This paper describes the use of a radial basis function (RBF) neural network. It approximates the process parameters for the extrusion of a rubber profile used in tyre production. After introducing the problem, we describe the RBF net algorithm and the modeling of the industrial problem. The algorithm shows good results even using only a few training samples. It turns out that the „curse of dimensions“ plays an important role in the model. The paper concludes by a discussion of possible systematic error influences and improvements

Brause, Rüdiger W.

Pietruschka, Ulf

English

Hochschulschriftenserver - Universität Frankfurt am Main

Using RBF-Nets in Rubber Industry Process ControlU. Pietruschka, R. BrauseJ.W. Goethe-Universität, Frankfurt a.M., Germanybrause@informatik.uni-frankfurt.deAbstractThis paper describes the use of a radial basis function (RBF) neural network. It approxi-mates the process parameters for the extrusion of a rubber profile used in tyre production.After introducing the problem,  we describe  the RBF net algorithm and the modeling ofthe industrial problem. The algorithm shows good results even using only a few trainingsamples. It turns out that the „curse of dimensions“ plays an important role in the model.The paper concludes by a discussion of possible systematic error influences and improve-ments.1 IntroductionProcess control in rubber industry has the smell of a „dirty“ industrial branch.This comes not only from the often very dull and dusty rubber and tyre productionrooms where the products are „baked“ by heat and steam, but also from the fact thatthe macromolecular proportions of rubber are hard to predict due to their nonlinearcharacter. In the extruder (the melting and form-giving machine) the rubber mixtureis heated up to 110°-140°C, compressed with 70-140 bar by a screw conveyor andpressed through a metal mask. On leaving the extruder, the rubber relaxes, that is itexpands or shrinks, depending on the mixture, changing therefore its shape in a non-linear manner by 10%-20% up to 50%. Figure 1 shows the basic layout for our ex-ample of tyre profile production.Fig. 1 The tire rubber profile extrusionThe task of process control consists of estimating the necessary extrusion parameters(i.e. the shape of the extrusion metal mask) for an acceptable rubber product afterrelaxation. The modeling has to reflect the following facts:· The rubber expansion pressure and flow within the profile heavily dependwhether the neighbor parts of the profile have a high level, or if the neighborparts are low-leveled. This causes the rubber profile to be also a function of theprofile height of the neighbored points.Proc. Int. Conf. on Art. Neural Networks ICANN-96, Lecture Notes inComputer Science LNCS 1112, Springer Verlag 1996, pp. 605-610· Additionally, the extruded rubber profile heights depend nonlinearly on the rub-ber mixture G, the pressure P by the screw conveyor, on the temperature T, onthe extruder type E and on the weight w per meter of the band.· By the nonlinear form of the screw conveyor the pressure along the profile maskdecreases nonlinearly. Therefore, the rubber profile does also depend on the ab-solute position along the metal mask.Nevertheless, the whole system is deterministic: the same rubber mixture G with thesame mask g(x), temperature T and pressure P result in the same rubber profile r(x),even on a different extruder machine of the same type E.Up to now, due to the nonlinear nature of the macromolecular mixture this taskcan not be solved analytically. Instead, specialized people estimate the profile of theoriginal metal mask by their experience with the subject and correct their estimatesafter experience. This gives a trial-and-error production cycle that causes severedisadvantages for the production business:· The start for a new product is delayed by the time for 2-3 cycles. Each one takes4-5 days to make a new mask, install it on the extruder, make an extrusion try,measure the obtained rubber profile and estimate a new metal mask. This delaydoes not only waste time, money and natural resources, but also increase the pro-duction overhead and impedes therefore the production flexibility severely.· The experienced employees are tied to this job (which they judge as „boring“)without the possibility of a change within the enterprise. Additionally, in the caseof illness of an employee or a change to another enterprise, the knowledge is nolonger accessible. This causes mayor obstacles for the production.Now, in this paper we will show that adaptive process control methods can over-come this kind of problems. They will update the parameters purely based on thefinal, measured outcome data.2 An RBF approximation networkIn this section we describe the methods for approximating the exact metal maskprofile f(x) at location x of the extruder by a two-layer network function F(x). Thisproduces the desired rubber profile r(x), see figure 2. It is well known