# The Fractional Kalman Filter-Based Asynchronous Multirate Sensor Information Fusion

Authors: Guangyue Xue; Yubin Xu; Jing Guo; Wei Zhao
Journal: Complexity
 (2018)

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## Abstract

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


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## Body

## 1. Introduction

Multisensor information fusion has been a key issue in sensor research since the 1970s and it has been applied in many fields, such as navigation, tracking, control, and wireless sensor networks. Due to the limitations of sensors, the single sensor cannot accurately estimate the system states; various asynchronous sensors with multiple sampling rates are used, such as visual sensors, position sensors, and inertial sensors. Moreover, with the improvements of the complexity of multisensor systems and design accuracy, the designs of integer-order estimators cannot also meet the existing requirements. Considering the high-accuracy estimation, fractional-order and multisensor fusion are studied by more and more scholars.Because systems can be accurately described by fractional calculus operator, fractional order has been widely applied in many fields, such as electromagnetism, thermal, electrochemical, robot, control system, and image processing [1]. Sierociuk and Dzieliski [2] proposed fractional-order Kalman filter to estimate the states and parameters of discrete fractional-order state models. In [3], the fractional-order Kalman filter was introduced to fuse the MEMS (microelectromechanical systems) sensor data, which was successfully applied in the estimation of motion problems. In [4], the extended fractional Kalman filter is utilized for state estimation strategy for fractional-order systems with noises and multiple time delayed measurements. In [5], the control, estimation, and stability analysis for fractional-order system were investigated. In [6–8], the stability theories of fractional-order systems were studied to provide theoretical basis for fractional-order systems state estimation methods. The discrete-time differential system modeling was improved in [9, 10]; fractional-order system modeling lays the foundation for the discrete filter design. Although fractional-order filters are used in some fields, the fractional-order-based asynchronous multirate sensor fusion is not considered.The state estimate plays an important role in practical application, such as tracking control [11, 12]; multiple sensors are utilized to achieve the higher estimate performance. Since multisensor fusion can get more comprehensive and refined information than any single sensor alone, the asynchronous multirate sensor information fusion becomes an important problem in the actual system. In [13], the asynchronous multirate information fusion was modeled, and Kalman filter-based information fusion algorithm was proposed. In [14], the Kalman filter-based asynchronous optimal estimation was presented for a class of 2D gauss Markov process. Xia et al. [15] designed multiple-lag out-of-sequence measurement filtering algorithms for fusing the network delayed data. The main drawback of the above methods is only a consideration for the integer Kalman filters; thus, the system states cannot be approximated accurately.In this paper, we present a novel fractional fusion algorithm to the asynchronous multirate sensor systems. The fractional multirate sensor system is addressed, and the fractional Kalman filter is used for asynchronous fusion algorithm, such that the fusion results achieve high-precision and economic storage space.


