XVI Congreso Español sobre Tecnologías y Lógica Fuzzy 
Valladolid, 1-3 de febrero de 2012A Neurofuzzy model of a mixing chamber pressure has been proposed. The process is a part of
a copper smelter plant. The principal component
analysis (PCA) method has been used to reduce
the inputs space for a recurrent fuzzy model. The
coupling among variables and their mutual influence between themselves, are taken into account
by the projection into the PCA axis. The model
have been validated with real data from the factory. The validation result shows that the model
is suitable for simulation.Ministerio de Ciencia e Innovación DPI2010-21589-C05-0

Bordons Alba, Carlos

Escaño González, Juan Manuel

English

idUS. Depósito de Investigación Universidad de Sevilla

NEUROFUZZY MODEL OF AN INDUSTRIAL PROCESS, REDUCINGCOMPLEXITY BY USING PRINCIPAL COMPONENT ANALYSISJuan Manuel Escaño, Carlos BordonsUniversidad de Sevilla. Escuela Superior de Ingeniería. Avda. de los Descubrimientos s/n. 41092 Sevilla,{jmesca,bordons}@cartuja.us.esAbstractA Neurofuzzy model of a mixing chamber pres-sure has been proposed. The process is a part ofa copper smelter plant. The principal componentanalysis (PCA) method has been used to reducethe inputs space for a recurrent fuzzy model. Thecoupling among variables and their mutual influ-ence between themselves, are taken into accountby the projection into the PCA axis. The modelhave been validated with real data from the fac-tory. The validation result shows that the modelis suitable for simulation.Keywords: Neurofuzzy modeling, Industrialprocess, Principal Component Analysis.1 IntroductionThe process studied in this work is a part of Atlantic CopperSmelter facilities in Huelva (Spain), whose anual produc-tion is around three hundred thousand tons of copper [1].This plant includes a Flash Furnace and four Pierce-Smithconverters, two of them blowing simultaneously. The threecurrents of gases generated in these processes are mixed inthe mixing chamber and sent to three acid plants operatingin parallel (see figure 2). It is very important to maintain thegas pressure in the mixing chamber at a desired value, al-ways bellow ambient pressure in order to avoid gas lossesto the atmosphere. That pressure depends on other vari-ables of the production line and it is very difficult to get anaccurate prediction of it. On the one hand, the causes ofthe pressure oscillations are hard to detect. Moreover, duethat there are different control system in the copper smelterand the acid plant, no clocks synchronization is possible,so there are considerable uncertainties when trying to mea-sure cause-effect delays. A suitable model for predictionone step ahead, has been developed in [1]. Other modelsFigure 1: General view of the copper smelter [1]have been made in [5] and [6]. All of them are predic-tion models, because the actual pressure value at time kPMC(k) is used to predict PMC(k+ 1). Taking into accountthat the converters operate on a batch mode, while those ofthe flash furnace and acid plants are continuous, extremelyhigh disturbances both in flow and SO2 concentration oc-cur at the acid plants inlet due to the converters’ operatingschedule. The existing control strategy, based on indepen-dent single loop PID controllers, is not able to cope withthose disturbances [1]. Advanced control schemes shouldbe applied, but it would be interesting to have a model suit-able for simulation. In this case, it is difficult to derive aprecise mathematical model, based on first principles. Be-sides, the computation of the solution of models obtainedthrough this methodology may require a large computa-tional effort making them useless for real time tasks likecontrol or optimization. Neurofuzzy modeling, which per-mits an easy way to derive successful models, is a goodalternative which can be employed to overcome such limi-tations [2],[3],[7] and [9]. Fuzzy Neural Networks (FNN)combine the capability