RESEARCH Open Access

Fuzzy probabilistic data association filterand its application to single maneuveringtargetXiaobin Li1,2, En Fan3,4*, Shigen Shen4,5, Keli Hu4 and Pengfei Li6

Abstract

The fuzzy recursive least squares-probabilistic data association (FRLS-PDA) filter is presented for tracking singlemaneuvering target in cluttered situations with unknown process noises. In the proposed filter, the associationprobabilities of the current valid measurements belonging to a motion target are calculated by the probabilisticdata association (PDA) algorithm. Then these probabilities are used to weight the valid measurements forgenerating a fused measurement, which is applied to determine the maneuvering characteristics of the movingtarget in real time including the current measurement residual and heading change. According to the abovecharacteristics calculated, the fuzzy recursive least squares (FRLS) filter is used to estimate the current state of thetarget. The proposed filter can provide the advantage of the FRLS filter, which relaxes the restrictive assumptions ofmotion models of a maneuvering target. Moreover, it can realize single maneuvering target tracking in clutteredsituations. The performance of the FRLS-PDA filter is evaluated by two experiments with the simulation data andreal data, and it is found to be better than those of the PDA filter, IMM-PDA filter, fuzzy adaptive α-β filter, and FRLSfilter in tracking accuracy.

Keywords: Maneuvering target tracking, Probabilistic data association, Recursive least squares filter, Fuzzy system

1 IntroductionDue to the uncertainty in tracking process, single maneu-vering target tracking (MTT) has become a difficult prob-lem in information fusion, particularly in clutter [1, 2].Generally, its tracking procedure can be divided into twosteps: data association and state estimation. Data associ-ation originated from the uncertainty of sensor detectionand observed environments, and state estimation isdifficult to obtain due to the uncertainty of the motionmodel of a target. For tracking single maneuvering target,it needs to determine which measurements are from thetarget and then utilize the measurements to detect themaneuver and estimate the target state.In the traditional data association methods, the nearest

neighbor (NN) association algorithm is simple andpractical one. It mainly utilizes the closest measurement to

update the filter, but may delete the possible measurementassociated with the target [3]. The multiple hypothesistracker (MHT) can establish the hypotheses of all possiblemeasurements and keep these hypotheses [4]. Unfortu-nately, it easily leads to the exponential growth of associ-ation combinations. The probabilistic data association(PDA) algorithm is well accepted as the classical associationmethod [5, 6]. It can delete the impossible hypotheses andestablish the possible hypotheses of the associated measure-ments. Then these measurements are used to calculate theprobabilities associated with the target. After data associ-ation, one needs to utilize the associated measurements toestimate a target’s state. Many tracking methods based onthe PDA algorithm have been successively developed [7, 8].The interacting multiple model (IMM) algorithm is a well-known filter for estimating the state of a maneuveringtarget, and its modified methods are broadly utilized invarious applications [9–11]. The combination of the PDAalgorithm and IMM algorithm is usually utilized in singlemaneuvering target tracking [12]. However, it is apt to bringincorrect tracking results when these tracking methods

* Correspondence: efan@usx.edu.cn3ATR Key Laboratory, Shenzhen University, Shenzhen, Guangdong 518060,China4Department of Computer Science and Engineering, Shaoxing University,Shaoxing, Zhejiang 312000, ChinaFull list of author information is available at the end of the article

EURASIP Journal on Advancesin Signal Processing

© 2016 The Author(s). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link tothe Creative Commons license, and indicate if changes were made.

