In this paper, a novel learning architecture based on neural networks is used for temperature
inverse modeling in microwave-assisted drying processes. The proposed design combines the
accuracy of the radial basis functions (RBF) and the algebraic capabilities of the matrix
polynomial structures by using a two-level structure. This architecture is trained by
temperature curves, TcðtÞ; previously generated by a validated drying model. The
interconnection of the learning-based networks has enabled the finding of electric field (E)
optimal values which provide the TcðtÞ curve that best fits a desired temperature target in a
specific time slo

Monzó Cabrera, Juan

Pedreño Molina, Juan Luis

Sánchez Hernández, David Agapito

English

Repositorio Digital de la Universidad Politécnica de Cartagena

U13579111315171921232527293133353739414345ARTICLE IN PRESS3B2v8:06a=w ðDec 5 2003Þ:51cXML:ver:5:0:1 NEUCOM : 10007Prod:Type:FTPftppp:128ðcol:fig::NILÞED:KandavelMPAGN:savi SCAN:omincNeurocomputing ] (]]]]) ]]]–]]]NC0925-2312/$ -doi:10.1016/jCorrespoE-mail adwww.elsevier.com/locate/neucomOFLettersA new predictive neural architecture for solvingtemperature inverse problems in microwave-assisted drying processesJ.L. Pedreño-Molina, J. Monzó-Cabrera, D. Sánchez-HernándezDepartment of Information Technologies and Communications, Technical University of Cartagena, CampusMuralla del Mar s/n, E-30.202, Cartagena, Murcia. SpainReceived 21 July 2004; received in revised form 24 November 2004; accepted 24 November 2004Communicated by R.W. Newcomb ORRECTED PRAbstractIn this paper, a novel learning architecture based on neural networks is used for temperatureinverse modeling in microwave-assisted drying processes. The proposed design combines theaccuracy of the radial basis functions (RBF) and the algebraic capabilities of the matrixpolynomial structures by using a two-level structure. This architecture is trained bytemperature curves, T cðtÞ; previously generated by a validated drying model. Theinterconnection of the learning-based networks has enabled the finding of electric field (E)optimal values which provide the T cðtÞ curve that best fits a desired temperature target in aspecific time slot.r 2005 Published by Elsevier B.V.Keywords: Learning-based predictive system; Electric field estimation; Neural network modeling;Microwave-assisted drying applications; Inverse problemOsee front matter r 2005 Published by Elsevier B.V..neucom.2004.11.026nding author. Tel.: +34968 326533; fax: +34 968325355.dress: juan.pmolina@upct.es (J.L. Pedreño-Molina).U13579111315171921232527293133353739414345ARTICLE IN PRESSNEUCOM : 10007J.L. Pedreño-Molina et al. / Neurocomputing ] (]]]]) ]]]–]]]2F1. IntroductionThe conventional study of drying processes is mainly based on the numericalresolution of the differential equations describing the physical phenomena. Severalsimplifying drying models providing relatively accurate results can be found in theliterature [2,4]. However, these models, generally based upon differential equations,present great limitations for solving the inverse problem. In contrast, thearchitectures based on neural networks inherently provide ways to solve theappearance of constraints. Additionally, most neural network structures applied tomodel drying processes involve solutions in which the output of the network isreduced to a set of values [1,3], but not to a time-dependent function. In this work,however, the proposed architecture is able to generate complete temperature curves,T cðtÞ; from only two numerical input parameters: the electric field (E) and the airflowtemperature (Tair). The neural architecture is configured in two levels by using radialbasis function (RBF) neural