Increasing dynamics in power systems on account of renewable integration, electric vehicle penetration and rising demands have resulted in the exploration of energy storage for potential solutions. Recent technology- and industry-driven developments have led to a drastic decrease in costs of these storages, further advocating their usage. This thesis compiles the author's research on optimal integration of energy storage. Unpredictability is modelled using random variables favouring the need of stochastic optimization algorithms such as Lyapunov optimization and stochastic approximation. Moreover, consumer interactions in a competitive environment implore the need of topics from game theory. The concept of Nash equilibrium is introduced and methods to identify such equilibrium points are laid down. Utilizing these notions, two research contributions are made. Firstly, a strategy for controlling heterogeneous energy storage units operating at different timescales is put forth. They strategy is consequently employed optimally for arbitrage in an electricity market consisting of day-ahead and real-time pricing. Secondly, energy storages owned by consumers connected to different nodes of a power distribution grid are coordinated in a competitive market. A generalized Nash equilibrium problem is formulated for their participation in arbitrage and energy balancing, which is then solved using a novel emph{weighted} Lyapunov approach. In both cases, we design real-time algorithms with provable suboptimality guarantees in terms of the original centralized and equilibrium problems. The algorithms are tested on realistic scenarios comprising of actual data from electricity markets corroborating the analytical findings. / Master of Science / Modern power system, which is responsible for generation and transport of electricity, is witnessing a lot of changes such as the increased adoption of wind and solar energy, promotion of electric vehicles, and ever increasing consumer demands. Amidst such developments, energy storage devices like batteries are being propagated as a necessary addition to the power system for its safe operation. This has been further supported by the decrease in prices of these devices over time. An effective assimilation of energy storage however, requires extensive mathematical studies on account of unpredictable renewable generation and consumer demands.This thesis focuses upon the preceding concern. To this note, two novel research contributions are made. In the first, an individual consumer is modeled who wishes to reduce his/her energy costs by simultaneously employing energy storages belonging to different technologies. In the latter, a more challenging multi-consumer interaction is reviewed where multiple end consumers wish to reduce costs while competing against each other over limited resources. In either of the cases, efficient algorithms are designed that are shown to produce desirable results over real-life data and have mathematically provable performance guarantees.

There are still many open questions regarding spatiotemporal chaos although many well developed theories exist for chaos in time. Rayleigh-B'enard convection is a paradigmatic example of spatiotemporal chaos that is also experimentally accessible. Discoveries uncovered using numerics can often be compared with experiments which can provide new physical insights. Lyapunov diagnostics can provide important information about the dynamics of small perturbations for chaotic systems. Covariant Lyapunov vectors reveal the true direction of perturbation growth and decay. The degree of hyperbolicity can also be quantified by the covariant Lyapunov vectors. To know whether a dynamical system is hyperbolic is important for the development of a theoretical understanding. In this thesis, the degree of hyperbolicity is calculated for chaotic Rayleigh-B'enard convection. For the values of the Rayleigh number explored, it is shown that the dynamics are non-hyperbolic. The spatial distribution of the covariant Lyapunov vectors is different for the different Lyapunov vectors. Localization is used to quantify this variation. The spatial localization of the covariant Lyapunov vectors has a decreasing trend as the order of the Lyapunov vector increases. The spatial localization of the covariant Lyapunov vectors are found to be related to the instantaneous Lyapunov exponents. The correlation is stronger as the order of the Lyapunov vector decreases. The covariant Lyapunov vectors are also computed using a spectral element approach. This allows an exploration of the covariant Lyapunov vectors in larger domains and for experimental conditions. The finite conductivity and finite thickness of the lateral boundaries of an experimental convection domain is also studied. Results are presented for the variation of the Nusselt number and fractal dimension for different boundary conditions. The fractal dimension changes dramatically with the variation of the finite conductivity. / Ph. D. / There are still many open questions regarding chaos. Rayleigh-Bènard convection is a type of natural convection which occurs when a fluid is placed between a hot bottom plate and a cold top plate. Rayleigh-Bènard convection is a classical model to explore chaos in space and time. The major application of Rayleigh-Bènard convection is weather prediction which is an extremely difficult problem of intense interest. The governing equations can only be solved using supercomputing resources. The main reason for this difficulty is the presence of a very large number of degrees of freedom that may influence the weather. To reduce the number of degrees of freedom by only including ones that contribute significantly is a difficult problem. In this thesis, vectors describing the growth of disturbances have been calculated for Rayleigh-Bènard convection. These vectors give us information about which regions in space are more important than others. For weather example, the knowledge of these vectors would tell us which regions are important. With this information, scientists and engineers can focus on the important regions and possibly improve their long term predictions. These vectors also yield the number of degrees of freedom to characterize a chaotic system, on average. In this thesis, this number is also explored for Rayleigh-Bènard convection. This thesis extends the calculation of these vectors to a realistic fluid model which gives us new insights into fundamental questions about chaos in space and time.