## 2. Problem Formulations

### 2.1. Discrete Linear System Model

Consider the state equation and measurement equation of integer systems as follows:(1)xk+1=Axk+Buk+wk,yik=Cixk+vk,where Ci denotes the measurement matrix; i=1 and i=2 represent fast and slow sampling rate measurements, respectively; vk is the measurement noise; wk is the process noise.Due to slow rate and delay, system states cannot be measured accurately by slow sensor. Thus, fast sensor is introduced to improve control performance of robotic systems. The relationship between slow sensor and fast sensor is depicted in Figure1, where l is a positive integer, and s1 and s2 are defined as the different rates of sensors, respectively, and satisfy the following equation:
(2)s1=ls2.Figure 1
The asynchronous measurement schematic diagram between different sampling rates.Because systems can be accurately described by fractional calculus, the fractional-order Grunwald-Letnikov difference is introduced to transfer the original systems to fractional systems.According to the definition of the fractional-order Grunwald-Letnikov difference,(3)Δnxk=1hn∑j=0k−1jnjxk−j,where n∈R is the order of fractional difference; R is the set of real numbers; h is the sampling interval, later assumed to be 1, and k is the number of samples for which the derivative is calculated. The factor can be obtained from
(4)nj=1j=0,nn−1⋯n−j+1j!j>0.Using the definition in (3), the traditional discrete linear stochastic state-space system can be rewritten as follows:
(5)Δϒxk+1=Adxk+Buk+wk,(6)xk+1=Δϒxk+1−∑j=1k+1−1jϒjxk+1−j,where
(7)ϒj=n1j0⋯00n2j⋯0⋮⋮⋱⋮00⋯nNj,(8)Δϒxk+1=Δn1x1k+1⋮ΔnNxNk+1,with Ad=A−I and n1,…,nN represent the orders of systems.According to (5)–(8), the system model is summarized as follows:
(9)xk+1=Adxk+Buk+wk−∑j=1k+1−1jϒjxk+1−j,(10)yik=Cixk+vk.Assumption 1.
The process noisewk and measurement noise vk are Gaussian noises, which are independent from each other and satisfy
(11)Ewk=0,Evk=0,EvjwTk=0,EwjwTk=Rkδjk,EvjυTk=Qkδjk,where R and Q are variances of wk and vk, respectively.Assumption 2.
The initial statex0 is irrelevant with process noise wk and measurement noise vk; moreover, it satisfies
(12)Ex0=μ0,Ex0−μ0x0−μ0T=P0.


### 2.2. Fractional Kalman Filter

Lemma 1 (see [2]).
Considering the fractional discrete state (9) and measurement (10), the fractional recursive Kalman filter is devised as follows:
(13)x^k+1∣k=Adx^k∣k+Buk−∑j=1k+1−1jϒjx^k+1−j,(14)Pk+1∣k=Ad+ϒ1Pk∣kAd+ϒ1T+Qk+∑j=1k+1ϒjPk+1−j∣k+1−jϒjT,(15)x^k+1∣k+1=x^k+1∣k+Kk+1yk+1−Cix^k+1∣k,(16)Pk+1∣k+1=I−Kk+1Pk+1∣k,(17)Kk+1=Pk+1∣kCiTCiPk+1∣kCiT+Rk+1−1,(18)x^0∣0=μ0,P0∣0=P0,where
(19)Pk+1∣k=Ex^k+1∣k−xk+1x^k+1∣k−xk+1T=Ad+ϒ1Ex^k∣k−xkx^k∣k−xkTAd+ϒ1T+EwkwkT+∑j=1k+1ϒjEx^k−j∣k−j−xk−j×x^k−j∣k−j−xk−jTϒjT=Ad+ϒ1Pk∣kAd+ϒ1T+Qk+∑j=1k+1ϒjPk+1−j∣k+1−jϒjT.From above, the prediction of the covariance error matrix depends on the values of the covariance matrices in previous time samples. This is the main difference in comparison with an integer-order Kalman filter.In the following section, the fractional Kalman filter-based asynchronous multirate sensor fusion algorithm is elaborated.


