A method of determining regions of attraction for a system with multiple nonlinearities is considered in this thesis. Application of the method involves finding the global minimum of a nonconvex Lyapunov function. This is done by finding a graphical solution using Lagrange multipliers and then applying the projected gradient method to determine the exact solution. A three machine power system example is included to illustrate the application. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

The threat of capsize in unpredictable seas has been a risk to vessels, sailors, and cargo since the beginning of a seafaring culture. The event is a nonlinear, chaotic phenomenon that is highly sensitive to initial conditions and difficult to repeatedly predict. In extreme sea states most ships depend on an operating envelope, relying on the operator's detailed knowledge of headings and maneuvers to reduce the risk of capsize. While in some cases this mitigates this risk, the nonlinear nature of the event precludes any certainty of dynamic vessel stability. This research presents the use of Lyapunov exponents, a quantity that measures the rate of trajectory separation in phase space, to predict capsize events for both intact and damaged stability cases. The algorithm searches backwards in ship motion time histories to gather neighboring points for each instant in time, and then calculates the exponent to measure the stretching of nearby orbits. By measuring the periods between exponent maxima, the lead-time between period spike and extreme motion event can be calculated. The neighbor-searching algorithm is also used to predict these events, and in many cases proves to be the superior method for prediction. In addition to the ship stability research, the Lyapunov exponents are used in conjunction with bifurcation analysis to determine regions of stable behavior in strange attractors when the system parameters are varied. The boundaries of stability are important for algorithm validation, where these transitions between stable and unstable behavior must be accounted for. / Master of Science

Made available in DSpace on 2014-06-11T19:27:07Z (GMT). No. of bitstreams: 0 Previous issue date: 2008-02-26Bitstream added on 2014-06-13T19:06:47Z : No. of bitstreams: 1 santos_ild_me_sjrp.pdf: 434711 bytes, checksum: 230caec3d969a14efac9b1700fd1dd97 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho introduzimos uma classe de sistemas dinâmicos descontínuos com espaço tempo contínuo e analisamos Teoremas que asseguram condições suficientes para a estabilidade de Lyapunov utilizando funções de Lyapunov. Além disso, consideramos também Teoremas de Recíproca, que sob algumas condições garantem uma determinada necessidade para esses Teoremas de estabilidade de Lyapunov. / In this work we introduce a class of discontinuous dynamical systems with time space continuous and we analyze Theorems that ensure sufficient conditions for the Lyapunov stability using Lyapunov functions. Moreover, we also consider Converse Theorems, which under some conditions guarantee a determined necessity for those Theorems of Lyapunov stability.

Vita. / Digitized by Kansas Correctional Industries

<p>The phenomenon of drag reduction by polymers in turbulent flow has beenstudied over the last 60 years. New insight have been recently gained by meansof numerical simulation of dilute polymer solution at moderate values of theturbulent Reynolds number and elasticity. In this thesis, we track elastic parti-cles in Lagrangian frame in turbulent channel flow at Reτ = 180, by tracking,where the single particle obeys the FENE (finite extendible nonlinear elastic)formulation for dumbbel model. The feedback from polymers to the flow is notconsidered, while the Lagrangian approach enables us to consider high valuesof polymer elasticity. In addition, the finite time Lyapunov exponent (FTLE)of the flow is computed tracking infinitesimal material elements advected bythe flow. Following the large deviation theory, the Cramer’s function of theprobability density function of the FTLE for large values of time intervals isstudied at different wall-normal positions. The one-way effect of the turbulentflow on polymers is investigated by looking at the elongation and orientation ofthe polymers, with different relaxation times, across the channel. The confor-mation tensor of the polymers deformation which is an important contributionin the momentum balance equation is calculated by averaging in wall-parallelplanes and compared to theories available in the literature.</p> / QC 20100706

