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

NEUROFUZZY MODEL OF AN INDUSTRIAL PROCESS, REDUCING COMPLEXITY BY USING PRINCIPAL COMPONENT ANALYSIS Juan Manuel Escaño,Carlos Bordons Universidad de Sevilla. Escuela Superior de Ingeniería. Avda. de los Descubrimientos s/n. 41092 Sevilla, {jmesca,bordons}@cartuja.us.es Abstract A 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. Keywords: Neurofuzzy modeling, Industrial process, Principal Component Analysis. 1 Introduction The process studied in this work is a part of Atlantic Copper Smelter facilities in Huelva (Spain), whose anual production is around three hundred thousand tons of copper [1]. This plant includes a Flash Furnace and four Pierce-Smith converters, two of them blowing simultaneously. The three currents of gases generated in these processes are mixed in the mixing chamber and sent to three acid plants operating in parallel (see figure 2). It is very important to maintain the gas pressure in the mixing chamber at a desired value, always bellow ambient pressure in order to avoid gas losses to the atmosphere. That pressure depends on other variables of the production line and it is very difficult to get an accurate prediction of it. On the one hand, the causes of the pressure oscillations are hard to detect. Moreover, due that there are different control system in the copper smelter and the acid plant, no clocks synchronization is possible, so there are considerable uncertainties when trying to measure cause-effect delays. A suitable model for prediction one step ahead, has been developed in [1]. Other models Figure 1: General view of the copper smelter [1] have been made in [5] and [6]. All of them are prediction models, because the actual pressure value at time k PMC(k)is used to predict PMC(k+1). Taking into account that the converters operate on a batch mode, while those of the flash furnace and acid plants are continuous, extremely high disturbances both in flow and SO2concentration occur at the acid plants inlet due to the converters’ operating schedule. The existing control strategy, based on independent single loop PID controllers, is not able to cope with those disturbances [1]. Advanced control schemes should be applied, but it would be interesting to have a model suitable for simulation. In this case, it is difficult to derive a precise mathematical model, based on first principles. Besides, the computation of the solution of models obtained through this methodology may require a large computational effort making them useless for real time tasks like control or optimization. Neurofuzzy modeling, which permits an easy way to derive successful models, is a good alternative which can be employed to overcome such limitations [2],[3],[7] and [9]. Fuzzy Neural Networks (FNN) combine the capability of uncertainty handling in information with learning skill. Recurrent Fuzzy Neural Network (RFNN) have proven to be an excellent choice in order to pressure Gassesoutputlines Acidplants PT chamber Mixing Figure 2: gas mixing. Acid plants get the dynamics of nonlinear and complex systems. They are systems which have the same advantages than recurrent neural networks [10]. RFNN are also known as Fuzzy Dynamical Systems (see figure 3) and extend the application domain of FNN to temporal problems. Feedback allows to capture dynamics and change. Principal Component AnalOutputs Inputs i Figure 3: Dynamical Neurofuzzy System ysis is a well know technique in the field of multivariate methods [8]. The use of PCA reduces the space of input variables and obtain new uncorrelated variables in order to simplify the Fuzzy system. This paper is organized as follows. In section 2, a formulation for dynamic neurofuzzy model is shown. An introduction to Principal Components Analysis is given in section 3. In section 4, the neurofuzzy model for the mixing chamber pressure is presented, giving the validation results. Conclusions are given in section 5. 