networks and polynomial learning structures, enablingthe prediction of E optimal values that force TcðtÞ to reach a desired temperaturetarget T c0 in a required time slot t0: ONCORRECTED PRO2. Structure of the neural architectureIn microwave-assisted drying processes, the evolution of T cðtÞ in the material ishighly dependent on the electric field (E) and the air temperature (Tair), providedthat the cavity structure and the internal conditions of the material do not vary [5].Additionally, T cðtÞ can present non-linear variations during all drying stages. Due tothis, the design of the proposed neural network architecture is based on learningstructures and focused on non-linear problems, such as the mentioned temperatureinverse problem, in order to predict the optimal E input variable. Thus, RBF neuralnetworks have been selected for temperature identification (level 1) and a learning-based polynomial network for mapping the RBF neuron weights obtained from eachtraining trial over the input variables E and Tair (level 2), as illustrated in Fig. 1,where ~Wk is the vector of neuron weights for each k trial. In the first level of theproposed architecture, the length interval of TcðtÞ; t ¼ ½0::T  1; is divided into Mtime slots. Also in this level, the T points of TcðtÞ are projected onto M neuronsðMoTÞ: The second level establishes the relationship between ~Wk and the ~Vkvector, for all the M neurons and all the k trials, by means of the matrix WW. Thecomponents of ~V are dependent on both E and Tair; which are the inputs variablesfor the drying process.The level 1 provides a solution to the interpolation of the non-linear functionT cðtÞ: For the kth pair (k ¼ ½1::N) of input variables ½E;Tairk; the estimation ofT ckðtÞ is given by~TckðtÞ ¼XMj¼1wjk exp t mj 2s2j !¼XMj¼1wjk  fjkðtÞ; (1)UOOF13579111315171921232527293133353739414345ARTICLE IN PRESSNEUCOM : 10007[ ]kairTE,  Validated  Model  [4] kkE  )(~tcT k)(tTc k  Polynomial Vector kairk TEfV→→→),( 3=WW)(tkφLEVEL 1 LEVEL 2 -+kV  )(tekkairT  Weight  Updating kW  kWkW+-)(tek→→→→WFig. 1. General scheme of the proposed architecture. The RBF neural network in level 1 generates theestimated temperature ~T ckðtÞ form E and Tair, while the polynomial network in level 2 establishes themapping between ~Wk and ~Vk for all the trials in level 1.J.L. Pedreño-Molina et al. / Neurocomputing ] (]]]]) ]]]–]]] 3NCORRECTED PRwhere the Gaussian fjkðtÞ is the j radial function, mj and sj are the center andstandard deviation of fjkðtÞ, N is the number of trials during the learning stage, andwjk is the value of the weight associated to fjkðtÞ for the kth trial. Transforming (1)into a matrix notation, results in~TckðtÞ ¼ ~Wk  ~fkðtÞT; (2)where ~Wk is the 1M dimension vector containing the RBF neuron weights for thekth trial and ~fkðtÞ the vector whose elements are the M Gaussian functions. In level2, the ~Wk obtained in each training trial is projected onto the input variables½E;Tairk; which have generated T ckðtÞ: By considering all the learning trials, thismapping generates the matrix WW, whose dimension is equal to 9M, 9 being thelength of the ~V vector according to Eq. (4), and M the number of RBF neurons. Theweights of the WW matrix are obtained by the minimization of the quadratic errorbetween ~Wk and W0k ¼~Vk WW: This mapping is carried out by a two-dimensional (2D) polynomial network whose order in each dimension is establishedin accordance to the dependence of each neuron with respect to E and Tair. From thesurface analyses in Fig. 2, one can observe that the proposed network has a third-order dependence of the weights in the RBF network on E in T cðtÞ curves. At thesame time, these surfaces also show a linear dependence