Modelagem de Sistemas Dinâmicos Não Lineares via RBF-GOBF.

Trata-se neste trabalho trata da modelagem e identificação de sistemas dinâmicos não lineares estáveis representáveis por modelos de Wiener por um estrutura formada por bases de funções ortonormais generalizadas (Generalized Orthonormal Basis Functions - GOBF) com funções internas e redes neurais com funções de base radial (Radial Basis Functions - RBF). Os modelos GOBF com funções internas são capazes de representar dinâmicas lineares intrincadas com uma parametrização que se vale apenas de valores reais, sejam os polos do sistema a ser representado complexos e/ou reais. Com informações de entrada e saída do sistema a ser identificado é possível obter um modelo GOBF-RBF inicial. Os clusters que determinam os parâmetros inciais das RBFs (centros das funções gaussianas e larguras ou spreads) são obtidos pelo método fuzzy C-means, o qual é inicializado com um número de centros pré-determinado, obtido pela técnica subtractive clustering, garantindo clusters com volume e densidade apropriados. São propostas duas técnicas para o ajuste dos parâmetros da estrutura. A primeira delas se baseia em um método de otimização não linear e os gradientes exatos da estrutura. Apresenta-se um procedimento para a obtenção dos cálculos analíticos dos gradientes de saída do modelo GOBF-RBF em relação a seus parâmetros (polos da base ortonormal, centros, larguras e pesos de saída da rede RBF). A segunda proposta se vale de um método metaheurístico chamado otimização por enxame de partículas com comportamento quântico. As metodologias são validadas com suas aplicações em três diferentes sistemas não lineares associados a modelos de processos práticos

Nonlinear sytems modeling based on ladder-strutured generalized orthonormal basis functions

Orientadores: Wagner Caradori do Amaral, Ricardo Jose Grabrielli Barreto CampelloTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de ComputaçãoResumo: Este trabalho enfoca a modelagem e identificação de sistemas dinâmicos não-lineares estáveis através de modelos fuzzy Takagi-Sugeno (TS) e/ou Volterra, ambos com estruturas formadas por bases de funções ortonormais (BFO), principalmente as bases de funções ortonormais generalizadas (GOBF - Generalized Orthonormal Basis Functions) com funções internas. As GOBF¿s com funções internas modelam sistemas dinâmicos com múltiplos modos através de uma parametrização que utiliza somente valores reais, sejam os polos do sistema reais e/ou complexos. Uma das principais contribuições desta tese concentra-se na proposta da otimização e ajuste fino dos parâmetros destes modelos não-lineares. Realiza-se a identificação dos modelos fuzzy TS-BFO utilizando-se de medidas dos sinais de entrada e saída do sistema a ser modelado. Os modelos fuzzy TS-BFO são inicialmente determinados utilizando-se uma técnica de agrupamento fuzzy (fuzzy clustering) e simplificados por algoritmos que eliminam eventuais redundâncias. Em sequência desenvolve-se o cálculo analítico dos gradientes da saída do modelo TS-BFO em relação aos parâmetros do modelo (polos da BFO, coeficientes da expansão da BFO e parâmetros das funções de pertinência). Utilizando-se técnicas de otimização não-linear e o valor dos gradientes, realiza-se a sintonia fina dos parâmetros dos modelos inicialmente obtidos. Para os modelos de Volterra-GOBF desenvolve-se uma nova abordagem utilizando-se GOBF com funções internas nos kernels dos modelos. São calculados os gradientes analíticos da saída do modelo de Volterra-GOBF, seja com kernels simétricos ou não simétricos, com relação aos parâmetros a serem determinados. Estes valores são utilizados em algoritmos de otimização que possibilitam a obtenção de modelos mais precisos do sistema sem nenhum conhecimento a priori de suas características. Além da identificação de sistemas não-lineares por modelos BFO, abordou-se também, nesta tese, uma nova metodologia para a otimização de modelos lineares BFO no domínio da frequência. Neste contexto, destaca-se como principal contribuição o desenvolvimento, no domínio da frequência, do cálculo analítico dos gradientes da resposta em frequência das funções de Kautz e Laguerre, com relação aos seus parâmetros de projeto. Os valores dos gradientes fornecem a direção de busca dos parâmetros dos modelos em processos de otimização não-linear. Também foram otimizados os modelos GOBF com funções internas, com o cálculo numérico dos seus gradientes, pois, ainda não foi possível estabelecer uma fórmula genérica para o cálculo analítico dos gradientes dos modelos GOBF, de qualquer ordem, em relação aos parâmetros a serem determinados. Exemplos ilustram a aplicação e eficiência dos métodos de identificação e otimização propostos na modelagem de sistemas lineares (domínio do tempo e da frequência) e não-lineares utilizando BFO¿s.Abstract: This work is concerned with the modeling and identification of stable nonlinear dynamic systems using Takagi-Sugeno fuzzy and Volterra models within the framework of orthonormal basis functions (OBF), mainly ladder-structured generalized orthonormal basis functions (GOBF). The ladderstructured GOBFs allows to model dynamic systems with multiple modes, real and/or complex poles, through a parameterization, which uses only real values. The main contribution of this thesis is the optimization and fine tuning of the parameters of OBF nonlinear models. The GOBF models identification are performed using only input and output measurements. The initial GOBF-TS fuzzy model is obtained using a fuzzy clustering technique and simplified by algorithms that eliminate any redundancies. Next, the analytical calculation of the gradients of GOBF-TS model concerning model parameters (GOBF poles, OBF expansion coefficients and the parameters of membership functions) is developed. A fine tuning of the model parameters is obtained by using a nonlinear optimization technique and the calculated gradients. For Volterra-GOBF models a new approach using kernels with ladder-structured GOBF is also proposed. Furthermore, Volterra-GOBF model optimization, with symmetrical or asymmetrical kernels, using an analytical gradients calculation of the output model regarding their parameters is presented. Following, a new approach for linear OBF models optimization, in frequency domain, is also addressed. In this context, the analytical calculation of the gradients of the Laguerre and Kautz frequency response concerning its parameters is presented The ladder-structured GOBF models optimization, in the frequency domain, is performed using only numerical calculation of its gradients, as it has not yet been possible to derive a generic analytical gradients. Examples illustrate the performance and effectiveness of identification methods proposed here in the modeling and optimization of linear (time domain and frequency) and non-linear systems.DoutoradoAutomaçãoDoutor em Engenharia Elétric

