Chapter 2 RobustnessAnalysis
© National Instruments Corporation 2-9 MATRIXx Xmath Robust Control Module
wcbode( )[WCMAG, NOMMAG] = wcbode (SysH, delb, {input, output,
graph})
The wcbode( ) function computes and plots the worst-case gain of a
closed-loop transfer function.
This function is useful for checking a system that already has been verified
to be robustly stable using smargin( ). For example, a system can have a
minimum stability margin of 4 dB, so it is robustly stable. If the worst-case
gain from a function input to the output it commands has a 20 dB peak, then
even though the system is robustly stable, the design is unacceptable. On
the other hand, if you verify that the perturbed closed-loop transfer function
increases only 2 dB over the nominal, then the design is probably
acceptable.
The wcbode( ) function computes and plots an approximation to
wcgain(ω), the largest possible magnitude of a perturbed closed-loop
transfer function that can be caused by uncertain transfer functions that
satisfy the magnitude bound. The wcbode( ) function is conservative:
it does not under-report the maximum of the perturbed transfer function.
A large value of wcbode( ) indicates instability: wcgain(ω) = ∞. In this
case, wcbode( ) returns a maximum value of ten times the maximum of
the nominal transfer function over all frequencies. Consequently, the
window is clipped at 20 dB above the maximum of the nominal transfer
function over all frequencies. The wcbode( ) function also plots the
nominal transfer function for reference.
Using wcbode( ) to Analyze Performance Degradation
The wcbode( ) function can be used to analyze performance degradation
for the system you have been using (Figure 2-3). The transfer function,
which should be small, is from reference to error (input 1 to output 1).
Figure 2-6 shows the results of the following function call:
[NOMMAG,WCMAG]=wcbode(H,delb,{input=1,output=1});
The performance degradation due to the uncertainties is small but not
negligible.