Table 4-1. types versus error measures – National Instruments NI MATRIXx Xmath User Manual
Page 81
Chapter 4
Frequency-Weighted Error Reduction
© National Instruments Corporation
4-11
•
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 C
r
(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 Table 4-1.
These error measures have certain interpretations, as shown in Table 4-2.
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(j
ω) – C
r
(j
ω) 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 C
r
(s).
Table 4-1. Types versus Error Measures
Type
Error Measure
"input stab"
"output stab"
"match"
"match spec"
"input spec"
C C
r
–
[
]P I CP
+
[
]
1
–
∞
I PC
+
[
]
1
–
P C C
r
–
[
]
∞
I PC
+
[
]
1
–
P C C
r
–
[
] I PC
+
[
]
1
–
∞
I PC
+
[
]
1
–
P C C
r
–
[
] I PC
+
[
]
1
–
V
∞
C C
r
–
[
]V
∞
V j
ω
( ) C jω
( ) C
r
j
ω
( )
–
[
]
∞