of uncertainty handling in informa-tion with learning skill. Recurrent Fuzzy Neural Network(RFNN) have proven to be an excellent choice in order topressureGassesoutputlinesAcid plantsPTchamberMixingFigure 2: gas mixing. Acid plantsget the dynamics of nonlinear and complex systems. Theyare systems which have the same advantages than recurrentneural networks [10]. RFNN are also known as Fuzzy Dy-namical Systems (see figure 3) and extend the applicationdomain of FNN to temporal problems. Feedback allows tocapture dynamics and change. Principal Component Anal-OutputsInputsiFigure 3: Dynamical Neurofuzzy Systemysis is a well know technique in the field of multivariatemethods [8]. The use of PCA reduces the space of inputvariables and obtain new uncorrelated variables in order tosimplify the Fuzzy system. This paper is organized as fol-lows. In section 2, a formulation for dynamic neurofuzzymodel is shown. An introduction to Principal ComponentsAnalysis is given in section 3. In section 4, the neurofuzzymodel for the mixing chamber pressure is presented, givingthe validation results. Conclusions are given in section 5.2 Dynamic Neurofuzzy Model formulationIn the neurofuzzy model proposed by Takagi-Sugeno(TS)[12], the structure of antecedent describes fuzzy re-gions in the inputs space, and the one of consequentpresents non-fuzzy functions of the model inputs. We canuse recurrent functions with NARMAX structure (Non-linear Auto Regressive Moving Average with eXogenous in-put), of the kind:yˆ(k+1) = f (y(k); :::;y(k m);u(k); :::;u(k n)) (1)Where u;y are respectively the inputs and outputs of thesystem, the Neurofuzzy system may be described, for eachrule, in the following way:R j :IF x1 is F1 j; :::; and xn is Fn j;THEN:y j = g j1x1+g j2x2+ :::+g jnxn+ c j (2)Where g ji and c j are constant terms, X =x1 x2 ::: xnT is the input vector of the neuro-fuzzy system, Fi j is the fuzzy set respective to xi on therule j, y j is the output of the model respective to theoperating region associated to the rule. xi can represent areal input to the system or any other variable, for instancethe previous values of inputs or outputs. Thus, we couldformulate the consequent (2) likey j(k) = a j(z 1)y(k 1)+b j(z 1)u(k d)+ c j (3)Where a j(z 1) = a1 j + a2 jz 1 + ::: + any jz (ny 1) andb j(z 1) = b0 j+b1 jz 1+b2 jz 2+ :::+bnu jz nud is the dead time. If mi j(k) is the membership degree ofx j(k) in the fuzzy set Fi j and the number of implications orrules is L, the RFNN complete model is described byy(k) =Låj=1w j(k)a j(z 1)y(k 1)+b j(z 1)u(k d)+x (k) (4)Wherew j(k) =m¯ j(k)åLj=1 m¯ j(k); m¯ j(k) =nÕi=1mi j(k)andx (k) =Låj=1w j(k)c jRewriting equation (4) asa¯(z 1)y(k) = b¯(z 1)u(k d)+x (k) (5)Wherea¯(z 1) = 1  a¯1z 1  a¯2z 2  :::  a¯nyz ny (6)b¯(z 1) = 1  b¯1z 1  b¯2z 2  :::  b¯nuz nu (7)a¯i =Låj=1w j(k)ai jz i (8)b¯i =Låj=1w j(k)bi jz i (9)In [4] a dynamic Neurofuzzy Model is used for simulation.We propose an improvement of that model, using a majornumber of inputs, including squares of variables, to providealso a non linear dependence for each rule. A PCA has beenused both in a model used in [4] as the one proposed here.In the first, the analysis is directed just to get uncorrelatedvariables, in the second also much more simplification isachieved in the FIS.3 Principal Components AnalysisDue to coupling between variables and the difficulty tomake experimental tests, the use of PCA provides two char-acteristics: to achieve new uncorrelated variables and con-densation of all the information in a smaller space, provid-ing a simpler Adaptive Neural Fuzzy Model (ANFIS). Itwill permit the addition of others input variables, withoutcomplexity increasing. The aim of PCA is to reduce systemdimension, minimizing missing information. The idea be-hind PCA is to form a minimum number of new variablesto describe the variation of the original data by using linearcombinations of the original variables.Let a be a vector of