AN OBJECT- ORIENTED MODEL FOR SIMULATION OF THE COLDROLLlNG MILL, USING NEURAL NETWORKS

L.E.Záratee H. HelmanDept" de Ciência da Computação, Pontifícia Universidade Cat61icade Minas Gerais, PUC-MGAv. Dom José Gaspar 500, Coração Eucarístico, CEPo30535-610 - Belo Horizonte - MG - Brasil

Depr' de Engenharia Metalúrgica e de Materiais, EEUFMG, DEP. 30160-030, Belo Horizonte -MG, Brasil.E-mail: zarate@brhs.com.br

E-mail: hhelman@demetufmg.br

Resumo Este artigo apresenta um novo modelo orientado aobjeto para simulação do processo de laminação a frio. Estemodelo introduz uma representação do processo de laminaçãona forma de objeto, onde o contexto é o comportamentocinemático do processo. O modelo inclui os sub-sistemasdesbobinadeira, bobinadeira de tensões à ré à frente e o sub-sistema laminador.

Para obter um modelo de simulação do laminador a frio, énecessário uma representação para calcular a espessura desaída e a carga de laminação. A expressão deve ser uma funçãodos outros parâmetros operacionais : espessura de entrada,tensões à frente e à ré, tensão média de escoamento, coeficientede atrito, entre outros. Neste trabalho, a representação ébaseado em redes neurais artificiais e desenvolvimentoorientado a objeto.

O paradigma orientado a objeto permite a descrição do modelode laminação numa forma modular e permite o reuso destesmodelos independentemente do ambiente nos quais estãoembutidos. Finalmente resultados e conclusões para mostrar aaplicação do modelo orientado a objeto são apresentados.

Palavras Chaves: Modelamento orientado a objeto, Processode laminação, Redes Neurais. Simulação .

Abstract This paper presents a new object-oriented model forlhe simulation of the coíd rolling processo It introduces arepresentation of the rolling process in object form, where. context is the cinematic behavior of the processo The modelincludes uncoiler and coiler subsystems, back and fronttensions and roIling mill subsystems.

In order to obtain a model for the simulation of a cold roIlingrnill, it is necessary a representation to calculate lhe outgoingthickness and the roIling load. The expression must be afunction of lhe other operational parameters: input thickness,back and front tensions, average yield stress, frictioncoefficient and others. In this paper, the representation is basedin Artificial Neural Networks and Object-orienteddevelopment

The object-oriented paradigm allows the description of roIlingrnodels in a modular fashion and permits the reuse of thesemodels independently of the environment in which they areembedded. Finally , ·simulation results and conclusions to showthe application of the object-oriented model are presented.

335

Keywords: Object-oriented modeling, Rolling mill process,Neural networks, Simulation.

1 INTRODUCTIONIn recent years, more than 90% of lhe rolling processoperations have been automated as the results of equipmentinvestment for automation. In this field, new technology isrequired for continuous processing to improve the quality andaccuracy ofproducts and also to enhance the productivity.

In the cold rolling process, requirements for enhanced qualityof products are everlasting and typical aspects being thicknessand shape. One means to attain these requirements is throughsupervision systerns, so that the output of on-line simulationcan be directly feedback into the process to restore theoutgoing thickness, when alterations in the rolling processoccur. The existent theoretical rriodels for simulation of therolling process demand a great computational effort (such isthe case of Alexander 1972). It prevents its use in on-linecontrol and supervision systems.

Is necessary, a new representation for lhe rolling process withless computational effort. Besides, lhe representation shouldpermit the reuse independently of the environment in whichthey are to be embedded.

Zárate (1998) proposed a representation for the cold rollingprocess based on Artificial Neural Network (ANN). Thesesnets use simple processing elements with high computationalefficiency and constant operation time. Some relevantcontributions in lhe area of neural networks applied in lhe steelindustry are lhe paper from Andersen et. aI. (1992), Srnartt(1992) and Sbarbaro et. alo(1993).

An object-oriented representations permit reusability and lesstime in the development of new systems as fault-diagnosis(Cellier and Elmqvist 1993 and Otter et. aI. 1996).

