Nonlinear model predictive control of the strip temperature in an annealing furnace

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This document contains a post-print version of the paper
Nonlinear model predictive control of the strip temperature in an
annealing furnace
authored by M. Niederer, S. Strommer, A. Steinboeck, and A. Kugi
and published in Journal of Process Control.
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Cite this article as:
M. Niederer, S. Strommer, A. Steinboeck, and A. Kugi, “Nonlinear model predictive control of the strip temperature
in an annealing furnace”, Journal of process control, vol. 48, pp. 1–13, 2016. doi: 10.1016/j.jprocont.2016.09.012
BibTex entry:
@ARTICLE{acinpaper,
AUTHOR = {Niederer, M. and Strommer, S. and Steinboeck, A. and Kugi, A.},
TITLE = {Nonlinear model predictive control of the strip temperature in an annealing furnace},
JOURNAL = {Journal of Process Control},
YEAR = {2016},
volume = {48},
pages = {1--13},
doi = {10.1016/j.jprocont.2016.09.012}
}
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published in M. Niederer, S. Strommer, A. Steinboeck, and A. Kugi, “Nonlinear model predictive control of the strip temperature in an
annealingfurnace”,Journal of process control,vol.48,pp.1–13,2016.doi: 10.1016/j.jprocont.2016.09.012
Nonlinear model predictive control of the strip temperature in an annealing furnace
M.Niederer,S.Strommer,A.Steinboeck,A.Kugi
∗
AutomationandControlInstitute,ViennaUniversityofTechnology,Gußhausstraße27–29,1040Vienna,Austria
Abstract
Anonlinearmodelpredictivecontrollerisdesignedforthestriptemperatureinacombineddirect-andindirect-firedstripannealing
furnace.Basedonatailoredfirst-principlesdynamicalmodelandtheestimatedcurrentsystemstate,therecedinghorizoncontroller
selectsoptimaltrajectoriesforboththefuelsupplyandthestripvelocityso thatthestriptemperatureiscontrolledtoitsdesired
targettemperature. Thecontrolleradditionallymaximizesthethroughputandminimizestheenergyconsumption. Inthecontrol
algorithm,thedynamicoptimizationproblemwithequalityconstraintsisnumericallysolvedbyusingtheGauss-Newtonmethod.
The gradient and the approximated Hessian matrix of the objective function are analytically computed using an adjoint-based
method.Thecapabilitiesoftheproposedcontrolleraredemonstratedforavalidatedhigh-fidelitysimulationmodelofanindustrial
annealingfurnace.
Keywords: steelindustry,annealingfurnace,direct-andindirect-firedsections,nonlinearmodelpredictivecontrol,receding
horizoncontrol,vector-valuedLagrangian,adjoint-basedmethod,Gauss-Newtonmethod
1. Introduction steady-statefurnaceoperation. Thefurnacecanbeconsidered
asacascadethermalsystem,wherethestripispartofthelast
1.1. Controlproblem
cascade. Since the thermal inertia of the strip is significantly
Inthesteelindustry,stripannealingfurnacesareusedforthe lowerthanthatofthefurnace,itisdifficulttoconsistentlyreal-
heattreatmentofsteelstripsinordertoachievethedesiredmet- izethedesiredtargettemperatureintransientoperationalsitua-
allurgicaland surfacepropertiesforsubsequentprocesssteps. tions.
Theannealingfurnaceconsideredinthispaper,cf.Fig.1,ispart Inadditiontotheproductquality,therearefurtherdemands
of a strip processing line of voestalpine Stahl GmbH in Linz, on the furnace operation like optimized energy consumption,
Austriaandcontains235msteelstrip. Thekeyparametersof material throughput,andCO emissions. A furnacetempera-
2
thefurnacearetabulatedinTab.1. Toensureacontinuousop- turecontrollerthatmeetsalltheserequirementsisstillanopen
erationoftheprocessingline,thestripsareweldedtogetherto researchissue. Thispaper,therefore,exploresthesuitabilityof
formanendlessstrip. advanced nonlinear control and optimization methods for the
consideredcontrolproblem.
Throughputofsteel 45.7t/h
Nominalheatingpower 15MW
Stripdimensions 1.2. Existingsolutions
Thickness 0.35 1.2mm
Width 800 1−640mm Intheliterature,differentconceptsforstriptemperaturecon-
−
Stripvelocities max.180m/min trol of annealing furnacescan be found. The following sum-
Numberofgasburners 48 maryshouldthereforeonlyserveasastartingpointforanin-
Numberofradianttubes 62
depthexploration.
Numberofheatingzones 7
Over decades, simple PID control concepts were used for
Table1:Nominalparametersofthestripannealingfurnace. striptemperaturecontrol.PIDcontroldoesnotrequireamathe-
maticalmodelandprovidesacceptableresultsforasteady-state
Tomeetthehighdemandsonthequalityofthefinalproduct, furnace operation. However, if the furnace is not operated at
thetemperatureevolutionofthestripisofimportance. While steady state, this concept is no longer suitable because of the
the strip moves through the furnace, it has to be heated to a largethermalinertiaofthefurnace.
predefined target temperature. This strip temperature control Occasionally,rule-basedexpertsystemsandfuzzylogiccon-
task isachallengemainlyforthefollowingreason: Anongo- trol concepts are used, see, e.g., [1, 2, 3, 4, 5]. Generally,
ingdiversificationoftheproductportfolioessentiallyprevents these semi-empirical concepts use a wide range of measure-
mentsfromtheplantinordertocharacterizetheoperationcon-
ditionsandadjusttheinputstoobtainthedesiredtemperature.
∗Corresponding author. Tel.: [phone], Fax: [phone] Basically,theymimicthehumanoperatorbuttheyperformthe
9376264.
Emailaddress:[email protected](M.Niederer) controltaskwithgreaterconsistencyandaccuracyandsmaller
PreprintsubmittedtoJournalofProcessControl October4,2016
Post-printversionofthearticle: M.Niederer,S.Strommer,A.Steinboeck,andA.Kugi,“Nonlinearmodelpredictivecontrolofthestrip
temperatureinanannealingfurnace”,Journal of process control,vol.48,pp.1–13,2016.doi: 10.1016/j.jprocont.2016.09.012
Thecontentofthispost-printversionisidenticaltothepublishedpaperbutwithoutthepublisher’sfinallayoutorcopyediting.
