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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
to form n physics constraints ( Eq. 7.98 ) and the two boundary constraints ( Eq. 7.99 ).
x i x i 1
h
x i ¼
_
(7.96)
x 1
2x i þ
x i 1
_
x i ¼
(7.97)
h
2
x 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.
The solution method used byWitkin and Kass [ 27 ] is a variant of sequential quadratic programming
(SQP) [ 10 ]. This method computes a second-order Newton-Raphson step in R , which is taken irres-
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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