Neural activity inspired asymmetric basis function TV-NARX model for the identification of time-varying dynamic systems

Inspired by the unique neuronal activities, a new time-varying nonlinear autoregressive with exogenous input (TV-NARX) model is proposed for modelling nonstationary processes. The NARX nonlinear process mimics the action potential initiation and the time-varying parameters are approximated with a series of postsynaptic current like asymmetric basis functions to mimic the ion channels of the inter-neuron propagation. In the model, the time-varying parameters of the process terms are sparsely represented as the superposition of a series of asymmetric alpha basis functions in an over-complete frame. Combining the alpha basis functions with the model process terms, the system identification of the TV-NARX model from observed input and output can equivalently be treated as the system identification of a corresponding time-invariant system. The locally regularised orthogonal forward regression (LROFR) algorithm is then employed to detect the sparse model structure and estimate the associated coefficients. The excellent performance in both numerical studies and modelling of real physiological signals showed that the TV-NARX model with asymmetric basis function is more powerful and efficient in tracking both smooth trends and capturing the abrupt changes in the time-varying parameters than its symmetric counterparts

Identificação e controle de processos via desenvolvimentos em séries ortonormais. Parte A: identificação

In this paper, an overview about the identification of dynamic systems using orthonormal basis function models, such as those based on Laguerre and Kautz functions, is presented. The mathematical foundations of these models as well as their advantages and limitations are discussed within the contexts of linear, robust, and nonlinear identification. The discussions comprise a broad bibliographical survey on the subject and a comparative analysis involving some specific model realizations, namely, linear, Volterra, fuzzy, and neural models within the orthonormal basis function framework. Theoretical and practical issues regarding the identification of these models are also presented and illustrated by means of two case studies related to a polymerization process.O presente artigo apresenta uma visão geral do estado da arte na área de identificação de sistemas utilizando modelos dinâmicos com estrutura desenvolvida através de bases de funções ortonormais, como as funções de Laguerre, Kautz ou funções ortonormais generalizadas. Discute-se as vantagens e possíveis limitações desse tipo de estrutura bem como os fundamentos matemáticos dos modelos correspondentes nos contextos de identificação linear, linear com incertezas paramétricas (identificação robusta) e não linear, incluindo uma revisão bibliográfica abrangente sobre o tema. Diferentes realizações de modelos com funções de base ortonormal, a saber, modelos lineares, de Volterra, fuzzy e neurais, são detalhadas e discutidas comparativamente em termos de capacidade de representação, parcimônia, complexidade de projeto e interpretabilidade. Aspectos práticos da identificação desses modelos são também apresentados e ilustrados através de dois casos de estudo envolvendo um processo simulado de polimerização isotérmica.301321Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq

