Rozprawy doktorskie na temat „Linear quadratic theory”

Linear Quadratic Regulatory theory (L.Q.R.) has received widespread application due to its simplicity and also due to the fact that the control provided this by theory is linear in form. These features make the implication of feedback control and easy task. In contrast, nonlinear regulation lack those attractive features enjoyed by the linear regulator. Moreover, in order to obtain the feedback control, one has to solve Hamilton-Jacobi-Bellman equation which is not an easy task. Also, if solution can be obtained, implementation is not always practical. In this work, we extend the Linear Quadratic Regulatory theory to the following; (I) LQR theory is modified for the case when there is no control contribution to the cost functional. (II) LQR is used to regulate or fine-tune a nonlinear system around a nominal trajectory through linearization of nonlinear systems. (III) Applying the LQR theory for the regulation of angular velocities of a three-axes satellite around a nominal point. (IV) Applying the LQR for the regulation of the movement of a robot around a time-optional trajectory. (V) The limitation of the control obtained through linearization is indicated.

This dissertation advances tools for the certification of unmanned aircraft system (UAS) flight controllers. We develop two thrusts to this goal: (1) the validation and improvement of an uncertain UAS framework based on integral quadratic constraint (IQC) theory and (2) the development of novel IQC theorems which allow the analysis of uncertain systems having time-varying characteristics. Pertaining to the first thrust, this work improves and implements an IQC-based robustness analysis framework for UAS. The approach models the UAS using a linear fractional transformation on uncertainties and conducts robustness analysis on the uncertain system via IQC theory. By expressing the set of desired UAS flight paths with an uncertainty, the framework enables analysis of the uncertain UAS flying about any level path whose radius of curvature is bounded. To demonstrate the versatility of this technique, we use IQC analysis to tune trajectory-tracking and path-following controllers designed via H2 or H-infinity synthesis methods. IQC analysis is also used to tune path-following PID controllers. By employing a non-deterministic simulation environment and conducting numerous flight tests, we demonstrate the capability of the framework in predicting loss of control, comparing the robustness of different controllers, and tuning controllers. Finally, this work demonstrates that signal IQCs have an important role in obtaining IQC analysis results which are less conservative and more consistent with observations from flight test data. With regards to the second thrust, we prove a novel theorem which enables robustness analysis of uncertain systems where the nominal plant and the IQC multiplier are linear time-varying systems and the nominal plant may have a non-zero initial condition. When the nominal plant and the IQC multiplier are eventually periodic, robustness analysis can be accomplished by solving a finite-dimensional semidefinite program. Time-varying IQC multipliers are beneficial in analysis because they provide the possibility of reducing conservatism and are capable of expressing uncertainties that have unique time-domain characteristics. A number of time-varying IQC multipliers are introduced to better describe such uncertainties. The utility of this theorem is demonstrated with various examples, including one which produces bounds on the UAS position after an aggressive Split-S maneuver.
Doctor of Philosophy

In this thesis, autopilots are developed for missiles with moderate dynamics and stationary targets. The aim is to use the designs in real applications. Since the real missile model is nonlinear, a linearization process is required to get use of systematic linear controller design techniques. In the scope of this thesis, the linear quadratic full state feedback approach is applied for developing missile autopilots. However, the limitations of measurement systems on the missiles restrict the availability of all the states required for feedback. Because of this fact, the linear quadratic design will be approximated by the use of projective control theory. This method enables the designer to use preferably static feedback or if necessary a controller plus a low order compensator combination to approximate the full state feedback reference. Autopilots are checked for the validity of linearization, robust stability against aerodynamic, mechanical and measurement uncertainties.

