Eigenvalue Perturbation Approaches for Robust Control Problems

Master's thesis, Koc University · Advisor: Emre Mengi

Abstract

This thesis work derives characterizations and algorithms via eigenvalue perturbation theory for several quantities of interest in robust control. The numerical estimation of the quantity in each part is based on the derived characterization, and involves the optimization of singular values of matrix-valued functions.

A rectangular polynomial matrix (a rectangular matrix-valued function with polynomial entries) generically has no eigenvalue. The first part focuses on the distance to a nearest rectangular polynomial matrix with an eigenvalue under general complex perturbations, as well as more restricted real perturbations. Controllability radii under complex and real perturbations for a linear time-invariant control system are special cases. Especially, the real perturbation case is a challenging open problem. We derive singular value optimization characterization and bounds for the complex distance and real distance, respectively. Furthermore, we devise algorithms to estimate the distance based on the derived singular value optimization problems expressed in polar coordinates. The algorithms exploit the level-sets and Lipschitz continuity of the objective singular value functions. Global convergence and quick rate-of-convergence results are shown formally for the algorithms.

In the second part, we focus on the estimation of the \(\epsilon\)-pseudospectral radius of a matrix and, more generally, an analytic matrix-valued function, which is the modulus of the outermost point in the \(\epsilon\)-pseudospectrum (the set consisting of eigenvalues of all nearby matrices and matrix-valued functions, respectively, at a prescribed distance \(\epsilon\) or closer). The estimation of the pseudospectral radius is motivated by robust stability and transient behavior considerations for discrete systems. All algorithms to date for large problems converge at best to locally outermost points. We put eigenvalue perturbation theory in use to obtain a first-order expansion of the pseudospectral radius, and even derive a second-order expansion in the matrix case. These expansions yield simple estimates for the \(\epsilon\)-pseudospectral radius for small \(\epsilon\). For larger \(\epsilon\), we propose fixed-point iterations, well-suited especially for large problems and likely to converge to the globally outermost points with initializations from perturbation theory.

The third part extends the approach in the second part to a structured \(\epsilon\)-pseudospectral abscissa (real part of the rightmost point in a structured \(\epsilon\)-pseudospectrum). The \(\mathcal{H}_\infty\)-norm is one of the widely employed norms for a control system, and turns out to be the reciprocal of \(\epsilon\) such that the structured \(\epsilon\)-pseudospectral abscissa is zero. Built on this characterization of the \(\mathcal{H}_\infty\)-norm, we propose a Newton's method based approach coupled with the fixed-point iteration for the structured \(\epsilon\)-pseudospectrum for estimating the \(\mathcal{H}_\infty\)-norm. All of our approaches are illustrated by performing numerical experiments on random examples and benchmark examples from real-life problems.

Software