Дисертації з теми "Modulating functions method"

Cette thèse développe la méthode des fonctions modulatrices pour des estimations non-asymptotiques et robustes pour des pseudo-états des systèmes nonlinéaires d'ordre fractionnaire, des systèmes linéaires d'ordre fractionnaire avec des accélérations en sortie, et des systèmes à retards d'ordre fractionnaire. Les estimateurs conçus sont fournis en termes de formules intégrales algébriques, ce qui assure une convergence non-asymptotique. Comme une caractéristique essentielle des algorithmes d'estimation conçus, les mesures de sorties bruitées ne sont impliquées que dans les termes intégraux, ce qui confère aux estimateurs une robustesse contre les bruits. Premièrement, pour les systèmes nonlinéaires d'ordre fractionnaire et partiellement inconnu, l'estimation de la dérivée fractionnaire du pseudo-état est abordée via la méthode des fonctions modulatrices. Grâce à la loi de l'indice additif des dérivées fractionnaires, l'estimation est décomposée en une estimation des dérivées fractionnaires de la sortie et une estimation des valeurs initiales fractionnaires. Pendant ce temps, la partie inconnue est estimée via une stratégie innovante de fenêtre glissante. Deuxièmement, pour les systèmes linéaires d'ordre fractionnaire avec des accélérations comme sortie, l'estimation de l'intégrale fractionnaire de l'accélération est d'abord considérée pour les systèmes mécaniques de vibration d'ordre fractionnaire, où seules des mesures d'accélération bruitées sont disponibles. Basée sur des approches numériques existantes qui traitent des intégrales fractionnaires, notre attention se limite principalement à l'estimation des valeurs initiales inconnues en utilisant la méthode des fonctions modulatrices. Sur cette base, le résultat est ensuite généralisé aux systèmes linéaires plus généraux d'ordre fractionnaire. En particulier, le comportement des dérivées fractionnaires à zéro est étudié pour des fonctions absolument continues, ce qui est assez différent de celui de l'ordre entier. Troisièment, pour les systèmes à retards d'ordre fractionnaire, l'estimation du pseudo-état est étudiée en concevant un système dynamique auxiliaire d'ordre fractionnaire, qui fournit un cadre plus général pour générer les fonctions modulatrices requises. Avec l'introduction de l'opérateur de retard et du changement de coordonnées généralisé bicausal, l'estimation du pseudo-état du système considéré peut être réduite à celle de la forme normale correspondante. Contrairement aux travaux précédents le schéma présenté permet une estimation directe du pseudo-état plutôt que d'estimer les dérivées fractionnaires de la sortie et un ensemble de valeurs initiales fractionnaires. De plus, l'efficacité et la robustesse des estimateurs proposés sont vérifiées par des simulations numériques dans cette thèse. Enfin, un résumé de ce travail et un aperçu des travaux futurs sont tirés
This thesis develops the modulating functions method for non-asymptotic and robust estimations for fractional-order nonlinear systems, fractional-order linear systems with accelerations as output, and fractional-order time-delay systems. The designed estimators are provided in terms of algebraic integral formulas, which ensure non-asymptotic convergence. As an essential feature of the designed estimation algorithms, noisy output measurements are only involved in integral terms, which endows the estimators with robustness against corrupting noises. First, for fractional-order nonlinear systems which are partially unknown, fractional derivative estimation of the pseudo-state is addressed via the modulating functions method. Thanks to the additive index law of fractional derivatives, the estimation is decomposed into the fractional derivatives estimation of the output and the fractional initial values estimation. Meanwhile, the unknown part is fitted via an innovative sliding window strategy. Second, for fractional-order linear systems with accelerations as output, fractional integral estimation of the acceleration is firstly considered for fractional-order mechanical vibration systems, where only noisy acceleration measurements are available. Based on the existing numerical approaches addressing the proper fractional integrals of accelerations, our attention is primarily restricted to estimating the unknown initial values using the modulating functions method. On this basis, the result is further generalized to more general fractional-order linear systems. In particular, the behaviour of fractional derivatives at zero is studied for absolutely continuous functions, which is quite different from that of integer order. Third, for fractional-order time-delay systems, pseudo-state estimation is studied by designing a fractional-order auxiliary modulating dynamical system, which provides a more general framework for generating the required modulating functions. With the introduction of the delay operator and the bicausal generalized change of coordinates, the pseudo-state estimation of the considered system can be reduced to that of the corresponding observer normal form. In contrast to the previous work, the presented scheme enables direct estimation for the pseudo-state rather than estimating the fractional derivatives of the output and a bunch of fractional initial values. In addition, the efficiency and robustness of the proposed estimators are verified by numerical simulations in this thesis. Finally, a summary of this work and an insight into future work were drawn

