Dissertations / Theses: 'Lyapunov stability'

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Gumus, Mehmet. "ON THE LYAPUNOV-TYPE DIAGONAL STABILITY." OpenSIUC, 2017. https://opensiuc.lib.siu.edu/dissertations/1421.

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In this dissertation we study the Lyapunov diagonal stability and its extensions through partitions of the index set {1,...,n}. This type of matrix stability plays an important role in various applied areas such as population dynamics, systems theory and complex networks. We first examine a result of Redheffer that reduces Lyapunov diagonal stability of a matrix to common diagonal Lyapunov solutions on two matrices of order one less. An enhanced statement of this result based on the Schur complement formulation is presented here along with a shorter and purely matrix-theoretic proof. We develop a number of extensions to this result, and formulate the range of feasible common diagonal Lyapunov solutions. In particular, we derive explicit algebraic conditions for a set of 2 x 2 matrices to share a common diagonal Lyapunov solution. In addition, we provide an affirmative answer to an open problem concerning two different necessary and sufficient conditions, due to Oleng, Narendra, and Shorten, for a pair of 2 x 2 matrices to share a common diagonal Lyapunov solution. Furthermore, the connection between Lyapunov diagonal stability and the P-matrix property under certain Hadamard multiplication is extended. Accordingly, we present a new characterization involving Hadamard multiplications for simultaneous Lyapunov diagonal stability on a set of matrices. In particular, the common diagonal Lyapunov solution problem is reduced to a more convenient determinantal condition. This development is based upon a new concept called P-sets and a recent result regarding simultaneous Lyapunov diagonal stability by Berman, Goldberg, and Shorten. Next, we consider various types of matrix stability involving a partition alpha of {1,..., n}. We introduce the notions of additive H(alpha)-stability and P_0(alpha)-matrices, extending those of additive D-stability and nonsingular P_0-matrices. Several new results are developed, connecting additive H(alpha)-stability and the P_0(alpha)-matrix property to the existing results on matrix stability involving alpha. We also point out some differences between these types of matrix stability and the conventional matrix stability. Besides, the extensions of results related to Lyapunov diagonal stability, D-stability, and additive D-stability are discussed. Finally, we introduce the notion of common alpha-scalar diagonal Lyapunov solutions over a set of matrices, which is a generalization of common diagonal Lyapunov solutions. We present two different characterizations of this new concept based on the well-known results for Lyapunov alpha-scalar stability [34]. The first one hinges on a general version of the theorem of the alternative, and the second one using Hadamard multiplications stems from an extension of the P-set property. Several illustrative examples and an application concerning a set of block upper triangular matrices are provided.

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Feng, Xiangbo. "Lyapunov exponents and stability of linear stochastic systems." Case Western Reserve University School of Graduate Studies / OhioLINK, 1990. http://rave.ohiolink.edu/etdc/view?acc_num=case1054928844.

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3

Grünvogel, Stefan Michael. "Lyapunov spectrum and control sets." Augsburg [Germany] : Wissner-Verlag, 2000. http://catalog.hathitrust.org/api/volumes/oclc/45796984.html.

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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’état
Modeling 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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England, Scott Alan. "Quantifying Dynamic Stability of Musculoskeletal Systems using Lyapunov Exponents." Thesis, Virginia Tech, 2005. http://hdl.handle.net/10919/44784.

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Increased attention has been paid in recent years to the means in which the body maintains stability and the subtleties of the neurocontroller. Variability of kinematic data has been used as a measure of stability but these analyses are not appropriate for quantifying stability of dynamic systems. Response of biological control systems depend on both temporal and spatial inputs, so means of quantifying stability should account for both. These studies utilized tools developed for the analysis of deterministic chaos to quantify local dynamic stability of musculoskeletal systems. The initial study aimed to answer the oft assumed conjecture that reduced gait speeds in people with neuromuscular impairments lead to improved stability. Healthy subjects walked on a motorized treadmill at an array of speeds ranging from slow to fast while kinematic joint angle data were recorded. Significant (p < 0.001) trends showed that stability monotonically decreased with increasing walking speeds. A second study was performed to investigate dynamic stability of the trunk. Healthy subjects went through a variety of motions exhibiting either symmetric flexion in the sagittal plane or asymmetric flexion including twisting at both low and high cycle frequencies. Faster cycle frequencies led to significantly (p<0.001) greater instability than slower frequencies. Motions that were hybrids of flexion and rotation were significantly (p<0.001) more stable than motions of pure rotation or flexion. Finding means of increasing dynamic stability may provide great understanding of the neurocontroller as well as decrease instances of injury related to repetitive tasks. Future studies should look in greater detail at the relationships between dynamic instability and injury and between local dynamic stability and global dynamic stability.
Master of Science

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Best, Eric A. "Stability assessment of nonlinear systems using the lyapunov exponent." Ohio University / OhioLINK, 2003. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1175019061.

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Tanaka, Martin L. "Biodynamic Analysis of Human Torso Stability using Finite Time Lyapunov Exponents." Diss., Virginia Tech, 2008. http://hdl.handle.net/10919/26580.

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Low back pain is a common medical problem around the world afflicting 80% of the population some time in their life. Low back injury can result from a loss of torso stability causing excessive strain in soft tissue. This investigation seeks to apply existing methods to new applications and to develop new methods to assess torso stability. First, the time series averaged finite time Lyapunov exponent is calculated from data obtained during seated stability experiments. The Lyapunov exponent is found to increase with increasing task difficulty. Second, a new metric for evaluating torso stability is introduced, the threshold of stability. This parameter is defined as the maximum task difficulty in which dynamic stability can be maintained for the test duration. The threshold of stability effectively differentiates torso stability at two levels of visual feedback. Third, the state space distribution of the finite time Lyapunov exponent (FTLE) field is evaluated for deterministic and stochastic systems. Two new methods are developed to generate the FTLE field from time series data. Using these methods, Lagrangian coherent structures (LCS) are found for an inverted pendulum, the Acrobot, and planar wobble chair models. The LCS are ridges in the FTLE field that separate two inherently different types of motion when applied to rigid-body dynamic systems. As a result, LCS can be used to identify the boundaries of the basin of stability. Finally, these new methods are used to find the basin of stability from time series data collected from torso stability experiments. The LCS and basins of stability provide a richer understanding into the system dynamics when compared to existing methods. By gaining a better understanding of torso stability, it is hoped this knowledge can be used to prevent low back injury and pain in the future. These new methods may also be useful in evaluating other biodynamic systems such as standing postural sway, knee stability, or hip stability as well as time series applications outside the area of biomechanics.
Ph. D.