## 2.1. Discrete Linear System Model

Consider the state equation and measurement equation of integer systems as follows:(1)xk+1=Axk+Buk+wk,yik=Cixk+vk,where Ci denotes the measurement matrix; i=1 and i=2 represent fast and slow sampling rate measurements, respectively; vk is the measurement noise; wk is the process noise.Due to slow rate and delay, system states cannot be measured accurately by slow sensor. Thus, fast sensor is introduced to improve control performance of robotic systems. The relationship between slow sensor and fast sensor is depicted in Figure1, where l is a positive integer, and s1 and s2 are defined as the different rates of sensors, respectively, and satisfy the following equation:
(2)s1=ls2.Figure 1
The asynchronous measurement schematic diagram between different sampling rates.Because systems can be accurately described by fractional calculus, the fractional-order Grunwald-Letnikov difference is introduced to transfer the original systems to fractional systems.According to the definition of the fractional-order Grunwald-Letnikov difference,(3)Δnxk=1hn∑j=0k−1jnjxk−j,where n∈R is the order of fractional difference; R is the set of real numbers; h is the sampling interval, later assumed to be 1, and k is the number of samples for which the derivative is calculated. The factor can be obtained from
(4)nj=1j=0,nn−1⋯n−j+1j!j>0.Using the definition in (3), the traditional discrete linear stochastic state-space system can be rewritten as follows:
(5)Δϒxk+1=Adxk+Buk+wk,(6)xk+1=Δϒxk+1−∑j=1k+1−1jϒjxk+1−j,where
(7)ϒj=n1j0⋯00n2j⋯0⋮⋮⋱⋮00⋯nNj,(8)Δϒxk+1=Δn1x1k+1⋮ΔnNxNk+1,with Ad=A−I and n1,…,nN represent the orders of systems.According to (5)–(8), the system model is summarized as follows:
(9)xk+1=Adxk+Buk+wk−∑j=1k+1−1jϒjxk+1−j,(10)yik=Cixk+vk.Assumption 1.
The process noisewk and measurement noise vk are Gaussian noises, which are independent from each other and satisfy
(11)Ewk=0,Evk=0,EvjwTk=0,EwjwTk=Rkδjk,EvjυTk=Qkδjk,where R and Q are variances of wk and vk, respectively.Assumption 2.
The initial statex0 is irrelevant with process noise wk and measurement noise vk; moreover, it satisfies
(12)Ex0=μ0,Ex0−μ0x0−μ0T=P0.


## 2.2. Fractional Kalman Filter

Lemma 1 (see [2]).
Considering the fractional discrete state (9) and measurement (10), the fractional recursive Kalman filter is devised as follows:
(13)x^k+1∣k=Adx^k∣k+Buk−∑j=1k+1−1jϒjx^k+1−j,(14)Pk+1∣k=Ad+ϒ1Pk∣kAd+ϒ1T+Qk+∑j=1k+1ϒjPk+1−j∣k+1−jϒjT,(15)x^k+1∣k+1=x^k+1∣k+Kk+1yk+1−Cix^k+1∣k,(16)Pk+1∣k+1=I−Kk+1Pk+1∣k,(17)Kk+1=Pk+1∣kCiTCiPk+1∣kCiT+Rk+1−1,(18)x^0∣0=μ0,P0∣0=P0,where
(19)Pk+1∣k=Ex^k+1∣k−xk+1x^k+1∣k−xk+1T=Ad+ϒ1Ex^k∣k−xkx^k∣k−xkTAd+ϒ1T+EwkwkT+∑j=1k+1ϒjEx^k−j∣k−j−xk−j×x^k−j∣k−j−xk−jTϒjT=Ad+ϒ1Pk∣kAd+ϒ1T+Qk+∑j=1k+1ϒjPk+1−j∣k+1−jϒjT.From above, the prediction of the covariance error matrix depends on the values of the covariance matrices in previous time samples. This is the main difference in comparison with an integer-order Kalman filter.In the following section, the fractional Kalman filter-based asynchronous multirate sensor fusion algorithm is elaborated.


