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Otsuki, Nobukazu. "Lyapunov-like functions and geodesic flows." 京都大学 (Kyoto University), 1985. http://hdl.handle.net/2433/86363.
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Elragig, Aiman Saleh. "On transients, Lyapunov functions and Turing instabilities." Thesis, University of Exeter, 2013. http://hdl.handle.net/10871/13789.
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Motivated by the papers [84, 85], this thesis considers the concepts of reactivity, Lyapunov stability and Turing patterns. We introduce the notion of P-reactivity, a new measure for transient dynamics. We extend a result by Shorten and Narendra [108] regarding joint dissipativity for second order systems. We derive an easy verifiable formula that determines systems P-reactivity with respect to a norm induced by the positive definite matrix P. An optimization problem aiming to determine the positive definite P with respect to which a stable system is most reactive is posed and solved numerically for second order systems. The stability radius is adopted as a measure of robustness of joint disspaptivity. We characterise the stability radius of joint dissipativity when the underlying systems are subject to certain specific perturbation structures. A detailed robustness analysis of the Shorten and Narendra conditions is also presented. Using the notion of common Lyapunov function we show that the necessary condition in [85] is a special case of a more powerful (i.e tighter) necessary condition. Specifically, we show that if the linearised reaction matrix and the diffusion matrix share a common Lyapunov function, then Turing instability is not possible. The existence of common Lyapunov functions is readily checked using semi-definite programming. We also further extend this to include more complicated movement mechanisms such as chemotaxis. Unlike the traditional techniques, this new necessary condition can be used to check Turing instability for systems with any dimension and any number of parameters. We apply our new conditions to various models in literature.
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Marikar, Mohamed Tariq. "Polyhedral Lyapunov functions and stabilization under polyhedral constraints." Thesis, Imperial College London, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.266114.
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Della, rossa Matteo. "Non smooth Lyapunov functions for stability analysis of hybrid systems." Thesis, Toulouse, INSA, 2020. http://www.theses.fr/2020ISAT0004.
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La Nature, dans ses multiples manifestations, nous fournit un grand nombre d’exemples pour lesquels il est nécessaire d’aller au-delà de la distinction classique entre modèles où le temps est décrit comme une entité continue et modèles où le temps est discret/discrétisé. En particulier, pour une multitude de systèmes en physique/ingénierie, ces deux aspects temporels sont fondamentalement liés, et nécessitent donc que ces deux paradigmes soient connectés et mis en relation, pour une meilleure précision et fidélité dans la représentation du phénomène. Cette famille de systèmes est souvent appelée ``systèmes hybrides’’, et différentes formalisations mathématiques ont été proposées.L’objectif de cette thèse est l’analyse et l’étude de la stabilité (asymptotique) pour certaines classes de systèmes hybrides, en proposant des conditions suffisantes à la Lyapunov. Plus spécifiquement, nous nous concentrerons sur des fonctions de Lyauponv non-lisses ; pour cette raison, les premiers chapitres de cette thèse peuvent être considérés comme une introduction générale de ce sujet, proposant les instruments nécessaires issus de l’analyse non-lisse.Tout d'abord, grâce à ces outils, nous pourrons étudier une classe de fonctions de Lyapunov construites par morceaux, avec une attention particulière aux propriétés de continuité des inclusions différentielles qui composent le système hybride considéré. Nous proposons des conditions qui doivent être vérifiées seulement sur un sous-ensemble dense, et donc allant au-delà de résultats existants.En négligeant les hypothèses de continuité, nous étudions ensuite comment les notions de dérivées généralisées se spécialisent en considérant des fonctions construites comme combinaisons de maximum/ minimum de fonctions lisses. Cette structure devient particulièrement fructueuse quand on regarde la classe des systèmes à commutation dépendant de l’état du système. Dans le cas où les sous-dynamiques sont linéaires, nous étudions comment les conditions proposées peuvent être vérifiées algorithmiquement.L’utilité des notions de dérivées généralisées est finalement explorée dans le contexte de la stabilité entrée-état (ISS) pour inclusions différentielles avec perturbations extérieures. Ces résultats nous permettent de proposer des critères de stabilité pour systèmes interconnectés, et notamment une application du synthèse de contrôleurs pour systèmes à commutation dépendant de l’étatModeling of many phenomena in nature escape the rather common frameworks of continuous-time and discrete-time models. In fact, for many systems encountered in practice, these two paradigms need to be intrinsically related and connected, in order to reach a satisfactory level of description in modeling the considered physical/engineering process.These systems are often referred to as hybrid systems, and various possible formalisms have appeared in the literature over the past years.The aim of this thesis is to analyze the stability of particular classes of hybrid systems, by providing Lyapunov-based sufficient conditions for (asymptotic) stability. In particular, we will focus on non-differentiable locally Lipschitz candidate Lyapunov functions. The first chapters of this manuscript can be considered as a general introduction of this topic and the related concepts from non-smooth analysis.This will allow us to study a class of piecewise smooth maps as candidate Lyapunov functions, with particular attention to the continuity properties of the constrained differential inclusion comprising the studied hybrid systems. We propose ``relaxed'' Lyapunov conditions which require to be checked only on a dense set and discuss connections to other classes of locally Lipschitz or piecewise regular functions.Relaxing the continuity assumptions, we then investigate the notion of generalized derivatives when considering functions obtained as emph{max-min} combinations of smooth functions. This structure turns out to be particularly fruitful when considering the stability problem for differential inclusions arising from regularization of emph{state-dependent switched systems}.When the studied switched systems are composed of emph{linear} sub-dynamics, we refine our results, in order to propose algorithmically verifiable conditions.We further explore the utility of set-valued derivatives in establishing input-to-state stability results, in the context of perturbed differential inclusions/switched systems, using locally Lipschitz candidate Lyapunov functions. These developments are then used in analyzing the stability problem for interconnections of differential inclusion, with an application in designing an observer-based controller for state-dependent switched systems
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Dos, Santos Paulino Ana Carolina. "Robust analysis of uncertain descriptor systems using non quadratic Lyapunov functions." Thesis, Strasbourg, 2018. http://www.theses.fr/2018STRAD049.
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Les systèmes descripteurs incertains sont convenables pour la représentation des incertitudes d’un modèle, du comportement impulsif et des contraintes algébriques entre les variables d’état. Ils peuvent décrire bien plus de phénomènes qu’un système dynamique standard, mais, en conséquence, l’analyse des systèmes descripteurs incertains est aussi plus complexe. Des recherches sont menées de façon à réduire le degré de conservatisme dans l’analyse des systèmes descripteurs incertains. L’utilisation des fonctions de Lyapunov qui sont en mesure de générer des conditions nécessaires et suffisantes pour une telle évaluation y figurent. Les fonctions de Lyapunov polynomiales homogènes font partie de ces classes, mais elles n’ont jamais été employées pour les systèmes descripteurs incertains. Dans cette thèse, nous comblons ce vide dans la littérature en étendant l’usage des fonctions de Lyapunov polynomiales homogènes du cas incertain standard vers les systèmes descripteurs incertainsUncertain descriptor systems are a convenient framework for simultaneously representing uncertainties in a model, as well as impulsive behavior and algebraic constraints. This is far beyond what can be depicted by standard dynamic systems, but it also means that the analysis of uncertain descriptor systems is more complex than the standard case. Research has been conducted to reduce the degree of conservatism in the analysis of uncertain descriptor systems. This can be achieved by using classes of Lyapunov functions that are known to be able to provide necessary and sufficient conditions for this evaluation. Homogeneous polynomial Lyapunov functions constitute one of such classes, but they have never been employed in the context of uncertain descriptor systems. In this thesis, we fill in this scientific gap, extending the use of homogeneous polynomial Lyapunov functions from the standard uncertain case for the uncertain descriptor one
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Chao, Chien-Hsiang. "Robust stabilization of linear time-invariant uncertain systems via Lyapunov theory." Diss., Virginia Polytechnic Institute and State University, 1988. http://hdl.handle.net/10919/53928.
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This dissertation is concerned with the problem of synthesizing a robust stabilizing feedback controller for linear time-invariant systems with constant uncertainties that are not required to satisfy matching conditions. Only the bounds on the uncertainties are required and no statistical property of the uncertainties is assumed.
The systems under consideration are described by linear state equations with uncertainties. I.e. x(t) = A̅(γ)x(t) +B̅(γ)u(t), where A̅(γ) is an n x n matrix and B̅(γ) is an n x m matrix. Lyapunov theory is exploited to establish the conditions for stabilizability of the closed loop system. We consider a Lyapunov function with an uncertain symmetric positive definite matrix P. The uncertain matrix P satisfies the Lyapunov equation ATP + PA + Q = 0, where the matrix A is in companion form and the matrix Q is symmetric and positive definite. In the solution of the Lyapunov equation, m rows of the matrix P are fixed in our approach of designing a robust controller. We derive necessary and sufficient conditions on these fixed m rows of the matrix P such that for given positive definite and symmetric Q the solution of the Lyapunov equation yields a positive definite matrix P and a companion matrix A that is Hurwitz. A discontinuous robust stabilizing controller is given.
Linear controller design is also investigated in this research. Under the same assumptions for the existence of a stabilizing discontinuous controller, we show that a linear robust stabilizing controller always exists.
