A new algorithm for L2 optimal model reduction

Spanos, John T.; Milman, Mark H.; Mingori, D. L.

doi:10.1016/0005-1098(92)90143-4

Cited by 111 publications

(74 citation statements)

References 15 publications

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“…This has allowed effective data-driven H 2 -optimal system approximation for a much larger class of functions, including many that are not necessarily rational such as arise with delay systems. For more details on optimal H 2 approximation, see [3,27,43,51,55] and references therein.…”

Section: Vector Fitting (Vf)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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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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Section: Vector Fitting (Vf)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 weighted model reduction problem is tackled most often using variants of frequency-weighted balanced truncation; see Varga and Anderson [2003], Anderson and Liu [2002], Schelfhout and De Moor [2002], Gugercin and Antoulas [2004], Enns [1984], Wang et al [2002], Lin and Chiu [1992], Wang et al [1999], Sreeram and Ghafoor [2005] and references therein. For approaches that seek to minimize a weighted-H 2 error, see Halevi [1992] and Spanos et al [1992].…”

Section: Weighted Model Reductionmentioning

confidence: 99%

Weighted Model Reduction via Interpolation

Beattie

Gugercin

2011

IFAC Proceedings Volumes

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“…The use of a reduced-order model makes it more efficient to implement system analyses, predictions and control designs. Model reduction problems have been discussed using a number of approaches [1][2][3][4][5][6][7][8][9][10][11]. One of the popular approaches is the L 2 optimal model reduction.…”

Section: Introductionmentioning

confidence: 99%

“…If an additional time delay is introduced into the reduced-order model, the approximation might be greatly improved with fewer parameters of the rational part. Although some model reduction methods with a time delay have been presented [1,[5][6][7][8][9][10], the time delay is known a priori [5], searched for one by one [6] or calculated by a gradient-based method [7,9].…”

Section: Introductionmentioning

confidence: 99%

Model Reduction with Time Delay Combining Least-Squares Method with Artificial Bee Colony Algorithm

Hachino

Sameshima

Takata

et al. 2013

Journal of Signal Processing

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In this paper we propose a novel method of model reduction with a time delay for single-input, single-output continuous-time systems by a separable least-squares (LS) approach. The reduced-order model is determined by minimizing the integral of the magnitude squared of the transfer function error. The denominator parameters and time delay of the reduced-order model are represented by the positions of the food sources of the employed bees and searched for by the artificial bee colony algorithm, while the numerator parameters are estimated by the linear LS method for each candidate of the denominator parameters and time delay. All the best parameters and the time delay of the reduced-order model are obtained through the search by the employed, onlooker and scout bees. Simulation results show that the accuracy of the proposed method is superior to that of the genetic algorithm (GA)-based model reduction algorithm.

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A new algorithm for L2 optimal model reduction

Cited by 111 publications

References 15 publications

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

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

Weighted Model Reduction via Interpolation

Model Reduction with Time Delay Combining Least-Squares Method with Artificial Bee Colony Algorithm

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