Image Processing Reference
In-Depth Information
4.3.2 S
TATE
E
QUATIONS OF
C
ONTINUOUS
-T
IME
S
YSTEMS
The state-space equations describing a MIMO continuous-time dynamic system have
a general form given by
d
x
(
t
)
d
t
¼
f
(
x
(
t
)
, u
(
t
)
, t
)
(
4
:
1
)
y
(
t
) ¼
h
(
x
(
t
)
, u
(
t
)
, t
)
(
4
:
2
)
where
x
(
t
) ¼
x
1
(
t
)
T
is the N
½
x
2
(
t
)
x
N
(
t
)
1 state vector
u
(
t
)
is the M
1 input vector
y
(
t
)
is the P
1 output vector
The vector functions f
(:)
are in general nonlinear functions of the state and
input vectors and are given by the following equations:
and h
(:)
2
3
f
1
(
x
1
(
t
)
, x
2
(
t
)
,
...
, x
N
(
t
)
, u
1
(
t
)
, u
2
(
t
)
,
...
, u
M
(
t
)
, t
)
4
5
f
2
(
x
1
(
t
)
, x
2
(
t
)
,
...
, x
N
(
t
)
, u
1
(
t
)
, u
2
(
t
)
,
...
, u
M
(
t
)
, t
)
f
(
x
(
t
)
, u
(
t
)
, t
) ¼
(
4
:
3
)
.
f
N
(
x
1
(
t
)
, x
2
(
t
)
,
...
, x
N
(
t
)
, u
1
(
t
)
, u
2
(
t
)
,
...
, u
M
(
t
)
, t
)
and
2
4
3
5
h
1
(
x
1
(
t
)
, x
2
(
t
)
,
...
, x
N
(
t
)
, u
1
(
t
)
, u
2
(
t
)
,
...
, u
M
(
t
)
, t
)
h
2
(
x
1
(
t
)
, x
2
(
t
)
,
...
, x
N
(
t
)
, u
1
(
t
)
, u
2
(
t
)
,
...
, u
M
(
t
)
, t
)
h
(
x
(
t
)
, u
(
t
)
, t
) ¼
.
(
4
:
4
)
h
P
(
x
1
(
t
)
, x
2
(
t
)
...
, x
N
(
t
)
, u
1
(
t
)
, u
2
(
t
)
...
, u
M
(
t
)
, t
)
,
,
The state equations given by Equations 4.3 and 4.4 are applicable to both linear and
nonlinear, time-invariant, and time-varying systems. If the system is linear, then the
state equations can be written as
d
x
(
t
)
d
t
¼
A
(
t
)
x
(
t
) þ
B
(
t
)
u
(
t
)
(
4
:
5
)
y
(
t
) ¼
C
(
t
)
x
(
t
) þ
D
(
t
)
u
(
t
)
(
4
:
6
)
where A
(
t
)
are time-varying matrices of appropriate dimensions. If
the system is LTI, then A, B, C,andD are constant matrices and the state equation
becomes
, B
(
t
)
, C
(
t
)
,andD
(
t
)
d
x
(
t
)
d
t
¼
Ax
(
t
) þ
Bu
(
t
)
(
4
:
7
)
The output equation is
y
(
t
) ¼
Cx
(
t
) þ
Du
(
t
)
(
4
:
8
)
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