>Magister Scientiae - MSc / The influx of infecteds into a population plays a critical role in HIV transmission. These infecteds are known to migrate from one region to another, thereby having some interaction with a host population. This interactive mobility or migration causes serious public health problems. In a very insightful paper by Shedlin et al. [51], the authors discover risk factors but also beneficial factors with respect to fighting human immunodeficiency virus (HIV) transmission, in the lifestyles of immigrants from different cultural backgrounds. These associated behavioral factors with cross-cultural migrations have not received adequate theoretical a attention. In this dissertation we use the compartmental model of Bhunu et al. [6] to form a new model of the HIV epidemic, to include the effect of infective immigrants in a given population. In fact, we first produce a deterministic model and provide a detailed analysis. Thereafter we introduce stochastic perturbations on the new model and study stability of the disease-free equilibrium (DFE) state. We investigate theoretically and computationally how cross-cultural migrations and public health education impacts on the HIV transmission, and how best to intervene in order to minimize the spread of the disease. In order to understand the long-time progression of the disease, we calculate the threshold parameter, known as the basic reproduction number, R0. The basic reproduction number has the property that if R0 is sufficiently small, usually R0 < 1, then the disease eventually vanishes from the population, but if R0 > 1, the disease persists in the population. We study the sensitivity of the basic reproduction number with respect to model parameters. In this regard, if R0 < 1, we show that the DFE is locally asymptotically stable. We also show global stability of the DFE using the Lyapunov method. We derive the endemic equilibrium points of our new model. We intend to counteract the negative effect of the influx of infecteds into a population with educational campaigns as a control strategy. In doing so, we employ optimal control theory to find an optimal intervention on HIV infection using educational campaigns as a basic input targeting the host population. Our aim is to reduce the total number of infecteds while minimizing the cost associated with the use of educational campaign on [0, T ]. We use Pontryagin’s maximum principle to characterize the optimal level of the control. We investigate the optimal education campaign strategy required to achieve the set objective of the intervention. The resulting optimality system is solved numerically using the Runge-Kutta fourth order method. We present numerical results obtained by simulating the optimality system using ODE-solvers in MATLAB program. We introduce randomness known as white noise into our newly formed model, and discuss the almost sure exponential stability of the disease-free equilibrium. Finally, we verify the analytical results through numerical simulations.

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The task of modeling and controlling the evolution of dynamical sys- tems is one of the main objectives in mechatronics engineering. When approaching the problem of controlling physical or digital systems, the dynamical models have been historically divided into continuous-time, described by differential equations, and discrete-time, described by difference equations. In the last decade, a new class of models, known as hybrid dynamical systems, has gained popularity in the control community because of its high versatility. This framework combines continuous-time and discrete- time evolution, thus allowing for both the description of a broader class of systems and the achievement of better-performing controllers, compared to the traditional continuous-time alternatives. After the first rigorous introduction of the framework, several Lyapunov-based results were published in the literature, and numerous application areas were shown to benefit from the introduction of a hybrid dynamics, like systems involving impacts or physical systems connected to digital controllers (cyber-physical systems). In this thesis, we use the hybrid framework to study different mechatronics-inspired control problems. The applications we consider are diverse, so we split the presentation into three parts. In the first part we further analyze a particular hybrid control strategy, known as reset control, providing some new theoretical guarantees, together with an application to adaptive control. In the second part we consider two applications of the hybrid framework to the network dynamics field, specifically we analyze the problems of distributed state estimation and of uniform synchronization of nonlinear oscillators. In the third part, we use a hybrid approach to study two applications where this framework has been rarely employed, or not at all, namely smart agriculture and trajectory tracking for a bipedal walking robot. We study these application-inspired problems from a theoretical point of view, giving robust Lyapunov-based stability guarantees. We complement the theoretical analysis with numerical results, obtained from simulations or from experiments.

