An Improved Global Harmony Search Algorithm for the Identification of Nonlinear Discrete-Time Systems Based on Volterra Filter Modeling

This paper describes an improved global harmony search (IGHS) algorithm for identifying the nonlinear discrete-time systems based on second-order Volterra model. The IGHS is an improved version of the novel global harmony search (NGHS) algorithm, and it makes two significant improvements on the NGHS. First, the genetic mutation operation is modified by combining normal distribution and Cauchy distribution, which enables the IGHS to fully explore and exploit the solution space. Second, an opposition-based learning (OBL) is introduced and modified to improve the quality of harmony vectors. The IGHS algorithm is implemented on two numerical examples, and they are nonlinear discrete-time rational system and the real heat exchanger, respectively. The results of the IGHS are compared with those of the other three methods, and it has been verified to be more effective than the other three methods on solving the above two problems with different input signals and system memory sizes

Identification of nonlinear interconnected systems

This thesis was submitted for the degree of Master of Philosophy and awarded by Brunel University.In this work we address the problem of identifying a discrete-time nonlinear system composed of a linear dynamical system connected to a static nonlinear component. We use linear fractional representation to provide a united framework for the identification of two classes of such systems. The first class consists of discrete-time systems consists of a linear time invariant system connected to a continuous nonlinear static component. The identification problem of estimating the unknown parameters of the linear system and simultaneously fitting a math order spline to the nonlinear data is addressed. A simple and tractable algorithm based on the separable least squares method is proposed for estimating the parameters of the linear and the nonlinear components. We also provide a sufficient condition on data for consistency of the identification algorithm. Numerical examples illustrate the performance of the algorithm. Further, we examine a second class of systems that may involve a nonlinear static element of a more complex structure. The nonlinearity may not be continuous and is approximated by piecewise a±ne maps defined on different convex polyhedra, which are defined by linear combinations of lagged inputs and outputs. An iterative identification procedure is proposed, which alternates the estimation of the linear and the nonlinear subsystems. Standard identification techniques are applied to the linear subsystem, whereas recently developed piecewise affine system identification techniques are employed for the estimation of the nonlinear component. Numerical examples show that the proposed procedure is able to successfully profit from the knowledge of the interconnection structure, in comparison with a direct black box identification of the piecewise a±ne system.Funding was obtained as a Marie Curie Early Stage Researcher Training fellowship, under the NET-ACE project (MEST-CT-2004-6724)

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

A note on the optimal expansion of Volterra models using Laguerre functions

This work tackles the problem of expanding Volterra models using Laguerre functions. A strict global optimal solution is derived when each multidimensional kernel of the model is decomposed into a set of independent orthonormal bases, each of which parameterized by an individual Laguerre pole intended for representing the dominant dynamic of the kernel along a particular dimension. It is proved that the solution derived minimizes the upper bound of the squared norm of the error resulting from the practical truncation of the Laguerre series expansion into a finite number of function,. This is an extension of the results in Campello, Favier and Amaral [(2004). Optimal expansions of discrete-time Volterra models using Laguerre functions. Automatica, 40, 815-822], where an optimal solution was obtained for the usual yet particular case in which a single Laguerre pole is used for expanding a given kernel along all its dimensions. It is also proved that the particular and extended solutions are equivalent to each other when the Volterra kernels are symmetric. (c) 2006 Elsevier Ltd. All rights reserved.42468969

A note on the optimal expansion of Volterra models using Laguerre functions

This work tackles the problem of expanding Volterra models using Laguerre functions. A strict global optimal solution is derived when each multidimensional kernel of the model is decomposed into a set of independent orthonormal bases, each of which parameterized by an individual Laguerre pole intended for representing the dominant dynamic of the kernel along a particular dimension. It is proved that the solution derived minimizes the upper bound of the squared norm of the error resulting from the practical truncation of the Laguerre series expansion into a finite number of function,. This is an extension of the results in Campello, Favier and Amaral [(2004). Optimal expansions of discrete-time Volterra models using Laguerre functions. Automatica, 40, 815-822], where an optimal solution was obtained for the usual yet particular case in which a single Laguerre pole is used for expanding a given kernel along all its dimensions. It is also proved that the particular and extended solutions are equivalent to each other when the Volterra kernels are symmetric

