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UNIVERSITY

OF TRENTO

DEPARTMENT OF INFORMATION AND COMMUNICATION TECHNOLOGY

38050 Povo – Trento (Italy), Via Sommarive 14

http://www.dit.unitn.it

SVM PERFORMANCE ASSESSMENT FOR THE CONTROL

OF INJECTION MOULDING PROCESSES

AND PLASTICATING EXTRUSION

Davide Anguita, Andrea Boni and Luca Tagliafico

January 2002

Technical Report # DIT-02-0035

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Moulding Processes and Plasticating Extrusion

Davide Anguita 

,Andrea Boni y

, and Luca Taglia co z Correspondingauthor: Andrea Boni

University of Trento,Via Mesiano, 77 I-38050Povo diMesiano (Trento){ Italy Phone/Fax: +39-0461-882440/+39-0461-881977

e{mail: andrea.boni@ing.unitn.it January22, 2002

Abstract

This paper presents the application of a new and promising learning algorithm based on kernel methods, i.e., support vector machines (SVMs), for the control of injectionmouldingprocessesandplasticatingextrusion. Inparticular, themain pur-pose of this work is to assess the e ectiveness of the method when applied to such kindsofindustrialprocesses,characterisedbyalargenumberofvariablesandstrictly correlatedbynonlinearrelationships. First,weanalysetheinjectionprocessby devel-opingasimpli edmodel, thenweidentifyitbyusingasupportvectormachine. The referenceofthecontrol systemistrackedthroughthedesignofacontrol blockbased onthestructureoftheSVM.



DavideAnguitaiswithDIBE,UniversityofGenova,Italy,e{mail: anguita@dibe.unige.it y

AndreaBoniiswiththeUniversityofTrento,Italy,e{mail: andrea.boni@ing.unitn.it z

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The extrusion process of molten polymeric materials is a well{established technique for the production of a large number of end products or components commonly found in a multitude of consumer goods. The wide variety of polymer blends and production techniques makes it quite diÆcult to de ne theoretically all the possible aspects of the controlof thelargenumberof process variablesinvolved (thermal and mechanical),fora systemwithahighlynonlinearbehaviour. Eachvariable(temperature, owrate,pressure, etc.), has a great impact on boththe nal qualityof a productand theproductionrate. Therefore, thechoice of thecontrolstrategyand thedevelopment of newregulationtools become crucial to have more eÆcient, energy saving and environmentally safe extrusion processes.

In thispaper, we proposetheuseof anew learningalgorithmto develop aframework (based on intelligent modules), to be inserted in the regulation chain of an extrusion process. Our maingoal isto verifyits actualvalidityon the basisof experimentscarried outby usingasimpli edthermaland uiddynamicmodelof theinjectionprocess.

The injection mouldingprocess (IMP) is one of the mostimportant and widely used extrusionprocesses(Rosato1995),andseveralnewcontrolapproacheshavebeenproposed in the past few years (Agrawal et al. 1987, Tsai and Lu 1998). In the IMP, the molten materialisinjectedinto themouldcavityat highspeedandhighpressurebymeansofan axiallymovingscrew,activated bya pistonmovingat a given axialspeed(Figure1).

Thequalityoftheextrudateishighlydependentonthetemperatureuniformityinside thepolymerduringtheinjection,ontheworkingpressure,onthescrewtranslationvelocity, on the homogeneity of the physical properties obtained by the mixing process in the metering section, and on the temperature pro le along the barrel. The temperature,in

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due to viscous dissipation. Each of these parameters must be carefully controlled to achieve asatisfactory qualityof extrudates. Oneof themainoperatingparameters is the polymertemperatureatthenozzleexit; thistemperatureisaverycomplicatedfunctionof materialproperties,processcontrolparameters,screwgeometry,durationsofthedi erent stages, residence time, and so on. It is indeed rather cumbersome to develop a global theoretical model able to describe the extruder behaviour in terms of the input-output system parameters. Therefore, the application of modern identi cationtechniques based onneuralnetworks(ina generalway), inparticular,onsupportvectormachines(SVMs), seems to be quite attractive, especially for regulation and control purposes. The SVM module we proposein thispaperactsas anidenti cationblock inthecontrolloop of the IMPprocess.

In general, an identi cationproblem can be de ned asthe process throughwhich an unknown function,  : <

d

! <, describing the behaviour of a dynamic system, is esti-mated on the basis of some of its samples, Z = f(x

i ;t

i )g

i=1:::np

. Usually, this problem can be tackled when the input/outputsignals of thesystem considered can be observed, but the system dynamics (i.e., the structure of ) is not known. Typical applications include: time{series forecasting, identi cation and control of nonlinear systems, signal and image processing, and others. Support Vector Machines (SVMs) (Cristianini and Shawe{Taylor 2000) area new paradigmthat have recently beenproposedto accomplish pattern{recognitionandfunction{approximationtasks, hencetheyrepresentanattractive approach to solve the injection mouldingproblem just described. There is a long list of theoreticalandpracticaladvantages ofSVMsoverotherconnectionistregressionmethods (e.g., Multi-Layer Perceptrons - MLPs (Bishop 1995)). Oneof themost appealing prop-erties is certainly the possibilityof solvinga quadratic programmingproblem subject to

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tionalgorithmintheMLPcase. Followingtheearlytheoreticalstudiesofthepropertiesof thisnew learningapproach,di erentapplicationsarestartingto emerge, especiallyinthe eld of optimal controlof nonlinearsystems, as described in recent works (e.g., Suykens et al. 2001).