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Schroll, Arno. "Der maximale Lyapunov Exponent." Doctoral thesis, Humboldt-Universität zu Berlin, 2020. http://dx.doi.org/10.18452/21994.

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Bewegungsstabilität wird durch die Fähigkeit des neuromuskulären Systems adäquat auf Störungen der Bewegung antworten zu können erreicht. Einschränkungen der Stabilität werden z. B. mit Sturzrisiko in Verbindung gebracht, was schwere Konsequenzen für die Lebensqualität und Kosten im Gesundheitssystem hat. Nach wie vor wird debattiert, wie eine geeignete Bewertung von Stabilität vorgenommen werden kann. Diese Arbeit behandelt den maximalen Lyapunov Exponenten. Er drückt aus, wie sensitiv das System auf kleine Störungen eines Zustands reagiert. Eine Zeitreihe wird zunächst mittels zeitversetzter Kopien in einen mehrdimensionalen Raum eingebettet. In dieser rekonstruierten Dynamik berechnet man dann die Steigung der mittleren logarithmischen Divergenz initial naher Punkte. Die methodischen Konsequenzen für die Anwendung dieser Systemtheorie auf Bewegungen sind jedoch bislang unzureichend beleuchtet. Der experimentelle Teil zeigt klare Indizien, dass es bei Bewegungen weniger um die Analyse eines komplexen Systemdeterminismus geht, sondern um verschieden hohe dynamische Rauschlevel. Je höher das Rauschlevel, desto instabiler das System. Anwendung von Rauschreduktion führt zu kleineren Effektstärken. Das hat Folgen: Die Funktionswerte der Average Mutual Information, die bisher nur zur Bestimmung des Zeitversatzes genutzt wurden, können bereits Unterschiede in der Stabilität zeigen. Die Abschätzung der Dimension für die Einbettung (unabhängig vom verwendeten Algorithmus), ist stark von der Länge der Zeitreihe abhängig und wird bisher eher überschätzt. Die größten Effekte sind in Dimension drei zu beobachten und ein sehr früher Bereich zur Auswertung der Divergenzkurve ist zu empfehlen. Damit wird eine effiziente und standardisierte Analyse vorgeschlagen, die zudem besser imstande ist, Unterschiede verschiedener Bedingungen oder Gruppen aufzuzeigen.
Reductions of movement stability due to impairments of the motor system to respond adequately to perturbations are associated with e. g. the risk of fall. This has consequences for quality of life and costs in health care. However, there is still an debate on how to measure stability. This thesis examines the maximum Lyapunov exponent, which became popular in sports science the last two decades. The exponent quantifies how sensitive a system is reacting to small perturbations. A measured data series and its time delayed copies are embedded in a moredimensional space and the exponent is calculated with respect to this reconstructed dynamic as average slope of the logarithmic divergence curve of initially nearby points. Hence, it provides a measure on how fast two at times near trajectories of cyclic movements depart. The literature yet shows a lack of knowledge about the consequences of applying this system theory to sports science tasks. The experimental part shows strong evidence that, in the evaluation of movements, the exponent is less about a complex determinism than simply the level of dynamic noise present in time series. The higher the level of noise, the lower the stability of the system. Applying noise reduction therefore leads to reduced effect sizes. This has consequences: the values of average mutual information, which are until now only used for calculating the delay for the embedding, can already show differences in stability. Furthermore, it could be shown that the estimation of the embedding dimension d (independently of algorithm), is dependent on the length of the data series and values of d are currently overestimated. The greatest effect sizes were observed in dimension three and it can be recommended to use the very first beginning of the divergence curve for the linear fit. These findings pioneer a more efficient and standardized approach of stability analysis and can improve the ability of showing differences between conditions or groups.

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Thomas, Neil B. "The analysis and control of nonlinear systems using Lyapunov stability theory." Thesis, This resource online, 1996. http://scholar.lib.vt.edu/theses/available/etd-08292008-063459/.

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McDonald, Dale Brian. "Feedback control algorithms through Lyapunov optimizing control and trajectory following optimization." Online access for everyone, 2006. http://www.dissertations.wsu.edu/Dissertations/Spring2006/D%5FMcDonald%5F050206.pdf.

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Nguyen, Bao. "Contribution to nonsmooth lyapunov stability of differential inclusions with maximal monotone operators." Tesis, Universidad de Chile, 2017. http://repositorio.uchile.cl/handle/2250/149077.

