Search

Let $ p_1 \lt p_2 \lt \cdots \lt p_n $ p 1 < p 2 < ⋯ < p n be positive numbers and r a non-negative real number. The Loewner matrix associated with the function $ x^{r+1} $ x r + 1 given by $ L_{r+1}=\begin {bmatrix}\frac {p_i^{r+1}-p_j^{r+1}}{p_i-p_j}\end {bmatrix} $ L r + 1 = [ p i r + 1 − p j r + 1 p i − p j ] and matrix $ P_r=[\begin {smallmatrix}{(p_i+p_j)^r}\end {smallmatrix}] $ P r = [ (p i + p j) r ] (the Hadamard inverse of rth Hadamard power of well-known Cauchy matrix) have same inertia. A question was left open in Inertia of Loewner matrices. Indiana Univ Math J. 2016;65(4):1251–1261 by Bhatia, Friedland and Jain to find a connection between these two matrix families. We aim to answer this question firmly in terms of a congruence relation between $ L_{r+1} $ L r + 1 and $ P_r $ P r . Indeed, a non-singular matrix X over $ \mathbb {R} $ R is explicitly obtained such that $ X'P_rX=L_{r+1} $ X ′ P r X = L r + 1 in this paper. [ABSTRACT FROM AUTHOR]

The exponential function maps the imaginary axis to the unit circle and for many applications this unitarity property is also desirable from its approximations. We show that this property is conserved not only by the |$(k,k)$| -rational interpolant of the exponential on the imaginary axis, but also by |$(k,k)$| -rational barycentric approximants that minimize a linearized approximation error. These results are a consequence of certain properties of singular vectors of Loewner-type matrices associated to linearized approximation errors. Prominent representatives of this class are rational approximants computed by the adaptive Antoulas–Anderson (AAA) method and the AAA–Lawson method. Our results also lead to a modified procedure with improved numerical stability of the unitarity property and reduced computational cost. [ABSTRACT FROM AUTHOR]

A state–space integer–order approximation of commensurate–order systems is obtained using a data–driven interpolation approach based on Loewner matrices. Precisely, given the values of the original fractional–order transfer function at a number of generalised frequencies, a descriptor–form state–space model matching these frequency response values is constructed from a suitable Loewner matrix pencil, as already suggested for the reduction of high–dimensional integer–order systems. Even if the stability of the resulting integer–order system cannot be guaranteed, such an approach is particularly suitable for approximating (infinite–dimensional) fractional–order systems because: (i) the order of the approximation is bounded by half the number of interpolation points, (ii) the procedure is more robust and simple than alternative approximation methods, and (iii) the procedure is fairly flexible and often leads to satisfactory results, as shown by some examples discussed at the end of the article. Clearly, the approximation depends on the location, spacing and number of the generalised interpolation frequencies but there is no particular reason to choose the interpolation frequencies on the imaginary axis, which is a natural choice in integer–order model reduction, since this axis does not correspond to the stability boundary of the original fractional–order system. [ABSTRACT FROM AUTHOR]

Let r be any real number and for any n let p1,...., pn be distinct positive numbers. A Kwong matrix is the n x n matrix whose (i, j) entry is (pri +prj)/(pi +pj). We determine the signatures of eigenvalues of all such matrices(The corresponding problem for the family of Loewner matrices [(pri +prj)/(pi - pj)] has been solved earlier. [ABSTRACT FROM AUTHOR]

The Loewner framework has recently been proposed for the system identification of mechanical systems, mitigating the limitations of current frequency domain fitting processes for the extraction of modal parameters. In this work, the Loewner framework computational performance, in terms of the elapsed time till identification, is assessed. This is investigated on a hybrid, numerical and experimental dataset against two well-established system identification methods (least-squares complex exponential, LSCE, and subspace state space system identification, N4SID). Good results are achieved, in terms of better accuracy than LSCE and better computational performance than N4SID. [ABSTRACT FROM AUTHOR]

