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