This study examines the characteristics of four methods for the control of flexible structures and investigates the control performances of each method. The investigation is concerned with various control performance measures, such as control gain magnitude, settling time and overshoot in transient response, actuator phase and gain margins, and stability in the presence of actuator failure. In conjunction with the system performance, a systematic approach to the choice of weighting matrices for optimal control is presented. The approach shows a relation between the weighting matrices and the closed-loop eigenvalues. Since the approach is based on a set of independent second·order modal dynamics, the dimensionality of the system is no longer a problem in obtaining the optimal control law. The newly developed Minimum Gain Pole Placement (MGPP) is an optimal method in the sense that it minimizes an objective function, where the objective function is taken as control gain magnitudes with constraints of exact pole placement for any set of modes. The robustness of Independent Modal Space Control (IMSC) is examined. In general, the parameters of the control system are usually approximated, so that the designed controller, based on a postulated model, will not perform on the actual system as expected. This study shows that when the IMSC method is used with collocated sensors and actuators, the modelling errors in the postulated system cannot lead to instability of the closed-loop system containing control modes and residual modes.However, in the case of coupled control (MGPP), this property cannot be shown. This points to the robustness of IMSC method with respect to modelling errors. The IMSC method requires the same number of actuators as the number of control modes. The method can be extended to the cases of fewer actuator and more actuator by using the pseudo-inverse of modal particification matrix, an approach referred to as pseudo-independent modal-space control (PIMC). It is shown that PIMSC also yields some form of optimal control and that it is robust as well. Modal filters are introduced to detect and identify failure of control components in large space structures. The failure mode is investigated in the modal space so that a simple failure detection and identification (FDI) based on modal dynamics is established. Moreover the information obtained from the modal analysis provides some guidelines for the identification of faulty components. The integral form of the modal filters provides the ability to mitigate the effects of noise, disturbances and parameter uncertainties pointing to the robustness of the method. The detection process proposed in this study reduces the computational effort and permits an assessment of the system stability. / Ph. D. / incomplete_metadata

Optimization of dynamic systems involving complex non-hermitian matrices is often computationally expensive. Major contributors to the computational expense are the sensitivity analysis and reanalysis of a modified design. The present work seeks to alleviate this computational burden by identifying efficient sensitivity analysis and approximate reanalysis methods. For the algebraic eigenvalue problem involving non-hermitian matrices, algorithms for sensitivity analysis and approximate reanalysis are classified, compared and evaluated for efficiency and accuracy. Proper eigenvector normalization is discussed. An improved method for calculating derivatives of eigenvectors is proposed based on a more rational normalization condition and taking advantage of matrix sparsity. Important numerical aspects of this method are also discussed. To alleviate the problem of reanalysis, various approximation methods for eigenvalues are proposed and evaluated. Linear and quadratic approximations are based directly on the Taylor series. Several approximation methods are developed based on the generalized Rayleigh quotient for the eigenvalue problem. Approximation methods based on trace theorem give high accuracy without needing any derivatives. Operation counts for the computation of the approximations are given. General recommendations are made for the selection of appropriate approximation technique as a function of the matrix size, number of design variables, number of eigenvalues of interest and the number of design points at which approximation is sought. / Ph. D.

The convergence rates for the method of Weinstein and a variant method of Aronszajn known as "truncation including the remainder" are derived in terms of the containment gaps between exact and approximating subspaces, using analytical techniques that arise in part in the convergence analysis of finite element methods for differential eigenvalue problems. An example of a one dimensional Schrodinger operator with a potential is presented which arises in quantum mechanics. Examples using the recent eigenvector-free (EVF) method of Beattie and Goerisch are considered. Since the EVF method uses finite element trial functions as approximating vectors, it produces sparse and well-structured coefficient matrices. For these large-order sparse matrix eigenvalue problems, we adapt a spectral transformation Lanczos algorithm for finding a few wanted eigenvalues. For a few particular examples of vibration in beams and plates, convergence behavior is experimentally evaluated. / Ph. D.

Low-density parity check (LDPC) codes are very popular among error correction codes because of their high-performance capacity. Numerous investigations have been carried out to analyze the performance and simplify the implementation of LDPC codes. Relatively slow convergence of iterative decoding algorithm affects the performance of LDPC codes. Faster convergence can be achieved by reducing the number of iterations during the decoding process. In this thesis, a new approach for faster convergence is suggested by choosing a systematic parity check matrix that yields lowest Second Smallest Eigenvalue Modulus (SSEM) of its corresponding Laplacian matrix. MATLAB simulations are used to study the impact of eigenvalues on the number of iterations of the LDPC decoder. It is found that for a given (n, k) LDPC code, a parity check matrix with lowest SSEM converges quickly as compared to the parity check matrix with high SSEM. In other words, a densely connected graph that represents the parity check matrix takes more iterations to converge than a sparsely connected graph.

