An optimal expansion of Volterra models using independent Kautz bases for each kernel dimension

A new solution for the problem of selecting poles of the two-parameter Kautz functions in Volterra models is proposed. In general, a large number of parameters are required to represent the Volterra kernels, although this difficulty can be overcome by describing each kernel using a basis of orthonormal functions, such as the Kautz basis. This representation has a Wiener structure consisting of a linear dynamic generated by the orthonormal basis followed by a non-linear static mapping represented by the Volterra series. The resulting Wiener/Volterra model can be truncated into fewer terms if the Kautz functions are properly designed. The underlying problem is how to select the arbitrary complex poles that fully parametrize these functions. This problem has been approached in previous research by minimizing an upper bound for the error resulting from the truncation of the kernel expansion. The present paper goes even further in that each multidimensional kernel is decomposed into a set of independent Kautz bases, each of which is parametrized by an individual pair of conjugate Kautz poles intended to represent the dominant dynamic of the kernel along a particular dimension. An analytical solution for one of the Kautz parameters, valid for Volterra models of any order, is derived. A simulated example is presented to illustrate these theoretical results. The same approach is then used to model a real non-linear magnetic levitation system with oscillatory behaviour

An optimal expansion of Volterra models using independent Kautz bases for each kernel dimension

A new solution for the problem of selecting poles of the two-parameter Kautz functions in Volterra models is proposed. In general, a large number of parameters are required to represent the Volterra kernels, although this difficulty can be overcome by describing each kernel using a basis of orthonormal functions, such as the Kautz basis. This representation has a Wiener structure consisting of a linear dynamic generated by the orthonormal basis followed by a non-linear static mapping represented by the Volterra series. The resulting Wiener/Volterra model can be truncated into fewer terms if the Kautz functions are properly designed. The underlying problem is how to select the arbitrary complex poles that fully parametrize these functions. This problem has been approached in previous research by minimizing an upper bound for the error resulting from the truncation of the kernel expansion. The present paper goes even further in that each multidimensional kernel is decomposed into a set of independent Kautz bases, each of which is parametrized by an individual pair of conjugate Kautz poles intended to represent the dominant dynamic of the kernel along a particular dimension. An analytical solution for one of the Kautz parameters, valid for Volterra models of any order, is derived. A simulated example is presented to illustrate these theoretical results. The same approach is then used to model a real non-linear magnetic levitation system with oscillatory behaviour.81696297

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. 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An Introduction To Models Based On Laguerre, Kautz And Other Related Orthonormal Functions - Part Ii: Non-linear 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. The first part of the paper approached issues related with linear models and models with uncertain parameters. Now, the mathematical foundations as well as their advantages and limitations are discussed within the contexts of non-linear system identification. The discussions comprise a broad bibliographical survey of the subject and a comparative analysis involving some specific model realisations, namely, Volterra, fuzzy, and neural models within the orthonormal basis functions framework. Theoretical and practical issues regarding the identification of these non-linear models are presented and illustrated by means of two case studies. 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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. 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