Дисертації з теми "Linearization of the mathematical model"

Robots have been a revolutionizing force in manufacturing in the 20th and 21st century but have proven too dangerous around humans to be used in many other fields including medicine. We describe a new control algorithm for robots developed by the Brigham Young University Robotics and Dynamics and Robotics Laboratory that has shown potential to make robots less dangerous to humans and suitable to work in more applications. We analyze the computational complexity of this algorithm and find that it could be a feasible control for even the most complicated robots. We also show conditions for a system which guarantee local stability for this control algorithm.

A feedback design procedure known as extended linearization consists in replacing a mathematical model of a nonlinear dynamical system with its family of linearizations, parametrized by the operating point, and then combining feedback gains designed for representatives of the family into a single nonlinear feedback law. The principles of the procedure, applicable both to lumped-parameter and distributed-parameter systems, are discussed at the outset. The development shows limits on feedback laws that can be designed, as well as nonuniqueness of solutions, inherent in the method.
Ph. D.

The paper is focused on the Predator-Prey model modified in the case of harvesting one or both populations. Firstly there is given a short description of the basic model and the sensitivity analysis. The first essential modification is percentage harvesting. This model could be easily converted to the basic one using a substitution. The next modification is constant harvesting. Solving this system requires linearization, which was properly done and brought valuable results applicable even for the basic or the percentage harvesting model. The next chapter describes regulation models, which could be used especially in applying environmental policies. All reasonable regulation models are shown after distinguishing between discrete and continuous harvesting. The last chapter contains an algorithm for maximizing the profit of a harvester using econometrical modelling tools.

The thesis describes various modifications of the predator-prey model. The modifications are considering several harvesting methods. At the beginning a solution and a sensitivity analysis of the basic model are provided. The first modification is the percentage harvesting model, which could be easily converted to the basic model. Secondly a constant harvesting including a linearization is derived. A significant part is devoted to regulation models with special a focus on environmental applications and the stability of the system. Optimization algorithms for one and both species harvesting are derived and back-tested. One species harvesting is based on econometrical tools; the core of two species harvesting is the modified Newton's method. The economic applications of the model in macroeconomics and oligopoly theory are expanded using the methods derived in the thesis.

Рассматривается задача линеаризации математических моделей, описывающих технологические процессы, с целью получения удобного инструмента для управления ими. Задача линеаризации решается с помощью геометрической теории управления (ГТУ). Привлекательность ГТУ связана с получением эквивалентных нелинейным моделям линейных моделей, которые удобно использовать для решения задач управления, получая структуры регуляторов или законы управления. После чего осуществляется обратный переход из пространства линейных систем в пространство исходной нелинейной системы. При этом основные аналитические преобразования автоматизированы с помощью специализированного программного обеспечения. Поиск функций преобразования, связывающих переменные линейной и нелинейной моделей, осуществляется с помощью нового конструктивного метода решения системы дифференциальных уравнений в частных производных.
The problem of linearization of mathematical models describing technological processes with the purpose of obtaining a convenient tool for managing them is considered. The problem of linearization is solved by means of a geometric control theory (GCT). The attractiveness of GCT is connected, first of all, with obtaining equivalent nonlinear linear models, which are convenient for solving management problems and receiving regulatory structures or control laws. After that performed the reverse transition from the space of linear systems to the space of the original nonlinear system. A wider application of the geometric control theory is hindered by cumbersome analytical transformations connected with the calculation of the derivatives and the Lie brackets, the definition of the involutivity of distributions, and so on, and also the problem of determining the transformation functions connecting the variables of linear models in the form of Brunovsky and initial non-linear models of control objects. The authors developed specialized software that automates the main analytical transformations of GCT. The search for the transformation functions connecting the variables of the linear and nonlinear models is carried out using a new constructive method for solving the system of partial differential equations.

Feedforward is known to be one of the best methods for power amplifier linearization due to its superior linearization performance and broadband stable operation. However feedforward systems have relatively poor power efficiency and are complicated due to the presence of two nonlinear amplifiers and the requirements of amplitude, phase and delay matching within two different loops. In this thesis stochastic characterization of a simple feedforward system with autocorrelation analysis has been presented for Code Division Multiple Access (CDMA) applications taking the amplitude and delay mismatches into consideration. It has been assumed that, the input signal can be represented as Gaussian noise, main and error amplifiers can be modeled with third order AM/AM nonlinearities and there exists no phase mismatch within the loops. Hence closed form expressions, which relate the main channel and distorted adjacent channel power at any point in the feedforward circuitry to the system parameters, have been obtained. Consequently, a mathematical handy tool is achieved towards specifying the circuit parameters rapidly for optimum linearity performance and efficiency. The developed analytical model has been verified by Radio Frequency (RF) and system simulations. An alternative approach towards modeling feedforward systems for arbitrary signals has also been brought into consideration and has been verified with system simulations.

