with guaranteed accuracy, by trying di erent
kH
k
1
values of see the Notes and References at the end of this chapter.
The result (5.41) can be understood as follows.
is true if and only if
kH
k
<
1
for all
, 2
( ) ( ) is invertible, or equivalently, the transfer matrix
!
2
R
I
;
H
j
!
H
j
!
( ) =
1
;
2
;
( )T ( ) ;
G
s
I
;
H
;s
H
s
has no
axis poles. We can derive a realization of as follows. A realization of
j
!
G
( ) (which is the adjoint system) is given by
T
H
;s
( )
1
;
;
;
;
T
= ; T
T
T
H
;s
B
sI
;
;A
;C
:
Using this and the block diagram of shown in gure 5.22, a realization of is
G
G
given by ( ) = (
) 1 + , where
;
G
s
C
sI
;
A
B
D
G
G
G
G
=
A
M
G
= B
B
G
0
= 0
1
;
T
B
C
G
=
D
I
:
G
Since
= , and it can be shown that this realization of is minimal, the
A
M
G
j
!
G
axis poles of are exactly the imaginary eigenvalues of
, and (5.41) follows.
G
M
+
r
y
u
+
q
( )
( )
T
H
s
H
;s
A realization of ( ) = 2( 2
( ) ( )) 1. has no
T
;
Figure
5.22
G
s
I
;
H
;s
H
s
G
imaginary axis poles if and only if the inequality specication kHk <
1
holds.
The condition that
not have any imaginary eigenvalues can also be expressed
M
in terms of a related algebraic Riccati equation (ARE),
+
+ 1
+ 1
= 0
(5.42)
T
;
T
;
T
A
X
X
A
X
B
B
X
C
C
:
(Conversely,
is called the Hamiltonian matrix associated with the ARE (5.42).)
M
This equation will have a positive de nite solution if and only if
has no
X
M
imaginary eigenvalues (in which case there will be only one such ). If
,
X
kH
k
<
1
we can compute the positive de nite solution to (5.42) as follows. We compute any
matrix such that
T
1
11 ~12
A
;
= ~A
T
M
T
0 ~22
A
5.6 STATE-SPACE METHODS FOR COMPUTING NORMS
123
where ~11 is stable (i.e., all of its eigenvalues have negative real part). (One good
A
choice is to compute the ordered Schur form of
see the Notes and References
M
at the end of this chapter.) We then partition as
T
=
11
12
T
T
T
21
22
T
T
and the solution is given by
X
= 21 1
;
11
X
T
T
:
The signi cance of is discussed in the Notes and References for chapter 10.
X
5.6.4
Computing the -Shifted
Norm
a
H
1
The results of the previous section can be applied to the realization
_ = ( + ) +
=
x
A
aI
x
B
w
z
C
x
of the -shifted transfer matrix (
). We nd that
holds if and
a
H
s
;
a
kH
k
<
1
a
only if the eigenvalues of all have real part less than
and the matrix
A
;a
+
1
;
T
A
aI
B
B
1
;
T
T
;
C
C
;A
;
aI
has no imaginary eigenvalues.
5.6.5
Computing the Entropy
To compute the -entropy of , we rst form the matrix
in (5.40). If
H
M
M
has any imaginary eigenvalues, then by the result (5.41),
and hence
kH
k
1
( ) = . If
has no imaginary eigenvalues, then we nd the positive de nite
I
H
1
M
solution of the ARE (5.42) as described above. The -entropy is then
X
( ) =
;
T
(5.43)
I
H
T
r
B
X
B
:
From (5.42), the matrix ~ =
satis es the ARE
X
X
~
2 ~
~
T
+ ~ + ;
T
+ T = 0
A
X
X
A
X
B
B
X
C
C
:
Note that as
, this ARE becomes the Lyapunov equation for the observability
!
1
Gramian (5.36), so the solution ~ converges to the observability Gramian obs.
X
W
From (5.43) and (5.35) we see again that ( )
22 as
(see (5.24) in
I
H
!
kH
k
!
1
section 5.3.5).
124