WSEAS TRANSACTIONS Zs on SYSTEMS Issue 2, Volume 3, April 2004 ISSN 1109-2777 http://www.wseas.org ‘CNN Models of Noolinear PDEs with Memory, Angela Slavova ‘Conditional Prediction of Markoy Processes Using uon Parametric Vicerbl Algorithm - Comparison with MLP and ML GRNW Models, PF Maras, 7 Monbet x6 smpetitive Strategies for Multilayer Perceptrons’ Training using Backpropagation and Parallel Processing, 382 RLS Abas, AC ML Alzuuorus, .D.Melo and A.D Dona Nao Pattern Recognition Using Advanced Error Back-propagation Methods, Joze Pihler 358 Economic and Minimum Emission Dispatch, Yeliz Demirel and Aysen Demiroren 364 Comparative Study On Neural Network Hased Role Extraction Algorithm And C&S, Makamed Sharkony, Mousa Shem Shea Kiter x69 A Conservative Approach to Perceptron Learning, Ramasubramanian Sundararajan, Asim K. Pal 315, ‘Ovtine Lentietion Based on Nearal Networks Using of Levenberg-Marguardt Method and Back propagation Algocithms, Per Pivonke, ir: Dokmal i 38 ‘AINew Universal Algorithm for Neural Rule Extraction, Jdhane Bowaia Smiani 386 ‘SEE: a Concept for an FPGA based Emulation Engine for Spiking Neurons with Adoptive Weights, Heik . Helimich, Heck Kar 380 Successive Role Refinement ina Hybrid Loteligent System, Nelson Marcos, Arm Azcoraga 386 Implementation of AWBT compensation using ANN, Addison Rios-Bolvr, Gloria Mowsll, Franckin Riva-Echeeria a ‘Adding a Reject Oplon toa Tralned Cassier, Ramasubramanian Sundararajan, Asim K. Pal 8 Nearo-uaxy Syotame A Survey, Jone Vira, Ferando Morgado Dias, Alecande Mota as Bootean Neural Networks, Roman Kohut, Berd Steinbach 20 ‘Misture Odor Clessifcaion using Fuzzy Neural Network and Its Optimization through Genetic Algorthi, Berarin Kusumopuro 26 Tomards «Volterra Series Representation from a Neural Network model, Geonjina Stegmayer, Maro Pros, Garcaro Orengo, Omar Chott cc Finding an Optimal Neural Network Structure Using Decision Trees, Rstisav Suharik, Ladiay Now, Alesandea Fara 8 Artifical Neurons Based on CMOS f-Driven Threshold Elements with Functional Inputs, Victor Varsha, Yeackesay Markos, Ua Levin “a Information Geometry of Gibbs Sampler, Kerja Takabotake oy A New Approach in Combining Fisher's Lincar Discriminant and Neutal Network for Face Detection, Homa Fashndi and 1M. Stateam Mole 455, Application of Sele Organizing Map Algorithm combined with Straturing Index to characterize strawberry variety aroma by SPMEIGCMS,1L. Gireudel,¥. De Boisheber, M. Monury 46 [New RBF Neural Network Classifier wth Optimized Hidden Neurons Number, Lari Behn, Ade Zour, Fabien Beir 47 ‘Optimization of Product Cade, Shinichi Meeda and Shin Ishi a ‘ASelf-Organized Neural Network fr 3D Surface Reconstruction, Agstino De Medeiros Brito Junior, ico Duarte Doria Neo, Jorge Dantas De Melo an ‘Neural Network Based Vision System for Micro Workpioces Manufacturing, Tatiana Baidyk, Est Kussul 43EDITORIN-CHIEF -MASTORAKIS N, Bitar Institutions of University Education, Hellenic Naval Acedemy, Piraeus, Greece HONORARY EDITOR ZADEH L,, University of Beezeley, USA ANTONIOU G., Montclair State Us van sc Us z State Uses, NS, USA, DATTELIS C, Uneriad Fao, Argento ERM Natya Teco ‘aiventy, Spore, KLUEV V, Univer o Airs Japan LLYIMING, Natal Chis ung Uierety, Hn, ates, NEDIAL 1, State University of Rio de Janeiro, Brazil, ZEMILIAK A., Puebla Autonomous University, Mexico. ft ‘TOPICS: Systems Theory, Control Systems, Robotics, Artificial