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The non-local stabilization of nonlinear systems by output feedback is a challenging problem that remains the subject of continuing investigation in control theory. In this thesis we develop two globally asymptotically stabilizing output feedback algorithms for multivariable nonlinear systems. Our first result is an extension a well-known output feedback method to a class of nonlinear systems whose dynamics can be written as a collection of subsystems that are dynamically coupled through output-dependent nonlinear terms. We show that this method must be modified to accommodate the dynamic coupling by introducing additional nonlinear damping terms into each control input. Our second contribution involves the application of observer backstepping to systems in a restricted block-triangular observer form. In this form, the nonlinearities entering each subsystem are allowed to depend on the output associated with the subsystem, and all upper subsystem states, including unmeasured ones. The proposed algorithm is demonstrated on a magnetically levitated ball. / Controls

Kite energy systems are an emerging renewable energy technology. Unlike conventional turbines, kite energy systems extract wind power using tethered kites which can move freely in the wind or underwater in an ocean current. Due to the mobility, kite power systems can harvest power from regions with higher and steadier power density by moving in high-speed cross flow motion. An airborne kite energy system harnesses wind power at an altitude higher than the conventional wind turbines, while an undersea kite energy system extracts power close to the ocean surface. In this dissertation, the physical limitation, mathematical modeling, and control system design of the kite energy systems are studied. First, three major physical effects that are acting on the kite energy systems are investigated, including potential force, steady aero-/hydro-dynamic force and added mass effects. Furthermore, the dissipativity of the steady aero-/hydro-dynamic forces with respect to the apparent velocity is established. Based on this analysis, the power generation limit of the kite energy systems is studied. A power limit formulation is given which generalize the two-dimensional result to three-dimensional case. The different physical phenomenon is modeled in different coordinate systems, the difference of the density, viscosity between air and water are significant, and the kite energy system can operate in two distinct modes. To combine different physical effects into a single simulation framework, the equivalences of the kite model in different coordinate systems are established through kinematic analysis. Using these equivalent relations, a unified simulation model for airborne and undersea kite energy systems are derived. The control system design of kite energy systems is also investigated. The resulting equations of motion of kite energy systems are highly nonlinear. Therefore, Lyapunov methods are used to analyze the system behavior. Three different techniques are reviewed, including Lyapunov analysis for autonomous and non-autonomous systems, the ultimate boundedness and input-to-state stability and passivity methods. For the fixed tether length kite energy systems, the ultimate boundedness of the kite translation is established through the dissipativity of the steady aero-/hydro-dynamic force. For the variable tether length kite energy system, the input-to-state analysis is used to design the tether tension that guaranteed the boundedness of the kite translation. In both cases, the Lyapunov based methods are used to design kite rotational control systems which result in PD type control signals. Although this control scheme generates consecutive power cycles for kite energy systems. It is shown that the kite aero-/hydro-dynamical performance is unstable in the simulation which could result in unsteady power generation. To provide a steadier performance in kite translation and power output, the relative dynamics of the kite translation is first proposed. In this model, the kite apparent speed and attitudes, the angle of attack and side-slip angle, are used to describe the kite translation. A nonlinear control scheme is designed to regulate the angle of attack and side-slip angle using back-stepping methods by using the kite angular velocity and control inputs. However, due to the magnitude limit of the angular velocity, the residual error of the apparent attitude tracking remain large for the large desired angle of attack and side-slip angle. To achieve a better power harvesting and aero-/hydro-dynamics performance, the geometric properties of kite angle of attack and side-slip angle are studied. A geometric attitudes trajectory is constructed to track given apparent attitudes. A rotational control system is designed based on the back-stepping and sliding mode methods for the desired geometric attitude, and the high gain observer is applied to acquire the information needed for the rotational control signal. Through the geometric apparent attitudes tracking control algorithm, the angle of attack and side-slip angle act as direct control inputs to the kite translational motion. The kite translational dynamics under the geometric apparent attitude tracking is studied. These dynamics give the possibility of controlling the kite translational motion only through the rotational control scheme.