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Doctor en Ciencias de la Ingeniería, Mención Modelación Matemática en cotutela con la Universidad de Limoges
In this PhD thesis, we make some contributions to nonsmooth Lyapunov stability of first-order differential inclusions with maximal monotone operators, in the setting of infinite-dimensional Hilbert spaces. We provide primal and dual explicit characterizations for parameterized weak and strong Lyapunov pairs of lower semicontinuous extended-real-valued functions, referred to as $a-$Lyapunov pairs, associated to differential inclusions with right-hand-sides governed by Lipschitz or Cusco perturbations $F$ of maximal monotone operators $A$, ẋ(t) ∈ F (x(t)) − A(x(t)), t ≥ 0, x(0) ∈ dom A. Equivalently, we study the weak and strong invariance of sets with respect to such differential inclusions. As in the classical Lyapunov approach to the stability of differential equations, the presented results make use of only the data of the differential system; that is, the operator $A$ and the multifunction $F$, and so no need to know about the solutions, nor the semi-groups generated by the monotone operators. Because our Lyapunov pairs and invariant sets candidates are just lower semicontinuous and closed, respectively, we make use of nonsmooth analysis to provide first-order-like criteria using general subdifferentials and normal cones. We provide similar analysis to non-convex differential inclusions governed by proximal normal cones to prox-regular sets. Our analysis above allowed to prove that such apparently more general systems can be easily coined into our convex setting. We also use our results to study the geometry of maximal monotone operators, and specifically, the characterization of the boundary of the values of such operators by means only of the values at nearby points, which are distinct of the reference point. This result has its application in the stability of semi-infinite programming problems. We also use our results on Lyapunov pairs and invariant sets to provide a systematic study of Luenberger-like observers design for differential inclusions with normal cones to prox-regular sets. The thesis is organized as follows: In chapter 1, we explain the main objectives of the thesis, the methodology that we follow, and we give a preview of the main results. We also make in this chapter a general overview of Lyapunov's theory, and present the main previous achievements on the subject. In Chapter 2, we present the main tools and preliminary results that we need in our analysis. In Chapter 3, we give the desired characterizations of Lyapunov pairs and invariant sets for differential inclusions with Lipschitz perturbations of maximal monotone operators, while in Chapter 4, we investigate differential inclusions with Lipschitz perturbations of proximal normal cones. This chapter includes the application to Luenberger-like observers design. In Chapter 5, we study differential inclusions with Lipschitz Cusco perturbations of maximal monotone operators. In Chapter 6, we give a result on the geometry of maximal monotone operators, and describe the boundary of their values. Finally, we give in Chapter 7 a resume of the results we obtained.
En esta tesis doctoral se realiza una contribución a la estabilidad de Lyapunov no suave de inclusiones diferenciales de primer orden con operadores maximales monótonos, en el con- texto de espacios de Hilbert de dimensión infinita. Se entregan caracterizaciones primales y duales explícitas para los pares de Lyapunov parametrizados débiles y fuertes de funciones inferiormente semicontinuas con valores extendedidos, referidas como pares a-Lyapunov, aso- ciados a inclusiones diferenciales con un lado derecho gobernado por perturbaciones F de tipo Lipschitz o Cusco de operadores maximales monótonos A, ẋ(t) ∈ F (x(t)) − A(x(t)), t ≥ 0, x(0) ∈ dom A. De manera equivalente, se estudian la invarianza débil y fuerte de conjuntos con respecto a tales inclusiones diferenciales. Tal como en el enfoque clásico de Lyapunov para estudiar la la estabilidad de ecuaciones diferenciales, los resultados presentados usan solamente la información del sistema; es decir, el operador A y la multiaplicación F , y, por lo tanto, no es necesario conocer las soluciones ni el semigrupo generado por el operador monótono. Dado que los pares de Lyapunov y conjuntos invariantes considerados aquí son, respectivamente, inferiormente semicontinuos y cerrados, se utiliza el análisis no-suave para proveer criterios de primer order utilizando subdiferenciales y conos lo suficientemente generales. Se realiza un análisis similar al caso de las inclusiones diferenciales no convexas gobernadas por conos normales proximales a conjuntos prox-regulares. Nuestro análisis permite demostrar que tales sistemas, aparentemente más generales, pueden ser fácilmente acuñados en nuestro con- texto. Además, nuestros resultados son utilizados para estudiar la geometría de operadores maximales monótonos, y específicamente, la caracterización de la frontera de los valores de tales operadores mediante sólo los puntos cercanos, diferentes del punto de referencia. Este resultado tiene aplicaciones en la estabilidad de problemas de programación semi-infinita. Además, nuestros resultados se utilizan en los pares de Lyapunov de conjuntos invariantes para realizar un estudio sistemático del diseño de observadores de tipo Luenberger para in- clusiones diferenciales con conos normales a conjuntos prox-regulares. La tesis está organizada de la siguiente manera: en el Capítulo 1, se explican los principales objetivos de la tesis, la metodología seguida, y se entrega una vista previa de los principales resultados. Además, en este capítulo, se da una visión general de la teoría de Lyapunov, y se presentan los resultados previos en el tema. En el Capítulo 2, se presentan las principales herramientas y los resultados preliminares necesarios en nuestro análisis. En el Capítulo 3, se entregan las caracterizaciones deseadas de los pares de Lyapunov y conjuntos invariantes para inclusiones diferenciales con perturbaciones Lipschitz de operadores maximales monótonos, mientras que en el Capítulo 4, se investigan las inclusiones diferenciales con perturbaciones Lipschitz de conos normales proximales. Este capítulo incluye una aplicación al disenño de observadores de tipo Luenberger. En el Capítulo 5, se estudian inclusiones diferenciales con perturbaciones Lipschitz Cusco de operadores maximales monótonos. En el Capítulo 6, se entrega un resultado sobre la geometría de los operadores maximales monótonos, y se describe la frontera de sus valores. Finalmente, en el Capítulo 7 se da un resumen de los resultados obtenidos.

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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 temps
In 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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Prado, Eder Flávio [UNESP]. "Existência da função de Lyapunov." Universidade Estadual Paulista (UNESP), 2010. http://hdl.handle.net/11449/94260.

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Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-02-19Bitstream added on 2014-06-13T18:47:53Z : No. of bitstreams: 1 prado_ef_me_sjrp.pdf: 346611 bytes, checksum: 28c34647c269c1cbaea17d3787faa4cf (MD5)
Neste trabalho vamos estudar equações diferenciais ordinárias e analisar seu comportamento ao longo de suas trajetórias, com o principal objetivo de encontar, caso possível, uma função de Lyapunov apropriada para o sistema, isto é, dar condição suficiente e necessária para a existência dessa função.
In this work we study ordinary differential equations and analyse the behavior along of trajectories. The main goal is to find Lyapunov functions for the system when possibel: i e, we want to find necessary and sufficient conditions for the existence of those.

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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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15

Story, William Robert. "Application of Lyapunov Exponents to Strange Attractors and Intact & Damaged Ship Stability." Thesis, Virginia Tech, 2009. http://hdl.handle.net/10919/33047.