The model for simulation of lhe cold rolling mill includesuncoiler and coiler subsystems, back and front tensions androlling mill subsystems. Where lhe rolling process isrepresented by ANN and object-oriented development.

This paper is organized in fivesections. In section two, object-oriented rolling process is described. In section three, lhemodel for a cold rolling mill is presented. In section four, the

(4)neural representation of the rolling process is presented. Insection five, the application of the object-oriented model to astrip rolling process is presented. Finally, conclusions arepresented.

2 OBJECT-ORIENTED REPRESENTATIONOFTHE ROLUNG PROCESS

!:!.h =!:!.g+ !:!.P.Wo M

By adequate manipulation of the Eqs. (3) and (4), theexpression to calculate the outgoing thickness, without the"algebraic loop'', Eq. (5) can be obtained. This equationrequires the calculation of the sensitivity factors.

In general, everything existing in nature might be representedas an object In this sense, the object-oriented modeling is aparadigm that seeks to represent a system through theidentification of its fundamental characteristics. The modelingprocedure consists of identifying the object attributes andactions. Attributes are usually known as data and actions asoperations.

(5)

The existent theoretical models for the rol1ing process allow tocalculate the rolling load by means of non-linear expressionsuch as Eqs . (2):

To obtain the data structure and transformation functions forthe object-oriented model, ís necessary an expression tocalculate the outgoing thickness and the rolling load, Eq. (I):

Normally, the sensitivity factors are calculated from equationsthat may have a difficult analytic soIution, such the case ofAlexander's model.

The transformationfunctions are the mathematical functions ofthe outgoing thickness (Eq 5) and of the rolling Ioad (Eq, 2).The third aspect, the events sequence, determines the tirning ofexecution of the functions.

Class rolling-process(<Attributes»Attribl: Nominal values of the stand rolling mill h;,

Considered the cold rolling process as the problem domain, theidentification process leads to the following attributes andoperations of the Class rolling processo

e', hj' , s' , f.l*, t;, t;, y*;Attrib 2: Current values of the stand rolling mill ho• P, hj, g, f.l ,

(1)

(2)

where: h j =input thickness; ho = output thickness; tb = backtension; tI = front tension; y = average yieId stress;

f.l =friction coefficient; E = Young modulus of the stripmaterial; R =roll radius; W =strip width; P =rolling load; M =rigidity rolling mill modulus; g = rol! gap.

The proposed object-oríented modeI includes three aspects ofthe object o representation: data structure, transformationfunctions and events seqllence. o o

(3)

3 MODEL FOR ROLUNG MILL SYSTEMTo analyze the behavior of the system in stationary state, thefollowing subsystems were identified: Uncoiler, Coiler, Standrolling mill and Interstand tensions subsystems.

Access:<Operations >Oper 1: inputloutput operations of dataOper 2: o CalcuIation of the sensibility factors through the

differentiation of the neural network for nominaloperation point;

Oper 3: Calculation of the output thickness (ho ) through Eq.(5);

\Oper 4: C álculation of the rolling load (P) through ANN;};

àP àp àpàth 'atl 'ày

th. 'r- y;3: Sensitivity Factors ap ap ap

o ah 'ah.' a 'o I Jl

Note in Eq . (I) and (2), the rolling load value depends on theoutput thickness value and vice versa . Therefore, an "algebraic loop" exists in Eq. (1) that prevent the analyticallycalculation of those parameters.

To solve this problem, a numerical algorithm involvingsuccessive interactions can be applied. This method demands agreat computational effort in order to find the new operationpoint when a disturbance takes place in the roll íng processo Thetiming of execution are not the same for different operationpoints . This prevents.the use of interaction solutions in on-linecontrol and supervision systems.

Another form to represent the rolling process isfuroughsensitivity factors (Zárate 1998), Eq. (3).

