Chapter 6 Tutorial
Xmath Model Reduction Module 6-12 ni.com
wtbalanceThe 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 ×10–1 8.420×10–2 5.869 ×1–2
1.999×10–2 1.382×10–2 7.198 ×10–3 6.336 ×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
tothe 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.
Vjω() s0.1+()s10+()
s1+()s1.4+()
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