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The threat of capsize in unpredictable seas has been a risk to vessels, sailors, and cargo since the beginning of a seafaring culture. The event is a nonlinear, chaotic phenomenon that is highly sensitive to initial conditions and difficult to repeatedly predict. In extreme sea states most ships depend on an operating envelope, relying on the operatorâ s detailed knowledge of headings and maneuvers to reduce the risk of capsize. While in some cases this mitigates this risk, the nonlinear nature of the event precludes any certainty of dynamic vessel stability. This research presents the use of Lyapunov exponents, a quantity that measures the rate of trajectory separation in phase space, to predict capsize events for both intact and damaged stability cases. The algorithm searches backwards in ship motion time histories to gather neighboring points for each instant in time, and then calculates the exponent to measure the stretching of nearby orbits. By measuring the periods between exponent maxima, the lead-time between period spike and extreme motion event can be calculated. The neighbor-searching algorithm is also used to predict these events, and in many cases proves to be the superior method for prediction. In addition to the ship stability research, the Lyapunov exponents are used in conjunction with bifurcation analysis to determine regions of stable behavior in strange attractors when the system parameters are varied. The boundaries of stability are important for algorithm validation, where these transitions between stable and unstable behavior must be accounted for.
Master of Science

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Cherifi, Abdelmadjid. "Contribution à la commande des modèles Takagi-Sugeno : approche non-quadratique et synthèse D -stable." Thesis, Reims, 2017. http://www.theses.fr/2017REIMS016/document.

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Ce travail de thèse traite de l’analyse de la stabilité et la stabilisation des systèmes non-linéaires représentés par des modèles T-S. L’objectif est de réduire le conservatisme des conditions de stabilité, obtenue par la méthode directe de Lyapunov, et écrites, dans la mesure du possible, sous forme de LMIs. Dans ce cadre, deux contributions principales ont été apportées. Tout d’abord, nous avons proposé de nouvelles conditions de synthèse non-quadratique de lois de commande, strictement LMIs et sans restriction d’ordre, pour les modèles T-S via des FLICs. En effet, dans ce contexte, les résultats de la littérature ne sont valables que pour les modèles T-S d’ordre inférieur ou égal à 2. Afin de lever cette restriction, les conditions ont été obtenues grâce à la démonstration d’une propriété de dualité. Ensuite, peu de travaux traitant de la spécification des performances en boucle fermée, de nouvelles conditions LMIs (quadratiques et non-quadratiques) ont été proposées via le concept de D-stabilité. Dans un premier temps, la synthèse de lois de commande PDC et non-PDC D-stabilisantes a été proposée pour les modèles T-S nominaux. Ensuite, ces résultats ont été étendus au cas des modèles T-S incertains. De plus, nous avons mis en évidence, au travers d’un exemple de D-stabilisation en attitude d’un modèle de drone quadrirotor, que les modèles T-S incertains pouvaient être avantageusement considérés lorsque les non-linéarités d’un modèle non-linéaire dépendent à la fois de l’état et de l’entrée
This work deals with the stability analysis and the stabilisation of nonlinear systems represented by T-S models.The goal is to reduce the conservatism of the stability conditions, obtained through the direct Lyapunov methodand written, when it is possible, as LMIs. In this framework, two main contributions has been proposed. First ofall, we have proposed some new conditions based on FLICs, strictly LMIs and without any order restrictions, forthe non-quadratic design of control laws devoted to stabilize T-S models. Indeed, in this non-quadratic context,the existing works are only available for 2nd order T-S models. In order to unlock this restriction, the proposed conditions have been obtained based on the proof of a dual property. Then, starting from the fact that few worksdeals with the closed-loop performances specification, some new LMI conditions (quadratic and non-quadratic)have been proposed via the D-stability concept. As a first step, D-stabilizing PDC and non-PDC controller designhas been considered for nominal T-S models. Then, these results have been extended to uncertain T-S models.Moreover, it has been highlighted, from an example of the attitude D-stabilization of a quadrotor model, that wecan make use of uncertain T-S models to cope with nonlinear models involving nonlinearities depending on bothstate and input variables

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Prado, Eder Flávio. "Existência da função de Lyapunov /." São José do Rio Preto : [s.n.], 2010. http://hdl.handle.net/11449/94260.

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Orientador: Vanderlei Minori Horita
Banca: Isabel Lugão Rios
Banca: Claudio Aguinaldo Buzzi
Resumo: Neste trabalho vamos estudar equações diferenciais ordinárias e analisar seu comportamento ao longo de suas trajetórias, com o principal objetivo de encontar, caso possível, uma função de Lyapunov apropriada para o sistema, isto é, dar condição suficiente e necessária para a existência dessa função.
Abstract: In this work we study ordinary differential equations and analyse the behavior along of trajectories. The main goal is to find Lyapunov functions for the system when possibel: i e, we want to find necessary and sufficient conditions for the existence of those.
Mestre

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Stovall, Kazumi Niki. "Semidefinite Programming and Stability of Dynamical System." Digital Archive @ GSU, 2006. http://digitalarchive.gsu.edu/math_theses/4.

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In the first part of the thesis we present several interior point algorithms for solving certain positive definite programming problems. One of the algorithms is adapted for finding out whether there exists or not a positive definite matrix which is a real linear combination of some given symmetric matrices A1,A2, . . . ,Am. In the second part of the thesis we discuss stability of nonlinear dynamical systems. We search using algorithms described in the first part, for Lyapunov functions of a few forms. A suitable Lyapunov function implies the existence of a hyperellipsoidal attraction region for the dynamical system, thus guaranteeing stability.

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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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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 controllerâ€™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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嚴利興 and Li-hing Yim. "Some stability results for time-delay control problems." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2000. http://hub.hku.hk/bib/B31225482.

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Yim, Li-hing. "Some stability results for time-delay control problems." Hong Kong : University of Hong Kong, 2000. http://sunzi.lib.hku.hk/hkuto/record.jsp?B22926094.

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Vance, Katelynn Atkins. "Evaluation of Stability Boundaries in Power Systems." Diss., Virginia Tech, 2017. http://hdl.handle.net/10919/78322.

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Power systems are extremely non-linear systems which require substantial modeling and control efforts to run continuously. The movement of the power system in parameter and state space is often not well understood, thus making it difficult or impossible to determine whether the system is nearing instability. This dissertation demonstrates several ways in which the power system stability boundary can be calculated. The power system movements evaluated here address the effects of inter-area oscillations on the system which occur in the seconds to minutes time period. The first uses gain scheduling techniques through creation of a set of linear parameter varying (LPV) systems for many operating points of the non-linear system. In the case presented, load and line reactance are used as parameters. The scheduling variables are the power flows in tie lines of the system due to the useful information they provide about the power system state in addition to being available for measurement. A linear controller is developed for the LPV model using H₂/H∞ with pole placement objectives. When the control is applied to the non-linear system, the proposed algorithm predicts the response of the non-linear system to the control by determining if the current system state is located within the domain of attraction of the equilibrium. If the stability domain contains a convex combination of the two points, the control will aid the system in moving towards the equilibrium. The second contribution of this thesis is through the development and implementation of a pseudo non-linear evaluation of a power system as it moves through state space. A system linearization occurs first to compute a multi-objective state space controller. For each contingency definition, many variations of the power system example are created and assigned to the particular contingency class. The powerflow variations and contingency controls are combined to run sets of time series analysis in which the Lyapunov function is tracked over three time steps. This data is utilized for a classification analysis which identifies and classifies the data by the contingency type. The goal is that whenever a new event occurs on the system, real time data can be fed into the trained tree to provide a control for application to increase system damping.
Ph. D.

