Image Processing Reference
In-Depth Information
4.6.2 S
TATE
E
QUATIONS
The general expressions for state-space equations of a MIMO dynamic system are
given by
x
(
k
þ
1
) ¼
f
(
x
(
k
)
, u
(
k
)
, k
)
(
4
:
82
)
y
(
k
) ¼
h
(
x
(
k
)
, u
(
k
)
, k
)
(
4
:
83
)
where
x
(
k
) ¼
x
1
(
k
)
T
is the N
½
x
2
(
k
)
x
N
(
k
)
1 state vector
u
(
k
)
is the M
1 input vector
y
(
n
)
is the P
1 output vector
The functions f and h are nonlinear and are de
ned as
2
3
f
1
(
x
1
(
k
)
, x
2
(
k
)
,
...
, x
N
(
k
)
, u
1
(
k
)
, u
2
(
k
)
,
...
, u
M
(
k
)
, k
)
4
5
f
2
(
x
1
(
k
)
, x
2
(
k
)
,
...
, x
N
(
k
)
, u
1
(
k
)
, u
2
(
k
)
,
...
, u
M
(
k
)
, k
)
f
(
x
(
k
)
, u
(
k
)
, k
) ¼
.
f
N
(
x
1
(
k
)
, x
2
(
k
)
,
...
, x
N
(
k
)
, u
1
(
k
)
, u
2
(
k
)
,
...
, u
M
(
k
)
, k
)
(
:
)
4
84
2
4
3
5
h
1
(
x
1
(
k
)
, x
2
(
k
)
,
...
, x
N
(
k
)
, u
1
(
k
)
, u
2
(
k
)
,
...
, u
M
(
k
)
, k
)
h
2
(
x
1
(
k
)
, x
2
(
k
)
,
...
, x
N
(
k
)
, u
1
(
k
)
, u
2
(
k
)
,
...
, u
M
(
k
)
, k
)
h
(
x
(
n
)
, u
(
n
)
, n
) ¼
.
h
P
(
x
1
(
k
)
, x
2
(
k
)
,
...
, x
N
(
k
)
, u
1
(
k
)
, u
2
(
k
)
,
...
, u
M
(
k
)
, k
)
(
4
:
85
)
If the system is linear, then the state equations can be written as
x
(
k
þ
1
) ¼
A
(
k
)
x
(
k
) þ
B
(
k
)
u
(
k
)
(
4
:
86
)
y
(
k
) ¼
C
(
k
)
x
(
k
) þ
D
(
k
)
u
(
k
)
(
4
:
87
)
If the system is LTI, then matrices A, B, C, and D are constant matrices and the state
equations become
x
(
k
þ
1
) ¼
Ax
(
k
) þ
Bu
(
k
)
(
4
:
88
)
y
(
k
) ¼
Cx
(
k
) þ
Du
(
k
)
(
4
:
89
)
where A, B, C, and D are N
N, N
M, P
N, and P
M constant matrices,
respectively. The block diagram of an LTI control system in state-space form is
shown in Figure 4.6.
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