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Généralisation du lemme de Gronwall-Bellman pour la stabilisation des systèmes fractionnaires

Abstract : In this dissertation, we proposed sufficient conditions for the asymptotical stabilization of a class of nonlinear fractional-order systems based on the generalization of Gronwall-Bellman lemma. We extended these results for the asymptotical stabilization of nonlinear singular fractional-order systems and proposed sufficient conditions for the existence and asymptotic stability of the observation error for the nonlinear fractional-order systems and nonlinear singular fractional-order systems. For the nonlinear integer-order systems, the proposed generalization of Gronwall-Bellman lemma allowed us to obtain sufficient conditions for : - the static state feedback and the static output feedback exponential stabilizations, - the robust exponential stabilization with regards to parameter uncertainties, - the observer-based control. We treated three cases for the asymptotical stabilization of linear fractional-order systems : the static state feedback, the static output feedback and the observer-based output feedback. Then, we proposed sufficient conditions for the asymptotical stabilization of linear fractional-order systems with nonlinear uncertain parameters. Finally, we treated the observer design for the linear and nonlinear fractional-order systems and for the linear and nonlinear singular fractional-order systems. The stabilization technique based on the generalization of Gronwall-Bellman lemma is extended to nonlinear fractional-order systems and nonlinear singular fractional-order systems. Sufficient conditions for the asymptotical stabilization, the robust asymptotical stabilization and the observer-based control of a class of nonlinear fractional-order systems and nonlinear singular fractional-order systems were obtained. Furthermore, the observer design for the nonlinear fractional-order systems and nonlinear singular fractional-order systems is proposed. This approach is based on a parameterization of the solutions of generalized Sylvester equations. The conditions for the existence of these observers are given and sufficient conditions for their stability are derived using linear matrix inequalities (LMIs) formulation and the generalization of Gronwall-Bellman lemma. The advantage of this method is that, firstly, the observation error does not depend explicitly on the state and control system and, secondly, this method unifies the design of full, reduced and minimal orders observers
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Ibrahima N'Doye. Généralisation du lemme de Gronwall-Bellman pour la stabilisation des systèmes fractionnaires. Autre. Université Henri Poincaré - Nancy 1, 2011. Français. ⟨NNT : 2011NAN10007⟩. ⟨tel-01746396⟩

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