A cone K in a vector space X is a subset which is closed under addition, positive scalar multiplication and the only element with additive inverse is zero. The pair (X, K) is called an ordered vector space. In this study, we consider the characterizations of reflexive Banach spaces. This is done by considering cones with bounded and unbounded bases and the second characterization is by reflexive cones. The relationship between cones with bounded and unbounded bases and reflexive cones is also considered. We provide an example to show distinction between such cones. / Dissertation (MSc (Mathematics))--University of Pretoria 2020. / UCDP - 523 / Mathematics and Applied Mathematics / MSc (Mathematics) / Unrestricted

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published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Tsirelson space is a reflexive, separable Banach space constructed by Boris Tsirelson to be the first example of a Banach space that does not contain an isomorphic copy of the sequence spaces c_0 or l_p for 1\leqslant p

Programme in Advanced Mathematics of Finance, University of the Witwatersrand, Johannesburg. / Peng introduced a typical ltration consistent nonlinear expectation, called a g-expectation in [40]. It satis es all properties of the classical mathematical expectation besides the linearity. Peng's conditional g-expectation is a solution to a backward stochastic di erential equation (BSDE) within the classical framework of It^o's calculus, with terminal condition given at some xed time T. In addition, this g-expectation is uniquely speci ed by a real function g satisfying certain properties. Many properties of the g-expectation, which will be presented, follow from the speci cation of this function. Martingales, super- and submartingales have been de ned in the nonlinear setting of g-expectations. Consequently, a nonlinear Doob-Meyer decomposition theorem was proved. Applications of g-expectations in the mathematical nancial world have also been of great interest. g-Expectations have been applied to the pricing of contingent claims in the nancial market, as well as to risk measures. Risk measures were introduced to quantify the riskiness of any nancial position. They also give an indication as to which positions carry an acceptable amount of risk and which positions do not. Coherent risk measures and convex risk measures will be examined. These risk measures were extended into a nonlinear setting using the g-expectation. In many cases due to intermediate cash ows, we want to work with a multi-period, dynamic risk measure. Conditional g-expectations were then used to extend dynamic risk measures into the nonlinear setting. The Choquet expectation, introduced by Gustave Choquet, is another nonlinear expectation. An interesting question which is examined, is whether there are incidences when the g-expectation and the Choquet expectation coincide.

This dissertation follows the history of functional ideas and their pedagogy, illuminates with many examples the implementation of my updated system of Functional Analysis, and discusses the pedagogical implications that this updated system implies. The main goal is to update a system of labeling to be as pedagogically friendly as possible, in order to assist students and teachers of harmony to more easily and enjoyably learn, teach, and engage with common-practice tonal harmonic practice. Example syllabi, assignments, classroom demonstrations, and long projects are also included, and each aspect of the labeling is carefully discussed as it is presented. By surveying the history of functional thinking in music theory, we find that desire to analyze for function is not a new idea, and has been a goal of many theorists and harmony teachers for centuries. However, the current methods for instructing in function still leave students confused or baffled, as they struggle to match functional concepts to labels that do not exemplify their analysis goals and methods that insist on starting from tiny detail instead of coming from a more complete musical perspective. The elaboration of each detail of my Functional Analysis system shows how each part of Functional Analysis has been designed to help make harmonic analysis quicker, easier, more intuitive, and more personalized. The greater pedagogical implications on a larger scale involving courses and curricula are also covered, informed by my experience both as a teacher of today’s standard system and from teaching Functional Analysis in the classroom.