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A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior. Hybrid systems arise in a wide variety of important applications in diverse areas, ranging from biology to computer science to air traffic dynamics. The interaction of continuous- and discrete-time dynamics in a hybrid system often leads to very rich dynamical behavior and phenomena that are not encountered in purely continuous- or discrete-time systems. Investigating the dynamical behavior of hybrid systems is of great theoretical and practical importance. The objectives of this thesis are to develop the qualitative theory of nonlinear hybrid systems with impulses, time-delay, switching modes, and stochastic disturbances, to develop algorithms and perform analysis for hybrid systems with an emphasis on stability and control, and to apply the theory and methods to real-world application problems. Switched nonlinear systems are formulated as a family of nonlinear differential equations, called subsystems, together with a switching signal that selects the continuous dynamics among the subsystems. Uniform stability is studied emphasizing the situation where both stable and unstable subsystems are present. Uniformity of stability refers to both the initial time and a family of switching signals. Stabilization of nonlinear systems via state-dependent switching signal is investigated. Based on assumptions on a convex linear combination of the nonlinear vector fields, a generalized minimal rule is proposed to generate stabilizing switching signals that are well-defined and do not exhibit chattering or Zeno behavior. Impulsive switched systems are hybrid systems exhibiting both impulse and switching effects, and are mathematically formulated as a switched nonlinear system coupled with a sequence of nonlinear difference equations that act on the switched system at discrete times. Impulsive switching signals integrate both impulsive and switching laws that specify when and how impulses and switching occur. Invariance principles can be used to investigate asymptotic stability in the absence of a strict Lyapunov function. An invariance principle is established for impulsive switched systems under weak dwell-time signals. Applications of this invariance principle provide several asymptotic stability criteria. Input-to-state stability notions are formulated in terms of two different measures, which not only unify various stability notions under the stability theory in two measures, but also bridge this theory with the existent input/output theories for nonlinear systems. Input-to-state stability results are obtained for impulsive switched systems under generalized dwell-time signals. Hybrid time-delay systems are hybrid systems with dependence on the past states of the systems. Switched delay systems and impulsive switched systems are special classes of hybrid time-delay systems. Both invariance property and input-to-state stability are extended to cover hybrid time-delay systems. Stochastic hybrid systems are hybrid systems subject to random disturbances, and are formulated using stochastic differential equations. Focused on stochastic hybrid systems with time-delay, a fundamental theory regarding existence and uniqueness of solutions is established. Stabilization schemes for stochastic delay systems using state-dependent switching and stabilizing impulses are proposed, both emphasizing the situation where all the subsystems are unstable. Concerning general stochastic hybrid systems with time-delay, the Razumikhin technique and multiple Lyapunov functions are combined to obtain several Razumikhin-type theorems on both moment and almost sure stability of stochastic hybrid systems with time-delay. Consensus problems in networked multi-agent systems and global convergence of artificial neural networks are related to qualitative studies of hybrid systems in the sense that dynamic switching, impulsive effects, communication time-delays, and random disturbances are ubiquitous in networked systems. Consensus protocols are proposed for reaching consensus among networked agents despite switching network topologies, communication time-delays, and measurement noises. Focused on neural networks with discontinuous neuron activation functions and mixed time-delays, sufficient conditions for existence and uniqueness of equilibrium and global convergence and stability are derived using both linear matrix inequalities and M-matrix type conditions. Numerical examples and simulations are presented throughout this thesis to illustrate the theoretical results.

