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<p> In power systems, dynamic stability analysis is an important field of interest for both design and operation studies. This stability analysis requires the formulation of the linearized power system equations in the state-space form.</p> <p> In this thesis, the state-space matrices of multi-machine systems are constructed by implementing two matrix formulation techniques, the "PQR" and the direct elimination "ELIM" methods. Two computer programs have been devised to apply these formulation techniques. The programs are capable of handling systems up to a maximum order of 70, with available central memory of about 49,000 words (decimal). Another feature of these programs is their capability of accommodating generating units with different degrees of complexity, by allowing a variety of models for the sub-system components. Both programs have been applied to two test examples to illustrate their validity.</p> <p> The two formulation technique programs were compared from the point of view of computational time, storage requirements and eigenvalue sensitivity evaluation.</p> / Thesis / Master of Engineering (MEngr)

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In today's technology, with digitally controlled optic sensing devices, there exists a need for a fast and accurate calibration procedure. Typical display devices and optic fiber bundles are plagued with inaccuracies. There are many sources of error such as delay, time constants, pixel distortion, pixel bleeding, and noise. The calibration procedure must measure these inaccuracies, and compute a set of correction factors. These correction factors are then used in real time to alter the command data, such that the intended pixels are correctly commanded. This paper discusses a calibration procedure, which employs a special matrix inverse algorithm. This algorithm, which is only applicable to sparse symmetric band diagonal matrices, successfully inverts a 10,000 by 10,000 matrix in less than four seconds on a VAX-11/780. It is estimated that, when using conventional Gauss-Jordan matrix inverse techniques, 4800 hours are required to compute the same matrix inverse. This paper also documents the BlendI routines, which will be used as a calibration procedure for BlendI System.

The class of real nxn matrices M, known as K-matrices, for t which the linear complementarity problem w-Mz=q, w≥O, z≥O, w^tz=O has a solution whenever w-Mz=q, w≥O, z≥O has a solution is characterized for dimensions n<4. The characterization is finite and"practical". Several necessary conditions, sufficient conditions, and counterexamples pertaining to K-matrices are also given. A finite characterization of completely K-matrices (K-matrices all of whose principal submatrices are also K-matrices) is proved for dimensions < 4. / M.S.

Given the prime power factorization of a positive integer m, a method for calculating the number of all distinct n x n - involutory matrices (mod m) is derived. This is done by first developing a method for the construction and enumeration of involutory matrices (mod P^α), without duplication, for each prime power modulus P^α. Using these results, formulas for the number of distinct involutory matrices (mod P^α) of order n are given where p is an odd prime, p=2, α= 1 and α > 1. The concept of a fixed group associated with an involutory matrix (mod P^α) is used to characterize such matrices. Involutory matrices (mod P^α) of order n are considered as linear transformations on a vector space of n-tuples to provide uncomplicated proofs for the basic results concerning involutory matrices over a finite field. / Master of Science

In this thesis the right and left pole structure of a not necessarily regular rational matrix function W is described in terms of pairs of matrices-right and left pole pairs. The concept of orthogonality in Rⁿ is investigated. Using this concept, the right and left zero structure of a rational matrix function W is described in terms of pairs and triples of matrices-right and left null pairs and right and left kernel triples. The definition of a spectral triple of a regular rational matrix function over a subset σ of C is extended to the nonregular case. Given a rational matrix function W and a subset σ of C, the left null-pole subspace of W over σ is described in terms of a left kernel triple and a left σ-spectral triple for W. A sufficient condition for the minimality of McMillan degree of a rational matrix function H which is right equivalent to W on σ, that is a rational matrix function H of the same size and with the same left null-pole subspace over σ as W, is developed. An algorithm for constructing a rational matrix function W with a left kernel triple (A_κ B_κ D_κ) and left null and right pole pairs over σ⊂C (A_ζ, B_ζ) and (C_π, A_π), respectively, from a regular rational matrix function with left null and right pole pairs over σ (A_ζ, B_ζ) and (C_π, A_π) is described. Finally, a necessary and sufficient condition for existence of a rational matrix function W with a given left kernel triple and a given left spectral triple over a subset σ of C is established. / Ph. D.

