Os métodos diretos são adequados à análise de estabilidade transitória em sistemas de potência, já que não requerem a resolução, integração numérica, do conjunto de equações diferenciais que representam o sistema. Os métodos diretos utilizam as idéias de Lyapunov associadas ao princípio de invariância de LaSalle para estimar a área de atração dos sistemas de potência. A grande dificuldade dos métodos diretos está em encontrar uma função auxiliar V, denominada função de Lyapunov que satisfaça as condições estabelecidas pelo Teorema de Lyapunov. Neste trabalho é realizada uma revisão bibliográfica das funções de Lyapunov utilizadas para análise de estabilidade transitória em sistemas de potência. Analisa-se o problema da existência de funções de Lyapunov quando as condutâncias de transferência são consideradas. Utilizando-se de uma extensão do princípio de Invariância de LaSalle, apresenta-se uma nova função a qual é uma função de Lyapunov no sentido mais geral da extensão do princípio de invariância de LaSalle quando as condutâncias de transferência da matriz admitância da rede reduzida são consideradas. Estudou-se também a existência de funções de Lyapunov no sentido mais geral de extensão do princípio de invariância de LaSalle para modelos que preserva a estrutura da rede. Neste caso, infelizmente não encontramos uma função satisfazendo todas as hipóteses requeridas. / The direct methods are well-suited for transient stability analysis to power systems, since they do not require the solution of the set of differential equations of the system model. The direct methods use the Lyapunov\'s ideas related to the LaSalle\'s invariance principle to estimate the power system attraction area. The great difficulty of the direct methods is to find an auxiliar function V, called Lyapunov function, which satisfies the conditions of Lyapunov\'s theorem. In this work, a bibliographic review of the Lyapunov functions used in transient stability analysis of power systems is done. The problem of existence of Lyapunov functions, when the transfer conductances are considered, is analysed. Using LaSalle\'s invariance principle extension, a Lyapunov function considering the transfer conductances is presented. The existence of Lyapunov functions for models that preserv the network structure was studied using the LaSalle\'s invariance principle. Unfortunately, in these cases, we did not find a function satisfing all the required hypothesis.

Made available in DSpace on 2017-07-21T19:26:08Z (GMT). No. of bitstreams: 1 Moises Souza Santos.pdf: 6452966 bytes, checksum: 0cca031e2fe3425ee7f63672b4643f3b (MD5) Previous issue date: 2014-02-27 / Fundação Araucária de Apoio ao Desenvolvimento Científico e Tecnológico do Paraná / This work report on a study of the chimera states in networks composed by logistic maps whose interaction between them may have two types of couplings (nonlocal and power law). The Lyapunov exponents and a derivative of this quantity, the Kolmogorov-Sinai entropy, were used as analysis tool to calculate the average time of “life” or collapse time of chimera for nonlocal case with different sizes of networks. For coupling power law the results show that when it has nonlocal features it is also possible to see chimera states through the network. We also calculate the probable existence of regions of chimeras within the parameter space for both cases. Since the chimera dynamic states occurs when coexist regions that are chaotic and periodic in time, we obtained the region of the parameter space where this behavior occurs. We show the probable relations between the chimera states and the global order parameter that provides information on how the sites of a coupled network are related. / Neste trabalho foi realizado um estudo dos estados quimera em redes compostas por mapas logísticos cuja interação entre os mesmos pode ser de dois tipos de acoplamentos (não-local e lei de potência). Utilizamos como ferramenta de análise os expoentes de Lyapunov e uma quantidade derivada deste, a entropia de Kolmogorov-Sinai, para calcular o tempo médio de “vida” ou tempo de colapso de quimera para o caso não-local em redes de diferentes tamanhos. Para o acoplamento lei de potência mostramos que quando este possui características nâo-locais a rede também pode exibir estados quimera. Além disso calculamos as prováveis regiões de existência de quimeras dentro do espaço dos parâmetros para ambos os casos. Uma vez que a dinâmica de estados quimera ocorre quando coexistem regiões que são periódicas e caóticas no tempo, obtivemos a região, no espaço de parâmetros, onde este comportamento ocorre. Mostramos as relações prováveis que existem entre os estados quimera e o parâmetro de ordem global que fornece informações sobre o quão relacionados estão sítios de uma rede acoplada.

