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This Thesis considers Sliding Mode Control (SMC) for linear systems subjected to uncertainties and delays using output feedback. Delay is a natural phenomenon in many practical systems, the effect of delay can be the potential cause -of performance deterioration or even instability. To achieve better control performance, SMC with output feedback is considered for its inherent robustness feature and practicality for implementation. In highlighting the main results, firstly a novel output feedback SMC design is presented which formulates the problem into Linear Matrix Inequalities (LMIs). The efficiency of the design is compared with the the existing literature in pole assignment. eigenstructure assignment and other LMI methods, which either require more constraints on system structures or are computationally less tractable. For systems with timevarying Slate delay, the method is extended to incorporate the delay effect in the controUer synthesis. Both sliding surface and controller design are formulated as LMI problems. For systems with input/output delays and disturbances. the robustness of SMC is degraded with arbitrarily small delay appearing in the high frequency switching component of the controller. To solve the problem singular perturbation method is used to achieve bounded performance which is proportional to the magnitudes of delay, disturbance and switching gain. The applied research has produced two practical implementation studies. Firstly it relates to the pointing control of an autonomous vehicle subjected to external disturbances and friction resulting from the motion of the vehicle crossing rough terrain. The second implementation concerns the attitude control of a flexible spacecraft with respect to roil, pitch and yaw attitude angles.

In recent years there has been an increased interest in working with piece-wise linear caricatures of nonlinear models. Such models are often preferred over more detailed conductance based models for their small number of parameters and low computational overhead. Moreover, their piece-wise linear (PWL) form, allow the construction of action potential shapes in closed form as well as the calculation of phase response curves (PRC). With the inclusion of PWL adaptive currents they can also support bursting behaviour, though remain amenable to mathematical analysis at both the single neuron and network level. In fact, PWL models caricaturing conductance based models such as that of Morris-Lecar or McKean have also been studied for some time now and are known to be mathematically tractable at the network level. In this work we proceed to analyse PWL neuron models of conductance type. In particular we focus on PWL models of the FitzHugh-Nagumo type and describe in detail the mechanism for a canard explosion. This model is further explored at the network level in the presence of gap junction coupling. The study moves to a different area where excitable cells (pancreatic beta-cells) are used to explain insulin secretion phenomena. Here, Ca2+ signals obtained from pancreatic beta-cells of mice are extracted from image data and analysed using signal processing techniques. Both synchrony and functional connectivity analyses are performed. As regards to PWL bursting models we focus on a variant of the adaptive absolute IF model that can support bursting. We investigate the bursting electrical activity of such models with an emphasis on pancreatic beta-cells.

La Programmation Linéaire en Nombres Entiers (PLNE) est couramment utilisée pour modéliser des problèmes d'optimisation du monde industriel, de par la facilité à modéliser des problèmes complexes d'optimisation et par l’existence d’une résolution générique par l'algorithme de Branch&Bound (B&B). La résolution B&B est souvent limitée pour des problèmes de taille réelle, les méthodes heuristiques sont alors utilisées pour trouver des solutions de bonne qualité sans avoir de preuve d'optimalité. Cette thèse étudie les limites de la résolution exacte et des heuristiques sur des problèmes industriels d'EDF, en vue de leur insertion dans le processus décisionnel opérationnel. L'application principale concerne la planification des arrêts de maintenance et de rechargement des centrales nucléaires, sujet du Challenge ROADEF 2010. Nous avons aussi traité un problème de production journalière d'un parc thermique à flammes. La méthodologie suivie est analogue pour les deux cas. On modélise tout d'abord le problème avec une formulation compacte PLNE, pour en analyser les limites de la résolution frontale, avant d’envisager des méthodes de décomposition. On dérive ensuite les méthodes exactes en matheuristiques pour résoudre des instances de taille réelle. Dans cette optique, l'hybridation de Variable Neighborhood Search (VNS) avec des voisinages définis par PLNE a donné des résultats très probants sur les deux problèmes en termes de qualités de solutions. Le fait d'avoir travaillé avec des méthodes exactes a permis également de chiffrer l'impact d'hypothèses de résolutions, de répondre à des considérations opérationnelles, mais également d'obtenir des bornes inférieures. / Mixed Integer Linear Programming (MILP) is a very popular and useful framework to model industrial optimization problems. This success is due to the facility to model complex optimization problems, the work can be focused on modeling, with a black box generic resolution to optimality with Branch&Bound (B&B) algorithm, or with a specialized decomposition algorithm. If MILP made lots of progresses on the last decades, it is often not sufficient to tackle real world size instances. In such cases, heuristic methods are commonly used to find good quality solutions, without any guarantee to reach the optimum and any proven bound to the optimum. Our work focus on two complex optimization problems from energy management. First application is a discretized daily Unit Commitment Problem of thermal units with specific dynamic constraints. Second application comes from the EURO/ROADEF 2010 challenge, scheduling problem of nuclear power plants' outages for maintenances and refueling. In both cases the methodology was first to model efficiently the considered problem with a MILP compact formulation, and analyze the frontal resolution's limits with B&B. Decomposition methods could also be investigated, before the exact methods are derived in a matheuristic, to be able to tackle real size instances. In particular, Variable Neighborhood Search (VNS) with MILP neighborhoods gave outstanding results on our problems. Our work allowed to estimate the impacts of usual and natural hypothesis. Furthermore, we derived dual bounds for these optimizations problems.