A note on the optimal expansion of Volterra models using Laguerre functions

This work tackles the problem of expanding Volterra models using Laguerre functions. A strict global optimal solution is derived when each multidimensional kernel of the model is decomposed into a set of independent orthonormal bases, each of which parameterized by an individual Laguerre pole intended for representing the dominant dynamic of the kernel along a particular dimension. It is proved that the solution derived minimizes the upper bound of the squared norm of the error resulting from the practical truncation of the Laguerre series expansion into a finite number of function,. This is an extension of the results in Campello, Favier and Amaral [(2004). Optimal expansions of discrete-time Volterra models using Laguerre functions. Automatica, 40, 815-822], where an optimal solution was obtained for the usual yet particular case in which a single Laguerre pole is used for expanding a given kernel along all its dimensions. It is also proved that the particular and extended solutions are equivalent to each other when the Volterra kernels are symmetric

Volterra models : nonparametric and robust identification using Kautz and generalized orthonormal functions

Orientador: Wagner Caradori do Amaral, Ricardo José Gabrielli Barreto CampelloDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Enfoca-se a modelagem de sistemas não-lineares usando modelos de Volterra com bases de funções ortonormais (Orthonormal Basis Functions - OBF) distintas para cada direção do kernel. Os modelos de Volterra constituem uma classe de modelos polinomiais não-recursivos, modelos sem realimentação da saída. Tais modelos são parametrizados por funções multidimensionais, chamadas kernels de Volterra, e representam uma generalização do bem conhecido modelo de resposta ao impulso (FIR) para a descrição de sistemas não-lineares. Como os modelos de Volterra não possuem realimentação do sinal de saída, um número elevado de parâmetros é necessário para representar os kernels de Volterra, especialmente quando o comportamento não-linear do sistema depende fortemente do sinal de saída. No entanto, é possível contornar esta desvantagem por descrever cada kernel por meio de uma expansão em bases de funções ortonormais (OBF). Resultando num modelo que, em geral, possui um número menor de termos para representar o sistema. O modelo resultante, conhecido como modelo OBF-Volterra, pode ser truncado em um número menor de termos se as funções da base forem projetadas adequadamente. O problema reside na questão de como selecionar os polos livres que completamente parametrizam estas funções de forma a reduzir o número de termos a serem utilizados em cada base. Uma abordagem já utilizada envolve a otimização numérica das bases de funções ortonormais usadas para a aproximação de sistemas dinâmicos. Esta estratégia é baseada no cálculo de expressões analíticas para os gradientes da saída dos filtros ortonormais com relação aos polos da base. Estes gradientes fornecem direções de busca exatas para otimizar uma dada base ortonormal. As direções de busca, por sua vez, podem ser usadas como parte de um procedimento de otimização para obter o mínimo de uma função de custo que leva em consideração o erro de estimação da saída do sistema. Esta abordagem considerou apenas os modelos lineares e não-lineares cujas direções dos kernels foram todas parametrizadas por um mesmo conjunto de polos. Neste trabalho, estes resultados foram estendidos de forma a permitir o uso de uma base independente para cada direção dos kernels. Isto permite reduzir ainda mais o erro de truncamento quando as dinâmicas dominantes do kernel ao longo das múltiplas direções são diferentes entre si. As expressões dos gradientes relativas à base de Kautz e à base GOBF são obtidas recursivamente o que permite uma redução no tempo de processamento. Esta metodologia utiliza somente dados de entrada-saída medidos do sistema a ser modelado, isto é, não exige nenhuma informação prévia sobre os kernels de Volterra. Exemplos de simulação ilustram a