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A predictor–corrector type algorithm for the pseudospectral abscissa computation of time-delay systems

Authors
  • Gumussoy, Suat
  • Michiels, Wim
Type
Published Article
Journal
Automatica
Publisher
Elsevier
Publication Date
Jan 01, 2010
Accepted Date
Dec 17, 2009
Volume
46
Issue
4
Pages
657–664
Identifiers
DOI: 10.1016/j.automatica.2010.01.032
Source
Elsevier
Keywords
License
Unknown

Abstract

The pseudospectrum of a linear time-invariant system is the set in the complex plane consisting of all the roots of the characteristic equation when the system matrices are subjected to all possible perturbations with a given upper bound. The pseudospectral abscissa is defined as the maximum real part of the characteristic roots in the pseudospectrum and, therefore, it is for instance important from a robust stability point of view. In this paper we present an accurate method for the computation of the pseudospectral abscissa of retarded delay differential equations with discrete pointwise delays. Our approach is based on the connections between the pseudospectrum and the level sets of an appropriately defined complex function. The computation is done in two steps. In the prediction step, an approximation of the pseudospectral is obtained based on a rational approximation of the characteristic matrix and the application of a bisection algorithm. Each step in this bisection algorithm relies on checking the presence of the imaginary axis eigenvalues of a complex matrix, similar to the delay free case. In the corrector step, the approximate pseudospectral abscissa is corrected to any given accuracy, by solving a set of nonlinear equations that characterizes the extreme points in the pseudospectrum contours.

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