Dissertations / Theses on the topic 'Linear singularly perturbed problem'

Small aerial vehicles such as quad-rotors have been widely used commercially, for research and for hobby for the last decade with use still growing. The high interest is mainly due to the vehicles being small, simple, cheap and versatile. Among rigid body dynamics fast dynamics exist cohering to motors and other fast actuators. A linear quadratic control design technique is here investigated. The design technique suggests that the linear quadratic controller can be designed with penalties on the slow states only. The fast dynamics are modeled but the states are not penalised in the linear quadratic design. The design technique is here applied and evaluated. The results show that this in several cases is a suitable design technique for linear quadratic control design. MATLAB and Simulink have been widely used for design and implementation of control systems. With additional toolboxes these control systems can be compiled to and run on remote computers. Small, lightweight computers with high computational capacity are now easily accessible. In this thesis an avionics solution based on a small, powerful computer is presented. Simulink models can be compiled and transferred to the computer from the Simulink environment. The result is a user friendly way of rapid prototyping and evaluation of control systems.

The dissertation is made of two chapters. The first chapter is dedicated to the investigation of some properties of the layer potentials of a constant coefficient elliptic partial differential operator. In the second chapter, we focus our attention to the Lamè equations, which are related to the physic of an isotropic homogeneous elastic body. In particular, in the first chapter, we investigate the dependence of the single layer potential upon perturbation of the density, the support and the coefficients of the corresponding operator. Under some more restrictive assumptions on the operator, we prove a real analyticity theorem for the single layer potential and its derivatives. As a first step, we introduce a particular fundamental solution of a given constant coefficient partial differential operator. For this purpose, we exploite the construction of a fundamental solution given by John (1955). We have verified that, if the coefficients of the operator are constrained to a bounded set, then there exist a particular fundamental solution which is a sum of functions which depend real analytically on the coefficients of the operator. Such a result resembles the results of Mantlik (1991, 1992) (see also Tréves (1962)), where more general assumptions on the operator are considered. We observe that it is not a corollary. Indeed, we need a suitably detailed expression for the fundamental solution, which cannot be deduced by Mantlik's results. The next step is to introduce the support of our single layer potentials. It will be a compact sub-manifold of the the n-dimensional euclidean space parametrized by a suitable diffeomorphism defined on the boundary of a fixed domain. Then, we will be ready to state in Theorem 1.7 the main result of this chapter, which is a real analyticity result in the frame of Shauder spaces. The main idea of the proof stems from the papers of Lanza de Cristoforis & Preciso (1999) and by Lanza de Cristoforis & Rossi (2004, 2005) and exploits the Implicit Mapping Theorem for real analytic functions. Indeed, our main Theorem 1.7 is in some sense a natural extension of theorems obtained by Lanza de Cristoforis & Preciso (1999) and by Lanza de Cristoforis & Rossi (2004, 2005), for the Cauchy integral and for the Laplace and Helmholtz operators, respectively. Here we confine our attention to elliptic operators which can be factorized with operators of order 2. In the last section of the first chapter, we consider some applications of Theorem 1.7. In particular, we deduce a real analyticity theorem for the single and double layer potential which arise in the analysis of the boundary value problems for the Lamè equations and for the Stokes system. In the second chapter, we focus our attention to the Lamè equations. We consider some boundary value problems defined in a domain with a small hole. For each of them, we investigate the behavior of the solution and of the corresponding energy integral as the hole shrinks to a point. This kind of problem is not new at all and has been long investigated by the techniques of asymptotic analysis. It is perhaps difficult to give a complete list of contributions. Here we mention the work of Keller, Kozlov, Movchan, Maz'ya, Nazarov, Plamenewskii, Ozawa and Ward. The results that we present are in accordance with the behavior one would expect by looking at the above mentioned literature, but we adopt a different approach proposed by Lanza de Cristoforis (2001, 2002, 2005, 2007.) To do so, we exploit the real analyticity results for the elastic layer potentials obtained in the first chapter. We now briefly outline the main difference between our approach and the one of asymptotic analysis. Let d>0 be a parameter which is proportional to the diameter of the hole, so that the singularity of the domain appears when d=0. By the approach of the asymptotic analysis, we can expect to obtain results which are expressed by means of known functions of d plus an unknown term which is smaller than a positive power of d. Whereas, our results are expressed by means of real analytic functions of d defined in a whole open neighborhood of d=0 and by, possibly singular, but completely known functions of d, such as d^(2-n) or log d. Moreover, not only we can consider the dependence upon d, we can also investigate the dependence of the solution and the corresponding energy integral upon perturbations of the coefficients of the operator, and of the point where the hole is situated, and of the shape of the hole, and of the shape of the outer domain, and of the boundary data on the boundary of the hole, and of the boundary data on the boundary of the outer domain, and of the interior data. Also in this case we obtain results expressed by means of real analytic functions and completely known functions such as d^(2-n) and log d. The first boundary value problem we have studied is a Dirichlet boundary value problem with homogeneous data in the interior. Then, we turned to investigate a Robin boundary value problem with homogeneous data in the interior. In this case we have also described the behavior of the solution and the corresponding energy integral when both the domain and the boundary data display a singularity for d=0. Finally, we have studied a Dirichlet boundary value problem with non-homogeneous data in the interior.