O objetivo deste trabalho é investigar propriedades qualitativas de certas equações diferenciais funcionais retardadas de segunda ordem quando lhes são impostos controles de impulsos adequados. Os principais resultados dizem respeito a estabilidade e oscilação por impulsos. Mais especificamente, consideramos algumas equações e provamos que suas soluções triviais podem ser estabilizadas por impulsos. Em seguida, consideramos uma destas equações e provamos que suas soluções podem se tornar oscilatórias com a imposição apropriada de controles de impulsos. Apresentamos alguns exemplos que ilustram nossos resultados. Além do objetivo acima, procuramos produzir um texto que compreendesse a teoria fundamental das equações diferenciais funcionais retardadas impulsivas, teoria esta que, até então, não podia ser encontrada num único texto como este. Desenvolvemos e discutimos existência, unicidade, continuação de soluções, intervalo maximal de existência e dependência contínua de soluções dos valores iniciais para equações diferenciais retardadas impulsivas. / The purpose of this work is to investigate qualitative properties of certain second order delay differential equations when some proper impulse controls are added to them. The main results concern the stability and scillation by impulses. More specifically, we consider some equations and prove that their trivial solutions can be stabilized by impulses. We also consider one of these equations and prove that all solutions oscillate when proper impulse controls are imposed. We give some examples to illustrate our results. Because dealing with systems with both delays and impulses is a recent interest of some mathematicians we also considered producing a text that would encompass the fundamental theory of retarded functional differential equations with impulses. Up to now such theory could not be found in a single text as this one. Therefore we discuss and develop basic aspects of the theory as existence, uniqueness, continuability of solutions, maximal interval of existence and continuous dependence of solutions on initial values for impulsive retarded differential equations.

In the present climate of deregulation and privatisation, theutilities are often separated into generation, transmission anddistribution companies so as to help promote economic efficiencyand encourage competition. Also, environmental concerns,right-of-way and cost problems have delayed the construction ofboth generation facilities and new transmission lines while thedemand for electric power has continued to grow, which must bemet by increased loading of available lines. A consequence isthat power system damping is often reduced which leads to a poordamping of electromechanical power oscillations and/or impairmentof transient stability. The aim of this thesis is to examine theability of Controllable Series Devices (CSDs), such as Unified Power Flow Controller (UPFC)   Controllable Series Capacitor (CSC)   Quadrature Boosting Transformer (QBT)   for improving transient stability and damping ofelectromechanical oscillations in a power system. For these devices, a general model is used in power systemanalysis. This model is referred to as injection model which isvalid for load flow and angle stability analysis. The model isalso helpful for understanding the impact of the CSDs on powersystem stability. A control strategy for damping of electromechanical poweroscillations is also derived based on Lyapunov theory. Lyapunovtheory deals with dynamical systems without input. For thisreason, it has traditionally been applied only to closed-loopcontrol systems, that is, systems for which the input has beeneliminated through the substitution of a predetermined feedbackcontrol. However, in this thesis, Lyapunov function candidatesare used in feedback design itself by making the Lyapunovderivative negative when choosing the control. This controlstrategy is called Control Lyapunov Function (CLF) for systemswith control input. / QC 20100609

Many new materials used in mechanical and structural engineering exhibit viscoelastic properties, that is, stress depends on the past time history of strain, and vice versa. Investigating the behaviour of viscoelastic materials under dynamical loads is of great theoretical and practical importance for structural design, vibration reduction, and other engineering applications. The objective of this thesis is to find how viscoelasticity affects the stability of structures under random loads. The time history dependence of viscoelasticity renders the equations of motion of viscoelastic bodies in the form of integro-partial differential equations, which are more difficult to study compared to those of elastic bodies. The method of stochastic averaging, which has been proved to be an effective tool in the study of dynamical systems, is applied to simplify some single degree-of-freedom linear viscoelastic systems parametrically excited by wide-band noise and narrow-band noise. The solutions of the averaged systems are diffusion processes characterized by Itô differential equations. Therefore, the stability of the solutions is determined in the sense of the moment Lyapunov exponents and Lyapunov exponents, which characterize the moment stability and the almost-sure stability, respectively. The moment Lyapunov exponents may be obtained by solving the averaged Itô equations directly, or by solving the eigenvalue problems governing the moment Lyapunov exponents. Monte Carlo simulation is applied to study the behaviour of stochastic dynamical systems numerically. Estimating the moments of solutions through sample average may lead to erroneous results under the circumstances that systems exhibit large deviations. An improved algorithm for simulating the moment Lyapunov exponents of linear homogeneous stochastic systems is presented. Under certain conditions, the logarithm of norm of a solution converges weakly to normal distribution after suitably normalized. This property, along with the results of Komlós-Major-Tusnády for sums of independent random variables, are applied to construct the algorithm. The numerical results obtained from the improved algorithm are used to determine the accuracy of the approximate analytical moment Lyapunov exponents obtained from the averaged systems. In this way the effectiveness of the stochastic averaging method is confirmed. The world is essentially nonlinear. A single degree-of-freedom viscoelastic system with cubic nonlinearity under wide-band noise excitation is studied in this thesis. The approximated nonlinear stochastic system is obtained through the stochastic averaging method. Stability and bifurcation properties of the averaged system are verified by numerical simulation. The existence of nonlinearity makes the system stable in one of the two stationary states.

