Observers are used in the monitoring and control of dynamical systems to deduce the values of unmeasured states. Designing an observer requires having an accurate model of the plant — if the model parameters are characterized imprecisely, the observer may not provide reliable estimates. An interval observer, which comprises an upper and lower observer, bounds the plant's states from above and below, given the range of values of the imprecisely characterized parameters, i.e., it defines an interval in which the plant's states must lie at any given instant. We propose a linear programming-based method of interval observer design for two cases: 1) only the initial conditions of the plant are uncertain; 2) the dynamical parameters are also uncertain. In the former, we optimize the transient performance of the interval observers, in the sense that the volume enclosed by the interval is minimized. In the latter, we optimize the steady state performance of the interval observers, in the sense that the norm of the width of the interval is minimized at steady state. Interval observers are typically designed to characterize the widest interval that bounds the states. This thesis proposes an interval observer design method that utilizes additional, but still-incomplete information, that enables the designer to identify tighter bounds on the uncertain parameters under certain operating conditions. The number of bounds that can be refined defines a class of systems. The definition of this class is independent of the specific parameters whose bounds are refined. Applying robust optimization techniques, under a cardinality constrained model of uncertainty, we design a single observer for an entire class of systems. These observers guarantee a minimum level of performance with respect to the aforementioned metrics, as we optimize the worst-case performance over a given class of systems. The robust formulation allows the designer to tune the level of uncertainty in the model. If many of the uncertain parameter bounds can be refined, the nominal performance of the observer can be improved, however, if few or none of the parameter bounds can be refined, the nominal performance of the observer can be designed to be more conservative.

Observers are used in the monitoring and control of dynamical systems to deduce the values of unmeasured states. Designing an observer requires having an accurate model of the plant — if the model parameters are characterized imprecisely, the observer may not provide reliable estimates. An interval observer, which comprises an upper and lower observer, bounds the plant's states from above and below, given the range of values of the imprecisely characterized parameters, i.e., it defines an interval in which the plant's states must lie at any given instant. We propose a linear programming-based method of interval observer design for two cases: 1) only the initial conditions of the plant are uncertain; 2) the dynamical parameters are also uncertain. In the former, we optimize the transient performance of the interval observers, in the sense that the volume enclosed by the interval is minimized. In the latter, we optimize the steady state performance of the interval observers, in the sense that the norm of the width of the interval is minimized at steady state. Interval observers are typically designed to characterize the widest interval that bounds the states. This thesis proposes an interval observer design method that utilizes additional, but still-incomplete information, that enables the designer to identify tighter bounds on the uncertain parameters under certain operating conditions. The number of bounds that can be refined defines a class of systems. The definition of this class is independent of the specific parameters whose bounds are refined. Applying robust optimization techniques, under a cardinality constrained model of uncertainty, we design a single observer for an entire class of systems. These observers guarantee a minimum level of performance with respect to the aforementioned metrics, as we optimize the worst-case performance over a given class of systems. The robust formulation allows the designer to tune the level of uncertainty in the model. If many of the uncertain parameter bounds can be refined, the nominal performance of the observer can be improved, however, if few or none of the parameter bounds can be refined, the nominal performance of the observer can be designed to be more conservative.

Les travaux de recherche présentés dans ce mémoire portent sur la synthèse d'observateurs intervalles pour la commande tolérante aux fautes de systèmes incertains. La présence de défauts, d'incertitudes et de perturbations peut provoquer des réactions indésirables du système commandé. Dans ce contexte, nous avons développé deux approches de commande tolérante aux fautes basées sur des observateurs intervalles dans le cas où les défauts et les incertitudes sont inconnus mais bornés. La première approche, dite passive, permet de garantir la stabilité du système en boucle fermée y compris en présence de défauts actionneurs et/ou composants. La seconde approche, dite active, permet de compenser l'effet des défauts et d'assurer la stabilité et les performances désirées du système. Ces contributions sont validées par des simulations numériques. / The research work presented in this thesis focuses on the design of interval observers for fault-tolerant control of uncertain systems. The presence of faults, uncertainties and disturbances in automated systems often causes undesirable reactions. In this context, two approaches of fault tolerant control have been developed based on interval observers in the case where the faults and the uncertainties are unknown but bounded. The first approach is passive and consists in ensuring the closed loop system stability even in the presence of actuator and/or component faults. The second approach, an active one, compensates the fault effect and ensures the system stability and desired performances. These contributions are validated through numerical simulations.

