National Instruments Xmath

Chapter 4 Frequency-Weighted Error Reduction

3. Compute weighted Hankel Singular Values σi (described in more

detail later). If the order of Cr(s) is not specified a priori, it must be

input at this time. Certain values may be flagged as unacceptable for

various reasons. In particular nscr cannot be chosen so that

σnscr=σnscr +1

4. Execute reduction step on stable part of C(s), based on a modification

of redschur( ) to accommodate frequency weighting, and yielding

stable part of Cr(s).

5. Compute Cr(s) by adding unstable part of C(s) to stable part of Cr(s).

6. Check closed-loop stability with Cr(s) introduced in place of C(s),

atleast in case C(s) is a compensator.

More details of steps3 and 4, will be given for the case when there is an

input weight only. The case when there is an output weight only is almost

the same, and the case when both weights are present is very similar, refer

to [Enn84a] for a treatment. Let

be a stable transfer function matrix to be reduced and its stable weight.

Thus, W(s) may be P(I+CP)–1, corresponding to "input stab", and will

thus have been calculated in step2; or it maybe an independently specified

stable V(s). Then

The controllability grammian P satisfying

is written as

Cs() DcCcsI Ac

–()

1–Bc

WSs() DwCwsI Aw

–()

1–Bw

Css()Ws() DcDwCcDcCw

sI Ac

–BcCw

0sI Aw

–

1–BcDw

PAc

′0

′BC

′Aw

′

AcBcCw

0Aw

PBcDw

′Bc

′Bw

′

++ 0=

PPcc Pcw

Pcw

′Pww