Many new materials used in mechanical and structural engineering exhibit viscoelastic properties, that is, stress depends on the past time history of strain, and vice versa. Investigating the behaviour of viscoelastic materials under dynamical loads is of great theoretical and practical importance for structural design, vibration reduction, and other engineering applications. The objective of this thesis is to find how viscoelasticity affects the stability of structures under random loads. The time history dependence of viscoelasticity renders the equations of motion of viscoelastic bodies in the form of integro-partial differential equations, which are more difficult to study compared to those of elastic bodies. The method of stochastic averaging, which has been proved to be an effective tool in the study of dynamical systems, is applied to simplify some single degree-of-freedom linear viscoelastic systems parametrically excited by wide-band noise and narrow-band noise. The solutions of the averaged systems are diffusion processes characterized by Itô differential equations. Therefore, the stability of the solutions is determined in the sense of the moment Lyapunov exponents and Lyapunov exponents, which characterize the moment stability and the almost-sure stability, respectively. The moment Lyapunov exponents may be obtained by solving the averaged Itô equations directly, or by solving the eigenvalue problems governing the moment Lyapunov exponents. Monte Carlo simulation is applied to study the behaviour of stochastic dynamical systems numerically. Estimating the moments of solutions through sample average may lead to erroneous results under the circumstances that systems exhibit large deviations. An improved algorithm for simulating the moment Lyapunov exponents of linear homogeneous stochastic systems is presented. Under certain conditions, the logarithm of norm of a solution converges weakly to normal distribution after suitably normalized. This property, along with the results of Komlós-Major-Tusnády for sums of independent random variables, are applied to construct the algorithm. The numerical results obtained from the improved algorithm are used to determine the accuracy of the approximate analytical moment Lyapunov exponents obtained from the averaged systems. In this way the effectiveness of the stochastic averaging method is confirmed. The world is essentially nonlinear. A single degree-of-freedom viscoelastic system with cubic nonlinearity under wide-band noise excitation is studied in this thesis. The approximated nonlinear stochastic system is obtained through the stochastic averaging method. Stability and bifurcation properties of the averaged system are verified by numerical simulation. The existence of nonlinearity makes the system stable in one of the two stationary states.

In this dissertation we study the Lyapunov diagonal stability and its extensions through partitions of the index set {1,...,n}. This type of matrix stability plays an important role in various applied areas such as population dynamics, systems theory and complex networks. We first examine a result of Redheffer that reduces Lyapunov diagonal stability of a matrix to common diagonal Lyapunov solutions on two matrices of order one less. An enhanced statement of this result based on the Schur complement formulation is presented here along with a shorter and purely matrix-theoretic proof. We develop a number of extensions to this result, and formulate the range of feasible common diagonal Lyapunov solutions. In particular, we derive explicit algebraic conditions for a set of 2 x 2 matrices to share a common diagonal Lyapunov solution. In addition, we provide an affirmative answer to an open problem concerning two different necessary and sufficient conditions, due to Oleng, Narendra, and Shorten, for a pair of 2 x 2 matrices to share a common diagonal Lyapunov solution. Furthermore, the connection between Lyapunov diagonal stability and the P-matrix property under certain Hadamard multiplication is extended. Accordingly, we present a new characterization involving Hadamard multiplications for simultaneous Lyapunov diagonal stability on a set of matrices. In particular, the common diagonal Lyapunov solution problem is reduced to a more convenient determinantal condition. This development is based upon a new concept called P-sets and a recent result regarding simultaneous Lyapunov diagonal stability by Berman, Goldberg, and Shorten. Next, we consider various types of matrix stability involving a partition alpha of {1,..., n}. We introduce the notions of additive H(alpha)-stability and P_0(alpha)-matrices, extending those of additive D-stability and nonsingular P_0-matrices. Several new results are developed, connecting additive H(alpha)-stability and the P_0(alpha)-matrix property to the existing results on matrix stability involving alpha. We also point out some differences between these types of matrix stability and the conventional matrix stability. Besides, the extensions of results related to Lyapunov diagonal stability, D-stability, and additive D-stability are discussed. Finally, we introduce the notion of common alpha-scalar diagonal Lyapunov solutions over a set of matrices, which is a generalization of common diagonal Lyapunov solutions. We present two different characterizations of this new concept based on the well-known results for Lyapunov alpha-scalar stability [34]. The first one hinges on a general version of the theorem of the alternative, and the second one using Hadamard multiplications stems from an extension of the P-set property. Several illustrative examples and an application concerning a set of block upper triangular matrices are provided.

