Dynamical systems possess an interesting and complex behaviour that have attracted a number of researchers across different fields, such as Biology, Economics and most importantly in Engineering. The complex and unpredictability of nonlinear customary behaviour or the chaotic behaviour, makes it strange to analyse them. This thesis presents the analysis of the system of nonlinear differential equations of the so--called Lu--Chen--Cheng system. The system has similar dynamical behaviour with the famous Lorenz system. The nature of equilibrium points and stability of the system is presented in the thesis. Examples of chaotic dynamical systems are presented in the theory. The thesis shows the dynamical structure of the Lu--Chen--Cheng system depending on the particular values of the system parameters and routes to chaos. This is done by both the qualitative and numerical techniques. The bifurcation diagrams of the Lu--Chen--Cheng system that indicate limit cycles and chaos as one parameter is varied are shown with the help of the largest Lyapunov exponent, which also confirms chaos in the system. It is found out that most of the system's equilibria are unstable especially for positive values of the parameters $a, b$. It is observed that the system is highly sensitive to initial conditions. This study is very important because, it supports the previous findings on chaotic behaviours of different dynamical systems.

El diseño y análisis de sistemas dinámicos es una tarea desa-fiante debido fundamentalmente a la no linealidad requerida en las ecuaciones matemáticas de los modelos, con el fin de lograr representaciones realistas de los sistemas y procesos. En lo que a ingeniería química respecta, estas no linealidades se manifiestan en los diversos términos de las ecuaciones de balance y se potencian por la realimentación positiva intro-ducida por los reciclos de masa y energía y por las hoy en día rigurosas restricciones de calidad de producto, medio ambien-te y seguridad. En esta tesis, se proponen metodologías para el diseño y análisis de sistemas dinámicos no lineales emple-ando elementos de la Teoría de Estabilidad de Lyapunov con la aplicación de técnicas de optimización. La base principal de la investigación es la determinación de estimaciones de los dominios de atracción de los puntos de equilibrio asintótica-mente estables. En primer lugar, se desarrolla una metodología para la estimación de dominios de atracción haciendo uso de funciones de Lyapunov de tipo racional y aplicando técnicas de optimización global. Esta metodología consiste en determi-nar el mejor conjunto de nivel de una función de Lyapunov contenida en la región de definición negativa de su derivada temporal. Posteriormente, se desarrolla una metodología para el diseño de sistemas dinámicos no lineales con el objetivo de optimizar en forma indirecta la extensión de los dominios de atracción. Este enfoque de diseño, asegura simultáneamente estabilidad asintótica del punto de equilibrio resultante y un dominio de atracción óptimo en términos del criterio adoptado. Las metodologías propuestas se ilustran por medio de ejemplos y se aplican a procesos y sistemas de interés ingenieril. / Dynamic systems design and analysis is a challenging task due to the nonlinearity required by the mathematical modeling equations, in order to obtain a realistic representation of the systems and processes. Regarding the chemical engineering discipline, these complexities are found in the different terms of the balance equations and are boosted by the positive feedback introduced by mass and energy recycles and by the current restrictive constraints on product quality, environment and safety. In this thesis, different methodolo-gies are proposed for the design and analysis of non-linear dynamical systems based on elements of the Lyapunov stability theory with the application of optimization techniques. The calculation of estimations of the domains of attraction of asymptotic stable equilibrium points is the main basis of this research. In the first place, a methodology is developed for the estimation of the domains of attraction of asymptotically stable equilibrium points using Lyapunov functions of the rational type and applying global optimization techniques,. This methodology consists in finding the best level set of a Lyapunov function contained in the negative definite region of its temporal derivate. Secondly, a metho-dology for the design of non-linear dynamical systems in order to optimize in an indirect way the extension of the domain of attraction is developed. This approach simultaneously ensures asymptotic stability of the resulting equilibrium and an optimum domain of attraction in terms of the adopted criterion. The proposed methodologies are illustrated through examples and applied to processes and systems of engineering interest.

