Biomedical Engineering Reference
In-Depth Information
≤
t
≤
T
. This control can be extended arbitrarily beyond this interval, pro-
vided that
The optimal control for Problem 4.4 is uniquely defined only for 0
m
1
V
2
2
P
m
1
+
m
2
t
>
T.
|
x(t)
|≤
,
|
F(t)
|≤
P,
(4.102)
m
1
F
0
(t)
0 is chosen for
t
>
T
.
Substitute the data of (2) and (6) in Example 4.1 into (4.99), (4.101),
and (4.96) to obtain
In expression (4.99), a simple extension
=
−
6kN
for 0
≤
t
≤
25 ms
,
F
0
(t)
=
(4.103)
0
for
t >
25 ms
,
J
1
(F
0
)
−
J
1
( F
0
)
≤
J
1
( F
0
)
=
12
.
5cm
,
0
.
4cm
.
For the near-optimal control
F
0
(t)
calculated using (4.95) on the basis
of the control
F
0
(t)
that is optimal for the rigid model, when applied to the
system of Eqs. (4.1) and (4.2), the constraint (4.6) imposed on the force
acting between bodies 1 and 2 is satisfied. When the control law
F
0
(t)
is
used instead of
F
0
(t)
, this constraint is significantly violated.
Figures 4.7 - 4.9 present time histories of the displacements of bodies 1
and 2 relative to the base (
x
and
y
) and the force acting between these bodies
(
W
) for the system subject to the constant-force control of Eq. (4.103). The
force
W
is defined by Eq. (4.7) as
W
x)
. It is apparent
from these figures that the time histories of the displacements of bodies 1
and 2 virtually coincide with the displacement time history for the rigid
model. However, the time histories of the force acting between the bodies
are substantially different. For the rigid model, this force is a constant equal
to
=
C(
y
˙
−˙
x)
+
K(y
−
−
4 kN, whereas for the deformable model, to which the control force
optimal for the rigid model is applied, this force varies and has a peak
15
10
5
0
0
10
20
30
Time (ms)
FIGURE 4.7
Time history of the displacement of body 1 relative to the base for a
constant-force control.
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