Li et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:102 DOI 10.1186/s13634-016-0401-8

utilize mismatching sub-models generated by the uncer-tainty of motion models. Although the Kalman filter is usu-ally used to track a target moving within a constantvelocity, its tracking performance seriously degrades intracking maneuvering targets. Moreover, it requires knownprocess noises. A data-driven approach to tracking whichhas been presented in [13] uses the least squares fitting of amotion model to a segment of data. Considered processnoises, the motion models fitted cannot exactly representthe real motion models of the target. The particle filter iswildly applied in various applications, particularly inmaneuvering target tracking. An adaptive fuzzy particlefilter (AFPF) method proposed in [14] is adapted to generalobject tracking in the field of computer vision. In the AFPFmethod, particle filtering samples are weighted using fuzzymembership functions and are applied to geometric andappearance features. The particle filter based on modifiedgeneralized probabilistic data association (PF-MGPDA)proposed in [15] combines the advantage of particle filterand generalized probabilistic data association to trackmaneuvering multi-targets. However, because the trackingperformance of particle filter is combined in proportion tothe number of the corresponding particles in target track-ing, the maneuvering target tracking methods based onparticle filter are difficult to satisfy the real tracking require-ment of tracking systems. For this reason, here, the particlefilter is not utilized to track maneuvering target in airsurveillance. Although the H∞ filter and its modified formshave also been applied in MTT due to its main advantageson the insensitiveness to the exact knowledge of the exter-nal noise and more robustness to the parameter uncer-tainty, these filters need to assume the noise covariancesare bounded, and this assumption is not suitable for highmaneuver situations [16–18].The mentioned methods above are in the framework of

statistics theory, and they generally assume that themotion models of a maneuvering target are known. In realsituations, the above requirement is usually difficult tosatisfy [19–21]. Moreover, the real process noises andestimated errors are generally correlated, and meanwhile,there exists great uncertainty in tracking process. Basedon these facts, the statistical association methods becomecomplicated in real situations. Fuzzy theory can utilize theadvantages of processing uncertainty information and it isgenerally utilized in state estimation [22, 23]. By incorpor-ating fuzzy theory into the traditional tracking algorithmsfor solving the related uncertainty problems, it canimprove the tracking performance of these trackingalgorithms [24–26]. Combined the fuzzy PDA algorithm,the IMM fuzzy probabilistic data association algorithm(IMM-FPDA) proposed in [24] utilizes the predictionerror and its change as the fuzzy inputs to optimizeparameters of the fuzzy system, and the fuzzy system isused to determine the process noise covariance of the

Kalman filter. The hybrid fuzzy probabilistic data associ-ation filter (HF-PDAF) and hybrid fuzzy joint probabilisticdata association filter [25] utilize a modified fuzzy c-means (FCM) algorithm to reconstruct the membershipgrades between measurements to underlying tracks. TheHF-PDAF method utilizes the Kalman filter to estimatethe target’s state, too. With the measurement residual, anew tracking algorithm proposed in [26] utilizes the FCMalgorithm to extract the acceleration and estimate themotion of the target. Based on [26], a multiple-structuredsmart tracking algorithm developed in [27] separates theacceleration from the overall noise by fuzzy c-meansclustering, and it can recognize the maneuvering targetwith multiple structures. As analyzed above, the Kalmanfilter can only be used in situations with known processnoises. In situations of unknown process noises, the recur-sive least squares (RLS) filter can obtain the perfectestimated results and possess less calculation complexitycompared with the traditional Kalman filter. It is widelyapplied in target tracking, system identification, andautomatic control, etc. [28–31]. However, the RLS filter isonly suitable to track a target moving at constant velocity.For this reason, the fuzzy recursive least squares (FRLS)filter is proposed for tracking single maneuvering target[28]. Unfortunately, it cannot be directly applied incluttered situations.Based on the above analysis, the fuzzy recursive least

squares-probabilistic data association (FRLS-PDA) filteris proposed for single MTT in cluttered situations withunknown process noises. In the proposed filter, the PDAalgorithm is applied to calculate the probabilities of thevalid measurements belonging to the target. Then theseprobabilities are utilized to weight these measurementsfor constructing a fused measurement. According to thefused measurement, the measurement residual andheading change at current time are calculated, which actas the inputs of the fuzzy system designed while its out-put acts as the fuzzy fading factor of the FRLS filter.Hence, the proposed filter can estimate the current stateof a maneuvering target through utilizing the fuzzyfading factor to adjust the influence of its predictedinnovation on the predicted estimate. Furthermore, weanalyze the adjustment function of the fuzzy fadingfactor on the predicted innovation for the currentpredicted state and then compare the calculationcomplexity of the proposed filter with those of theother three filters.The remainder of this paper is organized as follows: in

