The problem of finding a reduced order model, optimal in the H² sense, to a given system model is a fundamental one in control system analysis and design. The addition of a H^∞ constraint to the H² optimal model reduction problem results in a more practical yet computationally more difficult problem. Without the global convergence of homotopy methods, both the H² optimal and the combined H²/H^∞ model reduction problems are very difficult. For both problems homotopy algorithms based on several formulations input normal form; Ly, Bryson, and Cannon's 2 X 2 block parametrization; a new nonminimal parametrization are developed and compared here. For the H² optimal model order reduction problem, these numerical algorithms are also compared with that based on Hyland and Bernstein's optimal projection equations. Both the input normal form and Ly form are very efficient compared to the over parametrization formulation and the optimal projection equations approach, since they utilize the minimal number of possible degrees of freedom. However, they can fail to exist or be very ill conditioned. The conditions under which the input normal form and the Ly form become ill conditioned are examined. The over-parametrization formulation solves the ill conditioning issue, and usually is more efficient than the approach based on solving the optimal projection equations for the H² optimal model reduction problem. However, the over-parametrization formulation introduces a very high order singularity at the solution, and it is doubtful whether this singularity can be overcome by using interpolation or other existing methods. / Master of Science

A structured program has been developed to track the equilibrium path of geometrically nonlinear space structures by the modified Riks/Wempner method. The sparse normal flow code in 'HOMP ACK' is used for tracking the equilibrium paths by the homotopy method. Two subroutines were written as required by HOMPACK. Four different structures were analyzed by these two programs. Comparison of the two methods has been carried out based on the number of Jacobian matrix evaluations and the CPU time used by the programs. The Riks/Wempner program successfully traced the equilibrium path through the limit points and bifurcation points for all the four structures analyzed while the homotopy program could not trace the complete path for two of the structures. It has been concluded that the sparse normal flow code of HOMPACK needs to be modified. / Master of Science

Let Γ be a crystallographic group acting on the n-dimensional Euclidean space. In this dissertation, the surgery obstruction groups of Γ are computed in terms of certain sheaf homology groups defined by F. Quinn, when Γ has no 2-torsion. The main theorem is : Theorem : If a crystallographic group Γ has no 2-torsion, there is a natural isomorphism a : H_*(Rⁿ /Γ; L(p)) → L_*^-∞(Γ). / Ph. D.