Journal of Software - Academy Publisher

JOURNAL OF SOFTWARE, VOL. 6, NO. 5, MAY 2011 939
⎧ k + = k + k
⎪
⎪
⎪Δ
k + = Δ k
⎨ΔVk
+ = ΔVk
⎪
= +
⎪ k k k
⎪⎩
yk
= h(
k − Δ k ) − ΔVk
+ vk
*
* *
e 1 e w
H 1 H
1
* *
x x e
x H
(5)
where equations Δ H k +1 = ΔH
k and Δ Vk +1 = ΔVk
are
the reflections of the assumption above, meaning that the
error components are constant in the area concerned.
The compact form of (5) is
⎧ k + = k + k
⎪
⎨ k = k + ⋅ k
⎪
⎩y
k = h(
k − Δ k ) − ΔVk
+ vk
*
'
e 1 e w
*
x x F e
x H
(6)
* ⎡ e ⎤ k
⎢ ⎥ ⎡1
where ek
= ⎢ΔHk
⎥ , F = ⎢
⎢ ⎥
⎣ΔV
⎣0
k ⎦
0
1
0
0
0
0
⎡wk
⎤
0⎤
' ⎥
0
⎥, w
⎢
k =
⎢
0 .
⎥
⎦
⎢⎣
0 ⎥⎦
III. RECURSIVE BAYESIAN ESTIMATION
According to Bayesian theory, a Bayesian estimation
problem is defined by the joint density of the parameters
and the observations, p ( x,
y)
= p(
y | x)
p(
x)
. The
estimator under the minimum mean square error criterion
is the posterior mean x ˆ MS = ∫ xp(
x | y)
dx [6]. So if
n
R
there exists some relationship between the observation
and the parameter, then the parameter can be estimated
by the observation. From the information theory’s point
of view, the observation contains the information of the
parameter when there exists stochastic relationship
between the two. So we can make use of the observation
to estimate the parameter.
Let Y k be the augmented measurement vector
consisting of all the measurements up to time step k.
From Bayesian formula [6] and the new system model (6),
we have the posterior probability density function update:
p(
yk
| ek,
Yk
−1)
⋅ p(
ek
| Yk
−1)
p(
ek
| Yk
) =
p(
y | Y )
(7)
k
k−1
−1
*
= αk
⋅ pv
( yk
−h(
xk
−ek
−ΔH
k)
+ ΔVk
) ⋅ p(
ek
| Yk
−1)
k
where
*
α k = ∫ pv ( yk
− h(
x
N k − ek
− ΔHk
) + ΔVk
) ⋅ p(
ek
| Yk
⋅ dek
R k − 1)
.
The priori probability density function update is
p(
ek
+ 1 | Yk
) = ∫ p(
ek
ek
Yk
p ek
Yk
de
N + 1 | , ) ⋅ ( | ) ⋅ k
R
. (8)
* *
= p ( e − e ) ⋅ p(
e | Y ) ⋅ de
∫
N w
R k
k+
1
Given the initial prior density p ( e0 | Y−
1)
= p(
e0)
,
we can recursively generate the posterior probability
density through equation (7) and (8). And the state
© 2011 ACADEMY PUBLISHER
k
k
k
k
estimate is e ˆ k = E[ ek
| Yk
] with covariance matrix
ˆ
T
ˆ ) ( ˆ
k E[( k k k k ) | Yk
] e e e e P − ⋅ − = .
The recursive Bayesian equations above are the
theoretical solution for model (6) and are intractable due
to the complexities of the posterior and priori probability
density function in the non-linear model. Usually the
numerical methods should be used for calculating the
result, such as point mass filter (PMF) or particle filter
(PF). This paper uses the PF to solve the model.
IV. PARTICAL FILTER
The fundamental of particle filter is to use particles
with weights for representing the probability density
function (PDF) and uses the recursion of the particle set
to replace the recursion of the posterior density function.
When the complex probability density function is
represented by the particle set the integration in α k and
(8) can be easily calculated according to Monte Carlo
integration theory.
i i M
Let { e k , wk} i=
1 be particle set for the posterior PDF
M
∑
i=
1
i
p( e k | Yk
) , where w = 1 , then
k
k
k
M
∑
i=
1
i
p(
| Y ) ≈ w ⋅ ( e − e )
e δ (9)
where δ (⋅)
is Dirac-Delta function. The estimator and its
covariance are
=
=
k
∑
eˆ
k
= E[
e
=
=
∫ ek
⋅ N ∑
R
M
∑
i=
1
k
| Y ]
i
k
k
M
i=
1
w ⋅e
i
k
w
i
k
T
∫ ( ek
−eˆ
k)
⋅(
ek
−eˆ
k)
⋅
N ∑
R
M
i=
1
⋅
i=
1
i i
i T
w ⋅(
e −eˆ
) ⋅(
e −eˆ
)
k
k
k
k
k
k
k
k
k
M
k
( e
T
Pˆ
= E[(
e −eˆ
) ⋅(
e −eˆ
) | Y ]
k
k
i
k
i
− e ) ⋅ de
δ (10)
k
k
i
i
w ⋅δ
( e −e
) ⋅de
(11)
The recursion of the particle set is simply explained
below:
i i M
Let { e k−1 , wk−1} i=
1 be the particle set at time k-1
which represents the posterior PDF p( e k −1 | Yk
−1)
. At
time k, we first draw sample i
e k from an easy sampling
i
distribution q( e k | ek
−1,
yk
) and then update its weight
using
w
i
k
∝ w
i
k −1
i i i
p(
yk
| ek
) ⋅ p(
ek
| ek
⋅
i i
q(
e | e , y )
k
k
k −1
k
k
k
−1
)
k
k
(12)