Дисертація на здобуття наукового ступеня доктора філософії (PhD) за спеціальністю 123 – Комп’ютерна інженерія – Національний технічний університет “Харківський політехнічний інститут”, Харків, 2020. Об’єктом дослідження є процеси управління тяговим рухомим складом за допомогою бортової комп’ютерної системи, що використовується у дизель-потягах серії ДЕЛ-02. Предметом дослідження є моделі, методи та відповідні програмні компоненти, які використовуються в комп’ютерній системі тягового рухомого складу, та розширюють область застосування геометричної теорії управління при синтезі оптимальних керувань рухомим складом, а також методи і засоби розробки сучасних програмних комплексів в рамках розробки комп’ютерної системи підтримки прийняття рішень машиніста дизель-потяга серії ДЕЛ-02. У вступі акцентовано увагу та обгрунтовано актуальність теми, що досліджується, показано зв’язок роботи з науковими програмами, планами та темами, наведено наукову новизну, а також, сформульовано практичне значення отриманих результатів. В першому розділі здійснено аналітичний огляд моделей, методів та програмних компонентах, що використовуються в комп’ютерних системах управління тяговим рухомим складом. Розглянуто особливості структури та роботи подібних систем на залізничному транспорті в Україні та світі (Китай, Індія, Німеччина, країни СНД). На прикладі роботи таких систем було розглянуто їх структуру, технічні характеристики, області застосування та особливості використання. В рамках першого розділу, також, було розглянуто математичну модель об’єкта управління, приклад методу лінеаризації даної математичної моделі, метод пошуку функцій перетворення, що пов’язують змінні лінійної та нелінійної математичної моделей. Також, було розглянуто можливості використання нейромережевої асоціативної пам’яті в системах управління та проаналізовано методи синтезу оптимальних систем управління. В результаті, було обрано основні напрямки досліджень та поставлено основні задачі дисертаційної роботи. В другому розділі було розглянуто питання перетворення нелінійних математичних моделей в еквівалентні лінійні математичні моделі в формі Бруновського. Також, було розглянуто методи спрощення аналітичних перетворень під час виконання процесу лінеаризації за рахунок перетворення до лінійного виду нелінійних систем з різною кількістю одночленів в правих частинах диференційних рівнянь початкового об’єкту, а також, відокремлення лінійного рівняння від системи в цілому. Дані методи було перевірено шляхом моделювання руху по відрізку шляху початкового об’єкту у вигляді нелінійної системи диференційних рівняннь та об’єкту перетвореного у лінійну форму Бруновського, з подальшим порівнянням отриманих результатів, які показали співпадіння, що свідчить про те, що у разі використання даного методу лінеаризації отримується лінійна математична модель, що є повністю еквівалентною початковій недінійній моделі. Додатково, було виконано лінеаризацію більш складної нелінійної математичної моделі, що описує роботу потяга з двома окремими двигунами, перевірка результатів моделювання лінійної моделі показала повну еквівалентність її початковій формі. Результати досліджень дозволили отримати ряд наукових результатів: − визначено залежність кількості та складності розрахунків під час проведення лінеаризації та пошуку функцій перетворень від кількості одночленів в правій частині рівнянь нелінійної математичної моделі; − запропоновано два нових методи пошуку функцій перетворення, що пов’язують змінні лінійної та нелінійної моделей, що дозволяють розширити область застосування ГТУ на об’єкти, праві частини диференційних рівнянь яких містять більше двох одночленів; − запропоновано метод зниження кількості обчислень при виконанні лінеаризації за рахунок відокремлення лінійного рівняння від системи; − виконано перевірку даного методу, який показа свою роботоспроможність на більш складних математичних моделях, зокрема на моделі, що описує роботу потяга з використанням двох еквівалентних двигунів. В третьому розділі розглянуто питання створення нового методу для пошуку функцій перетворення з використанням нейронних мереж. В рамках даного розділу запропоновано нову нейронну мережу, яка може бути використана для пошуку функцій перетворення. Наряду з цим в даному розділі було запропоновано новий табличний метод пошуку функцій перетворення, який є простим та наочним, там може використовуватися для швидкого отримання результатів при виконанні розрахунків. Дослідження, проведені в даному розділі дозволили отримати наступні наукові результати: − створено та запропоновано нову нейронну мережу для пошуку функцій перетворення, що пов’язують змінні нелінійної та лінійної моделей об’єкта управління, а це, в свою чергу, розширює область застосування геометричної теорії управління; − запропоновано новий табличний метод для пошуку функцій перетворення, який є досить простим для сприйняття та достатньо наочним. В рамках цього, запропоновано систему рівнянь в частинних похідних з обмеженнями у вигляді диференціальних нерівностей представляти у вигляді відповідної таблиці, яка дає змогу в наочному вигляді отримувати залежність функцій перетворень від аргументів, також формувати системи лінійних однорідних рівнянь, за допомогою яких можна буде звужувати область пошуку функцій перетворення. В четвертому розділі присвячено увагу програмним компонентом бортової комп’ютерної системи, а також розробленому програмному забезпеченню, що дозволяє розширити область застосування геометричної теорії управління. А саме, було розглянуто нові функціональні можливості розробленого програмного забезпечення, та описано його основні характеристики та структуру. В рамках опису розробленого програмного забезпечення особливу увагу приділено структурі та опису роботи окремих функціональних блоків програми, розробці структури інтерфейсу, надійності програмного забезпечення, компонентів для вирішення завдань управління за допомогою геометричної теорії управління, оцінці якості програмного забезпечення. Також, в даному розділі приведено приклад роботи розробленого програмного забезпечення. Крім того, в даному розділі приведено результати рішення завдання оптимального руху дизель-потягу по маршруту його прямування, в рамках чого було виконано моделювання руху потяга по маршруту та та порівняння отриманих даних з даними руху реального потяга, а також виконано спробу підвищити ефективність руху потяга за рахунок оптимізації окремих множин перегонів з урахуванням особливостей маршруту прямування. В рамках даного розділу були отримані наступні наукові результати: − розроблено нове програмне забезпечення, яке отримало подальший розвиток завдяки використанню можливостей сучасних мов програмування. Розроблене програмне забезпечення є більш стабільним завдяки блоку тестування, більш зручним завдяки створеному графічному інтерфейсу користувача, більш функціональним, адже воно може виконувати процес лінеаризації та пошуку функцій перетворення, але при цьому багато функціональних можливостей є автоматизованими, в вихідних даних наявні коментарі та пояснення, що збільшує рівень зручності користування даним програмним забезпеченням, крім того, характеристики програми відповідають вимогам стандарту з якості програмного забезпечення; − було виконано дослідження залежності кількості спожитого палива під час руху потяга від особливостей рельєфу місцевості, стилю ведення потяга та розкладу його руху. − було запропоновано та протестовано метод зниження кількості спожитого палива, використовуючи особливості рельєфу місцевості, допустимі відставання чи випередження графіку руху потяга, а також визначення оптимального стилю руху як для маршруту в цілому, так і для його окремих частин; − було виконано моделювання руху потяга по реальному маршруту, а результати порівняні з реальним потягом, що курсує цим маршрутом, результати показали правильність моделювання. Отже, дисертаційна робота присвячена розвязанню науково-прикладної задачі, а саме, розробки моделей, методів та програмних компонентів компютерної системи тягового рухомого складу, яка створена на основі узагальнених математичних моделей, розробленого програмного забезпечення, а також засобів оптимізації управління рухомими об’єктами з використанням нових методів, а також використання нової стуктури нейронних мереж для пошуку функцій перетворення, що дозволило розширити область застосування геометричної теорії управління, що створює передумови для розробки автоматичних систем управління потягом та дозволяє поліпшити характеристики, повязані з об’ємами споживання енергоресурсів. Вдосконалена модель дизель-потяга враховує основні види взаємодії потяга та профілю шляху, а саме, повороти, нахили, а також роботу двигунів потяга, що адекватно відображає протікаючі в реальному дизель-потязі процеси. Було створено спеціалізоване програмне забезпечення, що має графічний інтерфейс користувача, а також відповідає вимогам оцінки якості програмного забезпечення. Дане програмне забезпечення реалізує вдосконалену структуру людино-машинної системи, дає можливість виконати автоматизацію аналітичних перетворень геометричної теорії управління у формі Бруновського. Нова структура нейронних мереж, базується на нейронних мережах типу АРТ, що дозволяє вирішувати завдання, що мають декілька рішень. Це дозволило виконати розробку нового методу пошуку функцій перетворення, які зв’язують змінні нелінійних та лінійних моделей у формі Бруновського. Для збільшення ефективності процесу лінеаризації було запропоновано декілька методів спрощення процесу розрахунків за рахунок зменшення кількості елементів в правій частині початкової системи диференційних рівнянь, та за рахунок відокремлення першого рівняння, яке саме по собі вже є лінійним, від загальної системи в цілому. Виконані дослідження та розробки дозволили вдосконалити структуру бортової компютерної системи підтримки прийняття рішень машиніста дизель потяга, що дозволило, в реальних умовах руху динамічного об’єкту, під час змін дорожніх умов, виконувати перерахунки та видавати машиністу нові закони керування, які дозволять продовжити рух по маршруту з дотриманням графіку та мінімальними витратами паливо-енергетичних ресурсів. Проведені дослідження на реальному обєкті та математичних моделях. Результати досліджень підтвердили правильність використовуваних інструментів, методів та алгоритмів, на основі яких були запропоновані відповідні рішення, які лягли в основу розробленого програмного забезпечення.
The thesis is submitted to obtain a scientific degree of Doctor of Philosophy, specialty 123 – Computer Engineering – National Technical University “Kharkiv Polytechnic Institute” , Kharkiv, 2020. The object of the research is the processes of managing the traction rolling stock with the help of an on-board computer system used in the DEL-02 series diesel trains. The subject of research are models, methods and corresponding software components used in the computer system of traction rolling stock, which extend the using scope of geometric control theory for the synthesis of optimal controls of rolling stock, as well as methods and tools for the development of modern software complexes in the development of computer decision support systems of the diesel train driver of the DEL-02 series trains. The introduction focused and explained on the relevance of the topic being researched, shows the relationship with scientific programs, plans and topics, presents the scientific novelty, as well as formulates the practical significance of the results. The first section provides an analytical overview of models, methods and software components used in computerized decision support systems of the diesel train driver and train control systems. The peculiarities of the structure and peculiarities of using such systems on rail transport in Ukraine and in the world (China, India, Germany, CIS countries) are considered. On the example of the operation of such systems considered their structure, specifications, applications and features of use. The first section also deals with the mathematical model of a control object, an example of a method of linearization of a given mathematical model, a method of finding transform functions that relate variables of linear and nonlinear mathematical models. Also, the possibility of using neural network associative memory in control systems was considered and methods of synthesis of optimal control systems were analyzed. As a result, the main directions of research were selected and the main tasks of the dissertation were set. In the second section, the question of converting nonlinear mathematical models into equivalent linear mathematical models in the form of Brunovsky was considered. Also, methods of simplifying analytical transformations during the linearization process by converting to a linear kind of nonlinear systems with different numbers of monomials in the right-hand sides of the differential equations of the initial object, as well as separating the linear equation from the other part of the system of equations, were considered. These methods were verified by modeling the motion along the path of the initial object in the form of a nonlinear system of differential equations and the object transformed into a linear Brunovsky form, with further comparison of the results obtained, which showed coincidence, which indicates that in the case of using this the linearization method allows to obtain a linear mathematical model that is completely equivalent to the original non-linear model. Additionally, linearization of a more complex nonlinear mathematical model describing the operation of a train with two separate engines was performed, and the verification of the results of the linear model simulation showed complete equivalence to its original form. Research results have yielded a number of scientific results: − dependence of quantity and complexity of calculations during linearization and search of transformation functions on the number of monomials in the right part of equations of nonlinear mathematical model is determined; − two new methods of finding transform functions are proposed that relate variables of linear and nonlinear models that extend the scope of geometric control theory to objects whose right-hand sides of differential equations contain more than two monomials; − was proposed a method of reducing the number of calculations when performing linearization by separating a linear equation from the system; − this method was tested, which showed its workability on more complex mathematical models, in particular, on a model that describing the operation of a train using two equivalent motors. In the third section of the paper, the question of creating a new method for finding functions of transformation using neural networks was considered. In this section proposes a new neural network that can be used to search for conversion functions. In addition, this section proposes a new tabular method of finding conversion functions, which is simple and clear and can be used to get results when performing the calculation process. The studies conducted in this section have yielded the following scientific results: − a new neural network has been created and proposed for searching the conversion functions that relate variables to nonlinear and linear models of a control object, which in turn widens the scope of geometric control theory; − a new tabular method for finding conversion functions is proposed, which is simple enough to understand and sufficiently visual. In this context, it is proposed to present a system of partial differential equations with constraints in the form of differential inequalities in the form of a corresponding table, which allows to visualize the dependence of transformation functions on arguments, as well as to form systems of linear homogeneous equations by which it is possible to narrow the search area of conversion functions. The fourth section focuses on the software components of the on-board computer system, as well as the developed software that extends the scope of geometric control theory. Specifically, shows with new functionality of designed software and describes its main characteristics and structure. In the framework of the description of the developed software, special attention is paid to the structure and description of the operation of individual functional blocks of the program, the development of the interface structure, the reliability of the software, components for solving control problems using geometric control theory, evaluation of the quality of the software. Also, this section gives an example of how the developed software works. In addition, this section presents the results of solving the problem of optimal motion of the diesel train along the route of its direction, in which the simulation of the train movement along the route was performed and the comparison of the obtained data with the data of the movement of the real train, as well as an attempt to improve the efficiency of train movement due to the optimization of individual sets of routes, taking into account the features of the route. The following scientific results have been obtained within this section: − new software has been developed that has been further developed through the use of modern programming languages. The developed software is more stable due to the testing unit, more convenient due to the created graphical user interface, more functional, because it can perform the process of linearization and search of conversion functions, many of the functionality are automated, there are comments and an explanation that increases the ease of use of this software, in addition, the characteristics of the program meet the requirements of the standard of program quality; − the study of the dependence of the amount of fuel consumed during train movement on the features of terrain, the style of running the train and its schedule; − a method of reducing the amount of fuel consumed was proposed and tested, using terrain features, permissible lag or advance of the train timetable, as well as determining the optimal driving style for the route as a whole and for its individual parts; − the train simulation was performed on a real route, and the results showed that the simulation was correct, because it was compared to the real train running on this route. Therefore, the dissertation is devoted to the solution of the scientific-applied problem, namely, the development of models, methods and software components of the computer system of traction rolling stock, which is created on the basis of generalized mathematical models, developed software, as well as the means of optimizing the control of moving objects new methods, as well as the use of a new neural network structure to search for transformation functions, which made it possible to extend the scope of geometric control theory it breeds the preconditions for developing automatic train control systems and improves performance related to energy consumption. The advanced diesel train model takes into account the main types of interaction between the train and the track profile, namely, turns, slopes, as well as the performance of the train engines, which adequately reflects the processes in real diesel train. Specialized software has been created that has a graphical user interface and complies with software quality assessment requirements. This software implements an advanced structure of the human-machine system, makes it possible to perform automation of analytical transformations of geometric control theory to the form of Brunovsky. The new neural network structure is based on ART-type neural networks to solve multiple-choice tasks. This made it possible to develop a new method of finding transform functions that relate variables of nonlinear and linear models in the form of Brunovsky. To increase the efficiency of the linearization process, several methods have been proposed to simplify the calculation process by reducing the number of elements in the right-hand side of the initial differential equation system, and by separating the first equation, which itself is linear, from the general system of equations. The performed research and development allowed to improve the structure of the on-board computer system of decision support of the driver of the diesel train, which allowed, under real conditions of movement of the dynamic object, during changes of road conditions, to perform recalculations and to give the driver new control laws which will allow to continue the movement on the route adhering to the timetable and minimum cost of fuel and energy resources. Appropriate researches were conducted on real object and mathematical models. The results of the researches confirmed the correctness of the used tools, methods and algorithms, on the basis of which the appropriate solutions that formed the basis of the developed software were proposed.

