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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 737146 7 pageshttpdxdoiorg1011552013737146

Research ArticlePositioning Errors Predicting Method of Strapdown InertialNavigation Systems Based on PSO-SVM

Xunyuan Yin Yingbo Sun and Changhong Wang

Space Control and Inertial Technology Research Center Harbin Institute of Technology Harbin 150001 China

Correspondence should be addressed to Changhong Wang cwanghiteducn

Received 13 July 2013 Revised 27 July 2013 Accepted 27 July 2013

Academic Editor Hamid Reza Karimi

Copyright copy 2013 Xunyuan Yin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The strapdown inertial navigation systems (SINS) have been widely used for many vehicles such as commercial airplanesUnmanned Aerial Vehicles (UAVs) and other types of aircrafts In order to evaluate the navigation errors precisely and efficiently apredictionmethod based on support vectormachine (SVM) is proposed for positioning error assessment Firstly SINS errormodelsthat are used for error calculation are established considering several error resources with respect to inertial units Secondly flightpaths for simulation are designed Thirdly the 120576-SVR based prediction method is proposed to predict the positioning errors ofnavigation systems and particle swarm optimization (PSO) is used for the SVM parameters optimization Finally 600 sets of errorparameters of SINS are utilized to train the SVMmodel which is used for the performance prediction of new navigation systems Bycomparing the predicting results with the real errors the latitudinal predicting accuracy is 9273 while the longitudinal predictingaccuracy is 9164 and PSO is effective to increase the prediction accuracy compared with traditional SVMwith fixed parametersThis method is also demonstrated to be effective for error prediction for an entire flight process Moreover the prediction methodcan save 75 of calculation time compared with analyses based on error models

1 Introduction

Strapdown inertial navigation systems have been widelyutilized in a wide range of fields such as the navigation ofairplanes ships and vehicles and the guidance of missiles [1]Although the positioning accuracy of strapdown systems islower than that of the platform inertial navigation systemsthe strapdown systems have several advantages that cannotbe found in platform systems They have low cost lowweight small volume good reliability and simplemechanicalstructures [2] So far almost all the civil aviation airplanesmanufactured by Boeing and Airbus are equipped with LTN-92 or LTN-101 laser SINS [3 4] Other types of SINS thatconsist of different kinds of gyroscopes and accelerometersare used for the navigation of vehicles which require mediumor low accuracy [5]

Navigation accuracy is the main factor that is used toaccess the performance of SINS [6] To help the vehicles tocomplete the flight task or arrive at a desired destination

the strapdown systems which can meet the navigationrequirements should be selected [7] If an effective methodthat could be established to predict the velocity errors andpositioning errors by assessing error parameters a gooddeal of time for error analyses would be saved and theaircrafts would be more likely to accomplish the initial tasks[8 9] Currently some researchers have placed importanceon the error analyses and error compensation of SINS [10]A methodology based on the theory of artificial neuralnetworks has been put up to predict the positioning errorscaused by the drift error of each singe axis of gyroscope [11]But research that stresses predicting the performance of SINSby error sources can hardly be found

Nowadays several methodologies have been reported forthe classification or regression in various fields Artificialneural networks (ANNs) have been pervasively adopted andare able to achieve acceptable results in many applicationsChen developed an ANN-based model which is called Evo-lutionary Fuzzy Neural Inference Model to predict Estimate

2 Abstract and Applied Analysis

at Completion (EAC) [12] However it has disadvantages toaddress the proposed problem since it is shown by simulationthat the predicting accuracy might be low due to a localminimization problemAs a consequence it is not guaranteedfor all the models to converge to optimal solutions [13]Besides ANN is also vulnerable to the selection of networkstructure and it has a high computational expense in termsof training process Extended Kalman filter (EKF) approachwhich is also popular in industrial applications is able tomaintain relatively high accuracy in terms of estimationbut EKF estimator often requires high cost due to highcomputational complexity [14] Fuzzy logic method is alsoreported to be available for estimation For more details onstate estimation and on filtering approaches for complexsystems the reader can be referred to [15ndash19]