The dissertation includes three examples to illustrate the design procedures for robust controllers. Example 2 shows that the design procedure may be applied to time-varying nonlinear systems.Ph. D.
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Tan, Bin. "Invariant manifolds, invariant foliations and linearization theorems in Banach spaces." Diss., Georgia Institute of Technology, 1998. http://hdl.handle.net/1853/29421.
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Kuhn, Zuzana. "Ranges of vector measures and valuations." Diss., Georgia Institute of Technology, 1997. http://hdl.handle.net/1853/30875.
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Silva, Flávio Henrique Justiniano Ribeiro da. "Funções de Lyapunov para a análise de estabilidade transitória em sistemas de potência." Universidade de São Paulo, 2001. http://www.teses.usp.br/teses/disponiveis/18/18133/tde-07032016-111317/.
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Os métodos diretos são adequados à análise de estabilidade transitória em sistemas de potência, já que não requerem a resolução, integração numérica, do conjunto de equações diferenciais que representam o sistema. Os métodos diretos utilizam as idéias de Lyapunov associadas ao princípio de invariância de LaSalle para estimar a área de atração dos sistemas de potência. A grande dificuldade dos métodos diretos está em encontrar uma função auxiliar V, denominada função de Lyapunov que satisfaça as condições estabelecidas pelo Teorema de Lyapunov. Neste trabalho é realizada uma revisão bibliográfica das funções de Lyapunov utilizadas para análise de estabilidade transitória em sistemas de potência. Analisa-se o problema da existência de funções de Lyapunov quando as condutâncias de transferência são consideradas. Utilizando-se de uma extensão do princípio de Invariância de LaSalle, apresenta-se uma nova função a qual é uma função de Lyapunov no sentido mais geral da extensão do princípio de invariância de LaSalle quando as condutâncias de transferência da matriz admitância da rede reduzida são consideradas. Estudou-se também a existência de funções de Lyapunov no sentido mais geral de extensão do princípio de invariância de LaSalle para modelos que preserva a estrutura da rede. Neste caso, infelizmente não encontramos uma função satisfazendo todas as hipóteses requeridas.The direct methods are well-suited for transient stability analysis to power systems, since they do not require the solution of the set of differential equations of the system model. The direct methods use the Lyapunov\'s ideas related to the LaSalle\'s invariance principle to estimate the power system attraction area. The great difficulty of the direct methods is to find an auxiliar function V, called Lyapunov function, which satisfies the conditions of Lyapunov\'s theorem. In this work, a bibliographic review of the Lyapunov functions used in transient stability analysis of power systems is done. The problem of existence of Lyapunov functions, when the transfer conductances are considered, is analysed. Using LaSalle\'s invariance principle extension, a Lyapunov function considering the transfer conductances is presented. The existence of Lyapunov functions for models that preserv the network structure was studied using the LaSalle\'s invariance principle. Unfortunately, in these cases, we did not find a function satisfing all the required hypothesis.
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Marinósson, Sigurour Freyr. "Stability analysis of nonlinear systems with linear programming a Lyapunov functions based approach /." [S.l.] : [s.n.], 2002. http://deposit.ddb.de/cgi-bin/dokserv?idn=982323697.
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Zhang, Xiping. "Parameter-Dependent Lyapunov Functions and Stability Analysis of Linear Parameter-Dependent Dynamical Systems." Diss., Georgia Institute of Technology, 2003. http://hdl.handle.net/1853/5270.
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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.
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Mohammed, Najla Abdullah. "Grid refinement and verification estimates for the RBF construction method of Lyapunov functions." Thesis, University of Sussex, 2016. http://sro.sussex.ac.uk/id/eprint/65711/.
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Lyapunov functions are functions with negative orbital derivative, whose existence guarantee the stability of an equilibrium point of an ODE. Moreover, sub-level sets of a Lyapunov function are subsets of the domain of attraction of the equilibrium. In this thesis, we improve an established numerical method to construct Lyapunov functions using the radial basis functions (RBF) collocation method. The RBF collocation method approximates the solution of linear PDE's using scattered collocation points, and one of its applications is the construction of Lyapunov functions. More precisely, we approximate Lyapunov functions, that satisfy equations for their orbital derivative, using the RBF collocation method. Then, it turns out that the RBF approximant itself is a Lyapunov function. Our main contributions to improve this method are firstly to combine this construction method with a new grid refinement algorithm based on Voronoi diagrams. Starting with a coarse grid and applying the refinement algorithm, we thus manage to reduce the number of collocation points needed to construct Lyapunov functions. Moreover, we design two modified refinement algorithms to deal with the issue of the early termination of the original refinement algorithm without constructing a Lyapunov function. These algorithms uses cluster centres to place points where the Voronoi vertices failed to do so. Secondly, we derive two verification estimates, in terms of the first and second derivatives of the orbital derivative, to verify if the constructed function, with either a regular grid of collocation points or with one of the refinement algorithms, is a Lyapunov function, i.e., has negative orbital derivative over a given compact set. Finally, the methods are applied to several numerical examples up to 3 dimensions.
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Esterhuizen, Willem Daniël. "Stability and stabilization conditions for Takagi-Sugeno fuzzy model via polyhedral Lyapunov functions." Thesis, Boston University, 2012. https://hdl.handle.net/2144/12365.
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Thesis (M.S.)--Boston University
PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you.The Takagi-Sugeno (T-S) fuzzy model together with parallel distributed compensation forms a very effective framework for modeling, analysis and control design for nonlinear systems. A large body of theory exists that deals with this framework and most of the fundamental notions, such as stability, stabilizability, controller design, observer design, etc., have been studied extensively. A large number of the well-established results are based on quadratic Lyapunov functions. The main reason is that the stability and design conditions under quadratic Lyapunov functions are in the form of linear matrix inequalities which are easily solvable. However, the class of quadratic Lyapunov functions are conservative, in the sense that there are systems for which their stability cannot be established by quadratic Lyapunov functions. A natural question to ask is: are there other candidate Lyapunov functions that are less conservative? It turns out that the class of polyhedral Lyapunov functions are universal for the T-S fuzzy model stability problem, that is if a T-S fuzzy system is stable, there exists a polyhedral Lyapunov function that proves the stability. This thesis is a first look at the applicability of polyhedral Lyapunov functions to the T-S fuzzy model-based framework for the stability analysis and control design of nonlinear systems. First, two stability theorems are presented in this thesis. It is shown that the stability of a T-S fuzzy system via polyhedral Lyapunov functions can be established through linear programming. Next, the stabilization problem is investigated to find control laws that guarantee the stability of the closed-loop systems. Two stabilization theorems are presented which are derived from the stability theorems. The conditions of the stabilization theorems are in the form of nonconvex inequalities that are not readily solvable. Implementation examples are included in which solutions are found through either brute-force, or making the constraints convex in exchange for a loss of feasible space. Two relaxed stabilization theorems are also derived by taking advantage of certain aspects of the T-S fuzzy model.
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Trimboli, Sergio. "Approximate Explicit MPC and Closed-loop Stability: Analysis based on PWA Lyapunov Functions." Doctoral thesis, Università degli studi di Trento, 2012. https://hdl.handle.net/11572/368455.
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Model Predictive Control (MPC) is the de facto standard in advanced industrial automation systems. There are two main formulations of the MPC algorithm: an implicit one and an explicit MPC one. The first requires an optimization problem to be solved on-line, which is the main limitation when dealing with hard real-time applications. As the implicit MPC algorithm cannot be guaran- teed in terms of execution time, in many applications the explicit MPC solution is preferable. In order to deal with systems integrating mixed logic and dynam- ics, the class of the hybrid and piecewise affine models (PWA) were introduced and tackled by the explicit MPC strategy. However, the resulting controller complexity leads to a requirement on the CPU/memory combination which is as strict as the number of states, inputs and outputs increases. To reduce drasti- cally the complexity of the explicit controller while preserving the controller’s performance, a strategy combining switched MPC with discontinuous simpli- cial PWA models is introduced in this thesis. The latter is proven to be circuit implementable, e.g., in FPGA. To ensure that closed-loop stability properties are guaranteed, a stability analysis tool is proposed which exploits suitable and possibly discontinuous PWA Lyapunov-like functions. The tool requires solving offline a linear programming problem. Moreover, the tool is able to compute an invariant set for the closed-loop system, as well as ultimate boundedness and input-to-state stability properties.
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Trimboli, Sergio. "Approximate Explicit MPC and Closed-loop Stability: Analysis based on PWA Lyapunov Functions." Doctoral thesis, University of Trento, 2012. http://eprints-phd.biblio.unitn.it/823/1/PhD-Thesis-Trimboli.pdf.
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Model Predictive Control (MPC) is the de facto standard in advanced industrial automation systems. There are two main formulations of the MPC algorithm: an implicit one and an explicit MPC one. The first requires an optimization problem to be solved on-line, which is the main limitation when dealing with hard real-time applications. As the implicit MPC algorithm cannot be guaran- teed in terms of execution time, in many applications the explicit MPC solution is preferable. In order to deal with systems integrating mixed logic and dynam- ics, the class of the hybrid and piecewise affine models (PWA) were introduced and tackled by the explicit MPC strategy. However, the resulting controller complexity leads to a requirement on the CPU/memory combination which is as strict as the number of states, inputs and outputs increases. To reduce drasti- cally the complexity of the explicit controller while preserving the controller’s performance, a strategy combining switched MPC with discontinuous simpli- cial PWA models is introduced in this thesis. The latter is proven to be circuit implementable, e.g., in FPGA. To ensure that closed-loop stability properties are guaranteed, a stability analysis tool is proposed which exploits suitable and possibly discontinuous PWA Lyapunov-like functions. The tool requires solving offline a linear programming problem. Moreover, the tool is able to compute an invariant set for the closed-loop system, as well as ultimate boundedness and input-to-state stability properties.