Section 2, the simplified form of the FRLS filter isdeduced; in Section 3, the FRLS-PDA filter is proposedbased on constructing a fused measurement, and bothits fuzzy fading factor and computation complexity areanalyzed; Section 4 presents the experiment results andthe performance comparison with the PDA filter, IMM-

Li et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:102 Page 2 of 10

PDA filter, fuzzy adaptive α-β filter, and FRLS filter; andfinally, the conclusions are provided in Section 5.

2 Fuzzy recursive least squares filterAssumed that a target moving with constant velocityat k time, its motion and measurement model aregenerally expressed as follows (in the situation thatthere exists only a single target in air surveillanceand the measurement associated with the target isdetermined):

xkþ1 ¼ Φkxk þ Gkvk : ð1Þzk ¼ Hkxk þ wk : ð2Þ

Here, xk and zk denote an n-dimensional state vectorand an m-dimensional measurement vector, and Φk andHk denote an n × n state transition matrix and an m × nmeasurement transition matrix. Gk is generally assumedto be a known matrix in n × n dimension. The processnoise vk in n × 1 dimension is assumed to be the zero-mean Gaussian white noise with an n × n covariance Qk,while the measurement noise wk in m × 1 dimension isassumed to be the zero-mean Gaussian noise with anm ×m covariance Rk. Nevertheless, Qk and Rk are usuallyunknown in real situations.In [27], the FRLS filter is proposed to track single

maneuvering target in uncluttered environments. Basedon the recursive form of the FRLS filter, its simplifiedform is deduced as follows:

xk ¼ xk−1 þ KkV k : ð3Þ

Pk ¼ ~λ−1k Pk−1−KkSkK

Tk : ð4Þ

Here, Pk is the n × n filter covariance matrix; Vk, Sk,

Kk, and ~λk (0 < ~λk≤1) are respectively called the n × 1 orpredicted innovation, n × n residual covariance matrix,n × n gain matrix, and fuzzy fading factor, which can be,respectively, calculated by

Vk ¼ zk−HkΦk xk−1: ð5Þ

Sk ¼ ~λ−1HkPk−1H

Tk þ I: ð6Þ

Kk ¼ Pk−1HTk Sk−1 : ð7Þ

~λk ¼

XM

m¼1λml sup

γk ∈~Ami ; ϕk ∈~B

mj

min u~Amiγk� �

;u~Bmjϕkð Þ

� �XM

m¼1sup

γk ∈~Ami ;ϕk ∈~B

mj

min u~Amiγk� �

;u~Bmjϕkð Þ

� � :

ð8ÞHere, I is an n × n unit matrix, and M = 16 denotes

the number of fuzzy rules; γk and ϕk are the meas-urement residual and heading change, which can be

obtained by Eqs. (9) and (10); ~Ami and ~B

mj denote the

corresponding fuzzy sets of γk and ϕk while u~Ami

andu~B

mj

denote their membership degrees, respectively;~Cml and u~C

ml

are assumed to be the fuzzy set and themembership function of ~λk ; λ

ml is the corresponding value

when u~Cmlobtains the maximal value in the lth fuzzy set.

In addition, all these notations above are also concretelyexplained in [27].