O método de Lyapunov, também conhecido como método direto, é eficiente para análise de estabilidade transitória em sistemas de potência. Tal método possibilita a análise de estabilidade sem requerer o conhecimento das soluções das equações diferenciais que modelam o problema. A maior desvantagem da utilização dos métodos diretos, é sem dúvida encontrar uma função (V) que satisfaça as condições do Teorema de Lyapunov, ou seja, V > 0 e V \'< ou =\' 0. Durante muitos anos a inclusão das condutâncias de transferência na modelagem do sistema de potência, com a rede reduzida aos nós dos geradores, foi um assunto que despertou interesse em vários pesquisadores. Em 1989, Chiang provou a não existência de uma Função de Lyapunov para sistemas de potência quando as condutâncias de transferência são consideradas. Essas condutâncias de transferência são responsáveis por gerar regiões no espaço de estados onde tem-se V > 0, não satisfazendo as condições do Teorema de Lyapunov. Recentemente, Rodrigues, Alberto e Bretas (2000) apresentaram a Extensão do Princípio de Invariância de LaSalle, onde é permitido que a Função de Lyapunov possua, em algumas regiões limitadas do espaço de estados, a derivada positiva. Neste caso, estas funções passam a ser denominadas Funções de Lyapunov Estendidas (FLE). Neste trabalho, são utilizadas a Extensão do Princípio de Invariância de LaSalle e as Funções de Lyapunov Estendidas para a análise de estabilidade transitória, considerando o efeito das condutâncias de transferência na modelagem do problema. Para isto, são propostas Funções de Lyapunov Estendidas para modelos de sistemas de potência que não apresentam uma Função de Lyapunov no sentido usual. Essas FLE\'s são propostas tanto para sistemas de 1-máquina versus barramento infinito quanto para sistemas multimáquinas. Para a obtenção de boas estimativas do tempo de abertura, nos estudos de estabilidade transitória, é proposto um algoritmo iterativo. Este algoritmo fornece uma boa estimativa local da área de atração do ponto de equilíbrio estável de interesse. / The method of Lyapunov, one of the direct method, is efficient for transient stability analysis of power systems. The direct methods are well-suited for stability analysis of power systems, since they do not require the solution of the set of differential equations of the system model. The great difficulty of the direct methods is to find an auxiliary function (V) which satisfies the conditions of Lyapunov\'s Theorem V > 0 and V \'< or =\' 0. For many years the inclusion of the transfer conductances in the power system model, with the reduced network, is a issue of interest for several researchers. In 1989, Chiang studied the existence of energy functions for power systems with losses and he proved the non existence of a Lyapunov Function for power systems when the transfer conductance is taken into account. The transfer conductances are responsible for generating regions in the state space where the derivative of V is positive. Therefore, the function V is nor a Lyapunov Function, because its derivative is not semi negative definite. Recently, an Extension of the LaSalle\'s Invariance Principle has been proposed by Rodrigues, Alberto and Bretas (2000). This extension relaxes some of the requirements on the auxiliary function which is commonly called Lyapunov Function. In this extension, the derivative of the auxiliary function can be positive in some bounded regions of the state space and, for distinction purposes, it is called, as Extended Lyapunov Function. Inthis work, the Extension of the LaSalle\'s Invariance Principle and the Extended Lyapunov Function are used for the transient stability analysis of power systems with the model taking transfer conductances in consideration. For at purpose in this research, Extended Lyapunov Functions for power system models which do not have Lyapunov Functions in the usual sense are proposed. Extended Lyapunov Functions are proposed for a single-machine-infinite- bus-system and multimachine systems. For obtaining good estimates of the critical clearing time in transient stability analysis, an iterative algorithm is proposed. This algorithm supplies a good local estimate of the attraction area for the post fault stable equilibrium point.