This study develops a measure which allows visual and quantitative assessment of bass reproduction accuracy in professional studio monitors. This type of loudspeaker must present mix engineers in particular with a faithful impression of recordings; they can then create an optimum musical balance between instruments that will translate well to other reproduction systems. Inaccurate monitors can lead to expensive remixing or a degraded musical experience for the end consumers, especially if the fundamental rhythm section has been poorly adjusted. Existing work suggested that the Modulation Transfer Function (MTF) might be a more informative descriptor of bass reproduction accuracy than typical steady-state measures; it might therefore provide a grading system of performance that engineers could use when selecting suitable monitors for their work. The purpose of this project was to investigate the technique and i) develop an algorithm to summarise the critical aspects of monitor performance at low frequencies and ii) see whether it predicted the subjective impression of reproduction accuracy. An algorithm was developed, considering different calculation methods and parameters to optimise it for low-frequency application with musical signals. It was applied to groups of loudspeaker models, simulating the responses of real mix monitors; then listening tests were conducted with these models reproducing music. The subjective and objective results were compared to see whether the algorithm would be a useful measure of monitor performance. The algorithm successfully summarised behaviour of simulated and measured monitor responses; it described important factors such as extension and smoothness, and how the system responded to temporally-varying input signals. Based on ordinal data, the algorithm was found to predict all statistically significant judgements from listeners. These participants had demonstrated that they were accurate and consistent listeners, but found it difficult to reach consensus in some evaluations where the listening task required more skilled judgements of overall performance. It was concluded that the algorithm in its current form is effective and suitable for the intended application, but subjective evaluations of more complex alignments are needed from professional mix engineers; this might allow the numerical MTF scores to be graded against perceived bass reproduction accuracy, therefore enhancing the predictive power of the technique.

This thesis consists of two parts. In the first chapter, we give some results on modulation spaces. First the relationship between the classical spaces and the modulation spaces is established. It is proved that certain modulation spaces defined on R² lie in the BMO space. Another result is that the Zak transform, a discrete time-frequency transform, maps a modulation space into a higher dimensional modulation space. And by using these results, an uncertainty principle for Gabor frames via modulation spaces is obtained. In the second part, we deal with optimization of an increasing positively homogeneous functions on the unit simplex. The class of increasing positively homogeneous functions is one of the function classes obtained via min-type functions in the context of abstract convexity. The cutting angle method is used for the minimization of this type functions. The most important step of this method is the minimization of a function which is the maximum of a number of min-type functions on the unit simplex. We propose a numerical algorithm for the minimization of such functions on the unit simplex and we mathematically prove that this algorithm finds the exact solution of the minimization problem. Some experiments have been carried out and the results of the experiments have been presented.

