Biomedical Engineering Reference
In-Depth Information
Performance criterion
subjected to a constraint
J
1
Maximum magnitude of the
displacement of the head relative
to the initial reference frame
Constraint
D
Maximum allowable value for the
criterion
J
1
J
2
(u
0
)
Optimal solution
Optimal value of the performance
index
J
2
u
0
Optimal control
u
For the system governed by Eq. (7.6) subjected to the initial conditions
of Eq. (7.7), find an optimal control
u
u
0
(t)
in the class of integrable
functions to minimize the criterion
J
2
, provided that the peak displace-
ment
J
1
does not exceed a prescribed positive value
D
,thatis,
=
J
2
(u
0
)
=
min
u
{
J
2
(u)
|
J
1
(u)
≤
D
}
.
(7.55)
If the potentials of an impact isolation coating for reducing head injuries
due to impacts are analyzed, Problem 7.2 is that of evaluation of the lowest
injury level that can be provided by a coating the thickness of which does
not exceed a given value.
Problem 7.2 is dual to Problem 7.1 in the sense that knowing the solu-
tion of Problem 7.1 as a function of
H
, one can obtain the solution of
Problem 7.2. The values of
v
0
and
are fixed and the same for both prob-
lems. For the optimal controls of Problems 7.1 and 7.2 assign the indices
H
and
D
, respectively, to indicate the dependence of these controls on the
maximum value allowed for the constrained criterion; that is, instead of
u
0
and
u
0
,use
u
0
and
u
0
D
.Let
f(H)
denote the minimum deceleration
distance in Problem 7.1 as a function of the maximum value allowed for
the HIC, that is,
f(H)
J
1
(u
0
)
. From the solution of Problem 7.1 it fol-
lows that the function
f(H)
is defined for all positive
H
, is continuous,
and decreases monotonically from
=
+∞
to zero as
H
increases from zero
. Hence, this function has the inverse
f
−
1
(D)
, which is defined
on the half-line 0
<D<
to
+∞
∞
, is continuous, and decreases monotonically
from
to zero. In this case, as shown in Section 3.1.4, the solutions of
Problems 7.1 and 7.2 are related by
+∞
u
f
−
1
(D)
u
0
D
(t)
J
2
(u
0
D
)
f
−
1
(D).
=
,
=
(7.56)
0
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