2 Dynamic Neurofuzzy Model formulation In the neurofuzzy model proposed by Takagi-Sugeno (TS)[12], the structure of antecedent describes fuzzy regions in the inputs space, and the one of consequent presents non-fuzzy functions of the model inputs. We can use recurrent functions with NARMAX structure (Nonlinear Auto Regressive Moving Average with eXogenous input), of the kind: ˆy(k+1) = f(y(k), ..., y(k−m),u(k), ..., u(k−n)) (1) Where u,yare respectively the inputs and outputs of the system, the Neurofuzzy system may be described, for each rule, in the following way: Rj: IF x1is F1j, ..., and xnis Fn j, THEN: yj=gj1x1+gj2x2+... +gjnxn+cj(2) Where gji and cjare constant terms, X= [x1x2... xn]Tis the input vector of the neurofuzzy system, Fi j is the fuzzy set respective to xion the rule j,yjis the output of the model respective to the operating region associated to the rule. xican represent a real input to the system or any other variable, for instance the previous values of inputs or outputs. Thus, we could formulate the consequent (2) like yj(k) = aj(z−1)y(k−1)+bj(z−1)u(k−d)+cj(3) Where aj(z−1) = a1j+a2jz−1+... +anyjz−(ny−1)and bj(z−1) = b0j+b1jz−1+b2jz−2+... +bnujz−nu dis the dead time. If µ i j(k)is the membership degree of xj(k)in the fuzzy set Fi j and the number of implications or rules is L, the RFNN complete model is described by y(k) = L ∑ j=1 wj(k)[aj(z−1)y(k−1)+bj(z−1)u(k−d)] + ξ (k)(4) Where wj(k) = ¯ µ j(k) ∑L j=1¯ µ j(k),¯ µ j(k) = n ∏ i=1 µ i j(k) and ξ (k) = L ∑ j=1 wj(k)cj Rewriting equation (4) as ¯a(z−1)y(k) = ¯ b(z−1)u(k−d)+ ξ (k)(5) Where ¯a(z−1) = 1−¯a1z−1−¯a2z−2−... −¯anyz−ny(6) ¯ b(z−1) = 1−¯ b1z−1−¯ b2z−2−... −¯ bnuz−nu(7) ¯ai= L ∑ j=1 wj(k)ai jz−i(8) ¯ bi= L ∑ j=1 wj(k)bi jz−i(9) In [4] a dynamic Neurofuzzy Model is used for simulation. We propose an improvement of that model, using a major number of inputs, including squares of variables, to provide also a non linear dependence for each rule. A PCA has been used both in a model used in [4] as the one proposed here. In the first, the analysis is directed just to get uncorrelated variables, in the second also much more simplification is achieved in the FIS. 3 Principal Components Analysis Due to coupling between variables and the difficulty to make experimental tests, the use of PCA provides two characteristics: to achieve new uncorrelated variables and condensation of all the information in a smaller space, providing a simpler Adaptive Neural Fuzzy Model (ANFIS). It will permit the addition of others input variables, without complexity increasing. The aim of PCA is to reduce system dimension, minimizing missing information. The idea behind PCA is to form a minimum number of new variables to describe the variation of the original data by using linear combinations of the original variables. Let α be a vector of pvariables, α =[ α 1 α 2... α p] And A∈Rn×pa data matrix generated by the pvariables, where nis the number of data. The aim of PCA is to find a base δ =[ δ 1 δ 2... δ k]where k<p, which defines a new subspace retaining the maximum information of the original data. δ 1=w11 α 1+w12 α 2... +w1p α p δ 2=w21 α 1+w22 α 2... +w2p α p .................... δ k=wk1 α 1+wk2 α 2... +wkp α p Then the data matrix will be D∈Rn×k. It is demonstrated that µ D=E(D) = E(WTA) = WTE(A)(10) And the covariance matrix of Dis equal to ΣD=E{(D− µ D)(D− µ D)T}=WTΣAW(11) The goal is to get the maximum data variance in the new axes, that is maximum ΣD, imposing the orthonormality constraint on it: WTW=I(12) We have to maximize: WTΣAW− λ (WTW−I)(13) deriving and making it equal to zero, we have (ΣA− λ I)W=0 (14) The problem is just one of calculation of eigenvectors of ΣA. The associated components to greater eigenvalues of ΣAare the most meaningful to build the data space. They are named Principal Components of the system. In order to choose how many principal components to use, a criterium based on the weight of each eigenvalue with respect to the other, can be used: ∑l i=1 λ i ∑n i=1 λ i ≥n(15) being na measurement of desired information. PC1 PC2 Figure 4: Principal Components Table 1: List of variables Description Units Type Pressure in mixing chamber mbar Output Flow to plant 1 KNm3/h Manipulated Flow to plant 2 KNm3/h Manipulated Flow to plant 3 KNm3/h Manipulated Dilution flow to plant 1 Nm3/h Manipulated Dilution flow to plant 2 Nm3/h Manipulated Dilution flow to plant 3 Nm3/h Manipulated Reference for flash furnace feeding Tons/h Disturbance Flow control valve for flash furnace % Disturbance Fan speed in flash furnace rmp Disturbance Reference for fan speed line 1 rpm Disturbance Reference for fan speed line 2 rpm Disturbance 4 Neurofuzzy Model of Mixing Chamber After a preliminary study based on some experiments with steps on the variables[1], the evolution of the pressure in the mixing chamber (PMC), is influenced by others that are divided into two groups: control signals and disturbances. In table 1 a brief description of the considered variables is given, whereas in figure 5 a scheme with the situation of each of them is presented. In [4], the inputs are the current samples of the variables of Table 1 and the previous samples of them, forming a PCA Z-1 . . . . . . Z-1 . . . 