A Discussion of “Rational Approximation of Frequency Domain Responses by Vector Fitting”

Hendrickx, W.; Dhaene, Tom

doi:10.1109/tpwrs.2005.860905

Cited by 82 publications

(52 citation statements)

References 8 publications

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“…For this reason, VF may be thought of as a representation of SK iteration in a well-chosen basis [32]. One of the points we make in this paper is that VF is more than that.…”

Section: Vector Fitting (Vf)mentioning

confidence: 99%

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Quadrature-Based Vector Fitting for Discretized $\mathcal{H}_2$ Approximation

Drmač¹,

Gugercin²,

Beattie³

2015

SIAM J. Sci. Comput.

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Abstract. Vector Fitting is a popular method of constructing rational approximants designed to fit given frequency response measurements. The original method, which we refer to as VF, is based on a least-squares fit to the measurements by a rational function, using an iterative reallocation of the poles of the approximant. We show that one can improve the performance of VF significantly, by using a particular choice of frequency sampling points and properly weighting their contribution based on quadrature rules that connect the least squares objective with an H 2 error measure. Our modified approach, designated here as QuadVF, helps recover the original transfer function with better global fidelity (as measured with respect to the H 2 norm), than the localized least squares approximation implicit in VF. We extend the new framework also to incorporate derivative information, leading to rational approximants that minimize system error with respect to a discrete Sobolev norm. We consider the convergence behavior of both VF and QuadVF as well, and evaluate potential numerical ill-conditioning of the underlying least-squares problems. We investigate briefly VF in the case of noisy measurements and propose a new formulation for the resulting approximation problem. Several numerical examples are provided to support the theoretical discussion.Key words. least squares, frequency response, model order reduction, vector fitting, transfer function AMS subject classifications. 34C20, 41A05, 49K15, 49M05, 93A15, 93C05, 93C151. Introduction. In many engineering applications, the dynamics that govern phenomenae of interest may be inaccessible to direct modeling, yet there may be an abundance of accurate frequency response measurements available. In such cases, one may build up an empirical dynamical system model that fits the measured frequency response data. This empirical system may then be used as a surrogate to predict behavior or derive control strategies.In other settings, one may have complete access to the underlying dynamical system of interest at least in principle (e.g., it may be an analytically derived computational model), however the full system may be a complex aggregate of many large subsystems, each perhaps representing diverse physics, and so it may be of such complexity that direct manipulation of the dynamical system is infeasible; potentially only simulation results would be available. Here, one may wish to capture the dominant dynamic features of the full aggregate system and realize them with a derived dynamical system (presumably of lower order) that can replicate the response characteristics of the full aggregate system. As before, this derived dynamical system may then be used as an efficient surrogate for the full system in contexts where performance is sensitive to model order.A natural formulation of this task leads one to a data fitting problem using rational functions and this ultimately is our principal focus. For convenience, we assume that the system of interest is a single-input/single-output...

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“…For this reason, VF may be thought of as a representation of SK iteration in a well-chosen basis [32]. One of the points we make in this paper is that VF is more than that.…”

Section: Vector Fitting (Vf)mentioning

confidence: 99%

“…Many authors have applied, modified, and analyzed VF, see e.g. [29], [32], [18], [17], [20], [19]. Our motivation for studying VF came initially from a desire to articulate the relationship between VF and optimal rational approximation, in particular, with H 2 -optimal model order reduction.…”

mentioning

confidence: 99%

Quadrature-Based Vector Fitting for Discretized $\mathcal{H}_2$ Approximation

Drmač¹,

Gugercin²,

Beattie³

2015

SIAM J. Sci. Comput.

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“…The Vector Fitting (VF) algorithm [6][7] has already been introduced in [8] to identify high frequency models of potential transformers from magnitude data. However, complicated magnitude responses demand a more robust implementation in order to achieve accurate approximations.…”

Section: Robust Transfer Function Identification Via An Enhanced Magnmentioning

confidence: 99%

“…3. The failure with VF occurs when the spectral factorization is applied to (6) in order to get (7). In fact, in order to get an accurate magnitude approximation, spectral factorization requires the proper form of the magnitude square rational function, given by the magVF method (i.e.…”

Section: Test Case: Fitting Of a Rational Functionmentioning

confidence: 99%

Robust transfer function identification via an enhanced magnitude vector fitting algorithm

Tommasi

Gustavsen

Dhaene

2010

IET Control Theory &Amp; Applications

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“…It minimizes a weighted linear cost function by iteratively relocating the transfer function poles using a Sanathanan-Koerner iteration [8], [12], [13]. Numerical ill conditioning is avoided by using a set of orthonormal rational basis functions, which are based on a prescribed set of poles.…”

Section: Ovfmentioning

confidence: 99%

Stability and Passivity Enforcement of Parametric Macromodels in Time and Frequency Domain

Deschrijver

Dhaene

2008

IEEE Trans. Microwave Theory Techn.

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Abstract-This paper presents a robust technique for the macromodeling of time-domain and frequency-domain responses, which are parameterized by one or more design variables. The representation of the multivariate macromodel ensures that stability of the transfer function poles is enforced by construction. Passivity of the parametric macromodel can be enforced in a post-processing step by perturbation of barycentric weights.

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A Discussion of “Rational Approximation of Frequency Domain Responses by Vector Fitting”

Cited by 82 publications

References 8 publications

Quadrature-Based Vector Fitting for Discretized $\mathcal{H}_2$ Approximation

Quadrature-Based Vector Fitting for Discretized $\mathcal{H}_2$ Approximation

Robust transfer function identification via an enhanced magnitude vector fitting algorithm

Stability and Passivity Enforcement of Parametric Macromodels in Time and Frequency Domain

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