## 3. Asynchronous Multirate Sensor Information Fusion

According to the relationship between sensors shown in Figure1, it is known that the slow rate delay measurement is the state measurement at the time tκ=tlk−l. As shown in Figure 2, the key task of the fusion method is to utilize the delay measurement with slow sampling rate to calculate x^lk.Figure 2
The core purpose of the proposed method.The realization of the algorithm is mainly divided into the following two steps:Step 1
As shown in Figure3, the slow rate delay measurement and fast rate measurement at the time tlk are applied to reestimate the state at the time tκ by a weighted fusion method
Step 2
The fusion estimatexFκ∣lk is utilized to update the current state estimateFigure 3
Calculate thexFκ∣lk by using weighted fusion method.Considering the smoothing and the weighted optimal fusion, the weighted fusion method in [16] is introduced to fuse multirate sensor measurements for reestimating the state at the time tκ. Moreover, the optimal estimation at the time tκ is used to update the state at the time tlk.Lemma 2 (see [16]).
Considering the state at the timetκ, multisensor measurements are available simultaneously, and then, system fusion estimation is formulated as follows:
(20)x^Fκ=a1x^1κ+a2x^2κ,(21)PF=a12P1+a22P2+2a1a2,where
(22)Pm=1,2,F=Ex^m−xx^m−xΤ,P12=Ex^1−xx^2−xT,a1=trP2−trP12trP1+trP2−2trP12,a2=trP1−trP12trP1+trP2−2trP12,where tr⋅ denotes trace of matrix. trPF satisfies the following equation:
(23)trPF≤trP1,trPF≤trP2.Remark 1.
For convenience, the fusion weights can also be chosen as follows:(24)a1=trP2trP1+trP2,(25)a2=trP1trP1+trP2.In order to solve the fusion estimation (20), the frictional Kalman filters are introduced to obtain x^1κ and x^2κ.Firstly, considering the slow sensor, the state estimationx^2k is solved at the time tκ, where the corresponding state and measurement equations are denoted as follows:
(26)xlk=Adlxlk−l+Bulk+∑j=1lAjwlk−∑j=1lk−1jϒjxlk−j,(27)y2lk=C2xlk+vlk.According to (26) and (27), x^2k can be solved by (13)–(18) as follows:
(28)x^2κ∣κ−1=Adlx^2κ−1+Buκ−∑j=1κ−1jϒjx^2κ−j,(29)P2κ∣κ−1=Adl+ϒ1P2κ−1κ−1Adl+ϒ1T+∑j=0l−1Adj−1Qκ−1∑j=0l−1Adj−1T+∑j=1κϒjP2κ−j∣κ−jϒjT,(30)x^2κ∣κ=x^2κ∣κ−1+K2κy2κ−C2x^2κ∣κ−1,(31)P2κ∣κ=I−K2κP2κ∣κ−1,(32)K2κ=P2κ∣κ−1C2TC2P2κ∣κ−1C2T+Rκ−1,(33)x^20∣0=μ0,P20∣0=P0.In the following, based on a reverse filter algorithm, the state at the timetκ is estimated by fast rate measurement at the time tlk. The fractional Kalman filter (13)–(18) is adopted to obtain the estimation x^1lk∣lk at the time tlk.
(34)x^1lk∣lk−1=Adx^1lk−1+Bulk−1−∑j=1lk−1jϒjx^1lk−j,(35)Plk∣lk−1=Ad+ϒ1P1lk−1∣lk−1Ad+ϒ1T+Qlk−1+∑j=1lkϒjPlk−j∣lk−jϒjT,(36)x^1lk∣lk=x^1lk∣lk−1+K1lky1lk−C1x^1lk∣lk−1,(37)P1lk∣lk=I−K1lkP1lk∣lk−1,(38)K1lk=P1lk∣lk−1C1TC1P1lk∣lk−1C1T+Rlk−1,(39)x^10∣0=μ0,P10∣0=P0.All the state transition matrixes of actual systems are exponential matrixes, which are reversible, defined asAd−1. Reverse state transition equation from time tlk to tκ is shown as follows:
(40)x1κ=Ad−lx1lk−∑j=1l−1Adj−1w1lk−j,and the solution of x^1κ∣κ is revealed as follows:
(41)x^1κ∣κ=Ad−lx^1lk∣lk−∑j=1l−1Adj−1w^1lk−j,which can be transformed to
(42)x^1κ∣κ=Ad−lx^1lk∣lk−1+K1lky1lk−C1x^1lk∣lk−1.Substituting (30) and (42) into (20), we can yield the optimal fusion estimation at the time tκ.
(43)x^Fκ=a1Ad−lx^1lk∣lk−1+K1lky1lk−C1x^1lk∣lk−1+a2x^2κ∣κ−1+K2κy2κ−C2x^2κ∣κ−1.Definingx~k=xk−x^k, one can find
(44)x^Fκ−xκ=a1Ad−lx^1lk∣lk−1+K1lky1lk−C1x^1lk∣lk−1+a2x^2κ∣κ−1+K2κy2κ−C2x^2κ∣κ−1−a1+a2xκ,(45)x~1lk∣lk−1=Adx~κ+K1lky1lk−C1x^1lk∣lk−1+a1−1AdlK2κy2κ−C2x^2κ∣κ−1.It is known from (45) that there is relationship between the current prediction error (or estimation error) and delay measurement. Therefore, the slow rate delay measurement can be applied to reestimate the current state.Theorem 1.
Consider asynchronous multirate sensor system (9) and (10). The minimum variance unbiased estimator can be described as (46)–(48) when the delayed measurement arrives.
(46)x^∗lk∣lk=x^1lk∣lk+Wlk,κy~Fκ,(47)y~Fκ=y2κ−C2κx^Fκ∣lk,(48)Wlk,κ=122P1xxlk∣lk+P1xxlk,κ∣lk+P1xwlk,lkCTP1xxlk,κ∣lk+P1xxlk,κ∣lk+P1xxlk∣lk+a22P2κ+CQ1κ,lk+CP1xwlk,lkCT−Rκ−1,where
(49)C¯=a1C1κA1κ,lk,P1xxlk∣lk=covx1lk,P1xwlk,κ∣lk=covxlk,wlk,κ,P2κ=covx2κ,Q1lk,κ=covw2lk,κ.Proof 1.
In order to prove the unbiasedness of proposed fuse algorithm, the update of coefficientWlk,κ of unbiased estimator is derived as follows:
P2κ is calculated in (31), and P1xx can be given by (37); P1xwlk,κ∣lk and Q1lk,κ are given as follows:
(50)P1xwlk,κ∣lk=Q1lk,κ−K1lkC1κQ1lk,κ,Q1lk,κ=∑j=1l−1Adj−l−1Q1lk−jAdj−l−1T.Combining (31), (46), and (47), one has
(51)x~∗lk∣lk=x~1lk∣lk−Wlk,κy~Fκ,where x~∗lk∣lk=xlk∣lk−x^lk∣lk.For simplicity,Wlk,κ is rewritten as W and we yield
(52)x~∗lk∣lk=x~1lk∣lk−Wy2κ−C2κx^Fκ∣lk.Substituting the coefficient given by (24) and (25) yields
(53)x~∗lk∣lk=x~1lk∣lk−Wy2κ+WC2κa2x^2κ∣lk+a1x^1κ∣lk,which can be approximatively regard as follows:
(54)x~∗lk∣lk≈x~1lk∣lk−Wa2y2κ−C2κa2x^2κ∣lk+Wa1y1κ−C1κa1x^1κ∣lk.Moreover, one can obtain that(55)x~∗lk∣lk=x~1lk∣lk−Wa2x~2κ+υκ−Wa1C1κA1κ,lk⏟C¯x~1lk−w~1lk,κ.DefineC¯ as a1C1κA1κ,lk; thus, (55) can be represented as follows:
(56)x~∗lk∣lk=I−WC¯x~1lk∣lk−Wa2x~2κ−Wvκ+WC¯w~1κ,lk.According to (56), P∗lk∣lk=Ex~∗lk∣lkx~∗lk∣lkT can be deduced as follows:
(57)P∗lk∣lk=I−WC¯P1xxlk∣lkI−WC¯T−I−WC¯P1xxlk,κ∣lkWT−I−WC¯P1xwlk,lkWC¯T+a22WP2κWT+WRκWT+WC¯Q1κ,lkWC¯T.Then, the trace ofP∗lk∣lk is introduced to solve W in the sense of linear minimum variance, and one can obtain
(58)∂trP∗lk∣lk∂W=2WP1xxlk∣lk+2a22WP2κ−2WRκ+2WC¯WQ1κ,lkC¯T+2WP1xxlk,κ∣lk+2WC¯P1xwlk,lkC¯T−P1xxlk,κ∣lk−P1xwlk,lkC¯T.Let∂trP∗lk∣lk/∂W=0, then it can be found that2PNxxlk∣lk+PNxxlk,κ∣lk+PNxwlk,lkC¯T=2WPNxxlk∣lk+aV2PVκ+C¯QNκ,lk+PNxxlk,κ∣lk+C¯PNxxlk∣lkC¯T−Rκ. Moreover, W can be shown as follows:
(59)W=122P1xxlk∣lk+P1xxlk,κ∣lk+P1xwlk,lkC¯TP1xxlk,κ∣lk+P1xxlk∣lk+a22P2κ+C¯Q1κ,lk+C¯P1xwlk,lkC¯T−Rκ−1.With the equation ofW, P∗lk∣lk satisfies the following inequality:
(60)0≤P∗lk∣lk≤Pilk∣lk,where i=1,2.


