[pt] Existe uma crescente demanda por modelos dinâmicos precisos, parte impulsionada pelo paradigma da indústria 4.0 que introduz, dentre outros, o conceito de gêmeo digital no qual modelos dinâmicos possuem um papel importante. Idealmente, um modelo dinâmico apresenta um compromisso entre complexidade e precisão, enquanto proporciona informações sobre a física do sistema. Para melhorar a precisão de um modelo mantendo a interpretabilidade, a abordagem usual é modelar matematicamente todas não-linearidades, o que leva a um modelo muito complexo. Outra abordagem envolve identificação caixa-preta, uma abordagem onde um modelo matemático é ajustado para descrever a relação de entrada e saída do sistema, a qual pode fornecer um modelo preciso, porém não interpretável. Os avanços na capacidade de processamento computacional permitiram o florescimento da area de aprendizado de máquinas, a qual tem mostrado resultados interessantes em diferentes campos do conhecimento. Uma dessas aplicações é em identificação caixa-preta, onde o aprendizado de máquinas tem sido empregado com sucesso na modelagem de sistemas não-lineares, o que tem inspirado pesquisas sobre o tema. Apesar dos modelos baseados em aprendizado de máquina apresentarem elevada precisão, o que é suficiente para diversas aplicações, eles não são interpretáveis. Dessa forma, visando obter modelos que possuem ambas as características de precisão e interpretabilidade, enquanto mantém um compromisso com a complexidade, esta tese propõe uma metodologia de identificação híbrida que combina um modelo fenomenológico caixa cinza com um modelo caixa preta baseado em redes neurais artificiais. A metodologia proposta é aplicada em três casos de estudo de sistemas não lineares com dados experimentais, a saber, a dinâmica vertical de um veículo, um atuador com junta flexível baseado em elastômero e um sistema de posicionamento eletromecânico. Os resultados mostram que o modelo híbrido proposto é até 60 por cento mais preciso enquanto proporciona a interpretabilidade física do sistema, sem aumentar significativamente a complexidade do modelo. / [en] There is a growing demand for accurate dynamic models, driven by the Industry 4.0 paradigm that introduces, among others, the concept of the digital twin in which dynamic models play an important role. Ideally, a dynamic model presents a compromise between complexity and accuracy, while providing physical insight into the system. To improve a model accuracy while keeping interpretability, the usual approach is to mathematically model all the nonlinearities, which ultimately leads to an overcomplex model. Another approach involves a black-box identification, a data-driven approach where a mathematical model is adjusted to describe the system s input-output relation, which may provide an accurate model, but it does not provide interpretability. The developments in computational processing capacity have allowed the flourishing of the field of machine learning, which has shown interesting results in different fields of knowledge. One of these applications is black-box identification, where machine learning has successfully been employed in the modeling of nonlinear systems, which has inspired research on the topic. Even though the machine-learning-based models present enhanced accuracy, which for several applications is sufficient, they do not provide interpretability. Aiming at providing both accuracy and interpretability while keeping a compromise with model complexity, this work proposes a hybrid identification methodology that combines a gray-box phenomenological model with a black-box model based on artificial neural networks. The proposed methodology is applied in three case studies of nonlinear systems with experimental data, namely, the vertical dynamics of a vehicle, an elastomer-based series elastic actuator, and an electromechanical positioning system. The results show that the proposed hybrid model is up to 60 percent more accurate while providing the physical interpretability of the system, without significantly increasing the complexity of the model.