Support vector machine (SVM) is an effective mathe-matical method in prediction and this technique has greatlydeveloped in the past decades SVM which is based onstructural risk minimization principle [20] is adopted byresearchers to address classification and regression problems[21] So far it has been successfully used in a variety of fieldsFor instance it is used for noise estimation and the predictionof air passengers [22 23] it is also utilized for image analysesbiomedicine and bioinformatics as estimator tools

Inertial navigation systems are sophisticated nonlinearsystems [24] Therefore it is unrealistic to estimate theperformance of the navigation systems by the analyses of themodels In order to predict the navigation errors by errorcoefficients a method based on support vector machineis proposed in this paper Support vector machine unlikeother traditional methods is relatively effective in termsof combating nonlinear situations It is more robust as anestimator than least-square based method because it isinsensitive to small changes [25] Firstly the error modelof SINS is established Secondly a series of flight paths thatmeet the characteristics of real trajectories are designedThen 300 sets of random error parameters which obeyGaussian distribution are generated and results of systemerrors are obtained by simulationThese parameters are usedto train the model of SVM Finally the trained model isused to predict the system errors of SINS with different errorcoefficients This method is tested to be effective and efficientby comparing simulation results with the actual navigationerrors

The remainder of this paper is organized as follows Errormodel of SINS is given and fight path for simulation isdesigned in Section 2 In Section 3 SVM-based navigationerror estimation method is proposed Simulation verificationof proposed prediction method is given and error predictionof an entire flight path is completed in Section 4 FinallySection 5 gives the conclusions

2 Error Model of SINS

Error equations in terms of velocity errors attitude errorsand position errors should be established in order to analyzethe navigation errors caused by strapdown inertial navigationsystems [2]

21 Attitude Error Equations The attitude errors of the air-craft are represented by 120575120572 120575120573 and 120575120574 respectively asfollows

120575 = minus

120575119881119873

119877

+ (

119881119864

119877

tan120593 + 120596119894119890sin120593) 120575120573

minus (

119881119864

119877

+ 120596119894119890cos120593) + 120576

119864

120575120573 =

120575119881119864

119877

minus (

119881119864

119877

+ 120596119894119890sin120593) 120575120572

minus 120596119894119890sin120593120575120593 minus

119881119873

119877

120575120574 + 120576119873

120575 120574 =

120575119881119864

119877

tan120593 + (119881119864119877

sec2120593 + 120596119894119890cos120593) 120575120593

+ (

119881119864

119877

+ 120596119894119890cos120593) 120575120572 +

119881119873

119877

120575120573 + 120576120585

(1)

where 120576119873 120576119864 and 120576

120585represent the gyroscope drift errors of

three axes

22 Velocity Error Equations With respect to the navigationsystem which is analyzed in this passage some error sourcessuch as the ones related to acceleration can be ignored whenthe coefficients are relatively small Therefore velocity errorequations are established in the following

120575119864= (

119881119864

119877

tan120593 minus119881120585

119877

)120575119881119864+ (

119881119864

119877

tan120593 + 2120596119894119890sin120593) 120575119881

119873

minus (

119881119864

119877

+ 2120596119894119890cos120593) 120575119881

120585

+ (

119881119864119881119873

119877

sec2120593 + 2120596119894119890cos120593119881

119873+ 2120596119894119890sin120593119881

120585) 120575120593

+ 119860119873120574 minus 119860

120585120573 + Δ119860

119864

120575119873= minus 2 (

119881119864

119877

tan120593 + 120596119894119890sin120593) 120575119881

119864minus

119881120585

119877

120575119881119873minus

119881119873

119877

120575119881120585

minus (

119881119864

119877

sec2120593 + 2120596119894119890cos120593)119881

119864120575120593

+ 119860120585120572 minus 119860

119864120574 + Δ119860

119873

120575120585= 2 (

119881119864

119877

+ 120596119894119890cos120593) 120575119881

119864+ 2

119881119873

119877

120575119881119873minus 2119881119864120596119894119890sin120593120575120593

+ 119860119864120573 minus 119860

119873120572 + Δ119860

120585

(2)

where Δ119860119864 Δ119860119873 and Δ119860

120585represent the accelerometer drift

errors of three axes

23 Positioning Error Equations The positioning errors arethemain factors that are considered when evaluating the per-formance of different kinds of aircrafts such as airplanes and