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Bisoffi, Andrea. "Weak Lyapunov functions for hybrid dynamical systems: applications to electrical and mechanical systems." Doctoral thesis, Università degli studi di Trento, 2017. https://hdl.handle.net/11572/367690.
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In this work we propose a number of relevant engineering applications that exhibit both a continuous and a discrete evolution, and are therefore suitably described by a recent formalism for hybrid dynamical systems. More specifically, (i) we design observer schemes for a nonsmooth disturbance arising in AC/DC conversion, which we then cancel from a desirable signal; (ii) we show how reset actuation applied to nonlinear mechanical systems can at the same time sustain or damp oscillations; (iii) we study the feedback interconnection of a classical proportional-integral-derivative controller with a sliding mass under Coulomb friction through differential inclusions. In the context of dynamical systems, we analyze the properties of these applications in terms of asymptotic stability through Lyapunov functions tailored for hybrid systems. Instead of the standard Lyapunov conditions, we prove asymptotic stability through weaker, or relaxed, conditions that are compensated by additional (structural) properties that may be easier to verify.
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Bisoffi, Andrea. "Weak Lyapunov functions for hybrid dynamical systems: applications to electrical and mechanical systems." Doctoral thesis, University of Trento, 2017. http://eprints-phd.biblio.unitn.it/2541/1/PhDThesisAndreaBisoffiSENT.pdf.
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In this work we propose a number of relevant engineering applications that exhibit both a continuous and a discrete evolution, and are therefore suitably described by a recent formalism for hybrid dynamical systems. More specifically, (i) we design observer schemes for a nonsmooth disturbance arising in AC/DC conversion, which we then cancel from a desirable signal; (ii) we show how reset actuation applied to nonlinear mechanical systems can at the same time sustain or damp oscillations; (iii) we study the feedback interconnection of a classical proportional-integral-derivative controller with a sliding mass under Coulomb friction through differential inclusions. In the context of dynamical systems, we analyze the properties of these applications in terms of asymptotic stability through Lyapunov functions tailored for hybrid systems. Instead of the standard Lyapunov conditions, we prove asymptotic stability through weaker, or relaxed, conditions that are compensated by additional (structural) properties that may be easier to verify.
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Ghandhari, Mehrdad. "Control Lyapunov Functions : A Control Strategy for Damping of Power Oscillations in Large Power Systems." Doctoral thesis, KTH, Elkraftteknik, 2000. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3039.
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In the present climate of deregulation and privatisation, theutilities are often separated into generation, transmission anddistribution companies so as to help promote economic efficiencyand encourage competition. Also, environmental concerns,right-of-way and cost problems have delayed the construction ofboth generation facilities and new transmission lines while thedemand for electric power has continued to grow, which must bemet by increased loading of available lines. A consequence isthat power system damping is often reduced which leads to a poordamping of electromechanical power oscillations and/or impairmentof transient stability. The aim of this thesis is to examine theability of Controllable Series Devices (CSDs), such as Unified Power Flow Controller (UPFC) Controllable Series Capacitor (CSC) Quadrature Boosting Transformer (QBT) for improving transient stability and damping ofelectromechanical oscillations in a power system. For these devices, a general model is used in power systemanalysis. This model is referred to as injection model which isvalid for load flow and angle stability analysis. The model isalso helpful for understanding the impact of the CSDs on powersystem stability. A control strategy for damping of electromechanical poweroscillations is also derived based on Lyapunov theory. Lyapunovtheory deals with dynamical systems without input. For thisreason, it has traditionally been applied only to closed-loopcontrol systems, that is, systems for which the input has beeneliminated through the substitution of a predetermined feedbackcontrol. However, in this thesis, Lyapunov function candidatesare used in feedback design itself by making the Lyapunovderivative negative when choosing the control. This controlstrategy is called Control Lyapunov Function (CLF) for systemswith control input.QC 20100609
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Li, Huijuan [Verfasser], and Grüne [Akademischer Betreuer] Lars. "Computation of Lyapunov functions and stability of interconnected systems / Huijuan Li. Betreuer: Grüne Lars." Bayreuth : Universität Bayreuth, 2015. http://d-nb.info/1067485880/34.
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Lopez, Ramirez Francisco. "Control and estimation in finite-time and in fixed-time via implicit Lyapunov functions." Thesis, Lille 1, 2018. http://www.theses.fr/2018LIL1I063/document.
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Dans ce travail, on montre des nouveaux résultats pour l’analyse et la synthèse des systèmes stables en temps fini et fixe. Ce genre des systèmes convergent exactement à un point d’équilibre dans une quantité du temps qui est fini et, dans le cas de systèmes stables en temps fixe, dans un temps maximal constant qui ne dépend pas des conditions initiales du système.Les chapitres 2 et 3 portent sur des résultats d’analyse ; ce premier present des conditions nécessaires et suffisants pour la stabilité en temps fixe des systèmes autonomes continues tandis que ce dernier combine l’approche de la fonction implicite de Lyapunov avec des résultats de stabilisation ISS pour étudier la robustesse de ce genre de systèmes.Les chapitres 4 et 5 présentent des résultats pratiques liés á la procédure de synthèse des contrôleurs et des observateurs. Le chapitre 4 emploie la méthode de la fonction de Lyapunov implicite afin d’obtenir des observateurs convergents en temps fini et fixe pour les systèmes linéaires MIMO. Le chapitre 5 utilise des propriétés d’homogénéité et des fonctions de Lyapunov implicites pour synthétiser un contrôleur de sortie en temps fixe pour une chaîne d’intégrateurs. Les résultats obtenus ont été validés par des simulations numériques et le chapitre 4 contient des tests de performance sur un pendule rotatifThis work presents new results on analysis and synthesis of finite-time and fixed-time stable systems, a type of dynamical systems where exact convergence to an equilibrium point is guaranteed in a finite amount of time. In the case of fixed-time stable system, this is moreover achieved with an upper bound on the settling-time that does not depend on the system’s initial condition.Chapters 2 and 3 focus on theoretical contributions; the former presents necessary and sufficient conditions for fixed-time stability of continuous autonomous systems whereas the latter introduces a framework that gathers ISS Lyapunov functions, finite-time and fixed-time stability analysis and the implicit Lyapunov function approach in order to study and determine the robustness of this type of systems.Chapters 4 and 5 deal with more practical aspects, more precisely, the synthesis of finite-time and fixed-time controllers and observers. In Chapter 4, finite-time and fixed-time convergent observers are designed for linear MIMO systems using the implicit approach. In Chapter 5, homogeneity properties and the implicit approach are used to design a fixed-time output controller for the chain of integrators. The results obtained were verified by numerical simulations and Chapter 4 includes performance tests on a rotary pendulum
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Ahmadi, Amir Ali. "Non-monotonic Lyapunov functions for stability of nonlinear and switched systems : theory and computation." Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/44206.
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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Includes bibliographical references (p. 87-90).
Lyapunov's direct method, which is based on the existence of a scalar function of the state that decreases monotonically along trajectories, still serves as the primary tool for establishing stability of nonlinear systems. Since the main challenge in stability analysis based on Lyapunov theory is always to nd a suitable Lyapunov function, weakening the requirements of the Lyapunov function is of great interest. In this thesis, we relax the monotonicity requirement of Lyapunov's theorem to enlarge the class of functions that can provide certicates of stability. Both the discrete time case and the continuous time case are covered. Throughout the thesis, special attention is given to techniques from convex optimization that allow for computationally tractable ways of searching for Lyapunov functions. Our theoretical contributions are therefore amenable to convex programming formulations. In the discrete time case, we propose two new sucient conditions for global asymptotic stability that allow the Lyapunov functions to increase locally, but guarantee an average decrease every few steps. Our first condition is nonconvex, but allows an intuitive interpretation. The second condition, which includes the first one as a special case, is convex and can be cast as a semidenite program. We show that when non-monotonic Lyapunov functions exist, one can construct a more complicated function that decreases monotonically. We demonstrate the strength of our methodology over standard Lyapunov theory through examples from three different classes of dynamical systems. First, we consider polynomial dynamics where we utilize techniques from sum-of-squares programming. Second, analysis of piecewise ane systems is performed. Here, connections to the method of piecewise quadratic Lyapunov functions are made.
(cont.) Finally, we examine systems with arbitrary switching between a finite set of matrices. It will be shown that tighter bounds on the joint spectral radius can be obtained using our technique. In continuous time, we present conditions invoking higher derivatives of Lyapunov functions that allow the Lyapunov function to increase but bound the rate at which the increase can happen. Here, we build on previous work by Butz that provides a nonconvex sucient condition for asymptotic stability using the first three derivatives of Lyapunov functions. We give a convex condition for asymptotic stability that includes the condition by Butz as a special case. Once again, we draw the connection to standard Lyapunov functions. An example of a polynomial vector field is given to show the potential advantages of using higher order derivatives over standard Lyapunov theory. We also discuss a theorem by Yorke that imposes minor conditions on the first and second derivatives to reject existence of periodic orbits, limit cycles, or chaotic attractors. We give some simple convex conditions that imply the requirement by Yorke and we compare them with those given in another earlier work. Before presenting our main contributions, we review some aspects of convex programming with more emphasis on semidenite programming. We explain in detail how the method of sum of squares decomposition can be used to efficiently search for polynomial Lyapunov functions.
by Amir Ali Ahmadi.
