Asynchronous Data Fusion With Parallel Filtering Frame

Published Online June 2011 in MECS

In recent years, multisensor data fusion technology is paid great attention in many military and civil fields, and is extensively applied. At present, a lot of data fusion algorithms to different application backgrounds and constraints are presented [1-11]. For the research of classical data fusion, the synchronous multisensor system, in which every sensor has common sampling rate and sampling time is uniform, is none of main objects. But, in the practical system, these sensors in the multisensor system have often different sampling rates and sampling points because of different task requirement and different kinds of sensors. As a result, it is interesting to study asynchronous data fusion with different sampling rates, and has important theoretical sense and extensive application scene.

Up to now, some useful data fusion algorithms for

Identify applicable sponsor/s here. If no sponsors, delete this text box. (sponsors)

asynchronous system under the centralized frame have been presented [2,6,7,8,10,11]. The work in [2] is to firstly discretize the continuous system, secondly establish the relative measurement to current state by use of the relation between the states of local points and the fusion center, and afterwards use the centralized fusion to estimate the state of the target. But, this algorithm only adapts the case that every sensor only has one measurement in the fusion period, and the more complex case cannot be dealt with. An interesting work is also done in [6] by combing wavelet with Kalman filter. It can treat with the noise-reduction effectively but the multisensor case cannot be solved. In [7], the multisensor multiscale fusion was considered; nevertheless the given algorithm is complex. In addition, the design of data fusion algorithm for multirate sampling system were researched in [10,11]. Basically, they both adopt the remodeling idea to the state of sampling points in the fusion period. Thereby, these two algorithms are suboptimal in the sensor of linear minimum mean square error (LMMSE).

Aiming at the above-mentioned problems, this paper takes a kind of multisensor dynamic system with different sampling rates, and introduces the parallel filtering and sequential filtering to solve asynchronous data fusion with integer times sampling. Accordingly, a novel optimal asynchronous data fusion algorithm is proposed and the running steps are listed in this paper. Its main structure includes four aspects such as measurement mapping, parallel filtering, judgment of measurement number, and sequential filtering.

The rest of this paper is organized as follows. In Section II, it describes the multisensor system with integer times sampling and problem formulation. Section III proposes a novel parallel filtering fusion algorithm. Computer simulation is done in Section IV. Finally, we conclude in Section V.

• ii.    Problem Formulation

• A.    System Description

A kind of multisensor system which is composed of N sensors is considered. Every sensor observes the target state with different sampling rate, and the measurement is z i (k , +1) = Hi (k , +1) x (k , +1) + v , (k , +1) (1) where i = 1,2 l, N , H, (ki +1) is the measurement matrix. The corresponding state equation of i sensor is x (ki +1) = Ф (k +1, ki) x (ki) + w (ki)      (2)

where the sampling period of i sensor is T_i which a is an integer, and

T. /

7_T = L , j = 1,2, l , N - 1; L = 1,2, l

/ ^Tj + 1

(3) where L is also an integer. Suppose that the sampling period for which the sampling period is the biggest among them is the fusion period, then

T = T 1

So, there are M_i measurements for sensor i in a fusion period, then

T = MT i (5)

It easily knows that the sampling periods for all of sensors have the integer L times relation from Eq. (3) and Eq. (4).

Figure. 1 shows the multisensor system with L = 2 . kM. + 1. ( i = 1,2, l , N ; l i = 1,2, l , M i ) is l i th sampling time of sensor i in ( k + 1) th fusion period, and

M , = L - ¹, ( i = 1,2, l , N ) (6)

			( k + 1) M , k +1	( k + 1) M , ---о k + 2
1	0	k
		kM 2 +1	( k + 1) M 2	( k + 2) M 2
Sensor 2	o— -	—o---1—	—o----1	——о
	0	k	k +1	k + 2
		kM N 1 +1	( k + 1) MN -,	( k + 2) M N -
Sensor N-	O-...	—0—1-	—о—1—	... —Q
	0		k + 1	k + 2
		kMN +1	( k + 1) MN	( k + 2) M N
Sensor
		kMN + 4
	0	kN	k +1	k + 2
		Ln ---Fusion period—►

Figure.1 The sampling of multisensor system with L = 2

Then, the dynamic given by Eq.(1) and (2) can be z i (kM, + 1,) = Hi. (kM, + 1,) x (kM, + 1,) + v, (kM, + 1,)

x ( kMi + 1 i ) = Ф , ( kM i + l i , kM i + l i - 1) x ( kMi + l i - 1) + w ( kM , + 1 , - 1)