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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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Schönlein, Michael [Verfasser], and Fabian [Akademischer Betreuer] Wirth. "Stability and Robustness of Fluid Networks: A Lyapunov Perspective / Michael Schönlein. Betreuer: Fabian Wirth." Würzburg : Universitätsbibliothek der Universität Würzburg, 2012. http://d-nb.info/1024658392/34.

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29

Sun, Yuming. "Energy efficient stability control of a biped based on the concept of Lyapunov exponents." Springer, 2011. http://hdl.handle.net/1993/23264.

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Balance control is important for biped standing. Due to the time-varying control bounds induced by the foot constraints, and the lack of tools for analyzing stability of highly nonlinear systems, it is extremely difficult to design balance control strategies for a standing biped with a rigorous stability analysis in spite of large efforts. In this thesis, three important issues are fully considered for a standing biped: maintaining the postural stability, minimizing the energy consumption and satisfying the constraints between the biped feet and the ground. Both the theoretical and the experimental studies on the constrained and energy-efficient control are carried out systematically using the genetic algorithm (GA). The stability for the proposed balancing system is thoroughly investigated using the concept of Lyapunov exponents. On the other hand, the controlled standing biped is characterized by high nonlinearity and great complexity. For systems with such features, in general the Lyapunov exponents are hard to be estimated using the model-based method. Meanwhile the biped is supposed to be stabilized at the upright posture, indicating that the system should possess negative Lyapunov exponents only. However the accuracy of negative exponents is usually poor if following the traditional time-series-based methods. As it is nontrivial to examine the system stability for bipedal robots, the numerical accuracy of the estimated Lyapunov exponents is extremely demanding. In this research, two novel approaches are proposed based upon system approximation using different types of Radial-Basis-Function (RBF) networks. Both the proposed methods can estimate the exponents reliably with straightforward algorithms, yet no mathematical model is required in any newly developed method. The efficacies of both methods are demonstrated through a linear quadratic regulator (LQR) balancing system for a standing biped, as well as several other dynamical systems. The thesis as a whole, has set up a framework for developing more sophisticated controllers in more complex movement for robot models with less conservative assumptions. The systematic stability analysis shown in this thesis has a great potential for many other engineering systems.

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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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Everding, Vanessa Quigley. "Stability Analysis of Human Walking." Case Western Reserve University School of Graduate Studies / OhioLINK, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=case1232680311.

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Shifman, Jeffrey Joseph. "The control of flexible robots." Thesis, University of Cambridge, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.385838.

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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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34

Khorrami, Farshad. "Asymptotic perturbation and Lyapunov stability based approaches for control of flexible and rigid robot manipulators /." The Ohio State University, 1988. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487592050230916.

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35

McConley, Marc Wayne. "A computationally efficient Lyapunov-based procedure for control of nonlinear systems with stability and performance guarantees." Thesis, Massachusetts Institute of Technology, 1997. http://hdl.handle.net/1721.1/10755.

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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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37

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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Santos, Iguer Luis Domini dos [UNESP]. "Análise de estabilidade de sistemas dinâmicos descontínuos e aplicações." Universidade Estadual Paulista (UNESP), 2008. http://hdl.handle.net/11449/94290.

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Made available in DSpace on 2014-06-11T19:27:07Z (GMT). No. of bitstreams: 0 Previous issue date: 2008-02-26Bitstream added on 2014-06-13T19:06:47Z : No. of bitstreams: 1 santos_ild_me_sjrp.pdf: 434711 bytes, checksum: 230caec3d969a14efac9b1700fd1dd97 (MD5)
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
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.
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.

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Djaneye-Boundjou, Ouboti Seydou Eyanaa. "Particle Swarm Optimization Stability Analysis." University of Dayton / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=dayton1386413941.

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40

Kojima, Chiaki. "Studies on Lyapunov stability and algebraic Riccati equation for linear discrete-time systems based on behavioral approach." 京都大学 (Kyoto University), 2007. http://hdl.handle.net/2433/135968.

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41

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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42

Bocquillon, Benjamin. "Méthodes d'entraînement pour l'analyse de la stabilité d'un système complexe." Electronic Thesis or Diss., université Paris-Saclay, 2024. http://www.theses.fr/2024UPASG014.