Airintake Furnaceinteriorwithinertgasatmosphere
Postcombustionchamber PyrometerP2(zrth)
RTH RTS
SSttrriipp
z }| { z }| {

Preheater
Heatingzones
Gas-fired
radianttube
Gasflow
Burner PyrometerP3(zrts=235m)
Pyrometer
P1(zdff)
Deflectionroll
z=0
Transfersectionwithair-lock
Direct-firedfurnace Indirect-firedfurnace
| {z }| {z }
F igure 1 : C o m b i n edd i r e c t - a n d i n d i r e c t - fi r e d s t r i pan n e a l i n g f u r n a c e .
responsetimesthanhumans. Duetothecomplexityandwide furnacemodel is linearized. In addition to the heating power
variety of operation situations of an annealing furnace, these ofthefurnace,thecontrolleroptimizesthethroughput,i.e.,the
control concepts are, however, expensive to install, commis- linespeed.
sion,andtune.
1.3. Motivation,objectives,andcontributionsofthiswork
Ahierarchicalcontrolconceptispresentedin[6].Inthetop-
Nonlinear model predictive control is an appropriate con-
mostlayer,set-pointvaluesforthelinespeedandthezonetem-
peraturesneededtoachievethetargetstriptemperaturearede- cept for strip temperaturecontrolof the consideredannealing
termined. Intheintermediatelayer,theswitchingtimesofthe furnace. It is a versatile optimization-based and anticipative
controlmethodthatissuitableforcomplexnonlinearmultiple-
individualset-pointvaluesaredeterminedusingasimplemodel
ofthestriptemperatureandoptimizationbasedmethods.Inthe inputmultipleoutputsystems.Moreover,itallowstheincorpo-
rationofvariouscontrolobjectivesandthesystematicconsid-
inner-mostlayer,simplelinespeedandtemperaturecontrollers
erationofinputconstraints,stateconstraints,andknowndistur-
controlthestriptemperatureandensurethatitiswithindefined
bounds. bances,e.g.,stripchanges.
Most developed model predictive controllers for strip tem-
A hierarchically structured model-based control concept is
peraturecontrolofannealingfurnacesarebasedonlinearsys-
also presented in [7]. The underlying semi-analytic furnace
temmodels. Thoughalinearmodelsimplifiesthecontrollaw,
modelisbasedonphysicalprinciplesandmeasuredsystemdy-
it may limit the capability and the accuracy of the obtained
namics. In the higher control layer, reference trajectories for
controller. Moreover,existingstriptemperaturecontrollersare
boththestripvelocityandthestriptemperaturearegenerated.
mainly developedforindirect-firedannealingfurnaces. How-
Inthelowercontrollayer,unknownparametersofthemodelare
ever, the annealing furnace considered in this paper features
recursivelyestimatedandthemassflowsoffuelaredetermined
also a direct-fired section. Since the direct- and the indirect-
usinggeneralizedpredictivecontrol[8,9].
fired furnace sections are physically coupled by the moving
A control concept without a hierarchical structure is pre- steelstrip,anintegratedcontrollerdesignforbothfurnacesec-
sented in [10]. Based on a linear model of the strip temper- tionsisrecommendable.Forthisreason,existingcontrolstrate-
ature, a model predictive controller for the heating power of giescannotbedirectlytransferredtotheannealingfurnacecon-
thefurnaceisdeveloped. Italsotakesintoaccountinputcon- sideredhere. Thismotivatesthetailoreddesignofanonlinear
straints.Theconstrainedoptimizationproblemissolvedbyus- modelpredictivecontrollerfornon-steady-statefurnaceopera-
ingquadraticprogramming[11]. tion.Itshouldrealizethefollowingcontrolobjectives:
The controllerpresentedin [12]is similarto thatpresented Accuratestriptemperaturecontrol
•
in[10]butisbasedonasimplenonlinearfurnacemodel. For Maximizationofthethroughputofthesteelstrip
•
theutilizationinalinearmodelpredictivecontrolconcept,the Minimizationofthefuelconsumption
•
2
Post-printversionofthearticle: M.Niederer,S.Strommer,A.Steinboeck,andA.Kugi,“Nonlinearmodelpredictivecontrolofthestrip
temperatureinanannealingfurnace”,Journal of process control,vol.48,pp.1–13,2016.doi: 10.1016/j.jprocont.2016.09.012
Thecontentofthispost-printversionisidenticaltothepublishedpaperbutwithoutthepublisher’sfinallayoutorcopyediting.
Thecontrollerhastorespectseveralconstraints: Adetailedfirst-principlesmodeloftheconsideredannealing
Boundsonthestriptemperature furnacecanbefoundin[13,14,15]. Thismodelincludesdif-
•
Boundsonthecontrolinputs ferent submodels for the flue gas, the radiant tubes, the wall,
•
Tosatisfytheserequirements,thefollowingcontrolinputsare thestrip,andtherolls.However,thefurnacemodelusedinthis
available: paper is a reduced model in terms of complexity, dimension,
Massflowsoffueltotheheatingzones andcomputationaleffort.Inthefollowing,themaindifferences
•
Stripvelocity betweenthefurnacemodelsarebrieflyoutlined:
•
However,thecontrollerdesignpresentedinthispapercanalso Inthereducedfurnacemodel,acoarserspatialdiscretiza-
•
beusedwhenthestripvelocityisprescribedbysubsequentpro- tionisusedthroughoutthefurnace.Thisleadstoareduc-
cessstepsandisthereforenotavailableasacontrolinput. tionofthemodelorder.
The contributions of this paper can be summarized as fol- The modeling of the combustion process of fuel inside
•
lows: theheatingzonesofthedirect-firedfurnaceissimplified.
Nonlinearfirst-principlesmodeloftheplant Hence,thecomplexityofthemodelisreduced.
•
Tailoredreal-timecapablemodelpredictivecontrollerfor The reduced model of a radiant tube inside the indirect-
• •
a strip annealing furnace based on the Gauss-Newton firedfurnacefeaturesonlyonetemperaturestate.Thisalso
method in combinationwith theanalytical calculationof leadstoareductionofthemodelorder.
the gradient and the approximated Hessian matrix using The reduced furnace model includes only temperature
•
anadjoint-basedmethod statesofrepresentative(reduced)radianttubes. Thetem-
Systematicoptimizationofthematerialthroughput peratures of the remaining radiant tubes are determined
•
Thefeasibilityoftheproposedapproachisdemonstrated fromtheserepresentativevaluesbasedonidentifiedmap-
•
by simulationstudiesforavalidatedhigh-fidelitymathe- pings.Thisleadstoafurtherreductionofthemodelorder.
maticalmodelofanindustrialannealingfurnacecompris- Inthefollowing,themostfundamentalequationsofthereduced
ing adirect-and an indirect-firedstrip annealingfurnace furnacemodelaresummarized.
section
2.1. Fluegas
1.4. Contents In the heating zones of the direct-fired furnace, natural gas
(CH )isburntinafuelrichcombustionprocess,i.e.,theexcess
4
The paper is organized as follows: Section 2 gives an
aircoefficientλsatisfiesλ < 1. Thecorrespondingstationary-
overviewof the mathematicalfurnacemodelthat is used as a
reactionequationreadsas
basis forthemodelpredictivecontrollerdesign. InSection3,
thefurnacetemperaturecontrolsystemisdescribed. Aformal CH +2λ(O +3.76N )
4 2 2
description of the controlobjectives is given in Section 4. In −→
χc CO+χc H +χc CO +χc H O+χc N .
