Page 308 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

CONSISTENCY CHECKS 297

(temporarily) provided with extra measurement equipment that meas-
ures the states directly and with sufficient accuracy so that their outputs
can be used as a reference. Usually, such measurement devices are too
expensive and the designer has to rely on the other two tests. However,
the NEES test is applicable in simulations of the physical process. It can
be used to see if the design is very sensitive to changes in the parameters
of the model. The other two tests use the innovations. Since the innova-
tions are always available, even in a real design, these are of more
practical significance.
In order to check the hypothesis that a data set obeys a given distribu-
tion, we have to apply a distribution test. Examples are the
Kolmogorov–Smirnov test and the Chi-square test (Section 9.3.3). Often,
instead of these rigorous tests a quick procedure suffices to give an
indication. We simply determine an interval in which, say, 95% of the
samples must be. Such an interval [A,B] is defined by
F(B) F(A) ¼ 0:95. Here, F( ) is the probability distribution of the
random variable under test. If an appreciable part of the data set is
outside this interval, then the hypothesis is rejected. For chi-square
distributions there is the one-sided 95% acceptance boundary with
A ¼ 0, and B such that F 2 (B) ¼ 0:95. Sometimes the two-sided 95%

Dof
acceptance boundaries are used, defined by F 2 (A) ¼ 0:025 and

Dof
F 2 (B) ¼ 0:975. Table 8.2 gives values of A and B for various degrees

Dof
of freedom (Dof).
For state estimators that have reached the steady state, S(i) is constant,
and the whiteness property of the innovations implies that the power
2
z
z
spectrum of any element of ~ z(i) must be flat. Suppose that ~ z n (i)isan
2
element of ~ z(i). The variance of ~ z n (i), denoted by , is the n-th diagonal
z
z
n
element in S. The discrete Fourier transform of ~ z n (i), calculated over
z
i ¼ 0, .. . , I 1, is:
I 1 p
~ X j2 ki=I ffiffiffiffiffiffiffi
z
Z Z n ðkÞ¼ ~ z n ðiÞe k ¼ 0; ... ; I 1 j ¼ 1 ð8:55Þ
i¼0
~
2
The periodogram of ~ z n (i), defined here as P n (k) ¼jZ n (k)j /I, is an esti-
z
Z
mate of the power spectrum (Blackman and Tukey, 1958). If the power
2
spectrum is flat, then E[P n (k)] ¼ . It can be proven that in that case the
n
2
For estimators that are not in the steady state, the innovations have to be premultiplied by
S 1/2 (i) so that the covariance matrix of the resulting sequence becomes constant.