Vector Fitting.

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Vector Fitting.

Piero Triverio¹•Institutions (1)

23 Aug 2019-arXiv: Computational Physics-

TL;DR: The Vector Fitting algorithm for the creation of reduced-order models from the sampled response of a linear time-invariant system is introduced and how the reduced model can be made stable and converted to a variety of forms for use in virtually any modeling context is discussed.

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Abstract: We introduce the Vector Fitting algorithm for the creation of reduced-order models from the sampled response of a linear time-invariant system. This data-driven approach to reduction is particularly useful when the system under modeling is known only through experimental measurements. The theory behind Vector Fitting is presented for single- and multiple-input systems, together with numerical details, pseudocodes, and an open-source implementation. We discuss how the reduced model can be made stable and converted to a variety of forms for use in virtually any modeling context. Finally, we survey recent extensions of the Vector Fitting algorithm geared towards time-domain, parametric and distributed systems modeling.

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Citations

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Journal Article•DOI•

On the Acceleration of the Vector Fitting for Multiport Large-Scale Macromodeling

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Chiu-Chih Chou¹, Jose E. Schutt-Aine¹•Institutions (1)

University of Illinois at Urbana–Champaign¹

01 Jan 2021-IEEE Microwave and Wireless Components Letters

TL;DR: An alternative solution procedure is proposed, the complexity of which is 2.7 times smaller than the conventional approach in terms of floating-point operations (FLOPs) and presents a considerable saving for large-scale problems.

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Abstract: Vector fitting (VF) is a robust macromodeling method to construct rational models of a network based on tabulated frequency responses. When fitting a network with a large number of ports, the matrix equation in VF quickly grows into a formidable size. The conventional strategy is to break the equation into several small QR factorizations and then combine the results into a least-squares (LS) problem. In this letter, an alternative solution procedure is proposed, the complexity of which is 2.7 times smaller than the conventional approach in terms of floating-point operations (FLOPs). Test cases of 40–200-port data show that the actual computing time can be reduced by as much as 85%, which presents a considerable saving for large-scale problems.

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6 citations

Journal Article•DOI•

Global Momentum Estimation of an Electric Power System

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Angelo Brambilla, Davide del Giudice, Daniele Linaro, Federico Bizzarri

31 Mar 2023-arXiv.org

TL;DR: In this article , the authors present a data-driven approach to estimate the global momentum of an electric power system using a grid-forming converter-interfaced generator and the speed of its virtual rotor.

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Abstract: This work presents a technique to estimate on-line the global momentum of an electric power system. It exploits the footprint of the principal frequency system dynamics. Probing tones are injected by a grid-forming converter-interfaced generator and the speed of its virtual rotor is exploited to extrapolate the footprint. The proposed technique is entirely data driven and thus it does not need any model of the power system. We show that the proposed technique is adequate to accurately estimate the actual global momentum contributed by synchronous machines and provided by the controllers of converter-interfaced generators that emulate the behavior of synchronous machines. The technique is comprehensively tested on a modified version of the IEEE 39-BUS system and a dynamic version of the IEEE 118-BUS system, containing grid forming converter interfaced generators.

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An Incremental State Space Modelling Method of Power Systems Based on Measured Components Impedance Matrices in Local dq Frames

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Weihua Zhou, Yanbo Wang, Zhe Chen

01 Jun 2019

Journal Article•DOI•

Optimal readout scheme for multi-degree-of-freedom seismometers

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Denis Martynov, Zong-Hong Zhu, Haixing Miao

29 Jun 2023-Classical and Quantum Gravity

TL;DR: In this article , the authors proposed an optimal readout scheme by using optimal filters to combine the sensor outputs to improve the low-frequency performance of advanced gravitational-wave detectors, which is limited by the seismic noise and the associated control noise.

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Abstract: The low-frequency performance of advanced gravitational-wave detectors is limited by the seismic noise and the associated control noise. A six-degree-of-freedom seismometer has been proposed in (Mow-Lowry and Martynov 2019 Class. Quantum Grav. 36 245006) to improve the active isolation system. The standard readout scheme, which directly inverts the sensing matrix, gives dynamically independent but not statistically independent estimators for different degrees of freedom. This paper studies the optimal readout scheme by using optimal filters to combine the sensor outputs. The improvement is ultimately limited by the sensor noise and the ground motion, which decreases the correlation. For the real-time implementation, we have considered fitting the optimal filters numerically under the passive and causal constraint.

