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Vector Fitting for Matrix-valued Rational Approximation (CROSBI ID 243038)

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Drmač, Zlatko ; Gugercin, Serkan ; Beattie, Christopher Vector Fitting for Matrix-valued Rational Approximation // SIAM journal on scientific computing, 37 (2015), 5; A2346, 33. doi: 10.1137/15M1010774

Podaci o odgovornosti

Drmač, Zlatko ; Gugercin, Serkan ; Beattie, Christopher

engleski

Vector Fitting for Matrix-valued Rational Approximation

Vector Fitting VF) is a popular method of constructing rational approximants that provides a least squares fit to frequency response measurements. In an earlier work, we provided an analysis of VF for scalar-valued rational functions and established a connection with optimal ${; ; ; \mathcal{; ; ; H}; ; ; }; ; ; _2$ approximation. We build on this work and extend the previous framework to include the construction of effective rational approximations to matrix-valued functions, a problem which presents significant challenges that do not appear in the scalar case. Transfer functions associated with multi-input/multioutput (MIMO) dynamical systems typify the class of functions that we consider here. Others have also considered extensions of VF to matrix-valued functions and related numerical implementations are readily available. However, to the best of our knowledge, a detailed analysis of numerical issues that arise does not yet exist. We offer such an analysis including critical implementation details here. One important issue that arises for VF on matrix-valued functions that has remained largely unaddressed is the control of the McMillan degree of the resulting rational approximant ; the McMillan degree can grow very high in the case of large input/output dimensions. We introduce two new mechanisms for controlling the McMillan degree of the final approximant, one based on alternating least-squares minimization and one based on ancillary system-theoretic reduction methods. Motivated in part by our earlier work on the scalar VF problem as well as by recent innovations for computing optimal $\mathcal{; ; ; H}; ; ; _2$ approximation, we establish a connection with optimal $\mathcal{; ; ; H}; ; ; _2$ approximation, and are able to improve significantly the fidelity of VF through numerical quadrature, with virtually no increase in cost or complexity. We provide several numerical examples to support the theoretical discussion and proposed algorithms.

least squares, frequency response, model order reduction, MIMO vector fitting, transfer function

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Podaci o izdanju

37 (5)

2015.

A2346

33

objavljeno

1064-8275

1095-7197

10.1137/15M1010774

Povezanost rada

Matematika

Poveznice
Indeksiranost