Page 299 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

288 STATE ESTIMATION IN PRACTICE

turns into:

T
T
T
BðijiÞ BðijiÞ¼ B ðiji 1ÞBðiji 1Þ B ðijÞBðiji 1ÞH T
T T 1 T

HB ðiji 1ÞBðiji 1ÞH þC v Þ HB ðiji 1ÞBðiji 1Þ
T
T
¼ B ðiji 1ÞBðiji 1Þ B ðiji 1ÞM
T T
1
M M þ C v M Bðiji 1Þ
1
T T T
¼ B ðiji 1Þ I MM M þ C v M Bðiji 1Þ
ð8:41Þ
def T
where M ¼ B(iji 1)H is an M N matrix.
If we would succeed in finding a Cholesky factor of:
T T
1
I MM M þ C v M ð8:42Þ
then we would have an update formula entirely expressed in Cholesky
factors. Unfortunately, in the general case it is difficult to find such a
factor.
For the special case of only one measurement, N ¼ 1, a factorization
is within reach. If N ¼ 1, then H becomes a row vector h. The matrix M
T
becomes an M dimensional (column) vector m ¼ B(iji 1)h . Substitu-
tion in (8.42) yields:

1
T 2 T
I mm m þ v m ð8:43Þ
2
is the variance of the measurement noise.
v
2
T
2
Expression (8.43) is of the form I mm with ¼ 1/(kmk þ ).
v
Such a form is called a symmetric elementary matrix. The form can be
easily factored by:
T

I mm ¼ I mm T T I mm T
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !
2
kk
1 1 m 1 2 v ð8:44Þ
with ¼ 2 ¼ 2 1 2
m þ
m
m
kk kk kk 2 v
Substitution of (8.44) in (8.41) yields:
T
T

BðijiÞ BðijiÞ¼ B ðiji 1Þ I mm T T I mm T Bðiji 1Þ ð8:45Þ