The rststepindevelopingacontrolsystembasedonanSVMistoassessifSVMsare capable to identify and reproduce the dynamic behaviour of the IMP. The basic idea of thepresentpaperistodevelopasimpli ed,thermal, uiddynamicmodeloftheinjection stage, studying the SVM behaviour during the identi cation of the system for dynamic controlpurposes. Thepaperisorganisedasfollows: thenextsectiondescribesthethermal, uidand dynamicmodel used to test the SVM method, brie youtlined in Section 3. In Section4, we present theidenti cation{control modelsand scheme thatwe have builtto control the polymer temperature. The designs of the SVM and control blocks are also extensivelydiscussed. InSection5,wedealwiththenumericalexperimentscarriedouton each buildingblockof thewholecontrolsystem; theperformancesof thedynamicmodel, the SVM and each control block are reported and discussed. Finally, in Section 6, we summarisetheobtainedresultsand proposefurtherlinesof research.

Inthefollowingtext,weshallindicatethevectorsandthematriceswithlowercaseand uppercaseboldfaceletters, respectively; Tables1, 2 and 3give the variables and symbols usedinthispaper.

2 Thermal, uid anddynamic model of the injection process

The injection process can be summarised as shown in Figure 2. In the rst stage, the polymer is fed by using powder or pellets in the hopper: then it is pushed forward by

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the injectionchamber. The polymer is molten inside thecompression section by electric heating(barrelheaters)andviscousdissipation. Finally,itreachesauniformtemperature andthermophysicalpropertiesinthemeteringsection. Whilethemoltenpolymer llsthe injectionchamber, the screw goes slightly back in order to allow the desired quantity of polymerto accumulate inthenozzle, andnew materialisfed fromthehopper. When the right polymercharge is readyto be injected, thepistonpushes thescrew (which behaves likea ram)axiallyalongthebarrel,thusallowingthemouldto be lledwiththepolymer melt. A holdingpressureis kepton the backof thepiston untilthe wholemouldis lled and thematerial insideit starts to solidifyowingto external refrigerationsystems. After thepolymerat thenozzleexit (i.e.,mouldentrance) hassolidi ed,themouldis removed and the cycle starts again. The whole process can be subdividedinto three mainstages, after thedieremoval:

1. Charge lling

2. Injection(translating{ screw)

3. Mould llingandcooling(holding { screw)

A further, shortstage can be added, which includesthe removal of the lledmould and theinsertionof a new,emptymouldreadyforthenext cycle.

Neglecting, as a rst approximation, the uid compressibility and keeping in mind that the time-pressure pro le at the nozzle exit is mainly driven by the cavity{pressure patterninsidethemould,theoutletpressurehasbeenassumedtobeanindependent,given boundary condition, which may vary during the injection stage. For the same reason, the polymer introduced into the chamber by rotating the screw is assumed to be at a given, uniform temperature at the screw/polymer interface. The operating parameters

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theimposed displacement velocity of thescrew duringthe injectionstage, and theheat{ transfercoeÆcientsbetweenthepolymerandtheinnercylinderwalls,whichcanbeheated and keptat thedesired temperature.

Forthepresent simpli edapproach,aone{dimensional, thermal, uiddynamicmodel hasbeendeveloped,makinganexplicitmarching{time nite{volumeanalysistocalculate, at each timestep, thetemperatureand pressuredistributionsinside thescrew tip/nozzle assembly. All the di erent contributions to the energy balance have been introduced: conduction,convection, viscous{dissipatione ects and dynamicbehaviour. Furthermore, all the thermophysical properties have been assumed to vary with the temperature, the shear{stress and shear{rate, as required by the study of the ow behaviour of polymer melts,whicharestrongly non{Newtonian uids.

The owchannelhasbeensubdividedintosmall nite{volumeelements, withvariable crosssections( ow passages),dependingon thechannelgeometry,asshown inFigure3.

The mass, momentum, energy, and constitutive equations have been integrated over time, usingdi erent particularsolutions in each of the three mainsteps of theextrusion mouldingprocess, i.e., injection, holdingand mould removal, and llingof a new charge. Intheinjectionstage,the owbehaviourhasbeensimulatedbythemomentumequation, appliedto each volumeintheform:

p j i+1 =p j i +p j i ; i=1;:::;N j ; j=0;:::;J (1) where p j i

is the pressure on the left side in the element i, and p j i

is the variation in pressurealong the element i duringthe time interval j, as shown inFigure 4. N

j is the numberof spaceelements, andX

j

ischosenoverthetimeintervalj. Dependingon the kindof study beingdeveloped (simulation, control, design oroptimisation), the pressure

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ofthescrew{tipworkingconditions(interfacialsurfaceareaand pushingforceduringthe injectionstage). Iftheaxial forceF

j s

on thescrew is givenfor eachtime intervalj of the process (as usuallyhappens ingiven conterpressure{controlapproaches), thepressureon the rst polymerelement is:

p j 1 = F j s A s (2) A s

stands for the frontal surface area of the screw. The evaluation of p j i

has been performed by the simpli ed method of \representative viscosity," which allows the rela-tionships derived forNewtonian material ows to be applied also in the case of polymer melts. Asa result,thepressuredropinthechannelis given by:

p j i = _ V j  j i ~ K j i f pi (3) where f pi

is a cross-section coeÆcient that is 1 for a circular cross-section, _ V j

is the volumetric ow rate, 

j i

is the local viscosity, and ~ K j i

is the so-called die conductance, which takes onthe form:

~ K j i = A 3 i 2X j P 2 i (4) forconstant cross-section owpassages,and theform:

~ K j i =  A 3 i 2X j P 2 i  0 B @ 3  Rmax R min 1  1  R min Rmax  3 1 C A (5)

forconvergingdies. AandP arethelocalcross{sectionarea[m 2

]andthewettedperimeter [m], respectively. To take into account the possibilitythat the screw axial velocity,

_ X

s , maybevariedduringtheinjectionforcontrolpurposes,following agiven velocity{pro le in time, the volumetric ow rate is variable over each time interval j, according to the equation: _ V j = _ X j s A s  = _ X j s D 2 s 4 (6)

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accordingto: X j = _ X j s (7)

The local viscositycan be computed, as described in (Vinogradov and Malkin 1966), byusingtheuniversaltemperatureand pressureequation calculatedas:

( _)= 1+a 1 ( _ 0 ) a 3 +a 2 ( _ 0 ) 2a 3  0 (T;p) ! 1 (8) where _ [sec 1

] is the shear rate, the empirical coeÆcients a 1 , a 2 and a 3 are, to a great extent, constant for each polymer over wide ranges of temperature and pressure values, and 

0

(T;p) is the reference viscosity inthe extrapolated viscosity limit at a zero shear rate. For the particular polymerconsidered, the following numerical expressions can be given fortheviscosityand shear rate:

T s =130 Æ C T s (p)=T s +0:03p b [ÆC]  0 ( T s )=4:4810 6 [Pasec] (9)  0 (T;p)= 0 ( T s )e 8:86( Tc Ts(p)) 101:6+( T c T s (p)) [Pasec] (10) wherep b

isthe pressureinbarunits,T istheactualpolymertemperaturein Æ C,and: _ j i = 4 _ V j R 3 i 0:815 (11)

Theenergyequationhasbeenusedinitsgeneralformasabalancebetweentheinternal variationin the energy of the control volume (on theleft) and all possibleheat{transfer and mechanical{workcontributions(ontheright):

M j i c vi T j i = h j i  j i i   j i + j i (12)

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M i =A i X i j i =h c P i X j  T j pi T j i   j i = ~ k(T j i ) 2X j h (A i +A i 1 )  T j i T j i 1  +( A i +A i+1 )  T j i T j i+1  i  j i =A i X j p j i  j i =A i X j  j i c vi  T j i 1 T j i  i=1;:::;N j 1 j=0;:::;J (13) where, fori=N j

,theadiabaticboundaryconditionat theexit gives:

 j i = ~ k  T j i  2X j (A i +A i 1 )  T j i T j i 1  (14)

The variation in the temperature of the element i over the time interval j{j +1 is expressedbytheexplicitmarching{timeschema:

T j i =T j+1 i T j i (15)

In the above equations, M [kg] is the mass, c vi  J(kg Æ K) 1  is the constant{volume speci c heat,A i [m 2

]isthelocalductcross{section,   kg(m 3 ) 1 

isthepolymerdensity, i

[W] is the mean convective{heat ow rate,  i

[W] is the net conductive{heat ow rate inside the polymer in the axial direction, 

i

is the compression work, and  i

is the axial energy ow due to the polymermotion. In general, the barrel temperature T

p can be variable in space and time according to temperature{control strategies; therefore, a genericT

j pi

hasbeenusedinequation 13.

The proposed simpli ed approach is based on the idea of assuming that the nite volumeX

j

playsadoublerole,thatis,asbothaspaceintervalandascrewdisplacement overthetimeinterval. Assuminga constant,uniformtime intervalfortheentire ow simulation,the owchannelisdiscretisedeverytimewithadi erentspacesize,depending

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us to calculate a spline approximation for the temperature pro le at the time j and to re-evaluatetheinitialtemperaturedistributioninthenewdiscretisedspatialdomainover thetimeintervalj+1. The solutionoftheset ofequations (1){(15)follows acalculation schema of the type illustrated in Figure 5, i.e., by giving the temperature and pressure pro lesalong thechannelduringanarbitrary numberofinjectioncycles.

3 An overview of SVMs for function approximation

Let us outline here the SVM approach to function approximation. Usually, a typical function{approximationtaskisde nedasfollows: a setoftrainingpointsoftheunknown function () to be estimated are given after, for example, a design of experiments. Let us indicate such a set as: Z = f(x

i ;t i )g i=1:::n p , where x 2 < d

is an input vector and t

i

2<isthe corresponding output of(). Thisis also indicatedasa regression problem, and di erent classic or advanced techniques can be used (most applications adopt B{ splines, polynomial functions or di erent avours of neural networks, like, for example, the Multilayer Perceptron). The task in which a function is estimated on the basis of some of its samples is also called the learning problem. At the end of the '90s, a new paradigmforlearningbyexamples,calledSupportVector Machines, wasobtainedthanks to the research work by the Russian mathematician V. Vapnik and his group. SVMs are based on two di erent theories developed in the '60s and '70: Statistical Learning Theory, by Vapnik and Chervonenkis (Vapnik 1998), and the theory for Reproducing Kernel Hilbert Space (RKHS) (Wahba 1999). The RKHS framework plays a basic role in the eÆciency of SVMs because it makes possible to obtain nonlinear approximation functions. Basically,anon{linearSVMcanbebuiltbymappingeachinputpatternxinto

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transformation  : < d

! < D

;D >> d. In that space, the use of Mercer's theorem permits the interpretation of kernels as inner products, thus bypassing the need for a directknowledgeof ().

