Image Processing Reference
In-Depth Information
De
ne the two states of the system to be the mass displacement x(t) and its velocity
x(t). Then, we have
x
1
(t)
¼ x(t)
(4
:
29)
x
2
(t)
¼ x(t)
Differentiating both sides of Equations 4.29 results in
dx(t)
dt
¼ x
2
(t)
x
1
(t)
¼
(4
:
30)
d
2
x(t)
dt
2
¼
k
m
x(t)
1
m
u(t)
k
m
x
1
(t)
1
m
u(t)
x
2
(t)
¼
þ
¼
þ
The above equations can be written in matrix form as
2
4
3
5
2
4
3
5
u(t)
"#
¼
"#
þ
01
0
1
m
x
1
(t)
x
2
(t)
x
1
(t)
x
2
(t)
k
m
(4
:
31)
0
Example 4.4
As another example, consider the same ideal spring
mass system with friction that
is assumed to be proportional to velocity. Applying Newton
-
'
is law of motion to the
system results in
m
d
2
x(t)
dt
2
b
dx(t)
dt
¼ u(t)
kx(t)
(4
:
32)
Using the same states, the state equations become
2
4
3
5
2
4
3
5
u(t)
"#
¼
"#
þ
0
1
0
1
m
x
1
(t)
x
2
(t)
x
1
(t)
x
2
(t)
k
m
b
m
(4
:
33)
Example 4.5 is the classical nonlinear control problem known as inverted pendulum.
Example 4.5
Consider a cart with an inverted pendulum as shown in Figure 4.5. It is assumed
that the cart and the pendulum move in only one plane and the friction is
negligible. The goal is to maintain the pendulum at the vertical position.
The horizontal (F
x
) and vertical (F
y
) forces applied by the cart on the pendulum
are given by
d
2
dt
2
(x þ l sin
d
dt
x þ l u
x þ l u
u l u
2
sin
F
x
¼ m
u
)
¼ m
(
cos
u
)
¼ m(
cos
u
)
(4
:
34)
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