Computing generalized inverses using LU factorization of matrix product

An algorithm for computing {2, 3}, {2, 4}, {1, 2, 3}, {1, 2, 4} -inverses and the Moore-Penrose inverse of a given rational matrix A is established. Classes A(2, 3)s and A(2, 4)s are characterized in terms of matrix products (R*A)+R* and T*(AT*)+, where R and T are rational matrices with appropriate dimensions and corresponding rank. The proposed algorithm is based on these general representations and the Cholesky factorization of symmetric positive matrices. The algorithm is implemented in programming languages MATHEMATICA and DELPHI, and illustrated via examples. Numerical results of the algorithm, corresponding to the Moore-Penrose inverse, are compared with corresponding results obtained by several known methods for computing the Moore-Penrose inverse

On Small Degree Extension Fields in Cryptology

This thesis studies the implications of using public key cryptographic primitives that are based in, or map to, the multiplicative group of finite fields with small extension degree. A central observation is that the multiplicative group of extension fields essentially decomposes as a product of algebraic tori, whose properties allow for improved communication efficiency. Part I of this thesis is concerned with the constructive implications of this idea. Firstly, algorithms are developed for the efficient implementation of torus-based cryptosystems and their performance compared with previous work. It is then shown how to apply these methods to operations required in low characteristic pairing-based cryptography. Finally, practical schemes for high-dimensional tori are discussed. Highly optimised implementations and benchmark timings are provided for each of these systems. Part II addresses the security of the schemes presented in Part I, i.e., the hardness of the discrete logarithm problem. Firstly, an heuristic analysis of the effectiveness of the Function Field Sieve in small characteristic is given. Next presented is an implementation of this algorithm for characteristic three fields used in pairing-based cryptography. Finally, a new index calculus algorithm for solving the discrete logarithm problem on algebraic tori is described and analysed

Application of the Round-Trip theory to electro-dynamic sources of vibration

The Round-Trip Identity, developed by Moorhouse & Elliott at The University of Salford allows the reconstruction of mechanical frequency response functions at remote locations on a test structure, without the need to physically measure at that location. This makes the identity a powerful tool in the field of modal analysis, as it may be applied to structures where measurement at a desired location may be practically difficult or impossible.This work aims to extend the applications of the Round-Trip Identity, which was originally applied to solely mechanical systems. Work by Moorhouse & Elliott in 2013 suggested that the identity could be applied to any system that is linear and time invariant. Electro-mechanical systems can be found throughout the modern world. This project investigates potential electro-mechanical extensions of the identity, which would allow application of the identity to these hybrid systems.A thorough summary of electrical network theorems and modal analysis theories is presented. Theories from both fields are subsequently combined to derive the electro-mechanical Round-Trip Identity.The first objective of this study was to verify the mechanical Round-Trip identity. Following this, an electro-mechanical version of the identity is derived.A series of increasingly complex experiments is presented that aim to verify the electro-mechanical Round-Trip Identity, the simplest case being a single degree of Freedom on a simply supported beam.The design of further experiments that would aim to verify the identity for more complex, multi degree of freedom systems is proposed. These relate to the analysis of plates and electric motors. Finally, recommendations are made for how the identity could be further developed and implemented in an industrial context

Algorithmes et architectures pour la commande et le diagnostic de systèmes critiques de vol