Mathematical Methods, Modelling and Applications

This volume deals with novel high-quality research results of a wide class of mathematical models with applications in engineering, nature, and social sciences. Analytical and numeric, deterministic and uncertain dimensions are treated. Complex and multidisciplinary models are treated, including novel techniques of obtaining observation data and pattern recognition. Among the examples of treated problems, we encounter problems in engineering, social sciences, physics, biology, and health sciences. The novelty arises with respect to the mathematical treatment of the problem. Mathematical models are built, some of them under a deterministic approach, and other ones taking into account the uncertainty of the data, deriving random models. Several resulting mathematical representations of the models are shown as equations and systems of equations of different types: difference equations, ordinary differential equations, partial differential equations, integral equations, and algebraic equations. Across the chapters of the book, a wide class of approaches can be found to solve the displayed mathematical models, from analytical to numeric techniques, such as finite difference schemes, finite volume methods, iteration schemes, and numerical integration methods

Use of mathematical methods in the resolution of chemical engineering problems

This thesis consists of a compendium of five works that illustrate the utilization of selected mathematical methods to solve specific chemical engineering problems. Hence, the thesis is intended to cover both, a review of fundamental mathematical procedures for the solution of models raised from chemical phenomena, and a demonstration of their effectiveness to obtain useful novel significant results. The opening paper explores diverse global optimization algorithms to adjust both kinetic constants and the binary interaction parameters (BIPs) for the Peng-Robinson equation of state to the experimental data. Those parameters are essential to determine the model raised from the supercritical transesterification of triolein with methanol to produce biodiesel, with CO2 as cosolvent, consisting of three reversible reaction in series. Here, a novel model merging the ordinary differential equations system raised from kinetic mechanism and the time-dependent thermodynamic state of the complex mixture is presented for diverse operating conditions. Among all results obtained, novel binary interaction coefficients for the intermediate reaction species (dioleins and monooleins) highlight. The second and fourth papers included in this thesis are aimed at the study of lanolin extraction from raw wool, using 5% ethanol in CO2. The former explores solid lanolin extraction under near-critical conditions by means of a mass-transfer model based on the shrinking-core concept, while the latter is addressed at the liquid lanolin supercritical extraction. Both models result in a partial differential equations (PDEs) system determined by the solubility of multiphasic lanolin, Henry-type partition coefficient and the lanolin mass transfer coefficient. Hence, in each paper the raised PDEs system is solved through a different method: in the second paper orthogonal collocation method is employed, while in the fourth paper finite differences method is used combined with the numerical integration of an expression previously obtained by means of the Laplace transform. Finally, an optimization procedure is used in order to fit the extraction parameters to the experimental data, achieving coherent results that agree well with those previously reported. Between the cases exposed, liquid lanolin extraction is significantly complex to model because of the diffusion phenomena that may occur inside the two lanolin fraction mixture added to the diffusion of solvent in the interphase. Therefore, in the third work a nonlinear autoregressive exogenous neural network model is designed to predict the outcoming extracted fraction of lanolin at diverse temperatures, pressures, solvent mass flow rates, wool packing densities and times. The problem with the scarce data available for training of the neural network is overcome by augmenting experimental data using an empirical Weibull function, which correctly predicts the lanolin breakthrough at the extractor exit. This hybrid Weibull - Neural Network algorithm results in a low prediction error and conform a powerful tool for optimizing operating conditions, proved by the fast convergence of genetic algorithm procedure. This thesis closes with Molecular Dynamics simulations for peptide-folding studies, followed