Research ArticleThe Fractional Kalman Filter-Based Asynchronous MultirateSensor Information Fusion

Guangyue Xue ,1,2 Yubin Xu,1 Jing Guo,1 and Wei Zhao3

1China Academy of Civil Aviation Science and Technology, Chaoyang District, Beijing 100028, China2China Transport Telecommunications & Information Center, Chaoyang District, Beijing 100011, China3The Thirty-Second Research Institute of China Electronic Technology Group Corporation, Jiading District, Shanghai 201808, China

Correspondence should be addressed to Guangyue Xue; iviiivi@163.com

Received 12 July 2018; Accepted 4 September 2018; Published 5 December 2018

Guest Editor: Liang Hu

Copyright © 2018 Guangyue Xue et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A fractional Kalman filter-based multirate sensor fusion algorithm is presented to fuse the asynchronous measurements of themultirate sensors. Based on the characteristics of multirate and delay measurement, the state is reestimated at the time when thedelayed measurement occurs by using weighted fractional Kalman filter, and then the state estimation is updated at the currenttime when the delayed measurement arrives following the similar pattern of Kalman filter. The simulation examples are given toillustrate the effectiveness of the proposed fusion method.

1. Introduction

Multisensor information fusion has been a key issue in sensorresearch since the 1970s and it has been applied in manyfields, such as navigation, tracking, control, and wirelesssensor networks. Due to the limitations of sensors, the sin-gle sensor cannot accurately estimate the system states; var-ious asynchronous sensors with multiple sampling rates areused, such as visual sensors, position sensors, and inertialsensors. Moreover, with the improvements of the complex-ity of multisensor systems and design accuracy, the designsof integer-order estimators cannot also meet the existingrequirements. Considering the high-accuracy estimation,fractional-order and multisensor fusion are studied by moreand more scholars.

Because systems can be accurately described by fractionalcalculus operator, fractional order has been widely appliedin many fields, such as electromagnetism, thermal, elec-trochemical, robot, control system, and image processing[1]. Sierociuk and Dzieliski [2] proposed fractional-orderKalman filter to estimate the states and parameters of discretefractional-order state models. In [3], the fractional-orderKalman filter was introduced to fuse the MEMS (microelec-tromechanical systems) sensor data, which was successfully

applied in the estimation of motion problems. In [4], theextended fractional Kalman filter is utilized for state esti-mation strategy for fractional-order systems with noisesand multiple time delayed measurements. In [5], the con-trol, estimation, and stability analysis for fractional-ordersystem were investigated. In [6–8], the stability theories offractional-order systems were studied to provide theoreticalbasis for fractional-order systems state estimation methods.The discrete-time differential systemmodeling was improvedin [9, 10]; fractional-order system modeling lays the founda-tion for the discrete filter design. Although fractional-orderfilters are used in some fields, the fractional-order-basedasynchronous multirate sensor fusion is not considered.

The state estimate plays an important role in practicalapplication, such as tracking control [11, 12]; multiple sen-sors are utilized to achieve the higher estimate performance.Since multisensor fusion can get more comprehensive andrefined information than any single sensor alone, the asyn-chronous multirate sensor information fusion becomes animportant problem in the actual system. In [13], the asyn-chronous multirate information fusion was modeled, andKalman filter-based information fusion algorithm was pro-posed. In [14], the Kalman filter-based asynchronous opti-mal estimation was presented for a class of 2D gauss

HindawiComplexityVolume 2018, Article ID 1450353, 10 pageshttps://doi.org/10.1155/2018/1450353

http://orcid.org/0000-0001-7574-0888https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2018/1450353

Markov process. Xia et al. [15] designed multiple-lag out-of-sequence measurement filtering algorithms for fusing the net-work delayed data. The main drawback of the above methodsis only a consideration for the integer Kalman filters; thus, thesystem states cannot be approximated accurately.

In this paper, we present a novel fractional fusionalgorithm to the asynchronous multirate sensor systems.The fractional multirate sensor system is addressed, and thefractional Kalman filter is used for asynchronous fusion algo-rithm, such that the fusion results achieve high-precision andeconomic storage space.