Intelligence, Fury Systems, Neural Network, VLSTReaization of Neus [Networis, Computation Inteligence in Systems Theory, Knowledge Modelling, Decision Support Systems, Hierarchical Control Sytem ‘Acroepace Systems, ightrave Engineering Stochastic Sytems, Non-linear Sjtems, Telecommusication Stems, Infomation Stems Sigal Processing Stems, Muliimeosionsl SystemsMukarible systems, Hybrid Systems, Multinte Stems Speech aod lang Processiag. Systems, Disccte Evest Dynamic Systems, Manufacturing, Systems, Deceatralised Systems, Remote Scosng ‘Microclectomechanical Stems, Human-Machine Systems. Eovronment Modeling Sonar and underwater acoustic stems. Underse ‘Systems, Naviguion and Tracking Systems. Space Systems. Waveleis. Systems Techniques for Wireless Applications. Filer dese ‘Verification and Validation. Systems for Statstieal Signal and Aray Processing. ‘The Accessiblity Number of Identifying Graphs, Pinar Dundar 8 “Modeling and Control of Toree Phase Boost Rectifiers via Wavelet Based Neural Network, Farzan Rashi, Amir Hooshang Mazinan 4 ‘Arabic Character Recognition Using Neural Networks, Salah F. Sal wn Comparison Between Mahalanobis Distance and Kullbeck-Leibler Divergence in Chustering Analysis, Allan de ‘Medeir Martins, Adbiao Duarte Doria Neto, Jonge Dantas de Melo sot Development of Modified Feature Lines for Higher Recognition of a -D Face Recognition System, Lina and Benyarain Kusumopuro 506 ‘A Nearal Stiefel Learning Based on Gendesics Revisited, Yasunori Nishimori 313 ‘Automatic Document Marlsop using a Selé Organising Map, Sharia Ailuar, John Dunnion, Ronan G Reily S10 lana Geometric and Handprint Texture based Prototype for Identity Authentication, Anzai Picon, Extbaliz Garoxe, Begonia Suarez, Sia Remeria. a 'A Nearal Network for Detection of Orientation, Velocity and Direction of Movement, Based on Physiological Rules, GE Ta Cara, M. Ritorato, M, Ussino Locally Connected BSB Neurat Networks as Associative Memories Storing Grey-Scale Lmages, Giovanni Costantng ‘Massimo Cara, Daniele Case 50 Parallel Manner and Twist Measurement for Sel-Onginizing Maps, Michihars Maeda, Hiromi Miaima si8 ‘Automatic Clustering ith Sel Organizing Maps and Genetic Algocithins TI: an Improved Approach, Ange! Peano Kit Morales 5st Apply Autosssociative Learning to Recover the Motion and the Shape from Sequences of Scaled Orthographic Images, Jun Fuji, Takashi Takahashi, Tokio Kurta co ‘TeacherDitected Information Maximlzaton: Supervised Information-Theoretie Competitive Learning with Gaussian ‘Activation Functions, Rotaro Karsinura 63 Multilayer Spline Neural Networks for Speech Denoting In Frequency Domain, Giovanni Costin, Massimo Corte $89 ‘The Importance of Information System for Optimization with Genetic Algorithms, Igor Bem, Robert Leskovar, Mojea Berk, Mivoiub Ric s [Notes on Modeling Evolutionary Directionality, Thomas Fucher 5B “Measure of Quality in State Space, Masta Zanchi, Tamara Supuk 3a ‘Parallel hybrid Price Genetic Algorithm for Global Optimization, Margherita Bresco, Gincurto Raison! 