Stability analysis of networked dynamical systems has been of interest in many disciplines such as biology and physics and chemistry with applications such as LASER cooling and plasma stability. These large networks are often modeled to have a completely random (Erdös-Rényi) or semi-random (Small-World) topologies. The former model is often used due to mathematical tractability while the latter has been shown to be a better model for most real life networks. The recent emergence of cyber physical systems, and in particular the smart grid, has given rise to a number of engineering questions regarding the control and optimization of such networks. Some of the these questions are: How can the stability of a random network be characterized in probabilistic terms? Can the effects of network topology and system dynamics be separated? What does it take to control a large random network? Can decentralized (pinning) control be effective? If not, how large does the control network needs to be? How can decentralized or distributed controllers be designed? How the size of control network would scale with the size of networked system? Motivated by these questions, we began by studying the probability of stability of synchronization in random networks of oscillators. We developed a stability condition separating the effects of topology and node dynamics and evaluated bounds on the probability of stability for both Erdös-Rényi (ER) and Small-World (SW) network topology models. We then turned our attention to the more realistic scenario where the dynamics of the nodes and couplings are mismatched. Utilizing the concept of ε-synchronization, we have studied the probability of synchronization and showed that the synchronization error, ε, can be arbitrarily reduced using linear controllers. We have also considered the decentralized approach of pinning control to ensure stability in such complex networks. In the pinning method, decentralized controllers are used to control a fraction of the nodes in the network. This is different from traditional decentralized approaches where all the nodes have their own controllers. While the problem of selecting the minimum number of pinning nodes is known to be NP-hard and grows exponentially with the number of nodes in the network we have devised a suboptimal algorithm to select the pinning nodes which converges linearly with network size. We have also analyzed the effectiveness of the pinning approach for the synchronization of oscillators in the networks with fast switching, where the network links disconnect and reconnect quickly relative to the node dynamics. To address the scaling problem in the design of distributed control networks, we have employed a random control network to stabilize a random plant network. Our results show that for an ER plant network, the control network needs to grow linearly with the size of the plant network.

<p>This thesis considers the problem of fault diagnosis and fault-tolerant control (FTC) for chemical process systems with nonlinear dynamics. The primary objective of fault diagnosis discussed in this work is to identify the failed actuator or sensor by using the information embodied in a process model, as well as input and output data. To this end, an active fault isolation method is first proposed to identify actuator faults and process disturbances by utilizing control action and process nonlinearity. The key idea is to move the process to a region upon fault detection where the effect of each fault can be differentiated from others. The proposed method enables isolation of faults that may not be achievable under nominal operation. This work then investigates the problem of sensor fault isolation by exploiting model-based sensor redundancy through state observer design. Specifically, a high-gain observer is presented and the stability property of the closed-loop system is rigorously established. A method that uses a bank of high-gain observers is then proposed to isolate sensor faults, which explicitly accounts for process nonlinearity, and to continue nominal operation upon fault isolation. In addition to fault diagnosis, this work addresses the problem of handling severe actuator faults using a safe-parking approach and integrating fault diagnosis and safe-parking techniques in a unified fault-handling framework. In particular, several practical issues are considered for the design and implementation of safe-parking techniques, including changes in process dynamics, the network structure of a chemical plant, and actuators frozen at arbitrary positions. The advantage of this approach is that it enables stable process operation under faulty conditions, avoiding the partial or entire shutdown of a chemical plant and resulting economic losses. The efficacy of the proposed fault diagnosis and FTC methods is demonstrated through numerous simulations of chemical process examples.</p> / Doctor of Philosophy (PhD)

