Research Article Support Vector Regression-Based Adaptive ...downloads.hindawi.com/journals/jam/2014/139503.pdf · Stirling sinterpolationformula. ealgorithmusedStirling s interpolation

Research ArticleSupport Vector Regression-Based Adaptive Divided DifferenceFilter for Nonlinear State Estimation Problems

Hongjian Wang Jinlong Xu Aihua Zhang Cun Li and Hongfei Yao

College of Automation Harbin Engineering University Harbin 150001 China

Correspondence should be addressed to Jinlong Xu xujinlong 1983sinacn

Received 2 March 2014 Accepted 4 May 2014 Published 25 May 2014

Academic Editor Weichao Sun

Copyright copy 2014 Hongjian Wang et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We present a support vector regression-based adaptive divided difference filter (SVRADDF) algorithm for improving the lowstate estimation accuracy of nonlinear systems which are typically affected by large initial estimation errors and imprecise priorknowledge of process and measurement noises The derivative-free SVRADDF algorithm is significantly simpler to computethan other methods and is implemented using only functional evaluations The SVRADDF algorithm involves the use of thetheoretical and actual covariance of the innovation sequence Support vector regression (SVR) is employed to generate theadaptive factor to tune the noise covariance at each sampling instant when the measurement update step executes which improvesthe algorithmrsquos robustness The performance of the proposed algorithm is evaluated by estimating states for (i) an underwaternonmaneuvering target bearing-only tracking system and (ii) maneuvering target bearing-only tracking in an air-traffic controlsystem The simulation results show that the proposed SVRADDF algorithm exhibits better performance when compared with atraditional DDF algorithm

1 Introduction

The problem of state estimation for nonlinear systems hasbeen a subject of considerable research interest in recentyears but there is still no single solution that outperformsall other approachesMost proposed estimators are nonlinearextensions of the dominated Kalman filter (see [1]) and eachapproach provides a suboptimal trade-off between propertiessuch as numerical robustness computational burden andestimation accuracy The extended Kalman filter (EKF)which linearizes both nonlinear terms of a current estimatedstate trajectory is based on a first-order Taylor series and dis-plays poor performance if the system is highly nonlinearThelimitations of EKFs are enumerated in [2] Another improvedalgorithm is the iterated extended Kalman filter (IEKF)which linearizes the nonlinear model around an updatedstate rather than the predicted state (see [3]) AlthoughIEKFs have been proven to perform better than EKFs inaddition to globally guaranteeing convergence the algorithmstill requires a Jacobian matrix just like EKFs However nosolution exists for the Jacobian matrix in nonlinear systems

for some situations which limits the potential application ofboth EKFs and IEKFs

In recent years a new class of filter known as sigma-point Kalman filter (SPKF) has attracted a great deal ofattention In SPKFs the algorithm propagates a cluster ofpoints centered on the current state instead of linearizingthe system dynamics to improve the approximations of theconditional mean and covariance Unscented Kalman filters(UKF) and divided difference filters (DDF) are two kinds ofSPKFs

UKFs use a deterministic sampling technique to pick aminimal set of sample points around the mean to catch thehigher order statistics of the system so as to better estimationaccuracy and convergence characteristics (see [4]) In [5 6]a UKF for a class of nonlinear discrete-time systems withcorrelated noises was designed to deal with the problemof nonlinear filtering failure found in conventional UKFswhen system noise is correlated withmeasurement noiseTheproposed UKF breaks the limitation of conventional UKFsthat requires system noise and measurement noise to beuncorrelatedGauss white noises thus extending the potential

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 139503 10 pageshttpdxdoiorg1011552014139503

2 Journal of Applied Mathematics

application of conventional UKFs In [7] a UKF filteringalgorithm with colored measurement noise was proposedThe algorithm was first derived on the basis of augmentedmeasurement information and minimum mean square errorestimation and a filtering recursive formula of UKF withcolored noise then added by applying an unscented trans-formation to calculate the posterior mean and covarianceof the nonlinear state within the optimal framework Theproposed UKF effectively dealt with the fact that traditionalUKFs fail when measurement noise is colored In [8] a UKFwas applied to multiple target tracking with the proposedUKF shown to have improved performance versus previousEKF approaches In [9] the paper discussed an adaptivemultiuser receiver for CDMA systems in which the scaledunscented filter (SUF) and the square root unscented filter(SURF) were used for joint estimation and tracking of thecode delays andmultipath coefficients of the received CDMAsignals The proposed channel estimators were more near-far resistant than in conventional EKFs and presented lowercomplexity than conventional particle filter- (PF-) basedmethods Computer simulation results demonstrated thesuperior performance of the proposed channel estimatorsand the proposed estimators were shown to exhibit lowercomplexity relative to the PF-based method Although UKFshave undergone a significant amount of meaningful theoryinnovation and are nowused inmany fields [10ndash12] show thatUKF accuracy is lower than that of DDF while also having ahigher computational cost

The divided difference filter (DDF) first proposed byNoslashgaard et al (see [13]) linearizes the nonlinear termsbased on Stirlingrsquos interpolation polynomial approximationsformula rather than Taylorrsquos approximation of nonlinearterms in an EKF Conceptually the implementation principleresembles that of an EKF however the DDF is significantlysimpler as it does not need to calculate the Jacobian matrixand no derivatives are required The DDF that Noslashgaardet al developed works on general discrete-time nonlinearmodels in which the noises are not assumed to be additiveIn [14] the paper further formulated a DDF in terms of theinnovation vector approach the additive process and themeasurement noise sources In [15] the paper proposed a newfilter named the maximum likelihood-based iterated divideddifference filter (MLIDDF) which improved the low stateestimation accuracy of nonlinear state estimation that resultsfrom large initial estimation errors and the nonlinearity ofthe measurement equations Simulation results showed thatthe MLIDDF algorithm possessed better state estimationaccuracy and a faster convergence rate In [16] the authorsproposed a novel adaptive version of the DDF that wasapplicable to nonlinear systems with a linear output equationIn order to make the filter robust to modeling errors upperbounds on the state covariance matrix were derived Theparameters of the upper bound were then estimated usinga combination of offline tuning and online optimizationwith a linear matrix inequality constraint which ensuredthat the predicted output error covariance was larger thanthe observed output error covariance Simulation resultsdemonstrated the superior performance of the proposed filteras compared to the standard DDF Reference [17] presented

an ensemble-based approach that handled nonlinearity basedon a simplified divided difference approximation throughStirlingrsquos interpolation formula The algorithm used Stirlingrsquosinterpolation formula to evaluate the statistics of the back-ground ensemble during the prediction step employing theformula in an ensemble square root filter (EnSRF) at thefiltering step to update the background for analysis In thissense the algorithm is a hybrid of Stirlingrsquos interpolationformula and theEnSRFmethodwhile the computational costof the algorithm is less than that of EnSRF

Different studies have focused on the application ofDDFs to nonlinear state estimation problems In [18] timedelay and channel gain estimation for multipath fadingcode division multiple access (CDMA) signals using a DDFwere investigated and the simulation results showed thatthe DDF was simpler to implement and more resilient tonear-far interference in CDMA networks compared with anEKF In [19 20] the relative kinematic states of a reentryvehicle obtained from noisy seeker measurements using aDDF were examined The results were compared to thoseobtained using an EKF and a UKF and showed that theDDF was more accurate than estimators based on a Taylorapproximation like the EKF Reference [21] investigatedthe possibility of using a DDF for estimating the internalvariables of a synchronous generator such as the rotor anglewhere the data acquired is from a phasor measurementThe effectiveness of the method was tested on a singlemachine infinite bus system a nine-bus system and a 68-bus New England-New York interconnected system In [22]a DDF using orientation estimation was considered Thefourth element of the quaternion error vector was removedfrom the system states to alleviate estimated error covariancematrix divergence The measurement system was a MARGsensor which consisted of a triaxial rate gyro a triaxialaccelerometer and a triaxial magnetometer The nonlinearmeasurement model was obtained based on the principalsof operation of the magnetometer and accelerometer and theproperties of the quaternion vector spaceTheperformance ofthree filtersDDF EKF andUKFwas comparedwith differentsampling frequencies The work showed that the tested DDFand the UKF were more robust than the EKF under the sameinitial angle-error conditionsTheDDF also performed betterthan the UKF although the computational load for the UKFwas less In [23] a DDF-based data fusion algorithmwas pre-sented which utilized the complementary noise profile of rategyros and gravimetric inclinometers to extend their limitsand achieve more accurate attitude estimates In [24] a DDF-based ballistic target tracking system for the reentry phasewas proposed The paper compared DDF EKF and UKFalgorithms using a Monte Carlo simulation approach withthe simulation results showing that the DDF outperformedboth the EKF and UKF in terms of estimation accuracy andfiltering credibility In [25] a DDF with quaternion-baseddynamic process modeling was applied to global positioningsystem (GPS) navigation to increase navigation estimationaccuracy at high-dynamic regions while preserving precisionat low-dynamic regions Some properties and performancemetrics were assessed and compared to those using EKF andUKF approaches

