Дисертації з теми "Stabilité Input-To-State"

Cette thèse traite trois problèmes liés à la stabilisation des équations des ondes. Nous considérons différents cadres fonctionnels et nous utilisons des techniques de démonstration basées sur la méthode des multiplicateurs. Tout d'abord, nous étudions la stabilité de l'équation des ondes avec un amortissement non-linéaire et localisé dans un cadre hilbertien standard en dimension deux. La preuve est basée sur des travaux existants dans le cas d'un amortissement non-localisé. Rajoutons une localisation ainsi que des perturbations. Nous démontrons la stabilité exponentielle le long des solutions fortes en l'absence de perturbation ainsi qu'une sorte stabilité au sens Input-To-State par rapport aux perturbations considérées. Dans un deuxième travail, nous considérons un cadre fonctionnel plus général non forcément hilbertien, c-a-d un cadre L^p avec p appartient (1,infini). Nous étudions la stabilité L^p de l'équation des ondes avec un amortissement linéaire et localisé. Cette étude n'est effectuée qu'en dimension un car il n'est pas toujours possible de définir l'opérateur des ondes en dimensions supérieures lorsque p différent 2. Nous démontrons la stabilité exponentielle du problème en généralisant les multiplicateurs du cadre hilbertien dans notre cadre plus général, avec des preuves différentes suivant que 1 2. Nous démontrons également dans le même problème mais avec des cas particuliers d'un amortissement global et constant, une stabilité exponentielle dans le cas p=1 et p=infini. Dans un troisième travail, nous considérons la version non-linéaire du problème précédent: en nous basant sur une technique de linéarisation, nous nous ramenons à la preuve du problème linéaire pour démontrer la stabilité exponentielle du non-linéaire
This thesis focuses on three problems in the context of the stabilization of wave equations. We consider different frameworks and we use techniques based on the multipliers method. First, we study the stability of the wave equation with non-linear localized damping in a standard Hilbertian framework in two dimensions. The proof is based on the work already existing in the case of a non-localized damping. We add a localization as well as disturbances. We prove the exponential stability of strong solutions in the absence of disturbances and also a weak Input-To-State stability property with respect to the considered disturbances. We next consider a more general functional framework, namely an L^p framework with p in (1,infty). We study the L^p stability of the wave equation with a linear and localized damping in one dimension since it is not always possible to define the wave operator in higher dimensions when p = 2. We prove the exponential stability of the problem by generalizing the multipliers of the Hilbertian framework in this new general framework, with a different proof for 1 2. We also prove in the same problem but with particular cases of a global constant damping, an exponential stability in the case p=1 and p=infty. We consider next the nonlinear case of the previous problem: relying on a linearizing technique, we reduce that study to that of the linear problem case in order to prove the exponential stability of the non-linear problem

Dans cette thèse, on s'intéresse au contrôle du profil de facteur de sécurité dans un plasma tokamak. Cette variable physique est liée à plusieurs phénomènes dans le plasma, en particulier des instabilités magnétohydrodynamiques (MHD). Un profil de facteur de sécurité adéquat est particulièrement important pour avoir des modes d'opération avancés dans le tokamak, avec haut confinement et stabilité MHD. Pour cela faire, on se focalise sur la commande du gradient du profil de flux magnétique poloidal dans le tokamak. L'évolution de cette variable est donnée par une équation de diffusion avec des coefficients distribuées et temps-variants. En utilisant des techniques de type Lyapunov et les propriétés de stabilité entrée-état du système on propose une loi de commande robuste qui prend en compte des contraintes non-linéaires dans l'action imposées par la physique des actionneurs.

