guidance, flight mechanics and trajectory optimization

2.5.2 Problem Statement
Let the system be described by the differential equation
i =Ax+Gu (2.5-l)
where x is an n vector denoting the state of the system, u is an Yvector
denoting the control, r is an n vector denoting noise or disturbing
forces and A and G are nxn and nxr matrices, respectively. The
state of the system is not known initially. Rather, the initial state,
$0 , is a Gaussian random variable
v, ; that is,
with mean & and covariance matrix
E (x0) = x”,
& ((%,- & ) (?ix,- ;i,i ) = 4
(2.5.2)
where L denotes the expectation operator defined over the entire ensemble
of states. Alternately, the Gaussian random variable can be represented
by its density function
with
P 0x0) = p @‘a)xz, , . ” XmJ =
E (X0) = JTo P (x0) dx,
-g
~+~o)ob- ~,fP(Xe)dx,
Note that the case in which JL" is precisely specified can also be
included in this formulation by requiring that
E ((h - i.)(x,- &;) = vo = 0
where now ?a denotes the specified value of X0 In this case, the
density function in Eq. (2.5.3) b ecomes a produci of n Dirac delta
functions with
110
a}
@.5+3)
(2.5.4)
(2.5-5)