guidance, flight mechanics and trajectory optimization

The noise vector, 3 , which appears in the state equation is required
to be a Guassian white noise with zero mean and covariance matrix Z:(k) l
Thus
(2.507)
where S(ti- Y) is again the Dirac delta function denoting the 'white'
or uncorrelated property of the noise. Note that x(t) is a symmetric
matrix and will be positive definite in the case in which E is truly
an n vector. In the case in which f is not P dimensional, additional
components of zero mean and zero variance can be added to make it n
dimensional. In such cases, the nxn symmetrix matrix C(t) is only
positive semi-definite. An example of this will be given later.
The optimization problem is to determine the control action
such that the expected value of a certain functional J is a minimum; that
is,
(2-5.8)
where Q2 is a positive definite symmetric matrix and Q and /t are
positive semi-definite symmetric matrices. The admissibl: control set u
is the entire r dimensional control space. Thus, no restrictions are
placed on the control vector u . Also, it is assumed that no constraints
are placed on the terminal state.
This problem is quite similar to the linear quadratic cost problem
treated in Section (2.4.10). The state equations are the same except for
the disturbing force P , while
functional J has been replaced
or expected value of J .
the problem of minimizing a quadratic
by the problem of minimizing the average
To illustrate the type of physical situation that can be represented
by Eqs. (2.5.1) to (2.5.8), consider a stochastic version of the simple
attitude control problem treated in Section (2.4.8). Let the system
equation be [see Eq. (2.4.97) and Sketch (2.4,6)].
where 1 is the moment of inertia, F is the applied force, 1 is the lever
arm and f is a Gaussian white noise (one dimensional) with zero mean
and varia&e 5 ; that is,
* *As stated previously, only quadratic cost will be considered at this time.
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