The paper deals with a singularly perturbed reaction diffusion model problem. The focus is on reliable a posteriori error estimators for the H^1 seminorm that can be applied to anisotropic finite element meshes. A residual error estimator and a local problem error estimator are proposed and rigorously analysed. They are locally equivalent, and both bound the error reliably. Furthermore three modifications of these estimators are introduced and discussed. Numerical experiments for all estimators complement and confirm the theoretical results.

Dans cette thèse, nous nous sommes intéressés à l'analyse locale de systèmes différentiels linéaires singulièrement perturbés et de systèmes de Pfaff complètement intégrables et multivariés à croisements normaux. De tels systèmes ont une vaste littérature et se retrouvent dans de nombreuses applications. Cependant, leur résolution symbolique est toujours à l'étude. Nos approches reposent sur l'état de l'art de la réduction formelle des systèmes linéaires singuliers d'équations différentielles ordinaires univariées (ODS). Dans le cas des systèmes différentiels linéaires singulièrement perturbés, les complications surviennent essentiellement à cause du phénomène des points tournants. Nous généralisons les notions et les algorithmes introduits pour le traitement des ODS afin de construire des solutions formelles. Les algorithmes sous-jacents sont également autonomes (par exemple la réduction de rang, la classification de la singularité, le calcul de l'indice de restriction). Dans le cas des systèmes de Pfaff, les complications proviennent de l'interdépendance des multiples sous-systèmes et de leur nature multivariée. Néanmoins, nous montrons que les invariants formels de ces systèmes peuvent être récupérés à partir d'un ODS associé, ce qui limite donc le calcul à des corps univariés. De plus, nous donnons un algorithme de réduction de rang et nous discutons des obstacles rencontrés. Outre ces deux systèmes, nous parlons des singularités apparentes des systèmes différentiels univariés dont les coefficients sont des fonctions rationnelles et du problème des valeurs propres perturbées. Les techniques développées au sein de cette thèse facilitent les généralisations d'autres algorithmes disponibles pour les systèmes différentiels univariés aux cas des systèmes bivariés ou multivariés, et aussi aux systèmes d''equations fonctionnelles<br>In this thesis, we are interested in the local analysis of singularly-perturbed linear differential systems and completely integrable Pfaffian systems in several variables. Such systems have a vast literature and arise profoundly in applications. However, their symbolic resolution is still open to investigation. Our approaches rely on the state of art of formal reduction of singular linear systems of ordinary differential equations (ODS) over univariate fields. In the case of singularly-perturbed linear differential systems, the complications arise mainly from the phenomenon of turning points. We extend notions introduced for the treatment of ODS to such systems and generalize corresponding algorithms to construct formal solutions in a neighborhood of a singularity. The underlying components of the formal reduction proposed are stand-alone algorithms as well and serve different purposes (e.g. rank reduction, classification of singularities, computing restraining index). In the case of Pfaffian systems, the complications arise from the interdependence of the multiple components which constitute the former and the multivariate nature of the field within which reduction occurs. However, we show that the formal invariants of such systems can be retrieved from an associated ODS, which limits computations to univariate fields. Furthermore, we complement our work with a rank reduction algorithm and discuss the obstacles encountered. The techniques developed herein paves the way for further generalizations of algorithms available for univariate differential systems to bivariate and multivariate ones, for different types of systems of functional equations