aplicação dessas abordagens para a modelagem de sistemas não-lineares. Por último, apresentam-se resultados referentes à identificação robusta de modelos não-lineares sob a hipótese de erro desconhecido mas limitado, cujo objetivo é definir os limites superior e inferior dos parâmetros de modelos (intervalos de pertinência paramétrica). É analisado o caso em que se tem informação somente sobre a incerteza na saída do sistema, fornecendo-se o cálculo dos limitantes das incertezas para modelos OBF-Volterra. Estuda-se também os processos que possuem incerteza estruturada, i.e., os parâmetros do modelo, ou os kernels de Volterra, são definidos por meio de intervalos de pertinência e a ordem do modelo é conhecida. Apresenta-se uma solução exata para este problema, eliminando restrições impostas por metodologias anterioresAbstract: It focuses in the modeling of nonlinear systems using Volterra models with distinct orthonormal basis functions (OBF) to each kernel direction. The Volterra models are a class of nonrecursive polynomial models, models without output feedback. Such models are parameterized by multidimensional functions, called Volterra kernels, they represent a generalization of the well-known impulse response model and are used to describe nonlinear systems. As the Volterra models do not have output feedback, it is required a large number of parameters to represent the Volterra kernels, especially when the nonlinear behavior strongly depends of the output signal. However, such drawback can be overwhelmed by describing each kernel by un expansion in orthonormal basis functions (OBF). Resulting in a model that, in general, requires fewer parameters to represent the system. The resulting model, so-called OBF-Volterra, can be truncated into fewer terms if the basis functions are properly designed. The underlying problem is how to select de free-design poles that fully parameterize these functions in order to reduce the number of terms to be used in each bases. An approach, already used, involves the numeric optimization of orthonormal bases of function used for approximation of dynamic systems. This strategy is based on the computation of analytical expressions for the gradient of the orthonormal filters output with respect to the basis poles. Such gradient provides exact search directions for optimizing the poles of a given orthonormal basis. The search direction can, in turn, be used as part of an optimization procedure to locate the minimum of a cost-function that takes into consideration the estimation error of the system output. Although, that approach took in count only the linear models and nonlinear models which kernels directions were parameterized by a single set of poles. In this work, these results are extended in such a way to allows a use of an independent basis to each kernel direction. It can reduce even more the truncation error when dominant dynamics of the kernel are different along its directions. The gradient expressions to Kautz and GOBF bases are obtained in a recursive way which allows reducing the time processing. This methodology relies solely on input-output data measured from the system to be modeled, i.e., no previous information about the Volterra kernels is required. Simulation examples illustrate the application of this approach to the modeling of nonlinear systems. At last, it is presented some results about robust identification of nonlinear models under the hypothesis of unknown but bounded error, whose aim is to define the upper and lower bounds of the model parameters (parameter uncertainty interval). It is analyzed the case where the information available is about the uncertainty in the system output signal, providing the calculation for the uncertainty intervals to OBF-Volterra models. The process having structured uncertainty, i.e., the models parameters, or the Volterra kernels, are defined by intervals and the model order is known, is also studied. An exact solution to this problem is developed, eliminating restrictions imposed by previous approachMestradoAutomaçãoMestre em Engenharia Elétric