L’estimation d’état d’un système non linéaire est essentielle pour la réussite des objectifs importants tels que : la surveillance, l’identification et le contrôle. Les observateurs sont des algorithmes qui estiment l’état actuel en utilisant, entre autres informations, les mesures effectuées par des capteurs. Le problème de synthèse d’observateur pour les systèmes non linéaires est un sujet de recherche majeur traité depuis plusieurs décennies dans le domaine de la théorie du contrôle. Récemment, les recherches ont également porté sur la synthèse des observateurs pour des modèles de plus en plus réels, qui peuvent prendre en compte des perturbations, des capteurs non linéaires et des sorties discrètes. Dans ce contexte, le but de cette thèse concerne la synthèse d’observateurs robustes pour certaines classes de systèmes non linéaires. Dans ce manuscrit, nous distinguons trois parties principales. La première partie porte sur l’analyse des systèmes affines en état, affectés par le bruit, et l’estimation de l’état via le filtre de Kalman à grand gain. Les propriétés de convergence de cet observateur sont fortement influencées par deux variables : le paramètre de réglage du gain de l’observateur et l’entrée du système. Nous présentons un nouvel algorithme d’optimisation, basé sur une analyse de Lyapunov, qui adapte ces deux variables en fonction des perturbations affectant la dynamique et la sortie du système. La nouveauté de cette approche est qu’elle fournit une méthode systématique de réglage du gain et de sélection d’entrée simultanément ce qui améliore l’estimation de l’état en présence de telles perturbations et évite l’utilisation des méthodes basées de type essais-erreurs. La deuxième partie concerne le problème de “redesign” d’observateurs pour des systèmes non linéaires sous une forme générale dont les sorties sont transformées par des fonctions non linéaires. En effet, l’observateur risque alors de ne pas estimer correctement l’état du système si elle ne prend pas en compte les non-linéarités des capteurs. Nous présentons une refonte d’observateur qui consiste en l’interconnexion de l’observateur originel avec un estimateur de sortie basé sur une inversion dynamique, et nous démontrons sa convergence asymptotique via des résultats du petit-gain. Nous illustrons notre méthode en utilisant deux classes de systèmes non linéaires courant dans la littérature : les systèmes affines en l’état avec injection de sortie, et les systèmes avec non-linéarité sous la forme canonique. Enfin, la troisième partie étend notre approche présentée pour les systèmes continus aux systèmes dont les sorties sont non seulement transformées mais également discrétisées dans le temps. Cette propriété ajoutée introduit un défi important ; nous implémentons les techniques de “sample-and-hold” qui mènent à un gain de l’observateur basé sur des inégalités matricielles linéaires. La principale caractéristique des méthodes proposées est la possibilité d’adapter un grand nombre d’observateurs de la littérature à des scénarii plus réalistes. En effet, les capteurs classiques utilisés dans les applications d’ingénierie sont souvent non linéaires ou discrets, alors qu’une hypothèse récurrente dans la conception d’observateurs est la linéarité ou la continuité de la sortie
Estimating the state of a nonlinear system is an essential task for achieving important objectives such as: process monitoring, identification and control. Observers are algorithms that estimate the current state by using, among other information, sensor measurements. The problem of observer design for nonlinear systems has been a major research topic in control for many decades. Recently, there has been an increasing interest in the design of observers for more realistic models, which can include disturbances, sensor nonlinearities and discrete outputs. This thesis concerns the design of robust observers for selected classes of nonlinear systems and we can distinguish three main parts. The first part studies state-affine systems affected by noise, and analyses the state estimation via the so-called high-gain Kalman filter. The convergence properties of this observer are strongly influenced by two variables: its tuning parameter and the properly excited system input. We present a new optimization algorithm, based on Lyapunov analyses, that adapts these variables in order to minimize the effect of both dynamic and output disturbances. The novelty of this approach is that it provides a systematic method of simultaneous tuning and input selection with the goal of improving state estimation in the face of disturbances, and that it avoids the use of trial-and-error based methods. The second part studies the problem of observer redesign for general nonlinear systems whose outputs are transformed by nonlinear functions. Indeed, a given observer might not estimate the system state properly if it does not take into account sensor nonlinearities and, therefore, such an output mismatch needs to be addressed. We present an observer redesign that consists in the interconnection of the original observer with an output estimator based on a dynamic inversion, and we show its asymptotic convergence via small-gain arguments. We illustrate our method with two important classes of systems: state-affine systems up to output injection and systems with additive triangular nonlinearity. Finally, the third part extends our redesign method to systems whose outputs are not only transformed but also discretized in time. This added assumption introduces important challenges; we now implement sample-and-hold techniques leading to an observer gain based on linear matrix inequalities. The main feature of our redesign methods is the possibility to adapt a large number of observers from the literature to more realistic scenarios. Indeed, classical sensors in engineering applications are often nonlinear or discrete, whereas a recurrent assumption in observer design is the linearity or continuity of the output