that such atwo-layer neural network can approximate any continuos function to any degree,provided that we have enough neurons in the first layer, e.g. [Xu, Krzyzak, Yuille1994].The activity y = (y1,...,ym) of the firstlayer is defined byyi = Si (x,ci) = exp(-d2)       i=1..md2= |M (x-c)|2 = (x-c)T MT M (x-c)and the second layer withy0º1, w0 @ bias byF(x) = F(y(x)) = w y j jjm= å1 = wTyx1x2· ·X   · ·· ·xn···y1ym Fig. 2 The activity approximation networkF(x)using m nonlinear basis functions Si that depend only on the Mahalanobis distance dbetween the input x and a neuronal center ci. For adapting and scaling the ellipsoidalinput field, we used the scaling equation MNEW = (I - g(1-a)(aaT)) MOLD            a = (x-c)/ |x-c|with the scaling factor a and the learning rate g, see [Pietruschka, Kinder 1995].There are principally two approaches to train the network parameters: either wetrain the two layers separately or as a whole.The approach of treating the two layers separately, clustering the input spacefirst and then optimizing the weights of the second layer, is fast, but it has someflaws. This gives us a high sample density of output values where we have clustersof input samples, not where the output error is high.Therefore, we optimize both layers at the same time. To avoid the computationalproblems of the backpropagation approach we choose a different strategy. We  startwith the lowest possible complexity of the network and gradually increase the num-ber of neurons in the first layer until the error is sufficiently reduced. This was al-ready proposed for RBF nets, for example by [Schiøler, Hartmann 1992]. We insertthe neuron at location xk, the k-th sample with the maximal error, that has to becompensated by the new m-th neuron. We have to design the width Mm such that itfits the new basis function in the context of all neighbored neuron basis functions. Incontrast to the approach of [Platt, 1995] we do not use gradient descend technique torearrange all other neurons and adapt all their receptive fields: this is computation-ally intensive and is the source of new errors. Instead, we stop the adaptation proc-ess of the new neuron by the criterion of non-significant activation on a data point.Additionally, we reduce the long distance neighborhood influence by a learning rateg(d) which drops with increasing distance from cm, that is with decreasing activitylevel, see [Pietruschka, Kinder 1995].3 Approximating the extrusion process parametersTo apply the approximation algorithm that we developed in the previous sections wehave to model the industrial process for the example of tire production. The maintask consists of estimating the profile of a metal mask that extrudes the profile of arubber band. This band is then cut into a stripe of the perimeter length of a tire andthen glued to the casing. The raw tire is then „baked“ in a metal tire form for 20minutes, giving the preliminary profile the ultimate form.3.1 Modeling the process Although the extruded rubber profile is a temporary form its desired accuracy is0.1 mm. This settles the upper limit for our approximation error. In figure 3 a sam-ple profile is shown.Fig. 3 A rubber profile and the corresponding metal maskThe upper profile is the desired rubber profile; the lower one shows the correspond-ing rectangular metal mask. On the right hand side a cut through the metal (shadedarea) shows the form of the opening (not shaded). Where the rubber flows in, theprofile has a wider opening. This corresponds to the dotted line that encircles theprofile opening in the metal mask.The analytical treatment of the nonlinear dependencies is very difficult. Con-ventional assumptions about energy (i.e. enthalpy) conservation are not valid here.Also the direct measurement of the process parameters like temperature and pres-sure in the profile are practically limited. In contrast to this, our approach modelsthe system as a whole, avoiding all difference equations and constants which arehard to devise and to measure.We devided the whole centered profile, depending on the tyre width, into 170-270 points that are placed in the regular distance of d mm. Each point xi has a de-sired rubber profile height r(i). The intermediate points are interpolative generated,see figure 4.                   