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References

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Book•

Matrix computations

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Gene H. Golub

01 Jan 1983

34,729 citations

"Vector Fitting." refers background in this paper

...Furthermore, the monomial terms (ω) in (8) will result in Vandermonde matrices in the least squares problem to be solved, which are ill-conditioned [28]....
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...A different approach is preferred, where (6) is linearized into a linear least squares problem, which can be solved efficiently and robustly with the QR decomposition [28]....
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...Specifically, the matrix of the least-squares problem to be solve will contain Vandermonde blocks [28], which are known to be ill conditioned even for relatively modest values of n̄;...
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...For reasons that will become clear later on, the Gilbert realization process begins with the truncated singular value decomposition [28] of residues Rn Rn = UnΣnV H n for n = 1, ....
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...System (21) can be solved in least-squares sense with a QR decomposition of the coefficient matrix [28]....
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Book•

Robust and Optimal Control

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John Doyle¹•Institutions (1)

California Institute of Technology¹

17 Aug 1995

TL;DR: This paper reviewed the history of the relationship between robust control and optimal control and H-infinity theory and concluded that robust control has become thoroughly mainstream, and robust control methods permeate robust control theory.

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Abstract: This paper will very briefly review the history of the relationship between modern optimal control and robust control. The latter is commonly viewed as having arisen in reaction to certain perceived inadequacies of the former. More recently, the distinction has effectively disappeared. Once-controversial notions of robust control have become thoroughly mainstream, and optimal control methods permeate robust control theory. This has been especially true in H-infinity theory, the primary focus of this paper.

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6,945 citations

Additional excerpts

...This mechanism is coupled 1The McMillan degree [93] of a matrix transfer function H(s) is the order of a minimal state space realization of H(s), such as the order N of the Gilbert realization discussed in Sec....
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Journal Article•

Robust and Optimal Control

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Kemin Zhou, John Doyle, Keith Glover

01 Jan 1997-Automatica

4,506 citations

Journal Article•DOI•

Rational approximation of frequency domain responses by vector fitting

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Bjorn Gustavsen, Adam Semlyen¹•Institutions (1)

University of Toronto¹

01 Jul 1999-IEEE Transactions on Power Delivery

TL;DR: The paper describes a general methodology for the fitting of measured or calculated frequency domain responses with rational function approximations by replacing a set of starting poles with an improved set of poles via a scaling procedure.

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Abstract: The paper describes a general methodology for the fitting of measured or calculated frequency domain responses with rational function approximations. This is achieved by replacing a set of starting poles with an improved set of poles via a scaling procedure. A previous paper (Gustavsen et al., 1997) described the application of the method to smooth functions using real starting poles. This paper extends the method to functions with a high number of resonance peaks by allowing complex starting poles. Fundamental properties of the method are discussed and details of its practical implementation are described. The method is demonstrated to be very suitable for fitting network equivalents and transformer responses. The computer code is in the public domain, available from the first author.

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2,950 citations

"Vector Fitting." refers background or methods in this paper

...The minimization of (34) is a nonlinear least squares problem, which VF solves iteratively by working on the linearized error [42] ( e (i) SK )2 = 1 k̄q̄m̄ k̄ ∑...
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...It can be shown that such zeros can be calculated as the eigenvalues of [42] { p n } = eig ( A(i−1) − bw ( c w )T) , (18)...
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...We are now ready to present the complete VF algorithm [42, 35], with a pseudocode implementation available in Algorithm 3....
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...The Vector Fitting (VF) algorithm [42, 35] is one of the most successful techniques for creating reduced-order models for linear systems starting from samples of their response....
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...The VF algorithm, conceived by Gustavsen and Semlyen [42], addresses both problems with a simple yet brilliant solution....
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Journal Article•DOI•

Complex-curve fitting

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E. C. Levy¹•Institutions (1)

TRW Inc.¹

01 May 1959-Ire Transactions on Automatic Control

TL;DR: In this paper, a method of evaluation of the polynomial coefficients is presented based on the minimization of the weighted sum of the squares of the errors between the absolute magnitudes of the actual function and the Polynomial ratio, taken at various values of frequency (the independent variable).

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Abstract: The mathematical analysis of linear dynamic systems, based on experimental test results, often requires that the frequency response of the system be fitted by an algebraic expression. The form in which this expression is usually desired is that of a ratio of two frequency-dependent polynomials. In this paper, a method of evaluation of the polynomial coefficients is presented. It is based on the minimization of the weighted sum of the squares of the errors between the absolute magnitudes of the actual function and the polynomial ratio, taken at various values of frequency (the independent variable). The problem of the evaluation of the unknown coefficients is reduced to that of the numerical solution of certain determinants. The elements of these determinants are functions of the amplitude ratio and phase shift, taken at various values of frequency. This form of solution is particularly adaptable to digital computing methods, be-because of the simplicity in the required programming. The treatment is restricted to systems which have no poles on the imaginary axis; i.e., to systems having a finite, steady-state (zero frequency) magnitude.

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827 citations