In this dissertation, the robust stability of uncertain continuous and discrete singularly perturbed systems with multiple time delays is investigated. Firstly, the asymptotic stability for a class of linear continuous singularly perturbed systems with multiple time delays is investigated. A simple estimate of an upper bound of singular perturbation parameter is proposed such that the original system is asymptotically stable for any . Moreover, a delay-dependent criterion, but -independent, is proposed to guarantee the asymptotic stability of the original system. Secondly, we consider the robust stability problem of uncertain continuous singularly perturbed systems with multiple time delays. Two delay-dependent criteria are proposed to guarantee the robust stability of a class of uncertain continuous multiple time-delay singularly perturbed systems subject to unstructured perturbations. Thirdly, the robust D-stability of nominally stable discrete uncertain systems with multiple time delays is considered. Finally, the robust stability of nominally stable uncertain discrete singularly perturbed systems with multiple time delays subject to unstructured and structured perturbations is investigated. Some criteria, delay-dependent or delay-independent, will be proposed to guarantee the robust stability of the uncertain discrete multiple time-delay singularly perturbed systems. The improvements of our results over those in recent literature are also illustrated if the comparisons are possible. Some numerical examples will also be provided to illustrate our main results.

Ce travail de thèse traite de l’analyse de la stabilité et la stabilisation des systèmes non-linéaires représentés par des modèles T-S. L’objectif est de réduire le conservatisme des conditions de stabilité, obtenue par la méthode directe de Lyapunov, et écrites, dans la mesure du possible, sous forme de LMIs. Dans ce cadre, deux contributions principales ont été apportées. Tout d’abord, nous avons proposé de nouvelles conditions de synthèse non-quadratique de lois de commande, strictement LMIs et sans restriction d’ordre, pour les modèles T-S via des FLICs. En effet, dans ce contexte, les résultats de la littérature ne sont valables que pour les modèles T-S d’ordre inférieur ou égal à 2. Afin de lever cette restriction, les conditions ont été obtenues grâce à la démonstration d’une propriété de dualité. Ensuite, peu de travaux traitant de la spécification des performances en boucle fermée, de nouvelles conditions LMIs (quadratiques et non-quadratiques) ont été proposées via le concept de D-stabilité. Dans un premier temps, la synthèse de lois de commande PDC et non-PDC D-stabilisantes a été proposée pour les modèles T-S nominaux. Ensuite, ces résultats ont été étendus au cas des modèles T-S incertains. De plus, nous avons mis en évidence, au travers d’un exemple de D-stabilisation en attitude d’un modèle de drone quadrirotor, que les modèles T-S incertains pouvaient être avantageusement considérés lorsque les non-linéarités d’un modèle non-linéaire dépendent à la fois de l’état et de l’entrée. / This work deals with the stability analysis and the stabilisation of nonlinear systems represented by T-S models.The goal is to reduce the conservatism of the stability conditions, obtained through the direct Lyapunov methodand written, when it is possible, as LMIs. In this framework, two main contributions has been proposed. First ofall, we have proposed some new conditions based on FLICs, strictly LMIs and without any order restrictions, forthe non-quadratic design of control laws devoted to stabilize T-S models. Indeed, in this non-quadratic context,the existing works are only available for 2nd order T-S models. In order to unlock this restriction, the proposed conditions have been obtained based on the proof of a dual property. Then, starting from the fact that few worksdeals with the closed-loop performances specification, some new LMI conditions (quadratic and non-quadratic)have been proposed via the D-stability concept. As a first step, D-stabilizing PDC and non-PDC controller designhas been considered for nominal T-S models. Then, these results have been extended to uncertain T-S models.Moreover, it has been highlighted, from an example of the attitude D-stabilization of a quadrotor model, that wecan make use of uncertain T-S models to cope with nonlinear models involving nonlinearities depending on bothstate and input variables.

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Orientador: Edvaldo Assunção / Resumo: Neste trabalho são propostas metodologias com base na teoria de estabilidade segundo Lyapunov para projetar controladores gain scheduling usando realimentação de estado ou derivativa e o conceito de D-estabilidade. As condições de projeto são dadas por desigualdades matriciais lineares (do inglês, Linear Matrix Inequalities - LMIs). A metodologia é aplicada em sistemas lineares sujeitos a parâmetros variantes no tempo (do inglês, Linear Parameter Varying - LPV). A utilização do Lema de Finsler eliminou a necessidade de inverter uma matriz literal para projetar o ganho de realimentação. Com o objetivo de satisfazer requisitos práticos, foi feito uso da restrição de D-estabilidade no projeto de um controlador para um sistema de suspensão ativa. A implementação prática mostra a eficiência da metodologia proposta. / Abstract: In this work are proposed methologies based on Lyapunov stability theory for designing gain scheduling controller using state-derivative feedback or state feedback and considering D-stability constraint. The design conditions are given by Linear Matrix Inequalities. The methodology is applied on system with time-variant parameter. The use of Finsler’s Lemma eliminated the problem of inverting a symbolic matrix to calculate the feedback gain. The theory of D-stability allowed to get implementable controllers for an active suspension system. The practical implementation showed the efficiency of the proposed methodology. / Mestre

