Stability-Preserving Rational Approximation Subject to Interpolation Constraints

Karlsson, Johan; Lindquist, Anders

doi:10.1109/tac.2008.929384

Cited by 9 publications

(15 citation statements)

References 26 publications

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“…The modified problem obtained by removing the a priori bound on the interpolants has been studied in [29]- [30]. In fact, by allowing the upper bound to tend to infinity, the entropy optimization problem becomes an optimization problem, and the interpolants are then parameterized in terms of poles rather than in terms of spectral zeros.…”

Section: Discussionmentioning

confidence: 99%

“…Moreover, by (15), , and therefore where and are polynomials of degrees and , respectively, and is fixed. Since the sublevel set of the trigonometric polynomials and satisfying (26) is convex for each , the problem is quasiconvex [6] and can be solved as problem (21) in [30]. Then is obtained from the optimal and , where is any factor of of appropriate dimension.…”

Section: B Approximation Of Interpolantsmentioning

confidence: 99%

“…It is also interesting to note that when the bound is removed and only stability is required, as in [30], the approximation is better in the low-frequency band , but worse in the high-frequency band . It seems as if approximation with a bound puts more emphasis on the region where the interpolant is close to the bound (i.e., ) at the expense of the region where .…”

Section: A Sensitivity Minimization (Continued)mentioning

confidence: 99%

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The Inverse Problem of Analytic Interpolation With Degree Constraint and Weight Selection for Control Synthesis

Karlsson

Georgiou

Lindquist

2010

IEEE Trans. Automat. Contr.

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The minimizers of certain weighted entropy functionals are the solutions to an analytic interpolation problem with a degree constraint, and all solutions to this interpolation problem arise in this way by a suitable choice of weights. Selecting appropriate weights is pertinent to feedback control synthesis, where interpolants represent closed-loop transfer functions. In this paper we consider the correspondence between weights and interpolants in order to systematize feedback control synthesis with a constraint on the degree. There are two basic issues that we address: we first characterize admissible shapes of minimizers by studying the corresponding inverse problem, and then we develop effective ways of shaping minimizers via suitable choices of weights. This leads to a new procedure for feedback control synthesis.

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Section: Discussionmentioning

confidence: 99%

Section: B Approximation Of Interpolantsmentioning

confidence: 99%

Section: A Sensitivity Minimization (Continued)mentioning

confidence: 99%

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The Inverse Problem of Analytic Interpolation With Degree Constraint and Weight Selection for Control Synthesis

Karlsson

Georgiou

Lindquist

2010

IEEE Trans. Automat. Contr.

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“…Computing a frequency response for particular applications (e.g., modeling of electro-magnetic structures) can be even cheaper, than inverting a state-space matrix A, as shown in [7], [8], [9]. One of the main tools for frequency domain approximation is the interpolation techniques ( [10], [11], [12]). Another tool is convex optimization as in the method proposed in [13], [14].…”

Section: G(z) − P(z)/q(z) Hmentioning

confidence: 99%

Semidefinite Hankel-Type Model Reduction Based on Frequency Response Matching

Sootla

2013

IEEE Trans. Automat. Contr.

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This paper is dedicated to model order reduction of linear time-invariant systems. The main contribution of this paper is the derivation of two scalable stability-preserving model reduction algorithms. Both algorithms constitute a development of a recently proposed model reduction method. The algorithms perform a curve fitting procedure using frequency response samples of a model and semidefinite programming methods. Computation of these samples can be done efficiently even for large scale models.Both algorithms are obtained from a reformulation of the model reduction problem. One proposes a semidefinite relaxation, while the other is an iterative semidefinite approach. The relaxation approach is similar to Hankel model reduction, which is a well-known and established method in the control literature. Due to this resemblance, the accuracy of approximation is also similar to the one of Hankel model reduction. An appealing quality of the proposed algorithms is the ability to easily perform extensions, e.g., introduce frequency-weighting, positive-real and bounded-real constraints.

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“…modeling of electro-magnetic structures) can be even cheaper, than inverting the state-space matrix A, as shown in [6], [7], [8]. The approximation from the frequency data is related to the celebrated Nevanlinna-Pick interpolation problem, an extension of which to Hardy spaces can be found in [9], [10] with a recent progress in [11], [12]. In [13] another approach was developed to obtain an approximation.…”

Section: Introductionmentioning

confidence: 99%

Hankel-type model reduction based on frequency response matching

Sootla

2010

49th IEEE Conference on Decision and Control (CDC)

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General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Hankel-type Model Reduction Based on Frequency Response Matching Aivar SootlaDepartment of Automatic Control, Lund University, Lund, Sweden, Box 118 SE 221 00Email: aivar@control.lth.seAbstract-In this paper, a stability preserving model reduction algorithm for single-input single-output linear time invariant systems is presented. It performs a data fitting in the frequency domain using semidefinite programming methods. Computing the frequency response of a model can be done efficiently even for large scale models making this approach applicable to those. The relaxation used to obtain a semidefinite program is similar to one used in Hankel model reduction. Therefore accuracy of approximation is also similar to Hankel model reduction one. The approach can be easily extended to frequency-weighted and parameter dependent model reduction problems.

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Stability-Preserving Rational Approximation Subject to Interpolation Constraints

Cited by 9 publications

References 26 publications

The Inverse Problem of Analytic Interpolation With Degree Constraint and Weight Selection for Control Synthesis

The Inverse Problem of Analytic Interpolation With Degree Constraint and Weight Selection for Control Synthesis

Semidefinite Hankel-Type Model Reduction Based on Frequency Response Matching

Hankel-type model reduction based on frequency response matching

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