AP = a P Ah .+ a P Ah + a P Atah . I ah f d t r

I o b

ap ap . ã P -+ _ o-D.tf+ --Af.l+ -=Ayat f a f.l a y

Since the rolling mill structure is not perfectly rigid, theoutgoing thickness is governed by the elastic equations of therolling mill, and any variation in the roIl gap can be expressedas:

3.1 Uncoiler subsystemTo represent uncoiler and coiler subsystems, it is necessary to

o consider variations of the coil radius. The reduction of theradius of the input coil causes an increase in the speed of the

336

the foIlowing expression for interstand tension is obtained:

T(t,)= EhW (Vo -v,».+T(O)L

strip (for constant angular speed). This alters the back tensionon the stand that wilI cause variations in the outgoinglhickness.

Each decrease on the radius of the coil (Ór,lC) is equivalent to

an average input thickness hi , Therefore, in each rotation, lheradius ruc is reduced in ôr.uc =h, :

Considering:

Vit) =Vil forVo(t) = Vb for

O s t i,

O s t t.(11)

(12)

(6)

where: (JJ angular speed (rad.ls); rue input-coil radius.

To calculate the interstand tensíons; it is necessary to obtainexpressions for lhe input and output strip velocities Vj and

Vo respectively. TIlese are obtained as:

3.2 Coiler subsystem .Each increase in radius .af the coil (Óre ) is equivalent to

average output lhickness ho , Therefor, in each rotation, lheradius re (O) is increased in Llre =ho :

(13)

where : v.: e v: are nominal valuesSince lhe volume of lhe metal is practicaIly unaffected, and inwide-strip roIling, lhe lateral spread of lhe strip is very small,its width may be considered as constant, lherefore:

(f)-'"c(t) = '"c(o)+_ehst

27c(7) (14)

3.3 Stand rolling mill subsystem

Normally, the uncoiler and coiler systems are controIled tomaintain constant the back and front tensions andconsequently, lhe outgoing thickness.

The single stand rolIíng miII governing equation is a non-Iinearfunction on several parameters, Eq. (8). This equation isequivalent to Eq. (5) and wiIl be represented in lhe object formo

(15)

(16)

(17)

(18)

(19)

considering: LlVoLlho "" O

v», =Vo' h; +Vo' ho+ ÓVoh;

By constancy of volume :

Voh o =v:h;

:. V = V *( 2 h; - h o )o o h *

o

Equating Eqs. (16) and (17), it is obtained:

Vo'óho+ ÓVoh; = Ocombining Eq.(13) and (18):

Yo _ 2 hoy* - -f[o o

The expression to calculate output velocity is:

as:

(8)(ho)=f (P, W;g,M, y,hj,tb,t f,jl,E,R)

where: (j) angular speed (rad.ls); Te output-coil radius.

Any alteration on either of them: lhe entry thickness, the frontor back tensions, lhe average yield stress or lhe frictioncoeffícient, wilI cause aIterations on the roUing load and,consequently on the outgoing lhickness.

In Zárate (1998) a new representation for lhe cold rollingprocess based on sensitivity factors is presented. Therepresentation (Eq. (5» is obtained through lhe differentiationofan artificial neural network previously trained. .

The model for lhe simulation of the cold rolIíng miII system isgiven by lhe set of equations (6), (7), (8), (12), (19) and (20) .

3.4 Interstand tensions subsystemsPrevious works have used Eq. (9) to calculate lhe interstandtensions (Zárate, 1998).

dT = EhW CV _ li: ) (9)dt o L o b

The expression to calculate input velocity is:

:. V. = VohoI h.

I

(20)

where: T = Interstand tension; L = distance between standand coils ; h =strip thickness; Vo - Vb=variation of lhe stripvelocity. Note lhat T can be the back or front tensions.

In a period ti, the integral of the Eq. (9) is:

(10)

Note that in the equations of lhe model, the outgoing thicknessvalue depends on lhe interstand tensions, that depends on lhestrip velocity, that depends on the outgoing thickness.Therefore, another "algebraic loop" exists in the modeI.

