; Marlin & Hrymak,1997).Iftheseconditionsaresatisfied,convergenceto theplant optimum can beachieved in one iteration (Forbes & Marlin,1996). Yet,these conditions are rarely metin practice. Regardingthelattercondition,inparticular,thesituationis some-what similartothat found in theareaof systemidentification andcontrol,wherethetwotasksofidentificationandcontrolare typicallyconflicting(dualcontrolproblemAström&Wittenmark, 1995).

Hence, inthe realistic caseofplant-model mismatchand/or insufficientexcitation,whethertheschemeconverges,ortowhich pointitconverges,becomesanyone’sguess.Thisisduetothefact thattheobjectiveforparameteradaptationmightbeunrelatedto thecostandconstraintsthatdriveoptimalityintheoptimization problem.Hence,minimizingthemean-squareerroroftheplant outputs,ymodel(t)−yplant(t),maynothelpinourquestfor feasi-bilityandoptimality.

Ontheotherhand,convergenceunderplant-modelmismatch hasbeenaddressedbyBiegler,Grossmann,andWesterberg(1985)

and Forbes, Marlin, and MacGregor (1994). It has been shown that optimaloperationis reached ifmodel adaptation leadsto matchedKKTconditionsforthemodelandtheplant.Hence,the basicidea istoadjust thevalueof in sucha mannerthatthe NCOpredicted by the model match those of the plant, which naturallyleadstothenextsectionconcernedwithmodelingfor optimization.

3.3. Modelingforoptimization

The basicidea isthatthevalidityofa modeldepends onits intendeduse. This concepthasbeen studiedextensively inthe 1990sintheareaofsystemidentificationandcontrolunderthe label“identificationforcontrol”(Aström,1993;Gevers,2004).The proposedsolutionhasbeentousematchingcriteriaforthe identifi-cationandcontroltasks.Thisisobtainedbyfilteringtheprediction error(identification)tomakeitresembletheclosed-loop perfor-mancecriterion(control).

Similarly in optimization, one can consider” modeling for optimization”,whereby thesynergybetweenthemodeling and optimizationstepsisimprovedbyreconcilingtheobjective func-tionsofthetwoproblems.Inconcreteterms,theobjectivefunction of the identification problem can be set up to minimize the errorbetweentheplantand modelNCOs(Srinivasan&Bonvin, 2002).

Theideacanbeformulatedasfollows.LetNbeusedtorepresent thecomponentsoftheNCOin(10).Plantoptimalityunder plant-modelmismatchisgivenby,

0_=Nplant

objective = N

model

optimization +Nplant−Nmodel

identification . (12)

Plant optimality implies zeroing the necessary conditions, Nplant=0. For its part, numerical optimization of the model

enforces Nmodel=0. Hence, if the identification step can

mini-mizeJid_=N

plant−Nmodel,theidentifiedmodelwillbesuitedfor

optimization.Suchachoiceofobjectivefunctionisreferredtoas “modelingforoptimization”.

IfthecomponentsoftheNCOcanbemeasured,yNCO,thenthe identificationcriterionbasedontheminimizationoftheoutput erroraddressestheoptimizationobjectivedirectly.However,the evaluationof certaincomponentsof theNCO, suchastheplant gradients,isdifficultandrequiresspecialcarewithrespecttoboth excitationandnoisefiltering.

4. Optimizingcontrolusingasolutionmodel

As shown in Fig. 1, optimization can also be implemented via feedbackcontrol. NCOtracking is organizedaround a solu-tionmodel,thatis,qualitativeknowledgeoftheoptimalsolution. Thissolution model isused toidentifytheNCO and selectthe correspondingmeasurementsyNCO and inputelementsNCO.To illustrate the NCO-based input parameterization, consider the inputu1(t)inagiventimeinterval.CVPwouldparameterizeu1(t)

using a number of constant values, which are then optimized numerically.However,ifweknowthatthepathconstraintS1is

activeduringthatintervalandu1 istheinputthatpushesS1 to

itsbound,analternativeparameterizationwouldbeu1(t)=Gc(S1),

whereGcisanappropriatecontrollerthatkeepsS1active.Hence,

theparameterization looksfor manipulatedvariables (MVs,the inputparametersNCO)thatcanbeusedtotrackcontrolled vari-ables (CVs,theelementsof yNCO)soastosatisfy theNCO. The solutionmodelisnotuniquesincetheNCOdependonthe param-eterizationthatisused.Hence,thediversityinsolutionmodelscan beexploitedtoeaseupimplementation.