Abstract and Applied Analysis 3

39 40 41 42 43 44

116118

120122

100015002000250030003500

Latitude (deg)Longitude (deg)

Hei

ght (

m)

Figure 1 Flight path for analyses

UAVs Therefore the latitudinal and longitudinal errors areimportant factors for the assessment of navigation systems

120575 =

120575119881119873

119877

minus 120575ℎ

119881119873

1198772

120575120582 = (

120575119881119864

119877

+ 120575119871

119881119873

1198772tan119871) sec 119871 minus 120575ℎ119881119864sec 119871

1198772

120575ℎ = 120575119881

119880

(3)

24 Flight Path Design According to several papers thatfocus on error analysis of inertial navigation systems theresearch is mainly based on some simple flight paths such asuniform linear motion or uniform turning motion Howeverthese flight paths cannot involve all the flight modes As aconsequence the results of these simulations or analyses maynot represent the real performance of the aircrafts that areequipped with SINS

Therefore it is necessary to design some flight paths thatnot only include the characteristics of real paths of aircraftsbut are also able to ensure for each error source to be stimu-lated As shown in Figure 1 a typical flight path for simulationis designed

3 Navigation Error Prediction

It is unrealistic to use error equations to analyze the perfor-mance of SINS especially when there is a large variety ofsystems that should be tested in short term as it will cost agood deal of time to solve the differential error equations

In order to avoid complicated calculations support vectormachine with strong generalization ability is utilized topredict system errors by assessing each single error sourceHowever a noticeable problem is that the navigation errorsare time varying and closely associated with the flight pathsTherefore characteristic vectors related to certain flight pathsand ultimate positioning errors should be established toaccomplish the prediction because positioning errors are themost important data for navigation systems

31 Support Vector Regression Support vector machine(SVM) a method closely associated with optimization algo-rithms is an effectivemethodology to address data processing

problems [12] It is demonstrated to be eligible to overcomethe traditional obstacles with respect to multidimensionalproblems and over learning So far SVM is widely used inmany fields such as biological information voice recogni-tion failure identification and prognostics

SVM consists of support vector classification and sup-port vector regression To solve problems with respect toprediction support vector regression method can be used[21] Since the problem is nonlinear a transform 119909 = 120601(119909)

should be introduced By using a nonlinear mapping thatmaps the sample data into a high dimensional space 120601

119877119899rarr 119867 linear regression method can be conducted in

the high-dimensional space 119867 to accomplish the nonlinearprediction

A training set is given as

119879 = (1199091 1199101) (1199092 1199102) (119909

119897 119910119897) isin (119877

119899times 119910)119897 (4)

Due to the varying characteristics of the error parametersof inertial navigation systems a leading problem that shouldbe overcome is that all the significant parameters with respectto positioning errors should be preprocessed as characteristicquantities for estimation which is considered to be effectiveto improve the estimation accuracy [26] Therefore 15 errorparameters that have considerable impacts on position errorsof SINS are considered for the model training In (4) 119909

119894

represent the error sources of SINS which are the zero biaserrors random walk errors of gyros zero random walk ofaccelerometers random walk errors of accelerometers andscale factor errors of gyros 119910

119894represent navigation errors

which are latitudinal and longitudinal positioning errors ofthe training models Compared with ANNs SVM has adrawback that it can only generate one output while theANNs are able to generate multiple outputs [27] Conse-quently two separate training processes should be separatelyconducted for latitudinal and longitudinal errors Specificallythe error coefficients and positioning errors of the 600 setsof navigation systems are used to train the SVM modelThen a suitable kernel function is selected as radial BasisFunction (RBF) As there are many kernel functions availablefor analyses [28 29] RBF is demonstrated to be effective forthis problem by contrasting with other kernel functions