with respect to Tair, whichjustifies the selection of the polynomial structure of level 2. By applying thepolynomial network to the RBF neuron weights and considering the matrixformulation in (2), ~T cðtÞ can be generated from the input variables ½E;Tair by meansof Eq. (3)UF13579111315171921232527293133353739414345ARTICLE IN PRESSNEUCOM : 10007Fig. 2. (a) Dependence of the considered weight over the input variables E and Tair. (b) Error produced bythe approximation of the third-order polynomial network in level 2 of the proposed architecture.J.L. Pedreño-Molina et al. / Neurocomputing ] (]]]]) ]]]–]]]4NCORRECTED PROO~Tcðt;Tair;EÞ ¼ ~W  ~Fðt;mi;sÞT¼ ~V ðE;TairÞ WW  ~Fðt;mi; sÞT; ð3Þwhere~V ðE;TairÞ ¼ E;Tair;E2;E  Tair;Tair  E2;E3; 1 : (4)In order to apply this neural architecture for solving the temperature inverseproblem, in this work we have fixed Tair at 45 1C and a target value for the sampletemperature (T c0) which has to be reached within t0 seconds. With these conditions,the proposed model is able to estimate the optimal value for E that generates theT cðtÞ curve that fits to the desired target point ½T c0; t0: By particularizing (3) for t0and Tair, Eq. (5) is obtained:Tcðt0;EÞ ¼ ~V ðEÞ WW  ~Fðt0ÞT¼ T c0: (5)By solving (5) for the variable E; expressions (6)–(7) have been obtained. It mustbe pointed out that, in this case, an accurate solution for the inverse problem can bereached only if desired target T c0ðt0Þ belongs to the learned range for E and Tair.A6ðt0ÞE3 þ ðA3ðt0Þ þ TairA5ðt0ÞÞE2þ ðA1ðt0Þ þ TairA4ðt0ÞÞE þ A7ðt0Þ þ TairA2ðt0Þ ¼ T c0; ð6ÞAkðt0Þ ¼XMj¼1W ~Wkj  ~Fkðt0ÞT: (7)U13579111315171921232527293133353739414345ARTICLE IN PRESSNEUCOM : 10007J.L. Pedreño-Molina et al. / Neurocomputing ] (]]]]) ]]]–]]] 5ROOF3. ResultsIn order to test the ability of the proposed neural model for solving inverseproblems, the neural architecture has been previously trained with random values forthe input variables. The training intervals have been set to 1214oEo3565 and30,3oTairo69,8. For all the Gaussian functions of the RBF neural network mi ¼Ti=Mði 2 ½1::MÞ and s ¼ 3T=M: The initial conditions and parameters for the usedmicrowave-assisted drying model have been: microwave frequency f0 ¼ 245GHz;initial sample temperature T0 ¼ 26; 25 1C; initial moisture content X ¼ 0912 (drybasis); dry material and liquid specific heat cps ¼ 1600 J/Kg 1C and cpw ¼ 4180 J/Kg 1C, respectively. For other simulation parameters the reader should refer to [1].For all simulations the training trials number has been set to 50, T ¼ 600 s andM ¼ 15:Fig. 3 illustrates the temperature target, the optimum value for E and thetemperature curves provided by the drying and the neural model. As Fig. 3 shows,the matching error at the targets, Tc0 ¼ 50 1C and t0 ¼ 20, 30, 50, 100 and 150 s, isnegligible, while the predictive identification of the temperature curves is precise.Finally, the behavior of the proposed architecture has been analyzed for differentvalues of M and learning trials. Fig. 4 shows the accuracy of the architecture bycomparing the values of Tcðt0Þ provided by the drying model [4] and the magnitude~T cðt0Þ estimated by this architecture. From Fig. 4, it can be concluded that 10 trialsduring the learning stage and 12 RBF neurons are sufficient to obtain a goodNCORRECTED Ptime (sec.)0 100 200 300 400 500 600 700Sample Temperature Tc(t) (°C)20406080100120Tc (t) from [4]Tc1NN  (t ; t0 = 20)Tc2NN  (t ; t0 = 30)Tc3NN  (t ; t0 = 50)Tc4NN  (t ; t0 = 100)Tc5NN  (t ; t0 = 150)50 E1 (error)= 3.643,0  (0.63%)E2 (error)= 3.049,6  (0.02%)E3 (error)= 2.524,6  (0.78%)E4 (error)= 2.002,6  (0.71%)E5 (error)= 1.690,6  (0.98%)INVERSE PROBLEME1E2E5E3E4 Fig. 3. Estimation of E for several drying