Page 282 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

OBSERVABILITY, CONTROLLABILITY AND STABILITY 271

sensitivity to round-off errors. A careful implementation of the design
must prevent these errors. See Section 8.3.
The third cause for instability lies in the dynamics of the state
estimator itself. In order to study the dynamic stability of the state
estimator it is necessary to consider the estimator as a dynamic system
with as inputs the measurements z(i) and the control vectors u(i). See
Appendix D. The output consists of the estimates x(iji). In linear
systems, the stability does not depend on the input sequences. For the
stability analysis it suffices to assume zero z(i)and u(i). The equations
of interest are derived from (8.2):

xði þ 1 i þ 1Þ¼ I KðiÞHðiÞÞFði 1ÞxðijiÞ ð8:21Þ
j
ð
with:

T T 1
KðiÞ¼ PðiÞH ðiÞ HðiÞPðiÞH ðiÞþ C v ðiÞ

P(i) is the covariance matrix C(i þ 1ji) of the predicted state x(i þ 1ji).
P(i) is recursively defined by the discrete Ricatti equation:

T
Pði þ 1Þ¼ FðiÞPðiÞF ðiÞþ C w ðiÞ
T T 1 T T
FðiÞPðiÞH ðiÞ HðiÞPðiÞH ðiÞ þ C v ðiÞÞ HðiÞP ðiÞF ðiÞ
ð8:22Þ

def
The recursion starts with P(0) ¼ C x (0). The first term in (8.22) repre-
sents the absorption of uncertainty due to the dynamics of the system
during each time step (provided that F is stable; otherwise it represents
the growth of uncertainty). The second term represents the additional
uncertainty at each time step due to the process noise. The last term
represents the reduction of uncertainty thanks to the measurements.
For the stability analysis of a Kalman filter it is of interest to know
whether a sequence of process noise, w(i), can influence each element of
the state vector independently. The answer to this question is found by
writing the covariance matrix C w (i) of the process noise as
T
C w (i) ¼ G(i)G (i). Here, G(i) can be obtained by an eigenvalue/
T
eigenvector diagonalization of C w (i). That is C w (i) ¼ V w (i) w (i)V (i)
w
and G(i) ¼ V w (i) 1/2 (i). See Appendix B.5 and C.3. w (i)isa K K
w
diagonal matrix where K is the number of nonzero eigenvalues of