SVMs have recently provided very good performances in accomplishing classi cation and function{approximation tasks. The advantage of usingSVMs over other methods is twofold. The structure of the optimisation problem, which consists in the resolution of a constrained quadratic problem (CQP), overcomes many typical drawbacks of classical neural{networkapproaches; forexample,theplagueoflocalminimathata ectsthe back-propagationalgorithmis completelyavoided inSVM learning. Furthermore, theintrinsic properties of themethod for controllingthe complexityof the model (i.e.,the Structural RiskMinimisationinductiveprinciple (Vapnik 1998)),guarantee considerable generalisa-tion capacity.

Brie y,themaingoal of"-SV regressionis to nda function ^

()thathasat mostan "deviation from the targets t

i

forall the points and,at thesame time, that is as at as possible(for moredetailson theuseofSVMsforregression purposes see(Cristianiniand Shawe{Taylor 2000, Scholkopf and Smola 1998)). The function

^ () is given, in general form,by: ^ t= ^ (x;w)= n p X i=1 ( i  i )K(x i ;x)+b (16)

wherethefree parametersare foundby solvingthedualformulation:

minE( ) = 1 2 t H +c t (17)

under the constraints 0   C and t

y = 0. C measures the tradeo between the deviation of each target from the function and the atness of the function itself, and all

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H = 2 6 6 4 Q Q Q Q 3 7 7 5 = 2 6 6 4  3 7 7 5 c= 2 6 6 4 " t "+t 3 7 7 5 y= 2 6 6 4 1 1 3 7 7 5 (18)

The threshold b is found throughthe Karush-Kuhn-Tucker conditions at optimality; q ij =q ji =K( x i ;x j )," i =";8i, ,c,y2< 2n p , ,  2< n p

;1isavector withallonesof n

p

elements,whereastisthevectorofthetargets;K(;)isakernelfunctionintheRKHS. The use of di erent kernel functions changes themapping from the inputto the feature space; therefore, it modi es the SVM structure and the corresponding approximation surface. Among available kernels, the most widely used are linear (K(x

i ;x j ) = x i x j , where\"is a simpledot{product),polynomialand Gaussianof theform:

K( x i ;x j )=e k x i x jk 2 2 2 (19) where  controls the amplitude of the RBF and the generalisation ability of the SVM. ThedescribedCQPcanbesolved byusingtraditionaloptimisationtechniques(Bertsekas 1995), orspecialorientedalgorithms(Platt 1999, Joachims1999).

The modelsdeveloped sofarrequirethesetting of two kindsof di erent parameters: 1. the vector and b, called functional variables, which can be selected through the

solutionoftheCQP;

2. aset of structural variables (C, and" 1

).

Theprocessbywhichonesearchesforoptimalvaluesofthestructuralvariablesisalso known asa model{selection task, and di erent approaches, based onstatistical methods, have recently beensuggested (see, for examples, the method described in(Chapelleand Vapnik2000). It isknownthatthemostcritical parameteristhe variance ;here we use thestandardbootstraptechniquediscussedin(Anguitaetal. 1999)fortheselectionof.

1

Notethat,inequation(16),wehaveindicatedthecollectionofthefunctionalandstructuralvariables withthevectorw.

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Inthissection,weshowhowtheSVM algorithm describedpreviouslycan be usedfor:

1. reproducingthe behaviour of thethermal, uid,dynamic model de ned in Section 2(design ofthe SVM{block).

2. generatingthecommand{inputto themodel (designof thecontrol{block).

Here we usea typical identi cation +control approach, as sketched inthe schema of Figure 6,where:

 t is the actual temperature (at the nozzle) that, in our case, is generated by the model (in Section 2 we have indicated it with thesymbol T

j N

j

, thetemperature of the last element at the time j); usually, this is also referred to, in system control theory,asthe outputof a plant;

 ^

tis theoutputof theSVM (equation16);

 ^eistheerrorbetweentheoutput ofthemodeland theoutputof theSVM(it isthe identi cationerror,anddependsontheapproximationcapabilityoftheSVMitself);

 r (reference)is thedesired temperatureat thenozzle;

 eisthe controlerror;

 u is the control signal orthe command input to the system; it is the temperature of the barrel heater at the injectionstage (in Section 2, we have indicated it, in a generalway,as T

j pi

).

Thecontrol{blockprovidesthecommandutothesystem,onthebasisofthestructure oftheSVM{block,inordertotrackthereferencetemperaturerthroughtheminimisation oftheerrore. Notethattheactualsystemissupposedtobeunknown;therefore,theonly

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following,we describehowsuch a blockhasbeendesigned.