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

Optimization Of Volterra Models With Asymmetrical Kernels Based On Generalized Orthonormal Functions

An improved approach to determine exact search directions for the optimization of Volterra models based on Generalized Orthonormal Bases of Functions (GOBF) is proposed. The proposed approach extends the work in [7], where a novel, exact technique for optimizing the GOBF parameters (poles) for Volterra models of any order was presented. The proposed extensions take place in two different ways: (i) the formulation here is derived in such a way that each multidimensional kernel of the model is decomposed into a set of independent orthonormal bases (rather than a single, common basis), each of which is parameterized by an individual set of poles intended for representing the dominant dynamic of the kernel along a particular dimension; and (ii) a novel, more computationally efficient method to analytically and recursively calculate the search directions (gradients) for the bases poles is derived. A simulated example is presented to illustrate the performance of the proposed approach. A comparison between the proposed method, which uses asymmetric kernels with multiple orthonormal bases, and the original method, which uses symmetric kernels with a single basis, is presented. © 2011 IEEE.10521058Mediterranean Control AssociationBillings, S.A., Identification of nonlinear systems - A survey (1980) IEE Proc. Pt D, 127 (6), pp. 272-285Boyd Stephen, Chua Leon, O., Fading memory and the problem of approximating nonlinear operators with volterra series (1985) IEEE transactions on circuits and systems, CAS-32 (11), pp. 1150-1161Bokor, J., Schipp, F., Approximate identification in Laguerre and Kautz bases (1998) Automatica, 34 (4), pp. 463-468. , PII S000510989700201XBroome, P.W., Discrete orthonormal sequences (1965) Journal of the Association for Computing Machinery, 12 (2), pp. 151-168Campello, R.J.G.B., Amaral, W.C., Favier, G., Optimal expansions of discrete-time Volterra models using Laguerre functions (2004) Automatica, 40, pp. 815-822Campello, 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 S0005109806000069Da 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 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., (2008) Exact Search Directions for Optimization of Linear and Nonlinear Models Based on Generalized Orthonormal Functions, , http://www.icmc.usp.br/, Technical report, Department of Computer Sciences, University of São Paulo (USP) campelloDa 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. 962-975Doyle III, F.J., Pearson, R.K., Ogunnaike, B.A., (2002) Identification and Control Using Volterra Models, , Springer-VerlagDumont, 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. 367-382Eykhoff, P., (1974) System Identification: Parameter and State Estimation, , John Wiley & Sons, New YorkHeuberger, P.S.C., Van Den Hof, P.M.J., Bosgra, O.H., A generalized orthonormal basis for linear dynamical systems (1995) IEEE Transactions 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, , SpringerKautz, W.H., Transient synthesis in time domains (1954) IRE Trans. on Circuit Theory, 1 (3), pp. 29-39Levenberg, K., A method for the solution of certain nonlinear problems in least squares (1944) Quartely of Applied Mathematics, 2 (2), pp. 164-168Marquardt, D.W., An algorithm for the least-squares estimation of nonlinear parameters (1963) SIAM Journal of Applied Mathematics, 11 (2), pp. 431-441Nelles, O., (2001) Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models, , Springer- Verlag, BerlinNinness, B., Gustafsson, F., A unifying construction of orthonormal bases for system identification (1997) IEEE Transactions on Automatic Control, 42 (4), pp. 515-521. , PII S0018928697028080Oliveira, G.H.C., Amaral, W.C., Favier, G., Dumont, G.A., Constrained robust predictive controller for uncertain processes modeled by orthonormal series functions (2000) Automatica, 36 (4), pp. 563-571. , DOI 10.1016/S0005-1098(99)00179-XRugh, W.J., (1981) Nonlinear Systems Theory - The Volterra/Wiener Approach, , The Johns Hopkins University Press, BaltimoreSchetzen, M., (1989) The Volterra and Wiener Theories of Nonlinear Systems, , Robert Krieger Publishing CompanyTakenaka, S., On the orthogonal functions and a new formula of interpolation (1925) Japanese Journal of Mathematics, 2, pp. 129-145Den Hof, P.M.J.V., Heuberger, P.S.C., Bokor, J., System identification with generalized orthonormal basis functions (1995) Automatica, 31 (12), pp. 1821-1834Wahlberg, B., System identification using Kautz models (1994) IEEE Transactions on Automatic Control, 39 (6), pp. 1276-1282Wahlberg, 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)00198-0Wiener, N., (1942) Response of A Nonlinear Device to Noise, , Technical Report 129, Radiation Laboratory, M.I.T., AbrilWiener, N., (1966) Nonlinear Problems in Random Theory, , The MIT PressZiaei, 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

