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Dérivation numérique : synthèse, application et intégration

Abstract : Differentiation algorithms are numerical methods for the derivative estimation of measured signals. These algorithms are necessary in the case of nonlinear output feedback control design as they enable the estimation of signals that are used for the control law computing by differentiating other measurable ones. The most popular approaches to the problem are based on the observer theory. Thus, the signal to be differentiated is modeled as the output of a given dynamic system whose input is a known canonical signal. The signal derivatives are then available by the observation of the model state. Most of these approaches are based on deterministic or stochastic assumptions on the signal to be differentiated and/or the perturbations on it. In this work, we are interested in the study and the application of linear and nonlinear observation approaches for signal derivative estimation. In a linear framework, different filtering and observation approaches (high gain, Kalman, H2)are then introduced and applied following the previous goal. An alternative approach for linear observer is also introduced in this work and applied to the differentiator design problem. In the latter, an observer is designed by minimizing an H∞ norm and the problem boils down to a convex optimization problem involving Linear Matrix Inequalities (LMI). The interesting point of this approach is that, in the case of weighted H∞ norm minimization, it allows the Power Spectral Density (PSD) shaping of the estimation error signal. An additional investigation is also carried out in order to introduce some linear observer structures as alternative to the classic Luenberger one. The nonlinear observers based on the sliding modes theory present an alternative approach to the linear ones. As their robustness was widely commented and proved, the nonlinear observers can potentially improve the differentiator performances in addition to the fact that they allow to introduce a new convergence property, the finite time convergence. Due to their nonlinear behavior, the tuning of these algorithms is complicated and depends on the characteristics of both the signal to be differentiated and the noise level affecting it and thus, an optimal tuning for a given signal is no more an optimal one for another one. To overcome this difficulty, adaptive procedures are introduced where the parameters are tuned on line in real time. Thereby, an adaptive sliding mode observer is introduced and applied to the differentiation problem. Though, our study was not just limited to the theoretical background of the differentiation problem. It was extended to the experimental aspects. The goal is then to design an embedded low cost " software sensor " for mechatronic system state variable estimation for control design. These estimations are obtained by measured signal differentiation. Thus, a study on the effect of the limited computing and signal conversion precision of the embedded device on the differentiator performances is done and results are presented in thiswork.Abstract186Then, the differentiation algorithms digital implementation is performed on a microcontroller based numerical device. This solution is then validated in open loop for the estimation of signal derivative. Finally, the numerical derivative device is introduced on an electropneumatic system control loop where the digital differentiator is used for the estimation of some derivative signals that are considered in the control computing. Comparative results on different differentiation algorithms and system trajectories are then presented.
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Submitted on : Thursday, April 12, 2012 - 4:22:48 PM
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  • HAL Id : tel-00687239, version 1


Mehdi Dridi. Dérivation numérique : synthèse, application et intégration. Autre. Ecole Centrale de Lyon, 2010. Français. ⟨NNT : 2010ECDL0040⟩. ⟨tel-00687239⟩



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