2. Problem Formulations

2.1. Discrete Linear System Model. Consider the state equa-tion and measurement equation of integer systems as follows:

x k + 1 = Ax k + Bu k +w k ,

yi k = Cix k + v k ,1

where Ci denotes the measurement matrix; i = 1 and i = 2represent fast and slow sampling rate measurements,respectively; v k is the measurement noise; w k is theprocess noise.

Due to slow rate and delay, system states cannot bemeasured accurately by slow sensor. Thus, fast sensor isintroduced to improve control performance of robotic sys-tems. The relationship between slow sensor and fast sensoris depicted in Figure 1, where l is a positive integer, and s1and s2 are defined as the different rates of sensors, respec-tively, and satisfy the following equation:

s1 = ls2 2

Because systems can be accurately described by frac-tional calculus, the fractional-order Grunwald-Letnikov dif-ference is introduced to transfer the original systems tofractional systems.

According to the definition of the fractional-orderGrunwald-Letnikov difference,

Δnxk =1hn

〠k

j=0−1 j

n

jxk−j, 3

where n ∈ R is the order of fractional difference; R is the set ofreal numbers; h is the sampling interval, later assumed to be1, and k is the number of samples for which the derivative iscalculated. The factor can be obtained from

n

j=

1 j = 0,n n − 1 ⋯ n − j + 1j

j > 04

Using the definition in (3), the traditional discrete linearstochastic state-space system can be rewritten as follows:

Δϒx k + 1 = Adx k + Bu k +w k , 5

x k + 1 = Δϒx k + 1 − 〠k+1

j=1−1 jϒjx k + 1 − j , 6

where

ϒj =

n1

j0 ⋯ 0

0n2

j⋯ 0

⋮ ⋮ ⋱ ⋮

0 0 ⋯nN

j

, 7

tlk−1 tlk−1 tlk tlk+1 tlk+1 Time

lkth s2 measurement

(k−1)th s2measurement

kth s2measurement

⁎ s2 sensor measurements• System states

° s1 sensor measurements

(k−2)th s2measurement

Figure 1: The asynchronous measurement schematic diagram between different sampling rates.

2 Complexity

Δϒx k + 1 =Δn1x1 k + 1

⋮

ΔnN xN k + 1, 8

with Ad = A − I and n1,… , nN represent the orders ofsystems.

According to (5)–(8), the system model is summarizedas follows:

x k + 1 = Adx k + Bu k +w k

− 〠k+1

j=1−1 jϒjx k + 1 − j ,

9

yi k = Cix k + v k 10

Assumption 1. The process noise w k and measurementnoise v k are Gaussian noises, which are independent fromeach other and satisfy

E w k = 0,

E v k = 0,

E v j wT k = 0,

E w j wT k = R k δjk,

E v j υT k =Q k δjk,

11

where R and Q are variances of w k and v k , respectively.

Assumption 2. The initial state x 0 is irrelevant with processnoisew k andmeasurement noise v k ; moreover, it satisfies

E x 0 = μ0,E x 0 − μ0 x 0 − μ0T = P0 12

2.2. Fractional Kalman Filter

Lemma 1 (see [2]). Considering the fractional discrete state(9) and measurement (10), the fractional recursive Kalman fil-ter is devised as follows:

x̂ k + 1 ∣ k = Adx̂ k ∣ k + Bu k

− 〠k+1

j=1−1 jϒjx̂ k + 1 − j ,

13

P k + 1 ∣ k = Ad +ϒ1 P k ∣ k Ad +ϒ1 T +Q k

+ 〠k+1

j=1ϒjP k + 1 − j ∣ k + 1 − j ϒTj ,

14

x̂ k + 1 ∣ k + 1 = x̂ k + 1 ∣ k + K k + 1y k + 1 − Cix̂ k + 1 ∣ k ,

15

P k + 1 ∣ k + 1 = I − K k + 1 P k + 1 ∣ k , 16

K k + 1 = P k + 1 ∣ k CTi CiP k + 1 ∣ k CTi

+ R k + 1 −1,17

x̂ 0 ∣ 0 = μ0,

P 0 ∣ 0 = P0,18

where

P k + 1 ∣ k = E x̂ k + 1 ∣ k − x k + 1

x̂ k + 1 ∣ k − x k + 1 T

= Ad +ϒ1 E x̂ k ∣ k − x k

x̂ k ∣ k − x k T Ad +ϒ1 T

+ E wkwTk + 〠k+1

j=1ϒjE x̂ k − j ∣ k − j

− x k − j × x̂ k − j ∣ k − j − x k − j T ϒTj= Ad +ϒ1 P k ∣ k Ad +ϒ1 T +Q k

+ 〠k+1

j=1ϒjP k + 1 − j ∣ k + 1 − j ϒTj

19

From above, the prediction of the covariance error matrixdepends on the values of the covariance matrices in previoustime samples. This is the main difference in comparison withan integer-order Kalman filter.

In the following section, the fractional Kalman filter-basedasynchronous multirate sensor fusion algorithm is elaborated.

3. Asynchronous Multirate SensorInformation Fusion

According to the relationship between sensors shown inFigure 1, it is known that the slow rate delay measurement isthe state measurement at the time tκ = tlk−l. As shown inFigure 2, the key task of the fusionmethod is to utilize the delaymeasurement with slow sampling rate to calculate x̂ lk .

The realization of the algorithm is mainly divided intothe following two steps:

Step 1 As shown in Figure 3, the slow rate delay mea-surement and fast rate measurement at the timetlk are applied to reestimate the state at the timetκ by a weighted fusion method

Step 2 The fusion estimate xF κ ∣ lk is utilized to updatethe current state estimate

Considering the smoothing and the weighted optimalfusion, the weighted fusion method in [16] is introduced tofuse multirate sensor measurements for reestimating thestate at the time tκ. Moreover, the optimal estimation at thetime tκ is used to update the state at the time tlk.

3Complexity

Lemma 2 (see [16]). Considering the state at the time tκ, mul-tisensor measurements are available simultaneously, andthen, system fusion estimation is formulated as follows:

x̂F κ = a1x̂1 κ + a2x̂2 κ , 20

PF = a21P1 + a22P2 + 2a1a2, 21

where

Pm =1,2,F = E x̂m − x x̂m − x Τ ,

P12 = E x̂1 − x x̂2 − xT ,

a1 =tr P2 − tr P12

tr P1 + tr P2 − 2tr P12,

a2 =tr P1 − tr P12

tr P1 + tr P2 − 2tr P12,

22

where tr ⋅ denotes trace of matrix. tr PF satisfies the follow-ing equation:

tr PF ≤ tr P1 ,

tr PF ≤ tr P223

Remark 1. For convenience, the fusion weights can also bechosen as follows:

a1 =tr P2

tr P1 + tr P2, 24

a2 =tr P1

tr P1 + tr P225

In order to solve the fusion estimation (20), the frictionalKalman filters are introduced to obtain x̂1 κ and x̂2 κ .

Firstly, considering the slow sensor, the state estimationx̂2 k is solved at the time tκ, where the corresponding stateand measurement equations are denoted as follows:

x lk = Aldx lk − l + Bu lk + 〠l

j=1Ajw lk

− 〠lk

j=1−1 jϒjx lk − j ,

26

y2 lk = C2x lk + v lk 27

According to (26) and (27), x̂2 k can be solved by(13)–(18) as follows:

x̂2 κ ∣ κ − 1 = Aldx̂2 κ − 1 + Bu κ

− 〠κ

j=1−1 jϒjx̂2 κ − j ,

28

P2 κ ∣ κ − 1 = Ald +ϒ1 P2 κ − 1 κ − 1 Ald +ϒ1T

+ 〠l−1

j=0Aj−1d Q κ − 1 〠

l−1

j=0Aj−1d

T

+ 〠κ

j=1ϒjP2 κ − j ∣ κ − j ϒTj ,

29

x̂2 κ ∣ κ = x̂2 κ ∣ κ − 1 + K2 κy2 κ − C2x̂2 κ ∣ κ − 1 ,

30

P2 κ ∣ κ = I − K2 κ P2 κ ∣ κ − 1 , 31

K2 κ = P2 κ ∣ κ − 1 CT2C2P2 κ ∣ κ − 1 CT2 + R κ

−1,32

x̂2 0 ∣ 0 = μ0,

P2 0 ∣ 0 = P033

In the following, based on a reverse filter algorithm, thestate at the time tκ is estimated by fast rate measurement atthe time tlk. The fractional Kalman filter (13)–(18) is adoptedto obtain the estimation x̂1 lk ∣ lk at the time tlk.