Journal of Applied Mathematics 3

1

15

2

25

3

True value

y=ex

0 01 02 03 04 05 06 07 08 09 1

x

1st Stirling1st Taylor

(a)

minus05

05

Erro

r

1

0

0

01 02 03 04 05 06 07 08 09 1

x


(b)

Figure 1 Comparison of first-order Stirling series with first-order Taylor series results

Despite their recent popularity DDF algorithms requirethat both the system model and the stochastic informationmust be accurate However this condition cannot be satisfiedin many practical situations which forces the filter to adaptitself to changing conditions One of the problems with thisrequirement is that any change in the process introducesmea-surement noise covariance In this work we make use of thetheoretical and actual covariance of the innovation sequenceemploying SVR to generate the scale factor to tune the noisecovariance at each sampling instant when the measurementupdate step is executed to adapt the filtering algorithm Thispaper is organized as follows Section 2 briefly introducesDDF theory and the proposed SVR-based adaptive strategyPassive target tracking is then carried out to evaluate theperformance of DDF and SVRADDF algorithms using aMonte Carlo simulation in Section 3 Finally conclusions areprovided in Section 4

2 Development of the Support VectorRegression-Based Adaptive DividedDifference Filter

21 Divided Difference Filter Consider the nonlinear func-tion

y = f (x) (1)

where x isin R119899119909 and y isin R119899119910 If the function is analytic thenthe first-order Taylor series expanded about some point x = xbecomes

y = (f (x + Δx) = f (x) +DΔxf = f (x)

+ (Δ1199091

120597

1205971199091

+ Δ1199092

120597

1205971199092

+ sdot sdot sdot + Δ119909119899

120597

120597119909119899

) f (x))10038161003816100381610038161003816100381610038161003816x=x

(2)

with (2) truncated after the first-order term Note that (2)can achieve a better local approximation if more terms areincluded However such an expanded Taylor series requiresderivatives and cannot be fulfilled in some situations Stir-lingrsquos interpolation formula is based on a finite number of

evaluations of the function and does not require derivativeswith the first-order approximation yielding

y = f (x + Δx) = f (x) + DΔxf (x)

= f (x) + 1

ℎ

(

119899

sum

119894=1

Δ119909119894120583119894120575119894) f (x)

(3)

where ℎ denotes a selected interval length and 120575 and 120583 aredetermined by

120575119894f (x) = f (x + ℎ

2

e119894) minus f (x minus ℎ

2

e119894)

120583119894f (x) = 1

2

(f (x + ℎ

2

e119894) + f (x minus ℎ

2

e119894))

(4)

with e119894being the 119894th unit vector

Figure 1 compares the results found by using (2) and (3)The function example is 119891(119909) = 119890

119909 where ℎ = 056 Fromthe figure we can see that Stirlingrsquos interpolation providesbetter accuracy than the Taylor series under the same orderapproximations

We now assume that the variable x has a Gaussiandensity with mean x and covariance Px We can introducea transformation matrix Sx which we select as a squareCholesky factor of Px such that Px = SxS119879x To illustrate howothers can be derived we introduce the linear transformationof x

z = Sminus1x x (5)

This linear transformation results in a stochastic decou-pling of x as the elements of z become mutually uncorrelated(see [13]) This changes (3) to

y = f (Sxz) = f (z) = f (z) + DΔzf (6)

where DΔzf is determined by

DΔzf = 1

ℎ

(

119899

sum

119894=1

Δz119894120583119894120575119894) f (z) (7)


The mean y covariance Pyy and cross covariance Pxy ofy are obtained from

y = E [y] = E [f (z) + DΔzf] = E [f (z)] = f (x)

Pyy = E [(y minus y) (y minus y)119879]

= E [(DΔzf) (D

Δzf)119879

]

=

1

4ℎ

2

119899

sum

119894=1

(f (z + ℎe119894) minus f (z minus ℎe

119894))

times (f (z + ℎe119894) minus f (z minus ℎe

119894))

119879

=

1

4ℎ

2

119899

sum

119894=1

(f (x + ℎsx119894) minus f ((x minus ℎsx119894)))

times (f (x + ℎsx119894) minus f ((x minus ℎsx119894)))119879

Pxy = E [(x minus x) (y minus y)119879]

= E [SxΔz(DΔzf)119879

]

=

1

2ℎ

119899

sum

119894=1

sx119894(f (x + ℎsx119894) minus f (x minus ℎsx119894))119879

(8)

where sx119894 is the 119894th column of the matrix SxConsider the following nonlinear dynamic system with

states to be estimated

x119896+1

= f (x119896) + 120596119896

y119896= h (x

119896) + ^119896

(9)

where 120596119896and ^119896are assumed to be independent and identi-

cally distributed and independent of current and past statessuch that 120596

119896sim N(oQ

119896) and ^

119896sim N(0R

119896)

TheDDF takes the same predictor-corrector structures inthe EKF and can be described as follows

Step 1 (initialization) Suppose the state distribution at 119896

instant is x119896sim N(x

119896P119896) where x

119896and P

119896are obtained by

x119896= E [x

119896]

P119896= E [(x minus x

119896) (x minus x

119896)

119879

]

(10)

Step 2 (square Cholesky factorizations) Consider the follow-ing

P119896= SxS

119879

x

Sxx =1

2ℎ

f119894(x119896+ ℎsx119895) minus f

119894(x119896minus ℎsx119895)

(11)

where sx119894 is the 119895th column of Sx

Step 3 (state and covariance propagation) One has

xminus119896+1

= f (x119896) (12)

Pminus119896+1

= Sxx(Sxx)119879

+Q119896

= Sminusx (Sminus

x )119879

(13)

Syx =1

2ℎ

h119894(xminus119896+1

+ sminusx119895) minus h119894(xminus119896+1

minus sminusx119895) (14)

where xminus119896+1

is the predicted state and Pminus119896+1

is the predictedcovariance matrix

Step 4 (observation and innovation covariance propagation)Consider

yminus119896+1

= h (xminus119896+1

) (15)

P^^119896+1

= (Syx) (Syx)119879

+ R119896

(16)

Pxy119896+1

= Sminusx (Syx)119879

(17)

where yminus119896+1

is the predicted observation vector P^^119896+1

is theinnovation covariance matrix and Pxy

119896+1is the cross correla-

tion matrix

Step 5 (update) Consider the following

120581119896+1

= Pxy119896+1

(P^^119896+1

)

minus1

P+119896+1

= Pminus119896+1

minus 120581119896+1

P^^119896+1120581119879

119896+1

120592119896+1

= y119896+1

minus yminus119896+1

x+119896+1

= xminus119896+1

+ 120581119896+1120592119896+1

(18)

where 120581119896+1

is the gain P+119896+1

is the updated covariancematrix 120592

119896+1is the innovation vector and x+

119896+1is the updated

estimated state

22 Adaptive DDF Algorithm As stated previously the DDFalgorithm assumes a complete prior knowledge of the processand the measurement noise statistics Q

119896and R

119896 However

Q119896and R

119896are unknown in most applications and incorrect

priori noise statistics can lead to performance degradation oreven divergence for the solution One of the effective waysto overcome this weakness is to use an algorithm to adaptthe noise statistics In this paper we propose using a supportvector regression adaptive scheme of the DDF to adjust Q

119896

and R119896 respectively

221 Support Vector Regression The principles underlyingsupport vector regression (SVR) developed byVapnik and arepresented in several works (see [26 27])


Given the train set 119879119878 = (x1 1199101) (x

119898 119910119898) isin

(R119899 times R)119898 where x119894isin R119899 119910

119894isin R 119894 = 1 2 119898 the SVR

problem can be defined as solving for the nonlinear function119892(x) about x isin R119899 to construct a relationship between theoutput and an arbitrary input x

