National Instruments Xmath

Chapter 4 Frequency-Weighted Error Reduction

Xmath Model Reduction Module 4-4 ni.com

Most of these ideas are discussed in [Enn84], [AnL89], and [AnM89].

The function wtbalance( ) implements weighted reduction, with five

choices of error measure, namely EIS, EOS, EM, EMS, and E1 with arbitrary

V(jω). The first four are specifically for controller reduction, whereas the

last is not aimed specifically at this situation.

Several features of the algorithms are:

• Only the stable part of C is really reduced; the unstable part is copied

exactly into Cr.

• A modification of balanced realization truncation underpins the

algorithms, namely (frequency) weighted balanced truncation,

although to avoid numerical problems, the actual construction of

afrequency weighted balanced realization of C is avoided.

• Frequency weighted Hankel singular values can be computed,

and although no error bound formula is available (in contrast to the

unweighted problem), generally speaking there is little damage done in

reducing by a number of states equal to the number of (relatively) small

Hankel singular values.

The error measures themselves deserve certain comments:

• The two stability based measures EIS and EOS are derived from a

sufficiency condition for stability, rather than a necessity and

sufficiency condition, and so capture stability a little crudely.

• For any constant nonsingular N, the error measure EIS can be replaced

by and the robustness result remains

valid. Use of an N may improve or worsen the quality of the

approximation.

•Having T–Tr small normally ensures closeness of the closed-loop

impulse and step responses.

• In classical control especially, constraints on the loop gain can be

imposed (Minimum value of gain in one band, maximum value of gain

in another band, for example). None of the methods presented directly

addresses the task of retaining satisfaction of these constraints after

reduction of a high order acceptable controller. However, judicious use

of a weight V can assist. Suppose that above the closed-loop bandwidth

there is an overbound constraint on the loop gain, which is violated

when a controller reduction is performed (but not with the original

controller). At these frequencies, roughly PC and PCr are small, so that

. Introduction of a weight V in EMS penalizing

frequencies in the region in question will evidently encourage PCr to

better track PC.

NC C

–()PI CP+()

1–N1–

∞

–PC C

–()=