conditions. Tair ¼ 45 1C. In the figure, both temperaturefunction estimated by this neural architecture, TcNNðtÞ;and that generated by the drying model, TcðtÞ; [1]are represented.U13579111315171921232527293133353739414345ARTICLE IN PRESSNEUCOM : 10007Tc(t0)  real30 40 50 60 70Tc(t 0) estimated from the neural architecture3035404550556065706 Trials7 Trials8 Trials9 Trials10 Trialst0 = 60 sec.M = 12Tair = 45°CTc(t0) real35 40 45 50 55 60 65 70Tc(t 0) estimated by the neural architecture3540455055606570N° RBF = 5N° RBF = 6N° RBF = 7N° RBF = 11 t0 = 60 sec.Trials = 10 sec.Tair = 45°C(a) (b)Fig. 4. Behavior of the neural architecture for variations of (a) trial numbers and (b) RBF neuron number.Points appear only when Eq. (6) provides real roots in the learning interval of E.J.L. Pedreño-Molina et al. / Neurocomputing ] (]]]]) ]]]–]]]6Fprediction of TcðtÞ for the drying process and, consequently, to accurately solve theinverse problem. Dots appear only when Eq. (6) provides values for E within thelearning interval. ORRECTED PRO4. ConclusionsIn this paper, the capabilities of neural networks have been applied to the designof a novel architecture and tested for solving temperature inverse problems inmicrowave-assisted drying processes. Precise results are obtained by interconnectingthe adaptive characteristics of the RBF with the algebraic tools of polynomialstructures. As a result, the proposed architecture is able to obtain the optimal valuefor an input variable of the process, in this case the electric field intensity, whichgenerates the proper temperature function whatever the imposed temperaturecondition. The main advantage of the proposed learning-based model is to provide aclosed solution for the described inverse problem, which is difficult to be solved byconventional drying models based on differential equations. Additionally, theadaptive capabilities of neural networks could be used to extend the excellentperformance of the proposed model to other different drying conditions, materialsand techniques.NCOReferences[1] C. Kuroda, J. Kim, Neural network modeling of temperature behavior exothermic polymerizationprocess, Neurocomputing 43 (1–4) (2002) 77–89.[2] A.V. Luikov, Heat and Mass Transfer in Capillary-Porous Bodies, Pergamon Press, Oxford, 1966, pp.233–248.[3] G.S. Mittal, J. Zhang, Prediction of temperature and moisture content of frankfurters during thermalprocessing using neural network, Meat Sci. 55 (1) (2000) 13–24.U13579111315171921232527293133353739414345ARTICLE IN PRESSNEUCOM : 10007J.L. Pedreño-Molina et al. / Neurocomputing ] (]]]]) ]]]–]]] 7[4] J. Monzó-Cabrera, J.M. Catalá-Civera, A. Dı́az-Morcillo, D. Sánchez-Hernández, P.J. Plaza-González, E. de los Reyes, Microwave Opt. Technol. Lett. 32 (6) (2002) 465–469.[5] J.L. Pedreño-Molina, J. Monzó-Cabrera, A. Toledo-Moreo, D. Sánchez-Hernández, Estimation ofmicrowave-assisted drying parameters using adaptive optimization inverse techniques, Int. Commun.Heat Mass Transfer (2004), in press.Juan L. Pedreño-Molina, is Assistant Professor of Telecommunication Engineer-ing at the Technical University of Cartagena (UPCT), Spain. He received in 1984his B.A.Sc. from the Technical University of Madrid, Spain (UPM) and thePh.D. in Neurotechnology, Control and Robotics in 2000. Since 1999 he is withthe Department of Information Technologies and Communications at UPCT.His research interests are in Signal Processing applied to the Control of non-linear systems, Tactile and Vision Sensors for accurate tasks with processingbased on Neural Networks, and drying processes modeling.ROOFJuan Monzó-Cabrera was born in Elda (Alicante), Spain, on January 1973. Hereceived the Dipl. Ing. and Ph.D. degrees in