γk ¼ VkTVk

� �1=2=γmax ð9Þ

ϕk ¼ φk−φk−1j j=ϕmax ð10Þ

φk ¼ arctan yk−yk−1� �

= xk−xk−1ð Þ� � ð11Þ

φk−1 ¼ arctan yk−1−yk−2� �

= xk−1−xk−2ð Þ� � ð12Þ

where γmax and ϕmax are the corresponding maximumvalues of the measurement residual and heading change;Vk is the predicted innovation, which can be calculatedby Eq. (5); xk, xk−1 , and xk−1 are the correspondingcomponents of zk, xk−1, and xk−2 in the x-axis direction,while yk, ŷk − 1, and ŷk − 2 are the components of zk, xk−1 ,and xk−2 in the y-axis direction.Equations (3) and (4) are called the FRLS filter. Its

initial states are determined with the measurements z1and z2, given as follows:

x2 ¼ P2 HT1 ;H

T2

� �zT1 ; z

T2

� �T: ð13Þ

P2 ¼ HT1 ;H

T2

� �HT

1 ;HT2

� �T� �−1: ð14Þ

Here, both H1 and H2 are an m ×m measurementtransition matrix, and P2 is an n × n filter covariancematrix.

3 The proposed filter for MTTTo track a single maneuvering target in clutter environ-ments, this paper further proposes a FRLS-PDA filterbased on the PDA algorithm and the FRLS filter. Itsdesigned procedure mainly consists of three blocks,illustrated in Fig. 1. As shown in Fig. 1, firstly, the PDAalgorithm is utilized to calculate the probabilities of thevalid measurements associated with the target in block1; secondly, these probabilities are used to weight themeasurements for constructing a fused measurement inblock 2; finally, one can calculate the measurementresidual and the heading change according to the fusedmeasurement in block 3. Hence, the proposed filter canutilize the fuzzy system to determine the magnitude ofthe fading factor of the RLS filter, which can be used toadjust the influence of the predicted innovation on thecurrent predicted state.

Li et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:102 Page 3 of 10

3.1 Construct the fused measurementAssumed that the sets of the valid measurements and thecumulative measurement are, respectively, expressed by:

~Z kð Þ ¼ zi;k� mk

i¼1 ð15Þ

~Zk ¼ ~Z jð Þ� k

j¼1 ð16Þ

where zi,k is the ith valid measurement, and mk is thenumber of all valid measurements in the tracking gate attime k.Based on the PDA algorithm, the state estimate of the

target at time k can be regarded as the probability-weighted sum of local state estimates corresponding toall valid measurements. In the Poisson clutter model, theassociation probability on a measurement belonging tothe target can be calculated by [3]:

βi;k ¼ P θi;k j~Zkn o

¼ ei= bþXmk

j¼1

ej

!; i ¼ 0; 1;⋯;mk ð17Þ

where

θi;k ¼ zi;k belonging to the target�

i ¼ 1;⋯;mk

no measurement belonging to the targetf g i ¼ 0

ð18Þ

e0 ¼ 2πð ÞMo2 λ

ffiffiffiffiffiffiffiffiffiffiSi;k�� ��q

1−PDð Þ ð19Þ

ei ¼ PD exp −12VT

i;kS−1i;kV i;k

�ð20Þ

where Mo is the dimension of the measurement vector;PD is the detection probability; Vi,k is the predictedinnovation corresponding to zi,k, which can be calculatedby Eq. (5); Si,k is the covariance matrix of Vi,k; λ is theparameter of Poisson distribution. If the clutter distribu-tion is a uniform distribution, one only needs to resetthe relative parameters [3].In the PDA algorithm, the association probability may

be used to evaluate the effect of the correspondingmeasurement in state estimation when there exist several

measurements. Based on this fact, the influences of allvalid measurements together on the current estimate areequal to that of a fused measurement. Hence, the fusedmeasurement can be defined by

zk ¼Xmk

i¼0

βi;kzi;k : ð21Þ

Here, z0;k ¼ Hk xk−1 is called as the zero measurement,which means that there are no measurements obtained.In this case, one utilizes the current position estimate asthe current measurement.

3.2 FRLS-PDA filterBased on the above conclusions, the main procedures ofthe FRLS-PDA filter can be described as below:

(1) Set the initial values x2 and P2, and start therecursive formulas at k = 3.