Orthogonal frequency division multiplexing (OFDM) has become a very popular method for high-data-rate communication. However, it is well known that OFDM is plagued by a large peak-to-average power ratio (PAPR) problem. This high PAPR results in overdesigned power amplifiers, which amongst other things leads to inefficient amplifier usage, which is undesirable. Various methods have been recommended to reduce the PAPR of an OFDM transmission; however, all these methods result in a number of drawbacks. In this thesis, a novel method called offset modulation (OM-OFDM) is proposed to control the PAPR of an OFDM signal. The proposed OM-OFDM method does not result in a number of the drawbacks being experienced by current methods in the field. The theoretical bandwidth occupancy and theoretical bit error rate (BER) expression for an OM-OFDM transmission is derived. A newly applied power performance decision metric is also introduced, which can be utilised throughout the PAPR field, in order to compare various methods. The proposed OM-OFDM method appears to be similar to a well-known constant envelope OFDM (CE-OFDM) transmission. The modulation, structural and performance differences between an OM-OFDM and a CE-OFDM method are discussed. By applying the power performance decision metric, the OM-OFDM method is shown to offer significant performance gains when compared to CE-OFDM and traditional OFDM transmissions. In addition, the OM-OFDM method is able to accurately control the PAPR of a transmission for a targeted BER. By applying the power performance decision metric and complementary cumulative distribution function (CCDF), the proposed OM-OFDM method is shown to offer further performance gains when compared to existing PAPR methods, under frequency selective fading conditions. In this thesis, the OM-OFDM method has been combined with an existing active constellation extended (ACE) PAPR reduction method. To introduce a novel method called offset modulation with active constellation extension (OM-ACE), to control the PAPR of an OFDM signal. The theoretical BER expression for an OM-ACE transmission is presented and validated. Thereafter, by applying the decision metric and CCDF, the OM-ACE method is shown to offer performance improvements when compared to various PAPR methods. The use of OM-OFDM for cognitive radio applications is also investigated. Cognitive radio applications require transmissions that are easily detectable. The detection characteristics of an OM-OFDM and OFDM transmission are studied by using receiver operating characteristic curves. A derivation of a simplified theoretical closed-form expression, which relates the probability of a missed detection to the probability of a false alarm, for an unknown deterministic signal, at various signal-to-noise ratio (SNR) values is derived and validated. Previous expressions have been derived, which relate the probability of a missed detection to the probability of a false alarm. However, they have not been presented in such a generic closed-form expression that can be used for any unknown deterministic signal (for instance OFDM and OM-OFDM). Thereafter, an examination of the spectrum characteristics of an OM-OFDM transmission indicates its attractive detection characteristics. The proposed OM-OFDM method is further shown to operate at a significantly lower SNR value than an OFDM transmission, while still offering better detection characteristics than that of an OFDM transmission under Rician, Rayleigh and frequency selective fading channel conditions. In addition to its attractive PAPR properties, OM-OFDM also offers good detection characteristics for cognitive radio applications. These aspects make OM-OFDM a promising candidate for future deployment.
Thesis (PhD)--University of Pretoria, 2012.
Electrical, Electronic and Computer Engineering
unrestricted

This thesis investigates advanced control methods and strategies to enhance the transient stability of a power system which is affected by the electromechanical wave propagation phenomenon. The characteristics of the travelling wave phenomenon and its impact on system separation during first swing are investigated in detail. This thesis identifies four innovative control implications for power system stability and proposes an improved method to enhance power system transient stability. By designing better control of existing equipment, this research achieves increased maximum power transfer under contingencies in a cost efficient and readily implemented way.