A recent development in the design of control system for a jet engine is to use a suitable, fast and accurate model running on board. Development of linear models is particularly important as most engine control designs are based on linear control theory. Engine control performance can be significantly improved by increasing the accuracy of the developed model. Current state-of-the-art is to use piecewise linear models at selected equilibrium conditions for the development of set point controllers, followed by scheduling of resulting controller gains as a function of one or more of the system states. However, arriving at an effective gain scheduler that can accommodate fast transients covering a wide range of operating points can become quite complex and involved, thus resulting in a sacrifice on controller performance for its simplicity. This thesis presents a methodology for developing a control oriented analytical linear model of a jet engine at both equilibrium and off-equilibrium conditions. This scheme requires a nonlinear engine model to run onboard in real time. The off-equilibrium analytical linear model provides improved accuracy and flexibility over the commonly used piecewise linear models developed using numerical perturbations. Linear coefficients are obtained by evaluating, at current conditions, analytical expressions which result from differentiation of simplified nonlinear expressions. Residualization of the fast dynamics states are utilized since the fast dynamics are typically outside of the primary control bandwidth. Analytical expressions based on the physics of the aerothermodynamic processes of a gas turbine engine facilitate a systematic approach to the analysis and synthesis of model based controllers. In addition, the use of analytical expressions reduces the computational effort, enabling linearization in real time at both equilibrium and off-equilibrium conditions for a more accurate capture of system dynamics during aggressive transient maneuvers. The methodology is formulated and applied to a separate flow twin-spool turbofan engine model in the Numerical Propulsion System Simulation (NPSS) platform. The fidelity of linear model is examined by validating against a detailed nonlinear engine model using time domain response, the normalized additive uncertainty and the nu-gap metric. The effects of each simplifying assumptions, which are crucial to the linear model development, on the fidelity of the linear model are analyzed in detail. A case study is performed to investigate the case when the current state (including both slow and fast states) of the system is not readily available from the nonlinear simulation model. Also, a simple model based control is used to illustrate benefits of using the proposed modeling approach.

