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This thesis presents the integration of reproducing kernel Hilbert spaces (RKHSs) into the model reference adaptive control (MRAC) framework to enhance the control systems of quadcopters. Traditional MRAC systems, while robust under predictable conditions, can struggle with the dynamic uncertainties typical in unmanned aerial vehicle (UAV) operations such as wind gusts and payload variations. By incorporating RKHS, we introduce a non-parametric, data-driven approach that significantly enhances system adaptability to in-flight dynamics changes. The research focuses on the design, simulation, and analysis of an RKHS-enhanced MRAC system applied to quadcopters. Through theoretical developments and simulation results, the thesis demonstrates how RKHS can be used to improve the precision, adaptability, and error handling of MRAC systems, especially in managing the complexities of UAV flight dynamics under various disturbances. The simulations validate the improved performance of the RKHS-MRAC system compared to traditional MRAC, showing finer control over trajectory tracking and adaptive gains. Further contributions of this work include the exploration of the computational impact and the relationship between the configuration of basis centers and system performance. Detailed analysis reveals that the number and distribution of basis centers critically influence the system's computational efficiency and adaptive capability, demonstrating a significant trade-off between efficiency and performance. The thesis concludes with potential future research directions, emphasizing the need for further tests and implementations in real-world scenarios to explore the full potential of RKHS in adaptive UAV control, especially in critical applications requiring high precision and reliability. This work lays the groundwork for future explorations into scalable RKHS applications in MRAC systems, aiming to optimize computational resources while maximizing control system performance. / Master of Science / This thesis develops and tests an advanced flight control system for quadcopters, using a technique referred to as reproducing kernel Hilbert space (RKHS) embedded model reference adaptive control (MRAC). Traditional control systems perform well in stable conditions but often falter with environmental challenges such as wind gusts or changes in weight. By integrating RKHS into MRAC, this new controller adapts in real-time, instantly adjusting the drone's operations based on its performance and environmental interactions. The focus of this research is on the creation, testing, and analysis of this enhanced control system. Results from simulations show that incorporating RKHS into standard MRAC significantly boosts precision, adaptability, and error management, particularly under the complex flight dynamics faced by unmanned aerial vehicles (UAVs) in varied environments. These tests confirm that the RKHS-MRAC system performs better than traditional approaches, especially in maintaining accurate flight paths. Additionally, this work examines the computational costs and the impact of various RKHS configurations on system performance. The thesis concludes by outlining future research opportunities, stressing the importance of real-world tests to verify the ability of RKHS-embedded MRAC in critical real-world applications where high precision and reliability are essential.

Natural streams typically have highly complex flow patterns. Velocity gradients, circulation zones, transverse flows, and other flow patterns are created in the presence of topographic features (e.g. exposed boulders, bars). How flow complexity influences a stream's ecological health and morphological stability, as well as how flow complexity responds to changes in hydrologic conditions, is poorly understood. One-dimensional (1-D) hydraulic models and two-dimensional (2-D) models that do not explicitly incorporate meso-scale topographic features are not capable of adequately reproducing the flow patterns found in channels having complex topography. Moreover, point measurements of depth and velocity, which are used to describe hydraulic conditions in habitat suitability studies, cannot be used to characterize spatially varying flow patterns of biological importance. A general methodology for incorporating meso-scale topography into 2-D hydraulic models is presented. The method provides a means of adequately reproducing spatial flows of interest to riverine researchers. The method is developed using 2-D model simulations of a reach of the North Fork of the Feather River in California. Specifically, the site is modeled with and without bathymetry data on exposed boulders found within the site. Results show that the incorporation of boulder topography and an adequately refined mesh are necessary for reproducing velocity gradients, transverse flows, and other spatial flows. These simulations are also used to develop and evaluate three spatial hydraulic metrics designed to distinguish between locations having uniform and non-uniform flow conditions. The first two metrics describe local variations in energy/velocity gradients, while the third metric provides a measure of the flow complexity occurring within an arbitrary area. The metrics based on principles of fluid mechanics (kinetic energy, vorticity, and circulation) can be computed in the field or with 2-D hydraulic model results. These three metrics, used in conjunction with detailed 2-D hydraulic model results, provide engineers, biologist, and water resource managers a set of tools with which to evaluate the importance of flow complexity within rivers. A conceptual model describing how such a tool can be used to help design channels being restored, better evaluate stream habitat, and evaluate how hydrologic changes in a watershed impact hydraulic conditions and concomitant habitat conditions is provided. / Ph. D.

We present important results from Hilbert space and functional analysis for understanding the subject ofReproducing kernel Hilbert spaces. We then showcase the underlying theory and properties of Reproducingkernel Hilbert Spaces. Finally, we show how the theory of reproducing kernel Hilbert spaces is applicable inboth interpolation and machine learning.