119910 (x120596) =119898

sum

119894=1

120596119894119892 (x119894) + 119887 = (120596 sdot g (x)) + 119887 (19)

In [26] the author shows that changing the regressionestimate minimizes the risk functional by using the followingform

119910 (x120572) =119898

sum

119894=1

(120572

lowast

119894minus 120572119894) (119892 (x

119894) sdot 119892 (x)) + 119887

=

119898

sum

119894=1

(120572

lowast

119894minus 120572119894)119870 (x

119894 x) + 119887

(20)

where 120572

lowast

119894and 120572

119894are Lagrange multipliers that satisfy the

condition 120572119894 120572

lowast

119894ge 0 120572

119894120572

lowast

119894= 0 and 119870(x

119894 x) is a kernel

function that satisfies Mercerrsquos condition In this paperwe use a translation invariant Gaussian kernel that is =exp(minusx

119894minus x221205902)

222 Adaptive Scheme Based on SVR (Q119896Is Fixed) The

covariancematrixR119896represents the accuracy of themeasure-

ment instrument If we assume that the noise covariance Q119896

is completely known then we can derive the SVR algorithmto estimate the measurement noise covarianceR

119896by defining

an adaptive factor Δ119903119896to get the form

R119896= Δ119903119896R (21)

where R is the constant noise covariance matrixThis work uses such an SVR algorithm to derive the

adaptive factor at time instant 119896 so as to estimate the valueof R119896during the algorithmrsquos execution

The innovation sequence 120592119896+1

has a theoretical covarianceP^^119896+1

as (16) showsThe actual residual covarianceP^^119896+1

can beapproximated using its sample covariance by averaging insidea moving window of size119873 where

C119896=

1

119873

119896

sum

119894=119896minus119873+1

120592119896+1120592119879

119896+1 (22)

If C119896differs from P^^

119896+1 then the diagonal elements of

R119896can be adjusted to minimize the difference as much as

possible The size of this difference is given by

DOM119896= diag (P^^

119896+1minus C119896) isin R119899 (23)

where the function diag denotes the diagonal elements of thematrix If the elements of DOM

119896are rounded with zero the

covariance matrix R119896is likely accurate otherwise there may

be a large deviation between valuesTo adjust R

119896 we define the SVR train set 119879119878

Δ119903as 119879119878Δ119903

=

(DOM119896minus119873+1

Δ119903119896minus119873+1

) (DOM119896 Δ119903119896) With this form

we can then create a corresponding function about the

adaptive factor Δ119903119896andDOM

119896based on (19) such that when

DOM119896+1

comes then Δ119903119896+1

can be resolved by the functionTo avoid having any element within the train set

be too large to affect the accuracy of Δ119903119896 we use

a normalized function of the train set 119879119878 that is119879119878

1015840

Δ119903= (DOM1015840

119896minus119873+1 Δ119903

1015840

119896minus119873+1) (DOM1015840

119896 Δ119903

1015840

119896) where

DOM1015840119896and 119903

1015840

119896can be expressed as follows

DOM1015840119896=

1

119873

119873

sum

119894=1

DOM119896minus119894+1119895

119895 = 1 2 119899

Δ119903

1015840

119896=

1

119873

119873

sum

119894=1

Δ119903119896minus119894+1

(24)

223 Adaptive Scheme Based on SVR (R119896Is Fixed) Assum-

ing that the noise covariance R119896is completely known we can

derive an SVR algorithm to estimate the measurement noisecovariance Q

119896 From (13) (14) and (16) we can deduce that

a change in Q119896will affect the covariance matrix P^^

119896+1 if we

increaseQ119896 then P^^

119896+1also increasesWe can adjustQ

119896in the

SVR by deliberately mismatching P^^119896+1

and C119896+1

We first define an adaptive factor Δ119903

119896 where R

119896has the

following form

Q119896= Δ119902119896Q (25)

whereQ is the constant noise covariance matrixWe can then define the SVR train set 119879119878

Δ119902as 119879119878

Δ119902=

(DOM119896minus119873+1

Δ119903119896minus119873+1

) (DOM119896 Δ119903119896) At this point the

solution process is the same as for solving Δ119903119896

3 Monte Carlo SimulationResults and Discussion

In this section we report the experimental results obtainedby applying SVRADDF to the nonlinear state estimation ofa nonmaneuvering target in an underwater tracking controlscenario and a maneuvering target in an air-traffic controlscenario To demonstrate the performance of the SVRADDFalgorithm we compare its performance against a DDF algo-rithm

31 Underwater Nonmaneuvering Target Bearing-Only Track-ing Control Scenario We consider a bearing-only trackingcontrol scenario where an underwater target executes auniform motion in a horizontal plane but unknown velocitywhile a passive sonar platform performs a uniform circularmotion in a horizontal plane Figure 2 shows a representativetrajectory of the target and the passive sonar platform Thekinematics of the relative motion between the target and theplatform can be modeled using the following linear processequation

x119896+1

=

[

[

[

[

1 0 119879 0

0 1 0 119879

0 0 1 0

0 0 0 1

]

]

]

]

x119896+ w119896 (26)


0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500

TargetPlatform

I

F

X (m)

Y(m

)

Figure 2 Target trajectory (I-initial position and F-final position) and sonar trajectory

Here the state of the equation is x = [119909 119910 119910] where119909 and 119910 denote position and and 119910 denote velocity inthe 119909 and 119910 directions respectively and 119879 is the time intervalbetween two consecutive measurements and the processnoise w

119896sim 119873(0Q) with a nonsingular covariance where

Q = 1199021times

[

[

[

[

[

[

[

[

[

[

119879

2

2

119879

2

2

119879

119879

]

]

]

]

]

]

]

]

]

]

(27)

In (27) the parameter 1199021is related to process noise

intensities The measurement equation is written as follows

y119896= 120579119896= tanminus1 (

119910119896

119909119896

) + 120592119896 (28)

where the measurement noise 120592119896sim 119873(0 119877) with a nonsingu-

lar covarianceGiven the following initial conditions

119879 = 1 s1199021= 00001m2 sminus3

119877 = 002mrad

(29)

the true initial state is

x0= [0m 1500m 0msminus1 0msminus1]119879 (30)

and the associated covariance is

P0= diag [100m2 2000m2 1m2 sminus2 1m2 sminus2] (31)

The initial estimate state x0was chosen randomly from

119873(x0P0) in each run and the total number of scans per run

was 1000To provide a fair comparison we performed 50 inde-

pendent Monte Carlo runs To track the underwater targetwe used both the SVRADDF and the DDF algorithms andcompared their performance The adaptive factor was set toΔ119902119896= Δ119903119896= 01 Both of the filters were initialized with the

same initial conditions for each runPerformance metrics to compare the nonlinear perfor-

mance of the filters we used the root mean square error(RMSE) of the target position and velocityTheRMSE yields acombinedmeasure of the bias and variance of a filter estimateThe RMSE of the position at time 119896 was found using

RSMEpos (119896) = radic

1

119873

119873

sum

119894=1

((119909

119894

k minus 119909

119894

119896)

2

+ (119910

119894

119896minus 119910

119894

119896)

2

)(32)

where (119909119894119896 119910

119894

119896) and (119909

119894

119896 119910

119894

119896) are the true and estimated posi-

tions respectively in the 119894th Monte Carlo run The form forthe RMSE of the velocity is similar

Figures 3 and 4 show the estimated RMSE in targetposition and velocity The SVRADDF uses SVR to adjust theadaptive factor during algorithm execution which leads to amarginally better performance compared to the DDF as seenin the figures

32 Maneuvering Target Tracking in the Air-Traffic ControlScenario A typical air-traffic control scenario was consid-ered next where an aircraft executes a maneuvering turnin a horizontal plane at a constant and known turn rate ΩFigure 5 shows a representative trajectory of the aircraft


0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

Posit

ion

RMSE

DDFSVRADDF

Figure 3 RMSE in position for DDF and SVRADDF

0

2

4

6

8

10

12

Velo

city

RM

SE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF

Figure 4 RMSE in velocity for DDF and SVRADDF

The kinematics of the turning motion can be modeledusing the following nonlinear process equation

x119896=

[

[

[

[

[

[

[

[

[

[

[

1

sinΩ119879

Ω

0 minus

1 minus cosΩ119879

Ω

0 cosΩ119879 0 minus sinΩ119879

0

1 minus cosΩ119879

Ω

1

sinΩ119879

Ω

0 sinΩ119879 0 cosΩ119879

]