Many mechanical systems exhibit nonlinear movement and are subject to perturbations from a desired equilibrium state. These perturbations can greatly reduce the efficiency of the systems. It is therefore desirous to analyze the asymptotic stabilizability of an equilibrium solution of nonlinear systems; an excellent method of performing these analyses is through study of Jacobian linearization's and their properties. Two enlightening examples of nonlinear mechanical systems are the Simple Inverted Pendulum and the Inverted Pendulum on a Cart (PoC). These examples provide insight into both the feasibility and usability of Jacobian linearizations of nonlinear systems, as well as demonstrate the concepts of local stability, observability, controllability and detectability of linearized systems under varying parameters. Some examples of constant disturbances and effects are considered. The ultimate goal is to examine stabilizability, through both static and dynamic feedback controllers, of mechanical systems

Ce travail concerne la linéarisation des amplificateurs de haute puissance en utilisant la pré-distorsion numérique. L’amplificateur de haute puissance est un composant non-linéaire. La pré-distorsion numérique adaptative en bande de base est un technique efficace pour linéariser ses non-linéarités et ses effets de mémoire. Les modèles de la pré-distorsion numérique de basse complexité sont étudiés dans cette thèse. Un algorithme est proposé pour déterminer une structure optimale de modèle uni-étage ou multi-étage en prenant compte du compromis entre la précision de modélisation et la complexité. La structure cascadée, qui est avantageuse en complexité comparé avec celle d'uni-étage, est étudiée avec des méthodes d'identifications différentes. En termes d'implémentations expérimentales, l'étude d'impact des choix de gain différents est approfondie dans cette thèse. Toutes les études ont été évaluées par un amplificateur de puissance Doherty
This dissertation contributes to the linearization techniques of high power amplifier using digital predistortion method. High power amplifier is one of the most nonlinear components in radio transmitters. Unfortunately, for most current types of power amplifiers, a good efficiency is obtained at the price of a poor linearity especially with modern communication waveforms. Baseband adaptive digital predistortion is a powerful technique to linearize the power amplifiers and allows to push the power amplifier operation point towards its high efficiency region. Linearization of power amplifiers using digital predistortion with low complexities is the focus of this dissertation. An algorithm is proposed to determine an optimal model structure of single-stage or multi-stage predistorter according to a trade-off between modeling accuracy and model complexity. Multi-stage cascaded digital predistortions are studied with different identification methods, which have advantages on complexity of model identification compared with single-stage structure. The linearization performances are validated by experimental implementations on test bench. In terms of experimental implementations, this dissertation studies the impact of different gain choices on linearized power amplifier. All studies are evaluated with a Doherty power amplifier

First comes a description of a mathematical model of the geomagnetic field. Then some discussion of the classical non-uniqueness results of Backus. Further we look at more recent results concerning reconstruction of the geomagnetic field from intensity and the normal component of the field. New stability estimate for this reconstruction is obtained.

A l'heure de l'acquisition de donn'ees 'a haut d'ebit concernant le m'etabolisme cellulaire et son 'evolution, il est absolument n'ecessaire de disposer de mod'eles permettant d'int'egrer ces donn'ees en un ensemble coh'erent, d'en interpr'eter les variations m'etaboliques r'ev'elatrice, les 'etapes clefs o'u peuvent s'exercer des r'egulations, voire mˆeme d'en r'ev'eler des contradictions apparentes met- tant en cause les bases sur lesquelles le mod'ele lui-mˆeme est construit. C'est ce type de travail que j'ai entrepris 'a propos de donn'ees exp'erimentales obtenues dans le laboratoire biologique sur le m'etabolisme de cellules tu- morales en r'eponse 'a un traitement anti-canc'ereux. Je me suis attach'ee 'a la mod'elisation d'un point particulier de ce m'etabolisme. Il concerne le m'etabolisme des glyc'erophospholipides qui sont de bons marqueurs de la prolif'eration cellulaire. Les phospholipides constituent l'essentiel des mem- branes d'une cellule et l''etude de leur synth'ese (en particulier chez les cellules de mammif'eres) est de ce fait un sujet important. Ici, nous avons pris le parti de mettre en place un mod'ele math'ematique par 'equations diff'erentielles ordinaires, qui est essentiellement bas'e sur des 'equations hyperboliques (Michaelis-Menten), mais aussi sur des cin'etiques type loi d'action de masse et diffusion. Le mod'ele, compos'e de 8 'equations diff'erentielles, donc de 8 substrats d'int'erˆet, comporte naturellementdes param'etres inconnus in vivo, et certains d'ependents des conditions cellulaires (diff'erentiations de cellules, pathologies, . . .). Le mod'ele s'epare la structure du r'eseau m'etabolique, l''ecriture de la matrice de stoechiom'etrie, celles des 'equations de vitesse et enfin des 'equations diff'erentielles. Le mod'ele choisi est le mod'ele murin (souris/rat), parce qu'il est lui-mˆeme un mod'ele de l'homme. Plusieurs con- ditions sont successivement consid'er'ees pour l'identification des param'etres, afin d''etudier les liens entre la synth'ese de phospholipides et le cancer : - le foie sain du rat, - le m'elanome B16 et le carcinome de la lign'ee 3LL chez la souris, respectivement sans traitement, en cours de traitement 'a la Chloro'ethyl-nitrosour'ee et apr'es traitement, - enfin le m'elanome B16 chez la souris sous stress de privation de m'ethionine. En r'esum'e, ce tra- vail fourni une interpr'etation nouvelle des donn'ees exprimentales en mon- trant le rˆole essentiel de la PEMT et la nature superstable de l''etat sta- tionnaire de fonctionnement du r'eseau m'etabolique des phospholipides lors de la canc'erog'en'ese et du traitement des cancers. Il montre bien l'avantage de l'utilisation d'un mod'ele math'ematique dans l'interpr'etation de donn'ees m'etaboliques complexes.

This thesis deals with the mathematical modelling of nutrient uptake by plant roots. It starts with the Nye-Tinker-Barber model for nutrient uptake by a single bare cylindrical root. The model is treated using matched asymptotic expansion and an analytic formula for the rate of nutrient uptake is derived for the first time. The basic model is then extended to include root hairs and mycorrhizae, which have been found experimentally to be very important for the uptake of immobile nutrients. Again, analytic expressions for nutrient uptake are derived. The simplicity and clarity of the analytical formulae for the solution of the single root models allows the extension of these models to more realistic branched roots. These models clearly show that the `volume averaging of branching structure' technique commonly used to extend the Nye-Tinker-Barber with experiments can lead to large errors. The same models also indicate that in the absence of large-scale water movement, due to rainfall, fertiliser fails to penetrate into the soil. This motivates us to build a model for water movement and uptake by branched root structures. This model considers the simultaneous flow of water in the soil, uptake by the roots, and flow within the root branching network to the stems of the plant. The water uptake model shows that the water saturation can develop pseudo-steady-state wet and dry zones in the rooting region of the soil. The dry zone is shown to stop the movement of nutrient from the top of the soil to the groundwater. Finally we present a model for the simultaneous movement and uptake of both nutrients and water. This is discussed as a new tool for interpreting available experimental results and designing future experiments. The parallels between evolution and mathematical optimisation are also discussed.

