A review of recent literature and the research effort underlying this dissertation indicates that fractional order differential equations have significant potential to advance dynamical system methods broadly. Particular promise exists in the area of control and estimation, even for systems where fractional order models do not arise “naturally”. This dissertation is aimed at further building of the base methodology with a focus on robust feedback control and state estimation. By setting the mathematical foundation with the fractional derivative Caputo definition, we can expand the concept of the fractional order calculus in a way that enables us to build corresponding controllers and estimators in the state-space form. For the robust eigenstructure assignment, we first examine the conditioning problem of the closed-loop eigenvalues and stability robustnesss criteria for the fractional order system, and we find a unique application of an n-dimensional rotation algorithm developed by Mortari, to solve the robust eigenstructure assignment problem in a novel way. In contradistinction to the existing Fractional Kalman filter developed by using Gru ̈ndwald-Letnikov definition, the new Fractional Kalman filter that we establish by utilizing Caputo definition and our algorithms provide us with powerful means for solving practical state estimation problems for fractional order systems.

Neste trabalho, estudamos equações diferenciais em espaços de Banach. Duas questões são abordadas: a robustez da estabilidade assintótica, e a aproximação de soluções de sistemas periódicos por wavelets. Observa-se que a estabilidade exponencial do sistema x = A(t)x é qualitativamente preservada pelo sistema perturbado x=A(t)x+B(t)x se B(t) for integralmente pequeno. Consequentemente, tal propriedade é preservada por uma perturbação B(wt)x para w suficientemente grande, mesmo se B(t) pertence a uma classe mais geral de funções do que as funções quase-periódicas, aqui apresentada. Além disso, estudamos o efeito de aproximações de uma função periódica f (t) por wavelets periódicas na solução de um sistema periódico x = Ax+ f (t). Conclui-se que as soluções do problema inicial podem inclusive ser aproximadas utilizando a wavelet base não-periódica. / In this work, we study differential equations in Banach spaces. Two questions were considered: the robustness of the asymptotic stability, and the approximation of solutions of periodic systems by wavelets. It is observed that the exponential stability of the system x = A(t)x is qualitatively preserved by the perturbed system x = A(t)x+B(t)x if B(t) is integrally small. As a consequence, this property is preserved by a perturbation B(wt) for w sufficiently large, even if B(t) is in a class of functions which is more general than almost-periodic functions, presented here. Furthermore, we study the effect of approximating a periodic function f (t) by periodic wavelets in the solution of a periodic system x = Ax+ f (t). It is concluded that the solutions of the initial problem can even be approximated using the non-periodic base wavelet.

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Cette thèse concerne la récupération d'énergie vibratoire dans le but de proposer une alternative aux batteries conventionnelles pour l’alimentation de systèmes autonomes sans fil. Ceci permettrait d’améliorer leur compacité (moins d’énergie stockée), leur tenue dans des environnements sévères (forte température) et de réduire leur besoin d'entretien. Cette étude se concentre plus particulièrement sur les générateurs oscillants bistables, intéressants pour leur grande plage de fréquences utile comparée à celle offerte par les générateurs linéaires (limitée à la zone de résonance). Cette thèse se divise en quatre grandes parties. La première présente la construction du modèle mathématique permettant de prédire les différents comportements du générateur bistable (ces derniers pouvant coexister sur certaines plages de fréquences) incluant l'étude de la stabilité aux petites perturbations. Ce modèle met en évidence des comportements du générateur encore peu exploités pour la récupération d'énergie : les comportements sous-harmoniques dont la plage de fréquences permet d'agrandir la bande passante globale du générateur. Afin d’améliorer la précision du modèle, celui-ci est ensuite complété dans la deuxième partie par un critère semi-analytique : le critère de robustesse de stabilité qui caractérise la sensibilité du générateur aux perturbations extérieures (plus un comportement est robuste plus il sera facile à maintenir dans le temps). Le modèle ainsi obtenu ainsi que le système expérimentale montrent une grande plage de fréquences sur laquelle coexistent des comportements intéressants pour la récupération d’énergie (les orbites hautes) et des comportements non désirables (les orbites basses). La troisième partie de cette thèse présente donc différentes stratégies permettant de sauter des orbites basses vers les orbites hautes en jouant directement sur les paramètres du générateur. Enfin, la quatrième et dernière partie s’attarde sur l’influence du circuit d'interface AC-DC entre le générateur bistable et la charge en vue de son intégration future. / This thesis concerns vibratory energy harvesting in order to propose an alternative to conventional batteries for the power supply of autonomous wireless systems. This would improve their compactness (less stored energy), their resistance to harsh environments (high temperature) and reduce their need for maintenance. This study focuses in particular on bistable oscillating generators, which are interesting for their large useful frequency range compared to that offered by linear generators (limited to the resonance zone). This thesis is divided into four main parts. The first presents the construction of the mathematical model to predict the different behaviors of the bistable generator (these behaviors can coexist over certain frequency ranges) including the study of stability to small disturbances. This model highlights original behaviors for energy recovery: subharmonic behaviors whose frequency range allows increasing the overall generator bandwidth. In order to improve the accuracy of the model, a semi-analytical criterion is then added: the stability robustness criterion which characterizes the sensitivity of the different behaviors to external disturbances (the more robust a behavior, the easier to maintain over time). The model obtained and the experimental prototype show a wide frequency range on which the interesting behaviors (high orbits) and the undesirable behaviors (low orbits) coexist. The third part of this thesis therefore presents different strategies for jumping from low to high orbits by playing directly on the generator parameters. Finally, the fourth and last part focuses on the influence of the AC-DC interface circuit between the bistable generator and the load for future integration.