本文研究了多变量系统的非线性输出调节问题，并处理了机电和机械控制系统中的两个典型控制问题。 / 输出调节问题是控制中的核心问题之一。它同时处理轨迹跟踪和干扰抑制问题。参考输入和干扰是通过一定的动态系统产生，称为外部系统。因此输出调节问题和其它轨迹跟踪问题的区别在于它允许跟踪一类参考信号，而不需要精确的知道参考信号的值。而且，输出调节问题可应用于真实世界的很多控制问题，包括能源系统，交通和系统生物学。 / 众所周知，处理输出调节问题的基本框架包含两个步骤。首先把被控对象的鲁棒输 出调节问题转化成由被控对象和叫做内模的动态补偿器组成的增广系统的鲁棒镇定问题，然后鲁棒镇定增广系统。问题的关键在于如何设计一个合适的内模。这个内模不仅可以产生所有的稳态信息，而且使得增广系统可镇定。 / 虽然在过去的二十几年里关于非线性输出调节问题的研究取得了重大进展，大部分的结果只处理了单输入单输出的非线性系统。多变量非线性系统非线性输出调节问题有关的结果甚少。两大难点阻碍了多变量非线性系统非线性输出调节问题的进一步发展。一方面，由于被控对象的复杂结构，很难构造出一个合适的内模。这个内模既可以产生必需的稳态信息，又可以保证增广系统的镇定问题可解。另一方面，由于内模的引入，增广系统变成一个更加复杂的多输入多输出的非线性系统，其中既包含了动态不确定性，也包含了时变的静态不确定性。这个增广系统具有一定特殊的结构，并且从未出现在现有的文献中。增广系统的镇定问题很有挑战性。此外，关于实际系统非线性输出调节问题的研究甚少。因为输出调节方法对于未知的对象参数具有一定的鲁棒性，而且允许外部系统含有未知参数，通过输出调节方法我们可能得出更好的结果。基于以上的动机，我们将考虑具有特定结构的多变量系统的全局鲁棒输出调节问题，并解决两个典型控制问题，包括永磁同步电机的速度跟踪控制和球形倒立摆的轨迹跟踪控制。本文的贡献总结如下。 / 1. 本文考虑了一类多变量非线性系统的全周鲁棒输出调节问题。通过把内模附加到被控对象， 增广系统具有」定特殊的结构。它的镇定问题尚未有人处理过。我们把增广系统的镇定问题分解为多个单输入系统的镇定问题。然后利用changingsupply function 技术，我们通过法代的方法求解多个单输入控制问题，从而解决了这一镇定问题。理论结果可以解决面装式永磁同步电机的速度跟踪和负载干扰抑制问题。与现有的结果比较，我们的设计允许所有的电机参数未知。 / 2. 本文考虑了未知外部系统作用下的面装式永磁同步电机的速度跟踪和负载干扰抑制问题。由于外部系统未知，传统的鲁棒控制方法不能处理这一问题。而且，外部系统中的未知参数使得增广系统的镇定问题更具挑战性。因此我们需要把鲁棒控制和自适应控制相结合处理这一问题。利用changing supply function 技术和动态坐标变换技术相结合，我们用迭代的方法求解了两个单输入控制问题，进而解决了这一问题。 / 3. 本文考虑了用内模设计的方法处理永磁同步电机的速度跟踪和负载干扰抑制问题。当dq 轴电感相等时，这一问题己经得到了广泛的研究。然而dq 轴电感不等时，由于转速方程中的复杂非线性，这一问题的研究甚少。利用输出调节方法，电机参数可以具有更好的鲁棒性。由于增广系统不具有任何己知的特殊结构，我们研究出一种具体的工具来处理镇定问题。具体的说，我们研究出一种适用于non-ISS (输入到状态稳定)系统的广义changing supply function 技术。通过把这个技术与一个特殊的非线性内模相结合，这一问题得到了很好的解决。 / 4. 本文给出了一种新的方法用于处理球形倒立摆的近似输出调节问题。这一方法基于神经网络近似调节器方程的解。求解调节器方程近似解的问题被转化成一个参数优化问题，进而有助于计算机实现的方便。而且，与现有结果中的基于三阶多项式的控制比较，对于参考输入信号的不同幅值的情况，这一方法的稳态跟踪误差保持了更加平稳的性能。 / 5. 本文给出了一种新的方法用于处理球形倒立摆的近似输出调节问题。这一方法是基于神经网络的增强型设计，并采用输出反馈控制。与我们之前工作中的神经网络设计比较，我们给出了一种新的方法用于近似前馈控制，而不是调节器方程的解。求解前馈控制近似解的问题被转化成一个参数优化问题。我们采用Hooke-J eeeves 方法和Powell 方法解决了参数优化问题。当所有的状态可用的时候，我们的设计对于被控对象参数的变化具有一定的鲁棒性。 / In this thesis, we investigate nonlinear output regulation problem of multivariable systems and their applications to two typical control problems in electromechanical and mechanical control systems. / The output regulation problem is one of the central problems in control. It deals with trajectory tracking and disturbance rejection problem simultaneously. Both the reference inputs and disturbances are generated by some dynamical system called exosystem. Thus it is in contrast with some other traj ectory tracking problems in that it allows tracking of a family of reference signals which does not have to be known exactly. Moreover, it can be applied to many control problems in real world including energy systems, transportation and system biology. / It is well known that the general framework for tackling the output regulation problem consists of two steps. The first step is to convert the robust output regulation problem for the given plant into a robust stabilization problem for the so-called augmented system composed of the given plant and a dynamic compensator called internal model, and the second step aims to robustly stabilize the augmented system. The key issue lies in how to design a suitable internal model which can generate all the steady-state information and render the augmented system to be stabilizable. / Although the research on the nonlinear output regulation has made significant progress for over two decades, most of the results only deals with single-input, single-output nonlinear systems. There are few results on nonlinear output regulation of multivariable nonlinear systems. Two challenges hinder the further development of nonlinear output regulation of multivariable nonlinear systems. On the one hand, due to the complicated structure of the system, it is difficult to construct a suitable internal model which can generate the necessary steady-state information while ensuring the solvability of stabilization problem of the augmented system. On the other hand, due to the introduction of the internal