Thedevelopmentofasolutionmodelinvolvesthreesteps: (1)Theoptimalsolutionisfirstcharacterizedintermsofthetypes

andsequenceofarcs.Thissteptypicallyusestheavailableplant modelandnumericaloptimizationtocomputetheinput pro-filesusingCVP.

(2)AfinitesetofparametersNCOisselectedtorepresenttheinput profiles (MVs),thecorresponding NCOareformulated from whichtheCVsareobtained,andtheMVsandCVsarepaired toformamultivariablecontrolproblem.

(3)Arobustnessanalysisisperformedtoensurethatthestructure oftheoptimalsolutionisinvariantinthepresenceof uncer-taintyand thenominaloptimalsolutionarestructurallythe same.Ifthisisnotthecase,itisnecessarytomodifythe struc-tureofthesolutionmodelandrepeattheprocedure.

Thesolution modelconsiders thedifferentparts oftheNCO (pathconstraints,pathsensitivities,terminalconstraintsand ter-minalsensitivities)thatneedtobeenforcedforoptimality.The variousNCOelementscanbeimplementedwithvariousdegrees ofeaseordifficulty:apathconstraintisoftenenforcedon-linevia constraintcontrol;apathsensitivityismoredifficulttoimplement asitinvolvestheadjointvariables,whicharenotavailableon-line withouttheuseofaplantmodel;theterminalconstraintsand sen-sitivitiescallforprediction,whichrequiresamodel,orelsethey canbemetiterativelyoverseveralruns.Toeaseimplementation,

itisoftenpossibletoapproximatetheoptimalinputswith sim-plerprofiles.Thisrepresentsthestrengthoftheapproach,asthe approximationsintroducedatthesolutionlevelcanbeassessedin termsofoptimalityloss.

TheMVsofthecontrolproblemarethehandlesavailableto reachoptimality.TheCVsaretheNCOfortheselectedinput param-eterization.Bydefinition,thereareasmanyoptimalityconditions astherearedegreesoffreedom,thusresultinginasquarecontrol system.ThepairingofMVsandCVscanbedoneinacentralized (multivariablecontrol)ordecentralized(multi-loopcontrol) fash-ion.Notethattherearedifferentwaysofimplementingagiven solutionmodel,forexampleusingalternativeMVsviaachangeof variables,usingdifferentpairingsofMVsandCVs,orusingaplant modelforprediction,eachwaydefiningadifferentNCO-tracking option(Srinivasan&Bonvin,2007).

5. ChoiceofaDRTOscheme

ThechoiceofaDRTOschemeisintimatelylinkedtothe mea-surementsandtheinputparameterizationasillustratedinFig.1

anddiscussednext.

5.1. Measurements

Measurementavailabilityiskeyinreal-timeoptimizationand determinesthewaytheuncertainmodelelementscanbeupdated toaccountfortheobserveduncertainty:

(1)Iftheoutputsyplantaremeasured,theuncertaintyisaccounted forbyvaryingtheparametersplantoftheplantmodel.Note, however,thatoptimalitycannotingeneralbeguaranteedin thepresenceofstructuralplant-modelmismatch.

(2)IftheNCOelementsyNCO aremeasured,theuncertaintycan beexpressedasadditivedisturbancesthatrepresentthe devi-ationsoftheNCOfromtheiridealvaluesofzero.Hence,special effortisneededtomeasureorestimatetheNCO,whichrequires excitationandspecificmeasurements.

Whichmeasurementsshouldbeused?Ifbothyplant andyNCO areavailable, itseems easiertoimplement DRTO via modifier-adaptationorNCOtrackingsincethetwo-stepapproachrequires system excitation and numerical optimization to estimate the parameters,anothernumericaloptimizationinthere-optimization stepand,inmanycases,atime-scaleseparationforthetwoiterative steps,namely,identificationandoptimization,toconverge.Yet,the two-stepapproachhasbeenusedextensivelysinceitrepresents theonlyavailableoptionwhenyNCOisnotavailable.Furthermore, measuring,estimatingandapproximatingcertainelementsofthe NCOisnottrivialandrepresentsanopenfieldofresearch.Hence, forthetimebeing,measuringyplantandexpressingtheuncertainty throughparametricvariationsoftheplantmodelremainsthemain option.

