A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science in Engineering in the Control Research Group School of Electrical and Information Engineering, Johannesburg, 2017 / In this research, we observed a torque-related limitation in the swing-up control of underactuated mechanical systems which had been integrated with viscous damping in the unactuated joint. The objective of this research project was thus to develop a practical work-around solution to this limitation. The nth order underactuated robotic system is represented in this research as a collection of compounded pendulums with n-1 actuators placed at each joint with the exception of the first joint. This system is referred to as the PAn-1 robot (Passive first joint, followed by n-1 Active joints), with the Acrobot (PA1 robot) and the PAA robot (or PA2 robot) being among the most well-known examples. A number of friction models exist in literature, which include, and are not exclusive to, the Coulomb and the Stribeck effect models, but the viscous damping model was selected for this research since it is more extensively covered in existing literature. The effectiveness of swing-up control using Lyapunov’s direct method when applied on the undamped PAn-1 robot has been vigorously demonstrated in existing literature, but there is no literature that discusses the swing-up control of viscously damped systems. We show, however, that the application of satisfactory swing-up control using Lyapunov’s direct method is constrained to underactuated systems that are either undamped or actively damped (viscous damping integrated into the actuated joints only). The violation of this constraint results in the derivation of a torque expression that cannot be solved for (invertibility problem, for systems described by n > 2) or a torque expression which contains a conditional singularity (singularity problem, for systems with n = 2). This constraint is formally summarised as the matched damping condition, and highlights a clear limitation in the Lyapunov-related swing-up control of underactuated mechanical systems. This condition has significant implications on the practical realisation of the swing-up control of underactuated mechanical systems, which justifies the investigation into the possibility of a work-around. We thus show that the limitation highlighted by the matched damping condition can be overcome through the implementation of the partial feedback linearisation (PFL) technique. Two key contributions are generated from this research as a result, which iii include the gain selection criterion (for Traditional Collocated PFL), and the convergence algorithm (for noncollocated PFL). The gain selection criterion is an analytical solution that is composed of a set of inequalities that map out a geometric region of appropriate gains in the swing-up gain space. Selecting a gain combination within this region will ensure that the fully-pendent equilibrium point (FPEP) is unstable, which is a necessary condition for swing-up control when the system is initialised near the FPEP. The convergence algorithm is an experimental solution that, once executed, will provide information about the distal pendulum’s angular initial condition that is required to swing-up a robot with a particular angular initial condition for the proximal pendulum, along with the minimum gain that is required to execute the swing-up control in this particular configuration. Significant future contributions on this topic may result from the inclusion of more complex friction models. Additionally, the degree of actuation of the system may be reduced through the implementation of energy storing components, such as torsional springs, at the joint. In summary, we present two contributions in the form of the gain selection criterion and the convergence algorithm which accommodate the circumnavigation of the limitation formalised as the matched damping condition. This condition pertains to the Lyapunov-related swing-up control of underactuated mechanical systems that have been integrated with viscous damping in the unactuated joint. / CK2018

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The purpose of this thesis is to develop new stability conditions for several linear dynamic systems, including linear parameter-varying (LPV), time-delay systems (LPVTD), slow LPV systems, and parameter-dependent linear time invariant (LTI) systems. These stability conditions are less conservative and/or computationally easier to apply than existing ones. This dissertation is composed of four parts. In the first part of this thesis, the complete stability domain for LTI parameter-dependent (LTIPD) systems is synthesized by extending existing results in the literature. This domain is calculated through a guardian map which involves the determinant of the Kronecker sum of a matrix with itself. The stability domain is synthesized for both single- and multi-parameter dependent LTI systems. The single-parameter case is easily computable, whereas the multi-parameter case is more involved. The determinant of the bialternate sum of a matrix with itself is also exploited to reduce the computational complexity. In the second part of the thesis, a class of parameter-dependent Lyapunov functions is proposed, which can be used to assess the stability properties of single-parameter LTIPD systems in a non-conservative manner. It is shown that stability of LTIPD systems is equivalent to the existence of a Lyapunov function of a polynomial type (in terms of the parameter) of known, bounded degree satisfying two matrix inequalities. The bound of polynomial degree of the Lyapunov functions is then reduced by taking advantage of the fact that the Lyapunov matrices are symmetric. If the matrix multiplying the parameter is not full rank, the polynomial order can be reduced even further. It is also shown that checking the feasibility of these matrix inequalities over a compact set can be cast as a convex optimization problem. Such Lyapunov functions and stability conditions for affine single-parameter LTIPD systems are then generalized to single-parameter polynomially-dependent LTIPD systems and affine multi-parameter LTIPD systems. The third part of the thesis provides one of the first attempts to derive computationally tractable criteria for analyzing the stability of LPV time-delayed systems. It presents both delay-independent and delay-dependent stability conditions, which are derived using appropriately selected Lyapunov-Krasovskii functionals. According to the system parameter dependence, these functionals can be selected to obtain increasingly non-conservative results. Gridding techniques may be used to cast these tests as Linear Matrix Inequalities (LMI's). In cases when the system matrices depend affinely or quadratically on the parameter, gridding may be avoided. These LMI's can be solved efficiently using available software. A numerical example of a time-delayed system motivated by a metal removal process is used to demonstrate the theoretical results. In the last part of the thesis, topics for future investigation are proposed. Among the most interesting avenues for research in this context, it is proposed to extend the existing stability analysis results to controller synthesis, which will be based on the same Lyapunov functions used to derive the nonconservative stability conditions. While designing the dynamic ontroller for linear and parameter-dependent systems, it is desired to take the advantage of the rank deficiency of the system matrix multiplying the parameter such that the controller is of lower dimension, or rank deficient without sacrificing the performance of closed-loop systems.

