Image Processing Reference
In-Depth Information
Example 4.6
Consider the following SISO system:
d
3
y(t)
dt
3
5
d
2
y(t)
dt
2
9
d
2
u(t)
dt
2
2
dy(t)
7
du(t)
þ
þ
dt
þ
8y ¼
þ
dt
þ
13u(t)
(4
:
51)
Obtain a state-space representation of this system in (a) controllable and (b)
observable canonical forms.
S
OLUTION
(a) Controllable canonical form:
2
4
3
5¼
2
4
3
5
2
4
3
5þ
2
4
3
5u(t) and y(t)
2
4
3
5
x
1
(t)
x
2
(t)
x
3
(t)
01 0
001
x
1
(t)
x
2
(t)
x
3
(t)
0
0
1
x
1
(t)
x
2
(t)
x
3
(t)
¼
½
13 7 9
(4
:
52)
8
2
5
(b) Observable canonical form:
2
4
3
5
¼
2
4
3
5
2
4
3
5
þ
2
4
3
5
u(t) and
2
4
3
5
x
1
(t)
x
2
(t)
x
3
(t)
00
8
x
1
(t)
x
2
(t)
x
3
(t)
13
7
9
x
1
(t)
x
2
(t)
x
3
(t)
10
2
y(t)
¼
½
001
01
5
(4
:
53)
4.4.3 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 MIMO system be
d
x
(
t
)
d
t
¼
Ax
(
t
) þ
Bu
(
t
)
(
4
:
54
)
Taking Laplace transform from both sides of Equation 4.54 yields
sX
(
s
) ¼
AX
(
s
) þ
BU
(
s
)
(
4
:
55
)
Hence,
X
(
s
) ¼ (
sI
A
)
1
BU
(
s
)
(
4
:
56
)
The output equation in Laplace transform domain is
Y
(
s
) ¼
CX
(
s
) þ
DU
(
s
) ¼ [
C
(
sI
A
)
1
B
þ
D
]
U
(
s
)
(
4
:
57
)
Search WWH ::
Custom Search