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example,
1 =
3s
H
2 + + 3
s
s
2 =
3s
H
2 + 0 5 + 2
s
:
s
have 1 pk step = 1 18 and 2 pk step = 1 65, so that 1 pk step 2 pk step =
kH
k
:
kH
k
:
kH
k
kH
k
1 95, but 2 1 pk step = 2 80 1 95.
:
kH
H
k
:
>
:
On the other hand, 1 = 3, 2 = 6, and 2 1 = 15 3 18, since
kH
k
kH
k
kH
H
k
:
1
1
1
the
norm is the RMS gain.
H
1
1
2
w
z
H
H
The gain of two cascaded transfer matrices is no larger
Figure
5.20
than the product of the gains of the transfer matrices, i.e.:
2 1 gn
kH
H
k
2 gn 1 gn.
kH
k
kH
k
5.4.2
Gain of a Feedback Connection
+
r
w
1
z
H
+
q
2
H
Two systems connected in a feedback loop.
Figure
5.21
Consider the feedback connection shown in gure 5.21. Assuming that this feedback
connection is well-posed, meaning that det(
1 2) is not identically zero, the
I
;
H
H
transfer matrix from to is
w
z
= (
1 2) 1
1
(5.29)
;
1 = 1(
2 1);
G
I
;
H
H
H
H
I
;
H
H
:
A fact that we will need in chapter 10 is that, provided the product of the gains
of the two transfer matrices is less than one, the gain of can be bounded. More
G
precisely, if
1 gn 2 gn 1
(5.30)
kH
k
kH
k
<
holds, then the feedback connection is well-posed and we have
gn
1 gn
kH
k
(5.31)
kGk
1
1 gn 2 gn
;
kH
k
kH
k
5.5 COMPARING NORMS
117
(note the similarity to (5.29)).
The condition (5.30) is called the \small gain" condition, since, roughly speaking,
it limits the gain around the loop in gure 5.21 to less than one. For this reason,
the result above is sometimes called the small gain theorem.
The small gain theorem can be used to establish stability of a feedback connec-
tion, if the gain
gn is such that
gn
implies that is stable. The RMS
k
k
kH
k
<
1
H
gain, for example, has this property: the small gain condition 1
2
1
kH
k
kH
k
<
1
1
implies that the transfer matrix is stable, that is, all of its poles have negative
G
real part. Similarly, if the gain
gn is the -shifted
norm, then the small
k
k
a
H
1
gain condition 1
2
1 implies that the poles of have real parts less
kH
k
kH
k
<
G
1
a
1
a
than .
;a
The small gain theorem is easily shown. Suppose that the small gain condi-
tion (5.30) holds. The feedback connection of gure 5.21 means
= 1( + 2 ) = 1 + 1 2
z
H
w
H
z
H
w
H
H
z
:
Using the triangle inequality,
1
+ 1 2
kz
k
kH
w
k
kH
H
z
k
where
is the norm used for all signals. Using the de nition of gain and the
k
k
property (5.28), we have
1 gn
+ 1 2 gn
1 gn
+ 1 gn 2 gn
kz
k
kH
k
kw
k
kH
H
k
kz
k
kH
k
kw
k
kH
k
kH
k
kz
k
so that
(1
1 gn 2 gn)
1 gn
kz
k
;
kH
k
kH
k
kH
k
kw
k:
Using the small gain condition, we have
1 gn
kH
k
(5.32)
kz
k
1
1 gn 2 gn kwk:
;
kH
k
kH
k
Since (5.32) holds for all signals , (5.31) follows.
w
5.5
Comparing Norms
The intuition that di erent norms for LTI systems should generally agree about
which transfer matrices are \small" or \large" is false. There are, however, some
general inequalities that the norms we have seen must satisfy.
118