[ES] Se puede afirmar que la inmensa mayoría de los fenómenos de la naturaleza estudiados tienen un carácter no lineal. Muchos de estos problemas se pueden modelar utilizando ecuaciones diferenciales no lineales (EDNL) para cuya resolución no existe una amplia colección de herramientas como si podemos encontrar para la resolución de las ecuaciones diferenciales ordinarias. En otros contextos y en este particularmente, los métodos iterativos para la resolución de sistemas no lineales adquieren gran importancia ya que una ecuación diferencial no lineal se puede aproximar numéricamente resolviendo un sistema de ecuaciones no lineales equivalente tras un proceso de discretización. En la presente Tesis Doctoral se proponen dos nuevas clases de métodos iterativos de sexto orden de convergencia basados en funciones peso y se realizan los respectivos análisis de convergencia. El primero de ellos se compara con otros métodos de sexto orden mostrando, aunque formalmente, un mejor rendimiento. En ambos casos se realiza un análisis dinámico donde se indaga la estabilidad de los métodos propuestos en dependencia de la aproximación inicial escogida, estos análisis se hacen empleando polinomios vectoriales de estudio muy simples (polinomios de prueba). Un mal rendimiento con estos últimos, nos aconseja rechazar las aproximaciones iniciales que les han dado lugar. Los métodos propuestos han sido testeados en múltiples experimentos numéricos con problemas académicos y con otros aplicados tales como la resolución numérica de la ecuación de Volterra, la ecuación de Van der Pol y la ecuación de transmisión de calor en un medio no homogéneo, mostrando en todos ellos muy buenos resultados. Finalmente se proponen las líneas de investigación futuras: dos de ellas, basadas en un paradigma determinista, es una continuación directa del trabajo realizado. La otra línea que contempla un paradigma no determinista, se fundamenta en procesos probabilísticos adoptando un método Montecarlo para la resolución de sistemas de ecuaciones no lineales. Una vez diseñado el método Montecarlo en cuestión, se pretende realizar una comparación de rendimiento entre ambos paradigmas. / [CA] Es pot afirmar que la immensa majoria dels fenòmens de la naturalesa estudiats tenen un caràcter no lineal. Molts d'aquests problemes es poden modelar utilitzant equacions diferencials no lineals (EDNL) per a que la seua resoluci ón no existeix una àmplia col·lecció d'eines com si podem trobar per a la resolució de les equacions diferencials ordinàries. En altres contextos i en aquest particularment, els mètodes iteratius per a la resolució de sistemes no lineals adquireixen gran importància ja que una equació diferencial no lineal es pot aproximar numèricament resolent un sistema d'equacions no lineals equivalent després d'un procés de discretización. En la present Tesi Doctoral es proposen dues noves classes de mètodes iteratius de sisé ordre de convergència basats en funcions pese i es realitzen les respectives anàlisis de convergència. El primer d'ells es compara amb altres mètodes de sisé ordre mostrant, encara que formalment, un millor rendiment. En tots dos casos es realitza una anàlisi dinàmica on s'indaga l'estabilitat dels mètodes proposats en dependència de l'aproximació inicial triada, aquestes anàlisis es fan emprant polinomis d'estudi molt simples (polinomis de prova). Un mal rendiment amb aquests últims, ens aconsella rebutjar les aproximacions inicials que els han donat lloc. Els mètodes proposats han sigut testats en múltiples experiments numèrics amb problemes acadèmics i amb altres aplicats com ara la resolució numèrica de l'equació de Volterra, l'equació de Van der Pol i l'equació de transmissió de calor en un mitjà no homogeni, mostrant en tots ells molt bons resultats. Finalment es proposen les línies d'investigació futures: dos d'elles, basat en un paradigma determinista, és una continuació directa del treball realitzat. L'altra línia que contempla un paradigma no determinista, es fonamenta en processos probabilístics adoptant un mètode Montecarlo per a la resolució de sistemes d'equacions no lineals. Una vegada dissenyat el mètode Montecarlo en qüestió, es pretén realitzar una comparació de rendiment entre tots dos paradigmes. / [EN] It can be argued that the vast majority of natural phenomena studied are nonlinear in nature. Many of these problems can be modeled using nonlinear dierential equations (NLDEs) for the solution of which there is no large collection of tools as we can nd for the solution of nonlinear dierential equations. There is not a large collection of tools for solving NDEs as there is for solving ordinary dierential equations dierential equations. In other