As central topics in systems and control theory, the study of stability, robustness, and sensitivity to disturbances for nonlinear systems has gained with the Input-to-State Stability (ISS) paradigm one of its most powerful tools. Standard formulations of ISS and several related notions characterize stability properties in a global setting with respect to a single equilibrium in the origin and for systems defined in Euclidean space. This classical requirements make the basic theory not applicable for a global analysis of many dynamical behaviours of interest for applications, such as: multistability, periodic oscillations, and different class of attractors. The aim of this work is to extend and further characterize the variety of notions available in standard ISS literature to multistable systems evolving on Riemannian manifolds. The definition of global multistability in consideration is based on the existence of a finite number of compact, globally attractive, invariant sets satisfying a specific condition of acyclicity. Such definition allows for Lyapunov characterizations in hybrid systems and is used to generalize ISS, integral ISS, and Output-to-State Stability (OSS) to multistable systems on manifolds. In different modalities, the novel properties of ISS and integral ISS - respectively denoted as Input-to-State Multistability (ISM) and integral ISM - are shown to be preserved in cascade interconnection and with input shifts. In particular, under suitable conditions, we can establish semi-global practical ISM in perturbed flows and in singular perturbation models.

Cette thèse porte sur l’amélioration du laminage à chaud, un procédé de fabrication sidérurgique permettant de transformer une brame de métal (10m de long, 1.5m de large et 250mm d’épaisseur) en une bande de tôle bobinée (1000m de long, 1m de large et 2mm d’épaisseur). Afin d’obtenir certaines propriétés mécaniques et de faciliter la phase de laminage, la brame est réchauffée à 1300°C et dégrossi avant d’être envoyé vers le train finisseur où elle est laminée en passant successivement dans plusieurs cages (ensemble de cylindres qui écrasent le métal) et qui permettent de réduire l’épaisseur à la valeur finale souhaitée. Le produit est finalement refroidi puis bobiné avant d’être envoyé au client. La thèse se focalise sur l’amélioration du train finisseur en proposant un contrôle du frottement entre la bande et les cylindres de travail à l’aide d’une lubrification. La lubrification consiste à déposer de l’huile sur le cylindre en vaporisant une émulsion d’eau et d’huile. L’huile déposée modifie l’interface entre la bande et le cylindre et diminue le coefficient de frottement. Cette diminution du coefficient de frottement a plusieurs avantages : elle permet de réduire l’usure des cylindres, d’améliorer l’état de surface de la bande, de réduire l’effort nécessaire de laminage donc la consommation d’énergie et d’augmenter la capacité du train. A l’inverse, un frottement trop bas dû à une lubrification trop importante peut causer un patinage de la bande entrainant l’arrêt du train. Il est donc important de contrôler le niveau de frottement de manière sécurisée. La conception du contrôle s’est faite à travers deux principales étapes : La modélisation et l’identification de l’effet de la lubrification sur le coefficient de frottement, la conception du contrôle du frottement
This thesis is about the improvement of the hot rolling process. This steelmaking process turns a slab (10m long, 1.5m wide, 250mm thick) into a coiled strip (1000m long, 1m wide, 2mm thick). To obtain some metallurgical properties and to make the rolling easier, the slab is heated up to 1300 ° C and roughly rolled before going to the finishing mill. In the finishing mill the strip is rolled through successive stands (set of rolls) to reduce the thickness to its final desired value. The product is finally cooled down and coiled before shipping it to the customers. The thesis focuses on the enhancement of the finishing mill through a friction control between the strip and the work rolls using lubrication. The lubrication consists in building up oil on the rolls by spraying an emulsion of water and oil. The deposited oil changes the contact interface between the strip and the roll and decreases the friction coefficient. The reduction of the friction presents the advantages of: reduce the roll wear, enhance the strip surface quality, decrease the rolling force (reduce then the energy consumption) and increase the mill capability. In the other hand, an insufficient amount of friction due to an overabundance of lubrication can induce a slippage of the strip leading to a stop of the mill. It is important to control the amount of friction in a secure way. The design of the controller was done through two main steps: Modeling and identification of the effect of lubrication on the friction coefficient, designing the friction control

We develop and adapt absolute stability results for nonnegative Lur'e systems, that is, systems made up of linear part and a nonlinear feedback in which the state remains nonnegative for all time. This is done in both continuous and discrete time with an aim of applying these results to population modeling. Further to this, we consider forced nonnegative Lur'e systems, that is, Lur'e systems with an additional disturbance, and provide results on input-to-state stability (ISS), again in both continuous and discrete time. We provide necessary and sufficient conditions for a forced Lur'e system to have the converging-input converging-state (CICS) property in a general setting before specializing these results to nonnegative, single-input, single-output systems. Finally we apply integral control to nonnegative systems in order to control the output of the system with the key focus being on applications to population management.