Section5,theoptimalcontrolproblemisfirstspecifiedandthen CO H2 2 CO2 2 H2O 2 N2 2
a numericalsolutionalgorithmis proposed. An industrialap- Here, χc with ν = CO,CO ,H O,H ,N is the corre-
plication example in Section 6 demonstrates the capability of ν ∈ Sν { 2 2 2 2}
spondingnumberofmoles. Takingintoaccountthewater-gas-
thedevelopedcontroller.Finally,Section7givessomeconclu-
shiftreaction[16,17]
sions.
CO+H2O⇋CO2+H2,
2. Mathematicalfurnacemodel
whichreachesachemicalequilibriumif(TcinKelvin)
AsshowninFig.1,theconsideredannealingfurnaceconsists 4577.8 χc χc
ofadirect-andanindirect-firedsection,whicharephysically K(Tc)=exp 4.33 = CO H2O
− Tc χc χc
coupled by the moving strip. In the direct-fired furnace sec- ! CO2 H2
tion (DFF), there are four heating zones, each equipped with is satisfied, the unknowns χc can be determined from simple
a set ofburnersfornaturalgas. To preventthestrip fromox- molbalances. Tc denotestheνadiabaticflametemperature. Let
idation, thecombustionin thesezonesis controlledto befuel
M denotethemolarmassofacomponentν with =
ν ν κ κ
rich.Thus,thefluegascontainsunburntproductswhichareox- CH ,O ,N . Themassflowsofthecomb∈usStio∪nSproductSsm˙c
idizedinapostcombustionchamberbyaddingfreshair. The { 4 2 2} ν
withν readthereforeas
ν
heatreleasedinthispostcombustionisusedtopreheatthestrip ∈S
inthepreheater. Theindirect-firedfurnacesectionfeaturesan m˙c =m˙b χcνMν. (1)
inertgasatmospheretopreventthestripfromoxidationandis ν CH4MCH4
dividedintoaradianttubeheatingsection(RTH)andaradiant
tubesoakingsection(RTS).BothsectionsareequippedwithW- Moreover,themassflowsoffuelm˙b andthecombustionair
CH4
shapedgas-firedradianttubesthataregroupedintothreeheat- m˙b +m˙b arecoupledintheform
ing zones. The strip temperature in the indirect-fired furnace O2 N2
section is thereforemainlycontrolledby the surfacetempera- χbM
m˙b =m˙b κ κ (2)
tureoftheradianttubes. κ CH4MCH4
3
Post-printversionofthearticle: M.Niederer,S.Strommer,A.Steinboeck,andA.Kugi,“Nonlinearmodelpredictivecontrolofthestrip
temperatureinanannealingfurnace”,Journal of process control,vol.48,pp.1–13,2016.doi: 10.1016/j.jprocont.2016.09.012
Thecontentofthispost-printversionisidenticaltothepublishedpaperbutwithoutthepublisher’sfinallayoutorcopyediting.
withκ and(κ,χb)= (CH ,1),(O ,2λ),(N ,7.52λ) . transferinsidethefurnace,thetemperaturestatesoftheremain-
∈Sκ κ { 4 2 2 }
Inthedirect-firedfurnace,thestripisheatedbythehotflue ingpipes j=2,3,4oftheradianttubeareapproximatedby
gas that is mainly characterized by its mass and temperature.
For modeling the flue gas, the furnace is discretized into Ng Trj,i=aijTr,i.
volumezones,whereeachzoneisassumedtobeawell-stirred
i.e.,it is assumed thatthe temperatureratios ofthe individual
reactor. Thevolumezonesarenumberedinascendingorderin
radiant tube pipes are constant. The coefficients aj [0,1]
flowdirectionofthefluegas.Neglectingthefluegasdynamics i ∈
wereobtainedinsimulationstudies.
[13],themassbalanceforeachcombustionproductν and
ν
∈S
theenthalpybalanceofanindividualvolumezoneireadsas
2.3. Wall
0=m˙ +m˙c m˙ (3)
ν,i−1 ν,i− ν,i The furnace wall consists of J different layers and is dis-
0= m˙ν,i 1hν(Tg,i 1) m˙ν,ihν(Tg,i) cretizedintoNwwallsegmentsalongthefurnace.Tomodelthe
− − − heattransferthroughthewall, theGalerkin weightedresidual
νX∈Sν νX∈Sν
methodisemployedtosolvetheone-dimensionalheatconduc-
+ m˙b h (T )+Q˙ . (4)
κ,i κ κ,i g,i tion equationwith aDirichletboundaryconditionat theouter
κX∈Sκ surface and a Neumann boundary condition at the inner sur-
Here,m˙ andm˙ denotethemassflowofacomponentν face. Ifthestationarysolutionofamulti-layeredfurnacewall
ν,i 1 ν,i
thaten−tersorleavesthevolumezonei.Forcalculatedvalue∈s isusedastrialfunction,thelumped-parametermodelofthe(in-
ν
S
χcν,theunknownmassflowsm˙ν,iwithν∈Sνthatarerequiredin ner)surfacetemperatureTw,i ofawallsegmentireadsas[14]
(4)canbeeasilydeterminedbysolvingalinearsetofequations
thatconsistsof(1),(2),and(3).