contexts and in this one in particular, iterative methods for solving nonlinear systems are of great importance. nonlinear systems acquire great importance since a nonlinear dierential equation can be approximated numerically by solving a system of nonlinear equations. by solving an equivalent system of nonlinear equations after a discretization process. In this Doctoral Thesis, two new classes of sixth order iterative methods of convergence based on weight functions are proposed and the respective convergence analyses are performed. The rst one is compared with other sixth order methods showing a better performance. In both cases, a dynamic analysis where the stability of the proposed methods is investigated in dependence of the initial approximation chosen. These analyses are performed using very simple study polynomials (test polynomials). A bad The proposed methods have been tested for their stability in dependence on the initial approximation chosen. The proposed methods have been tested in multiple numerical experiments with academic problems and with other applied problems such as the numerical solution of the Volterra equation, the Van der Pol equation and the heat transfer equation in an inhomogeneous medium, showing very good results in all of them. Finally, future lines of research are proposed: two of them, based on a deterministic paradigm, are a direct continuation of the work carried out. The other line, which contemplates a non-deterministic paradigm, is based on probabilistic processes adopting a Monte Carlo method for the resolution of systems of nonlinear equations. Once the Monte Carlo method has been designed, a performance comparison between both paradigms will be made. / Capdevila Brown, RR. (2023). Procesos iterativos para la resolución de sistemas no lineales con amplios conjuntos de estimaciones iniciales convergentes [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/201560

Les observateurs et la régulation de sortie sont deux thèmes centraux de la théorie des système non linéaires. Bien que de nombreux chercheurs ont consacré leur attention à ces questions depuis plus de trente ans, il y a encore de nombreuses questions ouvertes. Dans la théorie des observateurs un rôle clé est joué par les observateurs à grand gain. Le but de la première partie de la thèse est d'etudier nouvelles techniques qui permettent de surmonter ou au moins d'atténuer les principaux problèmes qui caractérisent cette classe d'observateurs. Nous proposons une nouvelle classe d'observateurs à grand gain, appelé "low-power", qui permet de surmonter les problèmes numériques, d'éviter le phénomène de peaking et d'améliorer les propriétés de sensibilité aux bruit de mesure à haute fréquence. La deuxième partie de la thèse aborde du problème de la régulation de sortie, qui a été résolu pour les systèmes linéaires au cours des années 70, par Francis et Wonham qui ont énoncé le célèbre «principe de modèle interne". Des solutions constructives ont aussi été proposées dans le cadre non linéaire mais sous des hypothèses restrictives qui réduisent la classe des systèmes auxquels cette méthodologie peut être appliquée. Dans la thèse, nous nous concentrons sur le problème de la régulation de sortie en présence de perturbations périodiques, et nous proposons une nouvelle approche qui nous permet de considérer une classe plus large de systèmes non linéaires. La technique obtenué est robuste au sens défini par Francis et Wonham. / Observers and output regulation are two central topics in nonlinear control system theory. Although many researchers have devoted their attention to these issues for more than 30 years, there are still many open questions. In the observer theory a key role is played by the so called high-gain observers. The purpose of the first part of the thesis is to study novel techniques which allow to overcome or at least to mitigate some of the main drawbacks characterizing this class of observers. We propose a novel class of high-gain observers, denoted as ``low-power'', which allows to overcome numerical problems, to avoid the peaking phenomenon and to improve the sensitivity properties to high-frequency measurement noise. The second part of the thesis addresses the output regulation problem, solved for linear systems during the 70's by Francis and Wonham who coined the celebrated ``internal model principle''. Constructive solutions have also been proposed in the nonlinear framework but under restrictive assumptions that reduce the class of systems to which this methodology can be applied. In this thesis we focus on the output regulation problem in presence of periodic disturbances and we propose a novel approach which allows to consider a broader class of nonlinear systems. The resulting design is robust in the sense defined by Francis and Wonham.

