The ADI iteration for Lyapunov equations implicitly performs<b>H2</b>pseudo-optimal model order reduction

Wolf, Thomas; Panzer, Heiko K. F.

doi:10.1080/00207179.2015.1081985

Cited by 18 publications

(23 citation statements)

References 30 publications

(81 reference statements)

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“…Motivated by the connection of LR-ADI to rational Krylov subspaces [15,17,28,51,52] we can borrow the greedy shift selection strategy from [16] which was developed for the rational Krylov subspace method for (1). Let S ⊂ C − be the convex hull of the set of Ritz values Λ(H ℓ ) and ∂S its boundary.…”

Section: Convex Hull Based Shiftsmentioning

confidence: 99%

Approximate residual-minimizing shift parameters for the low-rank ADI iteration

Kürschner¹

2019

etna

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The low-rank alternating directions implicit (LR-ADI) iteration is a frequently employed method for efficiently computing low-rank approximate solutions of large-scale Lyapunov equations. In order to achieve a rapid error reduction, the iteration requires shift parameters whose selection and generation is often a difficult task, especially for nonsymmetric coefficients in the Lyapunov equation. This article represents a follow up of Benner et al. [ETNA, 43 (2014[ETNA, 43 ( -2015, pp. 142-162] and investigates self-generating shift parameters based on a minimization principle for the Lyapunov residual norm. Since the involved objective functions are too expensive to evaluate and, hence, intractable, compressed objective functions are introduced which are efficiently constructed from the available data generated by the LR-ADI iteration. Several numerical experiments indicate that these residual minimizing shifts using approximated objective functions outperform existing precomputed and dynamic shift parameter selection techniques, although their generation is more involved.

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Section: Convex Hull Based Shiftsmentioning

confidence: 99%

Approximate residual-minimizing shift parameters for the low-rank ADI iteration

Kürschner¹

2019

etna

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“…In [4] and [16] it was independently shown, that the residual can be factorized as R = B ⊥ B T ⊥ -with real B ⊥ ∈ R n×m , if the set S is closed under conjugation. One way to compute B ⊥ is…”

Section: The Adi Iterationmentioning

confidence: 99%

“…The notation B ⊥ stems from the fact, that a specific projection can be associated to the ADI iteration: B ⊥ is the residual after projecting B, and it fulfills the Petrov-Galerkin condition, see [16] for details. The low-rank formulation R = B ⊥ B T ⊥ allows to compute the induced matrix 2-norm of the residual R 2 by the maximum eigenvalue of the m-by-…”

Section: The Adi Iterationmentioning

confidence: 99%

“…As it is advisable to employ T-LR-ADI with adaptively selected shifts s i and tangential directions b i , a first procedure of this kind is our third contribution. Because the results require the recently established connection of the ADI iteration to Krylov subspace projection methods [16], we briefly review this in the following subsections.…”

Section: The Problemmentioning

confidence: 99%

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ADI iteration for Lyapunov equations: A tangential approach and adaptive shift selection

Wolf

Panzer

Lohmann

2016

Applied Numerical Mathematics

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A new version of the alternating directions implicit (ADI) iteration for the solution of large-scale Lyapunov equations is introduced. It generalizes the hitherto existing iteration, by incorporating tangential directions in the way they are already available for rational Krylov subspaces. Additionally, first strategies to adaptively select shifts and tangential directions in each iteration are presented. Numerical examples emphasize the potential of the new results.

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“…Our method benefits from both Krylov [1] and Lyapunov techniques [2], [3], [4]. The proposed method generates a reduced order models with similar behavior of the original system, minimizes the absolute error, the H ∞ norm error, and preserves the stability of reduced system.…”

Section: Introductionmentioning

confidence: 99%

Lyapunov-Global-Lanczos Algorithm for Model Order Reduction & Adaptive PI Controller of Large Scale Electrical Systems

2017

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Abstract-Mathematical modeling of complex electrical systems, has led us to linear mathematical models of higher order. Consequently, it is difficult to analyse and to design a control strategy of these systems. The order reduction is an important and effective tool to facilitate the handling and designing of a control strategy. In this paper we present, firstly, a reduction method which is based on the Krylov subspace and Lyapunov techniques, that we call Lyapunov-Global-Lanczos. This method minimizes the H∞ norm error, absolute error and preserves the stability of the reduced system. It also provides a better reduced system of order 1, with closer behaviour to the original system. This first order system is used to design PI (Proportional-Integral) controller. Secondly, we implement an adaptive digital PI controller in a microcontroller. It calculates the PI parameters in real time, referring to the error between the desired and measured outputs and the initial values of PI controller, that were determined from the first order system. Two simulation examples and a real time experimentation are presented to show the effectiveness of the proposed algorithms.

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The ADI iteration for Lyapunov equations implicitly performsH2pseudo-optimal model order reduction

Cited by 18 publications

References 30 publications

Approximate residual-minimizing shift parameters for the low-rank ADI iteration

Approximate residual-minimizing shift parameters for the low-rank ADI iteration

ADI iteration for Lyapunov equations: A tangential approach and adaptive shift selection

Lyapunov-Global-Lanczos Algorithm for Model Order Reduction & Adaptive PI Controller of Large Scale Electrical Systems

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