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Linear orbit identification

For the reasons discussed in Section 3, it is not always possible to use the linear identification theory based upon all 6 orbital elements. The question is how to use the same algorithm on a subset of the orbital elements? The answer is implicit in the arguments presented in [Paper I], Section 2.3, Case 2, which we are going to use without repeating the formal proofs.

Let us suppose that the vector of estimated parameters is split into
two components, along linear subspaces of the parameter space:

where
are elements of interest. The normal and covariance matrices
and
are decomposed as follows:

Then the uncertainty of
for arbitrary
can be described by the penalty, with respect to the minimum point

and by the *marginal covariance matrix* .

Note that the *marginal normal matrix*
is not
and that to obtain the penalty of the above formula as a function of ,
the value of
has to be changed with respect to the nominal solution
of the unrestricted problem, by an amount which is a function of :

Let us apply this restricted penalty formula to the restricted identification
problem. Let
and
be the nominal solutions for the two arcs considered separately, and
and
the corresponding marginal normal matrices. The variables
are given as function of
by:

(3) |

By the same formalism of the previous subsection:

with

Note that is not the same as the complete minimum penalty of the previous section. The estimate of the minimum penalty is obtained by assuming that in the computation of while in the computation of . Thus there is, in general, no complete solution with a single to be fit to the observations of both arcs with penalty , but such value is obtained by using in the first arc, in the second arc, being the proposed restricted identification.

We claim that : is the minimum of the penalty over the space of variables , while is the minimum of the same penalty over the same space but with the additional constraint , and the minimum of a function can only increase when constraints are added.

In conclusion, the proposed restricted identification is not a complete identification, and the corresponding minimum penalty is not the full penalty to be paid to achieve a full identification. This procedure is, however, a good way to filter the possible identifications because : if a couple can be discarded as a possible identification with the restricted computation, because is too large, then it does not need to be tested with the complete algorithm.

Note that it is also possible to define a *constrained identification*
algorithm, based upon the conditional covariance matrices and the algorithm
for constrained optimisation on linear subspaces, as outlined in [Paper
I], Section 2.3.

1999-05-20