]

]

]

]

]

]

]

]

]

]

times x119896minus1

+ w119896minus1

(33)

The state of the aircraft is given by x = [119909 119910 119910]

119879where 119909 and 119910 denote position and and 119910 denote velocityin the 119909 and 119910 directions respectively and 119879 is the time

interval between two consecutive measurements and theprocess noise w

119896sim 119873(0Q) with a nonsingular covariance

where

Q = 1199021times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119879

3

3

119879

2

2

0 0

119879

2

2

119879 0 0

0 0

119879

3

3

119879

2

2

0 0

119879

2

2

119879

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(34)

The parameter 1199021related to process noise intensities A

passive radar is fixed at the origin and equipped to measure


0

0

2000

4000

6000

2000 4000 6000

TargetRadar

X (m)

Y(m

)

minus2000

minus4000

minus6000

minus2000minus4000minus6000

minus8000

Figure 5 Aircraft trajectory and radar location

280

300

320

340

360

380

400

420

440

Posit

ion

RMSE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF


the bearing 120579 The measurement equation is written asfollows

y119896= 120579119896= tanminus1 (

119910119896

119909119896

) + 120592119896 (35)

where the measurement noise 120592119896sim 119873(0 119877)

The parameters used in this simulation were119879 = 1 s Ω = minus3

∘ sminus1 1199021

= 01m2 sminus3 and119877 = radic10mard The true initial state of the aircraftwas x

0= [1000m 300msminus1 1000m 0msminus1]119879

and the associated covariance matrix was P0

=

diag[100m2 10m2 sminus2 100m2 10m2 sminus2] The initialstate x

0for the filters was chosen randomly from 119873(x

0P0)

in each Monte Carlo run and the simulation time per runwas 1000

For a fair comparison we performed 100 independentMonte Carlo runs for each filter To track the maneuveringaircraftwe used both the SVRADDFand theDDFalgorithmsand compared their performanceThe adaptive factor was setto Δ119902119896= 03 and Δ119903

119896= 05 Both filters were initialized with

the same initial conditions for each runFigures 6 and 7 show the estimate RMSE in target

position and velocity for the SVRADDF and DDF filters Notsurprisingly both filters exhibit divergence due to amismatchbetween the initial filter design assumption and the Gaussiannoise nature of the problem The SVRADDF filter exhibitsmarginally better performance compared to the DDF since itwas able to adjust the statistical properties of the noise duringexecution


15

155

16

165

17

175

18

Velo

city

RM

SE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF


4 Conclusions

This work has proposed and developed an innovation-basedSVRADDF algorithm The algorithm introduces an adaptivefactor estimated using SVR which allows for estimation ofthe noise statistical characteristics of nonlinear stochasticsystems The SVRADDF algorithm avoids instability anddivergence in the solution which is caused by incorrectstatistical characteristics of the noiseMonteCarlo simulationresults of an underwater nonmaneuvering bearing-only tar-get tracking system and a maneuvering target bearing-onlytracking system in an air-traffic control setting showed thatthe SVRADDF algorithm provides better state estimationaccuracy than a traditional DDF algorithm

Although the SVRADDF algorithm has showed betterperformance under Monte Carlo simulation there are stillseveral challenging issues to be considered for future studyTo improve its feasibility and effectiveness in the complexenvironment more comprehensive and detailed studies arestill needed to solve under the nonlinear and non-Gaussiannoise conditions Since the algorithm has only been testedunder Monte Carlo simulation the following work might betesting the algorithm under trial data to approve its betterperformance online

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research work is supported by the National NaturalScience Foundation of China (Grant no E09100250979017)PhD Programs Foundation of Ministry of Education ofChina and Basic Technology (Grant no 20092304110008)and Harbin Science and Technology Innovation Talents of

Special Fund Project (Outstanding Subject Leaders) (no2012RFXXG083)

References

[1] Y Bar-Shalom and T Kirubarajan Estimation with Applicationsto Tracking and Navigation John Wiley amp Sons New York NYUSA 2001

[2] S J Julier J K Uhlmann and H F Durrant-Whyte ldquoA newapproach for filtering nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1628ndash1632 June 1995

[3] B M Bell and F W Cathey ldquoIterated Kalman filter updateas a Gauss-Newton methodrdquo IEEE Transactions on AutomaticControl vol 38 no 2 pp 294ndash297 1993

[4] K-H Kim G-I Jee C-G Park and J-G Lee ldquoThe stabilityanalysis of the adaptive fading extended kalman filter usingthe innovation covariancerdquo International Journal of ControlAutomation and Systems vol 7 no 1 pp 49ndash56 2009

[5] X-X Wang L Zhao Q-X Xia W Cao and L Li ldquoDesign ofunscented Kalman filter with correlative noisesrdquoControlTheoryamp Applications vol 27 no 10 pp 1362ndash1368 2010

[6] X-X Wang L Zhao Q Pan Q-X Xia and W Hong ldquoDesignof UKFwith correlative noises based onminimummean squareerror estimationrdquo Control and Decision vol 25 no 9 pp 1393ndash1398 2010

[7] X-X Wang Y Liang Q Pan C-H Zhao and H-Z LildquoUnscented Kalman filter for nonlinear systems with coloredmeasurement noiserdquo Acta Automatica Sinica vol 38 no 6 pp986ndash998 2012

[8] W F Leven and A D Lanterman ldquoUnscented Kalman filtersfor multiple target tracking with symmetric measurementequationsrdquo IEEE Transactions on Automatic Control vol 54 no2 pp 370ndash375 2009

[9] J-S Kim D-R Shin and E Serpedin ldquoAdaptive multiuserreceiver with joint channel and time delay estimation of CDMAsignals based on the square-root unscented filterrdquoDigital SignalProcessing vol 19 no 3 pp 504ndash520 2009

[10] W Fan and Y Li ldquoAccuracy analysis of sigma-point Kalmanfiltersrdquo in Proceedings of the Chinese Control and Decision


Conference (CCDC rsquo09) pp 2883ndash2888 Guilin China June2009

[11] T Lefebvre H Bruyninckx and J de Schutter ldquoKalman filtersfor non-linear systems a comparison of performancerdquo Interna-tional Journal of Control vol 77 no 7 pp 639ndash653 2004

[12] J Dunik and M Simandl ldquoPerformance analysis of derivative-free filtersrdquo in Proceedings of the 44th IEEE Conference onDecision and Control pp 1941ndash1946 Seville Spain 2005

[13] M Noslashrgaard N K Poulsen and O Ravn ldquoNew developmentsin state estimation for nonlinear systemsrdquo Automatica vol 36no 11 pp 1627ndash1638 2000

[14] L Deok-Jin Nonlinear Bayesian filtering with applications toestimation and navigation [PhD thesis] 2005

[15] C Wang J Zhang and J Mu ldquoMaximum likelihood-basediterated divided difference filter for nonlinear systems fromdiscrete noisy measurementsrdquo Sensors vol 12 no 7 pp 8912ndash8929 2012

[16] N Subrahmanya and Y C Shin ldquoAdaptive divided differencefiltering for simultaneous state and parameter estimationrdquoAutomatica vol 45 no 7 pp 1686ndash1693 2009

[17] X Luo I Hoteit and IMMoroz ldquoOn a nonlinear Kalman filterwith simplified divided difference approximationrdquo Physica DNonlinear Phenomena vol 241 no 6 pp 671ndash680 2012

[18] Z Ali M A Landolsi and M Deriche ldquoMultiuser parameterestimation using divided difference filter in CDMA systemsrdquoInternational Journal of Advanced Science and Technology vol5 pp 35ndash50 2009

[19] P G Bhale P N Dwivedi P Kumar and A BhattacharyaaldquoEstimation of ballistic coefficient of reentry vehicle withdivided difference filtering using noisy RF seeker datardquo inProceedings of the IEEE International Conference on IndustrialTechnology (ICIT rsquo06) pp 1087ndash1092Mumbai IndiaDecember2006