Identification And Control Of Processes Via Developments In The Orthonormal Series Part A: Identification _net 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.183301321Aguirre, L.A., (2004) Introdução à Identificação de Sistemas: Técnicas Lineares e Não Lineares Aplicadas a Sistemas Reais, , 2 edn, Editora UFMGAguirre, L.A., Correa, M.V., Cassini, C., Nonlinearities in NARX polynomial models: Representation and estimation (2002) IEE Proc. Control Theory and Applications, 149, pp. 343-348Alataris, K., Berger, T.W., Marmarelis, V.Z., A novel network for nonlinear modeling of neural systems with arbitrary point-process inputs (2000) Neural Networks, 13, pp. 255-266(2000) Nonlinear Model Predictive Control, , Allgower, F. and Zheng, A, eds, Birkhauser VerlagArto, V., Hannu, P., Halme, A., Modeling of chromatographic separation process with Wiener-MLP representation (2001) Journal of Process Control, 11, pp. 443-458Babuška, R., (1998) Fuzzy Modeling for Control, , Kluwer Academic PublishersBack, A.D., Tsoi, A.C., Nonlinear system identification using discrete Laguerre functions (1996) Journal of Systems Engineering, 6, pp. 194-207Balestrino, A., Caiti, A., Zanobini, G., Identification of Wiener-type nonlinear systems by Laguerre filters and neural networks (1999) Proc. 14th International Federation of Automatic Control (IFAC) World Congress, pp. 433-438. , Beijing/China, ppBillings, S.A., Identification of nonlinear systems - a survey (1980) IEE Proc. Pt D, 127 (6), pp. 272-285Boyd, S., Chua, L.O., Fading memory and the problem of approximating nonlinear operators with Volterra series (1985) IEEE Trans. on Circuits and Systems, 32 (11), pp. 1150-1161Braga, A.P., Carvalho, A.C.P.L.F., Ludemir, T.B., (2000) Redes Neurais Artificiais: Teoria e Aplicações, , LTCBroome, P.W., Discrete orthonormal sequences (1965) Journal of the Association for Computing Machinery, 12 (2), pp. 151-168Broomhead, D.S., Lowe, D., Multivariate functional interpolation and adaptive networks (1988) Complex Systems, 2, pp. 321-355Camacho, E.F., Bordons, C., (1999) Model Predictive Control, , Springer-VerlagCampello, R.J.G.B., (2002) Arquiteturas e Metodologias para Modelagem e Controle de Sistemas Complexos utilizando Ferramentas Clássicas e Modernas, , PhD thesis, DCA/FEEC/UNICAMP, Campinas/SP, BrasilCampello, R.J.G.B., Amaral, W.C., Equivalência entre modelos nebulosos e redes neurais (1999) Anais IV Simpósio Brasileiro de Automação Inteligente, pp. 208-212. , São Paulo/Brasil, ppCampello, R.J.G.B., Amaral, W.C., Takagi-Sugeno fuzzy models within orthonormal basis function framework and their application to process control (2002) Proc. 11th IEEE Internat. Conference on Fuzzy Systems, pp. 1399-1404. , Honolulu/USA, ppCampello, R.J.G.B., Amaral, W.C., Favier, G., (2001) Optimal Laguerre series expansion of discrete Volterra models, Proc. European Control Conference, pp. 372-377. , Porto/Portugal, ppCampello, R.J.G.B., Amaral, W.C., Favier, G., Desenvolvimento ótimo de modelos de Volterra em funções ortonormais de Laguerre (2002) Anais XIV Congresso Brasileiro de Automática, pp. 128-133. , Natal/Brasil, ppCampello, R.J.G.B., Amaral, W.C., Favier, G., A note on the optimal expansion of Volterra models using Laguerre functions (2006) Automatica, 42, pp. 689-693Campello, R.J.G.B., Favier, G., Amaral, W.C., Optimal expansions of discrete-time Volterra models using Laguerre functions (2003) Proc. of the 13th IFAC Symposium on System Identification, pp. 1844-1849. , Rotterdam/The Netherlands, ppCampello, R.J.G.B., Favier, G., Amaral, W.C., Optimal expansions of discrete-time Volterra models using Laguerre functions (2004) Automatica, 4 (0), pp. 815-822Campello, R. J. G. B., Meleiro, L. A. C. and Amaral, W. C. (2004). Control