Prof. Marcel Wild completed his PhD with Zurick University and this is a copy of the original works
The original works can be found at http://www.hbz.uzh.ch/
ABSTRACT: A linear representation of a modular lattice L is a homomorphism from L into the lattice Sub(V) of all subspaces of a vector space V. The representation theory of lattices was initiated by the Darmstadt school (Wille, Herrmann, Poguntke, et al), to large extent triggered by the linear representations of posets (Gabriel, Gelfand-Ponomarev, Nazarova, Roiter, Brenner, et al). Even though posets are more general than lattices, none of the two theories encompasses the other. In my thesis a natural type of finite lattice is identified, i.e. triangle lattices, and their linear representation theory is advanced. All of this was triggered by a more intricate setting of quadratic spaces (as opposed to mere vector spaces) and the question of how Witt’s Theorem on the congruence of finite-dimensional quadratic spaces lifts to spaces of uncountable dimensions. That issue is dealt with in the second half of the thesis.

Orientador: José Mario Martínez Pérez
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Neste trabalho, serão propostos modelos matemáticos para problemas de empacotamento não reticulado de esferas em regiões limitadas por quadráticas no plano e no espaço. Uma técnica para construir representações ou parametrizações será introduzida, mediante a qual será possível encontrar um sistema de desigualdades que determinam o empacotamento de um número fixo de esferas. Desta forma, resolvemos o problema de empacotamento de esferas através de uma sequência de sistemas de desigualdades. Finalmente, para obter resultados eficientes, minimizaremos a função de sobreposição, usando o método do Lagrangiano Aumentado
Abstract: In this work, we will propose mathematical models for not latticed packing of spheres problems in regions bounded by quadratic in the plane and in the space. A technique to construct representations or parameterizations will be introduced, by which it will be possible to find a system of inequalities which determine the packing of a fixed number of spheres. Thus, we solve the problem of packing spheres through a sequence of systems of inequalities. Finally, to obtain effective results, we will minimize the overlay function using the Augmented Lagrangian Method
Doutorado
Matematica Aplicada
Doutor em Matemática Aplicada

The interest in autonomous vehicles has never been higher and there are several components that need to function for a vehicle to be fully autonomous; one of which is the ability to perform a parking at the end of a mission. The objective of this thesis work is to develop and implement an automatic parking system (APS) for a heavy-duty vehicle (HDV). A delimitation in this thesis work is that the parking lot has a known structure and the HDV is a truck without any trailer and access to more computational power and sensors than today's commercial trucks. An automatic system for searching the parking lot has been developed which updates an occupancy grid map (OGM) based on measurements from GPS and LIDAR sensors mounted on the truck. Based on the OGM and the known structure of the parking lot, the state of the parking spots is determined and a path can be computed between the current and desired position. Based on a kinematic model of the HDV, a gain-scheduled linear quadratic (LQ) controller with feedforward action is developed. The controller's objective is to stabilize the lateral error dynamics of the system around a precomputed path. The LQ controller explicitly takes into account that there exist an input delay in the system. Due to minor complications with the precomputed path the LQ controller causes the steering wheel turn too rapidly which makes the backup driver nervous. To limit these rapid changes of the steering wheel a controller based on model predictive control (MPC) is developed with the goal of making the steering wheel behave more human-like. A constraint for maximum allowed changes of the controller output is added to the MPC formulation as well as physical restrictions and the resulting MPC controller is smoother and more human-like, but due to computational limitations the controller turns out less effective than desired. Development and testing of the two controllers are evaluated in three different environments of varying complexity; the simplest simulation environment contains a basic vehicle model and serves as a proof of concept environment, the second simulation environment uses a more realistic vehicle model and finally the controllers are evaluated on a full-scale HDV. Finally, system tests of the APS are performed and the HDV successfully parks with the LQ controller as well as the MPC controller. The concept of a self-parking HDV has been demonstrated even though more tuning and development needs to be done before the proposed APS can be used in a commercial HDV.