model, the augmented system becomes a more complicated multi-input, multioutput nonlinear system containing both dynamic uncertainty and time-varying static uncertainty. Such augmented system takes certain special structure which has never been encountered before. The stabilization problem of such augmented system is very challenging. Moreover, t he research on the nonlinear output regulation of practical systems is very rare. Because output regulation approach offers certain robust property to unknown plant parameters and also allows unknown parameters in exosystem, better results may be achieved by using output regulation approach. Therefore we are motivated to consider global robust output regulation problem of multivariable systems with certain structures and solve two typical control problems including speed tracking control of permanent magnet synchronous motor and trajectory tracking control of spherical invert d pendulum. The main contributions of the thesis are summarized as follows. / 1. The global robust output regulation problem for a class of multivariable nonlinear systems is considered. By attaching the internal model to the given plant, the augmented system takes certain special structure and the stabilization problem of such a system has never been handled. We decompose the stabilization problem of the augmented system into the stabilization problem of several single-input systems. Then we solve the problem by solving several single-input control problems via a recursive approach utilizing the changing supply function technique. The theoretical result is applied to the speed tracking control and load torque disturbance rejection problem of a surface-mounted PM synchronous motor. Compared with existing results, our design allows all the motor parameters to be uncertain. / 2. A speed tracking and load torque disturbance rejection problem of surface-mounted PM synchronous motor subject to an unknown exosystem is considered. Because the exosystem is unknown, traditional robust control approach cannot be used to tackle this problem. Moreover, the unknown parameters in the exosystem make the stabilization problem of augmented system even more challenging. Therefore we need to incorporate robust control with adaptive control to t ackle the problem. We solve the problem by solving two single-input control problems via a recursive approach utilizing the changing supply function and dynamic coordinate transformation techniques. / 3. A speed tracking and load torque disturbance rejection problem of PM synchronous motor by internal model design is considered. When dq axes inductances are equal, the problem has been extensively studied. However, when dq axes inductances are not equal, such problem has received little attention for the complicated nonlinearity in speed equation. By using output regulation approach, a better robust property with respect to motor parameters can be achieved. As the augmented system does not take any known special form, we develop a specific tool to deal with the stabilization problem. In particular, a generalized changing supply function technique applicable to non-ISS (input-to-state stable) systems is developed. This technique, in conjunction with a particular nonlinear internal model leads to an effective solution to the problem. / 4. An alternative approach to the approximate output regulation problem of spherical inverted pendulum is considered. This method is based on a neural network approximation of the solution of the regulator equations. The problem of seeking approximate solution of the regulator equations has been converted into a parameter optimization problem which has lent itself to a convenient implementation in computer. Moreover, compared with a third order polynomial based control in existing results, the steady-state tracking errors of the method maintains a more uniform performance over different