Two numerical methods for optimizing matrix stability
type: Item
instance of: scholarly article
title: Two numerical methods for optimizing matrix stability
zbMATH Open document ID: 1005.65041
DOI: 10.1016/S0024-3795(02)00260-4
author: Michael L. Overton
author: James V. Burke
author: A. S. Lewis
published in: Linear Algebra and its Applications
publication date: 2002-08-28T00:00:00Z
review text: The authors consider the affine matrix family \(A(x)=A_{0}+ \sum_{k=1}^{m}x_{k}A_{k}\) where \(x\in R^{m}\) and \(A_{k}\) are \( n\times n\) matrices and the question is to minimize its spectral abscissa, i.e. the largest real part of the eigenvalues. This is a common problem in control theory, which usually arises in the stabilization of dynamical systems by output feedback. Here two methods and their algorithms are proposed: the first is to find a local minimizer by a direction search and is based on the random gradient bundle method; The second method, the robust spectral abscissa, is based on a bilinear matrix inequality and looks at the largest eigenvalue as a functional over the set of positive definite matrices. The algorithms associated with both methods, which compute local minimizers, are described: these are the random gradient bundle with the line search and the Newton barrier method. Both algorithms are implemented in the numerical examples section.
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 93D09
zbMATH DE Number: 1790278
zbMATH Keywords: spectral abscissa
zbMATH Keywords: robust spectral abscissa
zbMATH Keywords: optimal stability
zbMATH Keywords: eigenvalues
zbMATH Keywords: stabilization
zbMATH Keywords: dynamical systems
zbMATH Keywords: output feedback
zbMATH Keywords: algorithms
zbMATH Keywords: random gradient bundle method
zbMATH Keywords: matrix inequality
zbMATH Keywords: Newton barrier method
zbMATH Keywords: numerical examples
reviewed by: Amin Boumenir
describes a project that uses: Mosek
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1790278