Due to making full use of information of all the measurements, all types of filtering algorithms (such as EKF, UKF, and PF)
should have less information losses based on the systems (1) and (4). But the measurement Eq. (4) with a high dimension
will bring the large computational burden, particularly in the large-scale WSNs. So it is significant to find the equivalent or
approximate fusion methods to reduce the computational cost. Our aim is to find a universal weighted measurement fusion
estimation algorithm for nonlinear systems.
The following Theorem 1 gives a simple WMF algorithm for nonlinear systems.
Theorem 1. For the systems (1) and (2), if there is a linear relationship among the nonlinear measurement equation h
(j)
(x(k), k),
j =1,2,..., L, that is to say, there is a public function hðxðkÞ; kÞ2R
p
that satisfies the relationship of h
(j)
(x(k), k)=H
(j)
(k)h(x(k), k)
with the matrix H
ðjÞ
ðkÞ2R
m
j
p
, the optimal measurement equation of the weighted measurement fusion system (WMFS) is
given as:
z
ðIÞ
ðkÞ¼H
ðIÞ
ðkÞhðxðkÞ; kÞþ
v
ðIÞ
ðkÞ ð9Þ
where
z
ðIÞ
ðkÞ¼½M
T
ðkÞR
ð0Þ1
MðkÞ
1
M
T
ðkÞR
ð0Þ1
z
ð0Þ
ðkÞð10Þ
v
ðIÞ
ðkÞ¼½M
T
ðkÞR
ð0Þ1
MðkÞ
1
M
T
ðkÞR
ð0Þ1
v
ð0Þ
ðkÞð11Þ
where R
(0)1
=(R
(0)
)
1
, then the covariance matrix of
v
ðIÞ
ðkÞ can be calculated as
R
ðIÞ
ðkÞ¼½M
T
ðkÞR
ð0Þ1
MðkÞ
1
ð12Þ
M(k) and H
(I)
(k) are the full rank decomposition matrices of H
(0)
(k)=[H
(1)T
(k), ..., H
(L)T
(k)]
T
, which can be calculated by Hermite
canonical form. Under the condition without causing any confusion, the time mark k of H
(j)
(k), H
(I)
(k), H
(0)
(k), M(k) and R
(I)
(k) will
be omitted for simplifying the presentation hereinafter.
Proof. Let the rank of matrix H
(0)
=[H
(1) T
, ..., H
(L)T
]
T
is r; r 6 min
P
L
i¼1
m
i
; p
no
. From matrix theory [7], there are a full-col-
umn rank matrix M and a full-row rank matrix H
(I)
such that
H
ð0Þ
¼ MH
ðIÞ
ð13Þ
i.e., M and H
(I)
are the full rank decomposition matrices of H
(0)
. Then, we have
z
ð0Þ
ðkÞ¼H
ð0Þ
hðxðkÞ; kÞþ
v
ð0Þ
ðkÞ¼MH
ðIÞ
hðxðkÞ; kÞþ
v
ð0Þ
ðkÞð14Þ
Since M is full-column rank, M
T
R
(0)1
M is nonsingular. Taking H
(I)
h(x(k), k) as the measurement object and applying
weighted least squares (WLS) criterion, the optimum Gauss–Markov estimate of H
ðIÞ
h(x(k), k) can be obtained as (9)–(11)
[14]. The proof is completed. h
Remark 1. For linear systems, WMF is functionally equivalent to CMF under certain conditions, i.e., WMF has the global
optimality [18,8,23]. Moreover, WMF can reduce the computational cost since the measurements of WMF have a lower
dimension than the one of CMF by the lossless compress. Here, Theorem 1 gives a compress method for nonlinear systems
by the full rank decomposition. It has similar advantage to the WMF for linear systems, which can be seen in the later text.
Remark 2. In fact, the full rank decomposition (13) is only implemented under r < min
P
L
i¼1
m
i
; p
no
. When
r ¼ min
P
L
i¼1
m
i
; p
no
, the decomposition is avoided. Namely, M = H
(0)
and H
(I)
= I if H
(0)
is a full-column rank matrix, i.e.,
r = p when measurements can be still compressed without information loss; and M = I and H
(I)
= H
(0)
if H
(0)
is a full-row rank
matrix, i.e., r ¼
P
L
i¼1
m
i
when measurements cannot be compressed without information loss. However, we generally have
P
L
i¼1
m
i
p with the increase of the number of sensors.
3. Universal WMF
Theorem 2. For the systems (1) and (2), its approximate measurement fusion equation is given as:
~
z
ðIÞ
ðkÞ¼H
ðIÞ
1
Dx
ðDxÞ
2
.
.
.
ðDxÞ
l
0
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
A
þ
v
ðIÞ
ðkÞ ð15Þ
G. Hao et al. / Information Sciences 299 (2015) 85–98
87
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