d 1 K
In (4),athtνh(eTt)emisptehreatsupreecTifi.cThenetfihrasltpytwooftearmcosminpo(4n)endtenνot∈e dtTw,i= K1q˙w,i− K12 Tw,i−To , (6a)
ν κ
S ∪S (cid:0) (cid:1)
the enthalpy flows associated with the incoming and the out- withtheabbreviations
going bulk flow in the respectivevolume zone. The enthalpy
flow of the fuel and combustion air is described by the third J d −2 J ρ c k J d 3 J d 3
w,j w,j w,j w,j w,l w,l
tirenorcnmlmu.deMenst.oarBlelyotvhseoerlr,vmtihnaeglnitnhetetehrnaeocanttiloiflnnoeswaroQfs˙egtth,ioeifnflteouqeuthagetaiorsenwsspi(et4hc)t,iivttsheeeznflovuniee- K1 =Xj=1 kw,j Xj=1 3 Xl=j kw,l −l=Xj+1kw,l(6b)
gastemperaturesTg,iofallvolumezonesi∈{1,...,Ng}canbe J dw,j −1
determined. K2 = k . (6c)
j=1 w,j
2.2. W-shapedradianttube Here,dXdenotesthethicknessandthematerialpropertiesc ,
w,j w,j
k ,andρ denotetheheatcapacity,theheatconductivity,and
Insidetheindirect-firedfurnacesection,N W-shapedradiant w,j w,j
r
themassdensityofthelayer j 1,...,J . Moreover,q˙ is
tubesaregroupedintoseveralheatingzones. Allradianttubes ∈ { } w,i
theheatfluxintotheinnerwallsurfaceduetothermalradiation
ofasingleheatingzonearesuppliedwiththesameamountof
andconvection. T isthetemperatureattheouterwallsurface
fuel m˙ that can be continuously adjusted between a mini- o
mum aCnHd4a maximum value. As shown in [14], the heat Q˙ whichisassumedtobeequaltotheambienttemperature.
c,i
releasedbythecombustionprocessinsideasingleradianttube
iandtransferredintothetubewallcanbedescribedbythestatic 2.4. Stripandrolls
mappingQ˙c,i = ψi(m˙CH4). Thisrelationcanbeobtainedbyan Assumethatthestripentersthefurnacewithambienttemper-
energybalancemodeloftheradianttubeandmeasurementdata atureT andmovesalongthedirectionzwithavelocityv >0.
o s
fromtherealplant. Here,zisthespatiallyfixedlengthcoordinate,cf.Fig.1.Inside
LetgiQ˙c,i,withtheweightingfactorgi >0,betheheatinput thefurnace,thestripisheatedtoadesiredtargettemperature.
thatisallocatedtothefirststraightpipeoftheradianttube,cf. Usingthefirstlawofthermodynamics,theevolutionofthestrip
[14].ThepipehasthesurfaceareaSrandthewallthicknessdr. temperatureTs(z,t)inanEulerianframeworkreadsas
Itsmaterialpropertiesarethespecificheatcapacityc andthe
r
mass density ρr. Using asimple heatbalance, thedifferential ∂Ts(z,t) = 2q˙s(z,t) v ∂Ts(z,t). (7)
equationofthetemperaturestateT ofthepipereadsas ∂t ρ c (T (z,t))d − s ∂z
r,i s s s s
d 1 Here, ds is the strip thickness, ρs the mass density, cs the
T = q˙ +q˙ (5)
dt r,i ρrcr(Tr,i)dr c,i r,i temperature-dependentspecific heat capacity, and q˙s the heat
fluxintothestripsurfaceduetoconduction,radiationandcon-
(cid:0) (cid:1)
withthe(local)heatfluxq˙ = gQ˙ /S attheinnerpipesur- vection.
c,i i c,i r
faceandtheradiativeheatfluxq˙ fromthefurnaceinteriorto Letthestripbespatiallydiscretizedinto N sectionsofuni-
r,i s
theouterpipesurface. Forthecalculationoftheradiativeheat form length ∆z with N +1 grid points z. Using the upwind
s i
4
Post-printversionofthearticle: M.Niederer,S.Strommer,A.Steinboeck,andA.Kugi,“Nonlinearmodelpredictivecontrolofthestrip
temperatureinanannealingfurnace”,Journal of process control,vol.48,pp.1–13,2016.doi: 10.1016/j.jprocont.2016.09.012
Thecontentofthispost-printversionisidenticaltothepublishedpaperbutwithoutthepublisher’sfinallayoutorcopyediting.
scheme[18]fordiscretizingthetransporttermin(7),thestrip
temperatureT (t)=T (z,t)atagridpointz readsas
s,i s i i d
T =f (τ,T ,T ,u) (11a)
d 2q˙ T T dτ Σ 1 Σ g
s,i s,i s,i 1
dtTs,i=ρscs(Ts,i)ds −vs −∆z − (8) 0=f2(τ,TΣ,Tg,u) (11b)
withT =T . with the initial conditions T (τ ) = T and T (τ ) = T .
s,0 o Σ 0 Σ,0 g 0 g,0
ThestripisguidedthroughthefurnacebymeansofN rolls Equation(11a)includestheordinarydifferentialequations(5),
d
thataredesignedashollowcylinderswiththesurfaceareaS (6), (8), and (9) for the radiant tubes, the wall, the strip, and
d
andthewallthicknessd .Therelevantmaterialparametersare therolls,respectively. Equation(11b)representsasetofnon-
d
the heatcapacity c and the mass density ρ . Using a simple linearequationsfordeterminingthe fluegastemperatures,cf.
d d
heatbalance,thelumped-parametermodeloftherolliwiththe (4). In (11), TΣ are system states and Tg are algebraic vari-
homogeneoustemperatureT readsas ables. Themainbenefitsofusingthetimetransformation(10)
d,i
areaddressedinmoredetailinSection2.6.2andSection5.1.
d 1 Sc Sc
T = 1 d,i q˙ + d,iq˙c . (9)
dt d,i ρ c (T )d − S d,i S d,i
d d d,i d d ! d !
2.6.2. Discrete-timesystem
Here,Scdenotesthecontactareabetweentherollandthestrip.
d Theimplementationofthefurnacemodelonacomputersys-
The heat fluxes q˙ and q˙c capture radiation and convection
d,i d,i temrequiresthediscretizationofthetimedomain. Letτk with
withthefurnaceinteriorandconductionwiththestrip,respec-
k N0bethesamplingpointsalongthe(normalized)timedo-
tively. ∈
main. Thecorrespondingsamplingperiodis∆τ = τ τ .