48inputvariablestoPCA 11inputvariables FIS 8variables afterPCA PMC x2. . . . . . . . . . . . Figure 6: Neurofuzzy scheme proposed FlashFurnace AcidPlant1 Scrubber Converter1 Converter2 ST ST ST ZT FT PT PT AcidPlant3 AcidPlant2 FT FT FT FT DilutionValve DilutionValve DilutionValve FanSpeed FanSpeed MixingChamberPressure MC Figure 5: Process, manipulated variables and disturbances 0 200 400 600 800 1000 1200 1400 1600 −6 −4 −2 0 seconds Pressure (mbar) Real Data Model 0 200 400 600 800 1000 1200 1400 1600 −2 −1 0 1 seconds Error (mbar) Figure 7: Validation of the model used in [4] 0 200 400 600 800 1000 1200 1400 1600 −6 −4 −2 0 seconds Pressure (mbar) Real Data Model 0 200 400 600 800 1000 1200 1400 1600 −1 −0.5 0 0.5 1 seconds Error (mbar) Figure 8: Validation of the proposed model number of 24 inputs. After a PCA using all the nonzero eigenvalues, 21 components are obtained. As mentioned at the end of the introduction, the addition of entries does not complicate the model when using PCA. For this work, we have taken into account the input variables listed in Table 1 and their squares, to give to the model, quadratic elements of the input variables of the system. Figure 6 presents the scheme followed to obtain the neurofuzzy model. The inputs are the variables presented in table 1, including Pressure in mixing chamber, their squares and the previous samples of all of them. To carry out the PCA, data have been used for nearly three hours of operation, sampled every 2 seconds. With a loss of information from 1%, the new principal components are 7. Using these new uncorrelated variables as inputs to the fuzzy system, and the next sampling pressure as output, an ANFIS is designed, using Subtractive clustering technique [3]. The performance of the models can be seen in figures 7 and 8, where it is validated using a real data set from the process. It is important to note that PMC(t−1)is generated by the models output in the previous sampling. Looking at the figures we see that the error does not grow indefinitely, this fact makes the models, appropriate to simulate the process. In figure 7, the mean error is 0.4099◦C, while the new model proposed in this work, the mean error is 0.1469◦C, obtaining an improvement. 5 Conclusions A Neurofuzzy model, suitable for simulation, of a mixing chamber pressure has been designed. The PCA method has been used to reduce the inputs space for a recurrent fuzzy model. Using this technique, a previous increment of input has been added, including quadratic elements and previous samplings. The coupling among variables and their mutual influence between themselves is 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. In comparison with other simulation models of the same plant, an improvement has been got. Acknowledgements The 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 Project DPI2010-21589-C05-01 of the Spanish Ministry of Science and Innovation and FEDER funds. References [1] C. Bordons, M. R. Arahal, E. F. Camacho, J.M. Tejera. “ Energy Saving in a Copper Smelter by means of Model Predictive Control". Identification and Control, Chap. 3. pp. 63-65. Springer London 2007. [2] Chi-Huang L., Ching-Chih T. “Generalized predictive control using recurrent fuzzy neural networks for industrial processes". Journal of Process Control. 2007 [3] Chiu, S., “Fuzzy Model Identification Based on Cluster Estimation," Journal of Intelligent & Fuzzy Systems, Vol. 2, No. 3, Spet. 1994 [4] J.M. Escaño, C. Bordons, F. Dorado. “Pca Based Pressure Control of a Gas Mixing Chamber". 2009 IEEE Conference on Emerging Technologies & Factory Automation (ETFA 2009), pp. 1-6. Palma de Mallorca. IEEE. 2009. [5] J.M. Escaño, C. Bordons, A. 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