## 4. The Stability Analysis of Fractional Kalman Filter

The stability of fractional Kalman filter is analyzed based on Lyapunov stability theory in the section.Theorem 2.
Considering asynchronous multirate sensor system (9) and (10), the fractional Kalman filter is given by (13)–(18). Then, the error of estimation is exponentially bounded in sense of mean square.Proof 2.
The Lyapunov function is chosen as follows:(61)Vk=x~TkP−1kx~k,where x~k=xk−x^k is error of estimation, and Pk denotes covariance matrix.Due to the principle of stability, whenk increases, if ΔVk+1=Vk+1−Vk remains negative besides x~k=0, then x~k converges to 0.
(62)ΔVk+1=Vk+1−Vk=x~Tk+1P−1k+1x~k+1−x~TkP−1kx~k.It is obtained from the fractional Kalman filter equation that(63)x~k+1=AdI−Kk+1C⏟A¯x~k−∑j=1k+1−1jϒjx~k+1−j,where C is the measurement matrix; AdI−Kk+1C is written as A¯.Moreover,ΔVk+1 satisfies
(64)ΔVk+1<x~TkA¯TP−1k+1A¯−P−1k⏟ℙx~k.ℙ<0 is a sufficient condition for ΔVk+1<0. One has
(65)A¯TP−1k+1A¯−P−1k<0.It is deduced from (65) that
(66)P−1k+1−A¯−TP−1kA¯−1<0.Moreover, multiplyingPk+1 on both sides of (66) obtains
(67)I−Pk+1A¯−TP−1kA¯−1<0.According to (14) and (20), one can get
(68)Pk+1=A¯PkA¯T+AdKkRkKTkAdT+Qk+∑j=1k+1ϒjPk+1−j∣k+1−jϒjT.Substituting (68) to (67), we can derive
(69)−AdKkRkKTTkAdT+Qk+∑j=1k+1ϒjPk+1−j∣k+1−jϒjT×A¯−TP−1kA¯−1<0.Due toA¯−TP−1kA¯−1>0, (69) satisfies
(70)AdKkRkKTkAdT+Qk+∑j=1k+1ϒjPk+1−j∣k+1−jϒjT>0.SincePk+1 is positive definite, it is found that
(71)Pk+1=AdPkAdT−AdKkCPkAdT+Qk+∑j=1k+1ϒjPk+1−j∣k+1−jϒjT>0.According to (70), to make sure the inequality (69) is satisfied, one obtains that
(72)AdPkAdT−AdKkCPkAdT<AdKkRkKTkAdT,where P⋅ represents the positive definite error variance matrix, and ϒjPk+1−j∣k+1−jϒjT is positive definite.Due to the positive definite ofRk, it can be derived from (72) that
(73)AdKkRkKTk+KkCPk−PkAdT>0,KkRkKTk⏟>0+KkC−I⏟>0Pk>0.Thus,KkC−I>0 can ensure the stability of fractional Kalman filter.
(74)KkC−I>0.According to the fractional Kalman filter gain matrixKk, it can be obtained that
(75)KkC−I=PkCTCPkCT+Rk−1C−I>PkCTCPkCT−1C−I⏟=0,where Rk is positive definite to guarantee the stability of fractional Kalman filter.