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Due to the demand for high-performance equipments and the rising cost of energy, the industrial sector is developing equipments to attend minimization of the theirs operational costs. The implementation of these requirements generate a demand for projects and implementations of high-performance control systems. The optimal control theory is an alternative to solve this problem, because in its design considers the normative specifications of the system design, as well as those that are related to the operational costs. Motivated by these perspectives, it is presented the study of methods and the development of algorithms to the approximated solution of the Equation Hamilton-Jacobi-Bellman, in the form of discrete Riccati equation, model free and dependent of the dynamic system. The proposed solutions are developed in the context of adaptive dynamic programming that are based on the methods for online design of optimal control systems, Discrete Linear Quadratic Regulator type. The proposed approach is evaluated in multivariable models of the dynamic systems to evaluate the perspectives of the optimal control law for online implementations.
Devido a demanda por equipamentos de alto desempenho e o custo crescente da energia, o setor industrial desenvolve equipamentos que atendem a minimização dos seus custos operacionais. A implantação destas exigências geram uma demanda por projetos e implementações de sistemas de controle de alto desempenho. A teoria de controle ótimo é uma alternativa para solucionar este problema, porque considera no seu projeto as especificações normativas de projeto do sistema, como também as relativas aos seus custos operacionais. Motivado por estas perspectivas, apresenta-se o estudo de métodos e o desenvolvimento de algoritmos para solução aproximada da Equação Hamilton-Jacobi-Bellman, do tipo Equação Discreta de Riccati, livre e dependente de modelo do sistema dinâmico. As soluções propostas são desenvolvidas no contexto de programação dinâmica adaptativa (ADP) que baseiam-se nos métodos para o projeto on-line de Controladores Ótimos, do tipo Regulador Linear Quadrático Discreto. A abordagem proposta é avaliada em modelos de sistemas dinâmicos multivariáveis, tendo em vista a implementação on-line de leis de controle ótimo.

This project aims to analyze Nordic companies active in the sector of Information and Communications Technology (ICT), and does this in two parts. Part I entails analyzing public companies to construct a valuation model aimed at predicting the enterprise value of private companies. Part II deals with analyzing private companies to determine if there are opportunities providing excess returns as compared to investments in public companies. In part I, a multiple regression approach is utilized to identify suitable valuation models. In doing so, it is revealed that 1-factor models provide best statistical results in terms of significance and prediction error. In descending order, in terms of prediction accuracy, these are (1) total assets, (2) turnover, (3) EBITDA, and (4) cash flow. Part II uses model (1) and finds that Nordic ICT private equity does provide opportunities for positive alphas, and that it is possible to construct portfolio strategies that increase this alpha. However, with regards to previous research, it seems as though the returns offered by the private equity market analyzed does not adequately compensate investors for the additional risks related to investing in private equity.
Det här projektet analyserar nordiska bolag aktiva inom Informations- och Kommunikationsteknologi (ICT) i två delar. Del I behandlar analys av publika bolag för att konstruera en värderingsmodell avsedd att förutsäga privata bolags enterprise value. Del II analyserar privata bolag för att undersöka huruvida det finns möjligheter att uppnå överavkastning jämfört med investeringar i publika bolag. I del I utnyttjas multipel regressionsanalys för att identifiera tillämpliga värderingsmodeller. I den processen påvisas att modeller med enbart en faktor ger bäst statistiska resultat i fråga om signifikans och förutsägelsefel. I fallande ordning, med avseende på precision i förutsägelser, är dessa modeller (1) totala tillgångar, (2) omsättning, (3) EBITDA, och (4) kassaflöde. Del II använder modell (1) och finner att den nordiska marknaden för privata ICT-bolag erbjuder möjligheter för överavkastning jämfört med motsvarande publika marknad, samt att det är möjligt att konstruera portföljstrategier som ökar avkastningen ytterligare. Dock, med hänsyn till tidigare forskning, verkar det som att de möjligheter för avkastning som går att finna på marknaden av privata bolag som undersökts inte kompenserar investerare tillräckligt för de ytterligare risker som är relaterade till investeringar i privata bolag.