In this PhD thesis we study some weighted regular Sturm-Liouville problems in which the weight function takes on both positive and negative signs in an appropriate interval [a,b]. With such  problems there is the possible existence of non-real eigenvalues, unlike in the definite case (i.e. left or right definite) in which only real eigenvalues exist. This PhD thesis consists of five papers (papers A-E) and an introduction to this area, which puts these papers into a more general frame. In paper A we give some precise estimates on the Richardson number for the two turning point case, thereby complementing the work of Jabon and Atkinson from 1984 in an essential way. We also give a corrected version of their result since there seems to be a typographical error in their paper. In paper B we show that the interlacing property, which holds in the one turning point case, does not hold in the two turning point case. The paper consists of a detailed presentation of numerical results of the case in which the weight function is allowed to change its sign twice in the interval (-1, 2). We also present some theoretical results which support the numerical results. Moreover, a number of new open questions are raised. We also observe that the real and imaginary parts of a non-real eigenfunction either have the same number of zeros in the interval (-1,2) or the numbers of zeros differ by two. In paper C, we obtain bounds on real and imaginary parts of non-real eigenvalues of a non-definite Sturm-Liouville problem, with Dirichlet boundary conditions, thus complementing the results obtained in a paper byBehrndt et.al. from 2013 in an essential way. In paper D we obtain a lower bound on the eigenvalue of the smallest modulus associated with a Dirichlet problem in the general case of a regular Sturm-Liouville problem. In paper E we expand upon the basic oscillation theory for general boundary problems of the form   -y''+q(x)y=λw(x)y,  on  I = [a,b], where  q(x) and w(x) are real-valued continuous functions on [a,b] and y is required to satisfy a pair of homogeneous separated boundary conditions at the end-points. Already in 1918 Richardson proved that, in the case of the Dirichlet problem,  if w(x) changes its sign exactly once and the boundary problem is  non-definite, then the zeros of the real and imaginary parts of any non-real eigenfunction interlace. We show that, unfortunately, this result is false in the case of two turning points, thus removing any hope for a general separation theorem for the zeros of the non-real eigenfunctions. Furthermore, we show that when a non-real eigenfunction vanishes inside I, then the absolute value of the difference between the total number of zeros of its real and imaginary parts is exactly 2.

Parmi les outils les plus classiques de l'Analyse Multivariée, les Composantes Principales sont aussi un des plus anciens puisqu'elles furent introduites il y a plus d'un siècle par Pearson (1901) et redécouvertes ensuite par Hotelling (1933). Aujourd'hui, cette méthode est abondamment utilisée en Sciences Sociales, en Economie, en Biologie et en Géographie pour ne citer que quelques disciplines. Elle a pour but de réduire de façon optimale (dans un certain sens) le nombre de variables contenues dans un jeu de données.<p>A ce jour, les méthodes d'inférence utilisées en Analyse en Composantes Principales par les praticiens sont généralement fondées sur l'hypothèse de normalité des observations. Hypothèse qui peut, dans bien des situations, être remise en question.<p><p>Le but de ce travail est de construire des procédures de test pour l'Analyse en Composantes Principales qui soient valides sous une famille plus importante de lois de probabilité, la famille des lois elliptiques. Pour ce faire, nous utilisons la méthodologie de Le Cam combinée au principe d'invariance. Ce dernier stipule que si une hypothèse nulle reste invariante sous l'action d'un groupe de transformations, alors, il faut se restreindre à des statistiques de test également invariantes sous l'action de ce groupe. Toutes les hypothèses nulles associées aux problèmes considérés dans ce travail sont invariantes sous l'action d'un groupe de transformations appellées monotones radiales. L'invariant maximal associé à ce groupe est le vecteur des signes multivariés et des rangs des distances de Mahalanobis entre les observations et l'origine. <p><p>Les paramètres d'intérêt en Analyse en composantes Principales sont les vecteurs propres et valeurs propres de matrices définies positives. Ce qui implique que l'espace des paramètres n'est pas linéaire. Nous développons donc une manière d'obtenir des procédures optimales pour des suite d'experiences locales courbées. <p>Les statistiques de test introduites sont optimales au sens de Le Cam et mesurables en l'invariant maximal décrit ci-dessus.<p>Les procédures de test basées sur ces statistiques possèdent de nombreuses propriétés attractives: elles sont valides sous la famille des lois elliptiques, elles sont efficaces sous une densité spécifiée et possèdent de très bonnes efficacités asymptotiques relatives par rapport à leurs concurrentes. En particulier, lorsqu'elles sont basées sur des scores Gaussiens, elles sont aussi efficaces que les procédures Gaussiennes habituelles et sont bien plus efficaces que ces dernières si l'hypothèse de normalité des observations n'est pas remplie. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