Thesis (Ph.D.)--Indiana University, Dept. of Mathematics, 2007.
Title from dissertation home page (viewed Sept. 29, 2008). Source: Dissertation Abstracts International, Volume: 69-02, Section: B, page: 1045. Adviser: Victor W. Goodman.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Sloan School of Management, 2000.
Includes bibliographical references (p. 163-166).
This dissertation addresses the question of how musical pitches generate a tonal center. Being able to characterize the relationships that generate a tonal center is crucial to the computer analysis and the generating of western tonal music. It also can inform issues of compositional styles, structural boundaries, and performance decisions. The proposed Spiral Array model offers a parsimonious description of the inter-relations among tonal elements, and suggests new ways to re-conceptualize and reorganize musical information. The Spiral Array generates representations for pitches, intervals, chords and keys within a single spatial framework, allowing comparisons among elements from different hierarchical levels. Structurally, this spatial representation is a helical realization of the harmonic network (tonnetz). The basic idea behind the Spiral Array is the representation of higher level tonal elements as composites of their lower level parts. The Spiral Array assigns greatest prominence to perfect fifth and major /minor third interval relations, placing elements related by these intervals in proximity to each other. As a result, distances between tonal entities as represented spatially in the model correspond to perceived distances among sounding entities. The parameter values that affect proximity relations are prescribed based on a few perceived relations among pitches, intervals, chords and keys. This process of interfacing between the model and actual perception creates the opportunity to research some basic, but till now unanswered questions about the relationships that generate tonality. A generative model, the Spiral Array case; provides a framework on which to design viable and efficient algorithms for problems in music cognition. I demonstrate its versatility by applying the model to three different problems: I develop an algorithm to determine the key of musical passages that, on average, performs better than existing ones when applied to the 24 fugue subjects in Book I of Bach's WTC; I propose the first computationally viable method for determining modulations (the change of key); and, I design a basic algorithm for finding the roots of chords, comparing its results to those of algorithms by other researchers. All three algorithms were implemented in Matlab.
by Elaine Chew.
Ph.D.

Nella tesi è descritto un modello matematico per l'aura emicranica e, più precisamente, per lo scotoma scintillante (schema di fortificazione) e per la Cortical Spreading Depression, il fenomeno neuropatofisiologico alla base dell'aura. In particolare è spiegato un modello cinematico per l'evoluzione della CSD nella corteccia visiva primaria, considerata un mezzo debolmente eccitabile, la mappa retino-corticale e il modello, tramite fibrato, dei Pinwheels di V1.

Gene expression regulatory networks are molecular networks which describe interactions among gene products in terms of biochemical reactions. This helps us understand the molecular mechanisms underlying important biological processes as well as cell functioning as a whole. For instance, the phenomenon of bacterial competence, whereby a bacterium enters a transiently differentiated state, incorporating DNA fragments from its environment into its genome, has been studied with the help of such gene regulatory circuits (Suel et al., 2006; Maamar and Dubnau, 2005). As a result, a genetic circuit has been taken into account in order to describe the transition from a vegetative state to a transient state of competence and vice versa. In this work, we are going to study a genetic circuit presented by Suel et al. (2007) to describe this dynamical behaviour. The authors introduce model reduction techniques to study the behaviour of stochastic chemical system of X species by means of an adiabatic two dimensional model. While the adiabatic model helps us understand about the dynamics near the steady state, it gives an incorrect description of the time-scales of the competent state. For this reason, it is necessary to build up a model which better describes the system realistically. In the thesis, I propose an approximate two-dimensional model of the full high-dimensional system and from that, the dynamics of the system can be simulated more accurately compared to that of Suel et al. (2007). I then show how to put the noise back into the approximate model to be able come up with a stochastic model which can mathematically describe the dynamical behaviour of the original high dimensional system. I also found out that the evolution of the system is not well approximated by a Langevin process. This leads to a gap between the real behavior which is described by Gillespie's stochastic simulation and the Langevin approximation. To overcome this, I have fixed the stochastic Langevin model by incorporating empirically tunable noise into the model so as to obtain a similar behaviour as observed in the original system. I also introduce the chemical Fokker-Planck equation aimed to estimate the probability density function of species concentrations which are involved in the biochemical system.

Rotary kilns have numerous industrial applications including cement production. Frequent operational problems such as low thermal efficiency, refractory failure, and poor product quality have prompted extensive efforts to improve and optimize their design. Mathematical modeling and Computational Fluid Dynamics constitute effective tools recently used for these purposes. A cement kiln consists of three major parts: the hot flow, the bed, and the wall. A CFD code which had the capability of simulating the hot gas was developed further to simulate the kiln. In the present work, two 1-D mathematical models are proposed and implemented in the existing CFD code. The first model consists of the steady-state solution for the material and temperature evolution within the bed. The second one simulates tire combustion in the kiln. The tire burning model assumes that tire combustion occurs in two major successive steps, devolatization and char combustion. For the devolatization model, external heat and mass transfer, three parallel reactions, and enthalpy effects are considered the dominant phenomena. The char combustion model considers the enthalpy effect and the external mass transfer. With the aid of the developed model, full-scale industrial cement kilns under steady-state and realistic operational conditions are simulated. In addition, cement kilns with combustion of full scrap tires in the middle of them are mathematically modeled. The limits and feasibility of tire combustion are further explored by running numerical simulations with different tire flow rates and different injector locations. The flow field, temperature distribution and species distribution are presented. Analysis of the results indicates that, with the help of the proposed model, a better understanding of the important processes within cement kilns can be obtained. The model can be used for addressing operational problems and optimizing designs. It is also concluded that successful firing of tires can lead to a cheaper, longer lasting, and less polluting kiln.
Applied Science, Faculty of
Mechanical Engineering, Department of
Graduate

ITC/USA 2010 Conference Proceedings / The Forty-Sixth Annual International Telemetering Conference and Technical Exhibition / October 25-28, 2010 / Town and Country Resort & Convention Center, San Diego, California
This paper describes a model of how to configure settings on instrumentation. For any given instrument there may be 100s of settings that can be set to various values. However, randomly selecting values for each setting is not likely to produce a valid configuration. By "valid" we mean a set of setting values that can be implemented by each instrument. The valid configurations must satisfy a set of dependency rules between the settings and other constraints. The formalization provided allows for identification of different sets of configurations settings under control by different systems and organizations. Similarly, different rule sets are identified. A primary application of this model is in the context of a multi-vendor system especially when including vendors that maintain proprietary rules governing their systems. This thus leads to a discussion of an application user interface (API) between different systems with different rules and settings.

Pour un terminal mobile alimenté sur batterie, le rendement de l’amplificateur de puissance (AP) doit êtreoptimisé. Cette optimisation peut rendre non-linéaire la fonction d’amplification de l’AP. Pour compenser lesdistorsions introduites par le caractère non-linéaire de l’AP, un détecteur numérique fondé sur un modèle deVolterra peut être utilisé. Le comportement de l’AP et le canal étant modélisé par le modèle de Volterra, uneapproche par filtrage de Kalman (FK) permet d’estimer conjointement les noyaux de Volterra et les symbolestransmis. Dans ce travail, nous proposons de traiter cette problématique dans le cadre d’une liaison montantedans un contexte radio intelligente (RI). Dans ce cas, des contraintes supplémentaires doivent être prises encompte. En effet, étant donné que la RI peut changer de bande de fréquence de fonctionnement, les nonlinéaritésde l’AP peuvent varier en fonction du temps. Par conséquent, nous proposons de concevoir une postdistorsionnumérique fondée sur une modélisation par modèles multiples combinant plusieurs estimateurs àbase de FK. Les différents FK permettant de prendre en compte les différentes dynamiques du modèle.Ainsi, les variations temporelles des noyaux de Volterra peuvent être suivies tout en gardant des estimationsprécises lorsque ces noyaux sont statiques. Le cas d’un signal monoporteuse est adressé et validé par desrésultats de simulation. Enfin, la pertinence de l’approche proposée est confirmée par des mesures effectuéessur un AP large bande (300-3000) MHz
For a battery driven terminal, the power amplifier (PA) efficiency must be optimized. Consequently,non-linearities may appear at the PA output in the transmission chain. To compensatethese distortions, one solution consists in using a digital post-distorter based on aVolterra model of both the PA and the channel and a Kalman filter (KF) based algorithm tojointly estimate the Volterra kernels and the transmitted symbols. Here, we suggest addressingthis issue when dealing with uplink cognitive radio (CR) system. In this case, additionalconstraints must be taken into account. Since the CR terminal may switch from one subbandto another, the PA non-linearities may vary over time. Therefore, we propose to designa digital post-distorter based on an interacting multiple model combining various KF basedestimators using different model parameter dynamics. This makes it possible to track thetime variations of the Volterra kernels while keeping accurate estimates when those parametersare static. Furthermore, the single carrier case is addressed and validated by simulationresults. In addition, the relevance of the proposed approach is confirmed by measurementscarried on a (300-3000) MHz broadband PA