k k+1 k
−
The discrete-time representation of the semi-explicit differen-
2.5. Heattransfermechanisms tialalgebraicsystem(11)can,forinstance,beobtainedbyap-
plyinga1-stagehalf-explicitRunge-Kuttamethod[25]. Thus,
Theindividualsubmodelsoftheconsideredfurnace,i.e.,the
thediscrete-timesystemreadsas
modelsofthefluegas,theradianttubes,thewall,thestrip,and
the rolls, are interconnected by the heat transfer mechanisms
T =T +∆τ f (T ,T ,u ) (12a)
radiation, convection, and conduction. Due to the high tem- Σ,k+1 Σ,k k 1,k Σ,k g,k k
0=f (T ,T ,u ). (12b)
peraturesinsidethefurnace,thermalradiationisthedominant 2,k+1 Σ,k+1 g,k+1 k+1
modeofheattransfer. Assuggestedin [13,14], thezoneand
Fornumericalstabilityof(12),thenecessarycondition
thenet-radiationmethod[19,20,21,22]areusedforanalyzing
theheatexchangebyradiation.Bothmethodsuseaspatialdis- ∆τ L∆τk
cretizationofthefurnacetoobtainalinearrelationbetweenthe L k =v vs 1 (13)
localradiativeheatflowsandthefourthpowersoflocalsurface ∆z s ∆z ≤
andgastemperatures.Convectiveheattransferbetweentheflue
has to be satisfied. Condition (13) is the Courant-Friedrichs-
gasanditssurroundingsurfacesismodeledbyusingNewton’s
Lewy(CFL)condition[18]becauseL∆τ /v isthetimestepin
k s
law ofcoolinganddimensionlessnumbersfromfluid dynam-
the originaltime domain, cf. (10). TheCFL conditiongener-
ics[23,24]. Furthermore,theconductiveheatexchangeatthe
allydescribesthenexusbetweenthespatialandtemporaldis-
roll-stripcontactisdescribedbymeansofthermalresistances.
cretization of the advection equation (7) and the strip veloc-
ity. However, (13) includes no strip velocity due to the used
2.6. Assembleddynamicalsystem timetransformation(10). Foracomputationalefficientfurnace
model,therealizedtimestepsshouldalwaysbeaslargeaspos-
2.6.1. Continuous-timesystem
sible,i.e.,theequalitysignin(13)shouldholdifpossible.Note
Foracompactnotationoftheassembledfurnacemodel,the
thatagivennormalizedtimegridwhichsatisfiestheequalityof
surface temperatures of the radiant tubes, the wall, the strip,
(13)doesnotneedto beadaptedifthestripvelocitychanges.
and the rolls are summarized in the vector TΣ ∈ RNΣ with Inviewofmodelpredictivecontrol,wherethenumericaleffort
N = N +N +N +N = 60 andthefluegastemperatures
Σ r w s d should be minimized, this is a benefit of using a normalized
are summarized in the vector Tg ∈ RNg with Ng = 13. The time domain, especially because the strip velocity is an opti-
massflowsoffuelintotheheatingzonesaresummarizedinthe
mizationvariable.
vectorm˙. Thevectoru = [vs,m˙T]T ∈ RNu with Nu = 8 thus IntroducingthevectorT = TT ,TT T,(12)canberewrit-
containsallcontrolinputs. Inthefollowing,thetimetransfor- k Σ,k g,k
tenintheform
mation h i
L 0=F (T ,T ,u ,u ) (14a)
dt= dτ (10) k k+1 k k+1 k
v
s
with
withanarbitrarylengthL,thestripvelocityv ,andthenormal-
s
T T ∆τ f (T ,T ,u )
izedtimeτisused. Inthenormalizedtimedomain,theassem- Fk = Σ,k+1f− Σ(,Tk− ,kT1,k ,Σu,k )g,k k (14b)
bledcontinuous-timestate-spacemodelofthefurnacereadsas 2,k+1 Σ,k+1 g,k+1 k+1
" #
5
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temperatureinanannealingfurnace”,Journal of process control,vol.48,pp.1–13,2016.doi: 10.1016/j.jprocont.2016.09.012
Thecontentofthispost-printversionisidenticaltothepublishedpaperbutwithoutthepublisher’sfinallayoutorcopyediting.
1000 evolution of the strip matches well the measurement data. It
canbeinferredfromFig.2thatthefurnacemodelpresentedin
K) [13,14,15]ismoreaccuratecomparedto thereducedmodel.
( 900 However,thereducedmodelstill capturesthemostimportant
)
T(zsdff ndoarndlindeeaskrteoffpePctCs a(4ndGrHeqzu,i1re6sGoBnlyRA1.M8s) fCoPrUth-teimsiemounlaatiosntano-f
800 1h furnace operation. This is 40 times faster than the model
presentedin[15].Hence,thereducedfurnacemodelissuitable
1200 forrepeatedevaluationinanoptimizationalgorithm.
K) 1100 3. Furnacetemperaturecontrolsystem
(
T(z)srth 1000 sysTtehme misaitno oenbsjeucrteivtehaotfththeestceoenlsisdtreirpesdatreemhpeeartaetdurteocpornetdroe-l
fined target temperatures in order to achieve the desired ma-
terialpropertiesofthefinalproduct. AsoutlinedinFig.3,the
900
1200 temperaturecontrolsystemconsistsofthreehierarchicallayers.
Processdata
(K) 1100 Process Model u˜ Velocity u hs(T)
control predictive andfuel Furnace
T(z)srts 1000 system control control
Tˆk State
estimator
900
0 2 4 6 8 10 12
t (h) Figure3:Hierarchicalfurnacetemperaturecontrolsystem.
Furnacemodel[13,14,15] Measurement The topmost layer is the process control system that coor-
Reducedfurnacemodel dinates all tasks of the considered strip processing line. For
example, it schedules the production sequence depending on
Figure2:Striptemperaturesatthepyrometerpositionszdff,zrth,andzrts. priorityandeconomicvalueoftheindividualstrips. Moreover,
itprovidesthedatasetscharacterizingthestrips(processdata).
Thesedatasetincludetargetstriptemperatures,maximumstrip
andtheinitialconditionT =[TT ,TT ]T.
0 Σ,0 g,0 temperature limits, geometric dimensions, i.e., width, thick-
Themostimportantprocessquantityisthestriptemperature
ness,length,andmaterialparameterslikemassdensityandheat
which can be monitoredby means of threeradiation pyrome-
capacityofeachstrip.
ters,cf.Fig.1,atthestrippositionsz=z withi dff,rth,rts .
i ∈{ } In the intermediate control-layer, a model predictive con-
Theoutputfunction
troller selects reference trajectories u˜ for the system inputs u
so that the strip temperatureis controlled to its desired target
h (T ) (15)
s k
temperature. The design of the modelpredictivecontrolleris
provides the simulated strip temperature at these positions. themainobjectiveofthispaperandisdescribedindetailinthe
Equations (14) and (15) serve as the basis for the controller followingsections.
designinthefollowingsections. Mostsystemstatesthatarerequiredasinitialcondition,i.e.,
feedback,formodelpredictivecontrolarenotmeasured.There-
fore,T hastobeestimatedbymeansofastateestimatorthat
2.7. Validationofthereducedfurnacemodel k
is based onthefurnacemodel(11). Inadditionto the system
Since the most importantmodel output is the strip temper- inputs,theestimatorusesmeasurementdatafromtheradiation
ature at the pyrometer positions, the reduced furnace model pyrometerstoimprovethecurrentstateestimationTˆ .