(8) where i = 1,2, l , N ; 1 , = 1,2, l , M i ,and k >  0 is a dis crete time variable. x ( kM , + 1 , ) e R ” ^{x 1}is state vector, Ф , ( kMj + lj , kMi + ^ - 1) e R ” ^x ” is state transfer mat rix of sensor i . Process noise w ( kM , + 1 , - 1) e R ” ^{x 1}is a Gaussian white noise sequence, and satisfies

' E { w ( kM , + 1 , - 1) } = 0

* E { ( w ( kM i + l i - 1) )( w ( jM i + 1 , - 1) ) ^T }      (9)

= Q (kMi+ li-1) Skj where k, j > 0; 0 < 1, < M i.

z , ( kM_t + li ) e R p ^x is the measurement of sensor i to x , ( kM i + l , ) at kM i + l , , H , ( kM , + l , ) e R^p,x ” is the measurement matrix. Measurement noise v , ( kM , + 1 , ) e R^{p x 1} is also a Gaussian white noise and its statistical property is

/ E { v , ( kM , + 1 , ) } = 0

\ E { v , ( kM , + 1 , ) v T ( jM, + 1 , ) } = R , (kM , + li) 3 j

(10) where k , j >  0; 0 <  1 , <  M , . R , ( kM , + 1 , ) is a positive matrix, and there are correlative between process noise and measurement noises, namely

E {(w(kM,+1,-1))(v,(jM,+1,-1))}

= S,( kM,+1,-1) Skj where k, j > 0; 0 < 1, < M,. The original state x(0) is a random vector and satisfies

E {x (0)} = x о(12)

E {[ x (0) - x o][ x (0) - x o]T }= Po(13)

• B.    Problem Formulat on

In order to conveniently describe the proposed algorithm, it is necessary to transform the abovementioned multisensor dynamic system to a single one. Based on the sampling time of fusion center, the sampling points of all of sensors can be mapped to this reference axis, see Figure.2. we easily know: firstly, there is one measurement at least at the sampling point in a fusion period after they are mapped. And, the measurement number at every sampling point is different basically. Secondly, in a fusion period there are all measurements from sensor N at every sampling point.

( k + 1) M 1

( k + 1) M , - i - 1 ^{( k +} 1) M N - 1 kM N ^{+ 1}    kM N + 3   ^kMN ^{+ 5} ( k + 1) M N - 2 ( k + 1) M n

M -1

w * ( kM N + I n ) = E Ф^ w [ ( k + 1) M n + I n - I - 1 ] (18)

I =0

E^

kM_N + 2 kM N + 4           ( k + 1) M N - 1

k     ^kMN - 1 ^{+ 1} kM N - 1 ^{+ 2}                     k + 1

kM N - 2 + 1

-------------------Fusion period ----------------

Figure.2 The mapped multisensory dynamic sampling system

Then, the basic idea of parallel filtering fusion algorithm with integer times sampling is as follows: the sampling points of all of sensors are mapped to the reference axis on the basis of sampling time of fusion center. Afterwards, the time axis of sensor N can be taken as basis and the parallel filtering algorithm can be performed. Especially, when the sampling point has multiple measurements, the sequential filtering fusion in [9] can be used. Finally, we can get the fusion estimate based on the global information for every sampling point kM_N + l_N in the fusion period. In order to realize above-mentioned idea, the following problems must be solved: i) One is how to perform the parallel filtering algorithm based on sensor N . ii) The other is how to distinguish which sensor measurements every sampling time has in a fusion period. Next, the fusion algorithm is established in terms of solving above-mentioned two problems.

• iii.    Parallel Fitering Fusion Algorithm with Integer Times Sampling

• A. Parallel Filtering Fusion Algorithm

By considering i) this subsection presents the parallel filtering fusion algorithm in the case that there is only one measurement at every sampling point, namely only consider the measurement of sensor N .

From Eq.(7) and Eq.(8), state equation and measurement equation of sensor N are as follows: x ( kM N + I n ) = Ф x ( kM N + I n - 1) + w ( kM N + I n - 1) (14) z N ( kM N + I n ) = H N ( kM N + I n ) x ( kM N + I n )

+ VN (kMN + lN )

(15) So, one has the following theorem.