by a Principal Component Analysis (PCA) and clustering analysis to understand the Free Energy Landscape of the peptide (FEL). Those methods are aimed at assessing the conformational profile of bombesin, a peptide with interest in drug design as a possible novel agonist and/or antagonist in the fight against cancer. Results suggest that the peptide adopts mainly helical structures at the C-terminus and, to a lesser extent, hairpin turn structures at the N-terminus. Those results agree with those available from NMR in a 2,2,2-trifluoroethanol/water (30% v/v), and point out a suitable a-helix conformation for binding where Trp8 and His12 interaction has a significant role.Aquesta tesi consta d'un compendi de cinc treballs que il·lustren la utilització de mètodes matemàtics per resoldre problemes específics d'enginyeria química. Per tant, la tesi està destinada a ser una revisió dels procediments matemàtics fonamentals per a la solució de models derivats de fenòmens químics i, a més, una demostració de la seva efectivitat per obtenir resultats útils i innovadors. L'article que obre la tesi explora diferents algoritmes d'optimització global per ajustar tant les constants cinètiques com el Paràmetres d'Interacció Binària (PIB) per a l'equació d'estat de Peng Robinsos a les dades experimentals. Aquests paràmetres són essencials per determinar el model derivat de la transesterificació supercrítica de la trioleïna amb metanol per produir biodièsel, amb CO2 com a cosolvent, que consisteix en tres reaccions reversibles en sèrie. Aquí, es presenta un nou model que fusiona el sistema d'EDOs derivat del mecanisme cinètic i l'estat termodinàmic de la barreja per a condicions de funcionament diverses. Entre tots els resultats obtinguts, destaquen els nous PIBs trobats per a les espècies de reacció intermèdies. El segon i quart treball inclosos en aquesta tesi estan destinats a l'estudi de l'extracció de lanolina de llana crua amb 5% d'etanol en CO2. El primer explora l'extracció de lanolina sòlida en condicions gairebé crítiques mitjançant un model de transferència de massa basat en el concepte del nucli minvant, mentre que el segon s'adreça al cas de l'extracció supercrítica de lanolina líquida. Ambdós models donen com a resultat un sistema d'EDPs determinat per la solubilitat de la lanolina multifàsica, el coeficient de partició de Henry i el coeficient de transferències de massa. Per tant, a cada article el sistema d'EDPs obtingut es resol mitjançant un mètode diferent: en el article s'utilitza un mètode de col·laboració ortogonal, mentre que en el quart s'utilitza el mètode de diferències finites combinat amb la integració numèrica d'una expressió obtinguda mitjançant la Transformada de Laplace. Finalment, es porta a terme una optimització per ajustar els paràmetres d'extracció a les dades experimentals, aconseguint resultats coherents que coincideixen amb els reportats anteriorment. Entre els casos expotsats, l'extracció de lanolina líquida és significativament complexa de modelar a causa dels fenòmens de difusió que es poden produir a l'interior de les dues fraccions de lanolina a més de la difusió del dissolvent en la interfase. Per tant, en el tercer treball es dissenya un model de xarxa neuronal exògena no lineal autoregressiva per predir la fracció extreta de lanlina a diverses temperatures, pressions, cabals de dissolvent, densitats d'empaquetament i temps. El problema derivat de l'escassetat de dades disponibles per a l'entrenament de la xarxa neuronal es supera amb l'augment d'aquestes mitjançant una funció de Weibull empírica, que prediu correctament l'avanç de la lanolina a la sortida de l'extractor. Aquest algoritme híbrid Weibull - xarxa neuronal resulta en un baix error de predicció i conforma una potent eina per optimitzar les condicions operatives, demostrada per la ràpida convergència de l'algoritme genètic utilitzat. Aquesta tesi tanca amb simulacions de Dinàmica Molecular per a l'estudi del plegament de pèptids seguint d'un Anàlisi de Components Principals (ACP) i del "clustering" per a l'anàlisi del Paisatge d'Energia Lliure (PEL). L'objectiu és avaluar el perfil conformacional de la bombesina, un pèptid amb interès en el disseny de fàrmacs com a possible nou agonista i/o antagonista en la lluita contra el càncer. Els resultats suggereixen que el pèptid adopta estructures helicoïdals principalment al extrem C, i també en menor mesura estructures de forquilla al extrem N. Aquests resultats coincideixen amb els disponibles de RMN en 2,2-trifluoroetanol/aigua (30% v/v) i indiquen una conformació d’hèlix a adequada per a la unió on la interacció Trp8 i His12 té un paper important