k
is mainly validated by means of measurement data of these Thelowermostlayerincludesdecentralizedcontrolloopsfor
temperatures. The measurement data used for this validation realizingthe desired controlinputs, i.e.,the strip velocityand
stemfromameasurementcampaignconductedattherealplant. the mass flows of fuel. In the current furnace configuration,
Mostofthephysicalparametersusedinthefurnacemodelareat thistaskisaccomplishedmainlybymeansofPIcontrollers. It
leastroughlyknownfrommaterialhandbooksordesigndraw- is stressed that accurate fuel control is critical for safety rea-
ings. For the unknown strip emissivity, the estimated value sons as well as for the product quality. Since the model pre-
from[14]isused. dictivecontrollerprovidestrajectoriesoffuturesysteminputs,
Fig.2showsthesimulatedandmeasuredstriptemperatures efficienttwo-degrees-of-freedomcontrolconceptsmaybeused
at the positions z , z , and z . The simulated temperature for fuelcontrol, cf. [26, 27]. With such controlconcepts, the
dff rth rts
6
Post-printversionofthearticle: M.Niederer,S.Strommer,A.Steinboeck,andA.Kugi,“Nonlinearmodelpredictivecontrolofthestrip
temperatureinanannealingfurnace”,Journal of process control,vol.48,pp.1–13,2016.doi: 10.1016/j.jprocont.2016.09.012
Thecontentofthispost-printversionisidenticaltothepublishedpaperbutwithoutthepublisher’sfinallayoutorcopyediting.
furnacemightbeoperatedcloseratits limits. Forthefollow- ThevaluesT atthepositionsz andz canbedetermined
s±,k rth rts
ing,itisassumedthatu=u˜ holds,i.e.,theinnercontrolloops by using (17). Moreover, the strip temperatureat the pyrom-
areconsideredideal. eterposition z , i.e.,at theendofthedirect-firedfurnace,is
dff
constrainedby
4. Controltask T (z ) T (z ) T+ (z ) k (20)
s−,k dff ≤ s,k dff ≤ s,k dff ∀
Starting with the strip temperature, a formal description of in orderto avoid stripdamage,e.g.,heatbuckling,ifthestrip
thecontrolobjectivesandconstraintsisgiven. Forthedesired temperatureis raised too fast. Thelimits T (z ) are defined
s± dff
finalproductquality,thestripmustbeheatedtoitstargettem- similarlytothetemperaturelimitsshownin Fig.4. Forasin-
perature Td before reaching the end of the RTH section (py- glestripsection,atypicalheat-upcurvewiththedesiredtarget
s˜
rometer position zrth). This temperature level should then be temperature Tsd and corresponding temperature bounds Ts± is
maintainedup to theend oftheRTS section (pyrometerposi- showninFig.5.IntroducingthevectorT =[T (z)] ,
±s s± i i=dff,rth,rts
tion zrts). The strip temperature at the end of the direct-fired the constraints (18), (19), and (20) can be summarized in the
furnace (pyrometerposition zdff) is of minor importanceand, form
therefore,atargettemperaturedoesnotexist.
As indicated in Fig. 4, the target temperature Tsd˜ and the T−s,k ≤hs(Tk)≤T+s,k ∀k, (21)
boundsT onthestriptemperaturemayvaryfromstriptostrip
s±˜
andaredefinedusingLagrangiancoordinates. The(local)La- withtheoutputfunctionhs(Tk)from(15).
grangiancoordinatez˜isfixedforagivenmaterialpointwhereas At this point it should be noted that further control objec-
the Eulerian coordinate z is spatially fixed. The mapping be- tivescanbedefinedatanypositioninsidethefurnace. Thisis
tweenLagrangianandEuleriancoordinatesisgivenby becausethestateestimatorprovidesthefullsystemstate,inpar-
ticularthestriptemperatureprofilealongthefurnace.Ifprede-
t
finedheatingtrajectoriesaregivenforeachstrip,forexample,
z=z˜+ v (ξ)dξ. (16)
s
thedeviationbetweenthedesiredandtheestimatedtemperature
Zt0
profilecanbetakenintoaccountinthecontrolproblem.
Utilizingthetimetransformation(10),(16)resultsinthelinear
mapping
Ts(z) Ts+(zrth) Ts+(zrts)
z=z˜+L(τ τ ), (17)
− 0 Tsd Ts+(zdff)
which implies z = z˜ for τ = τ0. The target temperature Tsd,k Ts−(zrth) Ts−(zrts)
atagridpointτ describedinEuleriancoordinatescanthusbe
k
easilydeterminedbyusing(17). Ts−(zdff)
Ts˜ T+ z
Ts+˜,j−1 s˜,j Td Ts+˜,j+1 0 zdff zrth zrts
s˜,j DFF RTH RTS
Td
s˜,j+1
Td Ts−˜,j Figure5:Temperaturetrajectoryofastripsectionmovingthroughthefurnace
s˜,j 1
Ts−˜,j−−1 Ts−˜,j+1 z˜ switiitohntsarzgdefft,tzermthp,earnadtuzrretsT.sdandcorrespondingtemperaturelimitsTs±atthepo-
Strip j 1 Strip j Strip j+1
− Inadditiontotheprimaryobjectiveofaccuratestriptemper-
aturecontrol,themaximizationofthroughputofsteelstripand
Figure4:TargettemperaturesTsd˜ andtemperaturelimitsTs±˜ fordifferentstrips. theminimizationofenergyconsumptionaresecondarycontrol
objectives.Generally,themaximizationofthroughputρ d b v
In non-steady-state furnace operation, e.g., when a welded s s s s
istantamounttothemaximizationofthestripvelocity.Further-
jointthatconnectstwostripswithdifferenttargettemperatures
more,theminimizationoftheenergyconsumptionrequiresto
moves through the furnace, the desired target temperatures at
minimizethefuelflowsuppliedtotheheatingzonesofthean-
thepyrometerpositionsz andz generallycannotbereal-
rth rts nealingfurnace. Becausethesethreecontrolobjectivescanbe
ized simultaneously for both strips, at least not in the area of
antagonistic,theyareweightedintheoptimalcontrolproblem
theweldedjoint.However,thetemperaturesofbothstripsmust
accordingtotheirimportance.
remainwithincertainlimits, seeFig.4. Hence,thelocalstrip
Thecontrolinputs,i.e.,themassflowsoffuelandthestrip
temperatureshouldobey
velocity, are constrained in terms of both their absolutevalue
T (z ) T (z ) T+ (z ) k (18) andtheirslope.Hence,
s−,k rth ≤ s,k rth ≤ s,k rth ∀
Ts−,k(zrts)≤Ts,k(zrts)≤Ts+,k(zrts) ∀k. (19) u−k ≤uk ≤u+k, ∀k (22)
7
Post-printversionofthearticle: M.Niederer,S.Strommer,A.Steinboeck,andA.Kugi,“Nonlinearmodelpredictivecontrolofthestrip
temperatureinanannealingfurnace”,Journal of process control,vol.48,pp.1–13,2016.doi: 10.1016/j.jprocont.2016.09.012
Thecontentofthispost-printversionisidenticaltothepublishedpaperbutwithoutthepublisher’sfinallayoutorcopyediting.
v u u
u˙−k ≤ Ls,k k∆−τkk−1 ≤u˙+k, ∀k (23) fimorpionrdtiavnicdeuaolfstthreiprseescpteicotnivsebycotnhterowleoigbhjetcintigvemsactarincebseWad1j,uksatnedd
hu,k(uk,uk 1) W2,k.
