Page 288 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

COMPUTATIONAL ISSUES 277

As can be seen, the filter consists of two loops. If the Kalman filter is
stable, the second loop (the estimation loop) is usually not that sensitive
to round-off errors. Possible induced errors are filtered in much the same
way as the measurement noise. However, the loop depends on the
Kalman gains K(i). Large errors in K(i) may cause the filter to become
unstable. These gains come from the first loop.
In the first loop (the Ricatti loop) the prediction covariance matrix
P(i) ¼ C(i þ 1ji) is recursively calculated. As can be seen, the recursion
involves nonlinear matrix operations including a matrix inversion. Espe-
cially, the representation of these matrices (and through this the Kalman
gain) may be sensitive to the effects of round-off errors.
The sensitivity to round-off errors becomes apparent if an eigenvalue–
eigenvector decomposition (Appendix B.5 and C.3.2) is applied to the
covariance matrix P:

M
X T
P ¼ m v m v ð8:28Þ
m
m¼1
m are the eigenvalues of P and v m the corresponding eigenvectors. The
eigenvalues of a properly behaving covariance matrix are all positive (the
matrix is positive definite and non-singular). However, the range of the
eigenvalues may be very large. This finds expression in the condition
number max / min of the matrix. Here, max ¼ max ( m ) and
min ¼ min ( m ). A large condition number indicates that if the matrix
is inverted, the propagation of round-off errors will be large:

M 1
P 1 ¼ X v m v T ð8:29Þ
m
m¼1 m

In a floating point representation of P, the exponents are largely deter-
mined by max . Therefore, the round-off error in min is proportional to
max , and may be severe. It will result in large errors in 1/ min .
Another operation with a large sensitivity to round-off errors is the
subtraction of two similar matrices.
These errors can result in a loss of symmetry in the covariance
matrices and a loss of positive definiteness. In some cases, the eigenval-
ues of the covariance matrices can even become negative. If this occurs,
the errors may accumulate during each recursion, and the process may
diverge.