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( + )
jH
j
!
j
!
(a)
( + )
jH
j
!
j
(0 + )
jH
j
!
j
!
(1 + )
jH
j
!
j
kH
k
1
0
6
kH
k
1
;1
= 1
(b)
The magnitude of a transfer function is shown in (a). The
Figure
5.5
H
gain of the system is the peak magnitude of along the line = .
L
H
s
j
!
2
The exponentially weighted ( = 1)
gain of the system is the peak
a
;
L
2
magnitude of along the line = 1 + , as shown in (b).
H
s
j
!
5.2 NORMS OF SISO LTI SYSTEMS
103
a
1
+
r
1
+
s
s
(a)
(b)
A realization of (
) can be formed by taking each inte-
Figure
5.6
H
s
;
a
grator in a realization of ( ), shown in (a), and adding a feedback of , as
H
s
a
shown in (b).
5.2.8
Hankel Norm
The Hankel norm of a transfer function is a measure of the e ect of its past input on
its future output, or the amount of energy that can be stored in and then retrieved
from the system. It is given by
hankel
kH
k
(
1 2
)
= sup Z
Z
=
1
( )2
T
( )2
1 ( ) = 0
0 :
z
t
dt
0 w t dt
w
t
t
>
T
T
We can think of in this de nition as an excitation that acts over the time period
w
0
the response or ring of the system after the excitation has stopped is
t
T
( ) for
. An example of a past excitation and the resulting ring is shown in
z
t
t
>
T
gure 5.7.
It is useful to think of the map from the excitation ( ( ) for 0
) to
w
t
t
T
ring ( ( ) for
) as consisting of two parts: rst, the mapping of the excitation
z
t
t
T
into the state of the system at = and then, the mapping from the state of the
t
T
system at = (which \summarizes" the total e ect on the future output that the
t
T
excitation can have) into the output for
. This interpretation will come up
t
T
again when we describe a method for computing
hankel.
kH
k
5.2.9
Example 1: Comparing Two Transfer Functions
In this section we will consider various norms of the two transfer functions
(a)
44 1 3 + 334 2 + 1034 + 390
13 ( ) =
;
:
s
s
s
H
s
6 + 20 5 + 155 4 + 586 3 + 1115 2 + 1034 + 390
(5.17)
s
s
s
s
s
s
(b)
220 3 + 222 2 + 19015 + 7245
13 ( ) =
;
s
s
s
(5.18)
H
s
6 + 29 1 5 + 297 4 + 1805 3 + 9882 2 + 19015 + 7245:
s
:
s
s
s
s
s
These transfer functions are the I/O transfer functions ( ) achieved by the con-
T
trollers (a) and (b) in the standard plant example of section 2.4. The step
K
K
104