Neste trabalho apresentamos um assunto que tem contribuído em diversas áreas, o conhecido Problemas de Lurie. Para exemplificar sua aplicabilidade estudamos a Rede Neural de Hopfield e a relacionamos com o problema. Alguns teoremas são apresentados e um dos resultados do Problema de Lurie é aplicado ao modelo de Hopfield. / In the present work we show some properties of the so called Luries type equation. We treat particularly the stability conditions problem, and show how this theory is applied in a Hopfield neural network.

Les travaux de cette thèse portent sur la stabilisation des systèmes affines commutés. Ces systèmes appartiennent à la classe des systèmes dynamiques hybrides. Ils possèdent de plus la particularité d'avoir des points de fonctionnement non auto-maintenables : il n'existe pas de loi de commutations permettant de maintenir l'état du système en ce point. De ce fait, la stabilisation de ces systèmes en imposant à la loi de commutations une durée minimale entre chaque commutation aboutit à une convergence des trajectoires dans une région de l'espace d'état. Après avoir synthétisé différentes stratégies de commutations échantillonnées construites à partir d'une fonction de commande de Lyapunov en temps continu, nous cherchons à déterminer la région de l'espace dans laquelle converge asymptotiquement l'ensemble des trajectoires du système. Par la résolution d'un problème d'optimisation, une estimation de la taille de cette région est donnée et un lien avec les incertitudes du système y est établi. Un second problème de stabilisation est étudié dans cette thèse, en considérant une stratégie de commande basée observateur par retour de sortie. Cependant, du fait de la nature hybride du système, son observabilité est directement liée à la séquence de commutations. Il est alors nécessaire de garantir à la fois l'observabilité, par une condition algébrique, et la convergence du système vers un point de fonctionnement, par l'existence d'une fonction de commande de Lyapunov / This PhD thesis deals with the stabilization of switched affine systems. These systems belong to the class of hybrid dynamical systems. They exhibit a particular behavior: no switching law exists such that the state can be maintained on a chosen operating point. Hence, assuming a dwell time condition on switchings exists, the stabilization of these systems leads to a convergence of the trajectories to a region of the state space. Based on a control Lyapunov function in continuous time, we synthesize several sampled-data switching strategies. The whole trajectories asymptotically converge to a region which we attempt to determine. Solving an optimization problem, an estimation of the size of this region is given. A link with the system uncertainties is also established. This PhD thesis is dedicated to a second stabilization issue: observer-based output-feedback synthesis. By its hybrid nature, the observability of the system is connected to the switching sequence. Therefore, the synthesis of the switching strategy must respect an observability condition and guarantee the convergence to the operating point. The observability is achieved thanks to an algebraic condition. The convergence property is based on the existence of a control Lyapunov function

Singularly perturbed systems with or without delays commonly appear in mathematical modeling of physical and chemical processes, engineering applications, and increasingly, in mathematical biology. There has been intensive work for singularly perturbed systems, yet most of the work so far focused on systems without delays. In this thesis, we provide a new set of tools for the stability analysis for singularly perturbed control systems with time delays. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2015. / FAU Electronic Theses and Dissertations Collection

O presente trabalho apresenta a teoria de sistemas dinâmicos dissipativos impulsivos. Apresentamos resultados suficientes e necessários para obtermos dissipatividade para sistemas impulsivos autônomos e não autônomos utilizando funções de Lyapunov. No que segue, desenvolvemos a teoria de estabilidade para a seção nula de um sistema dinâmico não autônomo com impulsos. Utilizando os resultados da teoria abstrata para sistemas não autônomos com impulsos, apresentamos o estudo da estabilidade de um modelo presa-predador com controle e impulsos. / The present work presents the theory of impulsive dissipative dynamical systems. We present necessary and sufficient conditions to obtain dissipativity for autonomous and non-autonomous impulsive dynamical systems via Lyapunov functions. In the sequel, we develop the theory of stability for the null section of non-autonomous dynamical systems with impulses. Using the results from the abstract theory we present the study of stability for a controlled prey-predator model under impulse conditions.