50 Bravonary Neural Network Learning Algorithms for Changing Environments, Miguel Roca, Paulo Conerand Jose Neves “Mobile Agents Based QoS Maicast Routing (MAQMB), Mohamed El Hachini, Abdelhfid Abouaiss, Pascal Lorenz 602 Decentralized Clustering Through a Swarm of Autonomous Agents, Giang Fone, Aging Foren, Giandomarica Spenano ‘CMOS Fuzzy Decision Diagram Implementation, Victor Vorsavaly, Iya Levi, Viacheslav Matcknoaty, Alr Ruderman, Navaly Kravelenko sisSpot Extraction n Low-Contrast Images, Tuan Pham Suing Fuzy Kaomledge and Fuzzy Metaknowledge In Relational Systems, Jase Galindo, Angetica Umut, Mario Pini on or ‘Active Learning Method to solve Bio Packing Problem, Tayebet Lo, Saeed Baghen Shouraki on (Creation Method of Funry Modeling with Variation Degree, Michihane Maeda, Lisin Ma, Hiromi Miyaja oo Linguistic Values on Attribute Subdomains in Vague Database Querying, Comlia Tudor 646 ‘An Improved Performance Phase Fuzzy ARTMAP Algorthin Fuzzy ARTVar) in Power Systems Applications, ‘Shahram Javad 6st ‘Prediction by Movable-rate Gradient RBP Function Neoral Network with Fuzzy Curves, Hung.Ching Lu and Tetung Hung 87 Integrating Fuzzy Knowledge Base to Genetically Evolved Certainty Neuron Fuzzy Cognitive Maps to Facilitate Decision-Making, Andreas Andreou, Nicas Mateou, George Zombanakis eo Modeling Dynamical Systems via the Takagi Sugeno Fuzzy Model, Nikos E. Mastorakis 668 ‘An Approach of Fuzay Modeling Towards Intllgible Modeling, Cartes Garign-Berga 076 [Kuowiedge Base and inference Motor for an Automated Management System for Developing Expert Systems and Fuzzy (hassiers, Jena Sanches, Franchi Rivas, Jose Agular 82 ory Expert Systems and Multiagents for Intelligent Bulldings, Elva C. Xelhuantzi, Jaime Munoz and Carla A Rees Garcia 688 ‘The System Parameter Fusion Principle and Parameter Fusion Based on Fuzzy Inference, Yunchong Song, Qlangauo Pu, ‘Mis Masrakis 9 Processing of Information With Uncertain Boundarles™ Fuzzy Sets and Vague Sets, Fucheng Xi, unyi Shen 71 PID Fonzy Logie Position Tracking Controller for Detuned Field-Oriented Induction Mator Servo Drive, Finer .M. El-Sousy, Maged N. . Nashed aon ‘etrcemen Teanga Nols Bovronnen Light-seeking Rob ayes FM. Ei Sou, Mage NF. Nosed 1 [ANovel Neuro-Fuzzy Classifier Based on Weightbess Neural Network, Rada A-Alawi mm ‘Neuro-Fuzzy Control for Anserobic Wastewater Treatment, Snejana Yordanova, Rusanka Perova and Valei Mladenov 724 [AYicure-Predictive Approach for Tuning Industrial Controllers, Camel Lazar, Draguna Vrabie, Sorin Caran, Din Ivana Tt, ‘Stability Analysis of a Neural Predictive Control Scheme for Linear Systems, Crisian M. Roman, Manuel 4. Duarte Mermoud, Alejandro M. Suarez 76 ‘NearocootroUler Design with Rule Extracted by using Genetic Based Machine Learning and Reinforcement Learning, Sytem, Hung. Ching [a and Ta Hsing Fag wa ‘ATool for System Monitoring Based on Artificial Neural Networks, Maria Grazia Di Bono, Gabriele Pier, Ovidio Saver 746 ‘Maldtayer Neural Network Verification of Mutnal Inductance choice in sensorless Induction Motor Vector Control System, M. Smajo and D. Vakadinovic 252 “Aplication of Neural Networks for HotAir System Control, Petr Pvonk, Vaclev Veda 187 eakElectric Lond Estimation: A-Ain City, Shamsun Ahmed 701 Knowledge Based Artificial Neora Network for Arabic City-Names Recognition, Farah Nadir, Suit Lato Sri Tu Selami Molar 107 FautsDeteion and Isolation compatational to} Using Neural Networks and State Observers, Glorie Mousall- aya, Jeu Cldoon Veima, Franchi Ria Echevera, Addison Rios Bolivar ms ‘nteraton of Role-Based Systems and Nevral Networks into Speech Recognlion System, Halima Bah, Mokltar Selmi 778 Simple fuzzy Adaptive Contra fora Class of Nonlinear Plats, Saso Bai Igor Stan, Drago Marko 1 Furry model-based