[20] M JingW Chang-yuan Z Juan and C Fang ldquoState estimationof reentry target based on divided difference filterrdquo Journal ofXirsquoan Technological University vol 31 no 4 pp 377ndash381 2011

[21] P Tripathy S C Srivastava and S N Singh ldquoA divide-by-difference-filter based algorithm for estimation of genera-tor rotor angle utilizing synchrophasor measurementsrdquo IEEETransactions on Instrumentation and Measurement vol 59 no6 pp 1562ndash1570 2010

[22] M Ahmadi A Khayatian and P Karimaghaee ldquoAttitudeestimation by divided difference filter in quaternion spacerdquoActaAstronautica vol 75 pp 95ndash107 2012

[23] P Setoodeh A Khayatian and E Farjah ldquoAttitude estimationby divided difference filter-based sensor fufsionrdquoThe Journal ofNavigation vol 60 no 1 pp 119ndash128 2007

[24] C Wu and C Han ldquoSecond-order divided difference filter withapplication to ballistic target trackingrdquo in Proceedings of the 7thWorld Congress on Intelligent Control and Automation (WCICArsquo08) pp 6336ndash6341 Chongqing China June 2008

[25] D-J Jwo M-Y Hsieh and S-Y Lai ldquoGPS navigation process-ing using the quaternion-based divided difference filterrdquo GPSSolutions vol 14 no 3 pp 217ndash228 2010

[26] VN VapnikTheNature of Statistical LearningTheory SpringerBerlin Germany 1995

[27] L Guozheng W Meng and Z Huajun An Introduction toSupport Vector Machines and Other Kernel-Based LearningMethods Publishing House of Electronics Industry BeijingChina 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


application of conventional UKFs In [7] a UKF filteringalgorithm with colored measurement noise was proposedThe algorithm was first derived on the basis of augmentedmeasurement information and minimum mean square errorestimation and a filtering recursive formula of UKF withcolored noise then added by applying an unscented trans-formation to calculate the posterior mean and covarianceof the nonlinear state within the optimal framework Theproposed UKF effectively dealt with the fact that traditionalUKFs fail when measurement noise is colored In [8] a UKFwas applied to multiple target tracking with the proposedUKF shown to have improved performance versus previousEKF approaches In [9] the paper discussed an adaptivemultiuser receiver for CDMA systems in which the scaledunscented filter (SUF) and the square root unscented filter(SURF) were used for joint estimation and tracking of thecode delays andmultipath coefficients of the received CDMAsignals The proposed channel estimators were more near-far resistant than in conventional EKFs and presented lowercomplexity than conventional particle filter- (PF-) basedmethods Computer simulation results demonstrated thesuperior performance of the proposed channel estimatorsand the proposed estimators were shown to exhibit lowercomplexity relative to the PF-based method Although UKFshave undergone a significant amount of meaningful theoryinnovation and are nowused inmany fields [10ndash12] show thatUKF accuracy is lower than that of DDF while also having ahigher computational cost

The divided difference filter (DDF) first proposed byNoslashgaard et al (see [13]) linearizes the nonlinear termsbased on Stirlingrsquos interpolation polynomial approximationsformula rather than Taylorrsquos approximation of nonlinearterms in an EKF Conceptually the implementation principleresembles that of an EKF however the DDF is significantlysimpler as it does not need to calculate the Jacobian matrixand no derivatives are required The DDF that Noslashgaardet al developed works on general discrete-time nonlinearmodels in which the noises are not assumed to be additiveIn [14] the paper further formulated a DDF in terms of theinnovation vector approach the additive process and themeasurement noise sources In [15] the paper proposed a newfilter named the maximum likelihood-based iterated divideddifference filter (MLIDDF) which improved the low stateestimation accuracy of nonlinear state estimation that resultsfrom large initial estimation errors and the nonlinearity ofthe measurement equations Simulation results showed thatthe MLIDDF algorithm possessed better state estimationaccuracy and a faster convergence rate In [16] the authorsproposed a novel adaptive version of the DDF that wasapplicable to nonlinear systems with a linear output equationIn order to make the filter robust to modeling errors upperbounds on the state covariance matrix were derived Theparameters of the upper bound were then estimated usinga combination of offline tuning and online optimizationwith a linear matrix inequality constraint which ensuredthat the predicted output error covariance was larger thanthe observed output error covariance Simulation resultsdemonstrated the superior performance of the proposed filteras compared to the standard DDF Reference [17] presented

an ensemble-based approach that handled nonlinearity basedon a simplified divided difference approximation throughStirlingrsquos interpolation formula The algorithm used Stirlingrsquosinterpolation formula to evaluate the statistics of the back-ground ensemble during the prediction step employing theformula in an ensemble square root filter (EnSRF) at thefiltering step to update the background for analysis In thissense the algorithm is a hybrid of Stirlingrsquos interpolationformula and theEnSRFmethodwhile the computational costof the algorithm is less than that of EnSRF

Different studies have focused on the application ofDDFs to nonlinear state estimation problems In [18] timedelay and channel gain estimation for multipath fadingcode division multiple access (CDMA) signals using a DDFwere investigated and the simulation results showed thatthe DDF was simpler to implement and more resilient tonear-far interference in CDMA networks compared with anEKF In [19 20] the relative kinematic states of a reentryvehicle obtained from noisy seeker measurements using aDDF were examined The results were compared to thoseobtained using an EKF and a UKF and showed that theDDF was more accurate than estimators based on a Taylorapproximation like the EKF Reference [21] investigatedthe possibility of using a DDF for estimating the internalvariables of a synchronous generator such as the rotor anglewhere the data acquired is from a phasor measurementThe effectiveness of the method was tested on a singlemachine infinite bus system a nine-bus system and a 68-bus New England-New York interconnected system In [22]a DDF using orientation estimation was considered Thefourth element of the quaternion error vector was removedfrom the system states to alleviate estimated error covariancematrix divergence The measurement system was a MARGsensor which consisted of a triaxial rate gyro a triaxialaccelerometer and a triaxial magnetometer The nonlinearmeasurement model was obtained based on the principalsof operation of the magnetometer and accelerometer and theproperties of the quaternion vector spaceTheperformance ofthree filtersDDF EKF andUKFwas comparedwith differentsampling frequencies The work showed that the tested DDFand the UKF were more robust than the EKF under the sameinitial angle-error conditionsTheDDF also performed betterthan the UKF although the computational load for the UKFwas less In [23] a DDF-based data fusion algorithmwas pre-sented which utilized the complementary noise profile of rategyros and gravimetric inclinometers to extend their limitsand achieve more accurate attitude estimates In [24] a DDF-based ballistic target tracking system for the reentry phasewas proposed The paper compared DDF EKF and UKFalgorithms using a Monte Carlo simulation approach withthe simulation results showing that the DDF outperformedboth the EKF and UKF in terms of estimation accuracy andfiltering credibility In [25] a DDF with quaternion-baseddynamic process modeling was applied to global positioningsystem (GPS) navigation to increase navigation estimationaccuracy at high-dynamic regions while preserving precisionat low-dynamic regions Some properties and performancemetrics were assessed and compared to those using EKF andUKF approaches


1

15

2

25

3

True value

y=ex

0 01 02 03 04 05 06 07 08 09 1

x


(a)

minus05

05

Erro

r

1

0

0

01 02 03 04 05 06 07 08 09 1

x


(b)





y = f (x) (1)


y = (f (x + Δx) = f (x) +DΔxf = f (x)

+ (Δ1199091

120597

1205971199091

+ Δ1199092

120597

1205971199092


120597

120597119909119899

) f (x))10038161003816100381610038161003816100381610038161003816x=x

(2)



y = f (x + Δx) = f (x) + DΔxf (x)

= f (x) + 1

ℎ

(

119899

sum

119894=1

Δ119909119894120583119894120575119894) f (x)

(3)


120575119894f (x) = f (x + ℎ

2


2

e119894)

120583119894f (x) = 1

2

(f (x + ℎ

2

e119894) + f (x minus ℎ

2

e119894))

(4)





z = Sminus1x x (5)


y = f (Sxz) = f (z) = f (z) + DΔzf (6)


DΔzf = 1

ℎ

(

119899

sum

119894=1

Δz119894120583119894120575119894) f (z) (7)





= E [(DΔzf) (D

Δzf)119879

]

=

1

4ℎ

2

119899

sum

119894=1


119894))


119894))

