Calculating the $H_{\infty}$-norm Using the Implicit Determinant Method

Freitag, Melina A.; Spence, A.; Dooren, Paul Van

doi:10.1137/130933228

Cited by 16 publications

(21 citation statements)

References 22 publications

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“…As discussed in [32], there is always the possibility to check whether H .˛ / has purely imaginary eigenvalues, but this global optimality certificate may become too expensive for large-scale systems. As discussed in [21], the described algorithm can be extended in a rather straightforward manner to H 1 norm computations, even for the general case of descriptor systems. For this purpose, one only needs to replace the .1; 1/-block in M .!…”

Section: The Implicit Determinant Methodsmentioning

confidence: 99%

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: The Implicit Determinant Methodsmentioning

confidence: 99%

Distance Problems for Linear Dynamical Systems

Kreßner

Voigt

2015

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

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“…For larger problems, several approaches have been proposed in recent years. For instance, the characterization of the L ∞ -norm via a Hamiltonian eigenvalue problem has been used to formulate an associated root-finding problem which can be solved using Newton's method [14]. This approach requires solutions of linear systems of size equal to the order of the system.…”

Section: Literaturementioning

confidence: 99%

“…Similarly, the Lipschitz constant γ 2 in (15) can be expressed in terms of η, ζ and an upper bound on H (j) (iω) 2 for j = 1, 2 and for all ω ∈ I. Finally, equation (14) and inequalities (15) yield…”

Section: Rate Of Convergence Analysismentioning

confidence: 99%

Large-Scale Computation of $\mathcal{L}_\infty$-Norms by a Greedy Subspace Method

Aliyev

Benner

Mengi

et al. 2017

SIAM J. Matrix Anal. & Appl.

131

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We are concerned with the computation of the L∞-norm for an L∞-function of the form H(s) = C(s)D(s) −1 B(s), where the middle factor is the inverse of a meromorphic matrix-valued function, and C(s), B(s) are meromorphic functions mapping to shortand-fat and tall-and-skinny matrices, respectively. For instance, transfer functions of descriptor systems and delay systems fall into this family. We focus on the case where the middle factor is large-scale. We propose a subspace projection method to obtain approximations of the function H where the middle factor is of much smaller dimension. The L∞-norms are computed for the resulting reduced functions, then the subspaces are refined by means of the optimal points on the imaginary axis where the L∞-norm of the reduced function is attained. The subspace method is designed so that certain Hermite interpolation properties hold between the largest singular values of the original and reduced functions. This leads to a locally superlinearly convergent algorithm with respect to the subspace dimension, which we prove and illustrate on various numerical examples.

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“…1 For a different approach to approximating r · C when n is large, namely the "implicit determinant" method, see [FSVD14].…”

Section: An Ordinary Differential Equationmentioning

confidence: 99%

Approximating the Real Structured Stability Radius with Frobenius-Norm Bounded Perturbations

Guglielmi¹,

Gürbüzbalaban²,

Mitchell³

et al. 2017

SIAM J. Matrix Anal. & Appl.

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We propose a fast method to approximate the real stability radius of a linear dynamical system with output feedback, where the perturbations are restricted to be real valued and bounded with respect to the Frobenius norm. Our work builds on a number of scalable algorithms that have been proposed in recent years, ranging from methods that approximate the complex or real pseudospectral abscissa and radius of large sparse matrices (and generalizations of these methods for pseudospectra to spectral value sets) to algorithms for approximating the complex stability radius (the reciprocal of the H∞ norm). Although our algorithm is guaranteed to find only upper bounds to the real stability radius, it seems quite effective in practice. As far as we know, this is the first algorithm that addresses the Frobenius-norm version of this problem. Because the cost mainly consists of computing the eigenvalue with maximal real part for continuous-time systems (or modulus for discrete-time systems) of a sequence of matrices, our algorithm remains very efficient for large-scale systems provided that the system matrices are sparse.

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Calculating the $H_{\infty}$-norm Using the Implicit Determinant Method

Cited by 16 publications

References 22 publications

Distance Problems for Linear Dynamical Systems

Distance Problems for Linear Dynamical Systems

Large-Scale Computation of $\mathcal{L}_\infty$-Norms by a Greedy Subspace Method

Approximating the Real Structured Stability Radius with Frobenius-Norm Bounded Perturbations

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