In essence, Qobs determines the importance of the
accuracy of each particular process state. For the LQ
control problem, an optimal weighting matrix is readily
found as CT
z QCz, which yields the global minimum for
the cost function. Unfortunately, there is no such parameterization
when the optimal estimator is considered.
In the Kalman filter, the weighting is made
based on the noise characteristics, which are assumed
to be known a priori. There are also some rules of
thumb for choosing suitable candidates for the weighting
matrix, some of which can be found, for example, in
Anderson and Moore (1989). It should be emphasized
that a poor choice of the weighting may result in very
poor control performance, as some of the states may be
over emphasized and others very biased.