predicive control for a CSTR with multiple sendy stat: A simulation study and «compar vith oer nonlinear MBBPC contol algolithms, go Sans Marko Lap, Jor Luis Figueroa, Sao Baie 78 General Fory Systems as Extensions ofthe TakaghSugeno Methodology, Nikos E. Masorkis 85 Modeling, Stability and Regulatlon of Fuzzy Roled-Based Systems, Zvi Retchkiman 01 Supao Fuazy Controller of Helium Evaporation, Pavel Js es ‘Design Considerations of Hardware Based Fuzzy Controllers, Dumjan Ose, Mika Mraz, Nikolaj Zimic BL ‘Farry BDI Modeling for Intelligent Agent, Jong Yih Kuo 8 ‘Fury Approach in Enquiry to Regional Data sources for Municipalities, Simonova Stanislav, Jan Capek 83 ‘A uy Logic Approach to evaluate Health Care Liability and Risk of Medical Malpractice, Borioni Stfeno axating Agents for Remote Learning, DinitarLakov, Margarita Saralieva cc Applying Genetic Algorithms tothe DlahA-Ride Problem with Time Windons, Claudio CubiloxF, Franco Guid-Polenco Claudia Demari eicinceermaar ‘A Layered Evolutionary Algorirthm to Design Artificial Neural Networks, Yansina Mohante-Ben-Ali as "eau omer Diath o Fores Sites ith a NO ion Coto vm Hrs Alt, Tak Bolten nda Sima 9 sbi Rotator Rosin of aman wit CompesBagroun and Varig mination Jing Zi, Evolutionary Algorithm for Measurement of Serew Parameters, Oleksandr Makcjev Systems with Antiport Rules for Evolution Rules, Rudolf Freund, Marion Oswald ‘A Fast Dynami ‘onary 5 ' SE ‘Ft Daa Eaoay lth or Malbec Mahal Component Dg ior Zou, Nong Wer [Analysis of Gene Expression Data Using Evolutionary MultiAgent System, Gregor Stic, Peter Koko! ‘Towards An Evolutionary Approach to Case Retrieval, Nabila Nowacwra-Amri, Yorine Mohamed Ben Al, i Mel Taye Laski ow Evolutionary Computation can be Introduced to Select and Optimize Scenarii Along a Product Design Process, (Claude Baron, Daniel Fstee, Samuel Rocet ‘Robust Control via Geaetic Algorithms and Integral Criteria Minimization, Catalin Nicolae Calis Oa Normed Space of Ordered Fuzzy Numbers, Witld Kasinski [A Hybrid Neuro-Fuzzy Approach to Intelligent Behavior Acquisition in Complex Multi-Agent Systems, ‘Ali Athavan Binghsi, Futaneh Taghiyareh, Amir Hossein Simjour, Amin Khajencjad ‘SADEX -A Fuzzy CBR System for Fault Diagnosis, Virato Marques, Tomes Farinka, Antonio Brito Evolation Strategy Programming (ESP), Eada A. Sarhan, Iraky H. Khalifa, Mohamed S. Emam Integration Interoperable Software Component Into Inteligence Software Spstems, Romo Sendelj and Danijela Milosevic Engine late Speed Control Using ANFIS Controller, Jala, M, Farokhi, H. Toabi High Pectormance Simple Postion Neuro-Controlle for Fiekd-Oriented Induction Motor Servo Drives, Foyer FM, Et-Sousy, MM Salem Backprepagation: In Search of Performance Parameters, And Kumar Enumulapaly, Lingpvo Bu, Khosrow Kaikhah “Exploration of Unknown Environment with Focused Sonar Sensor and Fuzzy Control, U. La Fata, K. Lenac, E. Mumola, M.Nolich ‘Text Summarizaion Using Neural Networks, Kicwow Kaithah ‘ANearo-Control Approach for Flexible AC Transmission, N. M:Honnoon, BPKarit and 0. M. Asinash Fuzzy Coordination of Mol-Rabot System for Audio Survelilane, U. La Fate, K Lenac, E Munola, M, Notch experimenting Fuzzy Control Strategies for Mobile Rabots on a Rap Prototyping Platform, Fronceco Cuperting, Li Define, Vincenzo Giordano, David Naso, Big Turchiano Fuze uearst neighbor sytem: An Application to Handwritten Arable Literal Amount Words