Neste trabalho apresentamos um assunto que tem contribuído em diversas áreas, o conhecido Problemas de Lurie. Para exemplificar sua aplicabilidade estudamos a Rede Neural de Hopfield e a relacionamos com o problema. Alguns teoremas são apresentados e um dos resultados do Problema de Lurie é aplicado ao modelo de Hopfield. / In the present work we show some properties of the so called Luries type equation. We treat particularly the stability conditions problem, and show how this theory is applied in a Hopfield neural network.

Este trabalho trata das medidas de máxima entropia para certos difeomorfismos em nilvariedades. Considere um difeomorfismo parcialmente hiperbólico f definido em T3, dinamicamente coerente com folheação central compacta. Suponha ainda que a aplicação induzida por f no espaço das folhas centrais é um homeomorfismo de Anosov transitivo em T2. Mostramos que o conjunto das medidas ergódicas hiperbólicas de máxima entropia é enumerável. Usando o princípio de invariância, mostramos que se o primeiro retorno de f à alguma folha periódica tem número de rotação irracional, então, f tem no máximo duas medidas ergódicas de máxima entropia e ter apenas uma medida de máxima entropia equivale a ser extensão de rotação. Se a aplicação de primeiro retorno à alguma folha central periódica é Morse-Smale, então existe um su-toro periódico, ou temos uma cota superior para o número de medidas ergódicas de máxima entropia que depende do número de atratores da dinâmica nessa folha. Além disso, estudamos a topologia da bacia das medidas ergódicas de máxima entropia para uma outra classe de difeomorfismos especiais que são genéricos no espaço dos difeomorfismos absolutamente parcialmente hiperbólicos e denotada por SPH1(M). / This work is about maximal entropy measures for certain diffeomorphisms on nilmanifolds. Consider a partially hyperbolic diffeomorphism f on T3 , C2 , dinamically coherent with compact center foliation which is a circle bundle. Assume that the map induced by f on the space of center leaves is a transitive Anosov homeomorphism. We show that the set of hyperbolic ergodic maximal entropy measures of f is countable. Using the invariance principle, we show that if the first return map to some periodic leaf has irrational rotation number then f has at most two ergodic maximal entropy measures and, in this case, if f has only one maximal entropy measure then f is a rotation extension. If the first return map to some periodic leaf is Morse-Smale then either there exists some periodic su-torus or an upper bound for the number of ergodic maximal entropy measure depending on the number of the attractors of the dynamics in this leaf. Moreover, we study the topology of basin of ergodic maximal entropy measures of another set of special diffeomorphisms that are generic in the space of absolutely partially hyperbolic systems and denoted by SPH1(M).

Neste trabalho nós estudamos os expoentes de Lyapunov de aplicações f : Td → Td homotópicas a uma aplicação Anosov linear e a continuidade absoluta de folheações. Nós mostramos para algumas classes de homotopia de aplicações que a soma dos expoentes de Lyapunov está limitado pela soma dos expoentes de Lyapunov da aplicação Anosov linear. Além disso, admitindo uma propriedade conhecida como densidade uniformemente limitada (UBD) nas folheações, mostramos uma igualdade entre a soma dos expoentes de Lyapunov de f e do Anosov linear. Também construímos um conjunto C1 aberto de difeomorfismos parcialmente hiperbólicos do toro T4, preservando volume, com folheação central bidimensional não compacta e não absolutamente contínua. Ainda construímos um exemplo parcialmente hiperbólico com folhas centrais bidimensionais, não compactas onde a desintegração do volume ao longo da folheação central não é nem Lebesgue nem atômica. / In this work we study the Lyapunov exponents of maps f : Td → Td homotopic to a linear Anosov map. We proof for some homotopic classes of maps which the sum of Lyapunov exponents is bounded by the sum of the Lyapunov exponents of the linear Anosov map. Moreover, by assuming a property known as uniformly bounded density (UBD) in the foliations, we show an equality between the sum of the Lyapunov exponents of f and the linear Anosov. We also construct an C1 open set of volume preserving partially hyperbolic diffeomorphisms with non compact two dimensional center foliation and non absolutely continuous. We still build an example of partially hyperbolic diffeomorphism with non compact bidimensional center leaves where the disintegration of volume along the center foliation is neither Lebesgue nor atomic.