telecommunications engineeringfrom the Universidad Politécnica de Valencia, Valencia, Spain, in 1998 and 2002,respectively. From 1999 to 2000 he was a Research Assistant with the MicrowaveHeating Group (GCM). In 2000, he joined the Departamento de Tecnologı́as dela Información y las Comunicaciones, Universidad Politécnica de Cartagena,Cartagena, Spain, as an Associate Lecturer. At present, he is an AssociateLecturer with the Departamento de Tecnologı́as de la Información y lasComunicaciones. His current research areas cover microwave-assisted heatingand drying processes, microwave applicator design and optimization andECTED Pnumerical techniques in electromagnetics.David Sánchez-Hernández, obtained his Dipl.-Ing. in TelecommunicationsEngineering from Universidad Politécnica de Valencia, Spain, in 1992 and hisPh.D. from King’s College, University of London, UK, in early 1996. From 1992to 1994 he was employed as a Research Associate for The British Council-CAMat King’s College London where he worked on active and dual-band microstrippatch antennas. In 1994 he was appointed EU Research Fellow at King’s College,London, working on several joint projects at 18, 38 and 60GHz related to printedand integrated antennas on GaAs, microstrip antenna arrays, sectorization anddiversity. In 1997 he returned to Universidad Politécnica de Valencia, Spain,where he was co-leader of the Antennas, Microwaves and Radar Research GroupNCORRand the Microwave Heating Group. In early 1999 he received the Readershipfrom Universidad Politécnica de Cartagena and was appointed Vice-Dean of the School forTelecommunications Engineering and Leader of the Microwave, Radiocommunications and Electro-magnetism Engineering Research Group (GIMRE). In late 1999 he was appointed Vice-Chancellor forInnovation & Technology Transfer at Universidad Politécnica de Cartagena and member of severalFoundations and Societies for promotion of R&D in the Autonomous Region of Murcia, in Spain. InMay 2001 Dr. Sánchez-Hernández was appointed official advisor in technology transfer and member ofThe Industrial Advisory Council of the Autonomous Government of the Region of Murcia, in Spain. He isalso an IEE Chartered Engineer (Stage 1) and is the recipient of the prestigious R&D J. LanghamThompson Premium, awarded by the Institution of Electrical Engineers. In July 2003 he was appointedDean of the Signal Theory and Radiocommunications Department at the Universidad Politécnica deCartagena. He has published over 35 scientific papers, has contributed with over 70 conferencepresentations and is a reviewer of several international scientific journals and conferences. His researchinterests encompass all aspects of the design and application of printed antennas and MMICs for mobileU1357ARTICLE IN PRESSNEUCOM : 10007J.L. Pedreño-Molina et al. / Neurocomputing ] (]]]]) ]]]–]]]8communications, microwave heating and electromagnetic dosimetry issues. Dr. Sánchez-Hernández is alsoa member of the Ampere Board, the European Society for Microwave and High Frequency Heating, andan active board member of the IEE RF and Microwave Circuits Professional Network, the EuropeanCommission JRC-EMF-CA action on human exposure to radiation from GSM and GPRS/UMTS basestations across Europe, IEEE, COIT and AEIT, among other international scientific societies. Recently,Dr. Sánchez has been appointed Spanish representative at the CENELEC CLC/TC106X ‘‘Electro-magnetic Fields in the Human Environment’’ Committee, as well as member of several AENOR SpanishCommittees.NCORRECTED PROOF

A new predictive neural architecture for solving temperature inverse problems in microwave-assisted drying processes

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