Submitted by MARIO ROLANDO RODRIGUEZ CADALSO null (mario.cadalso@gmail.com) on 2017-07-12T16:08:35Z No. of bitstreams: 1 Tese_Final.pdf: 2137231 bytes, checksum: 15875414251123bb7429e0efb684604c (MD5) / Approved for entry into archive by Monique Sasaki (sayumi_sasaki@hotmail.com) on 2017-07-14T17:05:32Z (GMT) No. of bitstreams: 1 rodriguez_cadalso_mr_me_ilha.pdf: 2137231 bytes, checksum: 15875414251123bb7429e0efb684604c (MD5) / Made available in DSpace on 2017-07-14T17:05:32Z (GMT). No. of bitstreams: 1 rodriguez_cadalso_mr_me_ilha.pdf: 2137231 bytes, checksum: 15875414251123bb7429e0efb684604c (MD5) Previous issue date: 2016-08-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho são propostas metodologias com base na teoria de estabilidade segundo Lyapunov para projetar controladores gain scheduling usando realimentação de estado ou derivativa e o conceito de D-estabilidade. As condições de projeto são dadas por desigualdades matriciais lineares (do inglês, Linear Matrix Inequalities - LMIs). A metodologia é aplicada em sistemas lineares sujeitos a parâmetros variantes no tempo (do inglês, Linear Parameter Varying - LPV). A utilização do Lema de Finsler eliminou a necessidade de inverter uma matriz literal para projetar o ganho de realimentação. Com o objetivo de satisfazer requisitos práticos, foi feito uso da restrição de D-estabilidade no projeto de um controlador para um sistema de suspensão ativa. A implementação prática mostra a eficiência da metodologia proposta. / In this work are proposed methologies based on Lyapunov stability theory for designing gain scheduling controller using state-derivative feedback or state feedback and considering D-stability constraint. The design conditions are given by Linear Matrix Inequalities. The methodology is applied on system with time-variant parameter. The use of Finsler’s Lemma eliminated the problem of inverting a symbolic matrix to calculate the feedback gain. The theory of D-stability allowed to get implementable controllers for an active suspension system. The practical implementation showed the efficiency of the proposed methodology.

La réalisation de tâches de manipulation dextres requiert une complexité aussi bien dans la conception de préhenseur robotique que dans la synthèse de leurs lois de commande. Une optimisation de la mécatronique de ces systèmes permet de répondre aux contraintes d'intégration fonctionnelle en se passant de capteurs de force terminaux. L'utilisation de mécanismes réversibles rend alors possible la détermination du positionnement du système dans l'espace libre et la détection de son interaction avec les objets manipulés, grâce aux mesures proprioceptives inhérentes aux actionneurs électriques. L'objectif de cette thèse est de parvenir synthétiser, dans le contexte articulaire (un degré-de-liberté), une commande adaptée à la manipulation en tenant compte de ces particularités mécaniques. La méthode proposée est basée sur une commande robuste par rapport aux non-linéarités structurelles dues aux effets gravitationnels et aux frottements secs d'une part et par rapport aux rigidités variables des objets manipulés. L'approche choisie nécessite la connaissance précise de la configuration du système étudié à chaque instant. Une représentation dynamique de son comportement permet de synthétiser un capteur logiciel pour l'estimation des grandeurs indispensables à la commande. Ces différentes étapes sont validées par des essais expérimentaux pour justifier la démarche choisie menant à une commande adaptée à la manipulation d'objets. / The realization of dexterous manipulation tasks requires a complexity in robotic hands design as well as in their control laws synthesis. A mecatronical optimization of these systems helps to answer for functional integration constraints by avoiding external force sensors. Back-drivable mechanics allows the free-space positioning determination of such system as far as the detection of its interaction with a manipulated object thanks to proprioceptives measures at electric actuator level. The objective of this thesis is to synthesize a control law adapted to object manipulation by taking into account these mechanical properties in a one degree-of-freedom case. The proposed method is based on a robust control according to structural non-linearities due to gravitational effects and dry frictions on the one hand and with regard to a variable rigidity of manipulated objects on the other hand. The chosen approach requires a precise knowledge of the system configuration at all time. A dynamic representation of its behavior enables a software sensor synthesis for the exteroceptives variables estimation in a control law application purpose. The different steps are experimentally validated in order to justify the chosen approach leading to object manipulation.