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Doctora en Ingeniería Eléctrica / El presente trabajo aborda el problema del análisis de estabilidad, convergencia y desempeño de los sistemas adaptables fraccionarios, utilizando el enfoque de los modelos de error, problema que no ha sido abordado ni reportado en la literatura técnica hasta la fecha. Los modelos de error fraccionarios surgen al introducir las derivadas de orden fraccionario en los esquemas adaptables clásicos, ya sea describiendo la planta a controlar o identificar, o bien en las leyes de ajuste de los parámetros. Como parte del desarrollo del trabajo, se estudiaron los cuatro modelos de error conocidos hasta el momento, pero desde el punto de vista fraccionario. En todos los casos, el primer paso fue realizar exhaustivos estudios por simulación, que permitieron tener un nivel de comprensión inicial del desempeño de estos modelos de error, en cuanto a estabilidad y convergencia de los errores. Para analizar la estabilidad de estos modelos de error, fue preciso generar resultados matemáticos generales, los que también constituyen un importante aporte de esta Tesis Doctoral. Estos resultados permitieron completar el análisis de los Modelos de Error Fraccionarios 1 y 4 en su totalidad, y para ciertos casos particulares de los Modelos de Error Fraccionarios 2 y 3. En relación a la demostración de convergencia del error de salida a cero, se obtuvieron resultados analíticos para casos particulares en el Modelo de Error Fraccionario 1, y se expusieron de manera concreta las principales dificultades que han impedido, hasta el momento, generalizar estos resultados a los demás casos. También se obtuvieron otros resultados analíticos válidos para los cuatro modelos de error, que permiten afirmar que el promedio del cuadrado de la norma del error de salida, tiene una tendencia decreciente. Esto puede resultar de utilidad en algunas aplicaciones desde el punto de vista práctico. Respecto de la convergencia del error paramétrico, se logró determinar que ella está relacionada con alguna forma de excitación persistente, particular para los sistemas adaptables fraccionarios, pero no se logró dar cabal respuesta a esta interrogante. Sin embargo, se obtuvieron resultados analíticos parciales para el caso del Modelo de Error Fraccionario 1 escalar, quedando los restantes casos como parte del trabajo futuro a desarrollar en esta línea de investigación. No obstante, se expusieron las conclusiones intuitivas al respecto, obtenidas de los estudios por simulación. Finalmente, este trabajo se complementó con el diseño, implementación y análisis de dos aplicaciones de controladores fraccionarios. El primero corresponde al control por referencia a modelo de orden fraccionario para un regulador automático de voltaje, mientras que el segundo es un compendio de tres estrategias de control fraccionario para el control de posición en un sistema de levitación magnética, conocido como Anillo de Thomson.

In this thesis I make extensive use of the Lyapunov analysis formalism to unravel fundamental mechanisms of instability in two different systems : the Kuramoto model of globally coupled phase-oscillators and the Lorenz 96 (L96) atmospheric "toy" model, portraying the evolution of a physical quantity along a latitude circle. I start by introducing the relevant theoretical background, with special attention on the main tools I have been using throughout this work : Lyapunov Exponents (LEs), which quantify the asymptotic growth rates of infinitesimal perturbations in a system, and by extension, its degree of chaoticity, and Covariant Lyapunov Vectors (CLVs), which indicate the phase space direction (or the geometry) associated with these growth rates. The Kuramoto model is central in the study of synchronization among oscillatory units characterized by their various natural frequencies, but little is known on its chaotic dynamics in the unsynchronized state. I thus investigate the scaling behavior of the first LE, upon different assumptions on the natural frequencies, and make use of educated structural simplifications to analyze the origin of chaos in the finite size model. On the other hand, the L96 model has been devised to gather the main dynamical ingredients of atmospheric dynamics, namely advection, damping, external (solar) forcing and transfers across different scales of motion, in a minimalist and functional way. It features two coupled dynamical layers : the large scale variables, representing synoptic scale atmospheric dynamics, and the small scale variables, faster and more numerous, associated with convective scale dynamics. The core of the study revolves around geometrical properties of CLVs, in the aim of understanding the processes underlying the observed multiscale chaoticity, and an exhaustive study of a non-trivial ensemble of CLVs featuring relevant projection on the slow subspace.

