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Modélisation et identification de systèmes non-linéaires à l'aide de modèles de volterra à complexité réduite

Abstract : The identification of the non-linear dynamic systems from a set of input/output data is of a fundamental importance for the practical applications since a lot of physical systems possess non linear characteristics. The structure of the Volterra model can be used to represent a general class of non linear systems. However, the use of such a representation is often limited because of the huge number of parameters relative to such a structure. To overcome this inconvenience, several solutions are proposed in this thesis. The first uses expansions of the different kernels on orthogonal bases functions. The second is based on the use of techniques calling on reduced-order decompositions of the tensors associated to the kernels of order superior or equal to three. Various bases of functions (Laguerre, Kautz and Generalized Orthogonal Bases (BOG)) are first studied then to use them for the modelling the linear systems then for the representation of the kernels of the Volterra models. The problem of identification includes several parts: determination of the poles of the orthogonal bases functions, the order of the kernel developments, the Fourier coefficients of the development and the relative uncertainty to these coefficients. A state representation associated to a development on a Generalized Orthogonal Basis is developed and then used for the construction of the output predictor of the system to be modelled. Then, several tensorial decompositions are studied. The PARAFAC decomposition is specially considered. Reduced complexity Volterra models inspired from this technique are proposed. While considering the quadratic Volterra kernel as a matrix and the other kernels as tensors of orders superior to two, we use a singular value decomposition for the quadratic kernel and the PARAFAC decomposition for the kernels of orders superior to two in order to construct a new model called SVD-PARAFAC based Volterra model. A new algorithm called ARLS (Alternating Recursive Least Squares) is presented. This algorithm essentially based on the technical RLS applied in an alternate manner estimates the parameters of such Volterra models. Finally, new methods of robust identification called bounded error techniques are presented. They are used for the identification of linear models based on an expansion on a GOB, this work aims to use the results found lately for uncertain linear systems in the case of the uncertain non linear systems. One of the techniques of identification, the polytopic approach is especially considered. This approach allows to estimate the uncertainty intervals of the Fourier coefficients of the expansion on the different GOBs studied. The polytopic approach is also used in order to identify the uncertainty intervals of the parameters of the SVD-PARAFAC based Volterra model. The proposed methods allows to achieve an important numeric complexity reduction and a considerable gain in time calculation.
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Submitted on : Thursday, August 31, 2006 - 4:53:53 PM
Last modification on : Thursday, August 4, 2022 - 4:53:03 PM
Long-term archiving on: : Thursday, September 20, 2012 - 10:06:03 AM


  • HAL Id : tel-00090557, version 1



Anis Khouaja. Modélisation et identification de systèmes non-linéaires à l'aide de modèles de volterra à complexité réduite. Automatique / Robotique. Université Nice Sophia Antipolis, 2005. Français. ⟨tel-00090557⟩



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