Let p 1 , p 2 , ... , p n be distinct positive real numbers and m be any integer. Every symmetric polynomial f (x , y) ∈ C [ x , y ] induces a symmetric matrix f (p i , p j) i , j = 1 n. We obtain the determinants of such matrices with an aim to find the determinants of P m = (p i + p j) m i , j = 1 n and B 2 m = (p i − p j) 2 m i , j = 1 n for m ∈ N (where N is the set of natural numbers) in terms of the Schur polynomials. We also discuss and compute determinant of the matrix K m = p i m + p j m p i + p j i , j = 1 n for any integer m in terms of the Schur and skew-Schur polynomials. [ABSTRACT FROM AUTHOR]

We study the Kraus matrices associated with the function f (x) = x x − ε x 2 on (0 , 1) for 1 2 n + 2 < ε ≦ 1 2 n + 1 to show that f (x) is matrix convex of order n but not of order (n + 1) as an application of our preceding results on matrix monotone functions. [ABSTRACT FROM AUTHOR]

The Loewner framework has recently been proposed for the system identification of mechanical systems, mitigating the limitations of current frequency domain fitting processes for the extraction of modal parameters. In this work, the Loewner framework computational performance, in terms of the elapsed time till identification, is assessed. This is investigated on a hybrid, numerical and experimental dataset against two well-established system identification methods (least-squares complex exponential, LSCE, and subspace state space system identification, N4SID). Good results are achieved, in terms of better accuracy than LSCE and better computational performance than N4SID.

Let f be a positive, continuously differentiable function on (0 , α). We consider matrices K f of the form [ (f (p i) + f (p j)) / (p i + p j) ]. We show sufficient conditions for K f to be conditionally negative definite. We also study the positivity of determinants for two kinds of matrices [ (f (ξ i) + f (η j)) / (ξ i − η j) ] and [ (f (ξ i) − g (η j)) / (ξ i − η j) ]. [ABSTRACT FROM AUTHOR]

Let p 1 < p 2 < ⋯ < p n be positive numbers, then for any integer m the Loewner matrix associated with the function x m is given by L m = (p i m − p j m) / (p i − p j) i , j = 1 n. A question was left open in a paper [Bhatia R, Friedland S, Jain T. Inertia of loewner matrices. Indiana Univ Math J. 2016;65:1251–1261] by Bhatia, Friedland and Jain to find formulas for the determinants of the matrices L m in the case n = 3. We aim to answer this question firmly in terms of the Schur polynomials in this paper. [ABSTRACT FROM AUTHOR]

A selection of algorithms for the rational approximation of matrix-valued functions are discussed, including variants of the interpolatory adaptive Antoulas--Anderson (AAA) method, the rational Krylov fitting (RKFIT) method based on approximate least squares fitting, vector fitting, and a method based on low-rank approximation of a block Loewner matrix. A new method, called the block-AAA algorithm, based on a generalized barycentric formula with matrix-valued weights, is proposed. All algorithms are compared in terms of obtained approximation accuracy and runtime on a set of problems from model order reduction and nonlinear eigenvalue problems, including examples with noisy data. It is found that interpolation-based methods are typically cheaper to run, but they may suffer in the presence of noise for which approximation-based methods perform better. [ABSTRACT FROM AUTHOR]

In this article, we study the determinant of the matrix 2 a i + a j - ε and the positive definiteness/semidefiniteness of this matrix. As an application, we show the positivity gap of Kwong matrices for the function f (x) = 1 - ε x and that of Loewner ones for the function g (x) = x - ε x . [ABSTRACT FROM AUTHOR]

The article proposes a computationally efficient and robust method for estimating impulse or frequency response in the context of black-box multiple-input-multiple-output linear model identification of large power systems from simulations. The proposed approach is validated in conjunction with two realization algorithms – ERA method that is based on the established Ho-Kalman procedure for minimal realization, and the relatively new Loewner matrix based method. Practical aspects of identification in presence of numerical noise and nonlinearities are discussed. Study cases presented in the article include a linear model equivalent of a large-scale power system and the full-scale nonlinear transient model of the Eastern Interconnection. [ABSTRACT FROM AUTHOR]

This paper presents a theoretical review and comparisons between vector fitting, matrix-pencil-method, and Loewner matrix techniques for the fitting of frequency-domain functions. Firstly, the theoretical fundaments of each technique are briefly reviewed. Secondly, their computational performances and fitting accuracy are compared through different case studies. As for Loewner Matrix method, a novel implementation is proposed for a fair comparison with the other two techniques. Moreover, it is demonstrated that this novel implementation has some advantages over the traditional one. Finally, global remarks and recommendations are specified to take advantage of the capabilities exhibited by each technique. [ABSTRACT FROM AUTHOR]