119870(119909119894 119909119895) = exp(minus

10038171003817100381710038171003817119909119894minus 119909119895

10038171003817100381710038171003817

2

21205902

) (5)

In the next step following convex quadratic program-ming problem (5) is resolved to obtain that 120572(lowast) =

(1205721 120572lowast

1 120572

119897120572lowast

119897)119879 as follows

min 12

119897

sum

119894119895=1

(120572lowast

119894minus 120572119894) (120572lowast

119894minus 120572119894)119870 (119909

119894 119909119894)

+ 120576

119897

sum

119894=1

(120572lowast

119894+ 120572119894) minus

119897

sum

119894=1

119910119894(120572lowast

119894minus 120572119894)

(6)

If 120572119894is picked and then 119887 = 119910

119894minussum119897

119894=1(120572lowast

119894minus 120572119894)119870 + 120576 if 120572lowast

119896

is picked then 119887 = 119910119896minus sum119897

119894=1(120572lowast

119894minus 120572119894)119870 minus 120576

4 Abstract and Applied Analysis

Decision function is established as

119910 = 119892 (119909) =

119897

sum

119894=1

(120572lowast

119894minus 120572119894)119870 (119909

119894 119909) = 119887 (7)

32 PSO-Based Optimal SVR Parameters Selection Impor-tant factors of SVM are the constant 119862 the accuracy parame-ter 120576 and the kernel function As the kernel function has beenselected as RBF appropriate 119862 and 120576 should be selected inorder to increase the estimation accuracy Since itmay take anextremely long time to seek the best parameters and desiredresults may not be achieved by simply making differenttests an efficient method called particle swarm algorithm isadopted for the optimization

Particle swarm algorithm (PSO) is an algorithm thatis inspired by birdsrsquo foraging behavior and widely used toaddress optimization problems [30ndash32] The PSO algorithmis introduced to optimize the parameters of 120576 and 119862 ThePSO is initialized with random particles and then it worksto find optimal parameters by iterative methods Practicallythe initial parameters of 120576 and 119862 should be given and thenthe optimal values will be generated by calculation

4 Error Analyses for SINS

Before predicting the performance of certain inertial naviga-tion systems error analyses for SINS which are indispensablefor the process of error prediction should be conducted Asfor the strapdown navigation systems studied in this paperthe drift errors of the gyroscopes the white noise of thegyroscopes and the drift errors of accelerometers are consid-ered whereas the error coefficients related to acceleration andthe coefficients related to quadratic acceleration are ignoredsince such coefficients are comparatively small

Two criteria should be obeyed for the selection of trainingdata

(1) The training data should be not the same as the datafor test If there were no differences between thetraining data and testing data the prediction accuracywould be relatively high but biased

(2) The dimension of the inputs should be increased ifpossible If the training data had a high dimensionmore useful characteristics could be used for modeltraining

(3) The training data should represent different systemswith vastly different error coefficients With respectto this criterion suitable standard deviation for errorsources should be assigned

The error sources are assigned by the given parameterswhich are listed in Table 1 The navigation system is affectedby multiple error sources when it performs a navigationtask so it is necessary to assign all the error parametersthat are considered in this navigation system in order tomake the results adaptive for real flight situations To makepreparation for the performance prediction of SINS it issensible to carry out error analyses for 600 sets with different

Table 1The expected value and standard deviation of error sources

Error sources Expectedvalue

Standarddeviation

Zero bias errors of gyros 001∘h 0006∘hRandom walk errors of gyros 0001∘radich 00068∘radichAccelerometer zero bias errors 3120583g 05 120583gAccelerometer random walk errors 3120583gradicHz 5120583gradicHzScale factor errors of gyros 10 ppm 0

600 800 1000 1200 1400 1600 1800 2000

0

500

1000

1500

2000

Real longitudinal error (m)

Real

latit

udin

al er

ror (

m)

minus1000

minus500

Figure 2 Real distributing points of positioning errors

error parameters The assignment of different error sourcesabides by the Gaussian distribution Calculating with thesystem error functions the 600 sets of navigation errors ofthe navigation systems with different error coefficients areachieved

