Robust system analysis and nonlinear system model reduction

The aim of the first part of this thesis is to broaden the classes of linear systems and performance measures numerical tools for robustness analysis can be used for. First, we consider robustness problems involving uncertain real parameters and present several new approaches to computing an improved structured singular value {dollar}mu{dollar} lower bound. We combine these algorithms to yield a substantially improved power algorithm.; Then, we show that both the worst case {dollar}{lcub}cal H{rcub}sb{lcub}infty{rcub}{dollar} performance and the worst case {dollar}{lcub}cal H{rcub}sb2{dollar} performance of uncertain systems subject to a norm bounded structured LTI perturbations can be written exactly in terms of the skewed {dollar}mu.{dollar} The algorithm for the structured singular value lower bound computation, can be extended to computing skewed {dollar}mu{dollar} lower bound without significant loss of performance or accuracy.; We also demonstrate how a power algorithm can be used to compute a necessary condition for disturbance rejection of both discrete and continuous time nonlinear systems. For the general case of a system with a non-optimal controller this algorithm can provide us with knowledge of the worst case disturbance.; In the second part of this thesis we explore different approaches to systems model reduction. First, we show that balancing transformation and Galerkin projection commute. We also demonstrate that only if the balancing transformation matrix is orthogonal, balanced truncation and Galerkin projection commute.; Next, we pursue model reduction of nonlinear systems with rotational symmetry. We separate a movement of a wave from the evolution of the wave shape using the "centering procedure," and accurately approximate the shape of the wave with just few modes. The method may be viewed as a way of implementing the KLE on the space of solutions of the given PDE modulo a given symmetry group. The methodology is quite general and therefore should be useful in a variety of problems.