Cette thèse concerne la conception de commandes échantillonnées pour les systèmes non-linéaires en temps continu. Les systèmes échantillonnés sont des éléments inhérents aux systèmes contrôlés par ordinateur, les systèmes hybrides ou les systèmes embarqués. La conception et le calcul des contrôleurs numériques appropriés sont des taches difficiles car ils contiennent des composants à la fois continu et en temps discret. Ce travail s'inscrit dans une activité de recherche menée par S. Monaco et D. Normand-Cyrot dans le domaine des systèmes échantillonnés non-linéaires. L'idée de base est de concevoir des contrôleurs digitaux qui permettent de récupérer certaines propriétés en temps continu qui sont généralement dégradées par l'échantillonnage. Tel est le cas de l'émulation lorsque les contrôleurs en temps continu sont mis en ouvre en utilisant des bloqueurs d'ordre zéro. Cette thèse apporte des contributions dans trois directions complémentaires. La première concerne les développements théoriques: une nouvelle conception de type ``backstepping digital" est proposée pour les systèmes en forme ``strict-feedback". Cette méthode est comparée à d'autres stratégies proposées dans la littérature. La deuxième contribution est le développement d'un logiciel pour la synthèse des contrôleurs et d'une ``boîte à outils" pour simuler (en Matlab) les systèmes échantillonnés non-linéaires et leurs contrôleurs. Cette boîte à outils inclut plusieurs algorithmes pour la synthèse de contrôleurs échantillonnés tels que: commande de type multi-échelle, reproduction entrée-sortie/Lyapunov, backstepping digital, etc. La troisième contribution concerne plusieurs études de cas menées pour mettre en évidence les performances des contrôleurs échantillonnés, calculés avec l'aide du logiciel. Des résultats expérimentaux et des simulations sont décrits pour divers exemples réels dans les domaines électriques et mécaniques. / This thesis is concerned with the sampled-data control of non-linear continuous-time systems. Sampled-data systems are present in all computer controlled, hybrid or embedded systems. The design and computation of suitable digital controllers represent unavoidable tasks since both continuous and discrete-time components interact. The basic framework of this work takes part of a wide research activity performed by S. Monaco and D. Normand-Cyrot regarding non-linear sampled-data systems. The underlying idea is to design digital controllers that recover certain continuous-time properties that are usually degraded through sampling as it is the case when continuous-time controllers are implemented by means of zero-order holder devices (emulated control). This thesis brings contributions into three different directions. The first one regards theoretical developments: a new digital backstepping-like strategy design for strict-feedback systems is proposed. This method is compared with other strategies proposed in the literature. The second contribution is the development of a control designer and of a simulation toolbox (in Matlab) for non-linear sampled-data systems. This toolbox includes different digital design strategies such as: multi-rate control, input-output/Lyapunov matching, digital backstepping design, etc. The third contribution concerns several case studies conducted to highlight the performances of the sampled-data controller designs, computed by the means of the software toolbox. Experimental and simulation results are described for various real examples especially in the area of electrical and mechanical processes. / Teza de față se concentrează asupra studiului controlului eșantionat pentru sisteme neliniare în timp continuu. Sistemele eșantionate sunt componente indispensabile oricăror sisteme de control bazate pe dispozitive de calcul, sisteme hibride sau sisteme embedded. Sinteza și calculul comenzilor digitale, potrivite pentru astfel de sisteme, devine o sarcină dificilă o dată ce presupune existența de dinamici în timp discret respectiv în timp continuu.Cadrul de bază al acestei lucrări se regăsește în activitatea de cercetare realizată de Salvatore Monaco și Dorothée Normand-Cyrot în domeniul sistemelor eșantionate neliniare. Ideea care stă la bază este de a sintetiza comenzile digitale urmărind menținerea unor proprietăți impuse în timp continuu sub eșantionare. Aceste proprietăți sunt în general degradate sub eșantionare cum este cazul comenzilor emulate, când comenzile continue sunt implementate practic cu ajutorul extrapolatoarelor de ordin 0.Această teză își aduce aportul în 3 direcții complementare. Prima adresează dezvoltările teoretice unde o nouă sinteză de tip backstepping digital este propusă pentru sisteme în formă <strict-feedback>. Această metodă, dezvoltată în două versiuni, este comparată cu alte strategii similare propuse în literatură. A doua contribuție a tezei este legată de dezvoltarea unui toolbox software pentru sinteza de controllere digitale pentru sisteme nelinare sub eșantionare. Acest toolbox include strategii diferite pentru sinteza eșantionată precum: comandă de tip multi-rate, reproducere intrare-ieșire/Lyapunov, backstepping digital și alte soluții care sunt obiectul unor noi extensii. A treia contribuție este dată de studiile de caz dezvoltate pentru a scoate în evidență performanțele comenzilor eșantionate testate și calculate cu ajutorul aplicației software. Rezultatele experimentale și de simulare sunt obținute pentru diverse exemple reale din domeniul electric și mecanic.