In this work we design an iterative distributed optimization algorithm, based on the well-known distributed gradient tracking algorithm. We show the advantages of a system theoretical approach on a distributed optimization framework in which the cost function is given by a sum of quadratic functions. In other words, the main idea of the work consists in seeing the update equation characterizing an iterative optimization algorithm as the dynamics equation of a discrete-time system in which the decision variable plays the state variable role. The goal of the work is to show how system and control theory tools can be used to force this state variable to the optimum of the considered cost function even in presence of disturbances and/or uncertainties. So, the design of a distributed optimization algorithm is seen as the design of a controller able to solve the set point control problem in which the reference signal is given by the optimum of the considered cost function.

Marine craft at sea are affected by environmental disturbances including long-term ocean currents and relatively higher frequency wave disturbances. These disturbances impact on vessels resulting in wave-induced motion which reduces the performance of motion control systems and impacts on the safety of crew and cargo. This thesis investigates parameter estimation techniques for the online estimation of wave-induced motion models and platform control of marine craft in the presence of environmental disturbances.

Ubiquitous uncertainties exist in control systems. The concept of input-to-state stability (ISS) is one way to describe how uncertainties influence systems stability. The main purpose of this thesis is to develop some new ISS-based tools to design feedback control laws that are capable of counteracting specific uncertainties in nonlinear and network systems. In Part II, ISS cyclic-small-gain theorems are developed as a basic criterion to analyze the stability and robustness of nonlinear and network systems composed of interacting ISS subsystems. Part III presents a dissipativity-based switching adaptive control methodology, with which a control system could counteract uncertainties by appropriately switching the controller parameters. Based on the cyclic-small-gain theorem, the switching adaptive control law can be implemented in a decentralized manner for uncertain network systems. In Part IV, we propose cyclic-small-gain based tools for robust control design of nonlinear systems with disturbed measurements. With the proposed design tools, the controller can drive the control error of a nonlinear system to the level of the measurement disturbance. The efforts in Part V are devoted to quantized control of nonlinear systems. We will consider different quantizers and introduce new quantized control structures for nonlinear systems.

Research Doctorate - Doctor of Philosophy (PhD)
This thesis investigates stability analysis of discrete-time nonlinear dynamical systems. Stability, which plays a central role in nonlinear control system theory, has been studied extensively resulting in different frameworks such as l₂-gain stability, Input-to-State stability (ISS), convergent dynamics, and contraction analysis. All of these frameworks were derived differently, were motivated distinctly, and employ different mathematical toolsets. As a consequence, their mutual relationships are, in general, not fully understood. For instance, the relationships between contraction analysis, incremental stability, and convergent dynamics are not clear although they all characterize asymptotically convergent behaviours between solutions of dynamical systems. Therefore, this thesis aims at three goals. First, we systematically review and propose stability and convergence properties for discrete-time systems that have been unavailable in the literature. Second, we thoroughly study qualitative relationships between important discrete-time stability and convergence notions. Third, we also present computational methods for certain bounds used in stability properties. We begin by reviewing stability properties for systems without inputs. We then study and propose Lyapunov function characterizations for so-called convergence properties: incremental stability, convergent dynamics, and contraction analysis. Subsequently, we demonstrate that convergent dynamics and incremental stability are two distinct properties. However, contraction analysis, which is distinctly different from convergent dynamics, is equivalent to a subset of incremental stability. For nonlinear systems with inputs, we begin by reviewing the classical ISS and input-output l_p-gain stability properties and subsequently investigate two extensions of ISS: incremental ISS and ISS with respect to two measurement functions. We then examine relationships between ISS-based and l_p-based properties. In particular, we demonstrate that, via nonlinear changes of coordinates, ISS and integral ISS are qualitatively equivalent to linear and nonlinear l_p-gain properties, respectively. Illustrative examples with constructed changes of coordinates are provided to support the qualitative equivalence findings. Finally, we demonstrate two dynamic programming-based computational techniques including an extended real-valued terminal cost and a nonlinear "energy saturation" filter to estimate tight gain and transient bounds for the nonlinear l₂-gain properties. Numerical examples are provided to showcase the application of the proposed computational methods.

Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2007.
Source: Dissertation Abstracts International, Volume: 69-02, Section: B, page: 1230. Adviser: Daniel Liberzon. Includes bibliographical references (leaves 174-179) Available on microfilm from Pro Quest Information and Learning.

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