Biomedical Engineering Reference
In-Depth Information
7.2.3
Analysis and Discussion of the Results
The characteristic qualitative features of the behavior of the optimal control
of Eqs. (7.44) - (7.45) and the minimum deceleration distance of Eqs. (7.46)
and (7.47), depending on the dimensionless parameter
, will be listed.
The optimal control
u
0
(t)
is constant in sign (negative) on the decelera-
tion interval
(
0
,T
], is continuous on the intervals
k < t < (k
+
1
)
,and
−
3
/
5
has discontinuities at the points
t
. (The square
brackets with the missing upper horizontal bars denote the integer part of
the corresponding number.) The function
u
0
(t)
monotonically increases on
the intervals of continuity and tends to
=
k
,
k
=
0
,
1
,...,
on the right at the points of
discontinuity. The points of discontinuity occur with period
, beginning
from
t
=
−∞
0. The number of the discontinuity points on the deceleration
−
3
/
5
interval is
0.
Figures 7.2 - 7.4 show the time history of the optimal control on the
interval
(
0
,T
]for
+
1. This number increases without limit as
→
=
1 (Fig. 7.2),
=
0
.
5 (Fig. 7.3), and
=
0
.
25
(Fig. 7.4).
Equation (7.11) expresses the dimensionless parameter
[denoted by
in Eq. (7.11)] in terms of the primary dimensional parameters
,
H
,and
v
0
. The lower the
H
and/or the greater the
v
0
, the lower the
. Therefore,
a decrease in the maximum allowable value of the HIC and/or an increase
u
0
0
-1
-2
-3
-4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
t
FIGURE 7.2
Optimal control for
=
1.
u
0
0
-1
-2
-3
-4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
t
Optimal control for
=
0
.
5.
FIGURE 7.3
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