Le pseudospectre est un nouvel outil pour étudier les matrices et les opérateurs linéaires. L’outil traditionnel est le spectre. Celui-ci peut révéler des informations sur le comportement des matrices ou operateurs normaux. Cependant, il est moins informatif lorsque la matrice ou l’opérateur est non-normal. Le pseudospectre s’est toutefois révélé être un outil puissant pour les étudier. Il fournit une alternative analytique et graphique pour étudier ce type des cas. Le but de cette thèse est d’étudier le comportement d’une matrice non-normale A en se basant sur le pseudospectre. Il est bien connu que le théorème matriciel de Kreiss donne des estimations des bornes supérieures de [symbol] et [symbol] en fonction du pseudospectre. En 1999, Toh et Trefethen [31] ont généralisé ce célèbre théorème aux polynômes de Faber et aux matrices ayant des spectres dans des domaines plus généraux. En 2005, Vitse [34] a donné une généralisation du théorème aux fonctions holomorphes dans le disque unité. Dans cette thèse, on généralise le théorème matriciel de Kreiss aux fonctions holomorphes et aux matrices ayant des spectres dans des domaines plus généraux. Certaines conditions devraient cependant être vérifiées. L’étude du comportement d’une matrice au cas où la valeur exacte de la norme de la résolvante est connue a été aussi remise en question. Il est bien connu que si A et B sont des matrices à pseudospectres identiques, alors [symbol] Mais, qu’en est-il pour les puissances supérieures [symbol] En 2007, Ransford [21] a montré qu’il existe des matrices [symbol] avec des pseudospectres identiques et où peuvent prendre des valeurs aléatoires et indépendantes les unes des autres pour [symbol]. Serait-il aussi le cas pour n assez grand ? Par ailleurs, le pseudospectre est aussi utilisé pour étudier le semi-groupe [symbol], mais permet-il de déterminer [symbol] ? Cette thèse répond à toutes ces questions en démontrant des résultats plus généraux. Elle généralise l’inégalité (1) aux transformations de Möbius. Elle montre aussi que la condition de pseudospectre identique n’est pas suffisante pour déterminer le comportement d’une matrice. Cependant, la condition de pseudospectre super-identique pourrait l’être. / The theory of pseudospectra is a new tool for studying matrices and linear operators. The traditional tool is the spectrum. It reveals information on the behavior of normal matrices or operators. However, it is less informative as the matrix or the operator are non-normal. Pseudospectra have nevertheless proved to be a powerful tool to study them. They provide an analytical and graphical alternative to study this type of case. The purpose of this thesis is to study the behavior of a non-normal matrix A based on its pseudospectra. It is well known that the Kreiss matrix theorem provides estimates of upper bounds of [symbol] according pseudospectra. In 1999, Toh and Trefethen [31] generalized the celebrated theorem to Faber polynomials and matrices with spectra in more general domains. In 2005, Vitse [34] generalized the theorem for holomorphic functions in the unit disk. In this thesis, the Kreiss matrix theorem is generalized to holomorphic functions and matrices with spectra in more general domains. However, certain conditions should be imposed. The behavior of a matrix if the exact knowledge of the resolvent norm is assumed has also been questioned. It is well known that if A and B are matrices with identical pseudospectra, then [symbol] But what about higher powers [symbol] ? In 2007, Ransford [21] showed that there exist matrices [symbol] with identical pseudospectra and where [symbol] and [symbol] can take more or less arbitrary values for [symbol]. Is it also the case for large n? Moreover, pseudospectra are also used to study the semigroup [symbol], but do they allow us to determine [symbol] ? This thesis addresses all these issues by demonstrating more general results. It generalizes the inequality (2) to Möbius transformations. It also shows that the condition of identical pseudospectra is not sufficient to determine the behavior of a matrix. However, the condition of super-identical pseudospectra could do so.