The thesis consists of an introduction and four appended papers. In the introduction we give an overview of pulse-modulated systems and provide a few examples of such systems. Furthermore, we introduce the so-called dynamic phasor model which is used as a basis for analysis in two of the appended papers. We also introduce the harmonic transfer function and finally we provide a summary of the appended papers. The first paper considers stability analysis of a class of pulse-width modulated systems based on a discrete time model. The systems considered typically have periodic solutions. Stability of a periodic solution is equivalent to stability of a fixed point of a discrete time model of the system dynamics. Conditions for global and local exponential stability of the discrete time model are derived using quadratic and piecewise quadratic Lyapunov functions. A griding procedure is used to develop a systematic method to search for the Lyapunov functions. The second paper considers the dynamic phasor model as a tool for stability analysis of a general class of pulse-modulated systems. The analysis covers both linear time periodic systems and systems where the pulse modulation is controlled by feedback. The dynamic phasor model provides an $\textbf{L}_2$-equivalent description of the system dynamics in terms of an infinite dimensional dynamic system. The infinite dimensional phasor system is approximated via a skew truncation. The truncated system is used to derive a systematic method to compute time periodic quadratic Lyapunov functions. The third paper considers the dynamic phasor model as a tool for harmonic analysis of a class of pulse-width modulated systems. The analysis covers both linear time periodic systems and non-periodic systems where the switching is controlled by feedback. As in the second paper of the thesis, we represent the switching system using the L_2-equivalent infinite dimensional system provided by the phasor model. It is shown that there is a connection between the dynamic phasor model and the harmonic transfer function of a linear time periodic system and this connection is used to extend the notion of harmonic transfer function to describe periodic solutions of non-periodic systems. The infinite dimensional phasor system is approximated via a square truncation. We assume that the response of the truncated system to a periodic disturbance is also periodic and we consider the corresponding harmonic balance equations. An approximate solution of these equations is stated in terms of a harmonic transfer function which is analogous to the harmonic transfer function of a linear time periodic system. The aforementioned assumption is proved to hold for small disturbances by proving the existence of a solution to a fixed point equation. The proof implies that for small disturbances, the approximation is good. Finally, the fourth paper considers control synthesis for switched mode DC-DC converters. The synthesis is based on a sampled data model of the system dynamics. The sampled data model gives an exact description of the converter state at the switching instances, but also includes a lifted signal which represents the inter-sampling behavior. Within the sampled data framework we consider H-infinity control design to achieve robustness to disturbances and load variations. The suggested controller is applied to two benchmark examples; a step-down and a step-up converter. Performance is verified in both simulations and in experiments.
QC 20100628

Muchos tipos de señales que encontramos a diario pertenecen a la categoría de sinusoides no estacionarias. Una gran parte de esas señales son sonidos que presentan una gran variedad de características: acústicos/electrónicos, sonidos instrumentales harmónicos/impulsivos, habla/canto, y la mezcla de todos ellos que podemos encontrar en la música. Durante décadas la comunidad científica ha estudiado y analizado ese tipo de señales. El motivo principal es la gran utilidad de los avances científicos en una gran variedad de áreas, desde aplicaciones médicas, financiera y ópticas, a procesado de radares o sonar, y también a análisis de sistemas. La estimación precisa de los parámetros de sinusoides no estacionarias es una de las tareas más comunes en procesado digital de señales, y por lo tanto un elemento fundamental e indispensable para una gran variedad de aplicaciones. Las transformaciones de tiempo y frecuencia clásicas son solamente apropiadas para señales con variación lenta de amplitud y frecuencia. Esta suposición no suele cumplirse en la práctica, lo que conlleva una degradación de calidad y la aparición de artefactos. Además, la resolución temporal y frecuencial no se puede incrementar arbitrariamente debido al conocido principio de incertidumbre de Heisenberg. \\ El principal objetivo de esta tesis es revisar y mejorar los métodos existentes para el análisis de sinusoides no estacionarias, y también proponer nuevas estrategias y aproximaciones. Esta disertación contribuye sustancialmente a los análisis sinusoidales existentes: a) realiza una evaluación crítica del estado del arte y describe con gran detalle los métodos de análisis existentes, b) aporta mejoras sustanciales a algunos de los métodos existentes más prometedores, c) propone varias aproximaciones nuevas para el análisis de los modelos sinusoidales existentes i d) propone un modelo sinusoidal muy general y flexible con un algoritmo de análisis directo y rápido.
Many types of everyday signals fall into the non-stationary sinusoids category. A large family of such signals represent audio, including acoustic/electronic, pitched/transient instrument sounds, human speech/singing voice, and a mixture of all: music. Analysis of such signals has been in the focus of the research community for decades. The main reason for such intense focus is the wide applicability of the research achievements to medical, financial and optical applications, as well as radar/sonar signal processing and system analysis. Accurate estimation of sinusoidal parameters is one of the most common digital signal processing tasks and thus represents an indispensable building block of a wide variety of applications. Classic time-frequency transformations are appropriate only for signals with slowly varying amplitude and frequency content - an assumption often violated in practice. In such cases, reduced readability and the presence of artefacts represent a significant problem. Time and frequency resolu