An accurate framework for power system identification and model order reduction based on the Loewner matrix method is proposed. In this approach, frequency responses corresponding to non-symmetrical non-square MIMO transfer functions are introduced as tangential data in the Loewner interpolation process. Such responses can be either computed efficiently when explicit large sparse models are available or measured when only black-box models are at disposal. Numerical results are discussed for a couple of well-reported multi-machine test systems. The accuracy of the produced models and comparison against other state-of-the-art techniques are then discussed. [ABSTRACT FROM AUTHOR]

The Loewner framework for model reduction is extended to the class of bilinear systems. The main advantage of this framework over existing ones is that the Loewner pencil introduces a trade-off between accuracy and complexity. Furthermore, through this framework, one can derive state-space models directly from input-output data without requiring initial system matrices. The recently introduced methodology of Volterra series interpolation is also addressed. Several numerical experiments illustrate the main features of this approach. [ABSTRACT FROM AUTHOR]

This paper proposes a new approach for modelling frequency dependent power system network equivalents (FDNE) using tangential interpolation framework based on Loewner matrix (LM) pencil. The Loewner matrix based tangential interpolation technique has been recently proposed for modelling of large multi-port VLSI circuits. The LM approach fits accurate state space descriptor system models from the frequency response measurements. Singular value decomposition based LM approach has been investigated in this paper for modelling of FDNEs. The proposed method's performance is compared to the widely used vector fitting approach using four power system examples. A simple matrix implementation for LM formation and a MATLAB based stable model extraction approach is proposed. It has been shown that the data splitting step of the LM approach has significant impact on the accuracy of the fitting. The LM method is shown to be comparable to vector fitting in terms of accuracy, stability and passivity. It does not require any starting poles and has no convergence issues because it is a non-iterative method. It also gives an indication of the system order which is a unique advantage. [ABSTRACT FROM AUTHOR]

We introduce a new method for constructing reduced-order models that accurately approximate the frequency domain behavior of large-scale, parametrized, linear dynamical systems. Using multivariate rational interpolation, we compute reduced-order models that match transfer function measurements of the large-scale system. The main tools are a new barycentric formula and Loewner matrices formed directly from measurements. This data-driven approach introduces a new degree of freedom for parametrized model reduction--the ability to choose separate reduced orders for each parameter. More precisely, each reduced order is determined by computing the rank of appropriate Loewner matrices. Moreover, we show how to control the pointwise approximation error through a new formula involving the barycentric form and the smallest singular value of a Loewner matrix. Finally, we also give new state-space realizations for multivariate rational functions and conclude with several numerical examples showcasing the effectiveness of the new method. [ABSTRACT FROM AUTHOR]

In this paper we study positive operator monotone functions on (0,1) which have some differences from those on (0,∞): we show that for a concave operator monotone function f on (0,1), the Kwong matrices Kf (s1,..., sn) are positive semidefinite for all n and si ϵ (0,1), and f(sp)1/p for 0 < p ≤ 1 and s/f(s) are operator monotone. We also give a sufficient condition for the Kwong matrices to be positive semidefinite. [ABSTRACT FROM AUTHOR]

Abstract: We study upper bounds on the Schur multiplier norm of Loewner matrices for concave and convex functions. These bounds then immediately lead to upper bounds on the ratio of Schatten q-norms of commutators . We also consider operator monotone functions, for which sharper bounds are obtained. [Copyright &y& Elsevier]

Let f be a positive, continuously differentiable function on (0, ∞). We consider matrices K f of the form We show that for such a function f with f(0) = f ′(0) = 0, all K f are conditionally negative definite if and only if all are positive semidefinite (p.s.d.), and give the integral representation of f. We also study functions f with K f  (·, ·) conditionally negative definite or p.s.d. [ABSTRACT FROM AUTHOR]

Abstract: The goal of this contribution is to investigate interpolation of two-variable rational functions. The tool is the two-variable Loewner matrix, which is an extension of its single-variable counterpart. The main property of the Loewner matrix is that its rank encodes the information concerning minimal complexity interpolants. Both polynomial and generalized state-space (descriptor) realizations of interpolants are presented. Examples illustrate how two-variable rational functions can be recovered from appropriate measurements. [Copyright &y& Elsevier]

