2 research outputs found
Quadrature-Based Vector Fitting: Implications For H2 System Approximation
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 H2 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 H2 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
Transfer Function Estimation in System Identification Toolbox via Vector Fitting
This paper considers black- and grey-box continuous-time transfer function
estimation from frequency response measurements. The first contribution is a
bilinear mapping of the original problem from the imaginary axis onto the
unitdisk. This improves the numerics of the underlying Sanathanan-Koerner
iterations and the more recent instrumental-variable iterations. Orthonormal
rational basis functions on the unit disk are utilized. Each iteration step
necessitates a minimal state-space realization with these basis functions. One
such derivation is the second contribution. System identification with these
basis functions yield zero-pole-gain models. The third contribution is an
efficient method to express transfer function coefficient constraints in terms
of the orthonormal rational basis functions. This allows for estimating
transfer function models with arbitrary relative degrees (including improper
models), along with other fixed and bounded parameter values. The algorithm is
implemented in the tfest function in System Identification Toolbox (Release
2016b, for use with MATLAB) for frequency domain data. Two examples are
presented to demonstrate the algorithm performance.Comment: 20th IFAC World Congress, Toulouse, France, July 9-14, 201