To solve this problem, it is necessary to calculatethe outputthickness one interaction step ahead. Therefore, expression (8)becomes:

337

Note that, expression (21) is represented through of thesensitivity factors (Eq. 5) obtained by differentiating of aneural networks previously trained in the object formo

4 REPRESENTATION OF THEROLLlNGPROCESS THROUGH ANN ANDSENSITIVITY FACTORS

The "computational algorithmic implemented considers thefollowing sequences of calculation:

for: t=O;

(23)

(24a)

cmax N ; cmin N

emax N - cmin N

WJ.'N

cmax N - emin N

o

N h- (. r WljX j)(1 + exp • = o )2

-( WiI.X .)exp i = o 2. I

N 11-(.r W2 jX j)(1 + exp , o ) 2

N 11-( r W LiX. )cxp ; =0 '

-v. exp I

(smax I - smm I) _ V(l+exp 1 )2

ex - v2( smax 2 - srnin 2) P _ V

(I + exp 2 ) 2

. cxp-vM( smax M - snun M) _ V

(I + exp M)2

N h- (.r WLi X j )( I + exp .=0 )2

w1temax 1 - emin 1 cmax 2 - emin 2

wAemax 1 emin 1 cmax 2 ; emin 2

W W t2emax 1 - emin 1 cmax 2 - cmin 2

Ln = (Lo - Lmín) / (Lmax - Lrnin)

3).Lmín and Lmáx were computed asfollows:

Lmín = (4 x Limitelnf. - LimiteSup) / 3

where Ln is the normalized value, Lo value to normalize, Lmínand Lmáx are minimum and maximum the variable values,respectively.

Lmáx = (LimiteInf. - 0.8 x Lmín) / 0.2 (24b)

In Zárate (1998) equation (25) is presented, that allows toobtain the sensitivity factors of a Neural Network with N:inputs, M : outputs and L: number of neurons of the hiddenlayer. .

"(21)ho(t +1)=f (P(t), lv,g (t),M, y(t),

h, Ct),t b Ct),t f (t),J.l(t),E,R)

As already mentioned, the sensitivity factors from Eq. (5) arenormally calculated from equations that may have a difficultanalytic solution. In this item, the factors are obtained bydifferentiating the neural network output. In this case, theneural network considered is:

TucCt)=TucCO)- C;;; hjt01-

Tc Ct) =Tc (O)+_c hot2n

tit)= Ehit)W (Vit)-Vit)) t+tiO)" L

tit)= Eh/t)W (V;(t)-Vjt)) t+tiO). " Lho(t +1)=f (P(t), lv,g (t),M, y(t),

h, (t) ,tb (t),t f (t),J.l(t),E,R)

"Vo(t +1)=Vo*( 2h; - + 1))ho

V.(t + 1) = Vo(t + 1) h o(t + 1), . h/t)

Neural Network(22) Where:

(25)

In this case, a back-propagation neural network, with sixentries (N=6), two exits (M=2) and one hidden layer with 13neurons (2N+1), is used . A sigmoid function was selected asthe activating function.

Generally, the largest effort to get a neural network trairtedrelies on collecting and pre-processing neural network inputdata . The pre-processing consists on the data normalizationsuch that the inputs and outputs values must be within the Oto1 range. .

The following procedure was adopted to normalize the inputdata before using it in the ANN structure:1) In order to improve convergence of the ANN training thenormalization interval [O, 1] was decreased to [0.2.0.8] (noticethat for a sigmoid function, Oand 1 are values at infinity).

2) The data wasnormalized through lhe following formula :

Uj' i =O, ...,N are the inputs ofthe net and Uo =1 it is apolarization input

X j ' i =0,...,N are the normalized inputs and Xo =UowJ' i =1,...,L and j =O,...,N it contains the weight ofthe neuron i and input j

w; i 1....,M and j =O,...,L it contains the weight ofthe neuron i and input I, for the exit layer

Zj' i =1,...,M are the outputs of the net

e maxjvemin, k =1,o.,N are the maximum and rninimum. values of the inputs

338

The events sequence to execute object-oriented roIling processmodel are described next:

srnax.vsmín, k =1,..,M are the maximum andminimumvalues of the outputs.

