Biomedical Engineering Reference
In-Depth Information
replaced by
(t
2
−
t
1
)
,
dt
2
.
5
t
2
1
a(t)
g
HIC
=
max
t
1
,t
2
,t
2
−
t
1
≤
(7.48)
t
2
−
t
1
t
1
and time is measured in seconds. The diagram of Fig. 7.5 corresponds to
15 ms.
There are several regions on the diagram corresponding to different num-
bers of discontinuity points for the optimal control. The number of the
region indicates the number of the discontinuity points.
In accordance with Eqs. (7.46) and (7.47), the minimum (dimensionless)
deceleration distance
J
1
(u
0
)
monotonically increases from 0 to
=
3
8
as the
(dimensionless) parameter
increases from 0 to 1. As
continues to
increase beyond 1, the value
J
1
(u
0
)
remains equal to
3
8
. The plot of
J
1
(u
0
)
versus
for the interval 0
<
1 is shown in Fig. 7.6. The asymptotic
behavior of this curve for small values of
is determined by
≤
1
2
2
/
5
,
J
1
(u
0
)
∼
→
0
.
(7.49)
For applications, it is convenient to have an explicit relationship
between the minimum value of the performance index and the input
parameters of the problem represented in the primary dimensional
variables. To express the relations of Eqs. (7.46) and (7.47) in the
dimensional variables, multiply the right-hand sides of the cited relations
by
v
8
/
0
/H
2
/
3
and change
to
H
2
/
3
/v
5
/
0
in accordance with Eq. (7.11).
This representation shows that the minimum deceleration distance
in the optimal control problem monotonically increases as the initial
velocity,
v
0
, increases and monotonically decreases as the maximum value
allowed for the HIC functional,
H
, increases. Figure 7.7 shows
J
1
(u
0
)
as a function of
H
in the dimensional variables for
=
0
.
015 s and
v
0
=
40 (curve 1)
,
50 (curve 2)
,
and 60 (curve 3) km/h.
J
1
(
u
0
)
0.3
0.2
0.1
0.0
0.0
0.2
0.4
0.6
0.8
1.0
FIGURE 7.6
J
1
(u
0
)
as a function of
.
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