Image Processing Reference
In-Depth Information
T
, then
Now consider an arbitrary initial state x(0)
¼ x
1
(0)
½
x
2
(0)
x(0)
x
1
(0)
x
2
(0)
01
00
01
00
0
0
x(1)
¼
¼
¼
Therefore any arbitrary initial state is driven to zero in one step. This is called dead
beat control.
5.2.3 P
OLE
-P
LACEMENT
D
ESIGN OF
M
ULTIPLE
-I
NPUT
M
ULTIPLE
-O
UTPUT
(MIMO) S
YSTEMS
Consider an open-loop MIMO system in state-space form given by
x
(
k
þ
) ¼
Ax
(
k
) þ
Bu
(
k
)
y
(
k
) ¼
Cx
(
k
)
1
(
:
)
5
34
where
A
2
R
N
N
B
2
R
N
M
u
2
R
M
C
2
R
P
N
y
2
R
P
The state feedback control law is given by
u
(
k
) ¼
Kx
(
k
)
(
:
)
5
35
where the gain matrix K is M
N and is given as
2
3
K
11
K
12
K
1N
4
5
K
21
K
22
K
2N
K
¼
(
5
:
36
)
.
.
.
K
M1
K
M2
K
MN
The characteristic polynomial of the closed-loop system is
P
c
(l) ¼ l
I
A
þ
BK
j
j
(
5
:
37
)
On the other hand, the closed-loop characteristic polynomial in terms of the desired
poles is given by
P
c
(l) ¼ (l l
1
)(l l
2
) (l l
N
)
¼ l
N
N
1
N
2
þ a
1
l
þ a
2
l
þþa
N
(
5
:
38
)
Comparing Equations 5.37 and 5.38, we get N equations with M
N unknowns.
These unknowns are entries of the gain matrix K.
Therefore there are fewer equations than number of unknowns which means that
there are in
nite number of possible solutions. The MATLAB
1
function
file named
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