We give elementary proofs of the fact that the Loewner matrices $${[\frac{f(p_i) - f (p_j)}{p_i-p_j}]}$$ corresponding to the function f( t) = t on (0, ∞) are positive semidefinite, conditionally negative definite, and conditionally positive definite, for r in [0, 1], [1, 2], and [2, 3], respectively. We show that in contrast to the interval (0, ∞) the Loewner matrices corresponding to an operator convex function on (−1, 1) need not be conditionally negative definite. [ABSTRACT FROM AUTHOR]

• Investigates the applicability of MFTI for state space modeling of power system FDNEs. • Proposes a novel MATLAB® based implementation for Loewner matrix construction. • Impact of number of samples is studied for MFTI. • Properties of MFTI such as convergence speed, accuracy, stability and passivity are compared with vector fitting (VF) and VFTI. Vector format tangential interpolation (VFTI) framework for deriving controllable and observable descriptor state space model using Loewner matrix pencil from the transfer matrix data has been recently proposed for VLSI circuits. It uses data sampled directionally i.e. two columns or two rows of the transfer matrix, called tangential interpolation data. In VFTI the information contained in the frequency samples is not fully utilized and the accuracy gets affected with large number of samples. Matrix format tangential interpolation (MFTI) has been proposed as an enhancement to VFTI in the literature which uses the entire transfer matrix so that all the information contained in the samples is fully utilized. This paper investigates the applicability of MFTI for state space modeling of power system FDNEs and proposes a novel MATLAB® based implementation for Loewner matrix construction. Properties of MFTI such as convergence speed, accuracy, stability and passivity are compared with vector fitting (VF) and VFTI. Impact of number of samples is also studied. [ABSTRACT FROM AUTHOR]

• Review and numerical comparison of commonly used curve-fitting techniques: vector fitting, relaxed vector fitting, modal vector fitting, frequency-partitioning fitting (methods of Silveira, Campello, Noda), matrix pencil method and loewner method. • Review and numerical comparison of model order reduction (MOR) methods. • Three practical electrical systems used as case studies to investigate the performance of the methods. • Discussion and conclusions can be used to get a better institution on using the proper fitting method based on the frequency response of interest. It is a well-known practice to approximate the frequency-domain response of an electrical component or a subsystem with rational functions for electromagnetic transient (EMT) simulations of power systems. There are a variety of curve-fitting methods developed over time that offer different levels of accuracy, speed, and complexity. In some cases, the order of rational function may get very large to meet specified error criteria. Model order reduction (MOR) methods can be used to decrease the order of the function without a considerable deterioration of the approximation error. This paper presents a thorough review and comparison of the most popular curve-fitting methods, namely, the vector fitting (VF) method along with its later developments, the frequency-partitioning fitting (FpF) methods, matrix pencil method (MPM) and loewner matrix (LM) fitting technique. First, the fundamental theories of each one are briefly reviewed. Then, their accuracy and fitting order are compared together through three case studies. Lastly, the application of two different MOR methods to the resulted rational function is investigated. [ABSTRACT FROM AUTHOR]

Summary: We introduce a data‐driven model order reduction approach that represents an extension of the Loewner framework for linear and bilinear systems to the case of quadratic‐bilinear (QB) systems. For certain types of nonlinear systems, one can always find an equivalent QB model without performing any approximation whatsoever. An advantage of the Loewner framework is that information about the redundancy of the given data is explicitly available, by means of the singular values of the Loewner matrices. This feature is also valid for the proposed generalization. As for the linear and bilinear cases, these matrices can be directly computed by solving Sylvester equations. We begin by defining generalized higher‐order transfer functions for QB systems. These multivariate rational functions play an important role in the model order reduction process. We construct reduced‐order systems for which the associated transfer functions match those corresponding to the original system at selected tuples of interpolation points. Another benefit of the proposed approach is that it is data‐driven oriented, in the sense that one would only need computed or measured samples to construct a reduced‐order QB system. We illustrate the practical applicability of the proposed method by means of several numerical experiments. [ABSTRACT FROM AUTHOR]