The number of objects occupying orbital regimes beyond Geosynchronous Earth Orbit and cislunar space are expected to grow in the coming years; Especially with the Moon reemerging as latest frontier in the race for space exploration and technological superiority. In order to support this growth, new methods of autonomously navigating in cislunar space are necessary to reduce demand and reliance on ground based tracking infrastructure. Periodic orbits about the first libration point offer favorable vantage points for scientific or military spacecraft missions involving the Earth or Moon. This thesis develops a new autonomous spacecraft navigation method for cislunar space and analyzes its performance applied to Lyapunov and halo orbits around $L_1$. This method uses existing lunar ranging retroreflectors (LRRR) installed on the Moon's surface in the 1960s and 1970s. A spacecraft can make laser ranging measurements to the LRRR to estimate its orbit states. A simulation platform was created to test this concept in the circular restricted three body problem and evaluate its performance. This navigation method was found to be successful for a subset of Lyapunov and halo orbits when cycling the five measurement targets. Simulation data showed that sub-kilometer position estimation and sub 2 centimeter per second velocity accuracies are achievable without receiving any state updates from external sources. / Master of Science / The number of objects occupying the space between the Earth and Moon (cislunar space) is expected to grow in the coming years as the Moon regains popularity in the latest race for space exploration and technological superiority. In order to support this growth, new methods of determining a spacecraft's position and velocity while in this region of space are necessary to reduce demand and dependence on Earth based methods, which have historically relied upon. Repeating orbits around the equilibrium point between the Earth and Moon provide valuable observation points for scientific and military spacecraft missions. This thesis develops a new spacecraft navigation method for cislunar space and analyzes how well it performs in two different types of orbits, Lyapunov and halo orbits. This method uses existing laser reflector panels that were installed on the Moon's surface in the 1960s and 70s. A spacecraft can use these panels to make range or distance measurements in order to estimate its position and velocity. Software was written to simulate the motion of a spacecraft as it is acted on by gravity from the Earth and Moon. Different scenarios were then simulated and used to test this concept and evaluate its performance. Lunar laser ranging was found to be successful for a some Lyapunov and halo orbits when switching between the five different reflector panels on the Moon. Data generated from the simulations show that position can be estimated with errors less than SI{1}{kilo meter}, and velocity error on the order of a few centimeters per second, all without receiving any additional information from Earth based systems.

No description available.

The problem of reorienting a spacecraft to acquire a moving target is investigated. The spacecraft is modeled as a rigid body with N axisymmetric wheels controlled by axial torques, and the kinematics are represented by Modified Rodriques Parameters. The trajectory, denoted the reference trajectory, is one generated by a virtual spacecraft that is identical to the actual spacecraft. The open-loop reference attitude, angular velocity, and angular acceleration tracking commands are constructed so that the solar panel vector is perpendicular to the sun vector during the tracking maneuver. We develop a nonlinear feedback tracking control law, derived from Lyapunov stability and control theory, to provide the control torques for target tracking. The controller makes the body frame asymptotically track the reference motion when there are initial errors in the attitude and angular velocity. A spacecraft model, based on the X-ray Timing Explorer spacecraft, is used to demonstrate the effectiveness of the Lyapunov controller in tracking a given target. / Master of Science

Les contrôleurs à réinitialisation sont une classe de systèmes hybrides dont la valeur de tout ou partie des états peut être instantannément modifiée sous certaines conditions algébriques. Cette interaction entre dynamique temps-continu et temps-discret de ces contrôleurs permet souvent de dépasser les limites des contrôleurs temps-continu. Dans cette thèse, nous proposons des conditions constructives (sous forme d’Inégalités Matricielles Linéaires) pour analyser la stabilité et les performances de boucle de commande incluant un contrôleur à réinitialisation. En particulier, nous prenons en compte la présence de saturation en amplitude des actionneurs du système. Ces non-linéarités sont souvent source d'une dégradation des performances voir d’instabilité. Les résultats proposés permettent d’estimer le domaine de stabilité et un niveau de performance pour ces systèmes, en s’appuyant sur des fonctions de Lyapunov quadratiques ou quadratiques par morceaux. Au delà de l'aspect analyse, nous exposons deux approches pour améliorer la région de stabilité (nouvelle loi de réinitialisation et stratégie « anti-windup »). / Hybrid controllers are flexible tools for achieving system stabilization and/or performance improvement tasks. More particularly, hybrid controllers enrich the spectrum of achievable trade-offs. Indeed, the interaction of continuous- and discrete-time dynamics in a hybrid controller leads to rich dynamical behavior and phenomena not encountered in purely continuous-time system. Reset control systems are a class of hybrid controllers whose states are reset depending on an algebraic condition. In this thesis, we propose constructive conditions (Linear Matrix Inequalities) to analyze stability and performance level of a closed-loop system including a reset element. More particularly, we consider a magnitude saturation which could be the source of undesirable effects on these performances, including instability. Proposed results estimate the stability domain and a performance level of such a system, by using Lyapunov-like approaches. Constructive algorithms are obtained by exploiting properties of quadratic - or piecewise quadratic - Lyapunov functions. Beyond analysis results, we propose design methods to obtain a stability domain as large as possible. Design methods are based on both continuous-time approaches (anti-windup compensator) and hybrid-time approaches (design of adapted reset rules).

Four aspects of the dynamics of continuous-time dynamical systems are studied in this work. The relationship between the Lyapunov exponents of the original system and the Lyapunov exponents of induced Poincare maps is examined. The behavior of these Poincare maps as discriminators of chaos from noise is explored, and the possible Poissonian statistics generated at rarely visited surfaces are studied.