119879

=

1

4ℎ

2

119899

sum

119894=1





]

=

1

2ℎ

119899

sum

119894=1


(8)



x119896+1

= f (x119896) + 120596119896

y119896= h (x

119896) + ^119896

(9)



119896sim N(oQ

119896) and ^

119896sim N(0R

119896)




119896P119896) where x

119896and P


x119896= E [x

119896]


119896) (x minus x

119896)

119879

]

(10)


P119896= SxS

119879

x

Sxx =1

2ℎ

f119894(x119896+ ℎsx119895) minus f

119894(x119896minus ℎsx119895)

(11)



xminus119896+1

= f (x119896) (12)

Pminus119896+1

= Sxx(Sxx)119879

+Q119896

= Sminusx (Sminus

x )119879

(13)

Syx =1

2ℎ

h119894(xminus119896+1







yminus119896+1

= h (xminus119896+1

) (15)

P^^119896+1

= (Syx) (Syx)119879

+ R119896

(16)

Pxy119896+1


(17)





tion matrix


120581119896+1

= Pxy119896+1

(P^^119896+1

)

minus1

P+119896+1

= Pminus119896+1

minus 120581119896+1

P^^119896+1120581119879

119896+1

120592119896+1

= y119896+1


x+119896+1

= xminus119896+1

+ 120581119896+1120592119896+1

(18)

where 120581119896+1





estimated state


119896and R

119896 However

Q119896and R



119896





119898 119910119898) isin


119894isin R 119894 = 1 2 119898 the SVR


119910 (x120596) =119898

sum

119894=1

120596119894119892 (x119894) + 119887 = (120596 sdot g (x)) + 119887 (19)


119910 (x120572) =119898

sum

119894=1

(120572

lowast

119894minus 120572119894) (119892 (x

119894) sdot 119892 (x)) + 119887

=

119898

sum

119894=1

(120572

lowast

119894minus 120572119894)119870 (x

119894 x) + 119887

(20)

where 120572

lowast

119894and 120572


condition 120572119894 120572

lowast

119894ge 0 120572

119894120572

lowast

119894= 0 and 119870(x



119894minus x221205902)





119896by defining


R119896= Δ119903119896R (21)







C119896=

1

119873

119896

sum

119894=119896minus119873+1

120592119896+1120592119879

119896+1 (22)






119896+1minus C119896) isin R119899 (23)






Δ119903as 119879119878Δ119903

=

(DOM119896minus119873+1

Δ119903119896minus119873+1





DOM119896+1

comes then Δ119903119896+1




1015840

Δ119903= (DOM1015840

119896minus119873+1 Δ119903

1015840

119896minus119873+1) (DOM1015840

119896 Δ119903

1015840

119896) where

DOM1015840119896and 119903

1015840


DOM1015840119896=

1

119873

119873

sum

119894=1

DOM119896minus119894+1119895

119895 = 1 2 119899

Δ119903

1015840

119896=

1

119873

119873

sum

119894=1

Δ119903119896minus119894+1

(24)






119896+1 if we



119896in the


and C119896+1


119896 where R

119896has the

following form

Q119896= Δ119902119896Q (25)


Δ119902as 119879119878

Δ119902=

(DOM119896minus119873+1

Δ119903119896minus119873+1






x119896+1

=

[

[

[

[

1 0 119879 0

0 1 0 119879

0 0 1 0

0 0 0 1

]

]

]

]

x119896+ w119896 (26)


0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500

TargetPlatform

I

F

X (m)

Y(m

)




Q = 1199021times

[

[

[

[

[

[

[

[

[

[

119879

2

2

119879

2

2

119879

119879

]

]

]

]

]

]

]

]

]

]

(27)



y119896= 120579119896= tanminus1 (

119910119896

119909119896

) + 120592119896 (28)



119879 = 1 s1199021= 00001m2 sminus3

119877 = 002mrad

(29)












1

119873

119873

sum

119894=1

((119909

119894

k minus 119909

119894

119896)

2

+ (119910

119894

119896minus 119910

119894

119896)

2

)(32)

where (119909119894119896 119910

119894

119896) and (119909

119894

119896 119910

119894






0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

Posit

ion

RMSE

DDFSVRADDF


0

2

4

6

8

10

12

Velo

city

RM

SE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF



x119896=

[

[

[

[

[

[

[

[

[

[

[

1

sinΩ119879

Ω

0 minus

1 minus cosΩ119879

Ω


0

1 minus cosΩ119879

Ω

1

sinΩ119879

Ω

0 sinΩ119879 0 cosΩ119879

]

]

]

]

]

]

]

]

]

]

]

times x119896minus1

+ w119896minus1

(33)





where

Q = 1199021times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119879

3

3

119879

2

2

0 0

119879

2

2

119879 0 0

0 0

119879

3

3

119879

2

2

0 0

119879

2

2

119879

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(34)




0

0

2000

4000

6000

2000 4000 6000

TargetRadar

X (m)

Y(m

)

minus2000

minus4000

minus6000


minus8000


280

300

320

340

360

380

400

420

440

Posit

ion

RMSE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF



y119896= 120579119896= tanminus1 (

119910119896

119909119896

) + 120592119896 (35)



∘ sminus1 1199021




=



0P0)







15

155

16

165

17

175

18

Velo

city

RM

SE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF


4 Conclusions





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

15

2

25

3

True value

y=ex

0 01 02 03 04 05 06 07 08 09 1

x


(a)

minus05

05

Erro

r

1

0

0

01 02 03 04 05 06 07 08 09 1

x


(b)





y = f (x) (1)


y = (f (x + Δx) = f (x) +DΔxf = f (x)

+ (Δ1199091

120597

1205971199091

+ Δ1199092

120597

1205971199092


120597

120597119909119899

) f (x))10038161003816100381610038161003816100381610038161003816x=x

(2)



y = f (x + Δx) = f (x) + DΔxf (x)

= f (x) + 1

ℎ

(

119899

sum

119894=1

Δ119909119894120583119894120575119894) f (x)

(3)


120575119894f (x) = f (x + ℎ

2


2

e119894)

120583119894f (x) = 1

2

(f (x + ℎ

2

e119894) + f (x minus ℎ

2

e119894))

(4)





z = Sminus1x x (5)


y = f (Sxz) = f (z) = f (z) + DΔzf (6)


DΔzf = 1

ℎ

(

119899

sum

119894=1

Δz119894120583119894120575119894) f (z) (7)





= E [(DΔzf) (D

Δzf)119879

]

=

1

4ℎ

2

119899

sum

119894=1


119894))


119894))

119879

=

1

4ℎ

2

119899

sum

119894=1





]

=

1

2ℎ

119899

sum

119894=1


(8)



x119896+1

= f (x119896) + 120596119896

y119896= h (x

119896) + ^119896

(9)



119896sim N(oQ

119896) and ^

119896sim N(0R

119896)




119896P119896) where x

119896and P


x119896= E [x

119896]


119896) (x minus x

119896)

119879

]

(10)


P119896= SxS

119879

x

Sxx =1

2ℎ

f119894(x119896+ ℎsx119895) minus f

119894(x119896minus ℎsx119895)

(11)



xminus119896+1

= f (x119896) (12)

Pminus119896+1

= Sxx(Sxx)119879

+Q119896

= Sminusx (Sminus

x )119879

(13)

Syx =1

2ℎ

h119894(xminus119896+1







yminus119896+1

= h (xminus119896+1

) (15)

P^^119896+1

= (Syx) (Syx)119879

+ R119896

(16)

Pxy119896+1


(17)





tion matrix


120581119896+1

= Pxy119896+1

(P^^119896+1

)

minus1

P+119896+1

= Pminus119896+1

minus 120581119896+1

P^^119896+1120581119879

119896+1

120592119896+1

= y119896+1


x+119896+1

= xminus119896+1

+ 120581119896+1120592119896+1

(18)

where 120581119896+1





estimated state


119896and R

119896 However

Q119896and R



119896





119898 119910119898) isin


119894isin R 119894 = 1 2 119898 the SVR


119910 (x120596) =119898

sum

119894=1

120596119894119892 (x119894) + 119887 = (120596 sdot g (x)) + 119887 (19)