(2) Calculate the predicted innovation:

V i;k ¼ z i;k−HkΦk xk : ð22Þ(3) Calculate the covariance of the predicted

innovation:

Sk ¼ HkPk−1Hk þ I: ð23Þ(4) Calculate the gain matrix:

Kk ¼ Pk−1HkSk−1 : ð24Þ(5) Calculate the tracking gate: g2.(6) Calculate the fading factor:

~λk ¼

XM

m¼1λml sup

γk∈Aemi ;ϕk∈B

emj

min μ~Amiγk� �

; μ~Bmjϕkð Þ

� �XM

m¼1sup

γk∈Aemi ;ϕk∈B

emj

min μ~Amiγk� �

; μ~Bmjϕkð Þ

� � :

ð25Þ

(7) Calculate the FRLS filter:

Fig. 1 Flow chart of FRLS-PDA filter

Li et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:102 Page 4 of 10

xk ¼ xkjk−1 þ KkVk ð26Þwhere Vk ¼ zk−zkjk−1.

(8) Update the filter covariance:

Pk ¼ ~λ−1k Pkjk−1−~λ

−2k 1−β0� �

WkSkWTk

þ ~λk−2Wk

Xmk

i¼1

βi;kV i;k ;VTi;k−VkV

Tk

" #WT

k : ð27Þ

3.3 Analysis of the fading factor and computationcomplexityIn [28], the FRLS filter utilizes a measurement residualand a heading change as the inputs of the fuzzy systemdesigned and its output as the fading factor of the FRLSfilter. Then, the FRLS filter can adjust the influence ofthe predicted innovation on the current predicted state.In situations with unknown motion models, the FRLSfilter can obtain good estimate results. Similarly, theFRLS-PDA filter can provide the above advantage of theFRLS filter. Furthermore, it can solve the single MTTproblem in clutter by constructing a fused measurementaccording to all valid measurements at each time.To better evaluate the performance of the MTT

methods below, the computational complexity of theproposed filter is analyzed, which mainly denotes thetimes including multiplications, divisions, additions, andsubtraction for each run of these filters under no consid-eration of the procedures of generating, moving, andmeasuring for a moving target. Here, it is assumed thatthe state vector and measurement vector are n ‐dimen-sional and m-dimensional, generally m < n. In addition, lis the number of fuzzy rules utilized in Eq. (8), and r isthe number of sub-models in the IMM-PDA filter. Thecomputational complexity for four types of filters isapproximately given in Table 1.

4 Experiment results and analysisTwo experiments of simulation data and real data havebeen carried out to evaluate the performance of theproposed filter in comparison with the other four filters,including the PDA filter [3], IMM-PDA filter [12], fuzzyadaptive (FA) α-β filter [21], and FRLS filter [28] in

terms of the position root mean squared errors (RMSE).In the experiments, the detection probability and gateprobability set Pd = 1 and Pg = 0.99, respectively. Thefollowing experiments are conducted by using a com-puter with a dual-core CPU of Intel® 2.20 GHz, 8-GBRAM. The programs are performed by using MATLAB2009a version software.

4.1 Simulation data experimentFor easy comparison, we still adopt the similar simula-tion scenario in [28] but with clutter, concretelydesigned as follows. There exists a target moving in theair surveillance of 2 ‐D Cartesian xy ‐plane according tothe given trajectory in Fig. 2. Its initial state is given byx0 = [800 m, 100 m/s, 900 m, 173 m/s]T and its trajectoryis formed by 114 measurements collected by a radarlocated at (0 m, 0 m). The performance parameters ofthe radar are listed as follows: the sampling interval T =5 s, range error σr = 15 m, and azimuth error σβ = 0.01°.The trajectory can be divided into five phrases: constantvelocity motion, turn rate motion (−0.0873 rad/Ts), con-stant velocity motion, turn rate motion (0.0873 rad/Ts),and constant velocity motion. The turn model of themoving target is expressed as:

xkþ1 ¼1

sinωkTωk

0 −1− cosωkT

ωk0 cosωkT 0 − sinωkT

01− cosωkT

ωk1

sinωkTωk

0 sinωkT 0 cosωkT

2666664

3777775xk þ Gkvk ð28Þ

Gk ¼ T2=2; T ; 0; 00; 0; T 2=2; T

�Tð29Þ

where ωk is the turn rate, and v(k) is the zero-meanwhite Gaussian noise with an unknown covariance. Theclutter model is assumed to be of uniform distributionand the number of false measurements (clutters) is