x̂1 lk ∣ lk − 1 = Adx̂1 lk − 1 + Bu lk − 1

− 〠lk

j=1−1 jϒjx̂1 lk − j ,

34

Time

Arrival time ofmeasurements

tlk+ltlk(tk+1)

t(lk)

tlk−l(tk) tlk−l

C2x(lk − l) + v(lk − l)C1x(lk) + v(lk)

x(lk)ˆ

Figure 2: The core purpose of the proposed method.

Arrival time ofmeasurements

Timetlk+ltlk+ltlktlk−l(tk) tlk−l

x(lk − l)C1x(lk − l) + v(lk − l)

ˆC2x(lk − l) + v(lk − l)

Figure 3: Calculate the xF κ ∣ lk by using weighted fusion method.

4 Complexity

P lk ∣ lk − 1 = Ad +ϒ1 P1 lk − 1 ∣ lk − 1Ad +ϒ1 T +Q lk − 1

+ 〠lk

j=1ϒjP lk − j ∣ lk − j ϒTj ,

35

x̂1 lk ∣ lk = x̂1 lk ∣ lk − 1 + K1 lky1 lk − C1x̂1 lk ∣ lk − 1 ,

36

P1 lk ∣ lk = I − K1 lk P1 lk ∣ lk − 1 , 37

K1 lk = P1 lk ∣ lk − 1 CT1C1P1 lk ∣ lk − 1 CT1 + R lk

−1,38

x̂1 0 ∣ 0 = μ0,

P1 0 ∣ 0 = P039

All the state transition matrixes of actual systems areexponential matrixes, which are reversible, defined as A−1d .Reverse state transition equation from time tlk to tκ is shownas follows:

x1 κ = A−ld x1 lk − 〠l−1

j=1Aj−1d w1 lk − j , 40

and the solution of x̂1 κ ∣ κ is revealed as follows:

x̂1 κ ∣ κ = A−ld x̂1 lk ∣ lk − 〠l−1

j=1Aj−1d ŵ1 lk − j , 41

which can be transformed to

x̂1 κ ∣ κ = A−ld x̂1 lk ∣ lk − 1 + K1 lk y1 lk− C1x̂1 lk ∣ lk − 1

42

Substituting (30) and (42) into (20), we can yield the opti-mal fusion estimation at the time tκ.

x̂F κ = a1A−ld x̂1 lk ∣ lk − 1 + K1 lk y1 lk− C1x̂1 lk ∣ lk − 1 + a2 x̂2 κ ∣ κ − 1+ K2 κ y2 κ − C2x̂2 κ ∣ κ − 1

43

Defining x k = x k − x̂ k , one can find

x̂F κ − x κ = a1A−ld x̂1 lk ∣ lk − 1+ K1 lk y1 lk − C1x̂1 lk ∣ lk − 1+ a2 x̂2 κ ∣ κ − 1 + K2 κ y2 κ− C2x̂2 κ ∣ κ − 1 − a1 + a2 x κ ,

44

x1 lk ∣ lk − 1 = Adx κ + K1 lk y1 lk− C1x̂1 lk ∣ lk − 1+ a−11 A

ldK2 κ y2 κ

− C2x̂2 κ ∣ κ − 1

45

It is known from (45) that there is relationship betweenthe current prediction error (or estimation error) and delaymeasurement. Therefore, the slow rate delay measurementcan be applied to reestimate the current state.