S.M.
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Silva, Emerson Ravazzi Pires da [UNESP]. "Controle de sistemas lineares incertos via realimentação derivativa utilizando Funções de Lyapunov dependentes de parâmetros." Universidade Estadual Paulista (UNESP), 2012. http://hdl.handle.net/11449/100278.
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Este trabalho trata do problema de estabilização robusta de sistemas lineares contínuos no tempo sujeitos a incertezas do tipo politópicas no modelo. Todo o trabalho é fundamen- tado em leis de controle por realimentação da derivada do vetor de estado (realimentação derivativa). A motivação em utilizar a realimentação derivativa (u(t) = −Kd x(t)) em ̇ vez da realimentação do vetor de estado convencional é devido à facilidade de imple- mentação em algumas aplicações mecânicas, por exemplo, no controle de vibrações de sistemas mecânicos, nos quais sensores como acelerômetros têm sido utilizados para me- dir a derivada de segunda ordem (aceleração) de uma variável de estado (posição) desses sistemas. A metodologia apresenta condições suficientes na forma de desigualdades ma- triciais lineares (LMIs, acrônimo inglês para Linear Matrix Inequalities) para a síntese de controladores lineares robustos estáticos (Kd ), visando a princípio apenas a estabilização do sistema, na sequência a estabilização com restrição de taxa de decaimento (γ > 0) e por fim projetos que asseguram a D-estabilidade (alocação regional) robusta, restringindo os autovalores a uma determinada região do plano complexo. Os índices de desempenho de taxa de decaimento e D-estabilidade são adicionados no projeto dos controladores visto que, garantir apenas a estabilidade do sistema nem sempre é suficiente para um bom desempenho prático. As formulações LMIs são realizadas através de lemas largamente utilizados (Lema da Projeção Recíproca e Lema de Finsler) em análise de estabilidade e projetos de controladores para os mais diversos problemas. Estes lemas permitem o uso de uma função de Lyapunov dependente de parâmetros (PDLF, acrônimo inglês para Parameter-Dependent Lyapunov Function) para assegurar a estabilidade...
This work deals with the problem of robust stabilization of continuous-time linear sys- tems subjected to polytopic uncertainties in the plant. All our work is based on control techniques using only the state-derivative feedback. The motivation for the use of state- derivative feedback (u(t) = −Kd x(t)) instead of conventional state feedback is due to ease ̇ of implementation in some mechanical applications, for instance, in the vibration control of mechanical systems, in which sensors like accelerometers have been used to measure the second order derivative (acceleration) of one state variable (position) of these systems. The methodology presents sufficient conditions in the form of linear matrix inequalities (LMIs) for the synthesis of static linear robust controllers (Kd ), aiming at first only the system’s stability, followed by the system’s stability with decay rate (γ > 0) and finally designs that ensure the system’s robust D-stability (regional allocation), restricting the eigenvalues at a given region of the complex plane. The performance indexes of decay rate and D-stability are added in the controllers design since ensuring system’s stability only is not always sufficient for a good practical performance. The LMIs formulations are made through widely used lemmas (Reciprocal Projection Lemma and Finsler’s Lemma) in the stability analysis and in the controllers design for many problems. These lemmas allow the use of a parameter-dependent Lyapunov function (PDLF) to ensure the asymptotic stability of the systems in the sense of Lyapunov. Comparing with the existing litera- ture, in which the results consider classical LMIs formulations, based on the existence of a common quadratic Lyapunov function (CQLF) for the solution of the problems, the present work shows to be less conservative in most occasions. In many cases... (Complete abstract click electronic access below)
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Silva, Emerson Ravazzi Pires da. "Controle de sistemas lineares incertos via realimentação derivativa utilizando Funções de Lyapunov dependentes de parâmetros /." Ilha Solteira, 2012. http://hdl.handle.net/11449/100278.
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Orientador: Edvaldo AssunçãoBanca: Marcelo Carvalho Minhoto Teixeira
Banca: Rodrigo Cardim
Banca: Roberto Kawakami Harrop Galvão
Banca: Flávio Andrade Faria
Resumo: Este trabalho trata do problema de estabilização robusta de sistemas lineares contínuos no tempo sujeitos a incertezas do tipo politópicas no modelo. Todo o trabalho é fundamen- tado em leis de controle por realimentação da derivada do vetor de estado (realimentação derivativa). A motivação em utilizar a realimentação derivativa (u(t) = −Kd x(t)) em ̇ vez da realimentação do vetor de estado convencional é devido à facilidade de imple- mentação em algumas aplicações mecânicas, por exemplo, no controle de vibrações de sistemas mecânicos, nos quais sensores como acelerômetros têm sido utilizados para me- dir a derivada de segunda ordem (aceleração) de uma variável de estado (posição) desses sistemas. A metodologia apresenta condições suficientes na forma de desigualdades ma- triciais lineares (LMIs, acrônimo inglês para Linear Matrix Inequalities) para a síntese de controladores lineares robustos estáticos (Kd ), visando a princípio apenas a estabilização do sistema, na sequência a estabilização com restrição de taxa de decaimento (γ > 0) e por fim projetos que asseguram a D-estabilidade (alocação regional) robusta, restringindo os autovalores a uma determinada região do plano complexo. Os índices de desempenho de taxa de decaimento e D-estabilidade são adicionados no projeto dos controladores visto que, garantir apenas a estabilidade do sistema nem sempre é suficiente para um bom desempenho prático. As formulações LMIs são realizadas através de lemas largamente utilizados (Lema da Projeção Recíproca e Lema de Finsler) em análise de estabilidade e projetos de controladores para os mais diversos problemas. Estes lemas permitem o uso de uma função de Lyapunov dependente de parâmetros (PDLF, acrônimo inglês para Parameter-Dependent Lyapunov Function) para assegurar a estabilidade... (Resumo completo, clicar acesso eletrônico abaixo)
Abstract: This work deals with the problem of robust stabilization of continuous-time linear sys- tems subjected to polytopic uncertainties in the plant. All our work is based on control techniques using only the state-derivative feedback. The motivation for the use of state- derivative feedback (u(t) = −Kd x(t)) instead of conventional state feedback is due to ease ̇ of implementation in some mechanical applications, for instance, in the vibration control of mechanical systems, in which sensors like accelerometers have been used to measure the second order derivative (acceleration) of one state variable (position) of these systems. The methodology presents sufficient conditions in the form of linear matrix inequalities (LMIs) for the synthesis of static linear robust controllers (Kd ), aiming at first only the system's stability, followed by the system's stability with decay rate (γ > 0) and finally designs that ensure the system's robust D-stability (regional allocation), restricting the eigenvalues at a given region of the complex plane. The performance indexes of decay rate and D-stability are added in the controllers design since ensuring system's stability only is not always sufficient for a good practical performance. The LMIs formulations are made through widely used lemmas (Reciprocal Projection Lemma and Finsler's Lemma) in the stability analysis and in the controllers design for many problems. These lemmas allow the use of a parameter-dependent Lyapunov function (PDLF) to ensure the asymptotic stability of the systems in the sense of Lyapunov. Comparing with the existing litera- ture, in which the results consider classical LMIs formulations, based on the existence of a common quadratic Lyapunov function (CQLF) for the solution of the problems, the present work shows to be less conservative in most occasions. In many cases... (Complete abstract click electronic access below)
Doutor
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Silva, Flávio Henrique Justiniano Ribeiro da. "Funções de Lyapunov estendidas para análise de estabilidade transitória em sistemas elétricos de potência." Universidade de São Paulo, 2004. http://www.teses.usp.br/teses/disponiveis/18/18133/tde-13062017-112014/.
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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.
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Gimenes, Luciene Parron. "Estabilidade e oscilação de soluções de equações diferenciais com retardos e impulsos." Universidade de São Paulo, 2007. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-26042007-205409/.
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O objetivo deste trabalho é investigar propriedades qualitativas de certas equações diferenciais funcionais retardadas de segunda ordem quando lhes são impostos controles de impulsos adequados. Os principais resultados dizem respeito a estabilidade e oscilação por impulsos. Mais especificamente, consideramos algumas equações e provamos que suas soluções triviais podem ser estabilizadas por impulsos. Em seguida, consideramos uma destas equações e provamos que suas soluções podem se tornar oscilatórias com a imposição apropriada de controles de impulsos. Apresentamos alguns exemplos que ilustram nossos resultados. Além do objetivo acima, procuramos produzir um texto que compreendesse a teoria fundamental das equações diferenciais funcionais retardadas impulsivas, teoria esta que, até então, não podia ser encontrada num único texto como este. Desenvolvemos e discutimos existência, unicidade, continuação de soluções, intervalo maximal de existência e dependência contínua de soluções dos valores iniciais para equações diferenciais retardadas impulsivas.The purpose of this work is to investigate qualitative properties of certain second order delay differential equations when some proper impulse controls are added to them. The main results concern the stability and scillation by impulses. More specifically, we consider some equations and prove that their trivial solutions can be stabilized by impulses. We also consider one of these equations and prove that all solutions oscillate when proper impulse controls are imposed. We give some examples to illustrate our results. Because dealing with systems with both delays and impulses is a recent interest of some mathematicians we also considered producing a text that would encompass the fundamental theory of retarded functional differential equations with impulses. Up to now such theory could not be found in a single text as this one. Therefore we discuss and develop basic aspects of the theory as existence, uniqueness, continuability of solutions, maximal interval of existence and continuous dependence of solutions on initial values for impulsive retarded differential equations.
