Biomedical Engineering Reference
In-Depth Information
object's relative displacement
J
1
is used as the objective
function is
given by
⎨
V
U
≤
3
4
V
U
,
U
for
−
t<
−
3
4
V
U
≤
t<
V
u
0
(t)
=
(3.148)
−
U
for
−
2
U
,
⎩
V
2
U
,
0
for
t
≥
and provides the minimal magnitude of the peak displacement
V
2
16
U
.
J
1
(u
0
,t
∗
0
)
=
(3.149)
The pre-action time in this optimal control is
V
U
,
t
∗
0
=
(3.150)
which is twice the pre-action time for the control of Eq. (3.135) that min-
imizes the rattlespace. Unlike the optimal control for the rattlespace, the
optimal control of Eq. (3.148) for the peak displacement magnitude is not
a constant-force control.
The time histories of the coordinate
x
and the velocity
˙
x
of the object
corresponding to the optimal control are obtained by solving Eq. (3.128)
subject to the conditions of Eqs. (3.129) and (3.130) for the control law of
Eq. (3.148) and are given by
⎧
⎨
⎩
U
V
t
1
2
V
2
2
U
V
U
≤
3
4
V
U
,
+
for
−
t<
−
1
1
2
2
2
U
V
2
16
U
3
4
V
U
≤
−
V
t
+
for
−
t<
0
,
x(t)
=
1
V
t
2
2
U
1
2
V
2
16
U
16
U
t<
V
−
+
+
≤
V
t
for 0
2
U
,
V
2
16
U
V
2
U
,
for
t
≥
(3.151)
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