amplitudes of the reference input signals. / 5. An alternative approach to the approximate output regulation problem of spherical inverted pendulum is considered. This method has been based on the neural network enhanced design with output feedb ack control. Different from the neural network design in our previous work, we present a novel method to approximate the feedforward control instead of the solution of regulator equations. The problem of seeking approximate solutions of feedforward control is converted into a parameter optimization problem. We solve the parameter optimization problem by both Hooke-Jeeves method and Powell method. When all the states are available, our design also offers certain robustness to plant parameter variations. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Ping, Zhaowu. / "October 2011." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 160-170). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Abstract --- p.i / Acknowledgement --- p.VI / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Literature Overview --- p.1 / Chapter 1.2 --- Contribution of Thesis --- p.7 / Chapter 1.3 --- Organization of Thesis --- p.9 / Chapter 2 --- Background and Preliminaries --- p.10 / Chapter 2.1 --- Fundamentals of Nonlinear Systems --- p.10 / Chapter 2.1.1 --- Lyapunov Stability --- p.11 / Chapter 2.1.2 --- Input-to-state Stability --- p.13 / Chapter 2.2 --- Framework of Robust Output Regulation --- p.15 / Chapter 2.3 --- Stabilization of Nonlinear Systems --- p.19 / Chapter 2.3.1 --- Technical Lemmas for ISS Systems --- p.20 / Chapter 2.3.2 --- Technical Lemmas for Non-ISS Systems --- p.23 / Chapter 2.4 --- Feedforward Control of Nonlinear Output Regulation --- p.28 / Chapter 2.4.1 --- Polynomial Based Approximation Method --- p.31 / Chapter 2.4.2 --- Neural Network Based Approximation Method --- p.32 / Chapter 3 --- Global Robust Output Regulation for a Class of Multivariable Systems --- p.39 / Chapter 3.1 --- Introduction --- p.40 / Chapter 3.2 --- Preliminaries --- p.42 / Chapter 3.3 --- Main Results --- p.48 / Chapter 3.4 --- Application to Speed Control of Surface-Mounted PM Synchronous Motor --- p.56 / Chapter 3.5 --- Conclusion --- p.71 / Chapter 4 --- Speed Tracking Control of Surface-Mounted PM Synchronous Motor Subject to an Unknown Exosystem --- p.72 / Chapter 4.1 --- Problem Formulation and Preliminaries --- p.73 / Chapter 4.2 --- Solvability of The Motor Control Problem --- p.79 / Chapter 4.3 --- Evaluation of The Control Law --- p.87 / Chapter 4.4 --- Conclusion --- p.99 / Chapter 5 --- Speed Tracking Control of PM Synchronous Motor by Internal Model Design --- p.100 / Chapter 5.1 --- Problem Formulation and Preliminaries --- p.101 / Chapter 5.2 --- Solvability of the Motor Control Problem --- p.107 / Chapter 5.3 --- Evaluation of the Control Law --- p.110 / Chapter 5.4 --- Conclusion --- p.119 / Chapter 6 --- Approximate Output Regulation of Spherical Inverted Pendulum by Neural Network Control --- p.120 / Chapter 6.1 --- Introduction --- p.121 / Chapter 6.2 --- Neural Network Based Approximate Solution of the Regulator Equations --- p.124 / Chapter 6.3 --- Approximate Control Law --- p.131 / Chapter 6.4 --- Conclusion --- p.135 / Chapter 7 --- Neural Network Enhanced Output Regulation of Spherical Inverted Pendulum --- p.136 / Chapter 7.1 --- Neural Network Based Output Feedback Control Law --- p.137 / Chapter 7.1.1 --- Hooke-J eeves Method --- p.139 / Chapter 7.1.2 --- Powell Method --- p.143 / Chapter 7.2 --- Robustness Analysis --- p.149 / Chapter 7.2.1 --- Hooke-Jeeves Method --- p.149 / Chapter 7.2.2 --- Powell Method --- p.153 / Chapter 7.3 --- Conclusion --- p.156 / Chapter 8 --- Conclusions --- p.157 / Bibliography --- p.160 / Biography --- p.171

Typescript Includes bibliographical references

Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaves 187-110). Also available on the Internet.