Pour faire face aux mutations du paysage énergétique, les réseaux de distribution d'électricité sont soumis à des exigences de fonctionnement avec des indices de fiabilité à garantir. Dans les années à venir, de grands investissements sont prévus pour la construction des réseaux électriques flexibles, cohérents et efficaces, basés sur de nouvelles architectures et des solutions techniques innovantes, adaptatifs à l'essor des énergies renouvelables. En prenant en compte ces besoins industriels sur le développement des réseaux de distribution du futur, nous proposons, dans cette thèse, une approche reposant sur la théorie des graphes et l'optimisation combinatoire pour la conception de nouvelles architectures pour les réseaux de distribution. Notre démarche consiste à étudier le problème général de recherche d'une architecture optimale qui respecte l'ensemble de contraintes topologiques (redondance) et électrotechniques (courant maximal, plan de tension) selon des critères d'optimisation bien précis : minimisation du coût d'exploitation (OPEX) et minimisation de l'investissement (CAPEX). Ainsi donc, les deux familles des problèmes combinatoires (et leurs relaxations) ont été explorées pour proposer des résolutions efficaces (exactes ou approchées) du problème de planification des réseaux de distribution en utilisant une formulation adaptée. Nous nous sommes intéressés particulièrement aux graphes 2-connexes et au problème de flot arborescent avec pertes quadratiques minimales. Les résultats comparatifs de tests sur les instances de réseaux (fictifs et réels) pour les méthodes proposées ont été présentés
To cope with the changes in the energy landscape, electrical distribution networks are submitted to operational requirements in order to guarantee reliability indices. In the coming years, big investments are planned for the construction of flexible, consistent and effective electrical networks, based on the new architectures, innovative technical solutions and in response to the development of renewable energy. Taking into account the industrial needs of the development of future distribution networks, we propose in this thesis an approach based on the graph theory and combinatorial optimization for the design of new architectures for distribution networks. Our approach is to study the general problem of finding an optimal architecture which respects a set of topological (redundancy) and electrical (maximum current, voltage plan) constraints according to precise optimization criteria: minimization of operating cost (OPEX) and minimization of investment (CAPEX). Thus, the two families of combinatorial problems (and their relaxations) were explored to propose effective resolutions (exact or approximate) of the distribution network planning problem using an adapted formulation. We are particularly interested in 2-connected graphs and the arborescent flow problem with minimum quadratic losses. The comparative results of tests on the network instances (fictional and real) for the proposed methods were presented

Dans ce travail, nous avons développé une approche systématique traitant un problème particulier dans le domaine de la commande non linéaire. Il concerne le fait d'assurer une transition stable entre deux points opérationnels d'un système non linéaire. Cette approche emploie la stratégie de séquencement de gain et la notion d'incertitude bornée en norme pour approximer un système non linéaire à travers une famille de systèmes linéaires incertains à incertitude bornée en norme. Autour d'un ensemble de points d'équilibre, des lois de commande locale sont déterminées en garantissant quelques spécifications de performances locales. Le séquencement est déterminé de sorte que la stabilité est garantie. Par interpolation polynômiale continue, une loi de commande continue est établie à partir des points d'équilibre trouvés hors ligne et des correcteurs associés. Une stratégie de commande séquencée par retour d'état ainsi qu'une stratégie par retour de sortie dynamique ont été envisagées. Dans le premier cas, la commutation est faite lorsque l'état se trouve dans le bassin d'attraction du point d'équilibre ultérieur visé. Dans le deuxième, nous avons déterminé une loi de commande par retour de sortie dynamique o'u la politique de commutation est basée sur les états éstimés. Quelques exemples ont été donnés pour montrer l'efficacité de la méthode proposée.

by Chen Xi.
"July 2000."
Thesis (Ph.D.)--Chinese University of Hong Kong, 2000.
Includes bibliographical references (p. 107-112).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Mode of access: World Wide Web.
Abstracts in English and Chinese.

