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 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/multi-output (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 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 H 2 approximation, we establish a connection with optimal 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. 1. Introduction. Rational functions provide significant advantages over other classes of approximating functions, such as polynomials or trigonometric functions, that are important in the approximation of functions that occur in engineering and scientific applications. Matrix-valued rational functions offer substantial additional flexibility and broaden the domain of applicability by providing the potential for the interpolation and approximation of parameterized families of multi-dimensional observations. For example, in a variety of engineering applications, the dynamics arising from multi-input/multi-output (MIMO) dynamical systems may be inaccessible to direct modeling, yet input-output relationships often may be observed as a function of frequency, yielding an enormous amount of data. In such cases, one may wish to deduce an empirical dynamical system model nominally represented as a matrix-valued rational function, that fits the …