Lorsqu’on fait face à des systèmes non linéaires, les notions classiques de stabilité ne suffisent pas à garantir un comportement approprié vis-à-vis de problématiques telles que le suivi de trajectoires, la synchronisation et la conception d’observateurs. La stabilité incrémentale a été proposée en tant qu’outil permettant de traiter de tels problèmes et de garantir que le système présente des comportements qualitatifs pertinents. Cependant, comme c’est souvent le cas avec les systèmes non linéaires, la complexité de l’analyse conduit les ingénieurs à rechercher des relaxations, ce qui introduit du conservatisme. Dans cette thèse, nous nous intéressons à la stabilité incrémentale d’une classe spécifique de systèmes, à savoir les systèmes affines par morceaux, qui pourraient fournir un outil avantageux pour aborder la stabilité incrémentale de systèmes dynamiques plus génériques.Les systèmes affines par morceaux ont un espace d’états partitionné, et sa dynamique dans chaque région est régie par une équation différentielle affine. Ils peuvent représenter des systèmes contenant des non linéarités affines par morceaux, ainsi que servir comme des approximations de systèmes non linéaires plus génériques. Ce qui est plus important, leur description est relativement proche de celle des systèmes linéaires, ce qui permet d’obtenir des conditions d’analyse exprimées comme des inégalités matricielles linéaires qui peuvent être traités numériquement de façon efficace par des solveurs existants.Dans la première partie de ce document de thèse, nous passons en revue la littérature sur l’analyse des systèmes affines par morceaux en utilisant des techniques de Lyapunov et la dissipativité. Nous proposons ensuite de nouvelles conditions pour l’analyse du gain L2 incrémental et la stabilité asymptotique incrémentale des systèmes affines par morceaux exprimés en tant qu’inégalités matricielles linéaires. Ces conditions sont montrées être moins conservatives que les résultats précédents et sont illustrées par des exemples numériques.Dans la deuxième partie, nous considérons le cas des systèmes affines par morceaux incertains représentés comme l’interconnexion entre un système nominal et un bloc d’incertitude structuré. En utilisant la théorie de la séparation des graphes, nous proposons des conditions qui étendent le cadre des contraintes quadratiques intégrales afin de considérer le cas où le système nominal est affine par morceaux, à la fois dans les cas non incrémental et incrémental. Via la théorie de la dissipativité, ces conditions sont ensuite exprimées en tant qu’inégalités matricielles linéaires.Finalement, la troisième partie de ce document de thèse est consacrée à l’analyse de systèmes non linéaires de Lur’e incertains. Nous développons une nouvelle technique d’approximation permettant de réécrire ces systèmes de façon équivalente comme des systèmes affines par morceaux incertains connectés avec l’erreur d’approximation. L’approche proposée garantit que l’erreur d’approximation est Lipschitz continue avec la garantie d’une borne supérieure prédéterminée sur la constante de Lipschitz. Cela nous permet d’utiliser les techniques susmentionnées pour analyser des classes plus génériques de systèmes non linéaires. / When dealing with nonlinear systems, regular notions of stability are not enough to ensure an appropriate behavior when dealing with problems such as tracking, synchronization and observer design. Incremental stability has been proposed as a tool to deal with such problems and ensure that the system presents relevant qualitative behavior. However, as it is often the case with nonlinear systems, the complexity of the analysis leads engineers to search for relaxations, which introduce conservatism. In this thesis, we focus on the incremental stability of a specific class of systems, namely piecewise-affine systems, which could provide a valuable tool for approaching the incremental stability of more general dynamical systems.Piecewise-affine systems have a partitioned state space, in each region of which the dynamics are governed by an affine differential equation. They can represent systems containing piecewise-affine nonlinearities, as well as serve as approximations of more general nonlinear systems. More importantly, their description is relatively close to that of linear systems, allowing us to obtain analysis conditions expressed as linear matrix inequalities that can be efficiently handled numerically by existing solvers.In the first part of this memoir, we review the literature on the analysis of piecewise-affine