## 5. Simulation Results

To demonstrate the applicability of proposed method, the fractional Kalman filter-based fusion algorithm with various values is simulated. The system parameters and the system initial values are shown as follows:(76)Ad=00.1−0.035−0.01,n1n2=0.50.4,B=00.1,C1=0.30.4,C2=0.60.9,wk∼N0,00.10.10,υk∼N00.1,l=4,P0=10000100,x0=00.The control law is designed asu=ur−−0.0150.03x, and ur is a square signal with the period of 10s.Based on the conclusions in [2], the order of system equation should be defined previously and the control accuracy is infected by the value range of j in the sum term ∑j=1k+1−1jϒjxk+1−j. Thus, the upper bound of j must be defined previously in practice. In the simulation, as the width of a circular buffer of past state vectors for fractional-order difference, memory length L is chosen as 200 or 50.The fractional Kalman filter-based asynchronous multirate sensor information fusion results are described from Figures4–8. The reference input and output signal in the system as well as the multisensor measurement are shown in Figure 4. Moreover, the system state fusion estimation, measurement estimation results of sensor 1 and sensor 2 are shown in Figures 5 and 6, where sensors 1 and 2 represent fast sensor and slow sensor, respectively. The estimation errors of x1 and x2 are described by Figures 5 and 6 in detail.Figure 4
Input signal, output signal, and multisensor measurements (L=200).Figure 5
The estimation ofx1 (sensor 1, sensor 2, and fusion) (L=200).Figure 6
The estimation ofx2 (sensor 1, sensor 2, and fusion) (L=200).Figure 7
The estimate error ofx1 (sensor 1, sensor 2, and fusion) (L=200).Figure 8
The error ofx2 estimation (sensor 1, sensor 2, and fusion) (L=200).The simulation results ofnN=50 are depicted from Figures 9–13. Figure 9 provides input signal, output signal, and the measurement of output. The state estimation based on fusion method and a single sensor are provided by Figures 10 and 11, respectively. Finally, Figures 12 and 13 describe the error curves with different algorithms. From the simulation results, one can conclude that the proposed algorithm gives better control performance.Figure 9
Input signal, output signal, and multisensor measurement (L=50).Figure 10
The estimation ofx1 (sensor 1, sensor 2, and fusion) (L=50).Figure 11
The estimation ofx2 (sensor 1, sensor 2, and fusion) (L=50).Figure 12
The estimate error ofx1 (sensor 1, sensor 2, and fusion) (L=50).Figure 13
The estimate error ofx2 estimation (sensor 1, sensor 2, and fusion) (L=50).In simulation results, the effectiveness of the proposed fusion method is verified, and the system output value and fusion accuracy are compared. It can be obtained that high-precision and economic storage space requirement is satisfied asL=50.


## 6. Conclusion

The fractional Kalman filter-based fusion algorithm is presented to solve the problem of asynchronous multirate sensor information fusion. According to the memory performance of fractional order, which can accurately describe the essential characteristics of the system, the fractional filter is introduced to improve the estimation accuracy. Based on the relationship between slow rate delay measurement and the current state estimation, the minimum variance unbiased estimator is designed to update the current estimation. The simulation results prove the superiority of this method.



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*Source: 1450353-2018-12-05.xml*