of a bioprocess using orthonormal basis function fuzzy models, Proc. 13th IEEE Internat. Conference on Fuzzy Systems, Budapest/Hungary, pp. 801-806Campello, R.J.G.B., Oliveira, G.H.C., Modelos não lineares (2007) Enciclopédia de Automática, 3. , L. A. Aguirre, A. P. Alves da Silva, M. F. M. Campos and W. C. Amaral eds, Cap. 4, Edgard BlücherCampello, R. J. G. B., Oliveira, G. H. C., Von Zuben, F. J. and Amaral, W. C. (1999). Redes neurais com base de funções ortonormais, Anais IV Simpósio Brasileiro de Automação Inteligente, São Paulo/Brasil, pp. 213-218Campello, R.J.G.B., Von Zuben, F.J., Amaral, W.C., Meleiro, L.A.C., Maciel Filho, R., Hierarchical fuzzy models within the framework of orthonormal basis functions and their application to bioprocess control (2003) Chemical Engineering Science, 58, pp. 4259-4270Cho, K.B., Wang, B.H., Radial basis function based adaptive fuzzy systems and their applications to system identification and prediction (1996) Fuzzy Sets and Systems, 83, pp. 325-339(1994) Advances in Model Based Predictive Control, , Clarke, D. W, ed, Oxford University PressCorrêa, M.V., Aguirre, L.A., Identificação não-linear caixa-cinza - Uma revisão e novos resultados (2004) Controle & Automação, 15, pp. 109-126da Rosa, A., (2005) Desenvolvimento de modelos discretos de Volterra usando funções de Kautz, , Master's thesis, DCA/FEEC/UNICAMP, Campinas/SP, Brasilda Rosa, A., Amaral, W.C., Campello, R.J.G.B., Desenvolvimento de modelos de Volterra usando funções de Kautz e sua aplicação à modelagem de um sistema de levitação magnética (2006) Anais do Congresso Brasileiro de Automática, pp. 274-279. , Salvador/Brasil, ppda 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)da Silva, I.N., (1995) Estimação paramétrica robusta através de redes neurais artificiais, , Master's thesis, DCA/FEEC/UNICAMP, BrasilDoyle III, F.J., Ogunnaike, B.A., Pearson, R.K., Nonlinear model-based control using second-order Volterra models (1995) Automática, 31, pp. 697-714Doyle III, F.J., Pearson, R.K., Ogunnaike, B.A., (2002) Identification and Control using Volterra Models, , Springer-VerlagDumont, G.A., Fu, Y., Non-linear adaptive control via Laguerre expansion of Volterra kernels (1993) Int. J. Adaptive Control and Signal Processing, 7, pp. 367-382Espinosa, J., Vandewalle, J., Wertz, V., (2004) Fuzzy Logic, Identification and Predictive Control, , SpringerVerlagEykhoff, P., (1974) System Identification: Parameter and State Estimation, , John Wiley & SonsFavier, G. and Arruda, L. V R. (1996). Review and comparison of ellipsoidal bounding algorithms, in M. Milanese (ed.), Bounding Approaches to System Identification, Plenum Press, New York, pp. 43-68Favier, 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/TunisiaFu, Y., Dumont, G.A., An optimum time scale for discrete Laguerre network (1993) IEEE Trans. on Automatic Control, 38 (6), pp. 934-938Garcia, C.E., Prett, D.M., Morari, M., Model predictive control: Theory and practice - a survey (1989) Automatica, 25 (3), pp. 335-348Haykin, S., (1999) Neural Networks: A Comprehensive Foundation, , 2nd edn, Prentice HallHenson, M.A., Nonlinear model predictive control: Current status and future directions (1998) Computers and Chemical Engineering, 23, pp. 187-202Heuberger, 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, pp. 451-465Heuberger, P.S.C., Van den Hof, P.M.J., Wahlberg, B., (2005) Modelling and Identification with Rational Orthogonal Basis Functions, , SpringerHunt, K.J., Haas, R., Murray-Smith, R., Extending the functional equivalence of radial basis function networks and fuzzy inference systems (1996) IEEE Trans. Neural Networks, 7, pp. 776-781Kibangou, A., Favier, G., Hassani, M., Generalized orthonormal basis selection for expanding