O método de Lyapunov, também conhecido como método direto, é eficiente para análise de estabilidade transitória em sistemas de potência. Tal método possibilita a análise de estabilidade sem requerer o conhecimento das soluções das equações diferenciais que modelam o problema. A maior desvantagem da utilização dos métodos diretos, é sem dúvida encontrar uma função (V) que satisfaça as condições do Teorema de Lyapunov, ou seja, V > 0 e V \'< ou =\' 0. Durante muitos anos a inclusão das condutâncias de transferência na modelagem do sistema de potência, com a rede reduzida aos nós dos geradores, foi um assunto que despertou interesse em vários pesquisadores. Em 1989, Chiang provou a não existência de uma Função de Lyapunov para sistemas de potência quando as condutâncias de transferência são consideradas. Essas condutâncias de transferência são responsáveis por gerar regiões no espaço de estados onde tem-se V > 0, não satisfazendo as condições do Teorema de Lyapunov. Recentemente, Rodrigues, Alberto e Bretas (2000) apresentaram a Extensão do Princípio de Invariância de LaSalle, onde é permitido que a Função de Lyapunov possua, em algumas regiões limitadas do espaço de estados, a derivada positiva. Neste caso, estas funções passam a ser denominadas Funções de Lyapunov Estendidas (FLE). Neste trabalho, são utilizadas a Extensão do Princípio de Invariância de LaSalle e as Funções de Lyapunov Estendidas para a análise de estabilidade transitória, considerando o efeito das condutâncias de transferência na modelagem do problema. Para isto, são propostas Funções de Lyapunov Estendidas para modelos de sistemas de potência que não apresentam uma Função de Lyapunov no sentido usual. Essas FLE\'s são propostas tanto para sistemas de 1-máquina versus barramento infinito quanto para sistemas multimáquinas. Para a obtenção de boas estimativas do tempo de abertura, nos estudos de estabilidade transitória, é proposto um algoritmo iterativo. Este algoritmo fornece uma boa estimativa local da área de atração do ponto de equilíbrio estável de interesse. / The method of Lyapunov, one of the direct method, is efficient for transient stability analysis of power systems. The direct methods are well-suited for stability analysis of power systems, since they do not require the solution of the set of differential equations of the system model. The great difficulty of the direct methods is to find an auxiliary function (V) which satisfies the conditions of Lyapunov\'s Theorem V > 0 and V \'< or =\' 0. For many years the inclusion of the transfer conductances in the power system model, with the reduced network, is a issue of interest for several researchers. In 1989, Chiang studied the existence of energy functions for power systems with losses and he proved the non existence of a Lyapunov Function for power systems when the transfer conductance is taken into account. The transfer conductances are responsible for generating regions in the state space where the derivative of V is positive. Therefore, the function V is nor a Lyapunov Function, because its derivative is not semi negative definite. Recently, an Extension of the LaSalle\'s Invariance Principle has been proposed by Rodrigues, Alberto and Bretas (2000). This extension relaxes some of the requirements on the auxiliary function which is commonly called Lyapunov Function. In this extension, the derivative of the auxiliary function can be positive in some bounded regions of the state space and, for distinction purposes, it is called, as Extended Lyapunov Function. Inthis work, the Extension of the LaSalle\'s Invariance Principle and the Extended Lyapunov Function are used for the transient stability analysis of power systems with the model taking transfer conductances in consideration. For at purpose in this research, Extended Lyapunov Functions for power system models which do not have Lyapunov Functions in the usual sense are proposed. Extended Lyapunov Functions are proposed for a single-machine-infinite- bus-system and multimachine systems. For obtaining good estimates of the critical clearing time in transient stability analysis, an iterative algorithm is proposed. This algorithm supplies a good local estimate of the attraction area for the post fault stable equilibrium point.

Optimal control of autonomous systems is a fundamental and challenging problem, especially when many stringent safety constraints and tight control limitations are involved such that solutions are hard to determine. It has been shown that optimizing quadratic costs while stabilizing affine control systems to desired (sets of) states subject to state and control constraints can be reduced to a sequence of Quadratic Programs (QPs) by using Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs). Although computationally efficient, this method is limited by several factors which are addressed in this dissertation. The first contribution of this dissertation is to extend CBFs to high order CBFs (HOCBFs) that can accommodate arbitrary relative degree systems and constraints. The satisfaction of Lyapunov-like conditions in the HOCBF method implies the forward invariance of the intersection of a sequence of sets, which can then guarantee the satisfaction of the original safety constraint. Second, under tight control bounds, this dissertation proposes an analytical method to find sufficient conditions that guarantee the QP feasibility. The sufficient conditions are captured by a single state constraint that is enforced by a CBF and then added to the QP. Third, for complex safety constraints and systems in which it is hard to find sufficient conditions for feasibility, machine learning techniques are employed to learn the definitions of HOCBFs or feasibility constraints. Fourth, when time-varying control bounds and noisy dynamics are involved, adaptive CBFs (AdaCBFs) are proposed, which can guarantee the feasibility of the QPs if the original optimization problem itself is feasible. Finally, for systems with unknown dynamics, adaptive affine control dynamics are proposed to approximate the real unmodelled system dynamics which are updated based on the error states obtained by real-time sensor measurements. A set of events required to trigger a solution of the QP in order to guarantee safety is defined, and a condition that guarantees the satisfaction of the HOCBF constraint between events is derived. In order to address the myopic nature of the CBF method, a real-time control framework that combines optimal trajectories and the computationally efficient HOCBF method providing safety guarantees is also proposed. The HOCBFs and CLFs are used to account for constraints with arbitrary relative degrees and to track the optimal state, respectively. Eventually, an optimal control problem based on the proposed framework is always reduced to a sequence of QPs regardless of the formulation of the original cost function. Another contribution of the dissertation is to apply the above proposed methods to solve complex safety-critical optimal control problems, such as those arising in rule-based autonomous driving and optimal traffic merging control for Connected and Automated Vehicles (CAVs).

