Biomedical Engineering Reference
In-Depth Information
⎨
D
2
t
2
2
V
D
2
,
Vt
−
if 0
≤
t
≤
x
2
(t)
=
(4.150)
⎩
V
2
2
D
2
V
D
2
,
if
t >
V
2
2
D
2
.
J
min
=
max
t
|
x
2
(t)
|=
(4.151)
1
When applied to the case under consideration, Proposition 4.4 implies
that
⎧
⎨
⎩
2
D
3
D
2
V
2
0
if
≥
1
,
J
min
=
(4.152)
V
2
2
D
2
−
D
3
1
2
D
3
D
2
V
2
if
<
1
.
If 2
D
2
D
3
/V
2
<
1, the optimal control
u
1
(t)
is given by
⎧
⎨
⎩
2
D
2
D
3
V
2
V
D
2
,
2
D
2
D
3
V
V
δ(t)
−
if 0
≤
t
≤
u
1
(t)
=
−
+
(4.153)
V
D
2
.
if
t
>
0
2
D
2
D
3
/V
2
If
≥
1,
the
optimal
control
u
1
(t)
coincides
with
u
2
(t)
of
Eq. (4.149).
The control force
u
1
(t)
of Eq. (4.153) involves an impulsive component
represented by the term with the delta function. In accordance with this
control, body 1 (the container) should be impacted by the controller to
reduce its velocity from the initial value
V
instantaneously to the value
2
D
2
D
3
/V
and then subjected to the constant deceleration
2
D
2
D
3
/V
2
to come to a complete stop simultaneously with body 2 (the object to be
protected).
The optimal control
u
1
(t)
of Eq. (4.145) constructed for Problem 4.5 on
the basis of Proposition 4.4 in accordance with the solution of Problem 4.6
is not the only optimal control for Problem 4.5 in the general case. Other
control laws that provide the lower bound
J
mi
1
of Eq. (4.134) can also
exist. For instance, in the problem considered in the current section, the
control
u
1
(t)
of Eq. (4.153) can be replaced by the constant-force control
−
⎧
⎨
V
2
2
D
3
2
D
3
V
−
if 0
≤
t
≤
,
u
1
(t)
=
(4.154)
if
t >
2
D
3
V
⎩
0
.
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