Parameterized macromodeling of passive and active dynamical systems

L'abstract è presente nell'allegato / the abstract is in the attachmen

An Introduction To Models Based On Laguerre, Kautz And Other Related Orthonormal Functions - Part I: Linear And Uncertain Models

This paper provides an overview of system identification using orthonormal basis function models, such as those based on Laguerre, Kautz, and generalised orthonormal basis functions. The paper is separated in two parts. In this first part, the mathematical foundations of these models as well as their advantages and limitations are discussed within the context of linear and robust system identification. The second part approaches the issues related with non-linear models. The discussions comprise a broad bibliographical survey of the subjects involving linear models within the orthonormal basis functions framework. Theoretical and practical issues regarding the identification of these models are presented and illustrated by means of a case study involving a polymerisation process. Copyright © 2011 Inderscience Enterprises Ltd.1401/02/15121132Aguirre, L.A., Correa, M.V., Cassini, C., Nonlinearities in NARX polynomial models: Representation and estimation (2002) IEE Proc. Control Theory and Applications, 149, pp. 343-348Akcay, H., At, N., Membership set identification with periodic inputs and orthonormal regressors (2006) Signal Processing, 86 (12), pp. 3778-3786. , DOI 10.1016/j.sigpro.2006.03.025, PII S016516840600123X, Multimodal Human-Computer InterfacesAkcay, H., Ninness, B., Rational basis functions for robust identification from frequency and time-domain measurements (1998) Automatica, 34 (9), pp. 1101-1117. , PII S0005109898000521Allgower, F., Zheng, A., (2000) Nonlinear Model Predictive Control, , Birkhauser Verlag, Basel, SwitzerlandAraújo, H.X., Oliveira, G.H.C., An LMI approach for output feedback robust predictive control using orthonormal basis functions (2009) Proc. 48th Conference on Decision and Control, , Shanghai, ChinaBalbis, L., Ordys, A.W., Grimble, M.J., Pang, Y., Tutorial introduction to the modelling and control of hybrid systems (2006) International Journal of Modelling, Identification and Control, 2 (4), pp. 259-272Belt, H.J.W., Den Brinker, A.C., Optimality condition for truncated generalized Laguerre networks (1995) International Journal of Circuit Theory and Applications, 23 (3), pp. 227-235Biagiola, S.I., Figueroa, J.L., Wiener and Hammerstein uncertain models identification (2009) Mathematics and Computers in Simulation, 79 (11), pp. 3296-3313Bodin, P., Villemoes, L.F., Wahlberg, B., Selection of best orthonormal rational basis (2000) SIAM Journal of Control and Optimization, 38 (4), pp. 995-1032Broome, P.W., Discrete orthonormal sequences (1965) Journal of the Association for Computing Machinery, 12 (2), pp. 151-168Camacho, E.F., Bordons, C., (1999) Model Predictive Control, , Springer- Verlag, London, UKCampello, R.J.G.B., Amaral, W.C., Favier, G., Optimal Laguerre series expansion of discrete Volterra models (2001) Proc. European Control Conference, pp. 372-377. , Porto, PortugalCampello, R.J.G.B., Do Amaral, W.C., Favier, G., A note on the optimal expansion of Volterra models using Laguerre functions (2006) Automatica, 42 (4), pp. 689-693. , DOI 10.1016/j.automatica.2005.12.003, PII S0005109806000069Campello, R.J.G.B., Favier, G., Amaral, W.C., Optimal expansions of discrete-time Volterra models using Laguerre functions (2004) Automatica, 40 (5), pp. 815-822Clarke, D.W., (1994) Advances in Model Based Predictive Control, , Oxford University Press, New York, USAClowes, G.J., Choice of time-scaling factor for linear system approximation using orthonormal Laguerre functions (1965) IEEE Transactions on Automatic Control, 10 (4), pp. 487-489Da Rosa, A., (2009) Identification