In formulating discrete optimization problems, it is not only important to have a correct mathematical model, but to have a well structured model that can be solved effectively. Two important characteristics of a general integer or mixed-integer program are its size (the number of constraints and variables in the problem), and its strength or tightness (a measure of how well it approximates the convex hull of feasible solutions). In designing model formulations, it is critical to ensure a proper balance between compactness of the representation and the tightness of its linear relaxation, in order to enhance its solvability. In this dissertation, we consider these issues pertaining to the modeling of mixed-integer 0-1 programming problems in general, as well as in the context of several specific real-world applications, including a telecommunications network design problem and an airspace management problem. We first consider the Reformulation-Linearization Technique (RLT) of Sherali and Adams and explore the generation of reduced first-level representations for mixed-integer 0-1 programs that tend to retain the strength of the full first-level linear programming relaxation. The motivation for this study is provided by the computational success of the first-level RLT representation (in full or partial form) experienced by several researchers working on various classes of problems. We show that there exists a first-level representation having only about half the RLT constraints that yields the same lower bound value via its relaxation. Accordingly, we attempt to a priori predict the form of this representation and identify many special cases for which this prediction is accurate. However, using various counter-examples, we show that this prediction as well as several variants of it are not accurate in general, even for the case of a single binary variable. Since the full first-level relaxation produces the convex hull representation for the case of a single binary variable, we investigate whether this is the case with respect to the reduced first-level relaxation as well, and show similarly that it holds true only for some special cases. Empirical results on the prediction capability of the reduced, versus the full, first-level representation demonstrate a high level of prediction accuracy on a set of random as well as practical, standard test problems. Next, we focus on a useful modeling concept that is frequently ignored while formulating discrete optimization problems. Very often, there exists a natural symmetry inherent in the problem itself that, if propagated to the model, can hopelessly mire a branch-and-bound solver by burdening it to explore and eliminate such alternative symmetric solutions. We discuss three applications where such a symmetry arises. For each case, we identify the indistinguishable objects in the model which create the problem symmetry, and show how imposing certain decision hierarchies within the model significantly enhances its solvability. These hierarchies render an otherwise virtually intractable formulation computationally viable using commercial software. For the first problem, we consider a problem of minimizing the maximum dosage of noise to which workers are exposed while working on a set of machines. We next examine a problem of minimizing the cost of acquiring and utilizing machines designed to cool large facilities or buildings, subject to minimum operational requirements. For each of these applications, we generate realistic test beds of problems. The decision hierarchies allow all previously intractable problems to be solved relatively quickly, and dramatically decrease the required computational time for all other problems. For the third problem, we investigate a network design problem arising in the context of deploying synchronous optical networks (SONET) using a unidirectional path switched ring architecture, a standard of transmission using optical fiber technology. Given several rings of this type, the problem is to find a placement of nodes to possibly multiple rings, and to determine what portion of demand traffic between node pairs spanned by each ring should be allocated to that ring. The constraints require that the demand traffic between each node pair should be satisfiable given the ring capacities, and that no more than a specified maximum number of nodes should be assigned to each ring. The objective function is to minimize the total number of node-to-ring assignments, and hence, the capital investment in add-drop multiplexer equipments. We formulate the problem as a mixed-integer programming model, and propose several alternative modeling techniques designed to improve the mathematical representation of this problem. We then develop various classes of valid inequalities for the problem along with suitable separation procedures for tightening the representation of the model, and accordingly, prescribe an algorithmic approach that coordinates tailored routines with a commercial solver (CPLEX). We also propose a heuristic procedure which enhances the solvability of the problem and provides bounds within 5-13% of the optimal solution. Promising computational results that exhibit the viability of the overall approach and that lend insights into various modeling and algorithmic constructs are presented. Following this we turn our attention to the modeling and analysis of several issues related to airspace management. Currently, commercial aircraft are routed along certain defined airspace corridors, where safe minimum separation distances between aircraft may be routinely enforced. However, this mode of operation does not fully utilize the available airspace resources, and may prove to be inadequate under future National Airspace (NAS) scenarios involving new concepts such as Free-Flight. This mode of operation is further compounded by the projected significant increase in commercial air traffic. (Free-Flight is a paradigm of aircraft operations which permits the selection of more cost-effective routes for flights rather than simple traversals between designated way-points, from various origins to different destinations.) We begin our study of Air Traffic Management (ATM) by first developing an Airspace Sector Occupancy Model (AOM) that identifies the occupancies of flights within three dimensional (possibly nonconvex) regions of space called sectors. The proposed iterative procedure effectively traces each flight's progress through nonconvex sector modules which comprise the sectors. Next, we develop an Aircraft Encounter Model (AEM), which uses the information obtained from AOM to efficiently characterize the number and nature of blind-conflicts (i.e., conflicts under no avoidance or resolution maneuvers) resulting from a selected mix of flight-plans. Besides identifying the existence of a conflict, AEM also provides useful information on the severity of the conflict, and its geometry, such as the faces across which an intruder enters and exits the protective shell or envelope of another aircraft, the duration of intrusion, its relative heading, and the point of closest approach. For purposes of evaluation and assessment, we also develop an aggregate metric that provides an overall assessment of the conflicts in terms of their individual severity and resolution difficulty. We apply these models to real data provided by the Federal Aviation Administration (FAA) for evaluating several Free-Flight scenarios under wind-optimized and cruise-climb conditions. We digress at this point to consider a more general collision detection problem that frequently arises in the field of robotics. Given a set of bodies with their initial positions and trajectories, we wish to identify the first collision that occurs between any two bodies, or to determine that none exists. For the case of bodies having linear trajectories, we construct a convex hull representation of the integer programming model of Selim and Almohamad, and exhibit the relative effectiveness of solving this problem via the resultant linear program. We also extend this analysis to model a situation in which bodies move along piecewise linear trajectories, possibly rotating at the end of each linear translation. For this case, we again compare an integer programming approach with its linear programming convex hull representation, and exhibit the relative effectiveness of solving a sequence of problems based on applying the latter construct to each time segment. Returning to Air Traffic Management, another future difficulty in airspace resource utilization stems from a projected increase in commercial space traffic, due to the advent of Reusable Launch Vehicle (RLV) technology. Currently, each shuttle launch cordons off a large region of Special Use Airspace (SUA) in which no commercial aircraft are permitted to enter for the specified duration. Of concern to airspace planners is the expense of routinely disrupting air traffic, resulting in circuitous diversions and delays, while enforcing such SUA restrictions. To provide a tool for tactical and planning purposes in such a context within the framework of a coordinated decision making process between the FAA and commercial airlines, we develop an Airspace Planning Model (APM). Given a set of flights for a particular time horizon, along with (possibly several) alternative flight-plans for each flight that are based on delays and diversions due to special-use airspace (SUA) restrictions prompted by launches at spaceports or weather considerations, this model prescribes a set of flight-plans to be implemented. The model formulation seeks to minimize a delay and fuel cost based objective function, subject to the constraints that each flight is assigned one of the designated flight-plans, and that the resulting set of flight-plans satisfies certain specified workload, safety, and equity criteria. These requirements ensure that the workload for air-traffic controllers in each sector is held under a permissible limit, that any potential conflicts which may occur are routinely resolvable, and that the various airlines involved derive equitable levels of benefits from the overall implemented schedule. In order to solve the resulting 0-1 mixed-integer programming problem more effectively using commercial software (CPLEX-MIP), we explore the use of various facetial cutting planes and reformulation techniques designed to more closely approximate the convex hull of feasible solutions to the problem. We also prescribe a heuristic procedure which is demonstrated to provide solutions to the problem that are either optimal or are within 0.01% of optimality. Computational results are reported on several scenarios based on actual flight data obtained from the Federal Aviation Administration (FAA) in order to demonstrate the efficacy of the proposed approach for air traffic management (ATM) purposes. In addition to the evaluation of these various models, we exhibit the usefulness of this airspace planning model as a strategic planning tool for the FAA by exploring the sensitivity of the solution provided by the model to changes both in the radius of the SUA formulated around the spaceport, and in the duration of the launch-window during which the SUA is activated.
Ph. D.