Doctor en Ingeniería Eléctrica / La presente Tesis Doctoral explora el problema de la determinación de funciones de Lyapunov cuadráticas comunes (CQLF, por su sigla en inglés), en el marco de los sistemas conmutados, y el problema de la identificación en línea y control adaptable, en el marco de los sistemas adaptables basados en modelos de error. Ambos en el área de los sistemas dinámicos lineales y no lineales, y son resueltos aquí bajo el enfoque de la optimización basada en una herramienta llamada Optimización por Enjambre de Partículas (PSO, por su sigla en inglés). Los problemas anteriormente mencionados son de gran importancia y trascendencia en la actualidad, pues el primero entrega los elementos para la determinación de la estabilidad de sistemas lineales conmutados, y el segundo se relaciona con el control de plantas de parámetros desconocidos. Estos dos problemas poseen soluciones parciales, tanto desde el punto de vista de la optimización como de otros enfoques. Sin embargo, las soluciones existentes poseen beneficios demostrados, pero también limitaciones marcadas, que los siguen justificando como problemas abiertos. En cuanto al problema de la determinación de CQLFs, en la presente Tesis Doctoral se desarrollan dos nuevas metodologías: i) una metodología basada en PSO para la determinación de la no-existencia de una CQLF, y ii) una metodología basada en PSO para el cálculo de una CQLF. Ambas metodologías presentan evidentes mejoras comparativas respecto de las mejores soluciones actuales, con base en indicadores de desempeño objetivos. En el ámbito de los sistemas adaptables, el principal producto de la presente Tesis Doctoral es una metodología basada en PSO para el diseño de leyes de ajuste paramétrico en sistemas adaptables de tiempo discreto, representados por modelos de error. Desde este punto de vista, la investigación se centra en las propiedades de estabilidad que presenta el uso de PSO en sistemas adaptables, además de estudiar las ventajas comparativas respecto de técnicas tradicionalmente usadas como gradiente y mínimos cuadrados.

In maintaining power system stability; especially that of large interconnected systems, in the face of large disturbances it is desirable to have a non-linear control technique that is simple and inexpensive to implement. This thesis presents a non-linear control technique which relies on angle measurements taken at strategic points in the power system with the aid of the G.P.S. ( Global Positioning System ) timing signal. A method for estimating these bus angles which is faster than previous methods is developed as well as a technique for choosing the locations of these transducers. This transducer placement algorithm aims to place transducers at locations whose bus voltage response to the less well damped inter-area modes is maximised and whose response to the better damped local modes is minimised. Since the control techniques are based on aggregated classical models of coherent generators it is important to be able to estimate the internal voltages of these aggregate machines. The placement algorithm ensures maximally precise angle estimates in the presence of noise by minimising the condition number of the observation matrix relating transducer bus voltages to internal aggregated machine voltages. The non-linear control techniques presented rely on an energy function developed in this thesis which is based on the physical circuit energy of the system. One technique; the Direct Energy technique looks at maximising the negativity of the time rate of change of the energy function, assuming that the energy function is positive during the time frame of interest. It is shown that should the number of controllers be less than the number of modes, excluding the centre of area mode, then sustained oscillations appear which will only be damped by the natural damping of the system. This may be overcome by using techniques which rely on reducing the entire system energy over the time frame of interest. These so-called Lookahead techniques can rely on higher order time derivatives of the energy function or on co-states, the latter being the principal focus of this thesis. The Lookahead control technique developed is based on co-states which are estimated by the using the solution to the time independent Ricatti equation for a LQ model of the system. It is shown to produce good damping in a number of case studies. Furthermore it is shown to perform well in the presence of both static and dynamic load models. Also it is shown that the path dependent terms introduce some ambiguity as to whether or not the system will converge to a stable equilibrium point. It is shown that it is possible to put a bound on the region to which the power system can be assured to converge. Furthermore the addition of the above-mentioned control strategies has the effect of overcoming the effect of the path dependent terms and, should the control action be strong enough, completely swamping them and ensuring system convergence to a stable operating point. In any case the energy function could be directly monitored since all the data needed is being collected anyway for control purposes. / PhD Doctorate