Thethermal, uid,dynamicmodelpresentedinSection2isadiscrete{timedescription ( isthecorrespondingsampletime)ofthetemporalevolutionoftheinjectionmoulding process. Ingeneral, the output ofa discrete{time systemat the timek+1 can be repre-sentedasa functionof nprevioussamplesofthe outputitself and mprevioussamplesof thesignalinput. The relationshipis:

t(k+1)=(t(k);t(k 1);:::;t(k n+1);u(k);u(k 1);:::;u(k m+1) )=( x(k) ) x(k)=[ t(k);t(k 1);:::;t(k n+1);u(k);u(k 1);:::;u(k m+1) ]

T

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x 2 < d

, d = n+m. At the time k+1, the SVM gives an estimate of t(k+1), called ^

t(k+1),on thebasisofequation (16), thatis, ^

t(k+1)= 

( x(k);w);thisisalso known, inthe neural{network literatureas an identi cation{serial{parallel model (Narendra and Parthasarathy 1990). Note that, as in an on{line process the exact measurement of the output of a plant, at each sample time, is often a diÆcult task, a parallel identi cation model can be alternatively used. In this case, the input to the SVM depends on the previoussamplesof itsoutput, ratherthanon theoutput oftheactual system:

^ t(k+1 )=   ^ t(k);; ^ t( k n+1);u( k);;u( k m+1 );w  (21)

The formeridenti cationmodelis also preferredto thelatter because it guarantees a more accuratesystem stability,as statedin(Narendra andParthasarathy 1990).

In order to designthe SVM block(i.e., the setting of w), the followingsteps have to beexecuted insequence:

1. building a training set Z and setting the parameter vector w, as described in the previous section (note that functional and structural variables can be selected by usingthesame Z,asshown by theprocedure usedin(Anguita etal. 1999));

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T buildZ, we choose the curves in Figure 7; in this way, we are able to collect dif-ferent samples, representingtheevolutionof theprocess when di erentoperating param-eters are applied. In other words, we assume a given initial temperature (470 K), and then we set several values of the barrel heater (i.e., the signal u in our control system schema). Subsequently,we collect theoutput t, given bythemodel,at each sampletime: t(0);t(1);t(2);:::,etc. Thepairsf(x

i ;t

i

)g tobegiven tothetrainingalgorithmhavebeen simplydesigned inthefollowingway:

2 [(t(n 1); ;t(0);u(n 1);;u(n m));t(n) ]=( x 1 ;t 1 ) [(t(n);;t(n n+1);u(n);;u(n m+1) );t(n+1) ]=(x 2 ;t 2 ) [(t(n+1); ;t(2);u(n+1);;u(n m+2));t(n+2) ]=( x 3 ;t 3 ) . . . (22) and soon.

The job of the controlleris to providethe signal controlu in such a wayto minimise the control error e on the basis of the SVM{block. The idea is quite simple and based on the following criteria (given in (Noriega and Wang 1998)). A cost measuring the quadraticerrorbetweenthe referenceand theoutput of theSVM isde nedas:

= 1 2 e 2 (k+1 )= 1 2  r( k+1) ^ t(k+1)  2 (23)

The goalis to minimize by ndinga simplecontrollawfor u:

u( k+1 )=u( k)  @ @u(k)

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where isthe controldescent step to beproperlyset inorder to avoidundesired oscilla-tions. If a linear and a Gaussian kernelare supposed to be used,it ispossibleto obtain,

2

Note that, in fact, n and m are unknown parameters; their optimal values have to be established throughexperiments.

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Controller 1 (linearkernel): u( k+1)=u(k)+e( k+1) X i ( i  i )x i;n+1 (25)

Controller 2 (Gaussiankernel)

u( k+1 )=u( k)+   2 e( k+1 ) X i ( i  i )( x i;n+1 u(k))e kx i x(k )k 2 2 2 (26) byx i;n+1

we denote thecomponent n+1 of thevector x i

.

To complete this section, we provide some considerations about the stability of the proposedcontrolsystem. Letusobservethatthecontrolalgorithmjustdescribedisbased on an intelligent learningmethodology, which can be qualitativelyanalysed by usingthe Lyapunovtheory. Basically,nonlinearcontrolsystemsliketheonedescribedinthissection guarantee stablepointsas longasthefollowingconditions holdtrue:

 theSVM{blockconverges to theactualphysicalsystem;

 thecontrollaw stabilisestheSVM{block.

We can proceed as in (Noriega and Wang 1998). This is con rmed by the fact that thecase of a Gaussian kernelis a special caseof RBF neuralnetworks, discussedin that work. Furthermore,theuniversalapproximationpropertiesofSVMshavebeenextensively demonstrated byVapnik(Vapnik 1998).

5 Numerical results and SVM performances

Theexperimentsdescribedinthissection concernedeach blockin gure6. In particular, here we show the behaviour and the performance of the thermal, uid, dynamic model de nedinSection2,theprocedureadoptedtotraintheSVM-block,andthewaywechose theparameters ofthe control{block.

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inFigure3;itsthermophysicalproperties,geometricalandoperatingparametersaregiven inTables4and5. ThetimehistoryoftheoutlettemperatureispresentedinFigure8and some axial temperature pro les inside the injection chamber are shown in gure 9. The timeanalysisshowsclearlythateachinjectionstagewascharacterisedbyasharpincrease intemperature, dueto viscous dissipation,whilethe globaltrend wasa slow decrease in temperature due to the over{warmed polymer(initially, at 470K), as compared with to the heater{barrel temperature (T

p

= u = 460K). The axial temperature pro le of the polymerinsidethechamberpointsoutthat,asfarasthepolymer owed slowlyincontact withalowercylindertemperature,theaxialtemperaturedecreasedsteadily,whileonlyin thelastsharp convergencesection didthetemperatureincrease.

