Fast Approximation of the $H_\infty$ Norm via Optimization over Spectral Value Sets

Guglielmi, Nicola; Gürbüzbalaban, Mert; Overton, Michael L.

doi:10.1137/120875752

Cited by 47 publications

(66 citation statements)

References 9 publications

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“…"; A/ for a fixed value of ", we apply a root-finding algorithm to the nonlinear equation (20.28) in ", in order to determine " D r C .A/. In practice, Newton and Newton-bisection schemes turned out to be effective for this purpose [8,24]. Let .…”

Section: Lemma 1 ([57])mentioning

confidence: 99%

“…In [42], subspace acceleration techniques for both quantities were proposed for which, under certain mild conditions, locally superlinear convergence was proven. In practice, these subspace methods exhibit a quite robust convergence behavior, their local convergence is observed to be even locally quadratic, and they can be much faster than the methods from [24,31]. However, it is currently not clear how these methods can be extended to the more general situations discussed below.…”

Section: Lemma 1 ([57])mentioning

confidence: 99%

“…As shown in [24], such spectral value sets can again be realized by rank-1 perturbations, allowing for an extension of the algorithms discussed above. A further extension has been made in [8] for transfer functions of the form G.s/ D C.sE A/ 1 B CD.…”

Section: The H 1 Norm and Spectral Value Setsmentioning

confidence: 99%

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Distance Problems for Linear Dynamical Systems

Kreßner

Voigt

2015

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

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This chapter is concerned with distance problems for linear timeinvariant differential and differential-algebraic equations. Such problems can be formulated as distance problems for matrices and pencils. In the first part, we discuss characterizations of the distance of a regular matrix pencil to the set of singular matrix pencils. The second part focuses on the distance of a stable matrix or pencil to the set of unstable matrices or pencils. We present a survey of numerical procedures to compute or estimate these distances by taking into account some of the historical developments as well as the state of the art. IntroductionConsider a linear time-invariant differential-algebraic equation (DAE)with coefficient matrices E; A 2 R n n , a sufficiently smooth inhomogeneity f W OE0; 1/ ! R n , and an initial state vector x 0 2 R n . When E is nonsingular, (20.1) is turned into a system of ordinary differential equations by simply multiplying with E 1 on both sides. The more interesting case of singular E arises, for example, when imposing algebraic constraints on the state vector x.t/ 2 R n . Equations of this type play an important role in a variety of applications, including electrical circuits [54,55] and multi-body systems [48]. The analysis and numerical solution of more general DAEs (which also take into account time-variant coefficients, nonlinearities and time delays) is a central part of Volker Mehrmann's work, as witnessed by the monograph [43] and by the several other chapters of this book

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Section: Lemma 1 ([57])mentioning

confidence: 99%

Section: Lemma 1 ([57])mentioning

confidence: 99%

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Distance Problems for Linear Dynamical Systems

Kreßner

Voigt

2015

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

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“…In [3,4] authors propose a method for computing the distance of a stable polynomial to the set of unstable ones (both in the Hurwitz and in the Schur case). The method is based on the reformulation of the problem as the structured distance to instability of a companion matrix associated to a polynomial.…”

Section: Introductionmentioning

confidence: 99%

Stability markers in control theory

Polilov

Batiushina

2017

MATEC Web Conf.

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“…Therefore, in this paper we use another approach based on pseudospectra. Recently, also a similar method based on optimization over so-called spectral value sets was developed [16]. However, it does not take descriptor systems into account and uses a different optimization technique.…”

Section: Introductionmentioning

confidence: 99%

Numerical computation of structured complex stability radii of large-scale matrices and pencils

Benner

Voigt

2012

2012 IEEE 51st IEEE Conference on Decision and Control (CDC)

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Abstract-In this paper we discuss the problem of computing structured complex stability radii of large and sparse matrices and pencils. For this purpose we consider certain structured pseudospectra. To compute the structured complex stability radius we have to find the pseudospectrum which touches the imaginary axis. Therefore, we set up an iteration over the real part of the rightmost pseudoeigenvalue. For that we use a new fast iterative scheme which is based on certain rank-1 perturbations of the matrix or pencil. Finally, we illustrate the performance of our algorithm by using real-world example data.

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Fast Approximation of the $H_\infty$ Norm via Optimization over Spectral Value Sets

Cited by 47 publications

References 9 publications

Distance Problems for Linear Dynamical Systems

Distance Problems for Linear Dynamical Systems

Stability markers in control theory

Numerical computation of structured complex stability radii of large-scale matrices and pencils

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