5.3.4
RMS Gain
The RMS gain of a MIMO system is important for several reasons, one being that it
is readily computed from state-space equations. The RMS gain of a MIMO transfer
matrix is
rms gn =
= sup
( ))
kH
k
kH
k
s
1
0 max(H
<s>
the
norm of a transfer matrix (c.f. (5.11), the analogous de nition for transfer
H
1
functions). Thus,
if and only if the transfer matrix is stable. For
kH
k
<
1
H
1
stable , we can express the
norm as the maximum of the maximum singular
H
H
1
value over all frequencies:
= sup max( ( ))
kH
k
H
j
!
1
!
as shown in gure 5.17. Note that the other singular values do not a ect
.
kH
k
1
10
9
@
I
8
@
1
kH
k
1
;
7
;
6
5
4
3
2
2
;
3
;
;
1
;
0
0:1
1
10
100
!
The
norm of a stable transfer matrix is the maxi-
Figure
5.17
H
H
1
mum over frequency of the maximum singular value, 1. The other singular
values, 2 ... , do not aect the
norm.
H
n
1
5.3.5
Entropy of a System
In this section we describe a measure of the size of a MIMO system, which is not
a norm, but is closely related to the 2 norm and the
norm. For
0 we
H
H
>
1
5.3 NORMS OF MIMO LTI SYSTEMS
113
de ne the -entropy of the system with transfer matrix as
H
8
2 Z 1
2
( ) = <
logdet;
;
( ) ( )
if
;
2
I
;
H
j
!
H
j
!
d!
kH
k
<
1
I
H
;1
:
if
1
kH
k
1
(c.f. (5.21)). The -entropy can also be expressed in terms of the singular values as
8
1 Z
n
1
X
>
2
( ) = < 2
log;1 ( ( ( )) )2
if
;
;
H
j
!
=
d!
kH
k
<
i
1
I
H
=1
;1
i
>
:
if
1
kH
k
1
(c.f. (5.22)). This last formula allows us to interpret the -entropy of as a measure
H
of its size, that puts a weight 2 log(1 ( )2) on a singular value , whereas
;
;
=
the 2 norm uses the weight 2. This weight function is shown in gure 5.18, with
H
2 shown for comparison.
2 log;1 ( )2
;
;
=
@
@
R
@
I
@
2
2
0
0
The -entropy of is a measure of its size that puts a weight
Figure
5.18
H
2 log(1 ( )2) on a singular value , whereas the 2 norm uses the
;
;
=
H
weight 2.
Since these two weight functions are close when is small compared to , we
i
see that
lim q ( ) =
2
(5.24)
I
H
kH
k
:
!1
From gure 5.18 we can see that
q
( )
2
(5.25)
I
H
kH
k
114