Flight-Critical Systems such as Electromechanical Actuators driven by Engine Control Units (ECU) or Flight Control Units (FCU) are designed and developed regarding drastic safety requirements. In this study, an actuator control and monitoring ECU architecture based on analytic redundancy is proposed. In case of fault occurrences, material redundancies in avionic equipment allow certaincritical systems to reconfigure or to switch into a safe mode. However, material redundancies increase aircraft equipment size, weight and power (SWaP). Monitoring based on dynamical models is an interesting way to further enhance safetyand availability without increasing the number of redundant items. Model-base dfault detection and isolation (FDI) methods [58, 26, 47] such as observers and parity space are recalled in this study. The properties of differential flatness for nonlinear systems [80, 41, 73] and endogenous feedback linearisation are used with nonlinear diagnosis models. Linear and nonlinear observers are then compared with an application on hybrid stepper motor (HSM). A testing bench was specially designed to observe in real-time the behaviour of the diagnosis models when faults occur on the stator windings of a HSM.Les systèmes critiques de vol tels que les actionneurs électromécaniques ainsi que les calculateurs de commande moteur (ECU) et de vol (FCU),sont conçus en tenant compte des contraintes aéronautiques sévères de sureté defonctionnement. Dans le cadre de cette étude, une architecture calculateur pourla commande et la surveillance d’actionneurs moteur et de surfaces de vol est proposée et à fait l’objet d’un brevet [13]. Pour garantir ces mesure de sureté, les ECU et FCU présentent des redondances matérielles multiples, mais engendrent une augmentation de l’encombrement, du poids et de l’énergie consommée. Pour ces raisons, les redondances à base de modèles dynamiques, présentent un atout majeur pour les calculateurs car elles permettent dans certains cas de maintenir les exigences d’intégrité et de disponibilité tout en réduisant le nombre de capteurs ou d’actionneurs. Un rappel sur les méthodes de diagnostic par générateurs de résidus et estimateurs d’états [58, 26, 47] est effectué dans cette étude. Les propriétés de platitude différentielle et la linéarisation par difféomorphisme et bouclage endogène [80, 41, 73] permettent d’utiliser des modèles linéaires équivalents avec les générateurs de résidus. Un banc d’essai a été conçu afin de valider les performances des algorithmes de diagnostic

Time domain reflectometry imaging - A new moisture measurement technique for industry and soil science

This thesis describes the theoretical and practical aspects of a new technique for quantitative, non-invasive and non-destructive imaging of the near-surface moisture content distribution of composite materials. The technique relies on the alteration by a nearby composite material, of the propagation velocity of an electromagnetic pulse along a parallel transmission line, through distortion of the evanescent field. A set of measurements taken at different relative positions of the transmission line and composite material are, in conjunction with a forward model describing propagation velocity on the line, inverted to provide the image of moisture content distribution. Development of the technique, called 'Time Domain Reflectometry Imaging' (TDRI), involved four steps: 1. Instrumentation to obtain a set of measurements of propagation times; 2. A forward model; 3. An inverse procedure; and 4. Conversion of a calculated permittivity distribution to a moisture distribution. Critical to the success of the inverse method is a set of measurements of propagation velocity that provide pico-second propagation time accuracy, and are sufficiently linearly independent to enable discrimination of the permittivity of each discretised cell within the composite material. Using commercial time domain reflectometry (TDR) instruments, a switched reference measurement, waveform subtraction and intersecting waveform tangents, sufficient timing accuracy has been achieved. The forward model was developed using the moment method. The advantage of such an integral equation method is that recalculation is not required when changing the impressed field. Hence for a particular model of the composite material's moisture distribution, just one execution of the forward model provides predicted propagation velocities for all positions of the transmission line. A new pseudo 3-D variant of the volume integral equation approach was developed to suit the 2-D transmission line, and resulted in a 100 fold reduction in memory use, and a greater than 10 fold reduction in execution time. The forward solution uses the telegrapher's equation to predict propagation velocity from an arbitrary permittivity distribution surrounding the line. Inversion of the measured data was accelerated by the use of three novel tactics: a rapid electric field surrogate for the Jacobian; a dynamic method of determining the conjugate gradient weighting factor; and a new blocking technique that accelerated the convergence of buried cells that have only a small influence on propagation velocity. The final TDRI step is a numerical model to translate both the a priori moisture distribution data to a permittivity distribution, and conversely the solution permittivity distribution to moisture content. A dielectric model based on an earlier model of Looyenga was adapted to include both the different characteristic of tightly held water, and the Debye relaxation of free water. The intention was a model with applicability to a range of composite materials. It was tested with data for soil, bentonite clay and wood, and except for one free parameter, model parameters were set by measurable physical properties of the host material