A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior. Hybrid systems arise in a wide variety of important applications in diverse areas, ranging from biology to computer science to air traffic dynamics. The interaction of continuous- and discrete-time dynamics in a hybrid system often leads to very rich dynamical behavior and phenomena that are not encountered in purely continuous- or discrete-time systems. Investigating the dynamical behavior of hybrid systems is of great theoretical and practical importance. The objectives of this thesis are to develop the qualitative theory of nonlinear hybrid systems with impulses, time-delay, switching modes, and stochastic disturbances, to develop algorithms and perform analysis for hybrid systems with an emphasis on stability and control, and to apply the theory and methods to real-world application problems. Switched nonlinear systems are formulated as a family of nonlinear differential equations, called subsystems, together with a switching signal that selects the continuous dynamics among the subsystems. Uniform stability is studied emphasizing the situation where both stable and unstable subsystems are present. Uniformity of stability refers to both the initial time and a family of switching signals. Stabilization of nonlinear systems via state-dependent switching signal is investigated. Based on assumptions on a convex linear combination of the nonlinear vector fields, a generalized minimal rule is proposed to generate stabilizing switching signals that are well-defined and do not exhibit chattering or Zeno behavior. Impulsive switched systems are hybrid systems exhibiting both impulse and switching effects, and are mathematically formulated as a switched nonlinear system coupled with a sequence of nonlinear difference equations that act on the switched system at discrete times. Impulsive switching signals integrate both impulsive and switching laws that specify when and how impulses and switching occur. Invariance principles can be used to investigate asymptotic stability in the absence of a strict Lyapunov function. An invariance principle is established for impulsive switched systems under weak dwell-time signals. Applications of this invariance principle provide several asymptotic stability criteria. Input-to-state stability notions are formulated in terms of two different measures, which not only unify various stability notions under the stability theory in two measures, but also bridge this theory with the existent input/output theories for nonlinear systems. Input-to-state stability results are obtained for impulsive switched systems under generalized dwell-time signals. Hybrid time-delay systems are hybrid systems with dependence on the past states of the systems. Switched delay systems and impulsive switched systems are special classes of hybrid time-delay systems. Both invariance property and input-to-state stability are extended to cover hybrid time-delay systems. Stochastic hybrid systems are hybrid systems subject to random disturbances, and are formulated using stochastic differential equations. Focused on stochastic hybrid systems with time-delay, a fundamental theory regarding existence and uniqueness of solutions is established. Stabilization schemes for stochastic delay systems using state-dependent switching and stabilizing impulses are proposed, both emphasizing the situation where all the subsystems are unstable. Concerning general stochastic hybrid systems with time-delay, the Razumikhin technique and multiple Lyapunov functions are combined to obtain several Razumikhin-type theorems on both moment and almost sure stability of stochastic hybrid systems with time-delay. Consensus problems in networked multi-agent systems and global convergence of artificial neural networks are related to qualitative studies of hybrid systems in the sense that dynamic switching, impulsive effects, communication time-delays, and random disturbances are ubiquitous in networked systems. Consensus protocols are proposed for reaching consensus among networked agents despite switching network topologies, communication time-delays, and measurement noises. Focused on neural networks with discontinuous neuron activation functions and mixed time-delays, sufficient conditions for existence and uniqueness of equilibrium and global convergence and stability are derived using both linear matrix inequalities and M-matrix type conditions. Numerical examples and simulations are presented throughout this thesis to illustrate the theoretical results.