Ce doctorat est centré sur trois objectifs principaux. Le premier est le développement d’une stratégie de construction de retours d’état sans terme à retard qui stabilisent globalement asymptotiquement uniformément l’origine de systèmes variant dans le temps non linéaires ayant un retard distribué dans la commande. Le deuxième est l’adaptation de la célèbre technique de construction de lois de commande appelée backstepping au cas de systèmes non linéaires de forme feedback ayant dans la commande un retard ponctuel constant. Le troisième est la résolution d’un problème de stabilisation pratique pour une famille de systèmes non linéaires ayant un retard ponctuel dans l’entrée constant et ayant des commandes échantillonnées dont les instants d’échantillonnage dépendent de la valeur de la variable d’état du système. Chacun de ces trois objectifs est atteint au moyen de technique qui utilisent certains opérateurs d’un nouveau type. Des résultats de robustesse de type Input-to-State-Stability sont établis au moyen de fonctionnelles de Lyapunov-Krasovskii. / This PhD thesis is focused on three main objectives. The first goal is to develop a method of construction of state feedbacks without retarded terms that render the origin of timevarying nonlinear systems with a distributed input delay globally uniformly asymptotically stable.The second objective is the adaptation of the celebrated technique of construction of state feedbacks called ’backstepping’ to the case of nonlinear systems in feedback form with a pointwise constant delay in the input. The third aim of this thesis is to solve a problem of practical stabilization for a family of nonlinear systems with sampled retarded state feedbacks whose sampling instants depend on the value of the state variable of the system. Each of these three goals relies on the introduction of operators of a new type. Robustness results of Input-to-State-Stability type are established by using Lyapunov-Krasovskii functionals.

Nesta dissertação estudamos de um ponto de vista probabilístico, o comportamento assintótico de sistemas dinâmicos. Um exemplo simples de formular e profundo é o estudo de produto aleatório de matrizes (FURSTENBERG; KESTEN, 1960). Utilizaremos como ferramenta o estudo dos cociclos lineares, posteriormente mediante o Teorema de Furstenberg-Kesten definiremos o expoente de Lyapunov do cociclo, em seguida enunciamos e provamos o Teorema Ergódico Multiplicativo de Oseledets o qual nos permite entender o comportamento das órbitas típicas para um cociclo dado F : M x R2 → M x R2. O Teorema de Fusrtenberg-Kesten fornece informações sobre o crescimento das matrizes An(x), enquanto o Teorema de Oseledets descreve o comportamento assintótico dos vetores An(x).v. Finalmente provamos o teorema principal desta dissertação, o Teorema de Furstenberg o qual diz que na maioria dos casos o maior expoente de Lyapunov é positivo (FURSTENBERG, 1963). / In this thesis we study from a probabilistic point of view, the asymptotic behavior of dynamic systems, a deep and simple example is the random product of matrices (FURSTENBERG; KESTEN, 1960). We will use as a tool, the study of linear cocycles, later using the Furstenberg- Kesten Theorem we will define the Lyapunov exponent of the cocycle, then we enunciate and prove the Multiplicative Ergodic Theorem of Oseledets which allow us to understand the behavior of the typical orbits for a given cocycle F : M x R2 → M x R2. The Fusrtenberg- Kesten theorem provides information on the growth of the matrices A(x), while the theorems of Oseledets describe the asymptotic behavior of the vectors An(x).v. Finally we prove our main theorem, Furstenbergs Theorem which states that in most cases the greatest exponent of Lyapunov is positive (FURSTENBERG, 1963).

Nesta dissertação estudamos de um ponto de vista probabilístico, o comportamento assintótico de sistemas dinâmicos. Um exemplo simples de formular e profundo é o estudo de produto aleatório de matrizes (FURSTENBERG; KESTEN, 1960). Utilizaremos como ferramenta o estudo dos cociclos lineares, posteriormente mediante o Teorema de Furstenberg-Kesten definiremos o expoente de Lyapunov do cociclo, em seguida enunciamos e provamos o Teorema Ergódico Multiplicativo de Oseledets o qual nos permite entender o comportamento das órbitas típicas para um cociclo dado F : M x R2 → M x R2. O Teorema de Fusrtenberg-Kesten fornece informações sobre o crescimento das matrizes An(x), enquanto o Teorema de Oseledets descreve o comportamento assintótico dos vetores An(x).v. Finalmente provamos o teorema principal desta dissertação, o Teorema de Furstenberg o qual diz que na maioria dos casos o maior expoente de Lyapunov é positivo (FURSTENBERG, 1963). / In this thesis we study from a probabilistic point of view, the asymptotic behavior of dynamic systems, a deep and simple example is the random product of matrices (FURSTENBERG; KESTEN, 1960). We will use as a tool, the study of linear cocycles, later using the Furstenberg- Kesten Theorem we will define the Lyapunov exponent of the cocycle, then we enunciate and prove the Multiplicative Ergodic Theorem of Oseledets which allow us to understand the behavior of the typical orbits for a given cocycle F : M x R2 → M x R2. The Fusrtenberg- Kesten theorem provides information on the growth of the matrices A(x), while the theorems of Oseledets describe the asymptotic behavior of the vectors An(x).v. Finally we prove our main theorem, Furstenbergs Theorem which states that in most cases the greatest exponent of Lyapunov is positive (FURSTENBERG, 1963).