Graphics Reference
In-Depth Information
The function to be minimized is the fuel consumption, which here, for simplicity, is given as |
f
|
2
. For
a given time period
t
0
< t < t
1
, this results in
Equation 7.95
as the function to be minimized subject to
the time-space constraints and the motion equation constraint.
Z
t
1
2
R ¼
f
j
dt
(7.95)
t
0
In solving this, discrete representations of the functions
x
(
t
) and
f
(
t
) are considered. Time deriva-
tives of
x
are approximated by finite differences (
Eqs. 7.96 and 7.97
) and substituted into
Equation 7.93
x
i
x
i
1
h
x
i
¼
_
(7.96)
x
iþ
1
2x
i
þ
x
i
1
_
x
i
¼
(7.97)
h
2
x
iþ
1
2x
i
þ
x
i
1
p
i
¼ m
f
m
g
¼
0
(7.98)
h
2
c
a
¼
j
x
1
a
j ¼
0
(7.99)
c
b
¼
j
x
n
b
j ¼
0
If one assumes that the force function is constant between samples, the object function,
R
,becomesa
sum of the discrete values of
f
. The discrete function
R
is to be minimized subject to the discrete constraint
functions, which are expressed in terms of the sample values,
x
i
and
f
i
, that are to be solved for.
Numerical solution
The problem as stated fits into the generic form of constrained optimization problems, which is to “find
the
S
j
values that minimize
R
subject to
C
i
(
S
j
)
0.” The
S
j
values are the
x
i
and
f
i
. Solution methods can
be considered black boxes that request current values for the
S
j
values,
R
, and the
C
i
values as well as
the derivatives of
R
and the
C
i
with respect to the
S
j
as the solver iterates toward a solution.
pective of any constraints on the system. A first-order Newton-Raphson step is computed in the
C
i
to reduce the constraint functions. The step to minimize the objective function,
R
, is projected onto
the null space of the constraint step, that subspace in which the constraints are constant to a first-order
approximation. Therefore, as steps are taken to minimize the constraints, a particular direction for
the step is chosen that does not affect the constraint minimization and that reduces the objective
function.
Because it is first-order in the constraint functions, the first derivative matrix (the Jacobian) of the
constraint function must be computed (
Eq. 7.100
). Because it is second-order in the objective function,
the second derivative matrix (the Hessian) of the objective function must be computed (
Eq. 7.101
)
. The
first derivative vector of the objective function must also be computed,
@
R
/
@
S
j
.
J
ij
¼
@C
i
@S
j
¼
(7.100)
2
R
@S
i
@S
j
@
H
ij
¼
(7.101)
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