5.2. Inputparameterization

Theinputparameterizationisimportantasitaffectsthechoice oftheadaptationstrategy:

(1)Thesimplestandmostgenerallyvalidwayofparameterizing inputprofilesisviaCVP,followingwhichtheoptimalinputs aredeterminedbynumericaloptimization.

(2)ThealternativeistoconsidertheNCO-basedparameterization

NCO,whichlendsitselftothedesignofcontrolloopstoreject theuncertaintyintheNCOseenasadditivedisturbances.

Table1

Modelparametersandoperatingbounds.

m 1300 kg f 0.5 Ns2 m2 umin −8000 N umax 3600 N vmax 40 ms xdes 1000 m

Theparameterizationofchoicedependsontheproblemathand. TheNCOparameterizationisinitiallymoreinvolvedasitrequires numericaloptimizationtocomputethenominaloptimalsolution thatisusedtocharacterizetheNCO.Yet,thisparameterizationmay turnouttobeveryconvenientasoptimalitycanbereachedusing feedbackcontrol,thatis,withouthavingtorepeatoptimization on-line.ThesuccessoftheNCO-basedparameterizationdependsonits validitysinceatailor-madeparameterizationisseldomuniversally valid.

5.3. Choiceofascheme

Thechoiceofaschemedependsonseveralfactors:

(1) AretheNCOknownandcantheybemeasuredorestimated? (2) HavetheNCObeenverifiedtoberobustwithrespectto

uncer-tainty?

(3) Is a multivariable feedback control scheme easily imple-mentable?

Inthecaseofapositiveanswertoallthreequestions,NCOtracking isclearlyanoptionforimplementingDRTO.Modifieradaptationisa validoptionwhenonlyquestion(1)isverified.Otherwise,the two-stepapproachisstillthemethodofchoicebecauseofitsgenerality.

6. Illustrativeexample

Theuseofplantandsolutionmodelsforreal-timeoptimization willbeillustratedonthesimplecarexamplethatispresentednext:

•System:Movementofacarfromonepointtoanother.

•Uncertainty:Slopeoftheroad±5%.

•Objective:Minimizefinaltime.

•Manipulatedinput:Accelerating/brakingforce.

•Pathconstraints:Inputbounds;speedlimit.

•Terminalconstraints:Zerovelocityatfinaltime;coveratleastthe prescribeddistance.

6.1. Formulationoftheoptimizationproblem 6.1.1. Variablesandparameters

x:position,

v

:velocity,u:accelerating/brakingforce,s:slopeof theroad,f:frictioncoefficient,g:gravitationalconstant,and m: massofthecar.Thenumericalvaluesofthemodelparametersand operatingboundsaregiveninTable1.

6.1.2. Modelequations

˙

x=

v

x(0)=0, (13)

˙

v

=u−_mf

v

2 −s(x)g,

v(0)

=0. (14)

Thenominalmodelassumeszeroslope,thatiss(x)=0,whilein realitytheunknownelevationprofile,whichistheintegralofthe slopeprofile,isasshowninFig.2.

Fig.2.Elevationprofile.

6.1.3. Optimizationproblem

Theoptimizationproblemcanbeformulatedmathematicallyas follows:

min

u(t),tf

J=tf

s.t. dynamic system (13)and (14)

umin≤u(t)≤umax

v(

t)≤

v

max

v(

t_f)₌0

x(tf)≥xdes.

(15)

6.2. Characterizationoftheoptimalsolution

Fig.3showsthattheoptimalsolutionconsistsofthreearcs,with thesuccessiveinputsumax,upathandumin:

•Thefirstarcumaxcorrespondstomaximumaccelerationinorder toreach

v

maxasquicklyaspossible.Thedurationofthisarc,t1,

depends ontheslope,which isuncertain.However,t1 canbe

determinedimplicitlyuponreaching

v

max,thatis

v(

t1)=

v

max.