Distributed Rainfall-Runoff models are gaining widespread acceptance; yet, a fundamental issue that must be addressed by all users of these models is definition of an acceptable level of watershed discretization (geometric model complexity). The level of geometric model complexity is a function of basin and climatic scales as well as the availability of input and verification data. Equilibrium discharge storage is employed to develop a quantitative methodology to define a level of geometric model complexity commensurate with a specified level of model performance. Equilibrium storage ratios are used to define the transition from overland to channel -dominated flow response. The methodology is tested on four subcatchments in the USDA -ARS Walnut Gulch Experimental Watershed in Southeastern Arizona. The catchments cover a range of basins scales of over three orders of magnitude. This enabled a unique assessment of watershed response behavior as a function of basin scale. High quality, distributed, rainfall -runoff data was used to verify the model (KINEROSR). Excellent calibration and verification results provided confidence in subsequent model interpretations regarding watershed response behavior. An average elementary channel support area of roughly 15% of the total basin area is shown to provide a watershed discretization level that maintains model performance for basins ranging in size from 1.5 to 631 hectares. Detailed examination of infiltration, including the role and impacts of incorporating small scale infiltration variability in a distribution sense, into KINEROSR, over a range of soils and climatic scales was also addressed. The impacts of infiltration and channel losses on runoff response increase with increasing watershed scale as the relative influence of storms is diminished in a semiarid environment such as Walnut Gulch. In this semiarid environment, characterized by ephemeral streams, watershed runoff response does not become more linear with increasing watershed scale but appears to become more nonlinear.

Engine control systems aim to ensure satisfactory output performance whilst adhering to requirements on emissions, drivability and fuel efficiency. Model predictive control (MPC) has shown promising results when applied to multivariable and nonlinear systems with operational constraints, such as diesel engines. This report studies the torque generation from a mean-value heavy duty diesel engine with exhaust gas recirculation and variable-geometry turbocharger using state feedback linearization based MPC (LMPC). This is accomplished by first introducing a fuel optimal reference generator that converts demands on torque and engine speed to references on states and control signals for the MPC controller to follow. Three different MPC controllers are considered: a single linearization point LMPC controller and two different successive LMPC (SLMPC) controllers, where the controllers are implemented using the optimization tool CasADi. The MPC controllers are evaluated with the World Harmonized Transient Cycle and the results show promising torque tracking using a SLMPC controller with linearization about reference values.

In this work we present a geometric model of motor cortex that generalizes an already existing model of visual cortex. The thesis opens by recalling the notions of fiber bundles, principal bundles, Lie groups and sub-Riemannian geometry. In particular, we enunciate Chow’s theorem which ensures that if the Hörmander condition holds, the space connectivity property is satisfied. Then we recall the visual cortex model proposed by Citti-Sarti, which describes the set of simple cells as a Lie group with sub-Riemannian metric. The original part of the thesis is the extension to the motor cortex. Based on neural data, collected by Georgopoulos, we study the set of motor cortical cells and we describe them as a principal bundle. The fiber contains the movement direction and shapes the hypercolumnar structure measured. Finally we determine the intrinsic coordinates of the motor cortex, studying the cellular response to the motor impulse.

Have you thought of why you get tired or why you get hungry? Something in your body keeps track of time. It is almost like you have a clock that tells you all those things.

And indeed, in the suparachiasmatic region of our hypothalamus reside cells which each act like an oscillator, and together form a coherent circadian rhythm to help our body keep track of time. In fact, such circadian clocks are not limited to mammals but can be found in many organisms including single-cell, reptiles and birds. The study of such rhythms constitutes a field of biology, chronobiology, and forms the background for my research and this thesis.

Pioneers of chronobiology, Pittendrigh and Aschoff, studied biological clocks from an input-output view, across a range of organisms by observing and analyzing their overt activity in response to stimulus such as light. Their study was made without recourse to knowledge of the biological underpinnings of the circadian pacemaker. The advent of the new biology has now made it possible to "break open the box" and identify biological feedback systems comprised of gene transcription and protein translation as the core mechanism of a biological clock.

My research has focused on a simple transcription-translation clock model which nevertheless possesses many of the features of a circadian pacemaker including its entrainability by light. This model consists of two nonlinear coupled and delayed differential equations. Light pulses can reset the phase of this clock, whereas constant light of different intensity can speed it up or slow it down. This latter property is a signature property of circadian clocks and is referred to in chronobiology as "Aschoff's rule". The discussion in this thesis focus on develop a connection and also a understanding of how constant light effect this clock model.

To ensure that the conservation regulations which govern fishing gears are effective, they must be based on an understanding of the process by which fish are selected. The region where most fish selection is considered to take place is the cod-end, the aftmost part of a trawl net and the region where the catch accumulates. In recent years, it has become increasingly apparent that fish selection in the cod-end is dependent on a range of physical, environmental and fish behavioural parameters. Essential to a study of any of these parameters is a knowledge of the cod-end geometry which is determined by the interaction of the water flow, the catch size and the design and physical characteristics of the netting. In this thesis a continuum model of the deformation of a class of axisymmetric networks is developed where the mesh elements are reflection symmetric, the mesh bars are extensible and where arbitrary membrane forces act in the plane of the net, normal to the edges of the mesh elements. When applied to the fishing industry this provides a continuum model of the geometry of an axisymmetric trawl cod-end made from netting of a generalized mesh shape. It is shown how mesh shapes that are of interest to the fishing industry can be investigated, and the geometry of cod-ends made from diamond shaped mesh under the influence of various types of pressure loads is examined in detail. A qualitative description of the hydrodynamic forces that act on the cod-end catch is presented and it is shown that the predictions based on this description are consistent with experimental results from a series of wind tunnel trials. Using this description of the hydrodynamic forces the effect on cod-end geometry of mesh resistance to opening which arises as a result of twine flexural rigidity is examined.