Theorem 1 According to (14) and (15), one can get the following new multisensor system

x((k+1) Mn + In ) = Ф< x (kMN + In ) + w *( kMN + In )

z N (kMN + lN ) = HN x (kMN + lN ) + V N (kMN + lN )

(17) where

|f w " ( kM N IV V N ( kM N

and

^Ф N

<

^{+ l}N

+ lN )7

)    ⁽ w * T ( j ^MN ⁺ к ), V N ( jM N ^{+ l}

ГQN s T

V ° N

SR

N

5 _kJ

N 7

N

_Q * _N

M -1        /

= E **N QN Ф N )

l = 0

= Ф N (kMN + lN , kMN + lN - 1)

_ H N = H N (kMN + lN )QN = Q(kMN + In -1)_RN = RN (kMN + lN ), °N = °N (kMN + lN )

<

■

Proof . The derivations of Eq. (16) to Eq. (22) can be finished easily.

In terms of (14), we have

x((k +1)Mn + In )

= ^Ф N "  x N ^{( ( k +} 1) ^M N ^{+ l} N ^{- 1 )+} w N ^{( ( k +} 1) ^M N ^{+ l} N ^- 1 )

= ФN {ФN • xN ((k + 1)Mn +)lN - 2

^{+ w} N ^{( (} k ⁺ 1) ^M N ^{+ l} N ^{- 2 )}+} w N ^{( (} k ⁺ 1) ^M N ^{+ l} N ^- 1 )

= ^Ф N ■ x _N ^{( ( k +} 1) ^{M +} i ^- 2 ^{)+ Ф} _N ■ w _N ^{( ( k +} 1) ^M _N ^{+ l} _N ^{- 2 )}

⁺ w N ^{( ( k +} 1) ^M N ^{+ l} N ^- 1 )

= Ф^1 "{ФN " xN ((k + 1)MN + lN - 3)+ wN ((k + 1)MN + lN - 3)}

+ E ФN • wN ((k +1)Mn + In -1 -1)

= ФN • xN ((k +1)Mn + In - 3)

+ E ФN • wN ((k +1)Mn + In -1 -1)

= ФN • {фN • xn ((k +1)Mn + In - 4) + wN ((k +1)Mn + In - 4)}

+ zE ФN • wN ((k +1)Mn + In -1 -1)

= ФN • xN ((k +1)Mn + In - 4)+ФN • wN ((k +1)Mn + In - 4)

+ zE ФN • wn ((k +1)Mn + In -1 -1)

= ФN • xN ((k +1)Mn + In - 4)

+ ]T фN • wn ((k +1)Mn + In -1 -1)

l =0

M -1

= ^ N • x N ^{( kM + l} N ) ⁺ E ^Ф N • w N ^{( ( k +} 1) ^M N ^{+ l} N ^{- l -} 1 ) l =0

( A .1)

So, one has x N ((k + 1) MN + -N ) = ФN - x N (kMN + -N )

+ w N ⁽ kM N + l N )

z N ( kM + - , ) = H , x , ( kM N + I n )

+ v N ( kM N + lN ) where w n (kM , + lN )

_<^ФN = ^Ф N ^S N ^R N ^H N ^{, Q} N = ^Q N ^S N ^R N ^S N

z ( kM N + - n — 1) = z N ( kM N + - n — 1)

2) One step state predict between the fusion periods xlN —1 ((k +1) Mn + -n — 1|kMN + lN — 1)

= ^ф M x - N — I ⁽ k^M N ^{+ -} N ^{— 11 kM} N ^{+ -} N ^— ! )

M -¹   ,                                    x ( A .4)

= ^ ^Ф N ' w N ^{( ( k +} 1) ^M N ⁺ l N - l - 1 )

- = 0

^{+ S} N R N

f z(kMN + -n — 1)

_к ^— H N x l_N — 1 ^{( kM} N ^{+ l} N

'

— 1 1 kMN + -N — 1)J

and

E { w N ^{( kM} N

^{+ -} N

) - [ w N ^{( kM +} l N

= E ^

M - 1

^ Ф, - wN ((k + 1)MN + -N — l — 1)

. - = 0

Φ

2 ^Ф N ^- w N ^{( ( k + 1)} M N ^{+ -} N ^{— l — 1 )} l = 0

' M - 1

= E i E ^ф N ^X

„ I = 0

w n [( k + 1) M N + I n — l — 1]

X w N [( k + 1) M n + I n — - — 1]