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Master of Science / Department of Mechanical and Nuclear Engineering / Warren N. White / Mechanical systems play an integral part in our everyday lives. A subset of these systems can be described as underactuated; the defining characteristic of underactuated mechanical systems is that they have fewer control inputs than degrees of freedom. Airplanes, rockets, helicopters, overhead crane loads, surface vessels, and underwater vehicles are all examples of such systems. The control challenges associated with these systems arise from both the underactuation of the control input and the nonlinear nature of the dynamic equations describing the system’s motion. In this work, a control method for stabilization and tracking based on Lyapunov stability theory is presented. The remarkable result of this tracking controller development is that we arrive at three matching equations that are (with the exception of ) identical to matching equations developed for stabilization as shown in White et al. (2006, 2007, 2008). Asymptotic stabilization of the tracking errors (s) is not obtained. However, the norm of s (||s||) will decrease until an ultimate bound is reached, then it will stay within this bound. A lemma is provided for estimating this bound and it is shown that the magnitude of the bound depends upon the eigenvalues and norms of certain matrices in the Lyapunov formulation. Three examples are presented to illustrate the effectiveness of the direct Lyapunov approach. Two examples of holonomic systems are presented. The first is an inverted pendulum cart which is used to illustrate the formulations performance to tracking a desired path on the cart position or actuated axis. The second example is a ball and beam system in which a desired path is tracked by the ball or unactuated axis. The tracking control technique is also applied to an example of a nonholonomic system, a rolling wheel. The control technique is applied in two alternate manners. Finally, the controller is implemented on a laboratory inverted pendulum cart system in hard real time. A desired trajectory for the cart position is tracked and the control law is used to define the desired pendulum trajectory.

Doctor en Ciencias de la Ingeniería, Mención Modelación Matemática / En este trabajo se aborda el problema de encontrar soluciones regulares para algunas EDPs elípticas e hipo-elípticas no lineales y estudiar sus propiedades cualitativas. En una primera etapa, se considera la ecuación $$ -\Delta u = \lambda e^u, $$ $\lambda > 0$, en un dominio exterior con condición de Dirichlet nula. Un esquema de reducción finito-dimensional permite encontrar infinitas soluciones regulares cuando $\lambda$ es suficientemente pequeño. En la segunda parte se estudia la existencia de soluciones de la ecuación no local $$ (-\Delta)^s u = u^{p \pm \epsilon}, u > 0, $$ en un dominio acotado y suave, con condición de Dirichlet nula; donde $s > 0$ y $p:=(N+2s)/(N-2s) \pm \epsilon$ es cercano al exponente crítico ($\epsilon > 0$ pequeño). Para hallar soluciones, se utiliza un esquema de reducción finito-dimensional en espacios de funciones adecuados, donde el término principal de la función reducida se expresa a partir de las funciones de Green y de Robin del dominio. La existencia de soluciones dependerá de la existencia de puntos críticos de este término principal y de una condición de no degeneración. Por último, se considera un problema no local en el grupo de Heisenberg $H$. En particular, se buscan propiedades de rigidez para soluciones estables de $$ (-\Delta_H)^s v = f(v) en H, $$ $s \in (0,1)$. Como paso fundamental, se prueba una desigualdad del tipo Poincaré en conexión con un problema elíptico degenerado en $R^4_+$. Esta desigualdad se usará en un procedimiento de extensión para dar un criterio bajo el cual los conjuntos de nivel de las soluciones del problema anterior son superficies mínimas en $H$, es decir, tienen $H$-curvatura media nula.

Os modelos metapopulacionais são uma ferramenta muito importante nos estudos de habitats fragmentados. Sendo a natureza bastante diversificada, a análise de ambientes heterogêneos e primordial para a construção de uma dinâmica mais próxima da realidade. Com isso, buscou-se construir um modelo metapopulacional heterogêneo de múltiplas espécies, cujo objetivo e encontrar um critério de estabilidade assintótica de orbitas de sincronização parcial. Para tanto e descrito um ambiente com n patches ou sítios conectados por movimentos de migração divididos em conjuntos, que apresentam diferentes características de sobrevivência e reprodução de cada espécie. Obteve-se uma representação para matriz Jacobiana do sistema, al em de um critério para o cálculo do expoente de Lyapunov. Sendo possível, então, uma generalização para um modelo metapopulacional heterogêneo de múltiplas espécies. / The metapopulational models are an important appliance in the fragmented habitats studies."The nature is very diversi ed, so the heterogeneous environments analysis is primordial for close construction of dynamics realities. Therefore, this research aimed to construct a metapopulational heterogeneous model of multiple species in order to nd an asymptotic stability standard of partial synchronization of orbits. Hence an environment with n patches or connected sites by migration movements were described, whose were divided into groups with di erent survival and reproduction characteristics of each species. A Jacobian matrix of system representation was obtained, as well as a Lyapunov exponent calculation criteria. Thus, a generalization for a heterogeneous metapopulational model of multiple species was possible.

