guidance, flight mechanics and trajectory optimization

which form the columns of F must be found. This process is difficult in
general; thus, sometimes it is preferable to make a transformation of variables
to simplify the matrix A before attempting to find the fundamental matrix F.
For the present equations, however, the solutions are known; and the fundamental
matrix can be written in terms of real functions as
The inverse matrix is
Now, since
d(f"f)
09
F--k9 = l iant 41 cmnt
cmnt -h&d )
=o , it is easy to show that
f (f-9 = +-‘A
for all problems of this kind. In order to obtain the solution, it is conven-
ient to obtain a set of constants for equations with no forcing (a, =O). This
is accomplished by the substitution
which is seen to satisfy the differential equation
1.32
1.33
1.34
3 = F%)5 1.35
i =‘F-‘(t) B o3 U)//l, 1.36
This equation is integrated to give Z - Z, and is then multiplied by F(t)
on the left to obtain the original variable, 5 . The result is
The state transition matrix for these two variables is F(t)F -l (t 1. When the
coefficients in the matrix A are constant, it is possible to wriee
F(t) F -‘Cl, ) = G (t - 4,)
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