Theorem 1. Consider asynchronous multirate sensor system(9) and (10). The minimum variance unbiased estimatorcan be described as (46)–(48) when the delayed measure-ment arrives.

x̂∗ lk ∣ lk = x̂1 lk ∣ lk +W lk, κ yF κ , 46

yF κ = y2 κ − C2 κ x̂F κ ∣ lk , 47

W lk, κ =12

2P1xx lk ∣ lk + P1xx lk, κ ∣ lk

+ P1xw lk, lk CT P1xx lk, κ ∣ lk+ P1xx lk, κ ∣ lk + P1xx lk ∣ lk+ a22P2 κ + CQ1 κ, lk

+ CP1xw lk, lk CT − R κ−1,

48

where

C = a1C1 κ A1 κ, lk ,

P1xx lk ∣ lk = cov x1 lk ,

P1xw lk, κ ∣ lk = cov x lk ,w lk, κ ,

P2 κ = cov x2 κ ,

Q1 lk, κ = cov w2 lk, κ

49

Proof 1. In order to prove the unbiasedness of proposedfuse algorithm, the update of coefficient W lk, κ of unbi-ased estimator is derived as follows:

P2 κ is calculated in (31), and P1xx can be given by (37);P1xw lk, κ ∣ lk and Q1 lk, κ are given as follows:

P1xw lk, κ ∣ lk =Q1 lk, κ − K1 lk C1 κ Q1 lk, κ ,

Q1 lk, κ = 〠l−1

j=1Aj−l−1d Q1 lk − j A

j−l−1d

T 50

Combining (31), (46), and (47), one has

x∗ lk ∣ lk = x1 lk ∣ lk −W lk, κ yF κ , 51

where x∗ lk ∣ lk = x lk ∣ lk − x̂ lk ∣ lkFor simplicity, W lk, κ is rewritten as W and we yield

x∗ lk ∣ lk = x1 lk ∣ lk −W y2 κ − C2 κ x̂F κ ∣ lk 52

Substituting the coefficient given by (24) and (25) yields

x∗ lk ∣ lk = x1 lk ∣ lk −Wy2 κ+WC2 κ a2x̂2 κ ∣ lk + a1x̂1 κ ∣ lk ,

53

5Complexity

which can be approximatively regard as follows:

x∗ lk ∣ lk ≈ x1 lk ∣ lk −W a2y2 κ − C2 κ a2x̂2 κ ∣ lk+W a1y1 κ − C1 κ a1x̂1 κ ∣ lk

54

Moreover, one can obtain that

x∗ lk ∣ lk = x1 lk ∣ lk −Wa2 x2 κ + υ κ−Wa1C1 κ A1 κ, lk

C

x1 lk −w1 lk, κ 55

Define C as a1C1 κ A1 κ, lk ; thus, (55) can be repre-sented as follows:

x∗ lk ∣ lk = I −WC x1 lk ∣ lk −Wa2x2 κ−Wv κ +WCw1 κ, lk

56

According to (56), P∗ lk ∣ lk = E x∗ lk ∣ lk x∗ lk ∣ lk Tcan be deduced as follows:

P∗ lk ∣ lk = I −WC P1xx lk ∣ lk I −WCT

− I −WC P1xx lk, κ ∣ lk WT

− I −WC P1xw lk, lk WC T

+ a22WP2 κ WT +WR κ WT

+WCQ1 κ, lk WCT

57

Then, the trace of P∗ lk ∣ lk is introduced to solve W inthe sense of linear minimum variance, and one can obtain

∂ tr P∗ lk ∣ lk∂W

= 2WP1xx lk ∣ lk + 2a22WP2 κ

− 2WR κ + 2WCWQ1 κ, lk CT

+ 2WP1xx lk, κ ∣ lk+ 2WCP1xw lk, lk CT

− P1xx lk, κ ∣ lk − P1xw lk, lk CT

58

Let ∂ tr P∗ lk ∣ lk /∂W = 0, then it can be found that2PNxx lk ∣ lk + PNxx lk, κ ∣ lk + PNxw lk, lk CT = 2W

PNxx lk ∣ lk + a2VPV κ + CQN κ, lk + PNxx lk, κ ∣ lk + CPNxx lk ∣ lk CT − R κ Moreover, W can be shown as fol-lows:

W =12

2P1xx lk ∣ lk + P1xx lk, κ ∣ lk

+ P1xw lk, lk CT P1xx lk, κ ∣ lk+ P1xx lk ∣ lk + a22P2 κ + CQ1 κ, lk

+ CP1xw lk, lk CT − R κ−1

59

With the equation ofW, P∗ lk ∣ lk satisfies the followinginequality:

0 ≤ P∗ lk ∣ lk ≤ Pi lk ∣ lk , 60

where i = 1, 2

4. The Stability Analysis of FractionalKalman Filter

The stability of fractional Kalman filter is analyzed based onLyapunov stability theory in the section.