p variables, a =a1 a2 ::: apAnd A 2 Rnp a data matrix generated by the p variables,where n is the number of data. The aim of PCA is to finda base d =d1 d2 ::: dkwhere k < p, which definesa new subspace retaining the maximum information of theoriginal data.d1 = w11a1+w12a2 ::: +w1papd2 = w21a1+w22a2 ::: +w2pap::::::::::::::::::::dk = wk1a1+wk2a2 ::: +wkpapThen the data matrix will be D 2Rnk.It is demonstrated thatmD = E(D) = E(WTA) =WTE(A) (10)And the covariance matrix of D is equal toSD = Ef(D mD)(D mD)Tg=W TSAW (11)The goal is to get the maximum data variance in the newaxes, that is maximum SD, imposing the orthonormalityconstraint on it:WTW = I (12)We have to maximize:WTSAW  l (WTW   I) (13)deriving and making it equal to zero, we have(SA l I)W = 0 (14)The problem is just one of calculation of eigenvectors ofSA. The associated components to greater eigenvalues ofSA are the most meaningful to build the data space. Theyare named Principal Components of the system. In order tochoose how many principal components to use, a criteriumbased on the weight of each eigenvalue with respect to theother, can be used:åli=1liåni=1li n (15)being n a measurement of desired information.PC1PC2Figure 4: Principal ComponentsTable 1: List of variablesDescription Units TypePressure in mixing chamber mbar OutputFlow to plant 1 KNm3/h ManipulatedFlow to plant 2 KNm3/h ManipulatedFlow to plant 3 KNm3/h ManipulatedDilution flow to plant 1 Nm3/h ManipulatedDilution flow to plant 2 Nm3/h ManipulatedDilution flow to plant 3 Nm3/h ManipulatedReference for flash furnace feeding Tons/h DisturbanceFlow control valve for flash furnace % DisturbanceFan speed in flash furnace rmp DisturbanceReference for fan speed line 1 rpm DisturbanceReference for fan speed line 2 rpm Disturbance4 Neurofuzzy Model of Mixing ChamberAfter a preliminary study based on some experiments withsteps on the variables[1], the evolution of the pressure inthe mixing chamber (PMC), is influenced by others that aredivided into two groups: control signals and disturbances.In table 1 a brief description of the considered variables isgiven, whereas in figure 5 a scheme with the situation ofeach of them is presented.In [4], the inputs are the current samples of the variablesof Table 1 and the previous samples of them, forming aPCAZ-1......Z-1...48input variables to PCA11inputvariablesFIS8 variablesafter PCAPMCx2...... ... ...Figure 6: Neurofuzzy scheme proposedFlashFurnaceAcid Plant 1ScrubberConverter 1Converter 2STSTSTZTFTPTPTAcid Plant 3Acid Plant 2FTFTFTFTDilution ValveDilution ValveDilution ValveFan SpeedFan SpeedMixing Chamber PressureMCFigure 5: Process, manipulated variables and disturbances0 200 400 600 800 1000 1200 1400 1600−6−4−20secondsPressure (mbar)  Real DataModel0 200 400 600 800 1000 1200 1400 1600−2−101secondsError (mbar)Figure 7: Validation of the model used in [4]0 200 400 600 800 1000 1200 1400 1600−6−4−20secondsPressure (mbar)  Real DataModel0 200 400 600 800 1000 1200 1400 1600−1−0.500.51secondsError (mbar)Figure 8: Validation of the proposed modelnumber of 24 inputs. After a PCA using all the nonzeroeigenvalues, 21 components are obtained. As mentioned atthe end of the introduction, the addition of entries does notcomplicate the model when using PCA. For this work, wehave taken into account the input variables listed in Table 1and their squares, to give to the model, quadratic elementsof the input variables of the system. Figure 6 presents thescheme followed to obtain the neurofuzzy model. The in-puts are the variables presented in table 1, including Pres-sure in mixing chamber, their squares and the previoussamples of all of them. To carry out the PCA, data havebeen used for nearly three hours of operation, sampled ev-ery 2 seconds. With a loss of information from 1%, thenew principal components are 7. Using these new uncorre-lated variables as inputs to the fuzzy system, and the nextsampling pressure as output, an ANFIS is