More on pseudospectra for polynomial eigenvalue problems and applications in control theory

Higham, Nicholas J. and Tisseur, Françoise (2002) More on pseudospectra for polynomial eigenvalue problems and applications in control theory. Elsevier, Linear Algebra and its Applications, 351-35. pp. 435-453. ISSN 0024-3795

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Abstract

Definitions and characterizations of pseudospectra are given for rectangular matrix poly-nomials expressed in homogeneous form: P(α,β)=α^dA_d+α^{d−1}βA_{d−1}+...+β^dA_0. It is shown that problems with infinite (pseudo)eigenvalues are elegantly treated in this framework. For such problems stereographic projection onto the Riemann sphere is shown to provide a convenient way to visualize pseudospectra. Lower bounds for the distance to the nearest nonregular polynomial and the nearest uncontrollable dth order system (with equality for standard state-space systems) are obtained in terms of pseudospectra, showing that pseudospectra are a fundamental tool for reasoning about matrix polynomials in areas such as control theory. How and why to incorporate linear structure into pseudospectra is also discussed by example.

Item Type: Article
Uncontrolled Keywords: Polynomial eigenvalue problem; λ-Matrix; Matrix polynomial; Homogeneous form; Pseudospectrum; Stereographic projection; Riemann sphere; Nearest nonregular polynomial; Nearest uncontrollable system; Structured perturbations
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Ms Lucy van Russelt
Date Deposited: 27 Jun 2006
Last Modified: 20 Oct 2017 14:12
URI: http://eprints.maths.manchester.ac.uk/id/eprint/325

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