quadratic Volterra filters (2003) Proc. IFAC Symposium on System Identification, p. 11191124. , Rotterdam/The Netherlands, ppKibangou, A.Y., Favier, G., Hassani, M.M., Iterative optimization method of GOB-Volterra filters (2005) Proc. 16th International Federation of Automatic Control (IFAC) World Congress, , Prague/Czech RepublicKibangou, A.Y., Favier, G., Hassani, M.M., Laguerre-Volterra filters optimization based on Laguerre spectra (2005) EURASIP Journal on Applied Signal Processing, 17, pp. 2874-2887Kibangou, A.Y., Favier, G., Hassani, M.M., Selection of generalized orthonormal bases for second-order Volterra filters (2005) Signal Processing, 85, pp. 2371-2385Kosko, B., (1992) Neural Networks and Fuzzy Systems: A Dynamical Systems Approach to Machine Intelligence, , Prentice HallKosko, B., (1997) Fuzzy Engineering, , Prentice HallLeontaritis, I., Billings, S.A., Input-output parametric models for nonlinear systems - Parts I and II (1985) Int. Journal of Control, 41 (6), pp. 303-344Ljung, L., (1999) System Identification: Theory for the user, , 2nd edn, 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. 1285-1301Masnadi-Shirazi, M.A., Ahmed, N., Optimum Laguerre networks for a class of discrete-time systems (1991) IEEE Transactions on Signal Processing, 39 (9), pp. 2104-2108Medeiros, A.V., (2006) Modelagem de sistemas dinâmicos não lineares utilizando sistemas fuzzy, algoritmos genéticos e funções de base ortonormal, , Master's thesis, DCA/FEEC/UNICAMP, Campinas/SP, BrasilMedeiros, A.V., Amaral, W.C., Campello, R.J.G.B., GA optimization of generalized OBF TS fuzzy models with global and local estimation approaches (2006) Proc. 15th IEEE Internat. Conference on Fuzzy Systems, pp. 8494-8501. , Vancouver/Canada, ppMedeiros, A.V., Amaral, W.C., Campello, R.J.G.B., GA optimization of OBF TS fuzzy models with linear and non linear local models (2006) Anais Simpósio Brasileiro de Redes Neurais (SBRN), , Ribeirão Preto/BrasilMilanese, M., Belforte, G., Estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors : Linear families of models and estimators (1982) IEEE Trans. on Automatic Control, 27 (2), pp. 408-414Moreira, V.D., (2006) Controle Preditivo Robusto de Sistemas Hibridos Incertos Integrando Restrições, Lógica e Dinâmica Baseada em Séries de Funções Ortonormais, , PhD thesis, DCA/FEEC/UNICAMPNelles, O., (2001) Nonlinear System Identification, , Springer-VerlagNinness, B., Gustafsson, F., Orthonormal bases for system identification (1995) Proc. 3rd European Control Conference, 1, pp. 13-18. , Rome/Italy, ppNinness, B., Gustafsson, F., A unifying construction of orthonormal bases for system identification (1997) IEEE Trans. on Automatic Control, 42, pp. 515-521Norton, J.P., Fast and robust algorithm to compute exact polytope parameter bounds (1990) Mathematics and Computers in Simulation, 32, pp. 481-493Oliveira e Silva, T. A. M. (1994). Optimality conditions for truncated Laguerre networks, IEEE Trans. Signal Processing 42: 2528-2530Oliveira e Silva, T. A. M. (1995a). On the determination of the optimal pole position of Laguerre filters, IEEE Trans. Signal Processing 43: 2079-2087Oliveira e Silva, T. A. M. (1995b). Optimality conditions for truncated Kautz networks with two periodically repeating complex conjugate poles, IEEE Trans. Automatic Control 40: 342-346Oliveira, G.H.C., (1997) Controle Preditivo para Processos com Incertezas Estruturadas baseado em Séries de Funções Ortonormais, , PhD thesis, DCA/FEEC/UNICAMP, Campinas/SP/BrasilOliveira, G. H. C. and Amaral, W. C. (2000). Identificação e controle preditivo de processos não lineares utilizando séries de Volterra e bases de funções ortonormais, Anais XIII Congresso Brasileiro de Automática, Florianópolis/Brasil, pp. 