Dans cette thèse on s'intéresse a l'aspect spatial dans la modélisation en épidémiologie, ainsi que des conditions menant a la stabilité des systèmes que nous présentons, en épidémiologie, a partir des modèles classiques de Ross et Mckendrick. Dans un premier temps, nous examinons les effets de l'indice de la différence normalisée de végétation (NDVI) dans un modèle de contamination du paludisme a Bankoumana, une région du Mali. A partir du système obtenu, nous trouvons le taux de reproduction de base. Deux points d'équilibre sont déduits dont, un point d'équilibre sans maladie et un point d'équilibre endémique. Ce dernier point d'équilibre ainsi que le taux de reproduction de base sont fonction de l'indice de végétation normalisée. Par la suite, nous construisons un modèle ayant des équations a retard, dans lequel est également incorporée le NDVI. Le taux de reproduction de base ainsi que les deux points d'équilibre qui découlent de notre système sont fonction des retards introduits. Nous montrons que la stabilité de nos points d'équilibre est, non seulement fonction du taux de reproduction de base, mais elle est aussi étroitement liée aux retards introduits. Dans une autre optique, nous fractionnons la région d'étude en zones dans lesquelles nous émettons l'hypothèse que le taux de contagion induite par les individus d'une zone sur les individus de la même zone, ainsi que celui des individus d'une zone sur ceux d'une autre zone, peut être différents. Nous obtenons un système qui nous permet de déterminer les points d'équilibre ainsi que les conditions qui nous permettent d'obtenir la stabilité au sens de Lyapunov. Puis, nous perturbons le système précédent au niveau de son unique point d'équilibre endémique, en introduisant un bruit additif. Par suite, les conditions permettant la stabilité au sens de Lyapunov, sur le nouveau système obtenu, sont également déduites . Dans un cadre similaire, nous élaborons un modèle multi-groupes, dans lequel nous introduisons des coordonnées spatiales. Les groupes sont formés selon une proximité dépendant du rayon d'un cercle, de manière aléatoire. Ici, le taux de contagion est supposé uniforme dans les groupes. Après avoir déterminé les points d'équilibre ainsi que le taux de reproduction de base,nous trouvons les conditions qui favorisent la stabilité au sens de Lyapunov dans le cadre général. A l'ordre 1, c'est-a-dire, lorsqu'on suppose que nous n'avons qu'un groupe, les conditions de stabilité sont obtenus par le critère de Routh-Hurwitz. / In this thesis, our interest is on the aspect in space of the establishment of a spatial model in epidemiology and the conditions leading to the stability of the systems that we present, in epidemiology, from the classical models by Ross and Mckendrick. Firstly, we intend to examine the eects of the Normalized DiFerence Vegetation Index(NDVI) in a model of contamination of malaria in Bankoumana, a region in Mali. From the system obtained, we willnd the basic reproduction rate. Then we deduce two point of equilibrium, among which one point of equilibrium without the disease and another one with an endemic point. The latter with the basic reproduction rate vary according to the indices of normalized vegetation. Then, we will build a model having equations delay, containing the NDVI. The rate of basic reproduction and the two points of equilibrium that come from our system depend upon the delay introduced. We will show that the stability of our points of equilibrium is not only dependent upon the basic reproduction rate, but also closely related to the delays introduced. In another way, we will divide the region of study in areas where we will set hypotheses that the rate of contamination brought about by individuals in an area of study on the others, can be dierent. It will permit us to obtain a system in which we will determine the points of equilibrium and the conditions that will lead us to obtain the stability according to Lyapunov. Then, we will disturb the previous system at the level of its unique endemic point of equilibrium, with the introduction of an additional noise. The conditions leading to stability according to Lyapunov, on the new system obtained, are generally deduced here. In a similar framework, we will elaborate a multigroups model, in which we will introduce spatial coordinates. The groups are formed according to a closeness depending to a radius of a circle at random. Here, the rate of contamination is supposed to be uniform in the groups. After having determined the point of equilibrium and the rate of basic reproduction, we will nd the conditions facilitating stability in as by Lyapunov in a global framework. In the order1, it means that supposing that we have only one group, the conditions of stability are obtained according to the Routh-Hurvitz criteria.