Gao Jianjun.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2005.
Includes bibliographical references (leaves 75-76).
Abstracts in English and Chinese.
Chapter 1 --- Introduction --- p.1
Chapter 2 --- Solution Framework Using Dynamic Programming --- p.7
Chapter 2.1 --- Difficulty of using dynamic programming --- p.8
Chapter 2.2 --- Scalar-state problems --- p.12
Chapter 2.3 --- Time-invariant system --- p.17
Chapter 2.4 --- Illustrative example of a scalar-state problem --- p.21
Chapter 3 --- Cardinality Constrained Quadratic Optimization --- p.26
Chapter 3.1 --- Reformulation --- p.27
Chapter 3.2 --- NP hardness --- p.31
Chapter 3.3 --- Solving CCQP with an efficient branch and bound method --- p.34
Chapter 3.3.1 --- Efficient branch and bound algorithm --- p.34
Chapter 3.3.2 --- Geometrical interpretation of the proposed ranking order --- p.48
Chapter 3.3.3 --- Additional algorithmic ideas for enhancing computational efficiency --- p.56
Chapter 3.4 --- Numerical example and computational results --- p.60
Chapter 4 --- Summary and Future Work --- p.73

碩士
國立海洋大學
機械與輪機工程學系
89
Abstract Linear Quadratic Gaussian / Loop Transfer Recovery (LQG/LTR) technique and Quantitative Feedback Theory technique (QFT) are adopted to design robust controllers for a control system which is subject to parameter uncertainty . Experimental tests and simulations are performed on an inverted pendulum servo system with different pendulums . The results show that the compensated system works well with both controllers designed under nominal condition . However , when parameter changes , only the QFT controller works. In addition , the LQG/LTR and QFT method are compared with control schemes designed by other technique , such as root-locus method , state variable method , and LQG method . The simulations and results of experimental tests both show that the QFT controller is superior to the others in terms of robustness. Key Word ： Robust Control、QFT、LQG / LER 。

Thesis (Ph. D.)--Florida State University, 2006.
Advisor: Xu-Feng Niu, Florida State University, College of Arts and Sciences, Dept. of Statistics. Title and description from dissertation home page (viewed Sept. 18, 2006). Document formatted into pages; contains ix, 116 pages. Includes bibliographical references.

Although cardinality constraints naturally arise in many applications, e.g., in portfolio selection problems of choosing small number of assets from a large pool of stocks or dynamic portfolio selection problems with limited trading dates within a given time horizon and in subset selection of the regression analysis, the state-of-the-art in cardinality constrained optimization has been stagnant up to this stage, largely due to the inherent combinatorial nature of such hard problems. We focus in this research on developing efficient and implementable solution algorithms for cardinality constrained optimization by investigating prominent structures and hidden properties of such problems. More specifically, we develop solution algorithms for four specific cardinality constrained optimization problems, including (i) the cardinality constrained linear-quadratic control problem, (ii) the optimal control problem of linear switched system with limited number of switching, (iii) the time cardinality constrained dynamic mean- variance portfolio selection problem, and (iv) cardinality constrained quadratic optimization problem. Taking advantages of a linear-quadratic structure of cardinality constrained optimization problems, we strive for analytical solutions when possible. More specifically, we derive an analytical solution for problem (iii) and obtain for both problems (i) and (ii) semi-analytical expressions of the solution governed by a family of Ricatti-like equations, which still suffer an exponentially growing complexity. To achieve high-performance of the solution algorithm, we devise algorithms of a branch and bound (BnB) type with various tight and computationally-cheap lower bounds achieved by identifying suitable SDP formulations and by exploiting geometric properties of the problem. We demonstrate efficiency of our proposed solution schemes evidenced from numerical experiments and present a firm step-forward in tackling this long-standing challenge of cardinality constrained optimization.
Gao, Jianjun.
Adviser: Duan Li.
Source: Dissertation Abstracts International, Volume: 72-11, Section: B, page: .
Thesis (Ph.D.)--Chinese University of Hong Kong, 2009.
Includes bibliographical references (leaves 134-142).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Abstract also in Chinese.