Fig.3.Nominaloptimalsolution(s=0):inputu/100[N],velocityv[m/s]andposition

x/20[m]profiles.

Table2

PairingofMVsandCVs.

Pathobjectives Terminalobjectives Constraints t1: v(t1)=vmax t2→x(tf)=xdes

upath(t): v(t)=vmax tf: v(tf)=0

Sensitivities – –

•Thesecond arckeepsthevelocity at

v

max,for whichthe

cor-respondinginput valueupath can bedeterminedfrom (14)as

upath=fv2max+s(x)mg.Thevalueupath isalsoafunctionofthe uncertainslope.

•The third arccorresponds to fullbraking in order toachieve

v(

tf)=0.Theswitchingtime t2 betweenthesecondandthird

arcs is chosen so that x(t_f)=x_des, that is, the desired dis-tancewillbeexactly coveredwhen thevelocitygoestozero. The final time is determined upon reaching zero velocity,

v(

tf)=0.

6.2.1. Remarks

(1)Thiscarexampledoesnotinvolvesensitivities,thatis,the opti-malsolutionisentirelydeterminedbyactiveconstraints.The reader isreferred toSrinivasanand Bonvin(2007)for cases wheresensitivitiesareinvolved.

(2)Thetypesandsequenceofarcs(umaxfollowedbyupathandthen

umin)holdforanycarregardlessof itsweightand accelera-tion/brakingpower.Theyevenholdgenericallyforabicycle,for whichthesecondarcvanishes(t1doesnotexist).Thisgeneric

aspectoftheoptimalsolutionprovidesmuchrobustnesstothe solution-modelapproach.

6.3. NCOtracking

Inputparameterizationisstraightforwardinthisproblem,with theinputparameters=[t1upatht2tf]T.ThepairingofMVs(t1,upath,

t2andtf)andCVs(v(t1)=

v

max,

v(

t)=

v

max,x(tf)=xdes,and

v(

tf)=0)

followsdirectlyfromthecharacterizationoftheoptimalsolution andisgiveninTable2.

Sincethereisapredictioninvolvedinthepairingt2→x(tf)=xdes, meetingthisconstraintwilleitherrequireapredictivemodelor beimplementedoverseveralruns.Next,wepresenttwocontrol strategiesthatcorrespondtodifferentwaysofadjustingt2tomeet

theterminalconstraintx(tf)=xdes.

6.3.1. Run-to-runadaptationoft2

Intheabsenceofamodeltopredictx(tf)duringtherun,NCO trackingwillencompasson-linecontroltoenforce

v(

t)=

v

maxin

thesecond arcand run-to-run controltoadapt t2 soasto

sat-isfy x(t_f)=x_des over severalruns.The timeinstantst1 and tf are determinedbythevelocityreaching

v

maxand0,respectively.Itis

assumedthattheplant(thecarwithvaryingunknownslope)will havethesametypesandsequenceofarcs,butdifferentvaluesof

t1,upath,t2andtf.1

6.3.2. On-lineadaptationoft2

Predictionofthefinal positionx(tf)toinitiatebreakingatt2

canbedoneduringtherunusingthenominalplantmodelthat assumess(x)=0.Sincet1,upathandtfcanbedeterminedfromtheir

1_{Thiscanbeverifiednumericallyoff-linebyperturbingthenominalmodeland} computingthecorrespondingoptimalinputs.

Table3

Constraintandcostvaluesforvariousoptimizationscenarios:Iv+=

t2

t1

v+(t)dt,wherev+(t)=v(t)−vmaxifv(t)>vmaxand0otherwise;vm:maximalvelocity;xf:final

position;tf:finaltime(cost);tfaug:costpenalizedfordeviationfromconstraints,eachdeviationbeingweighted5timesthecorrespondingLagrangemultiplier.