of Nonlinear Systems Using Volterra Models Based on Kautz Functions and Generalized Orthonormal Functions, , PhD thesis, School of Electrical and Computer Engineering of the State University of Campinas (FEEC/UNICAMP), Campinas-SP, Brazil, in PortugueseDa Rosa, A., Campello, R.J.G.B., Amaral, W.C., Choice of free parameters in expansions of discrete-time Volterra models using Kautz functions (2007) Automatica, 43 (6), pp. 1084-1091. , DOI 10.1016/j.automatica.2006.12.007, PII S0005109807000738Da Rosa, A., Campello, R.J.G.B., Amaral, W.C., An optimal expansion of Volterra models using independent Kautz bases for each kernel dimension (2008) International Journal of Control, 81 (6), pp. 962-975Da Rosa, A., Campello, R.J.G.B., Amaral, W.C., Exact search directions for optimization of linear and nonlinear models based on generalized orthonormal functions (2009) IEEE Transactions on Automatic Control, 54 (12), pp. 2757-2772Da Silva, I.N., (1995) Robust Parametric Estimation Using Artificial Neural Networks, , MSc thesis, School of Electrical and Computer Engineering of the State University of Campinas (FEEC/UNICAMP), Campinas-SP, Brazil, in PortugueseDen Brinker, A.C., Sarroukh, B.E., Pole optimisation in adaptive laguerre filtering (2004) Proc. IEEE Int. Conf. Acoustics, Speech and Signal Processing, pp. 649-652. , Montreal, CanadaDen Brinker, A.C., Optimality conditions for truncated kautz series (1996) IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 43 (2), pp. 117-122. , PII S105771309601511XDoyle III, F.J., Ogunnaike, B.A., Pearson, R.K., Nonlinear model-based control using second-order Volterra models (1995) Automatica, 31 (5), pp. 697-714Doyle III, F.J., Pearson, R.K., Ogunnaike, B.A., (2002) Identification and Control using Volterra Models, , Springer- Verlag, London, UKDumont, G.A., Fu, Y., Non-linear adaptive control via Laguerre expansion of Volterra kernels (1993) Int. J. Adaptive Control and Signal Processing, 7 (5), pp. 367-382Favier, G., Arruda, L.V.R., Review and comparison of ellipsoidal bounding algorithms (1996) Bounding Approaches to System Identification, pp. 43-68. , Milanese, M. (Ed.), Plenum Press, New YorkFavier, G., Kibangou, A.Y., Campello, R.J.G.B., Nonlinear system modelling by means of orthonormal basis functions (2003) Proc. 2nd IEEE Int. Conference on Signal, Systems, Decision and Information Technology, , Sousse, TunisiaFigueroa, J.L., Biagiola, S.I., Agamennoni, O.E., An approach for identification of uncertain Wiener systems (2008) Mathematical and Computer Modelling, 48 (1-2), pp. 305-315. , DOI 10.1016/j.mcm.2007.09.012, PII S0895717707003214Fu, Y., Dumont, G.A., Optimum time scale for discrete Laguerre network (1993) IEEE Transactions on Automatic Control, 38 (6), pp. 934-938. , DOI 10.1109/9.222305Garcia Carlos, E., Prett David, M., Morari Manfred, Model predictive control: Theory and practice - A survey (1989) Automatica, 25 (3), pp. 335-348. , DOI 10.1016/0005-1098(89)90002-2Hacioglu, R., Williamson, G.A., Reduced complexity Volterra models for nonlinear system identification (2001) Eurasip Journal on Applied Signal Processing, 2001 (4), pp. 257-265. , DOI 10.1155/S1110865701000324Henson, M.A., Nonlinear model predictive control: Current status and future directions (1998) Computers and Chemical Engineering, 23 (2), pp. 187-202. , DOI 10.1016/S0098-1354(98)00260-9, PII S0098135498002609Heuberger, P.S.C., Van Den Hof, P.M.J., Bosgra, O.H., A generalized orthonormal basis for linear dynamical systems (1995) IEEE Trans. on Automatic Control, 40 (3), pp. 451-465Heuberger, P.S.C., Van Den Hof, P.M.J., Wahlberg, B., (2005) Modelling and Identification with Rational Orthogonal Basis Functions, , Springer- Verlag, London, UKKautz, W.H., Transient synthesis in the time domain (1954) IRE Transactions on Circuit Theory, 1 (3), pp. 