systems using Lyapunov and dissipativity techniques. We then propose new conditions for the analysis of incremental L2-gain and incremental asymptotic stability of piecewise-affine systems expressed as linear matrix inequalities. These conditions are shown to be less conservative than previous results and illustrated through numerical examples.In the second part, we consider the case of uncertain piecewise-affine systems represented as the interconnection between a nominal system and a structured uncertainty block. Using graph separation theory, we propose conditions that extend the framework of integral quadratic constraints to consider the case when the nominal system is piecewise affine, both in the non-incremental and incremental cases. Through dissipativity theory, these conditions are then expressed as linear matrix inequalities.Finally, the third part of this memoir is devoted to the analysis of uncertain Lur’e-type nonlinear systems. We develop a new approximation technique allowing to equivalently rewrite such systems as uncertain piecewise-affine systems connected with the approximation error. The proposed approach ensures that the approximation error is Lipschitz continuous with a guaranteed pre-specified upper bound on the Lipschitz constant. This enables us to use the aforementioned techniques to analyze more general classes of nonlinear systems.

Estimar a região de estabilidade de um ponto de equilíbrio assintoticamente estável é importante em aplicações tais como sistemas de potência, economia e ecologia. A compreensão da estrutura qualitativa da fronteira da região de estabilidade é fundamental para estimar com eficiência a região de estabilidade. Caracterizações topológicas e dinâmicas da fronteira da região de estabilidade foram desenvolvidas ao longo das últimas décadas. Estas caracterizações foram desenvolvidas sob hipóteses de hiperbolicidade dos pontos de equilíbrio na fronteira e transversalidade. Para sistemas que dependem de parâmetros, a condição de hiperbolicidade pode ser violada em pontos de bifurcações. Estaremos interessados em estimar a região de estabilidade, para sistemas sujeitos a variações de parâmetros, onde ocorre a violação da condição de hiperbolicidade dos pontos de equilíbrio na fronteira da região de estabilidade devido ao aparecimento de uma bifurcação sela-nó do tipo zero nesta fronteira. Apresentaremos neste trabalho uma caracterização completa da fronteira da região de estabilidade na presença de um ponto de equilíbrio não hiperbólico sela-nó do tipo zero. Motivados também em oferecer um algoritmo conceitual para obter estimativas da região de estabilidade perturbada via conjunto de nível de uma dada função energia na vizinhança de um parâmetro de bifurcação sela-nó do tipo zero, buscaremos exibir resultados que permitam compreender o comportamento da região de estabilidade e de sua fronteira sob a influência das variações do parâmetro, incluindo variações do parâmetro próximo a um parâmetro de bifurcação sela-nó do tipo zero. / Estimating the stability region of an asymptotically stable equilibrium point is fundamental in applications such as power systems, economy and ecology. The knowledge of the qualitative structure of the stability boundary is essential to estimate with efficiency the stability region. Topological and dynamical characterizations of the stability boundary have been developed over the past decades. These characterizations were developed under assumptions of hyperbolicity of equilibrium points on the stability boundary and transversality. For systems that depend on parameters, the condition of hyperbolicity can be violated at points of bifurcations. We will be primarily interested in estimating the stability region, for systems subjected to parameter variations, when the condition of hyperbolicity of equilibrium points on the stability boundary is violated due to the appearance of a type-zero saddle-node bifurcation on the stability boundary. We will develop in this work, a complete characterization of the stability boundary in the presence of a type-zero saddle-node non-hyperbolic equilibrium point. Also, motivated to providing a conceptual algorithm to obtain estimates of the perturbed stability region via level sets of a given energy function in the neighborhood of a type-zero saddle-node bifurcation parameter, we offer results that explain the behavior of the stability region and its boundary under the influence of parameter variations, including variations of the parameter close to a type-zero saddle-node bifurcation parameter.