41 Simulation Verification of Prediction Method The errorcoefficients and the positioning error parameters of the 600sets of SINS are transformed into characteristic values whichare used to train the SVM model Other 300 sets of errorcoefficients are randomly generated and the correspondingsystem errors are calculated by error equations Then theaccuracy parameter is initially given as 120576 = 05 and thepenalty parameter is given as119862 = 60 PSO is used to generateoptimal parameters of 120576 = 00104 and 119862 = 4912

Both original SVM with fixed parameters and SVMmodel with optimal parameters generated by PSO are used inorder to predict the positioning errors The real distributionpoints of positioning errors are shown in Figure 2 while thepredicting distribution points of positioning error are shownin Figure 3

The prediction errors in terms of latitude and longitudewhich are predicted by SVMmodel with optimal parametersgenerated by PSO are shown in Figure 4

By comparing the results the predicting results are rela-tively satisfying Specifically the average north error is 343mwhile the expected value of north error by prediction is4558m The standard deviation of north error by prediction

Abstract and Applied Analysis 5

Longitudinal prediction error (m)

Latit

udin

al p

redi

ctio

n er

ror (

m)

600 800 1000 1200 1400 1600 1800 2000

0

500

1000

1500

2000

minus1000

minus500

Figure 3 Prediction distributing points of positioning errors

0 50 100 150 200 250 300

0

200

400

minus200Erro

r of l

ongi

tudi

nal

pred

ictio

n (m

)

Number of SINS for test n

(a)

0 50 100 150 200 250 300

0

100

minus100

Erro

r of l

atitu

dina

lpr

edic

tion

(m)

minus200

Number of SINS for test n

(b)

Figure 4 Prediction errors generated by SVM

is 457mThe average east error is 964m while the expectedvalue of east error by prediction is 11946m The standarddeviation of east error by prediction is 1037m

The predicting accuracy is defined as 119875 which is accessedwith (7) 119890

119901represents the predicting error and 119890

119903represents

the real error calculated by system error equations Letter 119899represents the number of navigation systems for test

119875 = 1 minus

119899

sum

119894=1

10038161003816100381610038161003816100381610038161003816

119890119901119894minus 119890119903119894

119890119903119894

10038161003816100381610038161003816100381610038161003816

times

1

119899

(8)

After calculation the predicting accuracy is seen inTable 2 the latitudinal predicting accuracy is 9273 whilethe longitudinal predicting accuracy is 9164 The accuracyof PSO-based method is noticeably higher than that oforiginal SVMThere is also a substantial decrease in the timewhich is spent assessing the SINS performance Specifically

Table 2 Prediction accuracy of different methods

Method Latitudinal accuracy Longitudinal accuracyOriginal-SVM 8639 8420PSO-SVM 9273 9164

0 500 1000 1500 2000 2500 3000 3500 4000

0

200

400

600

800

1000

1200

Long

itudi

nal e

rror

(m)

Time (s)

Real longitudinal errorLongitudinal error by estimation

minus200

Figure 5 Longitudinal error of a strapdown system with knownerror parameters

if 100 systems are analyzed the error model based methodtakes 2510 s whereas the SVM-basedmethod only takes 623 sCompared with analysis method based on error model thepredicting method based on SVM is able to save up to 75of calculating time Therefore it is effective and efficient toevaluate the navigation errors of SINS

42 Error Prediction of an Entire Flight Process It is demon-strated that the accuracy of proposed method is satisfyingin terms of positioning error prediction However it is onlyavailable for the prediction with fixed flight time So it isnecessary to seek a solution to conditions with different flighttimes

One strapdown system with known error parameters isanalyzed during 3600 s flight time 100 models with respectto this system are trained by the given parameters onemodel is generated for every 36 s flight So it is achievableto predict the positioning error during the entire flight Thelongitudinal error and latitudinal error are seen in Figures5 and 6 respectively which show that at the beginning ofthe flight when the positioning errors are relatively smallthe prediction error is considerably low and can be ignoredThe reason is that during the initial period the trainingparameters of real positioning errors are small and similarAs a result SVR-based prediction method tends to generateestimation outputs around that value