Partial Differential Equations (PDEs) are commonly used to model complex systems that arise for example in biology, engineering, chemistry, and elsewhere. The parameters (or coefficients) and the source of PDE models are often unknown and are estimated from available measurements. Despite its importance, solving the estimation problem is mathematically and numerically challenging and especially when the measurements are corrupted by noise, which is often the case. Various methods have been proposed to solve estimation problems in PDEs which can be classified into optimization methods and recursive methods. The optimization methods are usually heavy computationally, especially when the number of unknowns is large. In addition, they are sensitive to the initial guess and stop condition, and they suffer from the lack of robustness to noise. Recursive methods, such as observer-based approaches, are limited by their dependence on some structural properties such as observability and identifiability which might be lost when approximating the PDE numerically. Moreover, most of these methods provide asymptotic estimates which might not be useful for control applications for example. An alternative non-asymptotic approach with less computational burden has been proposed in engineering fields based on the so-called modulating functions. In this dissertation, we propose to mathematically and numerically analyze the modulating functions based approaches. We also propose to extend these approaches to different situations. The contributions of this thesis are as follows. (i) Provide a mathematical analysis of the modulating function-based method (MFBM) which includes: its well-posedness, statistical properties, and estimation errors. (ii) Provide a numerical analysis of the MFBM through some estimation problems, and study the sensitivity of the method to the modulating functions' parameters. (iii) Propose an effective algorithm for selecting the method's design parameters. (iv) Develop a two-dimensional MFBM to estimate space-time dependent unknowns which is illustrated in estimating the source term in the damped wave equation describing the physiological characterization of brain activity. (v) Introduce a moving horizon strategy in the MFBM for on-line estimation and examine its effectiveness on estimating the source term of a first order hyperbolic equation which describes the heat transfer in distributed solar collector systems.

Fractional calculus has been introduced as an e cient tool for modeling physical phenomena, thanks to its memory and hereditary properties. For example, fractional models have been successfully used to describe anomalous di↵usion processes such as contaminant transport in soil, oil flow in porous media, and groundwater flow. These models capture important features of particle transport such as particles with velocity variations and long-rest periods. Mathematical modeling of physical phenomena requires the identification of pa- rameters and variables from available measurements. This is referred to as an inverse problem. In this work, we are interested in studying theoretically and numerically inverse problems for space Fractional Advection-Dispersion Equation (FADE), which is used to model solute transport in porous media. Identifying parameters for such an equa- tion is important to understand how chemical or biological contaminants are trans- ported throughout surface aquifer systems. For instance, an estimate of the di↵eren- tiation order in groundwater contaminant transport model can provide information about soil properties, such as the heterogeneity of the medium. Our main contribution is to propose a novel e cient algorithm based on modulat-ing functions to estimate the coe cients and the di↵erentiation order for space FADE, which can be extended to general fractional Partial Di↵erential Equation (PDE). We also show how the method can be applied to the source inverse problem. This work is divided into two parts: In part I, the proposed method is described and studied through an extensive numerical analysis. The local convergence of the proposed two-stage algorithm is proven for 1D space FADE. The properties of this method are studied along with its limitations. Then, the algorithm is generalized to the 2D FADE. In part II, we analyze direct and inverse source problems for a space FADE. The problem consists of recovering the source term using final observations. An analytic solution for the non-homogeneous case is derived and existence and uniqueness of the solution are established. In addition, the uniqueness and stability of the inverse problem is studied. Moreover, the modulating functions-based method is used to solve the problem and it is compared to a standard Tikhono-based optimization technique.

