7
Controllable Decomposition
(proof)
•
Proof
Using the partitioned equation 0=
AP+PA*+BB*
to
get
•
A
11
P
1
+P
1
A
11
*+B
1
B
1
*=0,
P
1
A
21
*+B
1
B
2
*=0
, and
B
2
B
2
*=0
•
then we conclude that
B
2
= 0 and
A
21
= 0. Hence, part of
the realization is not controllable:
•
•
•
P
1
>0 if
A
11
is stable. So (
A
11
,
B
1
) is controllable.
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