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In recent years, considerable attention has been focused on the application of orthogonal expansions to system analysis, parameter identification, model reduction and control system design. However, little research has been done in applying their useful properties to Power System analysis and control. This research attempts to make some inroads in applying the so called " orthogonal expansion approach " to analysis and control of Power systems, especially the latter. A set of orthogonal functions commonly called Walsh functions in system science after it's discoverer J.L. Walsh [1923] have been successfully used for parameter identification in the presence of severe nonlinearities. The classical optimal control problem is applied to a synchronous machine infinite bus system via the orthogonal expansion approach and a convenient method outlined for designing PID controllers which can achieve prespecified closed loop response characteristics. The latter is then applied for designing a dynamic series capacitor controller for a single machine infinite bus system. / Graduation date: 1993

Nous nous proposons d'étudier les polynômes de Macdonald en les mettant en parallèle avec les polynômes orthogonaux classiques. En effet, ces deux types d'objets apparaissent comme fonctions propres d'opérateurs à significations physiques, les polynômes orthogonaux intervenant dans des situations décrites par une seule variable et les polynômes de Macdonald dans d'autres en demandant plusieurs. En développant chacune des deux théories, nous constaterons qu'elles s'élaborent de façon analogue. Notre but est ainsi de faire ressortir ces points communs tout en dégageant les différences entre les deux contextes. En mettant en lumière ce parallèle, nous constaterons toutefois qu'il manque un élément pour qu'il soit complet. En effet, les polynômes orthogonaux satisfont une récurrence à trois termes qui leur est caractéristique. Or rien d'analogue n'est présent dans la théorie des polynômes de Macdonald. Mais nous verrons qu'une conjecture portant sur une famille élargie de polynômes de Macdonald a été formulée qui permettrait de compléter le tableau. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : polynômes orthogonaux, fonctions symétriques, polynômes de Macdonald, conjecture n!.

In this thesis I study reconstruction of orthogonal polyhedral surfaces and orthogonal polyhedra from partial information about their boundaries. There are three main questions for which I provide novel results. The first question is "Given the dual graph, facial angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the dihedral angles?" The second question is "Given the dual graph, dihedral angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the facial angles?" The third question is "Given the vertex coordinates of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the edges and faces, possibly after rotating?" For the first two questions, I show that the answer is "yes" for genus-0 orthogonal polyhedra and polyhedral surfaces under some restrictions, and provide linear time algorithms. For the third question, I provide results and algorithms for orthogonally convex polyhedra. Many related problems are studied as well.

In this thesis I study reconstruction of orthogonal polyhedral surfaces and orthogonal polyhedra from partial information about their boundaries. There are three main questions for which I provide novel results. The first question is "Given the dual graph, facial angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the dihedral angles?" The second question is "Given the dual graph, dihedral angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the facial angles?" The third question is "Given the vertex coordinates of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the edges and faces, possibly after rotating?" For the first two questions, I show that the answer is "yes" for genus-0 orthogonal polyhedra and polyhedral surfaces under some restrictions, and provide linear time algorithms. For the third question, I provide results and algorithms for orthogonally convex polyhedra. Many related problems are studied as well.

As the forecasting models become more sophisticated in their physics and possible depictions of the nearshore hydrodynamics, they also become increasingly sensitive to errors in the inputs. These input errors include: mis-specification of the input parameters (bottom friction, eddy viscosity, etc.); errors in input fields and errors in the specification of boundary information (lateral boundary conditions, etc.). Errors in input parameters can be addressed with fairly straightforward parameter estimation techniques, while errors in input fields can be somewhat ameliorated by physical linkage between the scales of the bathymetric information and the associated model response. Evaluation of the errors on the boundary is less straightforward, and is the subject of this thesis. The model under investigation herein is the Delft3D modeling suite, developed at Deltares (formerly Delft Hydraulics) in Delft, the Netherlands. Coupling of the wave (SWAN) and hydrodynamic (FLOW) model requires care at the lateral boundaries in order to balance run time and error growth. To this extent, we use perturbation method and spatio-temporal analysis method such as Empirical Orthogonal Function (EOF) analysis to determine the various scales of motion in the flow field and the extent of their response to imposed boundary errors. From the Swirl Strength examinations, we find that the higher EOF modes are affected more by the lateral boundary errors than the lower ones.

The dielectric waveguide is an important component used in the optical communication system. In this thesis, we conduct basic research on the propagation constant and the characteristic of the dielectric waveguide. We develop a method to expand 2-D rectangular dielectric waveguide modes with simple orthogonal bases. Furthermore, we improve the convergent rate by expanding waveguide modes with tensor product of properly chosen guiding-mode bases. We first derive the coupled differential equations of the two transverse magnetic field components which satisfy the continuous boundary conditions across all material interfaces. Then we investigate and verify the accuracy of this method on 1-D rectangular waveguide so that we can apply the technique to 2-D rectangular waveguides. By means of linear combination of simple 2-D orthogonal bases, we expand the mode of rectangular dielectric waveguide. Through rigorous mathematical closed-form integration, we obtain the equivalent matrix whose eigenvalues and associated eigenvectors become the mode propagation constants and mode field distribution functions of the underlying 2-D dielectric waveguide. Whenever symmetry exists we can reduce the size of the problem by choosing appropriate boundary conditions in accordance to particular mode polarization desired. This method provides at least four significant digits of propagation constant and detailed field description of the rectangular dielectric waveguide. We believe that it is an effective method for modal analysis of 2-D complex dielectric-waveguides.

The dielectric ridge waveguide is an important passive component used in the optical communication system. Compared to its cousins¡Xthe buried ridge waveguides, it is less expensive to process but harder to design due to its inherent complex field structure (it has a larger index contrast between the core and the cladding). Therefore, it is crucial to develop an efficient and accurate method to analyze the modal characteristic of ridge waveguides. We began with the expansion of one-dimensional three-layer dielectric slab waveguide using simple orthogonal basis functions. We examined both the step-index profile and the graded-index profile waveguides to confirm the feasibility of this method and to understand the level of accuracy our technique can reach. We then proceeded to derive the vector formulation of two-dimension dielectric waveguides and compared our results against the exact numerical solutions of optical fiber modes using Bessel functions. Our 2-D Cartesian mode solver gave up to 6 significant digits of the fiber propagation constants. Finally, for rectangular dielectric waveguides, we use the tensor product of 1-D guiding-mode bases as an improvement over the simple orthogonal bases to speed-up the numerical convergence and cut the storage requirement by a factor of ten without loss of accuracy which is around 4 to 5 digits over a wide range of parameters and mode types. We will use these bases to solve for the mode field distribution of ridge-waveguides and other complex structures.