Wtbalance, Wtbalance -12 – National Instruments NI MATRIXx Xmath User Manual
Page 107
Chapter 6
Tutorial
6-12
ni.com
wtbalance
The next command examined is
wtbalance
with the option
"match"
.
[syscr,ysclr,hsv] = wtbalance(sys,sysc,"match",2)
Recall that this command should promote matching of closed-loop transfer
functions. The weighted Hankel singular values are:
1.486 4.513
Ч 10
–1
8.420
Ч 10
–2
5.869
Ч 1
–2
1.999
Ч 10
–2
1.382
Ч 10
–2
7.198
Ч 10
–3
6.336
Ч 10
–3
The relative magnitudes suggest that reduction to order 2 will produce less
of an approximation error here (in the closed-loop transfer function) than a
reduction to this order through
redschur( )
or
ophank( )
(where the
implicit criterion is the unweighted error in approximating the controller
transfer function). Examination of Figures 6-9, 6-10, and 6-11 reveals that
far better approximation is now obtained.
Violation of the specification is to be observed in the open-loop gain.
Notice though that:
•
The error measure for
wtbalance
does not reflect the open-loop gain;
it reflects the closed-loop gain.
•
While the error in dB looks large, as an absolute value it is not
extremely so;
wtbalance
works with additive, not multiplicative
error.
Hence, it cannot be concluded that the algorithm is not working. Use of the
option
"match spec"
with
wtbalance
might be conjectured as a device
for reducing the violation of the specification: one could introduce a weight
V(jw) emphasizing frequencies from 0.1 radians per second to 5 radians per
second.
For example,
This would tend to force the closed-loop transfer functions derived from
the full-order and reduced controller to match better over this range;
because their absolute value is small there, they are approximately equal
to the open-loop gains which, accordingly, may be close. The flaw in this
reasoning is that a second-order controller, with four independent
parameters only, can only do so much, and the totality of designer demands
cannot be fully met.
V j
ω
( )
s 0.1
+
(
) s 10
+
(
)
s 1
+
(
) s 1.4
+
(
)
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