guidance, flight mechanics and trajectory optimization

It is interesting to compare the value of the performance index for
the perfectly, observable and perfectly inobservable cases. Since more
information is available and is used in the perfectly observable case, the
performance index for the perfectly observable case is the smaller of
the two. Let the covariance of the initial state, V, is Eq. (2.5.31) be
taken as zero. Hence, the initial state is known exactly in both the
perfectly observable and inobservable cases and is given by
4
8-L = z. AT t-4
From Eq. (2.5.28) and the expression for @ in (2.5.24) and (2.5.26),
the performance index in the perfectly observable case is given by
while from Equation (2.5.52)
EfJ) = $5X0) Jo f ! tr (CS) dt (2.5.53A)
OBSIiUVABLE
4
IN OBSEK?VABLP
Since the matrix s satisfies
both cases, it follows that
ml
-E/-i{
4
~'s&,);, +jtr(I/Q,)dt +tr&A) (2.5.53B)
c
the same equation and boundary condition in
This difference can be shown to be positive by noting that s and V,from
Eqs. (2.5.49) and (2.5.36),satisfy
Integrating this expression with the condition that t/(f,)=O and
combining with (2.5.54) yields
123
(2-5=55)