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Ogundare, Babatunde Sunday. "Qualitative and quantitative properties of solutions of ordinary differential equations." Thesis, University of Fort Hare, 2009. http://hdl.handle.net/10353/244.
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This thesis is concerned with the qualitative and quantitative properties of solutions of certain classes of ordinary di erential equations (ODEs); in particular linear boundary value problems of second order ODE's and non-linear ODEs of order at most four. The Lyapunov's second method of special functions called Lyapunov functions are employed extensively in this thesis. We construct suitable complete Lyapunov functions to discuss the qualitative properties of solutions to certain classes of non-linear ordinary di erential equations considered. Though there is no unique way of constructing Lyapunov functions, We adopt Cartwright's method to construct complete Lyapunov functions that are required in this thesis. Su cient conditions were established to discuss the qualitative properties such as boundedness, convergence, periodicity and stability of the classes of equations of our focus. Another aspect of this thesis is on the quantitative properties of solutions. New scheme based on interpolation and collocation is derived for solving initial value problem of ODEs. This scheme is derived from the general method of deriving the spline functions. Also by exploiting the Trigonometric identity property of the Chebyshev polynomials, We develop a new scheme for approximating the solutions of two-point boundary value problems. These schemes are user-friendly, easy to develop algorithm (computer program) and execute. They compare favorably with known standard methods used in solving the classes of problems they were derived for
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Santos, Iguer Luis Domini dos. "Análise de estabilidade de sistemas dinâmicos descontínuos e aplicações /." São José do Rio Preto : [s.n.], 2008. http://hdl.handle.net/11449/94290.
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Resumo: Neste trabalho introduzimos uma classe de sistemas dinâmicos descontínuos com espaço tempo contínuo e analisamos Teoremas que asseguram condições suficientes para a estabilidade de Lyapunov utilizando funções de Lyapunov. Além disso, consideramos também Teoremas de Recíproca, que sob algumas condições garantem uma determinada necessidade para esses Teoremas de estabilidade de Lyapunov.Abstract: In this work we introduce a class of discontinuous dynamical systems with time space continuous and we analyze Theorems that ensure sufficient conditions for the Lyapunov stability using Lyapunov functions. Moreover, we also consider Converse Theorems, which under some conditions guarantee a determined necessity for those Theorems of Lyapunov stability.
Orientador: Geraldo Nunes Silva
Coorientador: Luis Antônio Fernandes de Oliveira
Banca: Luis Antônio Barrera San Martin
Banca: Adalberto Spezamiglio
Mestre
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González, Germán Iván Temoatzin. "Contributions to analysis and control of Takagi-Sugeno systems via piecewise, parameter-dependent, and integral Lyapunov functions." Doctoral thesis, Universitat Politècnica de València, 2018. http://hdl.handle.net/10251/101282.
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Esta tesis considera un enfoque basado en Lyapunov para el análisis y control de sistemas no lineales cuyas ecuaciones dinámicas son reescritas como un modelo Takagi-Sugeno o uno polinomial convexo. Estas estructuras permiten resolver problemas de control mediante técnicas de optimización convexa, más concretamente desigualdades matriciales lineales y suma de cuadrados, que son eficientes herramientas desde un punto de vista computacional. Después de proporcionar una visión general básica del estado actual en el campo de los modelos Takagi-Sugeno, esta tesis aborda cuestiones sobre las funciones de Lyapunov por trozos, dependiente de parámetros e integral de línea, con las siguientes contribuciones:
Un algoritmo mejorado para estimaciones del dominio de atracción de sistemas no lineales para sistemas de tiempo continuo. Los resultados se basan en funciones de Lyapunov por trozos, desigualdades matriciales lineales y argumentaciones geométricas; enfoques basados en conjuntos de nivel en la literatura previa se han mejorado significativamente.
Una función Lyapunov generalizada dependiente de parámetros para la síntesis de controladores para sistemas Takagi-Sugeno. El enfoque propone una ley de control multi-índice que retroalimenta la derivada del tiempo de las funciones de membresía del modelo Takagi-Sugeno para anular los términos que causan localidad a priori en el análisis de Lyapunov.
Una nueva función integral de Lyapunov para el análisis de estabilidad de sistemas no lineales. Estos resultados generalizan aquellos basados en funciones de Lyapunov integral de línea al marco polinomial; resulta que los requisitos de independencia del camino pueden ser anulados por una definición adecuada de una función Lyapunov con términos integrales.This thesis considers a Lyapunov-based approach for analysis and control of nonlinear systems whose dynamical equations are rewritten as a Takagi-Sugeno model or a convex polynomial one. These structures allow solving control problems via convex optimisation techniques, more specifically linear matrix inequalities and sum-of-squares, which are efficient tools from the computational point of view. After providing a basic overview of the state of the art in the field of Takagi-Sugeno models, this thesis address issues on piecewise, parameter-dependent and line-integral Lyapunov functions, with the following contributions: An improved algorithm to estimate the domain of attraction of nonlinear systems for continuous-time systems. The results are based on piecewise Lyapunov functions, linear matrix inequalities, and geometrical argumentations; level-set approaches in prior literature are significantly improved. A generalised parameter-dependent Lyapunov function for synthesis of controllers for Takagi-Sugeno systems. The approach proposed a multi-index control law that feeds back the time derivative of the membership function of the Takagi-Sugeno model to cancel out the terms that cause a priori locality in the Lyapunov analysis. A new integral Lyapunov function for stability analysis of nonlinear systems. These results generalise those based on line-integral Lyapunov functions to the polynomial framework; it turns out path-independency requirements can be overridden by an adequate definition of a Lyapunov function with integral terms.
Aquesta tesi considera un enfocament basat en Lyapunov per a l'anàlisi i control de sistemes no lineals les equacions dinàmiques dels quals són reescrites com un model Takagi-Sugeno o un de polinomial convex. Aquestes estructures permeten resoldre problemes de control mitjançant tècniques d'optimització convexa, més concretament desigualtats matricials lineals i suma de quadrats, que són eines eficients des d'un punt de vista computacional. Després de proporcionar una visió general bàsica de l'estat actual en el camp dels models Takagi-Sugeno, aquesta tesi aborda qüestions sobre les funcions de Lyapunov per trossos, dependent de paràmetres i integral de línia, amb les següents contribucions: Un algoritme millorat per a estimar el domini d'atracció de sistemes no lineals per a sistemes de temps continu. Els resultats es basen en funcions de Lyapunov per trossos, desigualtats matricials lineals i argumentacions geomètriques; enfocaments basats en conjunts de nivell en la literatura prèvia s'han millorat significativament. Una funció Lyapunov generalitzada dependent de paràmetres per a la síntesi de controladors per a sistemes Takagi-Sugeno. L'enfocament proposa una llei de control multi-índex que retroalimenta la derivada del temps de les funcions de membres del model Takagi-Sugeno per anul·lar els termes que causen localitat a priori en l'anàlisi de Lyapunov. Una nova funció integral de Lyapunov per a l'anàlisi d'estabilitat de sistemes no lineals. Aquests resultats generalitzen aquells basats en funcions de Lyapunov integral de línia al marc polinomial; resulta que els requisits d'independència del camí poden ser anul·lats per una definició adequada d'una funció Lyapunov amb termes integrals.
González Germán, IT. (2018). Contributions to analysis and control of Takagi-Sugeno systems via piecewise, parameter-dependent, and integral Lyapunov functions [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/101282
TESIS
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Möhlmann, Eike Verfasser], Oliver [Akademischer Betreuer] Theel, and Martin [Akademischer Betreuer] [Fränzle. "Automatic stability verification via Lyapunov functions: representations, transformations, and practical issues / Eike Möhlmann ; Oliver Theel, Martin Fränzle." Oldenburg : BIS der Universität Oldenburg, 2018. http://d-nb.info/1199537357/34.
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Leising, Sophie. "Nonlinear controller synthesis for complex chemical and biochemical reaction systems." Link to electronic thesis, 2005. http://www.wpi.edu/Pubs/ETD/Available/etd-050205-152657/.
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Thesis (M.S.) -- Worcester Polytechnic Institute.Keywords: model predictive control; discrete-time model; continuous-time model; nonlinear systems; Lyapunov design. Includes bibliographical references (p. 99-102).
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Espiritu, Ledesma Guido Gerson 1985. "Lyapunov graph in the study of Smale flows and Morse-Novikov flows = Grafo de Lyapunov no estudo dos fluxos de Smale e fluxos de Morse-Novikov." [s.n.], 2014. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307539.