Le retard est un phénomène naturel présent dans la majorité des systèmes physiques et dans les applications d’ingénierie, ainsi, les systèmes à retard ont été un domaine de recherche très actif en automatique durant les 60 dernières années. La conception d’observateur est un des sujets les plus importants qui a été étudié, ceci est dû à l’importance des observateurs en automatique et dans les systèmes de commande en absence de capteur pour mesurer une variable. Dans ce travail, l’objectif principal est de concevoir des observateurs pour différentes classes de systèmes à retard avec un retard arbitrairement large, et ce en utilisant différentes approches. Dans la première partie de cette thèse, la conception d’un observateur a été réalisée pour une classe de systèmes non linéaires triangulaires avec une sortie échantillonnée et un retard arbitraire. Une l’autre difficulté majeure avec cette classe de systèmes est le fait que la matrice d’état dépend du signal de sortie non-retardé qui est immesurable. Un nouvel observateur en chaine, composé de sous-observateurs en série est conçu pour compenser les retards arbitrairement larges. Dans la seconde partie de ce travail, un nouvel observateur a été conçu pour un autre type de systèmes non linéaires triangulaires, où le retard a été considéré, cette fois-ci, comme une équation aux dérivées partielles de type hyperbolique du premier ordre. La transformation inverse en backstepping et le concept de l’observateur en chaine ont été utilisés lors de la conception de cet observateur afin d’assurer son efficacité en cas de grands retards. Dans la dernière partie de cette thèse, la conception d’un nouvel observateur a été réalisée pour un type de système modélisé par des équations paraboliques non linéaires où les mesures sont issues d’un nombre fini de points du domaine spatial. Cet observateur est constitué d’une série de sous-observateurs en chaine. Chaque sous-observateur compense une fraction du retard global. L'analyse de la stabilité des systèmes d’erreur a été fondée sur différentes fonctionnelles Lyapunov-Krasovskii. Par ailleurs, différents instruments mathématiques ont été employés au cours des différentes preuves présentées. Les résultats de simulation ont été présentés dans le but de confirmer l'exactitude des résultats théoriques / Time-delay is a natural phenomenon that is present in most physical systems and engineering applications, thus, delay systems have been an active area of research in control engineering for more than 60 years. Observer design is one of the most important subject that has been dealt with, this is due to the importance of observers in control engineering systems not only when sensing is not sufficient but also when a sensing reliability is needed. In this work, the main goal was to design observers for different classes of nonlinear delayed systems with an arbitrary large delay, using different approaches. In the first part, the problem of observer design is addressed for a class of triangular nonlinear systems with not necessarily small delay and sampled output measurements. Another major difficulty with this class of systems is the fact that the state matrix is dependent on the un-delayed output signal which is not accessible to measurement. A new chain observer, composed of sub-observers in series, is designed to compensate for output sampling and arbitrary large delays.In the second part of this work, another kind of triangular nonlinear delayed systems was considered, where this time the delay was considered as a first order hyperbolic partial differential equation. The inverse backstepping transformation was invoked and a chain observer was developed to ensure its effectiveness in case of large delays. Finally, a new observer was designed for a class of nonlinear parabolic partial differential equations under point measurements, in the case of large delays. The observer was composed of several chained sub-observers. Each sub-observer compensates a fraction of the global delay. The stability analyses of the error systems were based on different Lyapunov-Krasovskii functionals. Also different mathematical tools have been used in order to prove the results. Simulation results were presented to confirm the accuracy of the theoretical results

Orientador: Carlos Roberto Minussi / Resumo: Apresenta-se um modelo matemático, baseado na análise de sensibilidade, para a realização de controle de segurança dinâmica para a melhoria da estabilidade transitória de sistemas elétricos de potência. O controle de segurança dinâmica implementado consiste na alteração da impe-dância do sistema elétrico pela retirada/inclusão de linhas de transmissão. A proposta consiste em determinar um modelo de sensibilidade da margem de segurança do sistema em relação à admitância (impedância) do elemento considerado. Deste modo, pode-se estimar o impacto que as alterações no sistema de transmissão podem causar sobre a estabilidade transitória do sistema. Com as devidas adaptações, os resultados aqui apresentados podem ser estendidos para o caso do uso de dispositivos FACTS. A análise da estabilidade é realizada, via uso do conceito de sensibilidade da margem de segurança do sistema, que é determinada pelo Méto-do Direto de Lyapunov, por meio da função de energia total do sistema. Trata-se de um resul-tado que visa dar maior suporte