Made available in DSpace on 2014-06-11T19:22:31Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-05-27Bitstream added on 2014-06-13T20:47:51Z : No. of bitstreams: 1 esteves_tt_me_ilha.pdf: 642482 bytes, checksum: 6d87652f7c370af770e00c245690a225 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Neste trabalho é realizada a análise da estabilidade de sistemas fuzzy Takagi-Sugeno (TS) contínuos no tempo, através de Funções de Lyapunov Fuzzy (FLF), Funções de Lyapunov Metzler (FLM) e de Funções de Lyapunov Fuzzy-Metzler (FLFM) introduzida nesta disser- tação. Novas propostas são feitas a partir destas análises, sendo apresentadas condições su- ficientes para a estabilidade assintótica destes sistemas no sentido de Lyapunov. As soluções obtidas são baseadas em desigualdades lineares matriciais (LMIs, do inglês Linear Matrix Ine- qualities) e dependem da solução de um conjunto de desigualdades de Lyapunov-Metzler, que podem ser de difícil solução. Então, foram apresentadas condições de estabilidade baseadas em uma subclasse de matrizes de Metzler que, quando factíveis, podem ser resolvidas através de LMIs com a necessidade de uma busca unidimensional. Foram propostos métodos que genera- lizam os já existentes na literatura, baseados em FLF, para a estabilidade assintótica dos sistemas fuzzy TS / This work addresses the stability analysis of Takagi-Sugeno (TS) fuzzy systems via Fuzzy Lyapunov Functions (FLF), Metzler Lyapunov Functions (MLF) and Fuzzy-Metzler Lyapunov Functions (FMLF) that was proposed in this dissertation. New proposals are made from these analyses, and sufficient conditions for asymptotic stability of these systems in the sense of Lyapunov are presented. The results obtained are based on LMIs (Linear Matrix Inequalities) and depend on the solutions of a set of Lyapunov-Metzler inequalities, that are usually difficult to solve. Then, conditions for stability based on a subclass of Metzler matrices that, when feasible, can be described by a set of LMIs with an unidimensional search, are presented. The proposed methods generalize the similar methods available in the literature, based on FLF, for the asymptotic stability of TS fuzzy systems

Made available in DSpace on 2016-12-12T20:15:48Z (GMT). No. of bitstreams: 1 capa_ate_sumario.pdf: 940719 bytes, checksum: 34845651ded8147831931a5314e46c27 (MD5) Previous issue date: 2012-02-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work, we numerically studied a four-dimensional Chua circuit model through bifurcation diagrams and parameter spaces. Our main objective here is to ex-tend the studies already realized in this system, showing a wider range of its behavior. For this purpose, we constructed the parameter spaces using the Lyapunov exponents spectrum through color scales, varying simultaneously two parameters of the system. With this procedure it was possible to discover where are the chaotic regions, the pe-riodic ones and the fixed points for the set of parameters. / Este trabalho tem como foco principal estudar, por métodos numéricos, um circuito eletrônico de Chua composto de quatro equações diferenciais através de diagramas de bifurcação e espaços de parâmetros. Nossa proposta aqui é ampliar os estudos numéricos já realizados neste sistema, revelando uma gama maior do seu comportamento. Para isso, realizamos construções dos espaços de parâmetros nos quais apresentam os valores dos expoentes de Lyapunov através de escalas coloridas, mediante a variação de dois parâmetros que compõem o circuito eletrônico. Com este procedimento é possível descobrir onde existem regiões caóticas, periódicas e pontos fixos para o conjunto de parâmetros do sistema.