Optimizationscenario Iv+[m] vm[m/s] xf[m] tf[s] tfaug[s]

Idealsolution 0 40 1000 35.31 35.31

Open-loopuseofnominaloptimalinput 34.57 43.87 1045.3 35.71 43.70

Re-optimization–noadaptation 1.85 40.87 1003.8 36.00 36.56

Re-optimization–adaptation 0.05 40.08 1004.4 35.44 35.90

NCOtracking–t2notadapted 0.003 40.001 1012.2 35.64 36.86

NCOtracking–run-to-runt2adaptation 0.002 40.001 1000 35.31 35.31

NCOtracking– on-linet2adaptation 0.002 40.001 996.5 35.22 35.57

correspondingNCOduringsimulation(integration)ofthedynamic model,theoptimizationproblem(15)canberewrittenas:

min

t2

J=tf

s.t.dynamic system(13)and (14)

u(t)=

⎧

⎪

⎨

⎪

⎩

umax for 0≤t<t1 fv2max for t1≤t<t2 umin for t2≤t<tf t1:

v(

t1)=

v

max tf :

v(

tf)=0. (16)

Problem (16) is simpler to solve than theoriginal problem

(15) for at least two reasons: (i) the number of degrees of freedom has been reduced from ∞ to 1 and (ii) the discon-tinuitiesat the switching instants can be handledmuch more easily and without oscillations (Schlegel & Marquardt, 2006a). For implementation, the current state information is used as initial conditions for re-optimization at each sampling time. Note that the slope is not updated. The optimal value of t2

is computed at each re-optimization instant, and breaking is implementedwhentherunningtimeequalsthevalueoft2

com-putedlast.Aswithrun-to-runadaptationoft2,thetimeinstants

t1 and tf are determined by the velocity reaching

v

max and 0,

respectively.

6.4. Optimizationresults

Theperformance ofvariousDRTOschemesissummarizedin

Table3anddiscussednext.Thecaseofnomeasurementnoiseis consideredheretobeabletofocusonthedistinguishingfeatures ofthevariousapproaches.

Thereferencescenarioistheidealsolution,whichiscomputed withknowledge ofthevaryingslopeand isshownin Fig.4.As expected,thisscenariomeetsexactlyallthepathandterminal con-straints.Incomparison,theopen-loopuseofthenominaloptimal inputleadstosignificantviolationofthespeedconstraintasthereis nopossibilitytocompensateforthefactthattheroadgoesdownhill initially.

ThefivesDRTOschemesanalyzedbelowcorrespondtodifferent re-optimizationandNCO-trackingchoices:

•Re-optimization – no adaptation: Uncertaintyis consideredas additivestatedisturbances,andCVPisusedfornumerical opti-mization. Re-optimization every two seconds without slope adaptationgivestheprofileshowninFig.5(top-left).Thereis considerablechatteringcausedbytheneedforadditional bra-kingintheinitialdownhillpartfollowedbyadditionalpowerin theuphillpart.Thepathconstraintisnotmetaccurately,asseen bytheIv+ valuesinTable3.Notethatthisschemecorresponds toeconomicnonlinearMPC(Rawlings&Amrit,2009)andshows

thenegativeeffectofnotbeingabletoadapt forplant-model mismatch.

•Re-optimization–adaptation:Uncertaintyisconsideredasboth parametricvariationsandadditivestatedisturbances,andCVP isusedfornumericaloptimization.Withadaptationoftheslope parameter,chatteringandthedeviationfrom

v

maxinthesecond

arcarereduced(comparethecorrespondingIv+valuesinTable3). Treatingtheuncertaintyasparametricvariationshelpstrackthe pathconstraints,butthereisstillsomeresidualerror.Notethat itisverydifficulttocontroltheterminalpositioninasinglerun becauseofmodelingerrors.Acomparisonofthetwoinput pro-filesforre-optimizationshowsthatparameteradaptationhelps approximatetheidealinput(compareFigs.4and5(toprow)).

•NCOtracking–t2notadapted:UncertaintyisconsideredasNCO

deviations,andNCOtrackingisimplemented.Withoutt2

adapta-tion,thepathconstraint

v(

t)=

v

maxandthefinalvelocity

v(

tf)=0

aremet,butnotx(tf)=xdes.

•NCOtracking–t2adaptedoverseveralruns:Similartotheprevious

case,butt2isnowadaptedonarun-to-runbasis.After5runs,

alltheconstraintsaremetandoptimalityisreached,whichin thisproblemisguaranteedbysatisfactionoftheconstraintson maximumvelocityandfinalposition.