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Ces dernières années, le domaine de la recherche en automatique a été transformé par l’émergence de technologies avancées telles que l’Intelligence Artificielle. Cette thèse se concentre sur l’utilisation de techniques d’entraînement et d’optimisation pour démontrer la stabilité des contrôleurs intelligents, un élément clé dans le processus de certification. Elle débute par une exploration des principes de la stabilité de Lyapunov, une théorie introduite à la fin du XIXe siècle, qui permet d’étudier la stabilité d’un système dynamique sans hypothèse structurelle. Elle passe en revue certaines méthodes théoriques, numériques et automatiques pour identifier une fonction de Lyapunov, soulignant les limites de certaines approches existantes. Cette recherche aboutit au développement de méthodes d’optimisation innovantes pour déterminer des fonctions de Lyapunov, qui se distinguent par leur adaptabilité et leur efficacité à maximiser le domaine de stabilité estimé. L’algorithme proposé est testé rigoureusement, démontrant son adaptabilité à divers scénarios industriels tout en le comparant à certaines méthodes reconnues de l’état de l’art. En conclusion, cette thèse établit les bases pour l’application pratique d’un algorithme dans des systèmes industriels complexes, ouvrant la voie à des applications futures plus larges, détaillées en fin de manuscrit
In recent years, the world of scientific research in automatic has experienced a technological breakthrough with the emergence of new technologies such as Artificial Intelligence. In the context of intelligent controllers, the justification of stability is fundamental as it constitutes a cornerstone in the overall certification process. In this thesis, we propose the use of training and optimization techniques to demonstrate the stability of intelligent control loops.To begin our study, we explore the theoretical notions of stability in the sense of Lyapunov. This approach is important to establish a solid understanding of the principles of stability and their application in dynamic systems. Our research reveals that some notions of stability can be quite restrictive. Therefore, we choose to use Lyapunov's theory. Introduced at the end of the nineteenth century, this theory represents a practical and effective way of studying the stability of an equilibrium point for a dynamic system, without requiring structural assumptions about the system itself.This thesis then includes an in-depth review of theoretical, numerical, and automatic methods for determining a Lyapunov function, highlighting the effectiveness and limitations of different approaches existing in the literature.Following this study, we develop and implement several innovative optimization methods to identify Lyapunov functions in various contexts. Each approach is distinguished not only by its generic and adaptable nature but also by its ability to maximize the estimated guaranteed stability domain of the Lyapunov function, a fundamental aspect in ensuring the stability of dynamic systems.Our algorithm, designed to be flexible and applicable in various industrial contexts, undergoes several tests. These tests involve not only the application of the algorithm to different systems in a variety of scenarios but also rigorous performance tests to evaluate its position relative to the current state of the art. These tests demonstrate the effectiveness of our approach and its ability to adapt efficiently to various system configurations, offering valuable insights for its application in real industrial environments.Finally, we set the basis for the use of this algorithm in a specific industrial context. This step represents a significant move towards the practical application of the algorithm and its integration into complex industrial systems. This advancement sets the stage for wider future applications, listed at the end of this manuscript

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Arhinful, Daniel Andoh. "Lorenzův systém: cesta od stability k chaosu." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2020. http://www.nusl.cz/ntk/nusl-417087.

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The theory of deterministic chaos has generated a lot of interest and continues to be one of the much-focused research areas in the field of dynamics today. This is due to its prevalence in essential parts of human lives such as electrical circuits, chemical reactions, the flow of blood through the human system, the weather, etc. This thesis presents a study of the Lorenz equations, a famous example of chaotic systems. In particular, it presents the analysis of the Lorenz equations from stability to chaos and various bifurcation scenarios with numerical and graphical interpretations. It studies concepts of non-linear dynamical systems such as equilibrium points, stability, linearization, bifurcation, Lyapunov function, etc. Finally, it discusses how the Lorenz equations serve as a model for the waterwheel (in detail), and the convection roll for fluid.

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Dierks, Travis. "Nonlinear control of nonholonomic mobile robot formations." Diss., Rolla, Mo. : University of Missouri-Rolla, 2007. http://scholarsmine.umr.edu/thesis/pdf/Dierks_09007dcc803c490d.pdf.

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Thesis (M.S.)--University of Missouri--Rolla, 2007.
Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed November 28, 2007) Includes bibliographical references.

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Carlu, Mallory. "Instability in high-dimensional chaotic systems." Thesis, University of Aberdeen, 2019. http://digitool.abdn.ac.uk:80/webclient/DeliveryManager?pid=240675.

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In this thesis I make extensive use of the Lyapunov analysis formalism to unravel fundamental mechanisms of instability in two diﬀerent systems : the Kuramoto model of globally coupled phase-oscillators and the Lorenz 96 (L96) atmospheric "toy" model, portraying the evolution of a physical quantity along a latitude circle. I start by introducing the relevant theoretical background, with special attention on the main tools I have been using throughout this work : Lyapunov Exponents (LEs), which quantify the asymptotic growth rates of inﬁnitesimal perturbations in a system, and by extension, its degree of chaoticity, and Covariant Lyapunov Vectors (CLVs), which indicate the phase space direction (or the geometry) associated with these growth rates. The Kuramoto model is central in the study of synchronization among oscillatory units characterized by their various natural frequencies, but little is known on its chaotic dynamics in the unsynchronized state. I thus investigate the scaling behavior of the ﬁrst LE, upon diﬀerent assumptions on the natural frequencies, and make use of educated structural simpliﬁcations to analyze the origin of chaos in the ﬁnite size model. On the other hand, the L96 model has been devised to gather the main dynamical ingredients of atmospheric dynamics, namely advection, damping, external (solar) forcing and transfers across diﬀerent scales of motion, in a minimalist and functional way. It features two coupled dynamical layers : the large scale variables, representing synoptic scale atmospheric dynamics, and the small scale variables, faster and more numerous, associated with convective scale dynamics. The core of the study revolves around geometrical properties of CLVs, in the aim of understanding the processes underlying the observed multiscale chaoticity, and an exhaustive study of a non-trivial ensemble of CLVs featuring relevant projection on the slow subspace.

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MEARS, MARK JOHN. "A STABLE NEURAL CONTROL APPROACH FOR UNCERTAIN NONLINEAR SYSTEMS." University of Cincinnati / OhioLINK, 2003. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1060371930.

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Liang, Weichao. "Feedback exponential stabilization of open quantum systems undergoing continuous-time measurements." Thesis, Université Paris-Saclay (ComUE), 2019. http://www.theses.fr/2019SACLS391.