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Orientador: Ketty Abaroa de RezendeTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: Neste trabalho, usamos os grafos de Lyapunov como uma ferramenta combinat{\'o}ria para obter classifica\c{c}{\~o}es completas de fluxos Smale sobre $\ss$ e fluxos Morse-Novikov sobre superf{\'i}cies orient{\'a}veis e n{\~a}o orient{\'a}veis. Esta classifica\c{c}{\~a}o consiste em obter condi\c{c}{\~o}es necess{\'a}rias e suficientes que devem ser satisfeitas por um grafo de Lyapunov abstrato de forma a ser associado a um fluxo Smale sobre $\ss$ ou um fluxo Morse-Novikov sobre uma superf{\'i}cie respectivamente. Assim nesta tese de doutorado obtemos os seguintes resultados: \begin{enumerate} \item As condições locais que devem ser satisfeitas por cada vértice do grafo de Lyapunov, assim como as condições globais que devem ser satisfeitas pelos grafos para estarem associados a um fluxo Smale sobre $\ss$ ou a um fluxo Morse-Novikov sobre uma superfície s{\~a}o determinadas. \item A realização destes grafos abstratos sujeita {\'a}s condições determinadas acima, como fluxos Smale sobre $\ss$ ou fluxos Morse-Novikov sobre superfícies respectivamente, são obtidas. \end{enumerate}
Abstract: In this work Lyapunov graphs are used as a combinatorial tool in order to obtain a complete classification of Smale flows on $\ss$ and Morse-Novikov flows on orientable and non-orientable surfaces. This classification consists in determining necessary and sufficient conditions that must be satisfied by an abstract Lyapunov graph so that it is associated to a Smale flow on $\ss$ or to a Morse-Novikov flow on a surface respectively.\\ In summary in this doctoral thesis we obtain the following results: \begin{enumerate} \item The local conditions that must be satisfied by each vertex on a Lyapunov graph is determinated as well as the global conditions on the graph in order for it to be associated to a Smale flow on $\ss$ or a Morse-Novikov flow on a surface. \item The realization of these graphs subject to the conditions found above as Smale flows on $\ss$ or as Morse-Novikov flows on surfaces respectively is obtained. \end{enumerate}
Doutorado
Matematica
Doutor em Matemática
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Peruzzi, Nelson Jose. "Dinamica não linear e controle de sistemas ideais e não-ideais periodicos." [s.n.], 2005. http://repositorio.unicamp.br/jspui/handle/REPOSIP/265398.
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Orientadores: Jose Manoel BalthazarTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica
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Resumo: Neste trabalho, apresenta-se um novo método numérico para aproximar matriz de transição de estados (STM) para sistemas com coeficientes periódicos no tempo. Este método, é baseado na expansão polinomial de Chebyshev, no método iterativo de Picard e na transformação de Lyapunov-Floquet (L-F) e aplica-se na análise da dinâmica e o controle de sistemas lineares e periódicos. Para o controle, aplicam-se dois projetos para eliminar o comportamento caótico de sistemas periódicos no tempo. O primeiro, usa o projeto de controle realimentado baseado na aplicação da transformação L-F, e o objetivo do controlador é conduzir a órbita do sistema para um ponto fixo ou para uma órbita periódica. No segundo, utiliza-se o controle não-linear para bifurcação, e o objetivo, neste caso, é modificar (atrasar ou eliminar) as características de uma bifurcação ao longo de sua rota para o caos. Como exemplo, aplicou-se, com sucesso, a técnica para análise e o controle da dinâmica: num pêndulo com excitação paramétrica, no oscilador de Duffing, no sistema de Rõssler e sistema pêndulo duplo invertido. O método, também, mostrou-se satisfatório na análise e controle de um sistema monotrilho não ideal
Abstract: In thiswork, a new numericalmethodto approximatestatetransitionmatrix(STM) for systems with time-periodic coefficients is presented. This method is based on the expansion Chebyshev polinomials,on the Picard iterationand on the Lyapunov-Floquet transfonnation(transfonnationL-F). It is applied to the dynamicalanalysis and control of linear periodic systems.For the control, two projectsto eliminatethe chaoticbehaviorof time periodic systemsare applied.The first one, uses the feedbackcontroldesignbased on the L-F transfonnation,and the controller'sobjectiveis to drive the orbit of the systemto an equilibriumpoint or a periodicorbit. fu the secondone, the non-lineal control for bifurcations used, and the objective,in this case, is to modify (to put back or to eliminate) the characteristicsof a bifurcation along its route to chaos. As example, the technique for dynamical analysis and control was applied, successfully, to a pendulum with parametric excitement, the Duffing's oscillator,the Rõssler's systemand the inverteddoublependulum The methodwas, also, to be shownsatisfactoryin the analysisand controlof a monorailnon-idealsystem
Doutorado
Mecanica dos Sólidos e Projeto Mecanico
Doutor em Engenharia Mecânica
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Panimadai, Ramaswamy Shweta Annapurani. "Formation control of car-like mobile robots." Diss., Rolla, Mo. : Missouri University of Science and Technology, 2008. http://scholarsmine.mst.edu/thesis/pdf/Panimadai_Ramaswamy_09007dcc804aeda1.pdf.
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Thesis (M.S.)--Missouri University of Science and Technology, 2008.Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed April 14, 2008) Includes bibliographical references (p. 119-121).
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Seyfried, Aaron W. "Stability of a Fuzzy Logic Based Piecewise Linear Hybrid System." Wright State University / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=wright1370017300.
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Loquen, Thomas. "Some Results on Reset Control systems." Thesis, Toulouse, INPT, 2010. http://www.theses.fr/2010INPT0117/document.
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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)
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Costa, Éder Rítis Aragão. "Sistemas gradientes, decomposição de Morse e funções de Lyapunov sob perturbação." Universidade de São Paulo, 2012. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-13042012-162303/.
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Neste trabalho investigamos a existência de uma função de Lyapunov associada a um sistema de tipo gradiente, semigrupos ou processos de evolução. Para isso, um estudo detalhado da teoria de Morse desempenha um papel decisivo. Como principal consequência deste estudo obtemos a estabilidade dos sistemas gradientes sob perturbação (autônoma ou não). A aplicabilidade dos resultados abstratos que aqui discutimos é exemplificada estudando-se sistemas de equações diferenciais em espaços de Banach com acoplamento unilateralIn this work we investigated the existence of a Lyapunov function associated to a gradient-like system, semigroups or evolution processes. For that, a detailed study of Morse theory plays a central role. As the main consequence of this study we obtain the stability of gradient systems under perturbation (autonomous or not). The applicability of the abstract results discussed here is exemplified by studying systems of differential equations in Banach spaces with unilateral coupling
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Lacerda, Márcio Júnior 1987. "Filtragem robusta de sistemas lineares invariantes no tempo por meio de funções de Lyapunov polinomiais." [s.n.], 2010. http://repositorio.unicamp.br/jspui/handle/REPOSIP/259734.
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Orientadores: Pedro Luis Dias Peres, Ricardo Coração de Leão Fontoura de OliveiraDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação
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Resumo: Este trabalho apresenta novas condições na forma de desigualdades matriciais lineares para a síntese de filtros robustos H2 e H¥ de ordem completa, para sistemas incertos, contínuos e discretos no tempo. Os parâmetros incertos invariantes no tempo pertencem a um politopo com vértices conhecidos. Graças à existência de um número maior de variáveis de folga e à utilização de relaxações baseadas em matrizes polinomiais homogêneas, desigualdades matriciais lineares podem ser obtidas das condições propostas para o projeto de filtros robustos, com desempenho superior aos métodos existentes. A superioridade e eficiência do método proposto para o projeto dos filtros robustos são ilustradas por meio de comparações numéricas e exemplos da literatura
Abstract: This work presents new convex optimization procedures for full order robust H2 and H? filter design for continuous and discrete-time uncertain linear systems. The time-invariant uncertain parameters are supposed to belong to a polytope with known vertices. Thanks to the use of a larger number of slack variables and homogeneous polynomial relaxations, linear matrix inequalities for the design of robust filters can be derived from the proposed conditions, outperforming the existingmethods. The superiority and efficiency of the proposed method for robust filter design are illustrated by means of numerical comparisons in benchmark examples from the literature
Mestrado
Automação
Mestre em Engenharia Elétrica
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Latorre, Hector. "Modeling and Control of VSC-HVDC Transmissions." Doctoral thesis, KTH, Elektriska energisystem, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-32313.
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Presently power systems are being operated under high stress level conditions unforeseen at the moment they were designed. These operating conditions have negatively impacted reliability, controllability and security margins. FACTS devices and HVDC transmissions have emerged as solutions to help power systems to increase the stability margins. VSC-HVDC transmissions are of particular interest since the principal characteristic of this type of transmission is its ability to independently control active power and reactive power. This thesis presents various control strategies to improve damping of electromechanical oscillations, and also enhance transient and voltage stability by using VSC-HVDC transmissions. These control strategies are based of different theory frames, namely, modal analysis, nonlinear control (Lyapunov theory) and model predictive control. In the derivation of the control strategies two models of VSC-HVDC transmissions were also derived. They are Injection Model and Simple Model. Simulations done in the HVDC Light Open Model showed the validity of the derived models of VSC-HVDC transmissions and the effectiveness of the control strategies. Furthermore the thesis presents an analysis of local and remote information used as inputs signals in the control strategies. It also describes an approach to relate modal analysis and the SIME method. This approach allowed the application of SIME method with a reduced number of generators, which were selected based on modal analysis. As a general conclusion it was shown that VSC-HVDC transmissions with an appropriate input signal and control strategy was an effective means to improve the system stability.QC 20110412
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Herzog, David Paul. "Geometry's Fundamental Role in the Stability of Stochastic Differential Equations." Diss., The University of Arizona, 2011. http://hdl.handle.net/10150/145150.