Thesis (M.Mus.)--University of Stellenbosch, 2006. / ENGLISH ABSTRACT: In South Africa there has been an increase over the last few years in audio engineering courses which include modules of study in audio post-production or even offer audio post-production as a major focus of study. From an academic standpoint however, and despite the growth in the local film industry, very little study of this field has been undertaken in South Africa until recently. In 2005, a MMus thesis was submitted at the University of KwaZulu-Natal entitled Acoustic Ambience in Cinematography: An Exploration of the Descriptive and Emotional Impact of the Aural Environment (Turner, 2005: online). The thesis briefly outlines the basic components of the soundtrack and focuses on describing and analysing the properties of ambience, a sub-section of sound effects. At Stellenbosch University, research has recently begun in the fields of film music and Foley (sound effects associated with human movement onscreen). The purpose of this thesis is to provide an overview of audio post-production and the contribution of sound to the film medium. It provides an outline of the processes involved in creating a soundtrack for film and includes a description of the components of the soundtrack and recommendations for practical application. / AFRIKAANSE OPSOMMING: Gedurende die afgelope paar jaar was daar ‘n toename in oudio-ingenieurskursusse, insluitend studiemodules in oudio post-produksie, en selfs ‘n aanbod vir modules in post-produksie as hoofstudierigting. Desnieteenstaande, en ten spyte van die groei in die plaaslike filmindustrie is tot onlangs min akademiese studies op dié terrein in Suid-Afrika onderneem. In 2005 is ‘n MMus-tesis aan die Universiteit van KwaZulu-Natal voorgelê, met die titel Acoustic Ambience in Cinematography: An Exploration of the Descriptive and Emotional Impact of the Aural Environment (Turner, 2005: aanlyn). Hierdie tesis gee ‘n basiese oorsig oor die basiese komponente van die klankbaan, en fokus op die beskrywing en analise van die eienskappe van ambience – ‘n onderafdeling van klankeffekte. By die Universiteit van Stellenbosch is onlangs ‘n begin gemaak met navorsing oor die terreine van filmmusiek en Foley, d.w.s. klankeffekte geassosieer met menslike bewegings op die skerm.. Hierdie tesis beoog om ‘n oorsig te gee van oudio post-produksie en die bydrae van klank tot die filmmedium. Dit verskaf ‘n oorsig oor die prosesse betrokke by die daarstelling van ‘n filmklankbaan en sluit ook in ‘n beskrywing van die komponente van die klankbaan en aanbevelings vir die praktiese toepassing daarvan.

This thesis aims to contribute to the area of kernel methods, which are a class of machine learning methods known for their wide applicability and state-of-the-art performance, but which suffer from high training and evaluation complexity. The work in this thesis utilizes the notion of reduced-set models to alleviate the training and testing complexities of these methods in a unified manner. In the first part of the thesis, we use recent results in kernel smoothing and integral-operator learning to design a generic strategy to speed up various kernel methods. In Chapter 3, we present a method to speed up kernel PCA (KPCA), which is one of the fundamental kernel methods for manifold learning, by using reduced-set density estimates (RSDE) of the data. The proposed method induces an integral operator that is an approximation of the ideal integral operator associated to KPCA. It is shown that the error between the ideal and approximate integral operators is related to the error between the ideal and approximate kernel density estimates of the data. In Chapter 4, we derive similar approximation algorithms for Gaussian process regression, diffusion maps, and kernel embeddings of conditional distributions. In the second part of the thesis, we use reduced-set models for kernel methods to tackle online learning in model-reference adaptive control (MRAC). In Chapter 5, we relate the properties of the feature spaces induced by Mercer kernels to make a connection between persistency-of-excitation and the budgeted placement of kernels to minimize tracking and modeling error. In Chapter 6, we use a Gaussian process (GP) formulation of the modeling error to accommodate a larger class of errors, and design a reduced-set algorithm to learn a GP model of the modeling error. Proofs of stability for all the algorithms are presented, and simulation results on a challenging control problem validate the methods.

Sampling theory is an active field of research that spans a variety of disciplines from communication engineering to pure mathematics. Sampling theory provides the crucial connection between continuous and discrete representations of information that enables one store continuous signals as discrete, digital data with minimal error. It is this connection that allows communication engineers to realize many of our modern digital technologies including cell phones and compact disc players. This thesis focuses on certain non-Fourier generalizations of sampling theory and their applications. In particular, non-Fourier analogues of bandlimited functions and extensions of sampling theory to functions on curved manifolds are studied. New results in bandlimited function theory, sampling theory on curved manifolds, and the theory of self-adjoint extensions of symmetric operators are presented. Besides being of mathematical interest in itself, the research contained in this thesis has applications to quantum physics on curved space and could potentially lead to more efficient information storage methods in communication engineering.

Sampling theory is an active field of research that spans a variety of disciplines from communication engineering to pure mathematics. Sampling theory provides the crucial connection between continuous and discrete representations of information that enables one store continuous signals as discrete, digital data with minimal error. It is this connection that allows communication engineers to realize many of our modern digital technologies including cell phones and compact disc players. This thesis focuses on certain non-Fourier generalizations of sampling theory and their applications. In particular, non-Fourier analogues of bandlimited functions and extensions of sampling theory to functions on curved manifolds are studied. New results in bandlimited function theory, sampling theory on curved manifolds, and the theory of self-adjoint extensions of symmetric operators are presented. Besides being of mathematical interest in itself, the research contained in this thesis has applications to quantum physics on curved space and could potentially lead to more efficient information storage methods in communication engineering.