29-39Kibangou, A.Y., Favier, G., Hassani, M.M., Generalized orthonormal basis selection for expanding quadratic Volterra filters (2003) Proc. 13th IFAC Symposium on System Identification, pp. 1119-1124. , Rotterdam, The NetherlandsKibangou, A.Y., Favier, G., Hassani, M.M., Selection of generalized orthonormal bases for second-order Volterra filters (2005) Signal Processing, 85 (12), pp. 2371-2385. , DOI 10.1016/j.sigpro.2005.02.020, PII S0165168405001258Ljung, L., (1999) System Identification: Theory for the User, , 2nd ed. Prentice HallMäkilä, P.M., Approximation of stable systems by Laguerre filters (1990) Automatica, 26 (2), pp. 333-345Maner, B.R., Doyle III, F.J., Ogunnaike, B.A., Pearson, R.K., Nonlinear model predictive control of a simulated multivariable polymerization reactor using second-order Volterra models (1996) Automatica, 32 (9), pp. 1285-1301. , DOI 10.1016/0005-1098(96)00086-6, PII S0005109896000866Masnadi-Shirazi Mohammad, A., Ahmed, N., Optimum Laguerre networks for a class of discrete-time systems (1991) IEEE Transactions on Signal Processing, 39 (9), pp. 2104-2108. , DOI 10.1109/78.134447Milanese Mario, Belforte Gustavo, Estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors: Linear families of models and estimators (1982) IEEE Transactions on Automatic Control, AC-27 (2), pp. 408-414Mo, S.H., Norton, J.P., Fast and robust algorithm to compute exact polytope parameter bounds (1990) Mathematics and Computers in Simulation, 32 (5-6), pp. 481-493. , DOI 10.1016/0378-4754(90)90004-3Moreira, V.D., (2006) Robust Model Based Predictive Control of Uncertain Hybrid Systems Integrating Constraints, Logic, and Dynamics Based on Orthonormal Series Function, , PhD thesis, School of Electrical and Computer Engineering of the State University of Campinas (FEEC/UNICAMP), Campinas-SP, Brazil, in PortugueseNelles, O., (2001) Nonlinear System Identification, , Springer- Verlag, London, UKNgia, L.S.H., Separable nonlinear least-squares methods for efficient off-line and on-line modeling of systems using Kautz and Laguerre filters (2001) IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 48 (6), pp. 562-579. , DOI 10.1109/82.943327, PII S1057713001075000Ninness, B., Gustafsson, F., Orthonormal bases for system identification (1995) Proc. 3rd European Control Conference, 1, pp. 13-18. , Rome, ItalyNinness, B., Gustafsson, F., A unifying construction of orthonormal bases for system identification (1997) IEEE Transactions on Automatic Control, 42 (4), pp. 515-521. , PII S0018928697028080Ninness, B., Hjalmarsson, H., Gustafsson, F., The fundamental role of general orthonormal bases in system identification (1999) IEEE Transactions on Automatic Control, 44 (7), pp. 1384-1406Silva Oliveira, E.T.A.M., Optimality conditions for truncated Laguerre networks (1994) IEEE Trans. Signal Processing, 42 (9), pp. 2528-2530Silva Oliveira, E.T.A.M., On the determination of the optimal pole position of Laguerre filters (1995) IEEE Trans. Signal Processing, 43 (9), pp. 2079-2087Silva Oliveira, E.T.A.M., Optimality conditions for truncated Kautz networks with two periodically repeating complex conjugate poles (1995) IEEE Trans. Automatic Control, 40 (2), pp. 342-346Oliveira, G.H.C., (1997) Predictive Control for Processes with Structured Uncertainty Based on Orthonormal Series Function, , PhD thesis, School of Electrical and Computer Engineering of the State University of Campinas (FEEC/UNICAMP), Campinas-SP, Brazil, in PortugueseOliveira, G.H.C., Amaral, W.C., Latawiec, K., CRHPC using Volterra models and orthonormal basis functions: an application to CSTR plants (2003) Proc. IEEE Conference on Control Applications, pp. 718-723. , Istanbul, TurkeyOliveira, G.H.C., Amaral, W.C., Favier, G., Dumont, G., Constrained robust predictive controller for uncertain processes modeled by orthonormal