Les travaux de cette thèse portent sur la stabilisation des systèmes affines commutés. Ces systèmes appartiennent à la classe des systèmes dynamiques hybrides. Ils possèdent de plus la particularité d'avoir des points de fonctionnement non auto-maintenables : il n'existe pas de loi de commutations permettant de maintenir l'état du système en ce point. De ce fait, la stabilisation de ces systèmes en imposant à la loi de commutations une durée minimale entre chaque commutation aboutit à une convergence des trajectoires dans une région de l'espace d'état. Après avoir synthétisé différentes stratégies de commutations échantillonnées construites à partir d'une fonction de commande de Lyapunov en temps continu, nous cherchons à déterminer la région de l'espace dans laquelle converge asymptotiquement l'ensemble des trajectoires du système. Par la résolution d'un problème d'optimisation, une estimation de la taille de cette région est donnée et un lien avec les incertitudes du système y est établi. Un second problème de stabilisation est étudié dans cette thèse, en considérant une stratégie de commande basée observateur par retour de sortie. Cependant, du fait de la nature hybride du système, son observabilité est directement liée à la séquence de commutations. Il est alors nécessaire de garantir à la fois l'observabilité, par une condition algébrique, et la convergence du système vers un point de fonctionnement, par l'existence d'une fonction de commande de Lyapunov / This PhD thesis deals with the stabilization of switched affine systems. These systems belong to the class of hybrid dynamical systems. They exhibit a particular behavior: no switching law exists such that the state can be maintained on a chosen operating point. Hence, assuming a dwell time condition on switchings exists, the stabilization of these systems leads to a convergence of the trajectories to a region of the state space. Based on a control Lyapunov function in continuous time, we synthesize several sampled-data switching strategies. The whole trajectories asymptotically converge to a region which we attempt to determine. Solving an optimization problem, an estimation of the size of this region is given. A link with the system uncertainties is also established. This PhD thesis is dedicated to a second stabilization issue: observer-based output-feedback synthesis. By its hybrid nature, the observability of the system is connected to the switching sequence. Therefore, the synthesis of the switching strategy must respect an observability condition and guarantee the convergence to the operating point. The observability is achieved thanks to an algebraic condition. The convergence property is based on the existence of a control Lyapunov function

Este trabalho trata das medidas de máxima entropia para certos difeomorfismos em nilvariedades. Considere um difeomorfismo parcialmente hiperbólico f definido em T3, dinamicamente coerente com folheação central compacta. Suponha ainda que a aplicação induzida por f no espaço das folhas centrais é um homeomorfismo de Anosov transitivo em T2. Mostramos que o conjunto das medidas ergódicas hiperbólicas de máxima entropia é enumerável. Usando o princípio de invariância, mostramos que se o primeiro retorno de f à alguma folha periódica tem número de rotação irracional, então, f tem no máximo duas medidas ergódicas de máxima entropia e ter apenas uma medida de máxima entropia equivale a ser extensão de rotação. Se a aplicação de primeiro retorno à alguma folha central periódica é Morse-Smale, então existe um su-toro periódico, ou temos uma cota superior para o número de medidas ergódicas de máxima entropia que depende do número de atratores da dinâmica nessa folha. Além disso, estudamos a topologia da bacia das medidas ergódicas de máxima entropia para uma outra classe de difeomorfismos especiais que são genéricos no espaço dos difeomorfismos absolutamente parcialmente hiperbólicos e denotada por SPH1(M). / This work is about maximal entropy measures for certain diffeomorphisms on nilmanifolds. Consider a partially hyperbolic diffeomorphism f on T3 , C2 , dinamically coherent with compact center foliation which is a circle bundle. Assume that the map induced by f on the space of center leaves is a transitive Anosov homeomorphism. We show that the set of hyperbolic ergodic maximal entropy measures of f is countable. Using the invariance principle, we show that if the first return map to some periodic leaf has irrational rotation number then f has at most two ergodic maximal entropy measures and, in this case, if f has only one maximal entropy measure then f is a rotation extension. If the first return map to some periodic leaf is Morse-Smale then either there exists some periodic su-torus or an upper bound for the number of ergodic maximal entropy measure depending on the number of the attractors of the dynamics in this leaf. Moreover, we study the topology of basin of ergodic maximal entropy measures of another set of special diffeomorphisms that are generic in the space of absolutely partially hyperbolic systems and denoted by SPH1(M).

