( ) = 0 for
()
w
t
t
T
t
;
;
0
w
( ) rings for
z
t
t
T
;
;
()t 0
z
;10
;8
;6
;4
;2
0
2
4
6
8
10
t
;
T
The Hankel norm of a transfer function is the largest possible
Figure
5.7
square root energy in the output for
, given a unit-energy excitation
z
t
T
that stops at = .
w
t
T
responses of (a)
(b)
13 and 13 are shown in gure 5.8 and their frequency response
H
H
magnitudes in gure 5.9. The values of various norms of (a)
(b)
13 and 13 are shown
H
H
in table 5.1.
From the rst row of table 5.1 we see that the peak of the response of (a)
13 to
H
a step input is about the same as (b)
13 . Thus, in the sense of peak step response,
H
(a)
(b)
13 is about the same size as 13 .
H
H
If (a)
(b)
(a)
13 and 13 are driven by white noise, the RMS value of the output of 13
H
H
H
is less than half that of (b)
13 (second row). Figure 5.10 shows an example of the
H
Norm
(a)
(b)
13
13
gure
H
H
pk step 1 36 1 40
5.8
k
k
:
:
2
1 17 2 69
5.10
k
k
:
:
wc
1 60 1 68
5.11
k
k
:
:
pk gn
1 74 4 93
5.12
k
k
:
:
1 47 3 72
5.9
k
k
:
:
1
hankel
1 07 2 04
5.13
k
k
:
:
The values of six dierent norms of (a) and (b).
T
able
5.1
H
H
13
13
5.2 NORMS OF SISO LTI SYSTEMS
105
1:6
(b)
= 1 40
kH
k
:
pk
step
13
;
;
1:4
H
Y
(a)
= 1 36
H
kH
k
:
pk
step
13
1:2
1
@
I
@
(b)
0:8
( )
s
t
13
0:6
@
I
@
0:4
(a)
( )
s
t
13
0:2
0
;0:2
0
1
2
3
4
5
6
7
8
9
10
t
The step responses of the transfer functions in (5.17) and (5.18).
Figure
5.8
Note that (a)
= 1.36, and (b)
= 1.40.
kH
k
kH
k
pk
step
pk
step
13
13
10
(b)
= 3 72
(a)
= 1 47
kH
k
:
1
kH
k
:
13
1
13
;
;
?
1
j
H
j
(b)
( )
jH
j
!
j
13
0:1
;
;
;
;
(a)
( )
jH
j
!
j
13
0:01
0:1
1
10
100
!
The magnitudes of the transfer functions in (5.17) and (5.18).
Figure
5.9
Note that (a) = 1.47, and (b) = 3.72.
kH
k
kH
k
1
1
13
13
106