Existem métodos de integração de equações algébrico diferenciais não lineares (DAEs) considerados clássicos pela literatura. Porém, neste trabalho, através uma abordagem geométrica, apresenta-se um método de integração de DAEs. Tal método é inspirado na teoria de desacoplamento de sistemas não lineares explícitos, quando se considera que as saídas são restrições algébricas. Neste caso, a DAE pode ser identificada como dinâmica zero. O resultado principal desta abordagem é que, dada uma DAE, sob certas condições, é possível a construção de um sistema explícito, de tal maneira, que as soluções desse sistema explícito convergem para as soluções da DAE. / Classical methods for numerical integration of diferential algebraic equations (DAEs) can be formal in the literature. In this work, using a diferential geometric approach, a numerical method of integration of DAEs is established. This method is inspired in the decoupling theory of nonlinear explicit systems, when one considers that the outputs are algebraic constraints. The main result is the construction of an explicit system, whose solutions converge to the solutions of the DAE.

Cette thèse porte sur la représentation T-S des systèmes non linéaires et les non-linéarités qui leur sont associées (saturation et paramètres variants dans le temps) pour la commande et le diagnostic. Ainsi, une nouvelle approche utilisant la transformation par secteurs non linéaires permet de ré-écrire le système sous forme polytopique en prenant en compte la présence de paramètres variants dans le temps. Cette forme polytopique est ensuite utile pour la synthèse d'observateurs assurant l'estimation simultanée de l'état et des paramètres du système. Une application au diagnostic est également considérée en comparant les valeurs des paramètres estimés en ligne avec leurs valeurs nominales supposées connues et représentatives du mode de fonctionnement non défaillant. Concernant la commande, la contrainte de saturation est représentée sous forme de modèle T-S et est intégrée au modèle du système. La synthèse de plusieurs lois de commande assurant la stabilité du système bouclé, en prenant en compte les limites de saturation est proposée. La poursuite de modèle de référence est également traitée avec la mise en évidence des conditions structurelles de poursuite pour les modèles non linéaires sous forme T-S. L'accent est mis sur les différents critères de choix de commande en fonction des buts recherchés / A first contribution of this thesis is to propose a systematic procedure to deal with the state and parameter estimation for nonlinear time-varying systems. It consists in transforming the original system into a T-S model with unmeasurable premise variables using the sector nonlinearity transformation. Then a joint state and parameter observer is designed and the convergence conditions of the joint state and parameter estimation errors are established. The second contribution of this thesis is the stabilization problem under saturation constraints. In fact, we aim to present a new approach for the saturation nonlinearity study, where the sector nonlinearity transformation is used to represent the nonlinear behaviour of a saturated actuator under a T-S form. The T-S representation of the saturation is used to integrate the limitation constraints into the control synthesis, such that the system stability is ensured and the controller gains are calculated according to the saturation level. The model reference tracking control problem is also addressed. It aims to highlight the encoutered difficulties and the proposed solutions to achieve the tracking objective. Through analytical studies, by presenting clear structural conditions and control strategies, we try to point and answer some major interogations, that are : "how the model reference is to be chosen ? " and "which tracking criterion to consider to achieve a certain objective ? ". The case of constrained control input is also considered with a special focus on the anticipation for the saturated control