En este trabajo se analizan la regularidad y estabilidad de los sistemas lineales con saltos markovianos (SLSM). Se asume que la cadena de Markov que gobierna estos sistemas es homogénea y que su espacio de estados es finito. Por su novedad, importancia teórica y utilidad práctica, estamos particularmente interesados en los sistemas singulares, es decir, en aquellos SLSM donde aparece una matriz singular en el lado izquierdo de la ecuación dinámica. Si esta matriz no aparece, el sistema se conoce como no singular. Varios conceptos de estabilidad estocástica son introducidos en el capítulo 1. Se prueba que ellos son equivalentes y se establecen resultados algebraicos implementables computacionalmente que permiten determinar la estabilidad de un SLSM no singular. El capítulo 2 está dedicado a los sistemas singulares. La mayoría de los resultados obtenidos en el capítulo 1 son extendidos aquí. Vale la pena mencionar que esta extensión no es trivial, pues la singularidad representa una valla técnica que es muy difícil de superar. La estabilidad casi segura, que es la noción más importante de estabilidad desde el punto de vista práctico, es analizada en el capítulo 3 para sistemas SLSM singulares. Con el propósito de hacer este trabajo auto contenido, se ha añadido un anexo al final de la tesis. / Tesis

Dissertação (mestrado)—Universidade de Brasília, Faculdade de Tecnologia, Departamento de Engenharia Elétrica, 2018. / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) e Fundação de Apoio à Pesquisa do Distrito Federal (FAPDF). / O Laboratório de Aplicação e Inovação em Ciências Aeroespaciais (LAICA) da Universidade de Brasília (UnB) está desenvolvendo uma plataforma de testes de nanossatélites capaz de simular condições ambientais vistas no espaço, especialmente no que diz respeito ao campo magnético da Terra em órbitas, o movimento rotational livre de atrito e o torque gravitacional baixo. Essa plataforma compreende vários subsistemas, tais como uma mesa com rolamento a ar, na qual nanossatélites são montados para teste de seus subsistemas; uma gaiola de Helmholtz, responsável por simular o campo magnético da Terra presente em vários tipos de órbita, especialmente órbitas de baixa altitude (LOE), que são as mais comuns para nanossatélites; sistemas de atuação, tais como rodas de reação e atuadores magnéticos, usados para estudar estratégias de controle de atitude, e sistemas de determinação de atitude, tais como aqueles baseados em telemetria embarcada ou visão computacional. A mesa com rolamento a ar é a parte responsável por fornecer o movimento livre de atrito com três graus de liberdade rotacionais. Ademais, para fornecer o requisito de torque gravitacional baixo, um método deve ser desenvolvido para balancear a mesa com rolamento a ar. Neste trabalho, foco é dado para a solução desse problema. Vários métodos para balanceamento da plataforma de testes do LAICA são apresentados, especialmente quanto às soluções de filtragem, como aquelas que utilizam o filtro de Kalman e suas variações, e esquemas de controle adaptativo, auxiliados pela teoria de Lyapunov. A performance dos métodos de balanceamento propostos é avaliada por meio de simulações e experimentos. / The Laboratory of Application and Innovation in Aerospace Science (LAICA) of the University of Brasília (UnB) is developing a nanosatellite testbed capable of simulating the environment conditions seen in space, specially regarding the Earth magnetic field in orbits, the frictionless rotational movement and the low gravitational torque. This testbed comprises various subsystems, such as an air bearing table, on which nanosatellites are mounted for testing its subsystems; a Helmholtz cage, responsible for simulating the Earth magnetic field present in various kinds of orbit, specially Low Earth Orbits, which is the most common for nanosatellites; actuation systems, such as reaction wheels and magnetorquers, used to study attitude control strategies, and attitude determination systems, such as those based on embedded telemetry or computer vision. The air bearing table is the part responsible for providing the frictionless movement with three rotational degrees of freedom. Also, for providing the low gravitational torque requisite, a method must be developed for balancing the air bearing table. In this work, focus is given for solving this problem. Various methods for balancing the LAICA testbed are presented, specially regarding filtering solutions, such as those using the Kalman Filter and its variations, and adaptive control schemes, aided by the Lyapunov theory. The performance of the proposed balancing methods is evaluated through simulations and experiments.