No
Purpose: Cardiac x-ray detectors are used to acquire moving images in real-time for angiography and interventional procedures. Detective quantum efficiency (DQE) is not generally measured on these dynamic detectors; the required “for processing” image data and control of x-ray settings have not been accessible. By 2016, USA hospital physicists will have the ability to measure DQE and will likely utilize the International Electrotechnical Commission (IEC) standard for measuring DQE of dynamic x-ray imaging devices. The current IEC standard requires an image of a tilted tungsten edge test object to obtain modulation transfer function (MTF) for DQE calculation. It specifies the range of edge angles to use; however, it does not specify a preferred method to determine this angle for image analysis. The study aimed to answer the question “will my choice in method impact my results?” Four different established edge angle determination methods were compared to investigate the impact on DQE. Methods: Following the IEC standard, edge and flat field images were acquired on a cardiac flat-panel detector to calculate MTF and noise power spectrum, respectively, to determine DQE. Accuracy of the methods in determining the correct angle was ascertained using a simulated edge image with known angulations. Precision of the methods was ascertained using variability of MTF and DQE, calculated via bootstrapping. Results: Three methods provided near equal angles and the same MTF while the fourth, with an angular difference of 6%, had a MTF lower by 3% at 1.5 mm−1 spatial frequency and 8% at 2.5 mm−1; corresponding DQE differences were 6% at 1.5 mm−1 and 17% at 2.5 mm−1; differences were greater than standard deviations in the measurements. Conclusions: DQE measurements may vary by a significant amount, depending on the method used to determine the edge angle when following the IEC standard methodology for a cardiac x-ray detector. The most accurate and precise methods are recommended for absolute assessments and reproducible measurements, respectively.
Funded by Philips Healthcare, NL, and a University of Leeds Career Development Bursery.

Detectors sensitive to thermal and reflected infrared radiation are widely used for night-vision, communications, thermography, and object tracking among other military, industrial, and commercial applications. System requirements for the next generation of ultra-high-performance infrared detectors call for increased functionality such as large formats (> 4K HD) with wide field-of-view, multispectral sensitivity, and on-chip processing. Due to the low yield of infrared material processing, the development of these next-generation technologies has become prohibitively costly and time consuming. In this work, it will be shown that physics-based numerical models can be applied to predictively simulate infrared detector arrays of current technological interest. The models can be used to a priori estimate detector characteristics, intelligently design detector architectures, and assist in the analysis and interpretation of existing systems. This dissertation develops a multi-scale simulation model which evaluates the physics of infrared systems from the atomic (material properties and electronic structure) to systems level (modulation transfer function, dense array effects). The framework is used to determine the electronic structure of several infrared materials, optimize the design of a two-color back-to-back HgCdTe photodiode, investigate a predicted failure mechanism for next-generation arrays, and predict the systems-level measurables of a number of detector architectures.

We develop a theory of vectorial wave coupling in cubic photorefractive crystals placed in an alternating ac-field to enhance the nonlinear response. First we analytically and numerically investigate the dependences of the first Fourier harmonics of the space-charge field, induced in an AC-biased sillenite crystal by a light-interference pattern, on the light contrast m. The data obtained was used to extend the vectorial beam-coupling theory on the whole contrast region. In particular, we proved in the general case that despite of essential differences between thediffusion and AC nonlocal responses the later keeps the light interference fringes straight during the interaction. This fundamental feature allows, under certain restrictions, to reduce the nonlinear problem of vectorial coupling to the known linear problem of vectorial Bragg diffraction from a spatially uniform grating, which admits an exact solution. As a result, the nonlinear vectorial problem can be effectively solved for a number of practically important cases.The developed theory was applied to describe the transformation of a momentary phase changes of one of the input beams into the output intensity modulation (so-called grating translation technique). In contrast to the previous studies, we take into account the change of the space-charge field amplitude across the crystal (the coupling effects). The theory developed is employed to optimize the conditions for the linear signal detection under polarization filtering for the transverse and longitudinal optical configurations. We also analyze the possibility of the linear detection without polarization filtering.Illumination of AC-biased photorefractive BTO crystals with a coherent light beam results in development of strong nonlinear scattering. We investigate the angular and polarization characteristics of the scattered light for the diagonal optical configuration and different polarization states of the pump.