ferramental aos planejadores e aos operadores dos sistemas de energia elétrica. Visando ilustrar os resultados auferidos com a aplicação do modelo proposto, são apresentados os resultados via simulações considerando-se dois sistemas de energia elétri-ca (sistema de 9 barras / sistema clássico Anderson & Fouad, e uma versão do sistema sul bra-sileiro). / Abstract: This work presents a mathematical model based on sensitivity analysis for the implementation of Security Dynamic Control for improvement of transient stability of electric power systems. The Security Dynamic Control performed corresponds to system impedance change by out-put/input of transmission lines (TL’s). The propose aims to determine a sensitivity model for the security margin of the system in relation to the impedance (susceptance) of the considered element. Thus, it is possible estimate the influence of TL’s on transient stability. Considering adaptations and some simplifications, the results obtained with this proposal can be used for FACTS devices. The stability analysis is achieved using the Security Margin concept deter-mined by the direct Lyapunov (energy) method; it is a supporting tool to the planners and op-erators of electric power systems. In order to illustrate the results obtained with the application of the proposed model, the simulation results are presented considering two electric power systems (9-bus system / Anderson & Fouad classic system, and one version of the South Bra-zilian system). / Doutor

Les systèmes non linéaires sont si diverses que des outils communs de contrôle sont difficiles à développer. La théorie du contrôle non linéaire nécessite une analyse mathématique rigoureuse pour motiver ses conclusions. Cette thèse aborde deux branches distinctes et bien importantes de la théorie du contrôle non linéaire: le contrôle des systèmes non-linéaires incertains et le contrôle des systèmes sous-actionnés.Dans la première partie, une classe de contrôleurs par mode glissant d’ordre supérieur (MGOS) robuste, basée sur la synthèse de Lyapunov, est développée pour le contrôle des systèmes non-linéaires incertains. Cette classe de contrôleurs est basée sur une classe de régulateurs qui stabilisent une pure chaîne d’intégrateurs en temps fini, et nécessite la connaissance a priori des bornes sur les incertitudes du système. Puis, afin d’éliminer la dépendance liée à la connaissance de ces bornes, un contrôleur par MGOS adaptatif est développé. Dans un deuxième temps, un contrôleur par MGOS homogène universel est développé où il est montré que le degré d’homogénéité peut être manipulé pour obtenir des avantages supplémentaires, tels que la bornitude de la commande, la garantie d’une amplitude minimale de la discontinuité de la commande et la convergence en temps fixe. Les performances des contrôleurs proposés ont été démontrées par des simulations et à travers des résultats expérimentaux sur un système pile à combustible.Dans la deuxième partie de la thèse, deux problèmes de commande de systèmes sous-actionnés sont étudiés. Le premier problème concerne le suivi de chemin global d’un robot mobile avec un point de visée. Le deuxième problème concerne la poursuite de trajectoire globale d’un bateau. Ces deux problèmes sont de nature distincte, cependant, ils sont soumis à des contraintes physiques similaires liées à la bornitude de la commande. Ainsi, les contrôleurs proposés sont basés sur l’utilisation de commandes saturées. Des simulations ont été effectuées pour démontrer les performances de ces contrôleurs. / Nonlinear systems are so diverse that generalized tools for control are difficult to develop. Nonlinear control theory requires rigorous mathematical analysis to justify its conclusions. This thesis addresses two distinct, yet important branches of nonlinear control theory: control of uncertain nonlinear systems and control of under-actuated systems.In the first part, a class of Lyapunov-based robust arbitrary higher order sliding mode (HOSM) controllers is developed for the control of uncertain nonlinear systems. This class of controllers is based on a class of controllers for finite-time stabilization of pure integrator chain, and requires the limits of the system uncertainty to be known a-priori. Then, in order to eliminate the dependence on the knowledge of these limits, an adaptive arbitrary HOSM controller is developed. Using this new class, a universal homogeneous arbitrary HOSM controller is developed and it is shown that the homogeneity degree can be manipulated to obtain additional advantages in the proposed controllers, such as bounded control, minimum amplitude of discontinuous control and fixed time convergence. The performance of the controllers has been demonstrated through simulations and experiments on a fuel cell system.In the next part, the control of two under-actuated systems is studied. The first control problem is the global path following of car-type robotic vehicle, using target-point. The second problem is the precise tracking of surface marine vessels. Both these problems are distinct in nature; however, they are subjected to similar physical constraints. The solutions proposed for these control problems use saturated controls, taking into account the physical bounds on the control inputs. Simulations have been performed to demonstrate the performance of these controllers.