Orientador: Renata Zotin Gomes de Oliveira / Banca: Magda da Silva Peixoto / Banca: Suzinei Aparecida Siqueira Marconato / Resumo: O objetivo principal deste trabalho é o estudo da teoria qualitativa de sistemas de equações diferenciais ordinárias visando sua aplicação na análise de dois modelos clássicos de Dinâmica Populacional: presa-predador e competição entre duas espécies. Analisamos também duas variações para modelo predador-presa / Abstract: The main objective of this work is to study the qualitative theory for systems of ordinary di erential equations in order to use in the analysis of two classical models of Population Dynamics: predator-prey and competition between two species. We also analyse two variations for predator-prey model / Mestre

Este trabalho aborda o problema de análise de estabilidade e estabilização de sistemas não-lineares racionais sujeitos a saturação. A abordagem utilizada neste estudo é baseada em representações algébricas diferenciais (DAR) de sistemas racionais e na versão modificada da condição de setor generalizada para lidar com a saturação. Inicialmente, métodos para caracterizar a estabilidade de sistemas em tempo discreto sujeitos a perturbações são propostos. Neste contexto, apresentam-se abordagens na forma de desigualdades matriciais lineares (do inglês, Linear Matrix Inequalities) para o cálculo de estimativas da região de atração do sistema, bem como limites para uma classe de perturbações admissíveis ℓ2 de forma a garantir que as trajetórias sejam limitadas e estimativas do ganho ℓ2 do sistema. Duas abordagens são consideradas: a primeira é baseada em uma única função de Lyapunov quadrática e a segunda considerando funções de Lyapunov quadráticas por partes. Em seguida, técnicas para síntese de compensadores anti-windup são propostas com o objetivo de aumentar a região de atração de sistemas em tempo contínuo. As condições são desenvolvidas e incorporadas em um algoritmo iterativo, sendo que a cada iteração é resolvido um problema de otimização convexa com restrições na forma de LMIs. Tais resultados são estendidos para lidar com sistemas incertos e sistemas sujeitos a perturbações. Com o objetivo de evitar métodos iterativos e facilitar a aplicação em sistemas multivariáveis propõe-se uma nova abordagem para sintetizar este tipo de compensador (diretamente na forma de LMIs). Extensões dos resultados são apresentadas para tratar sistemas em tempo discreto. Por fim, é apresentada uma abordagem para síntese de realimentação estática de estados. Estes métodos são baseados em condições de estabilização local permitindo, simultaneamente, calcular o ganho de realimentação de estados e uma função de Lyapunov que leva a uma estimativa maximizada da região de atração do sistema em malha fechada. Propõe-se também uma extensão dos resultados abordando sistemas em tempo discreto. Exemplos numéricos são apresentados com o objetivo de ilustrar a aplicação e verificar a eficiência dos métodos propostos. / This work addresses the problem of stability analysis and stabilization of nonlinear rational systems subject to saturation. The approach used in this study is based on the differential algebraic representation (DAR) of rational systems and on a modified version of the generalized sector condition to deal with saturation. First, methods to characterize the stability of discrete-time systems subject to disturbances are proposed. In this context, approaches based on linear matrix inequalities to compute estimates of the region of attraction of the system, as well as limits for a class of admissible ℓ2 disturbances to ensure bounded trajectories and estimates of the ℓ2-gain of the system are presented. Two approaches are considered: the first one based on a single quadratic Lyapunov function and the second one considering piecewise quadratic Lyapunov functions. Then, techniques for the synthesis of anti-windup compensators are proposed in order to enlarge the region of attraction of continuous-time systems. The conditions are developed and incorporated into an iterative algorithm, where at each iteration, a convex optimization problem with LMI constraints is solved. These results are extended to deal with uncertain systems and systems subject to disturbances. In order to avoid iterative methods and facilitate the application to multivariable systems, a new approach to synthesize this type of compensator (directly in terms of LMI) is proposed. Extensions of the results are also presented to deal with discrete-time systems. Finally, a method for the synthesis of static state feedback gains is proposed. This method is based on local stabilization conditions which allow to calculate the state feedback gain and a Lyapunov function leading to a maximized estimate of the region of attraction of the closed-loop system. The extension of these results for the case of discrete-time systems is also addressed. Numerical examples are presented in order to illustrate the application and to verify the efficiency of the proposed methods.

The main focus of my dissertation is the Modified Stochastic Sine-Gordon Equation: utt = 2uxx − ut − sin(|u|^(&gamma)) + b(u, du/dt)dW/dt where &gamma > 0 is the parameter of the power of non-linearity, &delta &ge 0 is the magnitude of non-linearity, &alpha> 0 be the damping parameter, and &sigma the diffusion intensity, on one dimensional domain. We analyze the properties of the solution of the SPDE by the eigenfunctions approach allowing us to truncate the infinite-dimensional stochastic system (i.e the SDEs of Fourier coefficients related to the SPDE), to control its energy, existence, uniqueness, continuity and stability. The analysis relies on the investigation of expected Lyapunov functional of the energy in terms of all system-parameters. We simulate the model with respect to all system-parameters to visualize our conclusions.