We consider a singularly perturbed reaction-diffusion problem and derive and rigorously analyse an a posteriori residual error estimator that can be applied to anisotropic finite element meshes. The quotient of the upper and lower error bounds is the so-called matching function which depends on the anisotropy (of the mesh and the solution) but not on the small perturbation parameter. This matching function measures how well the anisotropic finite element mesh corresponds to the anisotropic problem. Provided this correspondence is sufficiently good, the matching function is O(1). Hence one obtains tight error bounds, i.e. the error estimator is reliable and efficient as well as robust with respect to the small perturbation parameter. A numerical example supports the anisotropic error analysis.

A singularly perturbed convection-diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis.

The subject that concerns this thesis is the modelling and control of plasma equilibria in the RFX-mod device operating as shaped tokamak. The aim was to develop an overall model of the plasma-conductors-controller system of RFX-mod shaped tokamak configuration for electromagnetic control purposes, with particular focus on vertical stability. Thus, the RFX-mod device is described by models of increasing complexity and involving both theoretical and experimental data. The CREATE-L code is used to develop 2D linearized plasma response models, with simplifying assumptions on the conducting structures (axisymmetric approximations). Such models, thanks to their simplicity, have been used for feedback controller design. The CarMa0 code is used to develop linearized plasma response models, but considering a detailed 3D description of the conducting structures. These models provide useful hints on the accuracy of the simplified models and on the importance of 3D structures in the plasma dynamics. The CarMa0NL code is used to model the time evolution of plasma equilibria, by taking into account also nonlinear effects which can come into play during specific phases (e.g. disruptions, limiter-to-divertor transitions, L-H transition etc.). The activity can be divided into two main parts: the first one involves the modelling of numerically generated low-β plasmas, which are used as a reference for the design and implementation of the plasma shape and position control system; the second part is related to the results of the experimental campaigns on shaped plasmas from low-β to H-mode regime, with particular efforts on the development of a novel plasma response model for the new equilibrium regimes achieved. Several challenges and peculiarities characterize the project in both the modelling and control frameworks. Strong plasma shape and different plasma regimes (i.e. low-β to H-mode plasmas), deeply affect the modelling activity and require the development of several numerical tools and methods of analysis. From the control system point of view, non-totally observable dynamic and model order reduction requirements allowed a full application of the model based approach in order to successfully design the plasma shape and vertical stability control system. The first part is based on theoretical data generated by the MAXFEA equilibrium code and used to derive the linearized model through the CREATE-L code. Two reference models have been produced for the magnetic configurations interested in shaped operations: the lower single null (LSN) and the upper single null (USN). The CREATE-L models are the most simple in terms of modelling complexity, because the conducting structures are described within the axisymmetric approximation. On the other hand, the simple but reliable properties of the CREATE-L model led to the successful design of the RFX-mod plasma shape and control system, which has been successfully tested and used to increase plasma performances involved in the second part of the thesis. Then, an investigation on the possible 3D effects of the conducting structures on these numerically generated plasma configurations has been carried out by producing plasma linearized models with an increased level of complexity. A detailed 3D volumetric description of the conducting structures of RFX-mod has been carried out and included in the plasma linearized models through the CarMa0 code. A comparison between the accuracy of this model and the previous 2D one has been performed. The different assumptions and approximations of the various models allow a clear identification of the key phenomena ruling the evolution of the n=0 vertical instability in RFX-mod tokamak discharges, and hence, provide fundamental information in the planning and the execution of related experiments and in refining the control system design. Finally, the nonlinear evolutionary equilibrium model including 3D volumetric structures CarMa0NL has been used to model nonlinear effects by simulating a "fictitious" linear current quench. The second part involves a modelling activity strictly related to the results of the experimental campaigns. In particular, new linearized models for the experimental plasmas in USN configuration have been carried out for all the plasma regimes involved in the experimental campaign, i.e. from low-β to H-mode. An iterative procedure for the production of accurate linearized plasma response models has been realized in order to handle the experimental data. The new plasma linearized models allowed further investigations on vertical stability, including 3D wall effects, in the three different plasma regimes (i.e. low-β, intermediate-β, H-mode). Furthermore, the axisymmetric plasma linearized models (CREATE-L) have been analyzed in the framework of the control theory revealing peculiar features in terms of associated SISO transfer function for vertical stability control and in terms of full MIMO model for shaping control. The MIMO model has been used to investigate the plasma wall-gaps oscillations experimentally observed in some intermediate-β plasma shots. A non-linear time evolution of the plasma discharge for a low-β plasma has been carried out by using the evolutionary equilibrium code CarMa0NL. Finally, it was investigated the vertical instability for the experimental plasmas in terms of a possible relation between plasma parameters and the occurrence of it; for these purposes, the solution of the inverse plasma equilibrium problem for the production of numerically generated plasma equilibria with variations on the plasma parameters observed experimentally was performed. This involves a wide class of numerical methods that will be described in details. Then, statistical hypothesis test has been adopted to compare the mean values of the parameters of both experimental and numerically generated plasmas showing different behaviours in terms of vertical stability.