Recognition, Farsh Nod, Selmi Moldear ‘A Fozzy Programming Method for Optimizing Network's Configuration inthe EDS Supervision, Cicero R. Covati and EP. Feel ‘The Egyptian Stock Market Return Prediction: A Genetic Programming Approach, Mohammed EP-Tebary -Karzy IF-THEN Rates Extraction for Medical Diagnosis Using Genetie Algorithm, Alexander Rotsiin, Monon Poser, anna Rakypanska Robust and Adaptive Load Frequency Contre of Multiarea Power Systems with sytem parametric uncertainties via TDMLP, Mokonimad Hesein Aphdale, Mefran Rashid 1001 ‘Anew Genetic Method for Mobile Robot Navigation, Beat Frouzandeb, Seyed Ehsan Mahmoud, A Athawan Bhs, Alisa Merandt 1009 Yeak Stick RBE Network for Online System Identification, Hassin Mobuh, Fark JanabiSharft 101s HNL ISSN 4109-2777 gBB geeek 882 8 BR 8 $ §$ 8888 88 # g"3o_WSEAS [RANSACTIONS ON. 2. 109.2777 Stability Analysis of a Neural Predictive Control Scheme for Linear Systems Cristian M. Roman'. Manuel A. Duarte-Mermoud"" and Alejandro M. Suarez” ‘Electrical Engineering Department, University of Chile Av. Tupper 2007. Casilla 412-3, Santiago, CHILE email: mduartem@gvec.uchile ¢| Jectronies Engineering Department, Federico Santa Maria Technical University ‘Av. Espaiia 1680, Casilla 110V, Valparaiso, CHILE. € mail: asuarezid elo.uttsin.cl Abstract: A neural predictive control algorithm based on three neural networks is presented in this paper. One ‘neural network is used to identify the plant under control (identifier), other is used to predict the control error several step ahead (prediction) and the third one is used to compute the control signal (controller). The neural predictive coutrol scheme is analyzed from the theoretical point of view. The stability analysis of the control scheme for a first order linear plant is completely resolved, obtaining conditions under which the overall adaptive scheme result stable in the Lyapunov sense, Simulation results arc presented to illustrate the behavior of the proposed contro! strategy Key Words: Predictive control, neural control, adaptive neural control, neural predictive control, adaptive ‘control. linear control. 1. Introduction In the majority of the industrial processes to be controlled, the parameters are unknown or partially known. Amongst the variety of control techniques applied perhaps the most successful are those based ‘on artificial neural networks (ANN). Since 1990, the adgptive control using ANN has been one of the most active research field. One of the first works on ANN applied to control that can be cited are [1,2], existing currently numerous papers on the subject eg. (3.4). ‘One important point in any neural control scheme is the stability analysis. Sometimes due to the complexity of the resultant structure its analysis become complex and sometimes intractable. Nevertheless some interesting stability results have been obtained eg. (5). where robust identification schemes are developed base on ANN with radial basis functions (RBF) and using the Lyapunov method and the gradient method modified with a projection algorithm guarantees the weights to be bounded. In {6] an adaptive controller based on RBF is presented, where the weights are updated using a law derived by Lyapunov °s theory. * Author to whom all correspondence should be addressed. Another neural control schemes is that given it [7]. where the stability a direct controller is analyzed using two updating