The study on developing a mathematical model for thermal conductivity enhancement in nanofluids was based on formation of nanoparticles into nanoclusters, nanolayer thickness, Brownian motion and volume fraction of nanoclusters. An equation for the thermal conductivity of nanofluids was developed. The expression developed successfully explained the enhanced thermal conductivity of nanofluids and led to some important conclusions. It was found that in this study the nanoparticles tend to form nanoclusters and the volume fraction of the nanoclusters and the trapped fluid in the nanocluster was contributing to the overall thermal conductivity enhancement. Various types of cluster formation was analyzed and it was understood that the nanoparticles forming a spherical nanoclusters are more effective in thermal conductivity enhancement. The contribution of Brownian motion of nanoparticles to the overall thermal conductivity of nanofluids was found to be very small. The study investigated the size distribution of nanoparticles which has been suggested to be an important factor and it gave satisfactory results. The values of the thermal conductivity for different nanofluid combinations were calculated using the expression developed from this study and they agreed with published experimental data. The present model was tested against several nanofluid combinations. To understand the properties that influence the thermal conductivity of nanofluids, parametric studies of a number of nanofluids were carried out. The parameters that were scrutinized to understand themal conductivity enhancement were nanoparticle diameter, nanolayer thickness and brownian motion. From the study, it was observed that Brownian motion is significant only when the particle diameter is less than 10 nm. From the parametric studies the mathematical model derived in this study was validated. The major factor for the thermal conductivity enhancement in nanofluids is the formations of nanoclusters. The combination of the base fluid and nanoparticles to from nanoclusters will provide better cooling solution than the convention cooling fluids.

Dans cet étude, un modèle de complexité réduite des interactions et rétroactions du système couplé climat-économie-biosphère est construit avec le minimum de variables et d'équations nécessaires. Le Coupled Climate-Economy-Biosphere (CoCEB) est un modèle d’évaluation intégrée (IAM pour Integrated assessment model) du changement globale. Alors que beaucoup IAM traitent les coûts de réduction des émissions (abattement) simplement comme une perte non productive de revenu, cet étude considère également les activités d’abattement comme un investissement dans l'efficacité énergétique globale de l'économie et dans la diminution de l’ « intensité carbone » du système énergétique. L’étude montre que ces efforts aident à l’abattement du changement climatique et ont un effet positif sur l’économie. La plupart des IAM actuels se concentrent principalement sur le secteur énergétique pour les mesures d’abattement, et ne tiennent compte des émissions provenant de l'utilisation des terres que comme un forçage exogène. Le CoCEB a donc été étendu en ajoutant une équation pour la biomasse ses échanges de carbone. Cela permet d’étudier les aspects économiques de la séquestration de carbone du au contrôle du déboisement dans les forêts, et aussi à l’application généralisée des technologies de capture et stockage du carbone (CCS). L’étude confirme que ces mesures réduisent l’impact du changement climatique sur la croissance économique, mais ces résultats restent très dépendants des grandes incertitudes sur le cout des CCS et du contrôle de la deforestation. Ce modèle est un cadre formel qui représente de façon simple les différents éléments du système couplé et leurs interactions, il rassemble les différentes estimations des coûts afférents aux mesures de mitigation et permet de les comparer de façon cohérente
The goal of this study is to build a global reduced-complexity model of coupled climate-economy-biosphere interactions, which uses the minimum number of variables and equations needed to capture the fundamental mechanisms involved and can thus help clarify the role of the different mechanisms and parameters. The Coupled Climate-Economy-Biosphere (CoCEB) model takes an integrated assessment approach to simulating global change. While many integrated assessment models treat abatement costs merely as an unproductive loss of income, the study considered abatement activities also as an investment in overall energy efficiency of the economy and decrease of overall carbon intensity of the energy system. The study shows that these efforts help to abate climate change and lead to positive effects in economic growth. Due to the fact that integrated assessment models in the literature mainly focus on mitigation in the energy sector and consider emissions from land-use as exogenous, the global climate-economy-biosphere (CoCEB) model was extended by adding a biomass equation and the related exchanges of CO2 and used to investigate the relationship between the effects of using carbon capture and storage (CCS) and deforestation control, and the economy growth rate. These measures are found to reduce the impacts of climate change and positively affect the economy growth. These results remain nevertheless sensitive to the formulation of CCS costs while those for deforestation control were less sensitive. The model developed brings together and summarizes information from diverse estimates of climate change mitigation measures and their associated costs, and allows comparing them in a coherent way

A mathematical model of the nickel converter has been developed based on the assumption that the converting reactions pass through a finite series of equilibrium steps. The model predicts the bath temperature and the composition of the three phases present. Detailed data collected during in-plant trials are used to test the validity of the model predictions. The model is found to give relatively accurate predictions for the first blows of a converting charge, but overpredicts both temperature and iron removal during the last blows. The errors in the last blows are expected to be caused by the converting reactions coming under liquid phase mass transport control. An analysis of some of the more important variables in a converter operation indicates that, according to the assumption of equilibrium, there is very little that can be done to chemically improve the converting process.
Applied Science, Faculty of
Materials Engineering, Department of
Graduate

The purpose of this investigation was to develop and validate mathematical models of the sprint relay race. Two approaches, a classical exponential approach and a "new" polynomial approach, were chosen for implementation. Archival film of a 100 m sprint was used to gather displacement data for the first 60 m of the race. Filming had been performed with a single highspeed 16 mm cine camera (LOCAM) at 50 fps. The coordinates were digitized, scaled and filtered using a low pass, critically damped, 4$\sp{\rm th}$ order, zero-lag Butterworth digital filter with a 1 Hz cutoff frequency. Linear velocities were calculated using finite differences. A sprinter was modelled in two ways. The first was an Exponential Model which required as input a personal best time for the 100 m race and the sprinter's maximum constant velocity. The second was a Polynomial Model which required as input the parameters mentioned in the Exponential Model and, additionally, the displacement coefficients for the first 60 m of the 100 m sprint. Relay software was developed to piece the sprint relay together using the corresponding exponential or polynomial approach. The results indicated that the relay software reasonably simulated the kinematic and temporal quantities of a 4 x 100 m relay and can be used by coaches to gain insight into the sprint relay without risking injury to their athletes. Furthermore, the Exponential Model, using less information, described the sprinter's kinematics better than the Polynomial Model.

This dissertation focuses on the application of neural networks to financial model calibration. It provides an introduction to the mathematics of basic neural networks and training algorithms. Two simplified experiments based on the Black-Scholes and constant elasticity of variance models are used to demonstrate the potential usefulness of neural networks in calibration. In addition, the main experiment features the calibration of the Heston model using model-generated data. In the experiment, we show that the calibrated model parameters reprice a set of options to a mean relative implied volatility error of less than one per cent. The limitations and shortcomings of neural networks in model calibration are also investigated and discussed.

I develop an empirical model of housing demand which is based as closely as possible on a theoretical intertemporal model of consumer demand. In the empirical model, intertemporal behavior by households is incorporated in two ways. First, a household's expected length of occupancy in a dwelling is a parameter in the model; thus, households are able to choose when to move. Second, a household's decision to move and its choice of dwelling are based on the same intertemporal utility function. The parameters of the utility function are estimated using a switching regresion model in which the decision to move and the choice of housing quantity are jointly determined. The model has four other features: (1) a characteristics approach to housing demand is taken, (2) the transaction costs of changing dwellings are incorporated in the model, (3) sample data on household mortgages are employed in computing the user cost of owned dwellings, and (4) demographic variables are incorporated systematically into the household utility function. Rosen's two step proceedure is used to estimate the model. Cragg's technique for estimating regressions in the presence of heteroscedasticity of unknown form is used to estimate the hedonic regressions in step one of the proceedure. In the second step, the switching regression model, is estimated by maximum likelihood. A micro data set of 2,513 Canadian households is used in the estimations. The stage one hedonic regressions indicate that urban housing markets are not in long run equilibrium, that the errors of the hedonic regressions are heteroscedastic, and that simple functional forms for hedonic regressions may perform as well as more complex forms. The stage two estimates establish that a tight link between the theoretical and empirical models of housing demand produces a better model. My results show that conventional static models of housing demand are misspecified. They indicate that households have vastly different planned lengths of dwelling occupancy. They also indicate that housing demand is determined to a great extent by demographic factors.
Arts, Faculty of
Vancouver School of Economics
Graduate