
130 Feedback Control in Systems Biology
0 5 10 15 20
10
−1
10
0
Frequency (rad/sec)
|S|
−
+
FIGURE 4.10: Illustration of the area formula
then for ω ≥ ω
1
|S| ≤
1
1 − |L|
<
1
1 − ω
−2
=
ω
2
ω
2
− 1
and hence
Z
∞
ω
1
log|S(jω)|dω ≤
Z
∞
ω
1
log
ω
2
ω
2
− 1
dω
The integral on the rig ht-hand side of the above equation is finite, [6, 5], and
so the availa ble positive area a t frequencies above ω
1
is limited. Thus, if |S|
becomes smaller and smaller over some part o f the fr equency range from zero
to ω
1
, then the required positive area must eventually be g enerated by making
|S| lar ge at some other frequencie s below ω
1
.
In this sectio n, we have provided only a few simple examples of the many ...