Made available in DSpace on 2017-07-21T19:26:06Z (GMT). No. of bitstreams: 1 Marli Terezinha Van Kan.pdf: 11927221 bytes, checksum: 471c3fab90a79cbb9d68147d12d6f6f6 (MD5) Previous issue date: 2013-06-18 / Networks of coupled map are used as models to understand the spatiotemporal phenomena in spatially extended systems. In this work it is considered a network of coupled logistic maps in which the interaction among the elements decays according to a power law. In order to characterize the spatial distribution of the logistic map network state variables it was used diagnostic complex order parameter to quantify the synchronization of chaos. The synchronization and suppression of chaos were obtained in the parameter space through coupling values of the intensity and range of interaction among network elements. Others measures were calculated as the spectrum of Lyapunov, Lyapunov dimension and Kolmogorov-Sinai entropy. From the study of the coupled maps network, it was investigated the dynamics of a neuron network by means of collective behavior, and the synchronized state. Hindmarsh-Rose model is the model neuron chosen in this work, and it is described by a system of three first order differential equations coupled in the state variable that represents the membrane potential which shows a succession of alternating activity and rest state. The simulation allowed us to understand the case of a neuron Hindmarsh-Rose and its dynamic properties generation of pulses. The coupling case between two neurons in the master-slave configuration and synchronization in the network of neurons with non-local coupling were used in this work. The network of coupled logistic maps and the Hindmarsh-Rose neurons networks were investigated in the parameter spaces since the two networks constituents are different. / Redes de mapas acoplados são utilizados como modelos para entender fenômenos espaçotemporais em sistemas espacialmente estendidos. Neste trabalho considera-se uma rede de mapas logísticos acoplados onde a interação entre os elementos decai segundo uma lei de potência. Para caracterizar a distribuição espacial das variáveis de estado da rede de mapas logísticos acoplados foi utilizado o diagnóstico parâmetro de ordem complexo a fim de quantificar a sincronização de caos. A sincronização e a supressão de caos foram obtidas no espaço de parâmetros por meio de valores da intensidade de acoplamento e do alcance de interação entre os elementos da rede. Outras medidas foram calculadas sendo o espectro de Lyapunov, a dimensão de Lyapunov e a entropia de Kolmogorov-Sinai. A partir do estudo da rede de mapas acoplados, foi investigada a dinâmica de uma rede de neurônios por meio do comportamento coletivo e do estado sincronizado. O modelo Hindmarsh-Rose é o modelo de neurônio escolhido para estudo, descrito por um sistema de três equações diferenciais de primeira ordem acopladas em que a variável de estado que representa o potencial de membrana mostra uma sucessão de alternância entre estado de atividade e de repouso. A simulação possibilitou a compreensão do caso de um neurônio Hindmarsh-Rose e suas propriedades dinâmicas de geração de pulsos. O caso de acoplamento entre dois neurônios na configuração mestre-escravo e a sincronização na rede de neurônios com acoplamento não local foram tratados neste trabalho. A rede de mapas logísticos acoplados e a rede de neurônios Hindmarsh-Rose foram investigadas no espaço de parâmetros, uma vez que os constituintes das duas redes são diferentes.

Firstly, we establish the existence of a solution u epsilon concentrating along a curve Gammaepsilon near the non-degenerate Gamma, exponentially small in epsilon at any positive distance from the curve, provided epsilon is small and away from certain critical numbers. The concentrating curve Gammaepsilon will collapse to Gamma as epsilon → 0. / In this thesis, we consider the following problem 32Du-u+up= 0 and u>0 in W , 6u6n= 0 on 6W, where O is a bounded domain in R2 with smooth boundary, epsilon is a small positive parameter, nu denotes the outward normal of O and p > 1. Let Gamma be a straight line intersecting orthogonally with ∂O at exactly two points. We use the infinite dimensional Lyapunov-Schmidt reduction method, introduced by M. del Pino, M. Kowalczyk and J. Wei in [14], to deal with the non-invertibility caused by the critical eigenvalues of the linearized operator in the perturbed problems and then construct interior concentration layers near Gamma, which interact with the boundary. Moreover, the method of successive improvements of the approximation helps us decompose the interaction between the boundary and the interior layers. / Secondly, for any given integer N with N ≥ 2 and for small epsilon away from certain critical numbers, we construct another solution uepsilon exhibiting N concentration layers at mutual distances O(epsilon∣ ln epsilon∣), whose concentration set will approach the non-degenerate and non-minimal Gamma as epsilon → 0, provided that the exponent p ≥ 2. Asymptotic location of these layers is governed by a Toda type system. / Yang, Jun. / "July 2007." / Adviser: Juncheng Wei. / Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0357. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 129-136). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science in Engineering. Johannesburg, 2016 / An investigation and analysis of a collection of different techniques, for estimating the states of nonlinear systems, was undertaken. It was found that most of the existing literature on the topic could be organized into several groups of nonlinear observer design techniques, of which each group follows a specific concept and slight variations thereof. From out of this investigation it was discovered that a variation of the adaptive observer could be successfully applied to numerous nonlinear systems, given only limited output information. This particular technique formed the foundation on which a design procedure was developed in order to asymptotically estimate the states of nonlinear systems of a certain form, using only partial state information available. Lyapunov stability theory was used to prove the validity of this technique, given that certain conditions and assumptions are satisfied. A heuristic procedure was then developed to get a linearized model of the error transient behaviour that could form the upper bounds of the transient times of the observer. The technique above, characterized by a design algorithm, was then applied to three well-known nonlinear systems; namely the Lorenz attractor, the Rössler attractor, and the Van Der Pol oscillator. The results, illustrated through numerical simulation, clearly indicate that the technique developed is successful, provided all assumptions and conditions are satisfied. / MT2017