119910 (x120572) =119898

sum

119894=1

(120572

lowast

119894minus 120572119894) (119892 (x

119894) sdot 119892 (x)) + 119887

=

119898

sum

119894=1

(120572

lowast

119894minus 120572119894)119870 (x

119894 x) + 119887

(20)

where 120572

lowast

119894and 120572


condition 120572119894 120572

lowast

119894ge 0 120572

119894120572

lowast

119894= 0 and 119870(x



119894minus x221205902)





119896by defining


R119896= Δ119903119896R (21)







C119896=

1

119873

119896

sum

119894=119896minus119873+1

120592119896+1120592119879

119896+1 (22)






119896+1minus C119896) isin R119899 (23)






Δ119903as 119879119878Δ119903

=

(DOM119896minus119873+1

Δ119903119896minus119873+1





DOM119896+1

comes then Δ119903119896+1




1015840

Δ119903= (DOM1015840

119896minus119873+1 Δ119903

1015840

119896minus119873+1) (DOM1015840

119896 Δ119903

1015840

119896) where

DOM1015840119896and 119903

1015840


DOM1015840119896=

1

119873

119873

sum

119894=1

DOM119896minus119894+1119895

119895 = 1 2 119899

Δ119903

1015840

119896=

1

119873

119873

sum

119894=1

Δ119903119896minus119894+1

(24)






119896+1 if we



119896in the


and C119896+1


119896 where R

119896has the

following form

Q119896= Δ119902119896Q (25)


Δ119902as 119879119878

Δ119902=

(DOM119896minus119873+1

Δ119903119896minus119873+1






x119896+1

=

[

[

[

[

1 0 119879 0

0 1 0 119879

0 0 1 0

0 0 0 1

]

]

]

]

x119896+ w119896 (26)


0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500

TargetPlatform

I

F

X (m)

Y(m

)




Q = 1199021times

[

[

[

[

[

[

[

[

[

[

119879

2

2

119879

2

2

119879

119879

]

]

]

]

]

]

]

]

]

]

(27)



y119896= 120579119896= tanminus1 (

119910119896

119909119896

) + 120592119896 (28)



119879 = 1 s1199021= 00001m2 sminus3

119877 = 002mrad

(29)












1

119873

119873

sum

119894=1

((119909

119894

k minus 119909

119894

119896)

2

+ (119910

119894

119896minus 119910

119894

119896)

2

)(32)

where (119909119894119896 119910

119894

119896) and (119909

119894

119896 119910

119894






0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

Posit

ion

RMSE

DDFSVRADDF


0

2

4

6

8

10

12

Velo

city

RM

SE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF



x119896=

[

[

[

[

[

[

[

[

[

[

[

1

sinΩ119879

Ω

0 minus

1 minus cosΩ119879

Ω


0

1 minus cosΩ119879

Ω

1

sinΩ119879

Ω

0 sinΩ119879 0 cosΩ119879

]

]

]

]

]

]

]

]

]

]

]

times x119896minus1

+ w119896minus1

(33)





where

Q = 1199021times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119879

3

3

119879

2

2

0 0

119879

2

2

119879 0 0

0 0

119879

3

3

119879

2

2

0 0

119879

2

2

119879

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(34)




0

0

2000

4000

6000

2000 4000 6000

TargetRadar

X (m)

Y(m

)

minus2000

minus4000

minus6000


minus8000


280

300

320

340

360

380

400

420

440

Posit

ion

RMSE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF



y119896= 120579119896= tanminus1 (

119910119896

119909119896

) + 120592119896 (35)



∘ sminus1 1199021




=



0P0)







15

155

16

165

17

175

18

Velo

city

RM

SE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF


4 Conclusions





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













= E [(DΔzf) (D

Δzf)119879

]

=

1

4ℎ

2

119899

sum

119894=1


119894))


119894))

119879

=

1

4ℎ

2

119899

sum

119894=1





]

=

1

2ℎ

119899

sum

119894=1


(8)



x119896+1

= f (x119896) + 120596119896

y119896= h (x

119896) + ^119896

(9)



119896sim N(oQ

119896) and ^

119896sim N(0R

119896)




119896P119896) where x

119896and P


x119896= E [x

119896]


119896) (x minus x

119896)

119879

]

(10)


P119896= SxS

119879

x

Sxx =1

2ℎ

f119894(x119896+ ℎsx119895) minus f

119894(x119896minus ℎsx119895)

(11)



xminus119896+1

= f (x119896) (12)

Pminus119896+1

= Sxx(Sxx)119879

+Q119896

= Sminusx (Sminus

x )119879

(13)

Syx =1

2ℎ

h119894(xminus119896+1







yminus119896+1

= h (xminus119896+1

) (15)

P^^119896+1

= (Syx) (Syx)119879

+ R119896

(16)

Pxy119896+1


(17)





tion matrix


120581119896+1

= Pxy119896+1

(P^^119896+1

)

minus1

P+119896+1

= Pminus119896+1

minus 120581119896+1

P^^119896+1120581119879

119896+1

120592119896+1

= y119896+1


x+119896+1

= xminus119896+1

+ 120581119896+1120592119896+1

(18)

where 120581119896+1





estimated state


119896and R

119896 However

Q119896and R



119896





119898 119910119898) isin


119894isin R 119894 = 1 2 119898 the SVR


119910 (x120596) =119898

sum

119894=1

120596119894119892 (x119894) + 119887 = (120596 sdot g (x)) + 119887 (19)


119910 (x120572) =119898

sum

119894=1

(120572

lowast

119894minus 120572119894) (119892 (x

119894) sdot 119892 (x)) + 119887

=

119898

sum

119894=1

(120572

lowast

119894minus 120572119894)119870 (x

119894 x) + 119887

(20)

where 120572

lowast

119894and 120572


condition 120572119894 120572

lowast

119894ge 0 120572

119894120572

lowast

119894= 0 and 119870(x



119894minus x221205902)





119896by defining


R119896= Δ119903119896R (21)







C119896=

1

119873

119896

sum

119894=119896minus119873+1

120592119896+1120592119879

119896+1 (22)






119896+1minus C119896) isin R119899 (23)






Δ119903as 119879119878Δ119903

=

(DOM119896minus119873+1

Δ119903119896minus119873+1





DOM119896+1

comes then Δ119903119896+1




1015840

Δ119903= (DOM1015840

119896minus119873+1 Δ119903

1015840

119896minus119873+1) (DOM1015840

119896 Δ119903

1015840

119896) where

DOM1015840119896and 119903

1015840


DOM1015840119896=

1

119873

119873

sum

119894=1

DOM119896minus119894+1119895

119895 = 1 2 119899

Δ119903

1015840

119896=

1

119873

119873

sum

119894=1

Δ119903119896minus119894+1

(24)






119896+1 if we



119896in the


and C119896+1


119896 where R

119896has the

following form

Q119896= Δ119902119896Q (25)


Δ119902as 119879119878

Δ119902=

(DOM119896minus119873+1

Δ119903119896minus119873+1






x119896+1

=

[

[

[

[

1 0 119879 0

0 1 0 119879

0 0 1 0

0 0 0 1

]

]

]

]

x119896+ w119896 (26)


0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500

TargetPlatform

I

F

X (m)

Y(m

)




Q = 1199021times

[

[

[

[

[

[

[

[

[

[

119879

2

2

119879

2

2

119879

119879

]

]

]

]

]

]

]

]

]

]

(27)



y119896= 120579119896= tanminus1 (

119910119896

119909119896

) + 120592119896 (28)



119879 = 1 s1199021= 00001m2 sminus3

119877 = 002mrad

(29)












1

119873

119873

sum

119894=1

((119909

119894

k minus 119909

119894

119896)

2

+ (119910

119894

119896minus 119910

119894

119896)

2

)(32)

where (119909119894119896 119910

119894

119896) and (119909

119894

119896 119910

119894






0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

Posit

ion

RMSE

DDFSVRADDF


0

2

4

6

8

10

12

Velo

city

RM

SE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF



x119896=

[

[

[

[

[

[

[

[

[

[

[

1

sinΩ119879

Ω

0 minus

1 minus cosΩ119879

Ω


0

1 minus cosΩ119879

Ω

1

sinΩ119879

Ω

0 sinΩ119879 0 cosΩ119879

]

]

]

]

]

]

]

]

]

]

]

times x119896minus1

+ w119896minus1

(33)





where

Q = 1199021times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119879

3

3

119879

2

2

0 0

119879

2

2

119879 0 0

0 0

119879

3

3

119879

2

2

0 0

119879

2

2

119879

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(34)