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Dans cette thèse, nous nous intéressons à la stabilisation par rétroaction des systèmes quantiques ouverts soumis à des mesures imparfaites en temps continu. Tout d'abord, nous introduisons la théorie du filtrage quantique pour décrire l'évolution temporelle de l'opérateur de densité conditionnelle représentant un état quantique en interaction avec un environnement. Ceci est décrit par une équation différentielle stochastique à valeurs matricielles. Deuxièmement, nous étudions le comportement asymptotique des trajectoires quantiques associées à des systèmes de spin à N niveaux pour des états initiaux donnés, pour les cas avec et sans loi de rétroaction. Dans le cas sans loi de rétroaction, nous montrons la propriété de réduction de l'état quantique à vitesse exponentielle. Ensuite, nous fournissons des conditions suffisantes sur la loi de contrôle assurant une convergence presque sûre vers un état pur prédéterminé correspondant à un vecteur propre de l'opérateur de mesure. Troisièmement, nous étudions le comportement asymptotique des trajectoires de systèmes ouverts à plusieurs qubits pour des états initiaux donnés. Dans le cas sans loi de rétroaction, nous montrons la réduction exponentielle de l'état quantique pour les systèmes N-qubit avec deux canaux quantiques. Dans le cas particulier des systèmes à deux qubits, nous donnons des conditions suffisantes sur la loi de contrôle assurant la convergence asymptotique vers un état cible de Bell avec un canal quantique, et la convergence exponentielle presque sûre vers un état cible de Bell avec deux canaux quantiques. Ensuite, nous étudions le comportement asymptotique des trajectoires des systèmes quantiques ouverts de spin-1/2 avec les états initiaux inconnus soumis à des mesures imparfaites en temps continu, et nous fournissons des conditions suffisantes au contrôleur pour garantir la convergence de l'état estimé vers l'état quantique réel lorsque le temps tend vers l'infini. En conclusion, nous discutons de manière heuristique du problème de stabilisation exponentielle des systèmes de spin à N niveaux avec les états initiaux inconnus et nous proposons des lois de rétroaction candidates afin de stabiliser le système de manière exponentielle
In this thesis, we focus on the feedback stabilization of open quantum systems undergoing imperfect continuous-time measurements. First, we introduce the quantum filtering theory to obtain the time evolution of the conditional density operator representing a quantum state in interaction with an environment. This is described by a matrix-valued stochastic differential equation. Second, we study the asymptotic behavior of quantum trajectories associated with N-level quantum spin systems for given initial states, for the cases with and without feedback law. For the case without feedback, we show the exponential quantum state reduction. Then, we provide sufficient conditions on the feedback control law ensuring almost sure exponential convergence to a predetermined pure state corresponding to an eigenvector of the measurement operator. Third, we study the asymptotic behavior of trajectories of open multi-qubit systems for given initial states. For the case without feedback, we show the exponential quantum state reduction for N-qubit systems with two quantum channels. Then, we focus on the two-qubit systems, and provide sufficient conditions on the feedback control law ensuring asymptotic convergence to a target Bell state with one quantum channel, and almost sure exponential convergence to a target Bell state with two quantum channels. Next, we investigate the asymptotic behavior of trajectories of open quantum spin-1/2 systems with unknown initial states undergoing imperfect continuous-time measurements, and provide sufficient conditions on the controller to guarantee the convergence of the estimated state towards the actual quantum state when time goes to infinity. Finally, we discuss heuristically the exponential stabilization problem for N-level quantum spin systems with unknown initial states and propose candidate feedback laws to stabilize exponentially the system

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Kangas, M. (Maria). "Stability analysis of new paradigms in wireless networks." Doctoral thesis, Oulun yliopisto, 2017. http://urn.fi/urn:isbn:9789526215464.

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Abstract Fading in wireless channels, the limited battery energy available in wireless handsets, the changing user demands and the increasing demand for high data rate and low delay pose serious design challenges in the future generations of mobile communication systems. It is necessary to develop efficient transmission policies that adapt to changes in network conditions and achieve the target delay and rate with minimum power consumption. In this thesis, a number of new paradigms in wireless networks are presented. Dynamic programming tools are used to provide dynamic network stabilizing resource allocation solutions for virtualized data centers with clouds, cooperative networks and heterogeneous networks. Exact dynamic programming is used to develop optimal resource allocation and topology control policies for these networks with queues and time varying channels. In addition, approximate dynamic programming is also considered to provide new sub-optimal solutions. Unified system models and unified control problems are also provided for both secondary service provider and primary service provider cognitive networks and for conventional wireless networks. The results show that by adapting to the changes in queue lengths and channel states, the dynamic policy mitigates the effects of primary service provider and secondary service provider cognitive networks on each other. We investigate the network stability and provide new unified stability regions for primary service provider and secondary service provider cognitive networks as well as for conventional wireless networks. The K-step Lyapunov drift is used to analyse the performance and stability of the proposed dynamic control policies, and new unified stability analysis and queuing bound are provided for both primary service provider and secondary service provider cognitive networks and for conventional wireless networks. By adapting to the changes in network conditions, the dynamic control policies are shown to stabilize the network and to minimize the bound for the average queue length. In addition, we prove that the previously proposed frame based does not minimize the bound for the average delay, when there are shared resources between the terminals with queues
Tiivistelmä Langattomien kanavien häipyminen, langattomien laitteiden akkujen rajallinen koko, käyttäjien käyttötarpeiden muutokset sekä lisääntyvän tiedonsiirron ja lyhyemmän viiveen vaatimukset luovat suuria haasteita tulevaisuuden langattomien verkkojen suunnitteluun. On välttämätöntä kehittää tehokkaita resurssien allokointialgoritmeja, jotka sopeutuvat verkkojen muutoksiin ja saavuttavat sekä tavoiteviiveen että tavoitedatanopeuden mahdollisimman pienellä tehon kulutuksella. Tässä väitöskirjassa esitetään uusia paradigmoja langattomille tietoliikenneverkoille. Dynaamisen ohjelmoinnin välineitä käytetään luomaan dynaamisia verkon stabiloivia resurssien allokointiratkaisuja virtuaalisille pilvipalveludatakeskuksille, käyttäjien yhteistyöverkoille ja heterogeenisille verkoille. Tarkkoja dynaamisen ohjelmoinnin välineitä käytetään kehittämään optimaalisia resurssien allokointi ja topologian kontrollointialgoritmeja näille jonojen ja häipyvien kanavien verkoille. Tämän lisäksi, estimoituja dynaamisen ohjelmoinnin välineitä käytetään luomaan uusia alioptimaalisia ratkaisuja. Yhtenäisiä systeemimalleja ja yhtenäisiä kontrollointiongelmia luodaan sekä toissijaisen ja ensisijaisen palvelun tuottajan kognitiivisille verkoille että tavallisille langattomille verkoille. Tulokset osoittavat että sopeutumalla jonojen pituuksien ja kanavien muutoksiin dynaaminen tekniikka vaimentaa ensisijaisen ja toissijaisen palvelun tuottajien kognitiivisten verkkojen vaikutusta toisiinsa. Tutkimme myös verkon stabiiliutta ja luomme uusia stabiilisuusalueita sekä ensisijaisen ja toissijaisen palveluntuottajan kognitiivisille verkoille että tavallisille langattomille verkoille. K:n askeleen Lyapunovin driftiä käytetään analysoimaan dynaamisen kontrollointitekniikan suorituskykyä ja stabiiliutta. Lisäksi uusi yhtenäinen stabiiliusanalyysi ja jonon yläraja luodaan ensisijaisen ja toissijaisen palveluntuottajan kognitiivisille verkoille ja tavallisille langattomille verkoille. Dynaamisen algoritmin näytetään stabiloivan verkko ja minimoivan keskimääräisen jonon pituuden yläraja sopeutumalla verkon olosuhteiden muutoksiin. Tämän lisäksi todistamme että aiemmin esitetty frame-algoritmi ei minimoi keskimääräisen viiveen ylärajaa, kun käyttäjät jakavat keskenään resursseja

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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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Tang, Ying. "Stability analysis and Tikhonov approximation for linear singularly perturbed hyperbolic systems." Thesis, Université Grenoble Alpes (ComUE), 2015. http://www.theses.fr/2015GREAT054/document.