Theorem 2. Considering asynchronous multirate sensor sys-tem (9) and (10), the fractional Kalman filter is given by(13)–(18). Then, the error of estimation is exponentiallybounded in sense of mean square.

Proof 2. The Lyapunov function is chosen as follows:

V k = xT k P−1 k x k , 61

where x k = x k − x̂ k is error of estimation, and P kdenotes covariance matrix.

Due to the principle of stability, when k increases, if ΔVk + 1 =V k + 1 − V k remains negative besides x k = 0,then x k converges to 0.

ΔV k + 1 = V k + 1 −V k= xT k + 1 P−1 k + 1 x k + 1

− xT k P−1 k x k

62

It is obtained from the fractional Kalman filter equationthat

x k + 1 = Ad I − K k + 1 CA

x k − 〠k+1

j=1−1 jϒjx k + 1 − j ,

63

where C is the measurement matrix; Ad I − K k + 1 C iswritten as A.

Moreover, ΔV k + 1 satisfies

ΔV k + 1 < xT k ATP−1 k + 1 A − P−1 kℙ

x k 64

ℙ < 0 is a sufficient condition for ΔV k + 1 < 0. One has

ATP−1 k + 1 A − P−1 k < 0 65

It is deduced from (65) that

P−1 k + 1 − A−TP−1 k A−1 < 0 66

6 Complexity

Moreover, multiplying P k + 1 on both sides of (66)obtains

I − P k + 1 A−TP−1 k A−1 < 0 67

According to (14) and (20), one can get

P k + 1 = AP k AT + AdK k R k KT k ATd

+Q k + 〠k+1

j=1ϒjP k + 1 − j ∣ k + 1 − j ϒTj

68

Substituting (68) to (67), we can derive

− AdK k R k KTT k ATd +Q k

+ 〠k+1

j=1ϒjP k + 1 − j ∣ k + 1 − j ϒTj

× A−TP−1 k A−1 < 0

69

Due to A−TP−1 k A−1 > 0, (69) satisfies

AdK k R k KT k ATd +Q k

+ 〠k+1

j=1ϒjP k + 1 − j ∣ k + 1 − j ϒTj > 0

70

Since P k + 1 is positive definite, it is found that

P k + 1 = AdP k ATd − AdK k CP k ATd +Q k

+ 〠k+1

j=1ϒjP k + 1 − j ∣ k + 1 − j ϒTj > 0

71

According to (70), to make sure the inequality (69) issatisfied, one obtains that

AdP k ATd − AdK k CP k A

Td < AdK k R k K

T k ATd , 72

where P ⋅ represents the positive definite error variancematrix, and ϒjP k + 1 − j ∣ k + 1 − j ϒTj is positive definite.

Due to the positive definite of R k , it can be derived from(72) that

Ad K k R k KT k + K k CP k − P k ATd > 0,

K k R k KT k

>0

+ K k C − I>0

P k > 0 73

Thus, K k C − I > 0 can ensure the stability of fractionalKalman filter.

K k C − I > 0 74

According to the fractional Kalman filter gain matrix Kk , it can be obtained that

K k C − I = P k CT CP k CT + R k −1C − I

> P k CT CP k CT −1C − I=0

, 75

where R k is positive definite to guarantee the stability offractional Kalman filter.

