Biomedical Engineering Reference
In-Depth Information
Use the relations of Eqs. (3.141) and (3.144) to obtain a lower bound for
the rattlespace
J
1
=
max
t
≥−
t
∗
x(t)
−
min
t
≥−
t
∗
x(t)
≥
P(t
∗
),
(3.145)
where
⎧
⎨
1
2
V
2
2
U
U
V
t
∗
V
2
U
,
−
for
t
∗
<
P(t
∗
)
=
(3.146)
U
V
t
∗
2
⎩
V
2
2
U
V
2
U
.
for
t
∗
≥
The normalized plot of the function
P(t
∗
)
is shown in Fig. 3.9. The curve of
this figure presents the quantity
UP/V
2
V/(
2
U)
,
the function
P(t
∗
)
reaches the absolute minimum equal to
V
2
/(
8
U)
.Since
the inequality of Eq. (3.145) is valid for any
t
∗
, it is valid for
t
∗
=
versus
Ut
∗
/V
.For
t
∗
=
V/(
2
U)
and, hence,
V
2
8
U
.
J
1
≥
(3.147)
The control law of Eq. (3.135) provides the lower bound of Eq. (3.147) for
the rattlespace and therefore is optimal. This completes the proof.
Minimization of Peak Magnitude of Object's Displacement
The
optimal control for Problem 3.5, in which the peak magnitude of the
UP
V
2
0.60
0.40
0.20
0.13
0.00
0.0
0.2
0.4
0.6
0.8
1.0
Ut /V
FIGURE 3.9
Proof of the optimality of the control minimizing the rattlespace. Normal-
ized plot of the function
P(t
∗
)
.
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