Mathematical Problems in Engineering 3
wherein
(
,
)
=
1
(
,
)
2
(
,
)
.
.
.
𝑚
(
,
)
,
(
,
)
=
1
(
,
)
2
(
,
)
.
.
.
𝑚
(
,
)
,
(
,
)
=
1
(
,
)
2
(
,
)
.
.
.
𝑚
(
,
)
,
(4)
where, in formula (3),theobservedvalue
𝑗
(,) ∈
𝑃
𝑗
×1
represents the observed values of the th sensor on the scale
(1), = 1,2,...,,thesamplingrateofthe
sample between the two dimensions of the sensor vector
is 2 : 1, namely, (,) = (,2
𝑁−𝑖
),
𝑗
(,) ∈
𝑃
𝑗
×𝑛
is
observation matrix, and observation noise
𝑗
(,) ∈
𝑃
𝑗
×1
is Gauss white noise sequence. At the same time,
{
(
,
)}
=0
(
,
)
𝑇
(
,
)
=
(
,
)
𝑘𝑗
.
(5)
Here we assumed that (,), (,),andV(,) are
uncorrelated.
3. Unscented Kalman Filter and
Wavelet Transform
3.1. Unscented Kalman Filter. Unscented Kalman lter (UKF)
is a Gaussian lter which calculates the mean and covari-
ance of nonlinear transformation using unscented transform
(UT). Based on the principle that the approximate proba-
bility distribution is easier than the approximate arbitrary
nonlinear transformation, the UT (1) characterizes certain
characteristics, such as mean and covariance, of a probability
distribution by deterministically selecting a set of sample
points; (2)propagates this set of sample points through the
nonlinear transformation; and (3) calculates the mean and
covariance of propagated sample points. Estimation accuracy
of UKF depends on the accuracy of mean and covariance
calculated by UT. Compared to the extended Kalman lter
(EKF), UKF oers better performance at the same amount of
calculation and does not require the calculation of Jacobian
matrix. It can be interpreted as random linear regression,
thereby revealing the reason why UKF is superior to EKF.
Existing UKF algorithm is essentially a second-order UT-
based nonlinear ltering method, which can only match
the second-order Taylor expansion terms of nonlinear func-
tion, and therefore has a limited accuracy.
Implementation steps of the unscented Kalman lter are
as follows:
(1) Calculating weights
𝑖
corresponding to sample
points using
𝑘−1|𝑘−1
and
𝑘−1|𝑘−1
given previously,
𝑚
𝑖
=
𝑐
𝑖
=0.5/(+),here=1,2,...,2, is the
dispersion degree of sample points,
𝑚
𝑖
is the weight
coecient of the rst-order statistical properties, and
𝑐
𝑖
is a function of second-order statistical properties
of the required weight coecients.
(2) Propagation function of the state evolution equation
related to the sample points should be calculated as
follows:
𝑖
(+1|)=(
𝑖
(|)).
(3) e statistical characteristic functions
(+1|)and
(+1|)should be calculated assisted with forecast
sampling points
𝑖
(+1|)and weights
𝑖
,
(
+1|
)
=
2𝐿
𝑖=0
𝑚
𝑖
𝑖
(
+1|
)
,
(
+1|
)
=
(
+1
)
+
2𝐿
𝑖=0
𝑐
𝑖
𝑖
(
+1|
)
−
(
+1|
)
⋅
𝑖
(
+1|
)
−
(
+1|
)
𝑇
.
(6)
(4) Calculation of dispersion function derived from mea-
surement equation of sample points that are obtained
by utilizing UT transformation,
𝑖
(+1|)=
𝑖
(+
1|).
(5)epredictedvalueandthemeasuredvalueofthe
statistical characteristics should be calculated by
(+
1|)=
∑
2𝐿
𝑖=0
𝑚
𝑖
𝑖
(+1|),
𝑍𝑍
=
(
+1
)
+
2𝐿
𝑖=0
𝑐
𝑖
𝑖
(
+1|
)
−
(
+1|
)
⋅
𝑖
(
+1|
)
−
(
+1|
)
𝑇
,
𝑋𝑍
=
2𝐿
𝑖=0
𝑐
𝑖
𝑖
(
+1|
)
−
(
+1|
)
⋅
𝑖
(
+1|
)
−
(
+1|
)
𝑇
.
(7)
Here
𝑍𝑍
is the measurement error covariance matrix,
𝑋𝑍
isthecovariancematrixofthestatevectorandthe
measured values.