−
To highlight a major benefit of using the time transforma-
mustbesatisfied inth|edis{crzete-ti}medomain. Here, theslope
tion(10),considerforamomentthattheoptimizationproblem
isapproximatedbythedifferencequotient. Thescalingfactor
isformulatedwithoutusingthistransformation. Ifthecontrol
v /Lisduetothetimetransformation(10).
s,k objectiveswerethenprioritizeddifferentlyforindividualstrips,
theweightingmatricesW andW woulddependonthestrip
1,k 2,k
5. Modelpredictivecontrol velocity(anoptimizationvariable).Thisfactwouldcomplicate
thecalculationofthegradientoftheobjectivefunction. How-
Inthissection,amodelpredictivecontrollerforthestriptem- ever,ifthetimetransformationis used,eachgridpointofthe
peratureisdeveloped. Thebasicideaofmodelpredictivecon- normalizedtimehorizoncorrespondstoastripsectionandthe
trolistorecurrentlysolveadynamicoptimizationproblemde- weighting matrices are therefore independent of the strip ve-
finedonafinitetimehorizon. Atthebeginningofeverytime locity. The strip length considered in the prediction horizon
horizon,thecurrentsystemstateestimatedbythestateestima- (τk0,τk1]isequaltothevalueLusedinthetimetransformation
torisusedasinitialstateoftheoptimizationproblem. Thisis (10), i.e.,thenormalizedcontrolproblemis notformulatedin
theessentialsourceoffeedbackusedinmodelpredictivecon- termsofafixed(real)timehorizonbutintermsofafixedstrip
trol. length.
The challenge in the implementation of a model predictive Withtheobjectivefunction(24),thediscrete-timesystemdy-
controlleris generally the computationallyexpensivesolution namics(14),andtheconstraints(21),(22)and(23),thefurnace
ofthedynamicoptimizationprobleminreal-time. Thisisthe controlproblemreadsas
case especially for complex and high-dimensional nonlinear
systemslikeannealingfurnaces. Oneoptiontoovercomethis minimize r (h (T ),u ) + r (u ) (25a)
csohlavlelednigneoisrdthearttothreeodputciemtihzeatcioonmppruotbalteiomnitsimjues.taTphpirsoaxpipmraotaeclhy ukk∈RK¯Nu Xk∈K¯ k 1,k s k k kW1,k k 2,k k kW2,k
∀ ∈
leadstoasuboptimalsolutionoftheoptimizationproblemand subjectto 0=F (T ,T ,u ,u ), k K¯, (25b)
k 1 k k 1 k k 1
may jeopardizethe controlaccuracy. However,it facilitates a − − − ∀ ∈
T =T ,u =u , (25c)
real-timeimplementationofthemodelpredictivecontrollerand k0 0 k0 0
isthereforeusedinthispaper. T−s,k ≤hs(Tk)≤T+s,k,∀k∈K¯, (25d)
u u u+, k K¯ (25e)
−k ≤ k ≤ k ∀ ∈
5.1. Constrainedoptimizationproblem u˙ h (u ,u ) u˙+, k K¯, (25f)
−k ≤ u,k k k−1 ≤ k ∀ ∈
For the finite time horizon(τ ,τ ] with τ τ = 1, the
controlobjectivesareimplementke0dbky1 theobjek1ct−ivekf0unction withtheabbreviationK¯ = k0+1,...,k1 andNk =(k1 k0)Nu
{ } −
optimizationvariables.
k1 A challengein solving the controlproblem(25) are the in-
kr1,k(hs(Tk),uk)kW1,k +kr2,k(uk)kW2,k (24a) equalityconstraints(25d). Inliterature,numerousapproaches
k=Xk0+1 forsolvingthiskindofproblemcanbefound,see,e.g.,[11,28].
However,iftheinequalityconstraintsareimplementedasthey
with
are,i.e.,ashardconstraints,theconstrainedoptimizationprob-
lemmightbeinfeasibleanditssolutionisgenerallycomputa-
L Td (z ) T (z )
r1,k(hs(Tk),uk)= svs,k " Tssd,,kk(zrrtths)−−Tss,,kk(zrrtths) # (24b) toifonaanlluymeexrpiceanlssivoelu.tiTonhiaslgisormithaimnliythbaescatousbeeincheevcekreydiwtehraettihoenr
L 1 theinequaltiyconstraintsareactiveornot. Tocircumventthis
r2,k(uk)= svs,k  qm˙vks,k  (24c) icsosnuset,raininepdarctiocnutlraorlipnrovbielwemof(2a5)reiasl-ttriamnesfoimrmpleedmienntotaatinona,lttehre-
andthequadraticform r =rTWrwhereWisa(diagonal) native representation with equality constraints only by using
k kW anonlinearinputtransformationandadditionalpenaltyterms,
positive semi-definite weighting matrix. The factor L/v in
s,k i.e.,softconstraints,intheobjectivefunction.
each term of the objective function (24) is inherited from the
timetransformation(10)appliedto anoriginalformulationof
5.2. Approximationofinequalityconstraints
theobjectivefunctionintherealcontinuous-timedomain.
Thefirsttermoftheobjectivefunction(24)ensuresstriptem- Sinceitisimportanttokeepthenumberofoptimizationvari-
peraturetrackingbypenalizingtheerrorbetweenthestriptem- ables at a moderate level, the number of optimization vari-
perature and its target temperature at the pyrometer positions ables is reduced, i.e., the solution of the optimization prob-
zrthandzrts,cf.Fig.1. Moreover,thesecondtermensureshigh lem is confined to a lower dimensionalsubspace of RNk. Let
material throughput and low energy consumption by penaliz- Ul RNu be the input vector at a grid point Tl of the coarse
∈
inglowstripvelocitiesandhighfuelconsumption.Therelative timegrid(T ,T ,...,T ,T )withT =τ andT =τ .
l0 l0+1 l1−1 l1 l0 k0 l1 k1
8
Post-printversionofthearticle: M.Niederer,S.Strommer,A.Steinboeck,andA.Kugi,“Nonlinearmodelpredictivecontrolofthestrip
temperatureinanannealingfurnace”,Journal of process control,vol.48,pp.1–13,2016.doi: 10.1016/j.jprocont.2016.09.012
Thecontentofthispost-printversionisidenticaltothepublishedpaperbutwithoutthepublisher’sfinallayoutorcopyediting.