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We study dynamical systems in the complex plane under the effect of constant noise. We show for a wide class of polynomial equations that the ergodic property is valid in the associated stochastic perturbation if and only if the noise added is in the direction transversal to all unstable trajectories of the deterministic system. This has the interpretation that noise in the "right" direction prevents the process from being unstable: a fundamental, but not well-understood, geometric principle which seems to underlie many other similar equations. The result is proven by using Lyapunov functions and geometric control theory.
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Zhang, Xiling. "On numerical approximations for stochastic differential equations." Thesis, University of Edinburgh, 2017. http://hdl.handle.net/1842/28931.
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This thesis consists of several problems concerning numerical approximations for stochastic differential equations, and is divided into three parts. The first one is on the integrability and asymptotic stability with respect to a certain class of Lyapunov functions, and the preservation of the comparison theorem for the explicit numerical schemes. In general, those properties of the original equation can be lost after discretisation, but it will be shown that by some suitable modification of the Euler scheme they can be preserved to some extent while keeping the strong convergence rate maintained. The second part focuses on the approximation of iterated stochastic integrals, which is the essential ingredient for the construction of higher-order approximations. The coupling method is adopted for that purpose, which aims at finding a random variable whose law is easy to generate and is close to the target distribution. The last topic is motivated by the simulation of equations driven by Lévy processes, for which the main difficulty is to generalise some coupling results for the one-dimensional central limit theorem to the multi-dimensional case.
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Maghenem, Mohamed Adlene. "Stability and Stabilization of Networked Systems." Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLS186/document.
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Dans cette thèse, des méthodes dites de Lyapunov sont proposées afin de résoudre des problèmes liés à la coordination distribuée des systèmes multiagent, plus précisément, un groupe de systèmes (agents) non-linéaires formés de robots mobiles non-holonomes est considéré. Pour ce groupe de systèmes, des lois de commande distribuée sont proposées dans le but de résoudre des problèmes de type leader-suiveur en formation et aussi des problèmes de type formation sans-leader par une approche de consensus, sous différentes hypothèses sur le graphe de communication et surtout sur les vitesses du leader.L'originalité de ce travail est dans l'approche proposée pour l'étude de stabilité de la boucle fermée, cette approche consiste à transformer les deux derniers problèmes en des problèmes de stabilisation globale asymptotique d'un ensemble invariant. L’analyse de stabilité est basée sur la construction de fonction de Lyapunov et de fonction de Lyapunov-Karasovskii strictes pour des classes de systèmes non-linéaires variant dans le temps présentant des retards bornés et variant dans le tempsIn this thesis, we propose a Lyapunov based approaches to address some distributedsolutions to multi-agent coordination problems, more precisely, we consider a groupof agents modeled as nonholonomic mobile robots, we provide a distributed controllaws in order to solve the leader-follower and the leaderless consensus problems under different assumptions on the communication graph topology and on the leader’strajectories. The originality of this work relies on the closed-loop analysis approach, that is, it consists on transforming the last two problems into a global stabilization problem of an invariant set. The stability analysis is mainly based on the construction of strict Lyapunov functions and strict Lyapunov-Krasovskii functionals for a classes of nonlinear time-varying and/or delayed systems
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Montúfar, López Hernán Roberto. "Teoria de Conley para campos Gutierrez-Sotomayor." [s.n.], 2010. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307543.
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Orientador: Ketty Abaroa de RezendeTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Em [6] são apresentadas condições necessárias e suficientes para a estabilidade estrutural e o teorema de densidade para campos de vetores em 2-variedades com singularidades simples dos seguintes tipos: cone, guarda-chuva de Whitney, ponto duplo e ponto triplo. Nesta tese, estudamos os fluxos induzidos por estes campos de vetores, que denominamos fluxos Gutierrez-Sotomayor, do ponto de vista topológico utilizando a teoria de Conley. Apresentamos uma fórmula dinâmico-topológica que relaciona o índice de Conley de uma variedade com singularidades simples M que possui uma estratificação que a decompõe numa união disjunta da sua parte regular e da sua parte singular. Usando essa estratificação mostramos que se a singularidade está na parte singular S de M o seu índice pode ser calculado tanto com respeito a M como com respeito a S. Definimos uma função de Lyapunov, neste contexto, e mostramos sua existência para fluxos sem órbitas periódicas e sem ciclos singulares. Em seguida, por uma análise da seqüência de homologia longa exata de um par índice determinamos propriedades que um grafo de Lyapunov deve satisfazer para estar associado a um fluxo. Também abordamos a questão da realização de grafos de Lyapunov abstratos. Para isto, primeiramente apresentamos a igualdade de Poincaré-Hopf, para o caso bidimensional, que caracteriza a relação entre o primeiro número de Betti das 1-variedades ramificadas que são fronteiras de um bloco isolante com seu número de componentes de fronteira e o índice numérico de Conley. Em seguida, mostramos que dados números inteiros positivos que satisfaçam a condição de Poincaré-Hopf sempre é possível construir um bloco isolante que satisfaz estes dados dinâmicos e homológicos
Abstract: In [6] a characterization and genericity theorem for C1-structurally stable vector fields tangent to a 2-dimensional compact subset M of Rk are established. Also in [6], new types of structurally stable singularities and periodic orbits are presented. In this thesis we study the continuous flows associated to these vector fields, which we refer to as the Gutierrez-Sotomayor flows on manifolds M with simple singularities using Conley Index Theory. We consider a stratification of M which decomposes it into a union of its regular and singular strata. We prove certain Euler type formulas which relate topology of M and dynamics on the singular strata. We show the existence of a Lyapunov function for Gutierrez-Sotomayor flows without periodic orbits and singular cycles in this context. Using long exact sequence analysis of index pairs we determine necessary and sufficient conditions for a Gutierrez-Sotomayor flow to be defined on an isolating block. We organize this combinatorially with the aid of Lyapunov graphs and using a Poincar'e-Hopf equality we give necessary conditions for a Lyapunov graph to be associated to a Gutierrez-Sotomayor flow and we also prove these conditions are sufficient
Doutorado
Geometria e Topologia
Doutor em Matemática
43
Ferreira, Jaqueline da Costa. "Sistemas semidinâmicos dissipativos com impulsos." Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-07112016-155325/.
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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.
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Tognetti, Tais Calliero. "Controle de sistemas dinamicos : estabilidade absoluta, saturação e bilinearidade." [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/261053.
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Orientador: Pedro Luis Dias PeresTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação
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Resumo: Esta tese apresenta contribuições para a solução de problemas de análise de estabilidade e síntese de controladores por realimentação de estados de sistemas dinâmicos que possuem elementos não-lineares, por meio de condições na forma de desigualdades matriciais lineares e funções de Lyapunov. Para sistemas chaveados sujeitos a saturação nos atuadores, são fornecidas condições convexas para o cálculo de ganhos chaveados e robustos. A saturação é modelada como uma não-linearidade de setor e uma estimativa do domínio de estabilidade é determinada. Para sistemas lineares com incertezas politópicas e não-linearidades pertencentes a setores, são fornecidas condições convexas de dimensão finita para construir funções de Lur'e com dependência polinomial homogênea nos parâmetros. Se satisfeitas, as condições garantem a estabilidade para todo o domínio de incertezas e para todas as não-linearidades pertencentes ao setor e permitem o cômputo de controladores estabilizantes robustos por realimentação linear e não-linear. Para sistemas bilineares instáveis, contínuos e discretos no tempo, é proposto um procedimento para calcular um ganho estabilizante de controle por realimentação de estados. O método baseia-se na solução alternada de dois problemas de otimização convexa descritos por desigualdades matriciais lineares, fornecendo uma estimativa do domínio de estabilidade. Extensões para tratar controladores robustos e lineares variantes com parâmetros são também apresentadas.
Abstract: This thesis presents contributions to the solution of the problems of stability analysis and synthesis of state feedback controllers for dynamic systems with non-linear elements, by means of conditions based on linear matrix inequalities and Lyapunov functions. For switched systems subject to saturation in the actuators, convex conditions to design switched and robust controllers are presented. The saturation is modeled as a sector non-linearity and an estimate of the domain of stability is determined. For linear systems with polytopic uncertainties and sector non-linearities, convex conditions of finite dimension to build Lur'e functions with homogeneous polynomially parameter dependence are provided. If satisfied, the conditions guarantee the stability of the entire domain of uncertainty for all sector non-linearities, allowing the design of linear and non-linear robust state feedback stabilizing controllers. For continuous and discrete-time unstable bilinear systems, a procedure to design a state feedback stabilizing control gain is proposed. The method is based on the alternate solution of two convex optimization problems described by linear matrix inequalities, providing an estimate of the domain of stability. Extensions to handle robust and linear parameter varying controllers are also presented.
Doutorado
Automação
Doutor em Engenharia Elétrica
45
Pinheiro, Rafael Fernandes. "O problema de Lurie e aplicações às redes neurais." Universidade de São Paulo, 2015. http://www.teses.usp.br/teses/disponiveis/45/45132/tde-25042015-223201/.
