Extensions to the structured singular value

Citation

Khatri, Sven H. (1999) Extensions to the structured singular value. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/aw0t-7881. https://resolver.caltech.edu/CaltechETD:etd-02082008-161357

Abstract

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There are two basic approaches to robustness analysis. The first is Monte Carlo analysis which randomly samples parameter space to generate a profile for the typical behavior of the system. The other approach is fundamentally worst case, where the objective is to determine the worst behavior in a set of models. The structured singular value, [...], is a powerful frame work for worst case analysis. Where [...] is a measure of the distance to singularity using the co-norm.

Under the appropriate projection, the uncertainty sets in the standard [...] framework that admit analysis are hypercubes. In this work, [...] and the computation of the bounds is extended to spherical sets or equivalently measuring the distance to singularity using the 2-norm. The upper bound is constructed by converting the spherical set of operators into a quadratic form relating the input and output vectors. Using a separating hyperplane or the S-procedure, a linear matrix inequality (LMI) upper bound can be constructed which is tighter than and consistent with the standard p upper bound. This new upper bound has special structure that can be exploited for efficient computation and the standard power algorithm is extended to compute lower bounds for spherical [...]. The upper bound construction is further generalized to more exotic regions like arbitrary ellipsoids, the Cartesian product of ellipsoids, and the intersection of ellipsoids. These generalizations are unified with the standard structures. These new tools enable the analysis of more exotic descriptions of uncertain models.

For many real world problems, the worst case paradigm leads to overly pessimistic answers and Monte Carlo methods are computationally expensive to obtain reasonable probabilistic descriptions for rare events.

A few natural probabilistic robustness analysis questions are posed within the [...] framework. The proper formulation is as a mixed probabilistic and worst case uncertainty structure. Using branch and bound algorithms, an upper bound can be computed for probabilistic robustness. Motivated by this approach, a purely probabilistic [...] problem is posed and bounds are computed. Using the existing machinery, the branch and bound computation cost grows exponentially in the average case for questions of probabilistic robustness. This growth is due to gridding an n-dimensional surface with hypercubes.

A motivation for the extensions of [...] to other uncertainty descriptions which admit analysis is to enable more efficient gridding techniques than just hypercubes. The desired fundamental region is a hypercube with a linear constraint. The motivation for this choice is the rank one problem. For rank one, the boundary of singularity is a hyperplane, but the conventional branch and bound tools still result in exponential gridding growth.

The generalization of the [...] framework is used to formulate an LMI upper bound for [...] with the linear constraint on uncertainty space. This is done by constructing the upper bound for the intersection of an eccentric ellipsoid with the standard uncertainty set. A more promising approach to this computation is the construction of an implicit [...] problem where the linear constraints on the uncertainty can be generically rewritten as an algebraic constraint on signals. This may lead to improvements on average to the branch and bound algorithms for probabilistic robustness analysis.

Thesis Files