5. Simulation Results

To demonstrate the applicability of proposed method, thefractional Kalman filter-based fusion algorithm with variousvalues is simulated. The system parameters and the systeminitial values are shown as follows:

Ad =0 0 1

−0 035 −0 01,

n1

n2=

0 5

0 4,

B =0

0 1, C1 = 0 3 0 4 , C2 = 0 6 0 9 ,

w k ∼N 0,0 0 1

0 1 0, υ k ∼N 0 0 1 ,

l = 4, P0 =100 0

0 100, x0 =

0

0

76

The control law is designed as u = ur − −0 015 0 03 x,and ur is a square signal with the period of 10s.

Based on the conclusions in [2], the order of system equa-tion should be defined previously and the control accuracy isinfected by the value range of j in the sum term ∑k+1j=1 −1

j

ϒjx k + 1 − j . Thus, the upper bound of j must be definedpreviously in practice. In the simulation, as the width ofa circular buffer of past state vectors for fractional-order dif-ference, memory length L is chosen as 200 or 50.

The fractional Kalman filter-based asynchronous multi-rate sensor information fusion results are described fromFigures 4–8. The reference input and output signal in the sys-tem as well as the multisensor measurement are shown inFigure 4. Moreover, the system state fusion estimation, mea-surement estimation results of sensor 1 and sensor 2 areshown in Figures 5 and 6, where sensors 1 and 2 representfast sensor and slow sensor, respectively. The estimationerrors of x1 and x2 are described by Figures 5 and 6 in detail.

The simulation results of nN = 50 are depicted fromFigures 9–13. Figure 9 provides input signal, output signal,and the measurement of output. The state estimation basedon fusion method and a single sensor are provided byFigures 10 and 11, respectively. Finally, Figures 12 and 13describe the error curves with different algorithms. Fromthe simulation results, one can conclude that the proposedalgorithm gives better control performance.

7Complexity

8

6

4

2

0

0 500

Measurement y1Measurement y2

Output signalInput signal

1000Time

1500

−2

−4

−6

−8

Figure 4: Input signal, output signal, and multisensor measurements(L=200).

3

2

1

0

0 500

320310300

State x1Sensor 1

Sensor 2Fusion

1000Time

Stat

e esti

mat

ion

1500

−1

−2.7

−2.65

−2.6

−2.55

−2.5

−2

−3

Figure 5: The estimation of x1 (sensor 1, sensor 2, and fusion)(L=200).

4

3

2

1

1

0

0 500

State x2Sensor 1

Sensor 2Fusion

1000Time

Stat

e esti

mat

ion

1500

−2

−3

−4

Figure 6: The estimation of x2 (sensor 1, sensor 2, and fusion)(L=200).

0.3 0.30.05

0

800 805 810 815 820

−0.05

−0.1

0.25

2

1

1

0

0 500

Sensor 1Sensor 2

Fusion

1000Time

Estim

atio

n er

ror o

f sta

te x

1

1500

−2

−3

−4

Figure 7: The estimate error of x1 (sensor 1, sensor 2, and fusion)(L=200).

10.4

0.2

0

980 985 990 995 1000−0.2

0.5

0

0 500

Sensor 1Sensor 2

Fusion

1000Time

Estim

atio

n er

ror o

f sta

te x

1

1500

−0.5

Figure 8: The error of x2 estimation (sensor 1, sensor 2, and fusion)(L=200).

6

4

1

0

0 500

Measurement y1Measurement y2

Output signalInput signal

1000Time

1500

−2

−4

−6

Figure 9: Input signal, output signal, and multisensor measurement(L=50).

8 Complexity

In simulation results, the effectiveness of the proposedfusion method is verified, and the system output value andfusion accuracy are compared. It can be obtained thathigh-precision and economic storage space requirement issatisfied as L = 50.

6. Conclusion

The fractional Kalman filter-based fusion algorithm is pre-sented to solve the problem of asynchronous multirate sensorinformation fusion. According to the memory performanceof fractional order, which can accurately describe the essen-tial characteristics of the system, the fractional filter is intro-duced to improve the estimation accuracy. Based on therelationship between slow rate delay measurement and thecurrent state estimation, the minimum variance unbiasedestimator is designed to update the current estimation. Thesimulation results prove the superiority of this method.

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (no. U1533114).

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2.52

1.51

2

1.95

1.90.5

0

−1−0.5

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1000Time

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e esti

mat

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9Complexity

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10 Complexity

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