designed, usingSubtractive clustering technique [3]. The performance ofthe models can be seen in figures 7 and 8, where it is vali-dated using a real data set from the process. It is importantto note that PMC(t  1) is generated by the models outputin the previous sampling. Looking at the figures we seethat the error does not grow indefinitely, this fact makesthe models, appropriate to simulate the process. In figure7, the mean error is 0:4099C, while the new model pro-posed in this work, the mean error is 0:1469C, obtainingan improvement.5 ConclusionsA Neurofuzzy model, suitable for simulation, of a mixingchamber pressure has been designed. The PCA method hasbeen used to reduce the inputs space for a recurrent fuzzymodel. Using this technique, a previous increment of inputhas been added, including quadratic elements and previoussamplings. The coupling among variables and their mutualinfluence between themselves is taken into account by theprojection into the PCA axis. The model have been vali-dated with real data from the factory. The validation resultshows that the model is suitable for simulation. In com-parison with other simulation models of the same plant, animprovement has been got.AcknowledgementsThe authors acknowledge J.M. Tejera (Atlantic Copper),for the provision of real data to carry out the study.This work has been funded by the National Plan ProjectDPI2010-21589-C05-01 of the Spanish Ministry of Sci-ence and Innovation and FEDER funds.References[1] C. Bordons, M. R. Arahal, E. F. Camacho, J.M.Tejera. “ Energy Saving in a Copper Smelter bymeans of Model Predictive Control". Identificationand Control, Chap. 3. pp. 63-65. Springer London2007.[2] Chi-Huang L., Ching-Chih T. “Generalized predictivecontrol using recurrent fuzzy neural networks for in-dustrial processes". Journal of Process Control. 2007[3] Chiu, S., “Fuzzy Model Identification Based on Clus-ter Estimation," Journal of Intelligent & Fuzzy Sys-tems, Vol. 2, No. 3, Spet. 1994[4] J.M. Escaño, C. Bordons, F. Dorado. “Pca BasedPressure Control of a Gas Mixing Chamber". 2009IEEE Conference on Emerging Technologies & Fac-tory Automation (ETFA 2009), pp. 1-6. Palma deMallorca. IEEE. 2009.[5] J.M. Escaño, C. Bordons, A. Nuevo. “Dynamical& non-Dynamical neurofuzzy models of a mixingchamber pressure". Sice Annual Conference. Tokyo.Japón. 2008[6] J.M. Escaño, C. Bordons. “Modelo neuroborroso dela presión de la cámara de mezcla en una fundiciónde cobre". XXVIII Jornadas Técnicas de Automática.CEA-IFAC. 2006[7] Jhy-Shyng, Jang, R. “Neuro-Fuzzy Modeling andControl". Proceedings of the IEEE, Vol 83, No. 3,March 1995[8] D. E. Johnson. “Applied multivariate methods for dataanalysis". Brooks Cole Publishing CO and ITP CO.1998[9] K. Kiriakidis, “Fuzzy Model-Based Control of Com-plex Plants". IEEE Transactions On Fuzzy Systems,vol. 6, NO. 4, Nov.1998[10] Chao-Chee Ku, Kwang Y. Lee, “Diagonal recurrentneural networks for dynamic systems control", IEEETrans. Neural Networks, vol.6, pp. 144-156, Jan.1995.[11] Ching-Hung Lee, Ching-Cheng Teng, “Identifica-tion and Control of Dynamic Systems Using Recur-rent Fuzzy Neural Networks", IEEE Transactions OnFuzzy Systems, vol.8, no. 4, Aug. 2000[12] Takagi and Sugeno, “Fuzzy Identification of Systemsand Its Applications to Modelling and Control", IEEETrans. Syst. Man. Cyber., Vol. 15, pp. 116-132, 1985[13] A. Trabelsi, F. Lafont, M. Kamoun, G.Enea,“Identification of Nonlinear MultivariableSystems by Adaptative Fuzzy Takagi-SugenoModel". International Journal of ComputationalCognition (http://www.YangSky.com/yangijcc.htm).Volume 2. Number 3. Pages 137-153. 2004View publication stats

Neurofuzzy model of an industrial process, reducing complexity by using principal component analysis

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Neurofuzzy model of an industrial process, reducing complexity by using principal component analysis

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