0

0

2000

4000

6000

2000 4000 6000

TargetRadar

X (m)

Y(m

)

minus2000

minus4000

minus6000


minus8000


280

300

320

340

360

380

400

420

440

Posit

ion

RMSE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF



y119896= 120579119896= tanminus1 (

119910119896

119909119896

) + 120592119896 (35)



∘ sminus1 1199021




=



0P0)







15

155

16

165

17

175

18

Velo

city

RM

SE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF


4 Conclusions





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119898 119910119898) isin


119894isin R 119894 = 1 2 119898 the SVR


119910 (x120596) =119898

sum

119894=1

120596119894119892 (x119894) + 119887 = (120596 sdot g (x)) + 119887 (19)


119910 (x120572) =119898

sum

119894=1

(120572

lowast

119894minus 120572119894) (119892 (x

119894) sdot 119892 (x)) + 119887

=

119898

sum

119894=1

(120572

lowast

119894minus 120572119894)119870 (x

119894 x) + 119887

(20)

where 120572

lowast

119894and 120572


condition 120572119894 120572

lowast

119894ge 0 120572

119894120572

lowast

119894= 0 and 119870(x



119894minus x221205902)





119896by defining


R119896= Δ119903119896R (21)







C119896=

1

119873

119896

sum

119894=119896minus119873+1

120592119896+1120592119879

119896+1 (22)






119896+1minus C119896) isin R119899 (23)






Δ119903as 119879119878Δ119903

=

(DOM119896minus119873+1

Δ119903119896minus119873+1





DOM119896+1

comes then Δ119903119896+1




1015840

Δ119903= (DOM1015840

119896minus119873+1 Δ119903

1015840

119896minus119873+1) (DOM1015840

119896 Δ119903

1015840

119896) where

DOM1015840119896and 119903

1015840


DOM1015840119896=

1

119873

119873

sum

119894=1

DOM119896minus119894+1119895

119895 = 1 2 119899

Δ119903

1015840

119896=

1

119873

119873

sum

119894=1

Δ119903119896minus119894+1

(24)






119896+1 if we



119896in the


and C119896+1


119896 where R

119896has the

following form

Q119896= Δ119902119896Q (25)


Δ119902as 119879119878

Δ119902=

(DOM119896minus119873+1

Δ119903119896minus119873+1






x119896+1

=

[

[

[

[

1 0 119879 0

0 1 0 119879

0 0 1 0

0 0 0 1

]

]

]

]

x119896+ w119896 (26)


0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500

TargetPlatform

I

F

X (m)

Y(m

)




Q = 1199021times

[

[

[

[

[

[

[

[

[

[

119879

2

2

119879

2

2

119879

119879

]

]

]

]

]

]

]

]

]

]

(27)



y119896= 120579119896= tanminus1 (

119910119896

119909119896

) + 120592119896 (28)



119879 = 1 s1199021= 00001m2 sminus3

119877 = 002mrad

(29)












1

119873

119873

sum

119894=1

((119909

119894

k minus 119909

119894

119896)

2

+ (119910

119894

119896minus 119910

119894

119896)

2

)(32)

where (119909119894119896 119910

119894

119896) and (119909

119894

119896 119910

119894






0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

Posit

ion

RMSE

DDFSVRADDF


0

2

4

6

8

10

12

Velo

city

RM

SE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF



x119896=

[

[

[

[

[

[

[

[

[

[

[

1

sinΩ119879

Ω

0 minus

1 minus cosΩ119879

Ω


0

1 minus cosΩ119879

Ω

1

sinΩ119879

Ω

0 sinΩ119879 0 cosΩ119879

]

]

]

]

]

]

]

]

]

]

]

times x119896minus1

+ w119896minus1

(33)





where

Q = 1199021times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119879

3

3

119879

2

2

0 0

119879

2

2

119879 0 0

0 0

119879

3

3

119879

2

2

0 0

119879

2

2

119879

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(34)




0

0

2000

4000

6000

2000 4000 6000

TargetRadar

X (m)

Y(m

)

minus2000

minus4000

minus6000


minus8000


280

300

320

340

360

380

400

420

440

Posit

ion

RMSE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF



y119896= 120579119896= tanminus1 (

119910119896

119909119896

) + 120592119896 (35)



∘ sminus1 1199021




=



0P0)







15

155

16

165

17

175

18

Velo

city

RM

SE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF


4 Conclusions





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500

TargetPlatform

I

F

X (m)

Y(m

)




Q = 1199021times

[

[

[

[

[

[

[

[

[

[

119879

2

2

119879

2

2

119879

119879

]

]

]

]

]

]

]

]

]

]

(27)



y119896= 120579119896= tanminus1 (

119910119896

119909119896

) + 120592119896 (28)



119879 = 1 s1199021= 00001m2 sminus3

119877 = 002mrad

(29)












1

119873

119873

sum

119894=1

((119909

119894

k minus 119909

119894

119896)

2

+ (119910

119894

119896minus 119910

119894

119896)

2

)(32)

where (119909119894119896 119910

119894

119896) and (119909

119894

119896 119910

119894






0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

Posit

ion

RMSE

DDFSVRADDF


0

2

4

6

8

10

12

Velo

city

RM

SE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF



x119896=

[

[

[

[

[

[

[

[

[

[

[

1

sinΩ119879

Ω

0 minus

1 minus cosΩ119879

Ω


0

1 minus cosΩ119879

Ω

1

sinΩ119879

Ω

0 sinΩ119879 0 cosΩ119879

]

]

]

]

]

]

]

]

]

]

]

times x119896minus1

+ w119896minus1

(33)





where

Q = 1199021times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119879

3

3

119879

2

2

0 0

119879

2

2

119879 0 0

0 0

119879

3

3

119879

2

2

0 0

119879

2

2

119879

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(34)




0

0

2000

4000

6000

2000 4000 6000

TargetRadar

X (m)

Y(m

)

minus2000

minus4000

minus6000


minus8000


280

300

320

340

360

380

400

420

440

Posit

ion

RMSE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF



y119896= 120579119896= tanminus1 (

119910119896

119909119896

) + 120592119896 (35)



∘ sminus1 1199021




=



0P0)







15

155

16

165

17

175

18

Velo

city

RM

SE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF


4 Conclusions





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

Posit

ion

RMSE

DDFSVRADDF


0

2

4

6

8

10

12

Velo

city

RM

SE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF



x119896=

[

[

[

[

[

[

[

[

[

[

[

1

sinΩ119879

Ω

0 minus

1 minus cosΩ119879

Ω


0

1 minus cosΩ119879

Ω

1

sinΩ119879

Ω

0 sinΩ119879 0 cosΩ119879

]

]

]

]

]

]

]

]

]

]

]

times x119896minus1

+ w119896minus1

(33)





where

Q = 1199021times

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119879

3

3

119879

2

2

0 0

119879

2

2

119879 0 0

0 0

119879

3

3

119879

2

2

0 0

119879

2

2

119879

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(34)




0

0

2000

4000

6000

2000 4000 6000

TargetRadar

X (m)

Y(m

)

minus2000

minus4000

minus6000


minus8000


280

300

320

340

360

380

400

420

440

Posit

ion

RMSE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF



y119896= 120579119896= tanminus1 (

119910119896

119909119896

) + 120592119896 (35)



∘ sminus1 1199021




=



0P0)







15

155

16

165

17

175

18

Velo

city

RM

SE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF


4 Conclusions





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

0

2000

4000

6000

2000 4000 6000

TargetRadar

X (m)

Y(m

)

minus2000

minus4000

minus6000


minus8000


280

300

320

340

360

380

400

420

440

Posit

ion

RMSE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF



y119896= 120579119896= tanminus1 (

119910119896

119909119896

) + 120592119896 (35)



∘ sminus1 1199021




=



0P0)







15

155

16

165

17

175

18

Velo

city

RM

SE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF


4 Conclusions





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










15

155

16

165

17

175

18

Velo

city

RM

SE

0 100 200 300 400 500 600 700 800 900 1000

Time (s)

DDFSVRADDF


4 Conclusions





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article Support Vector Regression-Based Adaptive ...downloads.hindawi.com/journals/jam/2014/139503.pdf · Stirling sinterpolationformula. ealgorithmusedStirling s interpolation