La surveillance des systèmes industriels et/ou embarqués constitue une préoccupation majeure en raison de l’accroissement de leur complexité et des exigences sur le respect des profilsde mission. La détection d’anomalies tient une place centrale dans ce contexte. Fondamentalement,les procédures de détection à base de modèles consistent à comparer le fonctionnement réel dusystème avec un fonctionnement de référence établi à l’aide d’un modèle sans défaut. Cependant,les systèmes à surveiller présentent souvent des dynamiques non linéaires et difficiles à caractériserde manière exacte. L’approche retenue dans cette thèse consiste à englober leur influencepar des incertitudes bornées. La propagation de ces incertitudes permet l’évaluation de seuils dedécision visant à assurer le meilleur compromis possible entre sensibilité aux défauts et robustesseaux perturbations tout en préservant une complexité algorithmique raisonnable. Pour cela, unepart importante du travail porte sur l’extension des classes de modèles dynamiques à incertitudesbornées pour lesquels des observateurs intervalles peuvent être obtenus avec les preuves d’inclusionet de stabilité associées. En s’appuyant sur des changements de coordonnées variant dans letemps, des dynamiques LTI, LPV et LTV sont considérées graduellement pour déboucher sur desclasses de dynamiques Non Linéaires à Incertitudes Bornées continues (NL-IB). Une transformationdes modèles NL-IB en modèles LPV-IB a été utilisée. Une première étude sur les non-linéaritésd’une dynamique de vol longitudinal est présentée. Un axe de travail complémentaire porte surune caractérisation explicite de la variabilité (comportement aléatoire) du bruit de mesure dansun contexte à erreurs bornées. En combinant cette approche à base de données avec celle à basede modèle utilisant un prédicteur intervalle, une méthode prometteuse permettant la détection dedéfauts relatifs à la position d’une surface de contrôle d’un avion est proposée. Une étude portenotamment sur la détection du blocage et de l’embarquement d’une gouverne de profondeur. / The monitoring of industrial and/or embedded systems is a major concern accordingto their increasing complexity and requirements to respect the mission profiles. Detection of anomaliesplays a key role in this context. Fundamentally, model-based detection procedures consist incomparing the true operation of the system with a reference established using a fault-free model.However, the monitored systems often feature nonlinear dynamics which are difficult to be exactlycharacterized. The approach considered in this thesis is to enclose their influence through boundeduncertainties. The propagation of these uncertainties allows the evaluation of thresholds aimingat ensuring a good trade-off between sensitivity to faults and robustness with respect to disturbanceswhile maintaining a reasonable computational complexity. To that purpose, an importantpart of the work adresses the extension of classes of dynamic models with bounded uncertaintiesso that interval observers can be obtained with the related inclusion and stability proofs. Based ona time-varying change of coordinates, LTI, LPV and LTV dynamics are gradually considered tofinally deal with some classes classes of nonlinear continuous dynamics with bounded uncertainties.A transformation of such nonlinear models into LPV models with bounded uncertainties has beenused. A first study on nonlinearities involved in longitudinal flight dynamics is presented. A complementarywork deals with an explicit characterization of measurement noise variability (randombehavior of noise within measurement) in a bounded error context. Combining this data-drivenapproach with a model-driven one using an interval predictor, a promising method for the detectionof faults related to the position of aircraft control surfaces is proposed. In this context, specialattention has been paid to the detection of runaway and jamming of an elevator.

Cette thèse traite du problème d'observation et d'estimation des variables caractéristiques des systèmes dynamiques. Il s’agit d’une problématique fondamentale qui est au cœur de nombreux domaines relavant des sciences de l'ingénieur. Les travaux sont conduits dans un contexte ensembliste. Les techniques développées pour l’estimation de l’état et des variables d’entrées ont pour objectif final le contrôle de cohérence des systèmes non linéaires à temps continu. Une première approche conjugue les relations de parité et les différentiateurs à modes glissants pour l’estimation des entrées d’un système non linéaire. Les domaines des entrées compatibles avec les mesures sont alors reconstruits grâce à l’analyse par intervalles et aux techniques de satisfaction de contraintes. Il est montré que la relaxation des contraintes de stabilité/coopérativité pour la construction d’un observateur intervalle peut se faire grâce à des changements de base déterminés de différentes manières et pouvant être variants ou invariants dans le temps. Des simulations numériques illustrent les techniques proposées. Une application à un système aéronautique est également présentée à l’aide d’un jeu de données réelles. / This thesis deals with the problem of a dynamical system observation and the estimation of its characteristic variables; the latter point constitutes the core element in many engineering science fields. The final aim is to build a general framework for integrity control and fault detection of such systems within a bounded error context. The developments offered herein make use of parity relations, sliding mode differentiators, interval observers and constraint satisfaction problems. Input reconstruction techniques are developed for a general class of nonlinear continuous-time systems. Domains are reconstructed for the input values which are consistent with the measurements using interval analysis and constraint satisfaction techniques. It is shown that time-varying or invariant coordinate changes may relax the applicability conditions (stability/cooperativity) of the interval observer design methods. Sliding mode differentiators were also used to enhance interval observer accuracy. The proposed approaches are illustrated through computer simulations and they have been applied to aircraft servo loop control surface for robust and early detection of abnormal positions.