r(x1)          r(x4)       r(x5)        r(x6)        r(x9)                       r(x2)  r(x3)                               r(x7) r(x8)Fig. 4 The intermediate interpolation of the profile and a neighborhood windowSince the influence of the sample points is limited to the neighborhood for a certainrubber profile height r(i) we have only to consider n=2s+1 neighbor pointsg(i) = F(ri-s,...,ri,...,ri+s, i, G, E, P,w,..)By this model we implement a neighborhood window that uses n=2s+1 samplingpoints around location i. All values rk for the sampling points outside the profilelimits are set to zero.3.2 Simulation resultsFor the determination of the two parameters, the number n of neighborhood sam-pling points, and the distance d between the sampling points, we decided to simulatedifferent configurations in order to get an acceptable choice.We generated the training set by shifting a window (determined by d and n) by anincrement of 1 mm over the profile data of 5 profiles with the same values of G, Eand P. This generated 1346 training patterns. The sixth profile was used for thegeneration of test set of 271 test points. The simulation results generally showedonly a very small influence of the position i. So, let us regard other dependencies.For the expected absolute error for 100 neurons we got different results, de-pending on the type of network we used. The nets with growing, radially symmetricinput regions have in the average 10-90% more error than the growing elipsoidalnets. The best performance of the two types converged by training to the followingexpected absolute error, depending on the number of sampling points n and theinterpoint distance d.n \ d 3 mm 4mm 5mm7 0,178 0,159 0,1879 0,167 0,162 0,19711 0,165 0,206 0,226It is interesting to see that the error does not automatically decrease or increasewhen we increase the number of sampling points. There is a configuration of theparameters where we roughly meet the balance and the error becomes quite small.The best results are observed by n=9 and d=4 mm which corresponds to a windowsize of 32 mm with the expected absolute error of 0.16 mm and the maximal abso-lute error of 0.56 mm. In figure 5 the test profile, the result of the network and theresultingerrorestimated mask profiledesired profileFig. 5 The wanted profile and the profile produced by the net for n=9, d=4error is shown for this configuration. The y-axis is scaled up by the factor of three toenhance the visibility of the errors.4 Discussion and outlookIn the previous sections we have presented an adaptive solution for the problem ofunknown process parameters in tyre production. The learning algorithm uses nointernal process variables or other intrinsic knowledge but only the measurable ex-ternal process parameters as the weight per meter and the resulting rubber profile.This approach avoids many technical and economic disadvantages that were given inthe introduction.Nevertheless, our work also shows that there are still several problems to be solved:· The current modeling uses the data provided by the production as tuples (wantedrubber profile, successful metal profile). The successful metal profile used fortraining was obtained after several trials and corrections, that is the training isbased on an artificial and not on a real sample of the input/output mapping.This problem can be solved by measuring and using only the directly obtainedrubber profile data.An important key for the simulation performance turned out to be the two parame-ters, the number n of neighborhood sampling points, and the distance d between thesampling points. We can determine the proper choice by balancing the counteractinginfluences:· If we choose d too small, we increase the number of necessary sampling pointsfor a certain neighborhood and increase therefore the dimension of the inputspace. Since we have only a small limited number of training samples, the train-ing becomes very difficult since the input space becomes very sparse. This isknown as the „curse of dimensions“ [Huber 85]. On the other hand, if we choosed too big, we can lose important information due to undersampling the depend-ency function.· If we choose n too big, we meet the same problem of sparseness of the trainingsamples in the input space. Additionally, by increasing to much context informa-tion, the generalization ability of the network will be limited. On the other hand,if we limit the window too much, necessary context information which helps todistinguish between different situations is lost.From the theoretical point of view, this is an interesting situation. By the nature ofthe problem, we have not hundreds of sample profiles but just a few ones. Never-theless, we are not aware of an applicable method of determining the optimal d andn to solve the problem of optimal training. Here, the work of  theorists is welcome.5 ReferencesP. Huber: Projection Pursuit; The Annals of Statistics, Vol.13, No.2, pp. 435-475, 1985U. Pietruschka, M. Kinder: Elipsoidal Basis Functions for Higher-Dimensional Approxima-tion Problems; Proc. ICANN-95, Vol II, Paris 1995, pp.81-85J. C. Platt; Learning by Combining Memorization and Gradient Descent; NIPS; pp. 714-720;1992.H. Shiøler, U. Hartmann: Mapping Neural Network Derived from Parzen Window Estimator;Neural Networks, Vol.5, pp.903-909 (1992)Lei Xu, Adam Krzyzak, Alan Yuille: On Radial Basis Function Nets and Kernel Regression:Statistical Consistency, Convergence Rates, and Receptive Field Size; Neural Networks, Vol.7, No.4, pp.609-628, (1994)

Using RBF nets in rubber industry process control

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