•NCOtracking–on-lineadaptationoft2:Evenwhent2isadapted,

theterminalpositionconstraintx(tf)=xdesisnotmetexactlysince itisadaptedonthebasisofaninaccuratemodel.Suchterminal objectivescanonlybemetoverseveralruns.

Fig.4.Idealsolutionforknownvaryingslope:inputu/100[N],velocityv[m/s]and positionx/20[m]profiles.

0 10 20 30 40 −10000

−5000 0 5000

Reoptimization − no parameter adaptation

0 10 20 30 40

−10000 −5000 0 5000

Reoptimization − parameter adaptation

0 10 20 30 40

−10000 −5000 0 5000

NCO tracking − no update of t

2

0 10 20 30 40

−10000 −5000 0 5000

NCO tracking − on-line control of t

₂

Fig.5. OptimalinputobtainedwithfourDRTOschemes.ThecaseofNCOtrackingwitht2adaptedover5runsisnotshownhereastheconvergedsolutioncorrespondsto theidealsolutionshowninFig.4.

7. Conclusions

Thispaperhasaddressedthequalityofmodelsusedfor real-timeoptimization.Accordingtothequote“Allmodelsarewrong butsomeareuseful”(Box,1979),themodelisnotviewedasthe “truth”,butratherasatoolthatmustbetailoredtotheoptimization scheme.Modelingisreallyaboutmakingeducatedapproximations toarriveatamodelofacceptablecomplexitythatisadequatefor optimizationinthepresenceofuncertainty.Whendealingwitha realplant,itisimportanttofindagoodwayofintroducing approx-imations.Isitattheplant-modellevel,beforegoingthroughthe optimizationmachinery?Orisitattheimplementationlevel,when theusercanseetheimplicationsofselectedapproximations?Ithas beenarguedthattheparametersoftheplantmodeladjustedusing outputmeasurementsmightnotbethebesthandlestoadjustin realtime,sincetheymightonlybelooselyconnectedto optimal-ity.Bycontrast,iftheNCOelementsaremeasuredorestimated,the optimizationobjectivecanbeincorporatedintheparameter iden-tificationstep,therebyaccommodatingtheobjectiveof“modeling foroptimization”.

Thepaperhasalsodiscussedthevariouselementsofasolution model.Itsprimitiveversionistousepiecewise-constantprofiles. However,one coulddeviseamore appropriatelyparameterized solutionstructurebydissectingthenominaloptimalinputprofiles andrelatingtheirelementstodifferentpartsoftheNCO.Sincethe complexityofsolutionmodelsdependsonthenumberofinputs, andnotonthenumberofstatesorthenonlinearityoftheplant,the solutionmodeliseasiertoobtainforproblemswithonlyafewarcs (andthusalsoonlyafewinputelementsNCO),andthisregardless oftheorderofthesystem.Suchatailor-madeparameterizationhas beenshowntobemoreefficientthanCVP.

One strength of NCO tracking is the possibility of combin-ingoff-linetasks(numericaloptimizationbasedonthenominal plantmodeltodeterminethesetofactiveconstraints) and on-lineactivities(optimizingcontrolthatadjuststheinputs onthe basisofmeasurements).Anothernicefeatureisthepossibility,if necessary,ofintroducingapproximationsinthevariousprofilesto easeimplementation.Thisisparticularlyeffectiveindealingwith sensitivity-seekingarcs,whichareoftendifficulttocomputebut, atthesametime,contributeonlynegligiblytothecost.Instead ofbuildingamodelthatwillpredicttheplantperformancefrom scratch,NCOtrackingstartswitharobustparameterizedmodelof thesolutionandadjuststhefewinputparametersthatare inti-matelylinkedtoplantoptimality.Thesynergybetweenoff-line andon-linecomputationsforreal-timeoptimizationneedstobe studiedinmoredetails.Multi-parametricprogrammingandNCO trackingrepresenttwosuchinitialattempts.Abigpushcouldcome fromresearchersinthefieldofoptimizationoncetheyrealizethat theavailablemodelsarenotsufficientlyaccuratetodriveplantsto optimality.

Acknowledgments

Theauthorswouldliketoacknowledgeinsightfuldiscussions withBenoîtChachuatandGrégoryFranc¸ois.

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