The equation (25) provides a linear form for the equationprocess in the neighborhood of an operation point (Ui) and it isvalid for smalI variations in the process parameters. Thesensitivity factors are calculated directIy from the networkinputs and the network weights . .

with:

W20L· . .· . .· . .W;;L

5.25106.0337- 3.89675.15855.4216-1.0687

IV':" = - 7.25700.2414-3.2n7- 0.9663- 5.83853.7317- 3.0088

·3.1772 - 2.9064- 0.0878 . 0.0705- 0.0626 - 0.08840.1237 0.03060.0126 0.04010.0767 - 0.0340

IV' = 0.0418 - 0.0196- 2.4975 - 2.23200.0412 - 0.0437- 0.0568 0.03610.0205 0.0008- 0.1528 0.1381- 0.0468 0.0634

W. =[ '3.2107]'w 2.8861

Where, the index "o" means the product 01' the first element ofthe vector with the whole first line of the matrix, extended toalI the lines of the matrix.

5 SIMULATION RESULTSTo ilIustrate the performance of the object-oriented modelproposed, a numerical application to a roIling milI system, wilIbe presented. The chosen operation point is: hj=5.0 mm; h",=3.6 mm; g=1.846 mm.; JL = 0.12; !j = 9.098 kgf/mm",

tb=O.441kgf/mm": y=46.918 kgf/mm": W . = 500. 2mm; y = 26.138+ 47.742Eo.4275; E=20,400 kgf/rrim ;R= 292.1

mm; M=500,000 kgf/mm2 and P= 875.31tf. With

T"c (O)=2340 mm; rc (O)=508 'mm; Vc (O) =403.798 m/min;v"c (O)=560.830 m/min ; » ; (O)=4389.97 radls; »; =1839radls; L= 4.5 m. For simulation purposes, were assumedconstant times delays, equals to 0.669 s and 0.481s. Thesedelays are due to the movement of the strip, before reachingand after leaving the stand.

AIexander's model (Alexander 1972) was used to generate adatabase for the cold rolling processo To obtain the data sets forANN training (Eq. (22», the parameters variations were chosenas: h;=± 8%; !lo,= ± 3%;/1 =±20% !j =±30%; tb =±

30% and y =±10%. Three different values werechosen foreach parameter resuItingin 729 training sets. The load roIlingwas obtained through Alexander's model and the roIl gap bythe elasticequations for the rolling mill, Eq..(4).

The neural networks have been trained using an averageparameter variation (around ao equilibrium point) of 15%. Thefinal weights for the hidden and output layers with itspolarization weight are:

- 2.3472 0.7327 - 1.6876 - 0.0036 0.6561 - 2.60323.9055 3.0808 - 5.1249 - 2.8544 - 5.3232 - 5.6772- 8.4711 5.1593 6.1539 -1.5899 - 4.3587 3.4279-11 .7598 0.2278 - 0.9994 - 0.3439 0.1404 6.6674· 8.4249 - 9.2360 ·5.5525 1.1392 2.4098 - 5.1437-1.3988 1.6645 9.0436 0.0068 5.0951 -7.2585.

W = 3.4416 0.4525 8.6002 - 6.6928 5.2036 1.8066- 2.0861 0.6307 ·0.6206 0.0006 0.8530 -1.8555- 4.1920 ' 0.5245 - 4.9311 7.3975 1.3536 6.687410.5998 3.1857 - 4.8274 1.8625 - 3.3112 - 2.84623.8174 13.3982 - 3.9206 -1.5036 -1.7811 0.69473.9012 - 2.6002 0.5488 2.1261 - 5.7294 - 5.7072o ct-t t

339

1. Define the nominal inputs: [h j*,g*,,u*,t;,t;,YJ=[5.00;1.846; 0.12; 0.441;9.098;46.918] (Oper1)

2. Provide the nominal outputs: [h; ,P*] =[3.6; 875.31] (Oper 1)

3. Calculate through Equation 26 the sensitivity factors for theselected nominal point: [ap ap ap ap ap apJ =

dho ' õh, , dJ1. 'atb ' dtf ' dY[498.66; -631.16; 7909.25; -86.56 ; -5.83; 56.75J (Oper 2);