procedures; the back propagation momentum (BPM) and the recursive least squares (RLS) for identification and control NN, establishing, bounds on the learning rates and on the covariance matrix In this paper we analyze the stability of a predictive control scheme developed in [8], for the ease of linear first order plants, a as first step for a later extension 10 the case of linear n-th order pants and nonlinear plants. More discussion on this neural predictive control method can be found in [8] where a comparison with ‘other methods is performed and in [9] where an application was done for a copper mineral plant. tn this paper we will be mostly concerned with the theoretical aspects of the method which were not completely addressed in [8,9]. 2. Neural Predictive Control (NPC) ‘The method proposed in [8] uses a direct control scheme with a plant model and retropropagates737 WSEAS TRANSACTIONS ON SYSTEMS au 2. Neural Predictive Control (NPC) The method proposed in [8] uses a direct control scheme with a plant model and retropropagates a weighed signal of the actual and future control errors, sgiven by a prediction NN. This method utilizes the temporal backpropagation assuming a dynamical relationship between the ouput of the control NN and the input to the identification NN. The scheme is shown in Figure 1 Figure 1: Neural predictive control. ‘The control will be done leaving fix the identification NN which will be trained off-line, but updating the weights of the control NN and prediction NN according to some specific updating methods and the analysis will be focused on linear first order plants of the form eet) ar G@)= 5,48) 2.1 Network Structures ‘The structures of the NN’s will have no hidden layers and have linear activation functions. The structure of the identification NN is a s follows: Inputs: [u(k), yp(k), 1X. (he). Weights: {w1, Wan, b'=W, (K). Then, the output of the identi expressed as (bv ife) 0, (4) = met) mary (E48 = nil a(t) The control NN exhibits the following structure: tion NN can be a Inputs:{r(k).....r(k-nr# 1), yofk)sn ¥pCeny +1), 1] XK) Jesus 2. Vol3 . April 2004 ISSN. 109.2777 Weights: [We. ... bJ/=We (k). where nr is the number of delays in the reference present at the input of the control NN, ny is the number of delays in the plant output present at the input of the control NN and nce =nr+ ny. Then. the output of the control NN can be written as #8) = "Sm sa ele» % 4 +B meill)rp 2 i) see (k)= Welk)” xe 4) ‘The two step ahead prediction NN has the following structure: Inputs: [r(k), ... . 10k-p)s yolk), » Yo(k-p). UUK-D). u¢k-q-1)}=X (4) Weights: Wpilk)-{mpt(B. . Wpuap(K). byl! WpAK=[W 92.108), + Wpeap kK). Bysl where p is the number of delays of the reference and the plant output present at the inputs of the prediction NN and q is the number of delays of the plant input ‘present at the input of the prediction NN. ‘Then the outputs of the prediction NN are given by (EAB) pret > fo ¥p 1s Ore) + Bp inpes rp (9+ 7 Ey™ pt in2ps2 tu (E=!-1) +bp4 (&) o S(O +I) spear (4) = Fry 95 eC ) +E paivpet rp (9s “Sym paren Oe 6-2)2p0(8) which can be written in matrix from as 2 (E+) Uy rear (A= Hy (RY aoe) So (# +2) Ypres (KD = Hg (4 oe) 2.2 Weight Updating ‘The cost function for the control NN is defines as738_WSEAS TRANSACTIONS ON SYSTEMS. Issue 2, Vol.3 April 2004 issn. 1uus2777 eel) © where e2(k) the weighted average of current and future control errors i.e. FW nO2Wrn Wks Ly Wed 2) yk). 