Digital Signal Processing Reference
In-Depth Information
where cov
(
y
(
k
)
)
is the propagated covariance of
q
k
,
x
k
. This will seize to apply for non-LP
models. There exists no general non-linear summation rule for propagated covariance. A
method of summation can be given though, if different ensembles are combined as in RS.
To combine ensembles of parametric
(
q
)
and non-parametric models
(
x
)
, collect all parameters,
q
→
(
q
T
x
T
)
T
, and diagonalize the enlarged covariance matrix, cov
(
q
)
=
U
T
S
2
U
. Build
^
with
two blocks and use CRS (section 5.4) for the non-parametric model,
æ
ö
1
ˆ
-
1
+
c V
×
0
ç
÷
n m
´
ç
ˆ
÷
0
1
+ ×
c V
g
´
v
ç
÷
(
)
(
m
T
ˆ
ˆ
ˆ
)
ç
ˆ
,
÷
V
º
c
=
,
V
V
=
m v I
+ ×
.
(28)
0
1
+ ×
c V
(
) (
)
(
) (
) (
) (
)
n k
+
g
´ +
m v
n k
+
g
´ +
m v
n k
+
g
´ +
m v
v
ç
g
´
v
÷
ç
M
M
÷
ç
÷
ˆ
ç
÷
0
1
+ ×
c V
è
ø
g
´
v
The scaling 1+
c
±1
may cause a similar scaling problem as the factor
n
in the UKF (section
4.2). Using extended excitation matrices these factors can be eliminated,
ˆ
ˆ
æ
ö
A
B
n c
´
ç
÷
ç
÷
g
´
c
ˆ
æ
ö
A
(
)
ç
÷
T
( ) (
)
ˆ
ˆ
n c
´
ˆ
ˆ
ˆ
V
º
,
E
=
ç
÷
,
E
E
= ×
c I c
,
=
max ,
m v n
>
+
g
.
B
(29)
ç
÷
(
)
( )
( )
( )
ˆ
n k c
+
g
´
g
´
c
n
+ ´
g
c
ç
÷
n
+ ´
g
c
n
+ ´
g
c
B
ç
÷
è
ø
g
´
c
M
ç
÷
ç
÷
ˆ
B
è
ø
g
´
c
A disadvantage of this summation is that the same type of ensemble must be used for all
parameters. Both alternatives combine the statistics of the two models non-linearly. The
uncertainties are propagated and combined by evaluating the model for all samples and
calculating the desired statistics, just as if the combined ensemble described one model.
5.6. Selected ensembles
The standard (STD) ensemble employed in the UKF (as defined in section 4.2) utilizes the
perhaps simplest possible excitation matrix,
ˆ
(
)
V
=
n
I
×
-
I
,
m n
=
2 .
(30)
STD
n n
´
n n
´
While the ultimate simplicity is its main advantage, the long maximal(!) range
M
(
∞
)
is its main
disadvantage.
