5.3
Norms of MIMO LTI Systems
Some of the common norms for multiple-input, multiple-output (MIMO) LTI sys-
tems can be expressed in terms of the singular values of the
transfer matrix
n
n
z
w
, which, roughly speaking, give information analogous to the magnitude of a SISO
H
transfer function. The singular values of a matrix
n
n
are de ned by
z
w
M
2
C
( ) = ( (
))1 2=
= 1 ... min
(5.20)
M
M
M
i
fn
n
g
i
i
z
w
where ( ) denotes the th largest eigenvalue. The largest singular value (i.e., 1)
i
i
is also denoted max. A plot of ( ( )) is called a singular value plot, and is
H
j
!
i
analogous to a Bode magnitude plot of a SISO transfer function (an example is
given in gure 5.17).
5.3.1
RMS Response to a Particular Noise Input
Suppose is stable (i.e., each of its entries is stable), and
is the power spectral
H
S
w
density matrix of . Then
w
1
1 2
Z
=
1
rms =
( ) ( ) ( )
kH
k
T
r
H
j
!
S
j
!
H
j
!
d!
:
w
2
w
;1
5.3.2
Norm: RMS Response to White Noise
H
2
If ( )
for those frequencies for which ( ) is signi cant, then this norm is
S
!
I
H
j
!
w
approximately the 2 norm of a MIMO system:
H
1
1 2
Z
=
1
rms
2 =
( ) ( )
(5.21)
kH
k
kH
k
T
r
H
j
!
H
j
!
d!
:
w
2 ;1
The 2 norm of is therefore the RMS value of the output when the inputs are
H
H
driven by independent white noises.
By Parseval's theorem, the 2 norm can be expressed as
H
1 2
Z
=
1
2 =
( ) ( )T
kH
k
T
r
0 h t h t dt
1 2
!
=
= n n
z
w
X
X
22
kH
k
=1
ik
=1
i
k
where is the impulse matrix of . Thus, the 2 norm of a transfer matrix is
h
H
H
H
the square root of the sum of the squares of the 2 norms of its entries.
H
5.3 NORMS OF MIMO LTI SYSTEMS
111
The 2 norm can also be expressed in terms of the singular values of the transfer
H
matrix :
H
1 2
!
=
Z
n
1
X
2 =
1
( ( ))2
(5.22)
kH
k
2
H
j
!
d!
=1 i
;1
i
where = min
. Thus, the square of the 2 norm is the total area under
n
fn
n
g
H
z
w
the squared singular value plots, on a linear frequency scale. This is shown in
gure 5.16.
80
70
60
50
40
21 + 22 + 23
30
;
;
20
10
0
0
5
10
15
20
25
30
35
40
!
For a MIMO transfer matrix,
22 is proportional to the area
Figure
5.16
kH
k
under a plot of the sum of the squares of the singular values of (shown
H
here for min
= 3).
fn
n
g
z
w
5.3.3
Peak Gain
The peak gain of a MIMO system is
Z
n
w
1
X
pk gn = sup kHwk1 = max
( )
(5.23)
kH
k
=0
1
0
jh
t
j
dt:
ij
i
n
kw
k
z
1
=1
kw
k
6
1
j
For MIMO systems, the peak gain is not the same as the average-absolute gain
(c.f. (5.7)). The average-absolute gain is
aa gn = sup
aa
kH
w
k
= T pk gn
kH
k
kH
k
aa=0
aa
kw
k
kw
k
6
the peak gain of the LTI system whose transfer matrix is the transpose of .
H
112