7K) and ys(k) are time varying weighting factors satisfying 7,(k) + yAK) + yo(k) =1 and also 7K) 20, 7K) 20, ¥(K) 20. ce in this ease there is no delayed plant inputs at the identification NN, Le. q~0, the cost function is finally Eww = (8) a ‘The updating of the control NN weights is chosen as i id Stora | awe ene Sr eTtty no ME eM ees @ + rae ee (kez) -I} where errors e{kHT) and ¢(k#2) are given by the prediction NN. ie. é,(+I|k) and &,(k +2\e). ‘The prediction NN outputs are given by fg b-th-2) raat) = wou lk Deu 6-2) = py (kt) (e-2) eel b-2) | vpealt-2) sigs)! apla -2) 0) Sent (t-3)= The computation of the error for training the NN are obtained as. per 1) 6 (1) eee (Kk 2) pe (EY = ee (D> foe (FR 2) ‘The updating of the prediction NN weights is done as follows wp (4) = ry (41) Hp (K-1) ng (0) = Wg (1) aH p9 (K-21) where (10) ap 2H pi (A~)= Mp6 pet (ADAP (K -2) aw pal Y= npepea (t)se(t-2) 2.3 Ideal Weights In the case of the control NN it is assumed that the ‘error control can be expressed as a function of the error of the control NN weights We(k) =He(k) HW « ie. [10] e(k)= —w, Welk 1) Xe(k - 0) 3) For the prediction NN the training errors are given by (10). Replacing equation (9) in (10) we can write the prediction errors for training purposes as follows eailk)=e.(k—1)-Wo(k1Y Nols -2) enak)= 6 (E)-Wp,(k-1) Xp(k—2) If we now assume that there exist ideal weights for the prediction NN, Wp,” and Wp: such that with these weights the prediction error is Zero, we have na A) =e (ED) =" Xo(-2) 0 aay gn (8)=0,(K)—WpsTXp(k-2)=0 that is to say 6. (1) Wo Xo(-2) (16) ¢,(k) = Wp." Xp(k-2) Defining the weight errors for the prediction NN as Wplk)=Wpl(k)-Wp", the prediction error for training the prediction NN can be finally expressed as Cpr (k)=e.(k 1) 8, (k= 1-2) = = Wp! (k-1) Xp(k-2) ea (k) =e, (k) 8, (ke -2) = =Wpy'(k-1)Xp(k-2) any 3. Stability Analysis of the NPC Stability analysis is done taking into account the errors in the weights of the prediction and control NN, choosing a Lyapunov function candidate of the type [10]Re ere OMey on 1 (6) =the, (6) hp, (6) «eg (6) vipg (W) + ie(i)! Welk) =¥4 (6) +45 (6) win (8) =o (0) Ho (8) +08 (8) og (0 (8) = We(A)! Wee) ‘t can be shown that the yariation of the function V,(k) is given by (10) IN (= -Apieds (E+1)= A pret (R41) 9) with =n, 2-1, 40-0} y ham, P-1,)x0k-19} ‘Therefore, to assure AV,(k)<0 is sufficient to choose the prediction learning rate 1, as follows: é Be 2 0 2h Weclesi sheic (2) 4, Simulation Results Let us consider a linear first order plant defined by its transfer function ¥(z) _ 0.00995 2-099 UG) es) We will apply the NPC using the conditions obtained in (23) for the control learning rate and in (21) for the prediction learning rate. The identification NN was trained off-line and kept constant during the simulations. The reference input is a sinus of amplitude one and frequency 2 [rad/sec}. The characteristics of the NN used in the this study are as follows: Control NN Suucture : Vi, Inputs: [r(k).r(k-1),y(R),ye1)] Sampling period: 0.01 Training: 5000 iteration on-line. Identification NN Structure: N!, Inputs: {u(k) y(k)] Sampling period: 0.01 Prediction NN ‘Structure: Nj. Arputs: [4(k),-.