Thenumberofoptimizationvariablesis thusreducedto N = ϕ(Vl) r(ξk)
l
(l l )N < N . Theinputvectoru atthenodalpointτ of
1 0 u k k k
the−finetimegridcanbecomputedbylinearinterpolationofthe 1
gridvaluesU intheform
l
∆
l1 0
uk = Φl(τk)Ul (26a) Vl ξk− ξk+ ξk
Xl=l0
withtriangularfunctions Fmigauxr(e0,ξ6k−:+∆In−puξkt,ξtkra−nsξfk+or+m∆a)tiown(shϕo(wVln)foarnsdcalaprenqaulatnytititeesrmonlyr)(.ξk) =
Φl(τk)= 0τTTTklll++−−11TT−−ll−Tτ−11kl eiifflsττekk.∈∈([TTll−,1T,l+T1l]] (26b) f(cid:13)(cid:13)(cid:13)orTihnestraenmcea,inbiyngcliansesqicuaallipt(cid:13)(cid:13)(cid:13)ieensal(t2y5-do)rabnadrr(i2e8r-)mceatnhobdesr[e3p0l,ac1e1d].,
The reduced controlperformancedueto this alternativeinput tHeromwever, for the considered control problem, the additional
parameterizationisusuallyrewardedbylesscomputationalef-
fortforsolvingtheoptimizationproblem. k1 l1
r (h (T )) + r (V) , (31a)
Thecoarsetimegridforthecontrolinputssuggeststhatthe 3,k s k W3,k 4,l W4,k
second term of the objective function (24) is evaluated at the k=Xk0+1(cid:13) (cid:13) l=Xl0+1(cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
time instancesTl with thevaluesUl, wherel = l0 +1,...,l1. with (cid:13) (cid:13) (cid:13) (cid:13)
Moreover,theconstraints(25e)and(25f)arereplacedby
r (h (T ))=max 0,T +∆ h (T ),h (T ) T+ +∆
U U U+, l L¯ (27) 3,k s k −s,k s− s k s k − s,k s
−l ≤ l ≤ l ∈ (cid:16) (31(cid:17)b)
U˙ h (U,U ) U˙+, l L¯ (28)
−l ≤ U,l l l−1 ≤ l ∈ r4,l(V)=max 0,U˙−l +∆r−hU,l(φl−1(Vl−1),φl(Vl)),
Uw˙it,hatnhdetahsesoabcibarteevdiaftuinocntiL¯on=hlU,l+, t1h,e.a.s.s,olci.atedlimitsU±l and (cid:16) hU,l(φl−1(Vl−1),φl(Vl))−U˙+l +∆r ,
±l {0 1} (31(cid:17)c)
Theconstraints(27)aretakenintoaccountbytheinputtrans-
formation the assembled input vector V = [Vl]l=l0+1,...,l1, and φl0(Vl0) =
U = u ,isaddedtotheobjectivefunction(24). Anexample
l0 0
Ul =φl(Vl)=diag{U+l −U−l}ϕ(Vl)+U−l (29a) ofthechosenapproachisshowninFig.6forageneralquantity
ξ constrainedto[ξ ,ξ+]. Howexactlytheoriginalconstraints
k k− k
withthenonlinearmapping areadheredto,canbecontrolledtoacertainextentbythepa-
rameters∆ > 0withi s,r andtheweightingmatricesW
1 i ∈ { } i
ϕ(V)= (1+tanh(2V)) (29b) withi 3,4 . Notethatusingthisapproachmaycauseminor
l 2 l ∈ { }
violationsoftheoriginalrestrictions(25d)and(28). However,
andthenewunconstrainedinputV.Here,diag(U)denotesadi- for the consideredcontrol problem, this approach is perfectly
l
agonalmatrixcontainingtheelementsofU. Themathematical acceptable.
operationtanh()ismeanttobeindividuallyappliedtothere- Theobjectivefunctionfinallyincludestheterms(24), (30),
·
spectivevectorelement.Moreover,1∈RNu representsavector and(31a). LetT=[Tk]k=k0+1,...,k1 betheassembledtemperature
with unityelementsonly. Ascalarrepresentationofthemap- vector.Introducing
ping (29b) is shown in Fig. 6. Note that the constraints (27)
areautomaticallysatisfiedifUliscomputedaccordingto(29a). c1(T,V)= c1,k(Tk,V) k=k0+1,...,k1
Clearly,ϕ(V) [0,1]from(29b)representsanormalizedinput
l r (h (T ),ψ (V))
vector.Thema∈pping(29b)requirestheadditionalterm =(cid:2) 1,k s k(cid:3) k
r (h (T ))
" 3,k s k #k=k0+1,...,k1
ε l1 VTV (30) r2,l(Vl)
l=l0+1 l l c2(V)= r4,l(V)
X  V 
wtioitnh(a2n4)aribniotrradreyrvtaoluaveoεid>s0intgoublaeraadrcdse,dctfo.[t2h9e]o.bFjoerctaivceofnucnisce- thefinalobjectivefunctionlreadsl=als0+1,...,l1
notation,thefunction,cf.(26)and(29),
C(V,T)=C (V,T)+C (V)= c (V,T) + c (V)
l1 1 2 k 1 kWc1 k 2 kWc2
ψ (V)= Φ(τ )φ(V)
k l k l l
with diagonal weighting matrices W and W that includes
Xl=l0 the (diagonal)entries of the matricesc1W, i =c21,3 , and W,
i i
{ }
isintroduced,i.e.,u =ψ (V). i= 2,4 ,aswellasεfrom(30),respectively.
k k
{ }
9
Post-printversionofthearticle: M.Niederer,S.Strommer,A.Steinboeck,andA.Kugi,“Nonlinearmodelpredictivecontrolofthestrip
temperatureinanannealingfurnace”,Journal of process control,vol.48,pp.1–13,2016.doi: 10.1016/j.jprocont.2016.09.012
Thecontentofthispost-printversionisidenticaltothepublishedpaperbutwithoutthepublisher’sfinallayoutorcopyediting.