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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.
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Borges, Renato Alves. "Controle e filtragem de sistemas lineares variantes no tempo por meio de funções de Lyapunov dependentes de parametros." [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/261054.
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Orientador: Pedro Luis Dias PeresTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação
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Resumo: A principal contribuição desta tese é a proposta de condições de sintese de filtros e controladores lineares, tanto robustos quanto dependentes de parametros, para sistemas discretos variantes no tempo.Os controladores, ou filtros, são obtidos solucionando problemas de otimização formulados em termos de desigualdades matriciais bililineares, por meio de um metodo que se baseia na alternancia de problemas convexos descritos por desigualdades matriciais lineares. Para obtenção das condiçoes de sintese foram utilizadas tanto funções de Lyapunov afins nos parâmetros quanto ametros, alem de variáveis multi-afins em diferentes instantes de tempo dos parâmetros, alem de variaveis extras introduzidas pelo lema de Finsler. Nesse contexto, sao tratados problemas de sintese com custo garantido H, assegurando robustez em relação a incertezas não estruturadas. Simulaçoes numéricas ilustram a eficiencia dos metodos propostos em termos de desempenho H quando comparados com outros metodos da literatura
Abstract: The main contribution of this dissertation is to propose conditions for linear filter and controller design, considering both robust and parameter dependent structures, for discrete time-varying systems. The controllers, or filters, are obtained through the solution of optimization problems, formulated in terms of bilinear matrix inequalities, using a method that alternates convex optimization problems described in terms of linear matrix inequalities. Both affine and multi-affine in different instants of time (path dependent)Lyapunov functions were usedto obtain the design conditions, as wellas extra variables introduced bythe Finsler's lemma.Design problems that take into account an H guaranteed cost were investigated, providing robustness with respect to unstructured uncertainties. Numerical simulations show the effciency of the proposed methods in terms of H performance when compared with other strategies from the literature
Doutorado
Automação
Doutor em Engenharia Elétrica
47
Ferracini, Evelize Aparecida dos Santos [UNESP]. "Estabilidade de equações diferenciais ordinárias através de funções dicotômicas." Universidade Estadual Paulista (UNESP), 2011. http://hdl.handle.net/11449/94356.
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ferracini_eas_me_rcla.pdf: 410350 bytes, checksum: c3d18941dab120874c42b2bbec91afc7 (MD5)Neste trabalho apresentamos um estudo sobre estabilidade do equilíbrio nulo de equações diferenciais ordinárias autônomas através do Segundo Método de Liapunov e do Método das Funções Dicotômicas, que é uma extensão do Segundo Método de Liapunov
This work presents a study about stability of the null equilibrium of autonomous ordinary differential equations by Liapunov’s Second Method and Method of Dichotomic Maps, which is an extension of the Liapunov’s Second Method
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Li, Hongxing, and 李宏兴. "Optimal data dissemination in stochastic and arbitrary wireless networks." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2012. http://hub.hku.hk/bib/B4832971X.
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Data dissemination among wireless devices is an essential application in wireless networks. In contrast to its wired counterparts which have more stable network settings, wireless networks are subject to network dynamics, such as variable network topology, channel availability and capacity, which are due to user mobility, signal collision, random channel fading and scattering, etc. Network dynamics complicate the protocol design for optimal data disseminations. Although the topic has been intensively discussed for many years, existing solutions are still not completely satisfactory, especially for stochastic or arbitrary networks. In this thesis, we address optimal data dissemination in both stochastic and arbitrary wireless networks, using techniques of Lyapunov optimization, graph theory, network coding, multi-resolution coding and successive interference cancellation.
We first discuss the maximization of time-averaged throughput utility over a long run for unicast and multirate multicast, respectively, in stochastic wireless networks without probing into the future. For multi-session unicast communications, a utility-maximizing cross-layer design, composed of joint end-to-end rate control, routing, and channel allocation, is proposed for cognitive radio networks with stochastic primary user occupations. Then, we study optimal multirate multicast to receivers with non-uniform receiving rates, also making dynamic cross-layer decisions, in a general wireless network with both a timevarying topology and random channel capacities, by utilizing random linear network coding and multi-resolution coding. In both solutions, we assume users are selfish and prefer only to relay data for others with strong social ties. Such social selfishness of users is a new constraint in network protocol design. Its impact on efficient data dissemination in wireless networks is largely unstudied, especially under stochastic settings. Lyapunov optimization is applied in our protocol design achieving close-to-optimal utilities.
Next, we turn to latency-minimizing data aggregation in wireless sensor networks having arbitrary network topologies under the physical interference model. Different from our effort for stochastic networks where we target at time-averaged optimality over a long run, the objective here is to minimize the time-span to accomplish one round of aggregation scheduling for all sensors in an arbitrary topology. This problem is NP-hard, involving both aggregation tree construction and collision-free link scheduling. The current literature mostly considers the protocol interference model, which has been shown to be less practical than the physical interference model in characterizing the interference relations in the real world. A distributed solution under the physical interference model is challenging since cumulative interferences from all concurrently transmitting devices need to be well measured. In this thesis, we present a distributed aggregation protocol with an improved approximation ratio as compared with previous work. We then discuss the tradeoff between aggregation latency and energy consumption for arbitrary topologies when the successive interference cancellation technique is in force. Another distributed algorithm is introduced with asymptotic optimality in both aggregation latency and latency-energy tradeoff.
Through theoretical analysis and empirical study, we rigorously examine the optimality of our protocols comparing with both the theoretical optima and existing solutions.published_or_final_version
Computer Science
Doctoral
Doctor of Philosophy
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Barbosa, Peterson Taylor Castro. "A fisiologia do neuronio e os modelos de Hodgkin-Huxley e FitzHugh-Nagumo." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307152.
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Orientador: Alberto Vazquez SaaDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Apresentaremos neste trabalho descrições detalhadas da modelagem e conceitos fisiológicos associados aos modelos de Hodgkin-Huxley e Fitzhugh-Nagumo, para em seguida compararmos suas características. Tentaremos justificar a opção pelo modelo de Fitzhugh-Nagumo como objeto de estudo viável pra interações entre neurônios, e em seguida introduziremos um tipo de interação na qual o comportamento do sistema para dois neurônios apresenta fenômenos de comportamento irregular. Para isso, mostraremos alguns resultados referentes à esta interação, chamada acoplamento repulsivo, descrito por Yanagita et al. (2005), e analisaremos fatos relativos às características qualitativas e quantitativas das equações envolvidas, como os Expoentes de Liapunov e o Intervalo entre Disparos, a partir de alguns resultados numéricos pertinentes
Abstract: We present in this work some detailed descriptions concerning modeling and physiological concepts on Hodgkin-Huxley and Fitzhugh-Nagumo models, and after that to compare its main features. We will try to justify the choice for the Fitzhugh-Nagumo model as a viable study object for neuronal interactions, and than we introduce a particular kind of interaction in which the reaction of a two-neuron system shows irregular behavior phenomena. In order to achieve this, we also present some results associated to this interaction, called repulsive coupling, described by Yanagita et al. (2005), and afterward, we analyze some facts related to quantitative and qualitative characteristics of the involved equations, such as Liapunov Exponents and the Interspike Interval, based on numerical simulations of interest
Mestrado
Biomatematica
Mestre em Matemática Aplicada
50
Santos, Lucianna Helene Silva dos. "Teoria de controle ótimo com aplicações a sistemas biológicos." Universidade do Estado do Rio de Janeiro, 2012. http://www.bdtd.uerj.br/tde_busca/arquivo.php?codArquivo=4993.
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Fundação de Amparo à Pesquisa do Estado do Rio de JaneiroNeste trabalho apresentamos as etapas para a utilização do método da Programação Dinâmica, ou Princípio de Otimização de Bellman, para aplicações de controle ótimo. Investigamos a noção de funções de controle de Lyapunov (FCL) e sua relação com a estabilidade de sistemas autônomos com controle. Uma função de controle de Lyapunov deverá satisfazer a equação de Hamilton-Jacobi-Bellman (H-J-B). Usando esse fato, se uma função de controle de Lyapunov é conhecida, será então possível determinar a lei de realimentação ótima; isto é, a lei de controle que torna o sistema globalmente assintóticamente controlável a um estado de equilíbrio. Como aplicação, apresentamos uma modelagem matemática adequada a um problema de controle ótimo de certos sistemas biológicos. Este trabalho conta também com um breve histórico sobre o desenvolvimento da Teoria de Controle de forma a ilustrar a importância, o progresso e a aplicação das técnicas de controle em diferentes áreas ao longo do tempo.
This dissertation presents the steps for using the method of Dynamic Programming or Bellman Optimization Principle for optimal control applications. We investigate the notion of control-Lyapunov functions (CLF) and its relation to the stability of autonomous systems with control. A control-Lyapunov function must satisfy the Hamilton-Jacobi- Bellman equation (H-J-B). Using this fact, if a control-Lyapunov function is known, it is possible to determine the optimal feedback law, in other words, the control law which makes the system globally asymptotically controllable at an equilibrium state. As an application, we present a mathematical model suitable for an optimal control problem of certain biological systems. This dissertation also presents a brief historic about the development of the Control Theory in a way of illustrate the importance and the progress of the control techniques, specially where it can be applied, according to the diverse areas and different times that this techniques were discovered and used.