In the design of the SVM{block our main goal has been to assess the performances of the parallel and serial{parallel models, as described in the previous section. In the rst experiment, a uniform initial temperature ( xed at 470K) of the polymer inside the charge was assumed. Then we generated two di erent pro les of theoutlet polymer temperatureto buildthetrainingset Z (training 1 and training2 respectively,asshown in Figure 10). In the rst realisation (training 1), the temperature of the barrel heater was increased from 462K to 464K, and from 462K to 464K in the second one (training 2). Thetestwasperformedbysettingthevalueofthecommandinputat 463K.Figure10 shows theoutput of two di erent SVMs (parallel and serial{parallel), and the output of themodel{plant. Notethatweusedn=7samplesoftheoutputandm=1sampleofthe input. We used a Gaussiankernel with 

2

=0:05 and "=10 3

,following thebootstrap technique described in (Anguita et. al 1999), to minimise the generalisation error of the SVM.FromFigure10,itispossibleto observethattheoutputofthemodel{plantreached a stationary temperature of 466:3K, and waswell{identi edbythe serial{parallel SVM.

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worst performance was obtained. In this case, the SVM{block produced a signi cant error. Thanks to these experiments, we deduced that only two curves usedas a training set are insuÆcient information about the system behaviour, so the range of operating temperatures wasextended,asshown inFigure7.

Once the SVM has been designed, it is possible to set up the controller by using controller 1 or controller 2, as described in the previous section. Figure 11 gives an exampleof applicationofcontroller2basedonaparallel{SVM;we setthereferencevalue to 466.3K and 468K, respectively; we used a sample time of 0:2 [sec] and a descent step of =0:8. We carried outseveral numerical experimentsalso forcontroller 1. From the obtainedresults, we deducedthat thethermal, uid,dynamicmodelpresentedinsection 2can beidenti edand controlled,to agoodapproximation,alsobyusinga linearkernel. Figure12showstheresponsesofthesystemtodi erentvaluesof(thereferencevaluewas xedat466K),whichcontrolsthespeedofconvergenceoftheclosedlooptothereference value. Our choice is a good tradeo between the speed of convergence to a stationary temperatureand themagnitudes oftheoscillations.

6 Discussion

Inthispaperwehavereportedona rstpreliminarystudyoftheapplicabilityofmachine{ learning methods for the identi cation and control of injection moulding processes and plasticatingextrusion. TheobtainedresultshaveshowntheeÆciencyoftheSVMapproach when applied to such kinds of industrial processes. However, it is worth noting that the actualuseofthewholeframeworkmustbeprecededbythedesignofe ectiveexperiments inorder to collect real samples(x

i ;t

i

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we addresssome critical pointsthatneed to beinvestigatedin detail.

The thermal, uid, dynamic model discussed in Section 2 has been designed under two basic assumptions: rst, in all the experiments, we have supposed a uniform initial temperature( xedat470K)ofthepolymerinsidethecharge;inother words,wehavenot theoreticallymodelledthevery rststageinwhichthesolid{statepolymerispushedinside theextruderandreachesthedesiredplasticating temperature. Thisstage isvery diÆcult to de ne analytically, even by using approximate model. Second, the basic hypothesis for a correct behaviour of the model is incompressibility propertyof the polymer. Only under this strong assumption does a volume variation in the rst space element, due to theaxialmovement ofthescrew in,produceanequalvolumevariationinthepolymer atthenozzleexit. Bothassumptionscanbeobviouslyrevisedbyanalysingthebehaviour oftheinjectionmouldingprocess inmoredetail. Thiswouldinevitablycauseasigni cant change in the structure of the whole set of equations and, accordingly, an increase in theircomplexity. Asaconsequence, asimplesimulationandan easy interpretationof the results,whichhavebeenourmaingoalsforthedevelopmentofageneralframework based on anSVM, wouldbe more diÆcultto obtain.

As a nal remark, related to the SVM{block, we note that, if a reference value not included in the pre-de ned range is imposed, the general stability conditions considered intheprevious sectionswillnotbeveri ed any more. This isa typical behaviour of con-trol systems based on neural approaches. For example, Figure 13 has been obtained by settingthereferencevalueto 475K.Inthiscase, toimprovethesystemonecould proceed in two di erent ways: 1) designing a fault{tolerant block (which helps to recover a cor-rect functionality) throughthemeasurement ofthe identi cationerror ^e, asdescribed in (Noriega and Wang 1998); 2) trainingthe SVM{block by an on{linerecursive algorithm,

(22)

approachneeds furtherinvestigationsbecause upto nowmanypointshavenotbeen clar-i ed; forexample, it is not clearhow to keep track of previoussamples when a new one willarrive.

References

AGRAWAL, A. R. , et. al., 1987, Injection moldingprocess control. Polymer Engi-neering and Science, 27, (18), 17{32.

AIZERMAN, M.,BRAVERMAN, E., and ROZONOER, L., 1964, Theoretical foun-dationsofthepotentialfunctionmethodinpatternrecognitionlearning.Automation andRemote Control, 25, 821{837.

ANGUITA, D., BONI, A., and RIDELLA, S., 1999, Evaluating the generalization abilityof supportvector machinesthroughthebootstrap.Neural Processing Letters, 11,1{8.

BERTSEKAS, D. P.,1995, Nonlinear Programming (Belmont, Mass., Athena Scien-ti c).

BISHOP, C., 1995, Neural Networks for Pattern Recognition (Oxford, Clarendon Press).

CHAPELLE, O., and VAPNIK, V., 2000, Model selection for support vector ma-chines. In Solla, S. A., Leen, T. K., and Muller, K. R. (Eds.) Advances in Neural Information Processing Systems 12(MITpress).