2174-2179Oliveira, 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., Campello, R. J. G. B. and Amaral, W. C. (1999). Fuzzy models within orthonormal basis function framework, Proc. 8th IEEE Internat. Conference on Fuzzy Systems, Seoul/Korea, pp. 957-962Oliveira, G.H.C., Campello, R.J.G.B., Amaral, W.C., Identificação e controle de processos via desenvolvimentos em séries ortonormais. Parte B: Controle preditivo (2007) Controle & Automação, 18 (3), pp. 322-336Oliveira, 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. , Tunisia, ppPassino, K.M., Yurkovich, S., (1997) Fuzzy Control, , Addison WesleyRugh, W.J., (1981) Nonlinear System Theory: The Volterra/ Wiener Approach, , The Johns Hopkins University PressSchetzen, M., (1980) The Volterra and Wiener Theories of Nonlinear Systems, , John Wiley & SonsSentoni, G., Agamennoni, O., Desages, A., Romagnoli, J., Approximate models for nonlinear process control (1996) AIChE Journal, 42, pp. 2240-2250Sentoni, G.B., Biegler, L.T., Guiver, J.B., Zhao, H., State-Space nonlinear process modeling: Identification and universality (1998) AIChE Journal, 44, pp. 2229-2239Sjöberg, 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, pp. 1691-1724Soeterboek, R., (1992) Predictive Control: A Unified Approach, , Prentice HallSugeno, M., Kang, G.T., Fuzzy modelling and control of multilayer incinerator (1986) Fuzzy Sets and Systems, 18, pp. 329-346Sugeno, M., Kang, G.T., Structure identification of fuzzy model (1988) Fuzzy Sets and Systems, 28, pp. 15-33Sugeno, M., Tanaka, K., Successive identification of a fuzzy model and its applications to prediction of a complex system (1991) Fuzzy Sets and Systems, 42, pp. 315-334Takagi, T., Sugeno, M., Fuzzy identification of systems and its applications to modeling and control (1985) IEEE Trans. Systems, Man and Cybernetics, SMC-15, pp. 116-132Tanguy, N., Morvan, R., Vilbé, P., Calvez, L.C., Online optimization of the time scale in adaptive Laguerre-based filters (2000) IEEE Trans. on Signal Processing, 48, pp. 1184-1187Tanguy, N., Morvan, R., Vilbé, P., Calvez, L.C., Pertinent choice of parameters for discrete Kautz approximation (2002) IEEE Trans. on Automatic Control, 47, pp. 783-787Tanguy, N., Vilbé, P., Calvez, L.C., Optimum choice of free parameter in orthonormal approximations (1811) IEEE Trans. on Automatic Controls, , 1995Van den Hof, P.M.J., Heuberger, P.S.C., Bokor, J., System identification with generalized orthonormal basis functions (1995) Automatica, 31 (12), pp. 1821-1834Vázquez, M.A.A., Agamennoni, O.E., Approximate models for nonlinear dynamical systems and their generalization properties (2001) Mathematical and Computer Modelling, 33, pp. 965-986Wahlberg, 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., Mäkilä, P.M., Approximation of stable linear dynamical systems using Laguerre and Kautz functions (1996) Automática, 32 (5), pp. 693-708Walter, E., Lahanier, P., Estimation of parameter bounds from bounded-error data: A survey (1990) Mathematics and Computers in simulation, 32, pp. 449-468Wang, L.-X., Mendel, J.M., Fuzzy basis functions, universal approximation and orthogonal least squares learning (1992) IEEE Trans. Neural Networks, 3, pp. 807-814Wiener, N., (1958) Nonlinear Problems in Random Theory, , WileyYager, R.R., Filev, D.P., (1994) Essentials of Fuzzy Modeling and Control, , John Wiley & SonsZeng, X.-J., Singh, M.G., Approximation theory of fuzzy systems - SISO case (1994) IEEE Trans. Fuzzy Systems, 2, pp. 162-176Zeng, X.-J., Singh, M.G., Approximation theory of fuzzy systems - MIMO case (1995) IEEE Trans. Fuzzy Systems, 3, pp. 219-235Zervos, C.C., Dumont, G.A., Deterministic adaptive control based on Laguerre series representation (1988) Int. J. Control, 48 (6), pp. 2333-235