Fig. 2 Ideal target trajectory

Table 1 Computational complexity of four filters

Filter Computational complexity

PDA filter 3n3 + (6m + 1)n2 + (4m2 + 2m − 1)n +m3 + 2m

IMM-PDA filter 3n3 + (6m + 1)n2 + (4m2 + 2m − 1)n + 2m3 + 2m2 +2m + 6r

FRLS filter 14n3 + (4m − 1)n2 + (12m2 + 2m − 1)n + 2m3 −m2 + 3l

FRLS-PDA filter 14n3 + (4m − 1)n2 + (12m2 + 2m − 1)n + 2m3 +m2 +2m + 3l

Li et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:102 Page 5 of 10

assumed to be of Poisson distribution with knownparameter λ = 10 (the number of false measurements perunit of volume (km2)). To compare the performances ofall filters, 100 Monte Carlo runs have been performed.In addition, the related information is assumed to satisfythe run requirements of the PDA filter, IMM-PDA filter,FA α-β filter, and FRLS filter. Because the particle filtergenerally consumes large time, it is difficult to apply intracking system with high requirements in time and itscomparison is not given here.Figure 3 shows that the variation curves of γk, ϕk, and

~λk are obtained according to the target motion in the ideal,

uncluttered, and cluttered case, respectively. Figures 4 and5 provide the tracking results and the root-mean-square(RMS) position errors using the mentioned filters above.From Fig. 3, the magnitudes of their fading factors are closeto the values obtained in the ideal case, and it is verifiedthat the curves of γk and ϕk in the uncluttered and clutteredcase can reflect the maneuver changes exactly. From Figs. 4and 5, the RMS position errors of the FRLS-PDA filter aresmaller than the other four filters, concretely discussed asfollows. Because the FRLS-PDA filter can utilize mea-surement residuals and heading changes to detect thecurrent maneuver of the moving target and determine its

Fig. 3 Estimates of motion characteristics

Fig. 4 Tracking results

Li et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:102 Page 6 of 10

magnitude by the fuzzy system designed, it can obtain thecurrent state timely and accurately. On the other, due tothe absence of the prior information of the sub-models, theperformance of the IMM-PDA filter is undesirable. ThePDA filter performs well in constant velocity phase,but degrades seriously and even diverges in maneu-vering phase as shown in Fig. 4. Although the FRLSfilter obtains the good performance in constantvelocity and maneuvering phase, unfortunately, itcannot be directly utilized in clutter mentioned inSection 1. From the above analysis of the simulationdata experiment, it shows that the FRLS-PDA filter

can utilize the fused measurements to detect themaneuver in real time and consequently obtain betterperformance in tracking accuracy than other fourfilters.In a word, the proposed filter can provide the

following advantages of the PDA algorithm and theFRLS filter: firstly, it can construct a fused measurementthrough weighting the valid measurements in clutterenvironments by the association probabilities, which arecalculated by the PDA algorithm; secondly, it can relaxthe restrictive assumption of motion models throughusing the FRLS filter to estimate the current target state.