CAUWENBERGHS,G., and POGGIO,T.,2001, Incremental and decremental sup-port vector machine learning. In Solla, S.A., Leen, T. K., and Muller, K. R. (Eds.) Advances in Neural Information ProcessingSystems 13(MIT press).

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plasticmoldingprocess. IEEETransactions on Industry Applications, 34(2).

CRISTIANINI,N.,andSHAWE{TAYLOR,J.,2000,AnIntroductiontoSupportVe c-torMachines (CambridgeUniversityPress, Cambridge, UK).

JOACHIMS, T., 1999, Making large-scaleSVM learningpractical. In Scholkopf, B., Burges, C., and Smola, A. (Eds.) Advances in Kernel Methods { Support Vector Learning(MIT Press, MA,USA).

NARENDRA, K. S., and PARTHASARATHY, K., 1990, Identi cation and control ofdynamicalsystemsusingneuralnetworks.IEEETransactionsonNeuralNetworks, 1(1),4{27.

NORIEGA,J.R.,andWANG,H.,1998, Adirectadaptiveneural{networkcontrolfor unknown non-linear system and its application. IEEE Transactions on Neural Net-works,9 (1).

PLATT,J., 1999, Fast trainingof supportvector machinesusingsequentialminimal optimization.In Scholkopf, B.,Burges,C., andSmola, A.(Eds.)Advances in Kernel Methods { SupportVector Learning (MITPress, MA, USA).

ROSATO,D.V.,1995,InjectionMoldingHandbook: theComplete MoldingOperation Technology, Performance, Economics(2nded., New York).

SCH 

OLKOPF, B., and SMOLA, A., 1998, A tutorial on support vector regression. NeuroCOLT2 TechnicalReport Series,NC2{TR{1998{030.

SUYKENS,J.A.K.,VANDEWALLE,J.,andDEMOOR,B.,2001,Optimalcontrol byleast squaressupportvector machines.Neural Networks, 14, 23{35.

VAPNIK,V.,1998, Statistical Learning Theory (Wiley{Interscience Pub.).

VINOGRADOV, G. V., MALKIN, A. Ya., 1966, Rheological properties of polymer melts.Journal of Polymer and Science, Part A-2,4,135{154.

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therandomizedGACV.InScholkopf,B.,Burges, C.,andSmola, A.(Eds.)Advances in KernelMethods { SupportVector Learning (MITPress, MA,USA).

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1 Nomenclature(1). . . 25

2 Nomenclature(2). . . 26

3 Nomenclature(3). . . 27

4 Duct geometry(Figure3). . . 28 5 Polymerthermophysicalproperties andvaluesof theoperatingparameters. 28

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1 The extruder. . . 29

2 The injection process: injection(or mould lling) - holding (or packing) -coolingand dieremoval- llingof thecharge chamber.. . . 30

3 Screw-channel assembly and nite volume discretization: i is the discrete axial co-ordinate; j isthediscrete{timeco-ordinate. . . 31

4 Pressurevariationintheelementi. . . 31

5 Calculationschema forthethermal, uid,dynamicmodel. . . 32

6 A typicalidenti cation{controlscheme. . . 33

7 Outletpolymertemperature,usedfortraining,at thenozzleexitfor di er-ent settingsof thebarrelheater. . . 34

8 Outlet polymertemperature at thenozzleexit. . . 35

9 Temperature distribution in the chamber for successive injection cycles of the polymerextrusion temperature (at thechamber inlet) T 0 =470K and theuniform barreltemperatureT p =460K. . . 36

10 Trainingpro les(training1andtraining2)andSVMoutputfortheparallel and serial{parallelmodels. . . 37

11 Closed{loop outlet polymer temperature at the nozzle exit (plant) and at theSVM{blockexit(SVM) fordi erentreference values. . . 38

12 Closed{loop outletpolymertemperaturefor di erentvaluesof the descent step parameter. . . 39

13 Example of system instability; outlet polymer temperature at the SVM{ blockexitfor areference valuebeyond therange. . . 40

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A crosssection C regularisationconstant D max =D min

max. and min. diameters

D feature{spacedimension

E quadraticfunction

H quadraticfunctionHessian matrix

F axialforce

J No. of discrete{timeelements

K(;) kernelfunction(innerproductinfeature space) ~

K dieconductance

L

conv=end=tot

screwlengths (Figure3)

N No. of axialelements

P wetted perimeters

Q quadraticfunctionHessian matrixblock R

max=min

max. and min. radii T=T p =T 0 polymer/barrel/polymer{extrusiontemperatures _

V volumetric owrate

X s

screwaxial co-ordinate _

X s

screwaxial velocity

(28)

a 1=2=3

empiricalcoeÆcients

b threshold

c quadraticfunctionlinear term c

v

constant{volume speci c heat d inputspace dimension

e controlerror ^ e identi cationerror f p cross{sectioncoeÆcient h c

polymer/walltransfer coeÆcient

i=j=k discrete{axial/discrete{time(model)/discrete{time(controlloop)counters ~

k meanpolymerthermalconductivity m No. of signal{controlsamples

n No. of signaloutput (plant/SVM)samples n

p

No. of trainingpatterns(n p

=dim(Z))

p pressure

r controlloop reference t plant output

^

t SVM output

t targetsvector

u control{signalinthecontrolloop w SVM freeparameters vector x SVM inputvector

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() themapping< d

!< D control{loop functionalcost quadraticfunctionfreeparameters

n

p

{dimension component vector of

 n

p

{dimension component vector of " lossfunctioninsensitivityparameter () plant I/Orelationship