Cette thèse est dédiée à l'étude du problème de la stabilisation des systèmes affines à commutations. L'objectif est de concevoir des lois de commutation dépendantes de l'état qui stabilisent le système en boucle fermée. Premièrement, un aperçu de quelque résultat existant dans la littérature est présenté. Ensuite, un résultat général permettant la synthèse de lois de commutations pour la stabilisation des systèmes nonlinéaires affines en l'entré est proposé. La particularisation de ce résultat aux cas des systèmes affines à commutations et des systèmes linéaires à temps invariant avec une commande à relais a permis de synthétiser des lois de commutations garantissant leur stabilité asymptotique locale ou globale en boucle fermée. Grace à l'utilisation des fonctions de Lyapunov commutées une méthode numérique basée sur des LMIs permettant la conception de surfaces de commutations nonlinéaires est proposée. Une méthode permettant la synthèse de lois de commutations robustes vis-à-vis des perturbations sur les mesures est également développée pour assurer la stabilisation des systèmes affines à commutations. Le résultat est ensuite particularisé au cas des systèmes linéaires temps invariant avec commande à relais robuste. Enfin, le problème de la synthèse de lois de commutations basée-observateur est considéré. Des surfaces de commutations linéaires et nonlinéaires sont proposées en utilisant des fonctions de Lyapunov quadratiques et non-quadratiques. Des conditions de stabilisation asymptotique locale et globale sont développées. Les lois de commutations conçues dépendent de l'état reconstruit en utilisant un observateur de type Luenberger. De plus, le principe de séparation est démontré pour les systèmes affines à commutations ainsi que pour les systèmes linéaires temps invariant avec une commande à relais. / This thesis is dedicated to the study of the stabilization problem of switched affine systems with state-dependent switching laws. First, an overview of some existing results is proposed. In order to define the closed-loop system's solutions and to analyze its behavior over the switching surfaces the Filippov formalism is used. The stabilization problem is addressed using a Lyapunov approach which allows to derive numerical approaches based on LMIs. Throughout this thesis both switched affine systems and LTI systems with relay controllers are considered. Using a general framework for the class of nonlinear input-affine systems, a full state-dependent switching controller is designed in order to ensure both local and global asymptotic stability of the closed-loop system. Thanks to switching (Lur'e type) Lyapunov functions, a numerical approach based on LMIs that allows to derive a nonlinear stabilizing switching law is proposed. Moreover, a design approach of robust state-dependent switching laws for switched affine systems stabilization are proposed. The robustness property is studied with respect to bounded exogenous disturbances that affect the state measurements which are used for the design of the switching laws. Finally, observer-based switching controllers are designed to guarantee both local and global asymptotic stability of the closed-loop system. Using both quadratic and non-quadratic Lyapunov functions, linear and nonlinear switching surfaces are designed. The derived switching surfaces depend on the estimated state which is computed by a Luenberger observer. For both switched affine systems and LTI systems with relay controller the separation principle is proved.