is a minimal realization of the stable transfer matrix , i.e.,
H
( ) = (
) 1
;
H
s
C
sI
;
A
B
:
The references cited at the end of this chapter give the generalization of the H1
norm computation method to the case with a feed-through term ( =
+
).
z
C
x
D
w
5.6.1
Computing the
Norm
H
2
Substituting the impulse matrix ( ) =
of into
At
h
t
C
e
B
H
2
Z
1
2 =
( )T ( )
kH
k
T
r
0 h t h t dt
we have
2
Z
1
2 =
T
T
A
t
T
At
kH
k
T
r
B
0 e
C
C
e
dt
B
= ;
(5.35)
T
obs
T
r
B
W
B
where
Z
1
obs =
T
A
t
T
At
W
0 e
C
C
e
dt
is the observability Gramian of the realization, which can be computed by solving
the Lyapunov equation
+
= 0
(5.36)
T
obs + obs
T
A
W
W
A
C
C
(see the Notes and References).
The observability Gramian determines the total energy in the system output,
starting from a given initial state, with no input:
Z
(0)
1
T
obs (0) =
( )T ( )
x
W
x
0 z t z t dt
where _ = , = .
x
Ax
z
C
x
Since ( ) = ( ), the above derivation can be repeated to give an alter-
T
r
R S
T
r
S
R
nate formula
1 2
2 = ; ;
contr
=
(5.37)
T
kH
k
T
r
C
W
C
where
Z
1
contr =
T
At
T
A
t
W
0 e BB e
dt
5.6 STATE-SPACE METHODS FOR COMPUTING NORMS
121
is the controllability Gramian, which can be found by solving the Lyapunov equation
contr + contr
+
= 0
(5.38)
T
T
AW
W
A
B
B
:
The controllability Gramian determines which points in state-space can be reached
using an input with total energy one:
_ =
+
(0) = 0 ( ) =
R
T
( ) ( )
1
T
x
Ax
B
w
x
x
T
x
0 w t w t dt
d
for some and
T
w
(
)
1
;
1
T
contr
x
W
x
<
:
d
d
Thus, the points in state-space that can be reached using an excitation with total
energy one is given by an ellipsoid determined by contr. (See chapter 14 for more
W
discussion of ellipsoids.)
5.6.2
Computing the Hankel Norm
The Hankel norm is readily computed from the controllability and observability
Gramians via
2
hankel =
1 2
1 2
max(
=
contr obs
=
contr) 1=
kH
k
W
W
W
= ( max( obs contr))1 2=
(5.39)
W
W
where max( ) denotes the largest eigenvalue. Roughly speaking, the Gramian obs
W
measures the energy that can be \retrieved" in the output from the system state,
and contr measures the amount of energy that can be \stored" in the system state
W
using an excitation with a given energy. These are the two \parts" of the mapping
from the excitation to the resulting ring that we mentioned in section 5.2.8 (5.39)
shows how the Hankel norm depends on the \sizes" of these two parts.
5.6.3
Computing the
Norm
H
1
There is a simple method for determining whether the inequality speci cation
is satis ed. Given
0 we de ne the matrix
kH
k
<
>
1
=
1
;
T
A
B
B
1
(5.40)
M
:
;
T
T
;
C
C
;A
Then we have
has no imaginary eigenvalues.
(5.41)
kH
k
<
(
)
M
1
Hence we can check whether the speci cation
is satis ed by forming
kH
k
<
and computing its eigenvalues. The equivalence (5.41) can be used to devise
M
122