guidance, flight mechanics and trajectory optimization

The developments in the preceding paragraphs, while algebraically
complex, considerably simplify the terminal constraint problem, Sub-
stituting the definition of Eq. (2.5.93) into the performance index of
(2.5.92) provides
The problem is now one of selecting the control U to minimize
subject to the new state equation
t = H46U + n4r
(2.5.98)
l
and where t. is a Gaussian random variable given by Eq. (2.5.96). The
elements
particular
(2.5.87)
of the diagonal matrix A are to be selected so that the
terminal constraint specified by one of the equations in
is satisfied. The number of independent or free diagonal
elements in A is equal to the number of constraints contained in Eq.
(2.5.87) For example, if Eq. (2.5.87A) is imposed, (i.e., one constraint)
then all the diagonal elements of A are equal with their particular value
chosen so that (2.5.87A) is satisfied. If Eq. (2.5.87B) is imposed, then
the first
are zero.
f, diagonal element are independent and the remaining p-p,
Since the form of the expectation operator in the performance index
depends on the type of observations taken, the perfectly observable,
perfectly inobservable and partially observable case must be treated
separately. This treatment follows in the next three sections.
2.5.3.1 Perfectly Observable Case
In the perfectly observable case , perfect knowledge of the state x
is available at each instant of time. Since z and X are related by the
deterministic transformation of Eq. (2.5.93), the vector e is also known
at each instant. Hence, the problem is one of minimizing EM) where
subject to the differential equation
g.=. H46u+ ff@g
(2.5.100)
It is assumed that t. is know? initially, or alternately, that 20 is
a Gaussian variable with mean t, and variance zero.
This problem is the same as that treated in Section (2.5.2.1) except
that A is not known; rather, this matrix must be selected to satisfy a
terminal condition. However, the analysis is essentially the same once A
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