<br>La presente tesi tratta la modellazione e il controllo di plasmi in equilibrio, a sezione non circolare e relativi all’esperimento RFX-mod operante come tokamak. L’obiettivo è di sviluppare un modello complessivo di RFX-mod (includendo plasmaconduttori- controllore) con finalità di controllo elettromagnetico del plasma. L’esperimento RFX-mod è stato descritto con modelli caratterizzati da un crescente livello di complessità, coinvolgendo sia dati teorici che sperimentali. Il codice CREATE-L è stato usato per lo sviluppo di modelli linearizzati di risposta di plasma, con ipotesi semplificative sulla rappresentazione delle strutture conduttrici (approssimazione assialsimmetrica). Questi modelli, grazie alla loro semplicità, sono stati utilizzati per la progettazione del sistema di controllo. Il codice CarMa0 è stato usato per sviluppare modelli analoghi ma con una rappresentazione tridimensionale delle strutture conduttrici; questi permettono di verificare l’accuratezza dei modelli semplificati e indagare l’importanza delle strutture tridimensionali sulla dinamica del sistema. Il codice CarMa0NL ha permesso la trattazione di fenomeni evolutivi nel tempo e nonlineari (e.g. disruzioni, transizioni limiter-divertor, transizioni L-H etc.). L’attività può essere suddivisa in due parti: la prima riguarda la modellizzazione di plasmi a basso β teorici, non ottenuti sperimentalmente, usati come riferimento per la progettazione e l’implementazione del sistema di controllo della forma e della posizione verticale del plasma; la seconda parte, è legata ai risultati delle campagne sperimentali sui plasmi a sezione non circolari in diversi regimi, dal basso β al modo H, con particolare attenzione allo sviluppo di un nuovo modello linearizzato di risposta di plasma per i nuovi regimi di equilibrio raggiunti. L’attività di ricerca è caratterizzata da molteplici problematiche e peculiarità sia in termini di modellazione che di controllo. La pronunciata non circolarità della forma di plasma e i diversi regimi coinvolti hanno influenzato fortemente l’attività di modellazione che ha richiesto, infatti, lo sviluppo di molteplici strumenti computazionali e di analisi dati. Per quanto concerne il controllo, la non completa osservabilità della dinamica del sistema e la necessità di ridurre l’ordine del modello sono solo alcuni degli aspetti che hanno determinato la progettazione del sistema di controllo di forma e di posizione verticale. La prima parte è basata su dati teorici generati dal codice di equilibrio MAXFEA e poi utilizzati per derivare il modello linearizzato attraverso il codice CREATE-L. In questo contesto, sono stati prodotti due modelli di riferimento per le configurazioni magnetiche relative a plasmi non circolari: il singolo nullo inferiore (LSN) e il singolo nullo superiore (USN). I modelli CREATE-L sono i più semplici in termini di complessità di modellazione, in quanto le strutture conduttive della macchina sono descritte nell’approssimazione assialsimmetrica. D’altro canto, le proprietà semplici ma affidabili del modello CREATE-L hanno portato alla progettazione del sistema di controllo di forma e posizione verticale del plasma di RFX-mod, che è stato in seguito testato e utilizzato con successo per aumentare le prestazioni del plasma. Successivamente, è stata condotta un’analisi sui possibili effetti 3D delle strutture conduttrici sulle due configurazioni di plasma di riferimento, producendo dunque modelli linearizzati caratterizzati da un sempre maggiore livello di complessità. Una dettagliata descrizione volumetrica (3D) delle strutture conduttrici di RFX-mod è stata eseguita e inclusa nei modelli linearizzati di plasma attraverso il codice CarMa0. Successivamente, è stato eseguito un confronto tra l’accuratezza di questo modello e quello precedente 2D. Le diverse ipotesi e approssimazioni dei vari modelli consentono una chiara identificazione dei fenomeni chiave che governano l’evoluzione dell’instabilità verticale n = 0 in scariche RFX-mod tokamak e quindi forniscono informazioni fondamentali nella pianificazione ed esecuzione di esperimenti correlati oltre che nella raffinazione del progetto del sistema di controllo. Infine, il modello di equilibrio evolutivo non lineare CarMa0NL, che comprende le strutture volumetriche 3D, è stato utilizzato per modellare gli effetti non lineari simulando una variazione di corrente lineare "fittizia". La seconda parte è costituita da un’attività di modellazione strettamente correlata ai risultati delle campagne sperimentali. In particolare, sono stati eseguiti nuovi modelli linearizzati per i plasmi sperimentali nella configurazione USN per tutti i regimi di plasma coinvolti, cioè dal basso β fino al modo H. È stata ideata e sviluppata una procedura iterativa per la produzione di modelli linearizzati di risposta di plasma estremamente accurati, al fine di riprodurre al meglio i dati sperimentali. I nuovi modelli hanno consentito ulteriori studi sulla stabilità verticale, inclusi gli effetti della parete 3D, nei tre diversi regimi studiati (basso β, β intermedio, modo H). I modelli linearizzati assialsimmetrici (CREATE-L) sono stati analizzati dal punto di vista della teoria dei controlli, rilevando caratteristiche peculiari in termini di funzione di trasferimento SISO associata al controllo della stabilità verticale e in termini di modello completo MIMO relativo al controllo di forma. Il modello MIMO è stato utilizzato per indagare le oscillazioni nella forma del plasma osservate sperimentalmente in alcune scariche a β intermedio. L’evoluzione temporale non lineare della scarica di plasma, per plasmi sperimentali a regimi a basso β, è stata effettuata usando il codice di equilibrio evolutivo CarMa0NL. Infine, è stata studiata l’instabilità verticale per i plasmi sperimentali in termini di un possibile rapporto tra i parametri del plasma e il suo verificarsi; a tal fine è stata eseguita la soluzione del problema inverso per la produzione di equilibri di plasma teorici di riferimento, prodotti come variazioni sui parametri dei plasmi osservati sperimentalmente, il che comporta una vasta gamma di metodi numerici descritti in dettaglio. Successivamente, è stato adottato un test di ipotesi statistica per confrontare i valori medi dei parametri di plasma, sia sperimentali che teorici, associati a due diversi comportamenti in termini di stabilità verticale.