-s(K-2),9(K) yon 2) A) pnt K-2)] Sampling period: 0.01 Prediction steps: 2 Training: $000 iterations on-line In Figure 2 are shown the results of the NPC scheme during the transient and steady state regimes for the €88E Xn =2 aNd X, age = 2. In Figure 3 the same results are shown for the case xpn,=5 and cma. = 5. In both cases it is shown the evolution af the weighting factor +,740 ‘ vol 3 a ® ) Figure 2: PNC with xX, =2 ¥ Xpmmx =2+ Figure 3: PNC with X05, =5 and Xpu =5- 4) Transient stage. b) Steady state regime. a) Transient stage. b) Steady state regime.TL_WSEAS TRANSACTIONS ON SYSTEMS 1 has been observed that the value of 7; increases as ‘compared with the case of one step prediction (not shown here for the sake of space) indicating that the current control error has more influence than the future control errors to assure stability. 5. Conclusions The stability analysis of a the neural predictive control scheme presented in {8] has been presented in this paper. The stability proof assures that all the signals remains bounded and the plant output asymptotically track the desired trajectory. ‘Simulation results indicate that the proposed scheme behave according to the theoretical results he main conclusion that can be drawn from this study is that in order to preserve the stability, the learning rates of the neural networks involved in the scheme have to be adjusted variably and differently of one 0 each other. This fact makes the transient behavior oscillatory and slow though in the steady state the control objectives are met. Currently work is undergoing to extend the results shown here to the case of linear systems of higher order than one and prediction of control error of more than two steps. Acknowledgements The results reported in this paper have been supported by CONICYT-CHILE through grant FONDECYT N° 1030962. ssue 2. Vol.3.. April 2004 155 309.2777 References a (2) fo) {9} a (8) io] 9} KS. Narendra and K. Parthasarathy. “Identification and control of dynamical systems using neural networks”, EEE Transactions on Neural Networks. Vol. |. March 1990, pp. 4-27. D, Psaltis, A. Sideris and A. Yamamura, “A multilayered neural network controller IEEE Control Systems Magazine. Vol. 8 April 1988, pp. 17-21, M.Norgaard, O.Ravn, N.K. Poulsen and L.K. Hansen, Neural networks for modeling and control of dynamic systems. Springer-Verlag, London, 2000. M, Hagan and H.Demuth, “Neural networks for control", American Control Conference 1999, San Diego. June 1999, pp. 1642-1656. Polycarpou and P. A. loannou, "Identification and control of nonlinear systems using neural network models: Design and stability analysis," Univ. Southern California, Los Angeles, Electrical Engineering Systems Rep, 91-09-01, Sept. 1991. R. Sanner and J.Stlotine, “Stable adaptive control and recursive identification using radial gaussian networks”. IEEE Transactions on Neural Networks. Vol. 3. No 6, November 1992, pp. 837-863. G.W.Neg, Application of neural necworks 10 adaptive control of nonlinear systems. A Research Studies Press BookUMITS. UK. 1997, A. Suirez, New architecture of predictive control for nonlinear systems using neurul networks (In Spanish). Ph.D. Thesis. Department of Electrical Engineering. University of Chile, Santiago, 1998. M. Duarte, A. Suérez and D. Bassi, “Cont! of grinding plants using predictive multivariable neural control”, Power Technology. Vol. 115, 2001, pp. 193-206. C. Romain, Adaptive control of nonlinear plants by means of neural networks \n Spanish). E.E. Thesis, Department of Electrical Engineering, University of Chile, Santiago, 2003,