^

() SVMI/O relationship _

shearrate  localviscosity

 control{block descent step  localdensity

meanconvective heat owrate  netconductive heat owrate  compressionwork

 axialenergy ow 

2

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D max 0.05[m] max. diameter D min 0.008 [m] min. diameter L tot

0.39[m] total charge length L

conv

0.15[m] convergingsection length L

end

0.02[m] terminationductlength

Table 5: Polymer thermophysicalpropertiesand valuesof theoperating parameters. c

v

2000 [J=(kgK)] constant{volume speci c heat

 780 [kg=m

2

] polymerdensity

k 0.16[W=mK] meanpolymerthermalconductivity  I 2[s] injection  M 2[s] holding  R 6[s] lling

 0.2[s] timestep interval T

0

470 [K] inlettemperature(extrusionscrew output) T p 465 [K] barreltemperature _ X s

0.11 [m=s] screw axialvelocity p

init

1.5 10 8

[Bar] polymer/screwinterfacepressure h

c

100 [W=m 2

K] polymer/wallheat transf. coe . (injection) h

cm

30[W=m 2

K] polymer/wallheat transf. coe . (holding) h

cr

30[W=m 2

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B a r r e l w i t h h e a t e r s

M o l d

N o z z l e

F e e d

S c r e w

P i s t o n

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I n j e c t i o n

H o l d i n g

F i l l i n g

D i e r e m o v i n g

Figure2: Theinjectionprocess: injection(ormould lling)-holding(orpacking)-cooling and dieremoval- llingof thechargechamber.

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E l e m e n t  i

j

XD

j

N

j

T

j

F

j

X 

t o t

L

c o n v

L

L

e n d

m a x

D

m i n

D

Figure 3: Screw-channelassemblyand nitevolume discretization: iis thediscreteaxial co-ordinate; j isthe discrete{timeco-ordinate.

j

i

p

j

i

p

1+

V o l u m e i

a t t i m e j

P o l y m e r

f l o w

j

XD

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S t a r t

j = 0

K n o w n i n i t i a l

c o n d i t i o n

m o m e n t u m

e q u a t i o n

e n e r g y

e q u a t i o n

j = j + 1

j = J

P r o f i l e s

S t o p

j

j

s

t o t

j

j

s

j

j

p i

j

s

j

s

j

s

X

X

L

N

X

X

T

X

X

F

D

-=

D

=

D





t

,

,

,

N o Y e s

S t o r i n g

j

j

i

T

(

)

1

)

(

+j

j

i

T

1

)1

(

j

+

j

+

i

T

j

xD

D

x

j

1+

( )

x

T

j

j

j

i

p

(

)

1

)

(

+j

j

i

p

1

)1

(

+

+

j

j

i

p

S t o r i n g o f s a m p l e s a n d d i s c r e t i z a t i o n s t e p

E n e r g y e q u a t i o n

R e - e v a l u a t i n g

(35)

C o n t r o l

( M o d e l )

P l a n t

S V M

+

-r

+

-P a r a m e t e r s

t

u

e

(36)

465

466

467

468

469

470

471

472

0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000

Temperature (K)

Samples

training 1 (462K - 464K)

training 2 (464K - 462K)

training 3 (466K - 462K)

training 4 (468K - 462K)

Figure 7: Outlet polymer temperature, used for training, at the nozzle exit fordi erent settingsof thebarrelheater.

(37)

462

463

464

465

466

467

468

469

470

471

0

10

20

30

40

50

60

70

80

90

100

Temperature (

K

)

Time (s)

Nozzle Temp.

(38)

C y c l e s

Figure 9: Temperature distribution in the chamber for successive injectioncycles of the polymerextrusion temperature(atthechamberinlet) T

0

=470K and theuniform barrel temperatureT

p

(39)

465

465.5

466

466.5

467

467.5

468

468.5

469

469.5

470

0

1000

2000

3000

4000

5000

6000

Temperature (

K)

Samples

Parallel SVM (463K)

Training 1 (462K - 464K)

Training 2 (464K - 462K)

Serial-Parallel SVM (463K)

Plant (463K)

Figure 10: Training pro les (training1 and training2) and SVM output forthe parallel and serial{parallelmodels.

(40)

462

463

464

465

466

467

468

469

470

471

0

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Temperature (

K

)

Time (sec)

SVM

Plant

Control

Figure 11: Closed{loop outlet polymertemperatureat thenozzle exit(plant)and at the SVM{block exit(SVM)fordi erent reference values.

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464.5

465

465.5

466

466.5

467

467.5

468

468.5

469

469.5

470

0

200

400

600

800

1000

Temperature (

K

)

Time (sec)

mu=0.5

mu=1.0

mu=1.5

mu=2.5

Figure12: Closed{loopoutletpolymertemperaturefordi erentvaluesofthedescentstep parameter.

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400

500

600

700

800

900

1000

1100

0

200

400

600

800

1000 1200 1400 1600 1800 2000

Temperature

Time (sec)

Control

SVM

Figure 13: Exampleof systeminstability;outlet polymertemperature at theSVM{block exitfora reference value beyond therange.

Figura

Figure 1: The extruder.
Figure 2: The injection process: injection (or mould lling) - holding (or packing) - cooling and die removal - lling of the charge chamber.
Figure 3: Screw-channel assembly and nite volume discretization: i is the discrete axial co-ordinate; j is the discrete{time co-ordinate.
Figure 5: Calculation schema for the thermal, 
uid, dynamic model.
+7

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