Ao, Weiwei.<br>"August 2010."<br>Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.<br>Includes bibliographical references (leaves 92-97).<br>Abstracts in English and Chinese.<br>Abstract --- p.ii<br>Acknowledgement --- p.v<br>Chapter 1 --- Introduction --- p.1<br>Chapter 2 --- Spikes on Single Line-Segments --- p.12<br>Chapter 2.1 --- Ansatz and sketch of the proof --- p.12<br>Chapter 2.2 --- Linear theory --- p.15<br>Chapter 2.3 --- The non linear projected problem --- p.20<br>Chapter 2.4 --- Projection of the error and proof of Theorem 1.0.1 --- p.24<br>Chapter 3 --- The triple junction solutions --- p.33<br>Chapter 3.1 --- Approximate solutions --- p.33<br>Chapter 3.2 --- linear and nonlinear projected problem --- p.35<br>Chapter 3.3 --- Error estimates and the proof of theorem 1.0.2 --- p.35<br>Chapter 4 --- Layer concentrations in three-dimensional domain --- p.45<br>Chapter 4.1 --- Preliminaries and setting up the problem --- p.45<br>Chapter 4.1.1 --- A linear model problem --- p.45<br>Chapter 4.1.2 --- Setting up the problem in suitable coordinates --- p.53<br>Chapter 4.2 --- The gluing procedure --- p.62<br>Chapter 4.3 --- The invertibility of L2 --- p.65<br>Chapter 4.4 --- Solving the nonlinear projected problem --- p.67<br>Chapter 4.5 --- Estimates of the projection against ∇w and Z --- p.72<br>Chapter 4.5.1 --- estimates for the projection of the error --- p.73<br>Chapter 4.5.2 --- projection of terms involving φ --- p.78<br>Chapter 4.5.3 --- projection of errors on the boundary --- p.80<br>Chapter 4.6 --- "The system for (f1, f2, e):proof of the theorem" --- p.81