A análise de estabilidade de sistemas não-lineares surge em vários campos da engenharia. Geralmente, esta análise consiste na determinação de conjuntos atratores estáveis e suas respectivas áreas de atração. Os métodos baseados no método de Lyapunov fornecem estimativas destes conjuntos. Entretanto, estes métodos envolvem uma busca não sistemática por funções auxiliares chamadas funções de Lyapunov. Este trabalho apresenta um procedimento sistemático, baseado no método de Lyapunov, para estimar conjuntos atratores e as respectivas áreas de atração para uma classe de sistemas não-lineares, aqui chamado de problema de Lure estendido. Este problema consiste de sistemas não-lineares que podem ser escritos na forma do problema de Lure, cuja função não-linear pode violar a condição de setor em torno da origem. O procedimento desenvolvido é baseado na extensão do princípio de invariância de LaSalle e usa as funções de Lyapunov genéricas do problema de Lure para estimar o conjunto atrator e sua respectiva área de atração. Os parâmetros das funções de Lyapunov são obtidos resolvendo um problema de otimização que pode ser colocado na forma de desigualdades matriciais lineares (LMIs). / The stability analysis of nonlinear systems is present in several engineering fields. Usually, the concern is the determination of stable attractor sets and their associated attraction areas. Methods based on the Lyapunov method provide estimates of these sets. However, these methods involve a nonsystematic search for auxiliary functions called Lyapunov functions. This work presents a systematic procedure, based on Lyapunov method, to estimate attractor sets and their associated attraction areas of a class of nonlinear systems, called in this work extended Lure problem. The extended Lure problem consists of nonlinear systems like those of Lure problem where the nonlinear functions can violate the sector conditions around the origin. The developed procedure is based on the extension of invariance LaSalle principle and uses the general Lyapunov functions of Lure problem to estimate the attractor set and their associated attraction area. The parameters of the Lyapunov functions are obtained solving an optimization problem write like a linear matrix inequality (LMI).

O presente trabalho apresenta a teoria de sistemas dinâmicos dissipativos impulsivos. Apresentamos resultados suficientes e necessários para obtermos dissipatividade para sistemas impulsivos autônomos e não autônomos utilizando funções de Lyapunov. No que segue, desenvolvemos a teoria de estabilidade para a seção nula de um sistema dinâmico não autônomo com impulsos. Utilizando os resultados da teoria abstrata para sistemas não autônomos com impulsos, apresentamos o estudo da estabilidade de um modelo presa-predador com controle e impulsos. / The present work presents the theory of impulsive dissipative dynamical systems. We present necessary and sufficient conditions to obtain dissipativity for autonomous and non-autonomous impulsive dynamical systems via Lyapunov functions. In the sequel, we develop the theory of stability for the null section of non-autonomous dynamical systems with impulses. Using the results from the abstract theory we present the study of stability for a controlled prey-predator model under impulse conditions.

Made available in DSpace on 2017-07-10T17:11:52Z (GMT). No. of bitstreams: 1 Texto completo - Rosiney.pdf: 7631119 bytes, checksum: 43c0461bb49060121b74d945a88d53d4 (MD5) Previous issue date: 2012-11-26 / The present work proposes a study of the dynamics of a gyroscope using simulated data of an analytical model by comparing with experimental data. Classical mechanical modeling approaches are used to identify the equilibrium points, stability and verification of the regions where the motion equations of the gyroscope can present regular or chaotic behavior. The Lyapunov exponents are identified through the standard method, Eckmann-Ruelle Method, Wolf method with time series and the 0-1 test. The results achieved illustrate the main advantages and drawbacks of each method and allow to observe qualitatively and quantitatively information about the motion of the gyroscope used. / Este trabalho propõe um estudo da dinâmica de um giroscópio usando dados de simulação de um modelo analítico comparando com dados experimentais. Verifica-se a modelagem usando mecânica clássica, estudo de pontos de equilíbrio, estabilidade e verificação de regiões onde o movimento do giroscópio pode ficar regular ou caótico. Os expoentes de Lyapunov são identificados usando o método padrão, método de Eckmann-Ruelle, método deWolf com séries temporais e o teste 0-1. Os resultados alcançados nesta dissertação permitiram comparar as principais vantagens e desvantagens de cada um dos métodos e extrair informações qualitativas e quantitativas sobre o movimento do giroscópio em estudo.