Although several estimating results with big errors withrespect to both longitudinal and latitudinal estimation aregenerated the overall trends are coherent with real errorcurves and the overall prediction accuracy is satisfyingTherefore the proposed method is demonstrated to be

6 Abstract and Applied Analysis

0 500 1000 1500 2000 2500 3000 3500 4000

0

200

400

600

800

1000

Latit

udin

al er

ror (

m)

Time (s)

Real latitudinal errorLatitudinal error by estimation

minus200

Figure 6 Latitudinal error of a strapdown systemwith known errorparameters

effective for error prediction of an entire flight process withmedium accuracy

5 Conclusion

An SVM-based predicting method for SINS positioningerrors is proposed which can be used to assess the perfor-mance of navigation systems Error functions of strapdownnavigation systems are established to provide necessary errorparameters which are not only used to train SVM model butalso utilized to make comparisons with the predicting resultsof extra systems RBF is selected to be the kernel function ofSVM and appropriate parameters of SVM are generated byPSO method

As shown in the numerical verifications the proposedprediction method is effective in terms of predicting nav-igation errors of strapdown systems with different errorparameters The accuracy of latitudinal prediction can reach9273 while the accuracy with respect to longitudinalprediction is 9164 which is considered to be high enoughfor application In addition this method compared witherror model analysis can save up to 75 of calculation timeFinally the proposed method is demonstrated to be effectivefor error prediction for an entire flight process which makesthe method more applicable

Therefore it enables the researchers to choose appropriatesystems for different trajectories or applications by assessingnavigation errors efficiently Since this method is able toevaluate the positioning errors precisely by assessing errorparameters of inertial measurement units it will be usefulin terms of error compensation of strapdown navigationsystems which are equipped on different kinds of aircrafts

References

[1] Y Wu X Hu M Wu and D Hu ldquoStrapdown inertial nav-igation using dual quaternion algebra error analysisrdquo IEEE

Transactions on Aerospace and Electronic Systems vol 42 no1 pp 259ndash266 2006

[2] Y Qin Inertial Navigation Science Press Beijing China 2009[3] X He W Wang and J Huang ldquoCharacteristics of gyro

error propagation on FOG-SINSrdquo Journal of Chinese InertialTechnology vol 15 no 4 pp 407ndash411 2007

[4] J S Stambaugh ldquoPropagation and system accuracy impact ofmajor sensor errors on a strapdown aircraft navigatorrdquo IEEETransactions on Aerospace and Electronic Systems vol 9 no 6pp 838ndash846 1973

[5] F Sun Y-Y Ben and W Gao ldquoApplication of spiral theory instrapdown inertial navigation algorithmrdquo Systems Engineeringand Electronics vol 29 no 9 pp 1532ndash1535 2007

[6] Y Hao J Gong W Gao and L Li ldquoResearch on the dynamicerror of strapdown inertial navigation systemrdquo in Proceedingsof the IEEE International Conference on Mechatronics andAutomation (ICMA rsquo08) pp 814ndash819 August 2008

[7] F Gomez-Estern and F Gordillo ldquoError analysis in strapdownINS for aircraft assembly linesrdquo in Proceedings of the 10thInternational Conference on Control Automation Robotics andVision (ICARCV rsquo08) pp 184ndash189 Hanoi Vietnam December2008

[8] W Gao B Cao Y Ben and B Xu ldquoAnalysis of gyrorsquos slope driftaffecting inertial navigation system errorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 3757ndash3762 Changchun China August 2009

[9] H Musoff and J H Murphy ldquoStudy of strapdown navigationattitude algorithmsrdquo Journal of Guidance Control and Dynam-ics vol 18 no 2 pp 287ndash290 1995

[10] J Wang and H Gu ldquoCompensation algorithm of device errorfor rate strapdown inertial navigation systemrdquo in Proceedingsof the 1st International Conference on Intelligent Networks andIntelligent Systems (ICINIS rsquo08) pp 667ndash670 Wuhan ChinaNovember 2008