4. In the presence of parameter variation, provide the current. inputs as: [hj , g ,J1.,tb,tf' YJ (Oper1)

5. Using Eq (5), calculate the current output: [hoJ (Oper 3).

6. Using ANN previously trained, determine the currentoutputs: [PJ (Oper 4).

The perturbation sequence considered in the simulation is:

without perturbation (WP) t<0.5s.input thickness+3.0% 0.5 <1.0

WP < 2.5inputthickness - 3.0 % 2.5 <3 .0

WP 4.5friction coeficient+5.0 % 4.5

.WP (26)

friction coeficient - 5.0 % t <7.0WP <8.5

average yield stress+ 3.0% 8.5 s t < 9.0WP

average yield stress - 3.0% 10.5 s t < 11.0WP 11.0 s t < 11.5

Figures 1 shows the output thickness and roIling load responsesfor the perturbation sequence. In this simulation, the existenceof a controIler for interstand tensions that keeps the back andfront tensions constants was considered. The adjustrnent of theangular velocities of the uncoiler and coiler systems tocompensate variations in the coils radius, was admitted.

. ." fl' . ' . ' .'.U'

Figures 2. Input and output velocity of the strip

REFERENCES

".2: :''0

lllí'H!tll}

Figures 1. Output thickness and rolling load for perturbationsequence respectively

1---,...-- - .........-·-..,---.,..",..,..-.-----";:;t .c;1· ;•. vi>,.- ,. f..,.. ; -.._ '

: i .. ' '. f

.. .:h-c:;.-+ -:

o :L,' o, . ' .

Observe that, the variations in the input thickness take place atthe time 0.67s. The output thickness increases when the inputthickness, the friction eoefficient or the average yield stressincrease. In this case, the model response keeps the error in theoutput thickness below 1% of the values obtained by usingAlexander's model, .

For simulation purpose, the rolI gap was assumed constant. Byconstancy of volume, when the outgoing thickness grows , thestrip output velocity falIs, Figtire 2. The value of the inputvelocity depends on the input and output thickness values (Eq.(20). If the input thickness is constant, there will be novariation in the input velocity.

Alexander, J.M., (1972). On the theory of Rolling. Proe. R.Soe. Lond. A. 326, pp. 535-563.

Andersen, K., Cook, G.E. and Barnett, R.J ., (1992). GasTungsten Arc Welding Process Control UsingArtificial Neural Networks, lnternational Trends ' inWelding, Scienee and Technology, ASM, pp. 877-1030.

Cellier, F.E. and EIrnqvist H. (1993). Automated FormulaManipulation Supports Object-Oríented Continuous-System Modeling. IEEE Control Systems, 13(2), pp.28-38

6 CONCLUSIONSIn this paper a object-oriented model based on a neural networkand sensitivity factors that perrnits to simulate the cold roIlingmill of a single stand is presented. The model considersuncoiler and coiler subsystems, back and front tensions androIling rnilI subsystems.

The object-oriented rolling process representation wasimplemented in software to simulate a stand roIling rnilI. Thiswork shows the possibility to mix object-oriented model, ANNand the sensibility factors by the cold rolling processo

The model was applied to a rolling process that showsreasonable agreement with other models. This model ean beused in on-line supervision systems since it uses sensitivityfactors to represent the rolling process, involving a minimumcomputational effort.

Otter, M.; Elmqvist, H. and Celier, F.E. (1996). Modeling ofMultibody Systems with the Pbjeet-Oriented ModelingLanguage Dymola. J. Nonlinear Dynamies, 9(1), pp.91-112.

Smartt, H.B ., (1992). Intelligent Sensing and Control of AreWelding, Intemational Trends in Welding, Scieneeand Technology, ASM, pp. 843-851.

Sbarbaro-Hofer, D., Neumerkel, D. and Hunt , K., (1993).Neural Control of a Steel RoIling Mill, IEEE ControlSystems, June, pp. 69-75.

Zárate, L.E., (1998). Tese de Doutorado, Universidade Federalde Minas Gerais, Belo Horizonte, MG, Brasil.

Observe that the representation based on sensitivity factors is alinear forrn of the processo

340