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Les dynamiques des systèmes modélisés par des équations aux dérivées partielles (EDPs) en dimension infinie sont largement liées aux réseaux physiques. La synthèse de la commande et l'analyse de la stabilité de ces systèmes sont étudiées dans cette thèse. Les systèmes singulièrement perturbés, contenant des échelles de temps multiples sont naturels dans les systèmes physiques avec des petits paramètres parasitaires, généralement de petites constantes de temps, les masses, les inductances, les moments d'inertie. La théorie des perturbations singulières a été introduite pour le contrôle à la fin des années $1960$, son assimilation dans la théorie du contrôle s'est rapidement développée et est devenue un outil majeur pour l'analyse et la synthèse de la commande des systèmes. Les perturbations singulières sont une façon de négliger la transition rapide, en la considérant dans une échelle de temps rapide séparée. Ce travail de thèse se concentre sur les systèmes hyperboliques linéaires avec des échelles de temps multiples modélisées par un petit paramètre de perturbation. Tout d'abord, nous étudions une classe de systèmes hyperboliques linéaires singulièrement perturbés. Comme le système contient deux échelles de temps, en mettant le paramètre de la perturbation à zéro, deux sous-systèmes, le système réduit et la couche limite, sont formellement calculés. La stabilité du système complet de lois de conservation implique la stabilité des deux sous-systèmes. En revanche un contre-exemple est utilisé pour illustrer que la stabilité des deux sous-systèmes ne suffit pas à garantir la stabilité du système complet. Cela montre une grande différence avec ce qui est bien connu pour les systèmes linéaires en dimension finie modélisés par des équations aux dérivées ordinaires (EDO). De plus, sous certaines conditions, l'approximation de Tikhonov est obtenue pour tels systèmes par la méthode de Lyapunov. Plus précisément, la solution de la dynamique lente du système complet est approchée par la solution du système réduit lorsque le paramètre de la perturbation est suffisamment petit. Deuxièmement, le théorème de Tikhonov est établi pour les systèmes hyperboliques linéaires singulièrement perturbés de lois d'équilibre où les vitesses de transport et les termes sources sont à la fois dépendant du paramètre de la perturbation ainsi que les conditions aux bords. Sous des hypothèses sur la continuité de ces termes et sous la condition de la stabilité, l'estimation de l'erreur entre la dynamique lente du système complet et le système réduit est obtenue en fonction de l'ordre du paramètre de la perturbation. Troisièmement, nous considérons des systèmes EDO-EDP couplés singulièrement perturbés. La stabilité des deux sous-systèmes implique la stabilité du système complet où le paramètre de la perturbation est introduit dans la dynamique de l'EDP. D'autre part, cela n'est pas valable pour le système où le paramètre de la perturbation est présent dans l'EDO. Le théorème Tikhonov pour ces systèmes EDO-EDP couplés est prouvé par la technique de Lyapunov. Enfin, la synthèse de la commande aux bords est abordée en exploitant la méthode des perturbations singulières. Le système réduit converge en temps fini. La synthèse du contrôle aux bords est mise en œuvre pour deux applications différentes afin d'illustrer les résultats principaux de ce travail
Systems modeled by partial differential equations (PDEs) with infinite dimensional dynamics are relevant for a wide range of physical networks. The control and stability analysis of such systems become a challenge area. Singularly perturbed systems, containing multiple time scales, often occur naturally in physical systems due to the presence of small parasitic parameters, typically small time constants, masses, inductances, moments of inertia. Singular perturbation was introduced in control engineering in late $1960$s, its assimilation in control theory has rapidly developed and has become a tool for analysis and design of control systems. Singular perturbation is a way of neglecting the fast transition and considering them in a separate fast time scale. The present thesis is concerned with a class of linear hyperbolic systems with multiple time scales modeled by a small perturbation parameter. Firstly we study a class of singularly perturbed linear hyperbolic systems of conservation laws. Since the system contains two time scales, by setting the perturbation parameter to zero, the two subsystems, namely the reduced subsystem and the boundary-layer subsystem, are formally computed. The stability of the full system implies the stability of both subsystems. However a counterexample is used to illustrate that the stability of the two subsystems is not enough to guarantee the full system's stability. This shows a major difference with what is well known for linear finite dimensional systems. Moreover, under certain conditions, the Tikhonov approximation for such system is achieved by Lyapunov method. Precisely, the solution of the slow dynamics of the full system is approximated by the solution of the reduced subsystem for sufficiently small perturbation parameter. Secondly the Tikhonov theorem is established for singularly perturbed linear hyperbolic systems of balance laws where the transport velocities and source terms are both dependent on the perturbation parameter as well as the boundary conditions. Under the assumptions on the continuity for such terms and under the stability condition, the estimate of the error between the slow dynamics of the full system and the reduced subsystem is the order of the perturbation parameter. Thirdly, we consider singularly perturbed coupled ordinary differential equation ODE-PDE systems. The stability of both subsystems implies that of the full system where the perturbation parameter is introduced into the dynamics of the PDE system. On the other hand, this is not true for system where the perturbation parameter is presented to the ODE. The Tikhonov theorem for such coupled ODE-PDE systems is proved by Lyapunov technique. Finally, the boundary control synthesis is achieved based on singular perturbation method. The reduced subsystem is convergent in finite time. Boundary control design to different applications are used to illustrate the main results of this work

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