Image Processing Reference
In-Depth Information
where
x
(
k
)
is the estimate of x
(
k
)
at time k
K
f
is the state feedback gain matrix obtained through pole placement
The combined controller and observer equations are given by
x
(
k
þ
1
) ¼
Ax
(
k
) þ
Bu
(
k
) ¼
Ax
(
k
)
BK
f
x
(
k
)
x
(
k
þ
1
) ¼
Ax
(
k
) þ
Bu
(
k
) þ
K
e
(
y
(
k
)
Cx
(
k
)) ¼
Ax
(
k
)
BK
f
x
(
k
) þ
K
e
(
y
(
k
)
Cx
(
k
))
(
5
:
101
)
or
x
(
k
)
x
(
k
)
x
(
k
þ
1
)
A
BK
f
¼
x
(
k
þ
1
)
K
e
CA
BK
f
K
e
C
(
5
:
102
)
y
(
k
) ¼
Cx
(
k
)
where K
e
is the observer gain matrix. The block diagram of the overall system is
shown in Figure 5.7.
The estimation error is controlled by the eigenvalues of A
K
e
C. If the
(
A, C
)
pair is observable, the eigenvalues can be assigned arbitrarily similar to the pole
x
(
k
+1)
x
(
k
)
y
(
k
)
Unit
delay
C
u
(
k
)
B
+
+
ˆ
x
(
k
)
A
ˆ
y
(
k
)
+
+
Unit
delay
C
-
+
+
ˆ
x
(
k
)
A
K
e
-
K
f
FIGURE 5.7
Combined state estimator and state feedback.
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