[11] Y-HQiao Y Liu B-K Su andMZeng ldquoTestmethod for errormodel coefficients of pendulous integrating gyro accelerometeron centrifugerdquo Journal of Astronautics vol 28 no 4 pp 854ndash931 2007

[12] T L Chen Estimate at completion for construction projectsusing evolutionary fuzzy neural inference model [MS thesis]Department of Construction Engineering National TaiwanUniversity of Science and Technology Taipei Taiwan 2008

[13] J A K Suykens J De Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 4 pp85ndash105 2002

[14] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packsmdashpart 2 modelingand identificationrdquo Journal of Power Sources vol 134 no 2 pp262ndash276 2004

[15] L Zhang ldquoHinfinestimation for discrete-time piecewise homoge-

neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009

[16] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

[17] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

Abstract and Applied Analysis 7

[18] L Zhang P Shi E-K Boukas and C Wang ldquoHinfin

modelreduction for uncertain switched linear discrete-time systemsrdquoAutomatica vol 44 no 11 pp 2944ndash2949 2008

[19] L Zhang E-K Boukas and A Haidar ldquoDelay-range-dependent control synthesis for time-delay systems withactuator saturationrdquo Automatica vol 44 no 10 pp 2691ndash26952008

[20] S R Gunn M Brown and K M Bossley ldquoNetwork perfor-mance assessment for neurofuzzy data modelingrdquo Advances inIntelligent Data Analysis Reasoning About Data vol 1280 pp313ndash323 1997

[21] T Hansen and C-J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005

[22] J Nong ldquoParameters selection and noise estimation of SVMregressionrdquo in Proceedings of the 5th International Joint Confer-ence on Computational Sciences and Optimization pp 379ndash381Harbin China 2012

[23] K-W Yan ldquoStudy on the forecast of air passenger flow basedon SVM regression algorithmrdquo in Proceedings of the 1st Inter-national Workshop on Database Technology and Applications(DBTA rsquo09) pp 325ndash328 Wuhan China April 2009

[24] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998

[25] T Hansen and C-J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005

[26] J C Alvarez Anton P J Garcia Nieto C Blanco Viejo and JA Vilan Vilan ldquoSupport vector machines used to estimate thebattery state of chargerdquo IEEE Transactions on Power Electronicsvol 28 no 12 pp 5919ndash5926 2013

[27] I Steinwart D Hush and C Scovel ldquoAn explicit descriptionof the reproducing Kernel Hilbert spaces of Gaussian RBFkernelsrdquo IEEE Transactions on Information Theory vol 52 no10 pp 4635ndash4643 2006

[28] Y-J Oyang S-C Hwang Y-Y Ou C-Y Chen and Z-WChen ldquoData classification with radial basis function networksbased on a novel kernel density estimation algorithmrdquo IEEETransactions on Neural Networks vol 16 no 1 pp 225ndash2362005

[29] G F Smits and EM Jordaan ldquoImproved SVM regression usingmixtures of kernelsrdquo in Proceedings of the International JointConference on Neural Networks (IJCNN rsquo02) pp 2785ndash2790Honolulu Hawaii USA May 2002

[30] H Wang Z Hu M Hu and Z Zhang ldquoShort-term predictionof wind farm power based on PSO-SVMrdquo in Proceedings of thePower and Energy Engineering Conference pp 1ndash4 ShanghaiChina 2012

[31] Y Bazi and F Melgani ldquoSemisupervised PSO-SVM regressionfor biophysical parameter estimationrdquo IEEE Transactions onGeoscience and Remote Sensing vol 45 no 6 pp 1887ndash18952007

[32] M Pastorino andA Randazzo ldquoThe SVM-based smart antennafor estimation of the directions of arrival of electromagneticwavesrdquo IEEE Transactions on Instrumentation and Measure-ment vol 55 no 6 pp 1918ndash1925 2006

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Stochastic AnalysisInternational Journal of