This thesis consists of two parts. In the first part, we study a class of special functions Aik (y), k = 2, 4, 6, ··· generalising the classical Airy function Ai(y) to higher orders and in the second part, we apply expressions and properties of Ai4(y) to spectral problem of a specific operator. The first part is however motivated by latter part. We establish regularity properties of Aik (y) and particularly show that Aik (y) is smooth, bounded, and extends to the complex plane as an entire function, and obtain pointwise bounds on Aik (y) for all k. Some analytic properties of Aik (y) are also derived allowing one to express Aik (y) as a finite sum of certain generalised hypergeometric functions. We further obtain full asymptotic expansions of Aik (y) and their first derivative Ai'(y) both for y > 0 and for y < 0. Using these expansions, we derive expressions for the negative real zeroes of Aik (y) and Ai'(y). Using expressions and properties of Ai4(y), we extensively study spectral properties of a non-local operator H whose physical interpretation is the massless relativistic quartic anharmonic oscillator in one dimension. Various spectral results for H are derived including estimates of eigenvalues, spectral gaps and trace formula, and a Weyl-type asymptotic relation. We study asymptotic behaviour, analyticity, and uniform boundedness properties of the eigenfunctions Ψn(x) of H. The Fourier transforms of these eigenfunctions are expressed in two terms, one involving Ai4(y) and another term derived from Ai4(y) denoted by Āi4(y). By investigating the small effect generated by Āi4(y) this work shows that eigenvalues λn of H are exponentially close, with increasing n Ε N, to the negative real zeroes of Ai4(y) and those of its first derivative Ai'4(y) arranged in alternating and increasing order of magnitude. The eigenfunctions Ψ(x) are also shown to be exponentially well-approximated by the inverse Fourier transform of Ai4(|y| - λn) in its normalised form.

Symmetric generalized eigenvalue problems arise in many physical applications and frequently only a few of the eigenpairs are of interest. Typically, the problems are large and sparse, and therefore traditional methods such as the QZ algorithm may not be considered. Moreover, it may be impractical to apply shift-and-invert Lanczos, a favored method for problems of this type, due to difficulties in applying the inverse of the shifted matrix. With these difficulties in mind, Golub and Ye developed an inverse free Krylov subspace algorithm for the symmetric generalized eigenvalue problem. This method does not rely on shift-and-invert transformations for convergence acceleration, but rather a preconditioner is used. The algorithm suffers, however, in the presence of multiple or clustered eigenvalues. Also, it is only applicable to the location of extreme eigenvalues. In this work, we extend the method of Golub and Ye by developing a block generalization of their algorithm which enjoys considerably faster convergence than the usual method in the presence of multiplicities and clusters. Preconditioning techniques for the problems are discussed at length, and some insight is given into how these preconditioners accelerate the method. Finally we discuss a transformation which can be applied so that the algorithm extracts interior eigenvalues. A preconditioner based on a QR factorization with respect to the B-1 inner product is developed and applied in locating interior eigenvalues.

This thesis considers two isoperimetric inequalities for the eigenvalues of the Laplacian on a family of spherically symmetric Riemannian manifolds. The Payne-Pólya-Weinberger Conjecture (PPW) states that for a bounded domain Ω in Euclidean space Rn, the ratio λ1(Ω)/λ0(Ω) of the first two eigenvalues of the Dirichlet Laplacian is bounded by the corresponding eigenvalue ratio for the Dirichlet Laplacian on the ball BΩof equal volume. The Szegö-Weinberger inequality states that for a bounded domain Ω in Euclidean space Rn, the first nonzero eigenvalue of the Neumann Laplacian μ1(Ω) is maximized on the ball BΩ of the same volume. In the first three chapters we will look at the known work for the manifolds Rn and Hn. Then we will take a family a spherically symmetric manifolds given by Rn with a spherically symmetric metric determined by a radially symmetric function f. We will then give a PPW-type upper bound for the eigenvalue gap, λ1(Ω) − λ0(Ω), and the ratio λ1(Ω)/λ0(Ω) on a family of symmetric bounded domains in this space. Finally, we prove the Szegö-Weinberger inequality for this same class of domains.

Diagonally dominant matrices arise in many applications. In this work, we exploit the structure of diagonally dominant matrices to provide sharp entrywise relative perturbation bounds. We first generalize the results of Dopico and Koev to provide relative perturbation bounds for the LDU factorization with a well conditioned L factor. We then establish relative perturbation bounds for the inverse that are entrywise and independent of the condition number. This allows us to also present relative perturbation bounds for the linear system Ax=b that are independent of the condition number. Lastly, we continue the work of Ye to provide relative perturbation bounds for the eigenvalues of symmetric indefinite matrices and non-symmetric matrices.