Fig. 5 Estimate errors for four filters

Fig. 6 Ideal trajectory of the target

Li et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:102 Page 7 of 10

4.2 Real data experimentTo illustrate the feasibility of the proposed filter, the realdata experiment in [28] is adopted here. The parametersof the clutter model are the same in Section 4.1. Thetrajectory of the target in Fig. 6 is formed by 32 mea-surements from some type of single radar and its per-formance parameters are given as follows: the samplinginterval T = 2 s, maximum detection range rmax = 21 km,range error σr = 20 m, and azimuth error σβ = 0.5°. Theinitial position of the target is z0 = [−10.72 km, −3.26 km]T. Figure 7 shows that the variation curves ofγk, ϕk, and ~λk are obtained in the ideal, uncluttered, and

cluttered case, respectively. From Fig. 7, γk and ϕk canstill reflect its changes correctly although the maneuverof the moving target is complicated in the real case.Consequently, the fading factor in uncluttered and clut-tered case can be approximated with the values obtainedin ideal case. Figures 8 and 9 show the tracking resultsand RMS position errors by using the above filters for100 Monte Carlo simulation runs. From Fig. 9, the RMSposition errors of the FRLS-PDA filter are smaller thanthe other three filters. From the real data experiment, it isverified that the proposed filter is feasible to track singlemaneuvering target in real environment. Therefore, the

Fig. 7 Estimates of motion characteristics

Fig. 8 Tracking results

Li et al. EURASIP Journal on Advances in Signal Processing (2016) 2016:102 Page 8 of 10

proposed filter can also estimate the maneuver parameterin actual situations and obtain better tracking perform-ance in clutter environments.

5 ConclusionsIn this paper, a FRLS-PDA filter is proposed based onstatistical and fuzzy theories for single MTT in clutteredsituations with unknown process noises. In the singleMTT process, measurement residuals and heading changesare two important characteristics on maneuvering motionmodels, which can effectively and exactly reflect maneuver-ing changes of the moving target. In the proposed filter, thePDA algorithm is utilized to calculate the associationprobabilities of the measurements that belong to the target,and then these probabilities are used to weighting the validmeasurements for constructing the fused measurement atcurrent time. Based on the fused measurement, we canfurther obtain the current measurement residual and head-ing change, which acts as the inputs of the fuzzy system todetermine the fading factor of the FRLS filter. Hence, theinfluence of the predicted innovations on the currentpredicted state can be adjusted through the fuzzy systemdesigned.The application of the characteristics of motion

models into the traditional MTT methods has beenproven to be an effective solution to improve their per-formances in situations with unknown process noisesand unknown motion models. The results of two experi-ments with the simulation data and real data show theeffectiveness and feasibility of the proposed filter, and itcan realize single MTT effectively. Hence, how to intro-duce more useful information on maneuvering targets

and design the fuzzy rules on the inputs and outputs ofthe designed fuzzy system is very important in the nextwork. In real situations, the proposed filter can also beextended to solve the real time MTT problem in sensornetworks with multipath or multisource measurements.In addition, the study of the applications of the proposedfilter is helpful for solving the MTT problem in sensornetwork for the future research, including the maneuver-ing multi-target tracking problems.

AcknowledgementsThis work is supported by the National Natural Science Foundation of China(No. 61603258 and No. 61272034), the Startup Project of Doctor ScienceResearch of Shaoxing University (No. 20145021 and No. 20155015), theScience Project of Shaoxing University (No. 2015LG1006), the Natural ScienceFoundation of Shenzhen Polytechnic (No. 601522 K21017), the LanzhouScience and Technology Project (No. 2015-4-3), and the Public Welfare Tech-nology Application Research-Industrial Project of Zhejiang Province (No.2016C31082).

Competing interestsThe authors declare that they have no competing interests.

Author details1College of Software Engineering, Lanzhou Institute of Technology, Lanzhou,Gansu 730000, China. 2Jožef Stefan International Postgraduate School,Ljubljana 1000, Slovenia. 3ATR Key Laboratory, Shenzhen University,Shenzhen, Guangdong 518060, China. 4Department of Computer Scienceand Engineering, Shaoxing University, Shaoxing, Zhejiang 312000, China.5College of Mathematics, Physics and Information Engineering, JiaxingUniversity, Jiaxing, Zhejiang 314001, China. 6Air Defense Forces Academy,Zhengzhou 450052, China.

Received: 1 August 2015 Accepted: 14 September 2016

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