Transfer Function Interpolation

Transfer function interpolation is an approach for parameter-preserving model order reduction which is based on a combination of balanced truncation (or any other model order reduction method for deterministic linear, time-invariant systems) at certain distinct parameter values (the interpolation points) with interpolation. The approach is quite flexible in allowing the use of numerous interpolation techniques like polynomial, Hermite, rational, sinc, and spline interpolation.

Description

The method will exemplarily be introduced for reducing a system given by the transfer function

$G(s,p) = C(sE - A1- p A2 )^{-1}B.$

The PMOR approach was originally proposed in using polynomial interpolation. It can simply be extended to a hybrid approach of balanced truncation applying different kinds of interpolation.

In the following rational interpolation will be employed for the interpolation of the locally reduced-order transfer functions.

It is assumed that for $k+1$ interpolation points $p_0, \dots, p_k$ somehow distributed over the parameter interval, the underlying non-parametric systems

$G(s,p_j) = C( sE - A1- p_j A2)^{-1} B, \quad j = 0,\dots k,$

are stable, i.e. all finite eigenvalues of the regular pencils $\lambda E - A1 - p_j A2$ lie in the open left half of the complex plane.

Then, balanced truncation can be applied to $G(s,p_j)$ leading to reduced-order systems of order $r_j$

$\hat G_j(s) := \hat G(s,p_j) = \hat C_j(s \hat E_j - \hat A1_j - \hat A2_j)^{-1}\hat B_j, \quad \textrm{for} \ j=0,\dots,k.$

The reduced-order transfer function over the whole parameter interval is obtained by rational interpolation, e.g. by use of the barycentric formula

$\hat G_I(s,p) = \frac{\textrm{num}(s,p)}{\textrm{den}(s,p)} = \frac{\sum\limits_{j=0}^k \frac{u_j}{p - p_j} \hat G_j(s) }{\sum\limits_{j=0}^{k} \frac{u_j}{p - p_j}}$

for real numbers $u_j\ne 0$ and with numerator and denominator of degree at most $k$ .

Higher dimensional parameter spaces

For systems including more than one parameter, the problem of finding a reduced-order interpolating function over the whole parameter space

is much more involved. The main reason for that is the exponentially growing number of interpolation points for higher dimensional parameter spaces.

This leads to a high computationally complexity since balanced truncation has to be applied many times. Furthermore, the order of the reduced-order

system grows exponentially as well.

A strategy for an effective and representative choice of parameter points in higher dimensional parameter spaces

comes through the use of sparse grids, see, e.g. , , .

This approach is based on a hierarchical basis and a sparse tensor product construction. Significantly less interpolation points are needed for obtaining a similar accuracy as interpolation in a full grid space.

A coupling of balanced truncation with piecewise polynomial interpolation using sparse grid points was described in .

Numerical results

Numerical results for the anemometer benchmark can be found in .

Contact

Ulrike Baur