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Chapter 2 Robustness Analysis
MATRIXx Xmath Robust Control Module 2-8 ni.com
Worst-Case Performance Degradation (wcbode)
Even if a system is robustly stable, the uncertain transfer functions still can
have a great effect on performance. Consider the transfer function from the
qth input, w
q
, to the pth output, z
p
. With δ
1
= ... = ...δ
k
= 0, you have the
nominal system, and this transfer function is the p,q entry of H
zw
. This is
called the nominal transfer function.
When the δ values are not zero, the transfer function from w
q
to z
p
is the p,q
entry of H
pert
given by the formula:
This is referred to as the perturbed transfer function. The perturbed transfer
function depends on the particular δ
1
, …, δ
k
.
If the magnitude bounds are small enough, then you expect the perturbed
transfer function H
pert
to be close to the nominal transfer function. Roughly
speaking, small perturbations should not significantly alter the closed-loop
transfer function from w
q
to z
p
.
The worst-case gain is defined as the largest magnitude of the perturbed
transfer function, considering all δ values that satisfy the magnitude bound.
More precisely:
(2-3)
wcgain(ω) is always larger than the nominal gain, |H
zw,pq
(jω)|. This is not
because the uncertain transfer functions only can increase the magnitude of
the transfer function from w
q
to z
q
. In fact, it is possible that for a lucky
choice of the δ values, the perturbed transfer function actually can be
smaller than the nominal transfer function over all frequencies. But in the
worst-case gain, you consider only the worst possible δ values, and these
always increase the perturbed gain over the nominal gain.
Intuitively, if the stability margin is large, then the uncertain transfer
functions should not greatly effect the gain from w
q
to z
p
, so that wcgain(ω)
should be not much larger than the nominal gain |H
zw,pq
(jω)|. If the stability
margin is small, however, wcgain(ω) could be much larger than the nominal
gain. An extreme case occurs if the stability margin is negative (in dB) at
the frequency δ. Then you have wcgain(ω) = ∞, although
wcbode( ) clips
the worst-case gain curve so that it never exceeds (the maximum nominal
gain) * 100, or +20 dB. Of course, instability is an extreme form of
performance degradation.
H
pert
H
zw
H
zr
Δ IH
qr
Δ–()
1–
H
qw
+=
wcgain ω() max H
pert,pq
Δ = diagonal δ
1
,...,δ
k
()δ
i
l
i
ω()≤,{}=