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).

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).

This work focuses on the safe and stable operation of an autonomous power system interconnecting an AC source with various types of power electronic loads. The stability of these systems is a challenge due to the inherent nonlinearity of the circuits involved. Traditionally, the stability analysis in this type of power systems has been approached by means of small-signal methodology derived from the Nyquist stability criterion. The small-signal analysis combined with physical insight and the adoption of safety margins is sufficient, in many cases, to achieve a stable operation with an acceptable system performance. Nonetheless, in many cases, the margins adopted result in conservative measures and consequent system over designs. This work studies the system stability under large-perturbations by means of three different tools, namely parameter space mapping, energy functions, and time domain simulations. The developed parameters space mapping determines the region of the state and parameter space where the system operation is locally stable. In this way stability margins in terms of physical parameters can be established. Moreover, the boundaries of the identified stability region represent bifurcations of the system where typical nonlinear behavior appears. The second approach, based on the Lyapunov direct method, attempts to determine the region of attraction of an equilibrium point, defined by an operation condition. For this a Lyapunov function based on linear matrix inequalities was constructed and tested on a simplified autonomous system model. In Addition, the third approach simulates the system behavior on a computer using a detailed system model. The higher level of model detail allows identifying unstable behavior difficult to observe when simpler models are used. Because the stability of the autonomous power system is strongly associated with the characteristics of the energy source, an improved voltage controller for the generator is also presented. The generator of an autonomous power system must provide a good performance under a wide variety of regimes. Under these conditions a model based controller is a good solution because it naturally adapts to the changing requirements. To this extent a controller based on the model of a variable frequency synchronous generator has been developed and tested. The results obtained show a considerable improvement performance when compared to previous practices. / Ph. D.

Resumen: En esta tesis doctoral se presentan tres modelos matemáticos que describen el comportamiento de dos enfermedades parasitarias con vector de transmisión; de los cuales dos modelos están dedicados a la Toxoplasmosis donde se explora la dinámica de la enfermedad a nivel de la población humana y de gatos domésticos. Los gatos juegan un papel de agentes infecciosos del Toxoplasma gondii. La dinámica cualitativa del modelo es determinada por el umbral básico de reproducción, R0. Si el parámetro R0 < 1, entonces la solución converge al punto de equilibrio libre de la enfermedad. Por otro lado, si R0 > 1, la convergencia es al punto de equilibrio endémico. Las simulaciones numéricas ilustran diferentes dinámicas en función del parámetro umbral R0 y muestra la importancia de este parámetro en el sector salud. Finalmente la Babesiosis bovina se modela a partir de cinco ecuaciones diferenciales ordinarias, que permiten explicar la influencia de los parámetros epidemiológicos en la evolución de la enfermedad. Los estados estacionarios del sistema y el número básico de reproducción R0 son determinados. La existencia del punto endémico y libre de enfermedad se expone, puntos que dependen del R0, ratificando la importancia del parámetro umbral en la salud publica. Objetivo: Construir modelos matemáticos epidemiológicos aplicados a enfermedades parasitarias (Toxoplasmosis y Babesiosis) con vector de transmisión. Metodología: Para la construcción de los modelos matemáticos epidemiológicos es necesario representar la enfermedad a partir de modelos de flujo, permitiendo ver la dinámica de la población entre los diferentes estadíos de la enfermedad, dichos movimientos son analizados a partir de sistemas dinámicos, análisis matemático y métodos numéricos; con estas herramientas es posible hacer un estudio detallado del modelo, permitiendo calcular parámetros umbrales que dominan la dinámica de la enfermedad y a su vez simular escenarios reales e hipotéticos. / Aranda Lozano, DF. (2011). Modeling of parasitic diseases with vector of transmission: toxoplasmosis and babesiosis bovine [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/11539