series functions (2000) Automatica, 36 (4), pp. 563-572Oliveira, G.H.C., Da Rosa, A., Campello, R.J.G.B., Machado, J.M., Amaral, W.C., An introduction to models based on Laguerre, Kautz and other related orthonormal functions - part II: Nonlinear models International Journal of Modelling, Identification and Control, , forthcomingOliveira, G.H.C., Favier, G., Dumont, G., Amaral, W.C., Uncertainties identification using Laguerre models with application to paper machine headbox (1998) Proc. of IEEE/IMACS Multiconference on Computational Engineering in Systems Applications - CESA, Symposium on Control Optimization and Supervision, 1, pp. 329-334. , TunisiaPatwardhan, S.C., Shah, S.L., From data to diagnosis and control using generalized orthonormal basis filters. Part I: Development of state observers (2005) Journal of Process Control, 15 (7), pp. 819-835. , DOI 10.1016/j.jprocont.2004.08.006, PII S0959152405000211Schetzen, M., (1980) The Volterra and Wiener Theories of Nonlinear Systems, , Krieger Publishing Company, Malabar, Florida, USASjöberg, J., Zhang, Q., Ljung, L., Benveniste, A., Delyon, B., Glorennec, P.-Y., Hjalmarsson, H., Juditsky, A., Nonlinear black-box modeling in system identification: A unified overview (1995) Automatica, 31 (12), pp. 1691-1724Soeterboek, R., (1992) Predictive Control: A Unified Approach, , Prentice Hall, Upper Saddle River, NJ, USATakenaka, S., On the orthogonal functions and a new formula of interpolation (1925) Japanese Journal of Mathematics, 2, pp. 129-145Tanguy, N., Morvan, R., Vilbe, P., Calvez, L.C., Online optimization of the time scale in adaptive Laguerre-based filters (2000) IEEE Transactions on Signal Processing, 48 (4), pp. 1184-1187. , DOI 10.1109/78.827551Tanguy, N., Morvan, R., Vilbe, P., Calvez, L.-C., Pertinent choice of parameters for discrete Kautz approximation (2002) IEEE Transactions on Automatic Control, 47 (5), pp. 783-787. , DOI 10.1109/TAC.2002.1000273, PII S0018928602047633Tanguy, N., Vilbé, P., Calvez, L.C., Optimum choice of free parameter in orthonormal approximations (1995) IEEE Trans. on Automatic Control, 40 (10), pp. 1811-1813Van Den Bosch, P.P.J., Van Der Klauw, A.C., (1994) Modeling Identification and Simulation of Dynamical System, , CRC Press Inc., Boca Raton, Florida, USAVan Den Hof, P.M.J., Heuberger, P.S.C., Bokor, J., System identification with generalized orthonormal basis functions (1995) Automatica, 31 (12), pp. 1821-1834Wahlberg, B., System identification using Laguerre models (1991) IEEE Trans. on Automatic Control, 36 (5), pp. 551-562Wahlberg, B., System identification using Kautz models (1994) IEEE Trans. on Automatic Control, 39 (6), pp. 1276-1282Wahlberg, B., Ljung, L., Hard frequency-domain model error bounds from least-squares like identification techniques (1992) IEEE Trans. on Automatic Control, 37 (7), pp. 900-912Wahlberg, B., Makila, P.M., On approximation of stable linear dynamical systems using Laguerre and Kautz functions (1996) Automatica, 32 (5), pp. 693-708. , DOI 10.1016/0005-1098(95)00198-0Walter, E., Piet-Lahanier, H., Estimation of parameter bounds from bounded-error data: A survey (1990) Mathematics and Computers in Simulation, 32 (5-6), pp. 449-468Wiener, N., (1958) Nonlinear Problems in Random, , Theory, MIT Press, Cambridge, MA, USAZervos, C.C., Dumont, G.A., Deterministic adaptive control based on Laguerre series representation (1988) Int. J. Control, 48 (6), pp. 2333-2359Zhu, Y., System identification for process control: Recent experience and outlook (2006) International Journal of Modelling, Identification and Control, 6 (2), pp. 89-103Ziaei, K., Wang, D.W.L., Application of orthonormal basis functions for identification of flexible-link manipulators (2006) Control Engineering Practice, 14 (2), pp. 99-106. , DOI 10.1016/j.conengprac.2004.11.020, PII S096706610500036