Image Processing Reference
In-Depth Information
4.7.3 T
RANSFER
F
UNCTION
(M
ATRIX
)
FROM
S
TATE
-S
PACE
E
QUATIONS
The transfer function (matrix) can be derived from state and output equations. For a
SISO system, the transfer function is 1
1 and for a MIMO system with M inputs
and P outputs, the transfer matrix is P
M. Let the state equations of a discrete
MIMO system be
x
(
k
þ
1
) ¼
Ax
(
k
) þ
Bu
(
k
)
(
4
:
95
)
Taking z-transform from both sides of Equation 4.95 yields
zX
(
z
) ¼
AX
(
z
) þ
BU
(
z
)
(
4
:
96
)
Hence,
X
(
z
) ¼ (
zI
A
)
1
BU
(
z
)
(
4
:
97
)
The output equation in z-domain is
Y
(
z
) ¼
CX
(
z
) þ
DU
(
z
) ¼ [
C
(
zI
A
)
1
B
þ
D
]
U
(
z
)
(
4
:
98
)
Therefore, the transfer function matrix is
H
(
z
) ¼
C
(
zI
A
)
1
B
þ
D
(
4
:
99
)
Example 4.11
Find the transfer function corresponding to the state equation given by
x
1
(k)
x
2
(k)
u(k)
x
1
(k þ
1)
0
1
2
3
¼
þ
x
2
(k þ
1)
0
:
6
0
:
5
x
1
(k)
x
2
(k)
y(k)
¼
½
1
2
S
OLUTION
The transfer function is
1
z
1
2
3
¼ C(zI A)
1
B þ D ¼
H(z)
½
1
2
0
:
6 z þ
0
:
5
or
2
3
1
z þ
0
:
51
H(z)
¼
½
1
2
0
:
6
z
z
2
þ
0
:
5z þ
0
:
6
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