Page 279 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

268 STATE ESTIMATION IN PRACTICE

According to (8.15), x(i) can be retrieved from a sequence
z(i), .. . , z(i þ M 1) if M is invertible; that is, if the rank of M
equals M.
The advantage of using the observability Gramian instead of the
observability matrix is that the former is more stable. Modelling errors
and round-off errors in the coefficients in both F and H could make the
difference between an invertible G or M and a noninvertible one. How-
ever, G is less prone to small errors than M is.
A more quantitative measure of the observability is obtained by using
the eigenvalues of the Gramian G. A suitable measure is the ratio
between the smallest eigenvalue and the largest eigenvalue. The system
is less observable as this ratio tends to zero. A likewise result can be
obtained by using the singular values of the matrix M (see singular value
decomposition in Appendix B.6).

Example 8.6 Observability of a second order system
Consider the system (F, H) given by:

" #
0:66 0:12 1 1
F ¼ H ¼
0:32 0:74 3 4

The rank of both the Gramian G and the observability matrix M
appear to be one, indicating that the system is not observable.
However, if the coefficients of F and H are represented in single
precision IEEE floating point format, the relative round-off error is
8
in the order of 10 . These round-off errors cause the ratio of
eigenvalues of G to be in the order of 10 16 instead of zero. The
9
comparable ratio of singular values of M is in the order of 10 .
Clearly, both ratios indicate that the observability is poor. However,
the one of M is overoptimistic. In fact, if MATLAB’s function rank()
is applied to G and M, the former returns one, while the latter
(erroneously) yields two. The corresponding MATLAB code, given in
Listing 8.1, uses functions from the Control System Toolbox. Espe-
cially, the function ss() is of interest. It creates a state space
model, i.e. a special structure array containing all the matrices of
a linear time-invariant system.