In this thesis, we consider a new generalization of the Fourier transform, depending on four complex parameters and all the powers of the Fourier transform. This new transform is studied in some Lebesgue spaces. In fact, taking into account the values of the parameters of the operator, we can have very different kernels and so, the corresponding operator is studied in different Lebesgue spaces, accordingly with its kernel. We begin with the characterization of each operator by its characteristic polynomial. This characterization serves as a basis for the study of the forthcoming properties. Following this, we present, for each case, the spectrum of the corresponding operator, necessary and sufficient conditions for which the operator is invertible, Parseval-type identities and conditions for which the operator is unitary and an involution of order n. After this, we contruct new convolutions associated with those operators and obtain the corresponding factorization identities and some norm inequalities. By using these new operators and convolutions, we construct new integral equations and study their solvability. In this sense, we have equations generated by the studied operators and also a class of equations of convolution-type depending on multi-dimensional Hermite functions. Furthermore, we study the solvability of classical integral equations, using the new operators and convolutions, namely a class of Wiener-Hopf plus Hankel equations, whose solution is written in terms of a Fourier-type series. For one case of this generalization of the Fourier transform, that only depends on the cosine and sine Fourier transforms, we obtain PaleyWiener and Wiener’s Tauberian results, using the associated convolution and a new translation induced by that convolution. Heisenberg uncertainty principles for the one-dimensional case and for the multi-dimensional case are obtained for a particular case of the introduced operator. At the end, as an application outside of mathematics, we obtain a new result in signal processing, more properly, in a filtering processing, by applying one of our new convolutions.
Nesta tese, consideramos uma nova generalização da transformação de Fourier, dependente de quatro parâmetros complexos e de todas as potências da transformação de Fourier. Esta nova transformação é estudada em alguns espaços de Lebesgue. De facto, tendo em conta os valores dos parâmetros, podemos ter núcleos muito diferentes e assim, o correspondente operador é estudado em diferentes espaços de Lebesgue, de acordo com o seu núcleo. Começamos com a caracterização de cada operador pelo seu polinómio característico. Esta caracterização serve de base para o estudo das propriedades seguintes. Seguindo isto, apresentamos, para cada caso, o espetro do correspondente operador, condições necessárias e suficientes para as quais o operador é invertível, identidades do tipo de Parseval e condições para as quais o operador é unitário e uma involução de ordem n. Depois disto, construímos novas convoluções associadas àqueles operadores e obtemos as correspondentes identidades de factorização e algumas desigualdades da norma. Usando estes novos operadores e convoluções, construímos novas equações integrais e estudamos a sua solvabilidade. Neste sentido, temos equações geradas pelos operadores estudados e também uma classe de equações do tipo de convolução dependendo de funções de Hermite multidimensionais. Além disso, estudamos a solvabilidade de equações integrais clássicas, usando os novos operadores e convoluções, nomeadamente uma classe de equações de Wiener-Hopf mais Hankel, cuja solução é escrita em termos de uma série do tipo de Fourier. Para um caso desta generalização da transformação de Fourier, que depende apenas das transformações de Fourier do cosseno e do seno, obtemos resultados de Paley-Wiener e resultados Tauberianos de Wiener, usando a convolução associada e uma nova translação induzida por essa convolução. Princípios de incerteza de Heisenberg para os casos unidimensional e multidimensional são obtidos para um caso particular do operador introduzido. No final, como uma aplicação fora da matemática, obtemos um novo resultado em processamento de sinal, mais propriamente, num processo de filtragem, por aplicação de uma das nossas novas convoluções.
Programa Doutoral em Matemática Aplicada