Chan Sik Kin.<br>Thesis (M.Phil.)--Chinese University of Hong Kong, 2004.<br>Includes bibliographical references (leaves 38-41).<br>Abstracts in English and Chinese.<br>Chapter 1 --- Introduction --- p.4<br>Chapter 2 --- Preliminary analysis --- p.11<br>Chapter 3 --- Liapunov-Schmidt Reduction --- p.19<br>Chapter 4 --- The reduced problem: A Minimizing Procedure --- p.32<br>Chapter 5 --- Proof of the theorem 1.2 --- p.35<br>Bibliography --- p.38

Xi-Cheng Lou, Alan S. Willsky, George C. Verghese.<br>Bibliography: p. 41.<br>Supported by the Air Force Office of Scientific Research under grant AFOSR-82-0258 Supported by the Army Research Offcie under grant DAAG-29-84-K-005

Yeung Wai Kong.<br>Thesis (M.Phil.)--Chinese University of Hong Kong, 2004.<br>Includes bibliographical references (leaves 51-55).<br>Abstracts in English and Chinese.<br>Chapter 1 --- Introduction --- p.5<br>Chapter 2 --- Some Preliminaries --- p.13<br>Chapter 3 --- "Approximate Function we,p" --- p.17<br>Chapter 4 --- "The Computation Of Je[we,p]" --- p.21<br>Chapter 5 --- The Signs of c1 And c3 --- p.30<br>Chapter 6 --- The Asymptotic Behavior of ue and Je[ue] --- p.35<br>Chapter 7 --- "The Proofs Of Theorem 1.1, Theorem 1.2 And Corol- lary 11" --- p.40<br>Appendix --- p.43<br>Bibliography --- p.51

碩士<br>國立成功大學<br>電機工程學系碩博士班<br>94<br>In this thesis, a tracker for the linear quadratic Nash game of multiparameter singularly perturbed sampled-data systems is newly established. A generalized cross-coupled multiparameter algebraic Riccati equation (GCMARE) for two quadratic cost functions is needed to be solved by applying the LQR design methodology for the tracker design. Firstly, the asymptotic expansions for the GCMARE are newly established, and the proposed algorithm based on the Newton’s method for solving the GCMARE guarantees the quadratic convergence. Then the low-gain sample-data controller with a high design performance is realized though the digital redesign method. Finally, for further improving the tracking performance, the chaos evolutionary programming algorithm (CEPA) is utilized to tune the parameters of the tracker. An example is presented to demonstrate the effectiveness on the proposed methodology