National Instruments Xmath

Chapter 4 Frequency-Weighted Error Reduction

• Reduce the order of a transfer function matrix C(s) through

frequency-weighted balanced truncation, a stable frequency weight

V(s) being prescribed.

The syntax is more accented towards the first use. For the second use,

the user should set S=0, NS= 0. This results in (automatically)

SCLR=NSCLR = 0. The user will also select the type="input

spec".

Let Cr(s) be the reduced order approximation of C(s) which is being

sought. Its order is either specified in advance, or the user responds to

a prompt after presentation of the weighted Hankel singular values.

Then the different types concentrate on (approximately) minimizing

certain error measures, through frequency weighted balanced

truncation. These are shown in Table4-1.

These error measures have certain interpretations, as shown in Table4-2.

In case C(s) is not a compensator in a closed-loop and the error measure

is of interest, you can work with type="input spec" and C', V' in lieu

of C and V.

There is no restriction on the stability of C(s) [or indeed of P(s)] in the

algorithm, though if C(s) is a controller the closed-loop must be stabilizing.

Also, V(s) must be stable. Hence all weights (on the left or right of

C(jω)–Cr(jω) in the error measures) will be stable. The algorithm,

however, treats unstable C(s) in a special way, by reducing only the stable

part of C(s) (under additive decomposition) and copying the unstable part

into Cr(s).

Table 4-1. Types versus Error Measures

Type Error Measure

"input stab"

"output stab"

"match"

"match spec"

"input spec"

–[]PI CP+[]

1–

∞

IPC+[]

1–PC C

–[]

∞

IPC+[]

1–PC C

–[]IPC+[]

1–

∞

IPC+[]

1–PC C

–[]IPC+[]

1–V∞

–[]V∞

Vjω()Cjω()Crjω()–[]

∞