Image Processing Reference
In-Depth Information
3
State
x
2
(
k
)
2.5
2
1.5
1
0.5
State
x
2
(
k
)
ˆ
0
0
2
4
6
8
10
12
14
k
FIGURE 5.9
Combined state estimator and state feedback (second state).
PROBLEMS
5.1
A dynamic system has a plant: H
(
z
) ¼
Y
(
z
)
10z
U
(
z
)
¼
(
z
þ
1
)(
z
þ
0
:
4
)(
z
þ
0
:
7
)
a. De
ne
the
state
variables
as
x
1
(
k
) ¼
y
(
k
)
,
x
2
(
k
) ¼
y
(
k
þ
1
)
,
and
. Derive state equation for this system.
b. By use of state feedback control u
¼
Kx, obtain the gain matrix K so that
the closed-loop poles are at
x
3
(
k
) ¼
y
(
k
þ
2
)
l
1
¼
0
:
1,
l
2
¼
0
:
4
þ
j0
:
3,
l
3
¼
0
:
4
j0
:
3
5.2
Consider the system de
ned by
x
1
(
k
)
x
2
(
k
)
u
(
k
)
x
1
(
k
þ
1
)
11
02
1
0
¼
þ
x
2
(
k
þ
)
1
a. Show that this system cannot be stabilized by the state feedback control
u
(
k
) ¼
Kx
(
k
)
for any choice of matrix K.
b. Is it controllable? If not, determine the uncontrollable mode.
5.3
Prove matrix inversion lemma which states that for any A, C, and D matrices of
appropriate dimensions, we have
(
A
þ
CD
)
1
¼
A
1
A
1
C
(
I
þ
DA
1
C
)
1
DA
1
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