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Hindawi Publishing CorporationApplied Computational Intelligence and Soft ComputingVolume 2013 Article ID 686345 7 pageshttpdxdoiorg1011552013686345

Research ArticleA Novel Feature Extraction Method for NonintrusiveAppliance Load Monitoring

Khaled Chahine1 and Khalil El Khamlichi Drissi2

1 Department of Electrical and Electronics Engineering Lebanese International University Mazraa Beirut 146404 Lebanon2 Pascal Institute UMR 6602 24 Avenue des Landais 63177 Aubiere Cedex France

Correspondence should be addressed to Khaled Chahine khaledchahineliuedulb

Received 30 March 2013 Accepted 12 April 2013

Academic Editor Baoding Liu

Copyright copy 2013 K Chahine and K El Khamlichi Drissi This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Improving energy efficiency by monitoring household electrical consumption is of significant importance with the climate changeconcerns of the present time A solution for the electrical consumption management problem is the use of a nonintrusive applianceload monitoring (NIALM) system This system captures the signals from the aggregate consumption extracts the features fromthese signals and classifies the extracted features in order to identify the switched-on appliances This paper focuses solely onfeature extraction through applying thematrix pencil method a well-known parametric estimation technique to the drawn electriccurrentThe result is a compact representation of the current signal in terms of complex numbers referred to as poles and residuesThese complex numbers are shown to be characteristic of the considered load and can thus serve as features in any subsequentclassification module In the absence of noise simulations indicate an almost perfect agreement between theoretical and estimatedvalues of poles and residues For real data poles and residues are used to determine a feature vector consisting of the contributionof the fundamental the third and the fifth harmonic currents to the maximum of the total load current The result is a three-dimensional feature space with reduced intercluster overlap

1 Introduction

The reason behind the drive for the installation of smartmeters in homes and businesses is that they facilitate for con-sumers to monitor their energy consumption thereby mak-ing it easier for them to save energy carbon emissions andmoney Tohelp customers aswell as utilities in themonitoringprocess researchers have been studying load disaggregationschemes for more than two decades

One method of load disaggregation is distributed directsensing which requires a sensor at each device or appliancein order tomeasure consumptionThe one-sensor-per-devicerequirement is both the blessing and the curse of thismethodfor it is highly accurate but expensive To overcome the limita-tions associated with the direct sensing approach researchershave explored methods to infer disaggregated energy usagevia a single sensor Pioneering work in this area is non intru-sive appliance load monitoring (NIALM) first introduced byHart in the late 1980s [1] In contrast to the direct sensing

methods NIALM relies solely on single-point measurementsof voltage and current on the power feed entering thehousehold NIALM consists of four steps data acquisitionevent detection feature extraction and event classificationThe raw current and voltage waveforms are transformed intoa feature vector that is a more compact and meaningfulrepresentation that may include real power reactive powercurrent-voltage phase difference and harmonics (eg [2])These extracted features aremonitored for changes identifiedas events (eg an appliance turning ldquoonrdquo or ldquooffrdquo) and clas-sified down to the appliance or device category level using aclassification algorithm which usually compares the featuresto a preexisting database of signatures Several reviews offeature extraction methods for electric loads in residentialand commercial buildings can be found in the literature [3 4]

Based on the degree of nonintrusiveness the literaturedraws a distinction between manual-setup NIALM (MS-NIALM) and automatic-setupNIALM(AS-NIALM) systemsWhile the former requires switching individual appliances on

2 Applied Computational Intelligence and Soft Computing

and off manually to learn their signatures the latter sets itselfup using prior information about potential appliances AS-NIALMhence extracts the signatures and labels themwithoutany sort ofmanual interventionwhichwould greatly facilitatemass installation of smart meters To the authorsrsquo knowledgeno AS-NIALM system has hitherto been implemented It ishence the main goal of this work to pave the way for such asolution

In this paper the matrix pencil method a well-knownparametric estimation technique is applied to the electriccurrent drawn by some elementary linear and nonlinear elec-tric loads driven by a sinusoidal voltage source as well as realloadsThe result is a compact representation of the current interms of complex numbers referred to as poles and residues[5 6]These complex numbers are shown to be characteristicof the considered load and thus can serve as features for thesubsequent classification phase [7] For both synthetic andreal data results indicate that poles and residues extracted bythe MPM allow an almost perfect reconstruction of drawnelectric currents Results obtained from a database of ahousehold indicate that the extracted features succeed inreducing the intercluster overlap of different appliances

The objectives of this paper are summarized in the follow-ing two points

(1) show that the reduced number of poles and residuesestimated byMPM enable an accurate reconstructionof synthetic and real signals

(2) show that the fundamental and higher harmoniccurrents determined from poles and residues yield afeature space with reduced intercluster overlap

The rest of the paper is organized as follows Section 2presents the signal model and the principle of the MPMSections 3 and 4 show the validation on simulated and realdata respectively Finally Section 5 provides the summaryand conclusion

2 Feature Extraction

21 Signal Model For a sinusoidal driving voltage of theform V(119905) = 119881radic2 sin(120596119905) the drawn electric current canbe modeled as a linear combination of 119889 cisoids (complex-valued sinusoidal signals) weighted by complex residuesaccording to the following signal model

119894 (119905) asymp

119889

sum119898=1

119903119898 exp (120572119898 + 1198952120587119891119898) 119905 + 119887 (119905) (1)

where 119903119898 is the residue of the119898th cisoid 120572119898 is its attenuationfactor 119891119898 is its frequency and 119887(119905) is additive white Gaussiannoise After sampling the time variable 119905 is replaced by 119905119896 =119896119905119904 where 119905119904 = 625times10

minus4 is the chosen sampling periodThediscrete current signal becomes

119894 (119896) asymp

119889

sum119898=1

119903119898119911119896

119898+ 119887 (119896) 119896 = 1 2 119873 (2)

where

119911119898 = exp (120572119898 + 1198952120587119891119898) 119905119904 119898 = 1 2 119889 (3)

is the119898th complex pole Undermatrix form the signalmodelis expressed by

i = Ar + b (4)

with the following notational definitions

i = [119894 (1) 119894 (2) 119894 (119873)]119879

A = [a1 a2 a119889]

a119898 = [119911119898 1199112

119898 119911119873

119898]119879

r = [1199031 1199032 119903119889]119879

b = [119887(1) 119887(2) 119887(119873)]119879

(5)

The superscript 119879 denotes the transpose operatorThe feature extraction problem can now be stated as fol-

lows Given the electric current data sequence 119894(119896)119873119896=1

usea feature extraction method to extract the complex poles119911119898119889

119898=1and residues 119903119898

119889

119898=1of the load

22Matrix Pencil Method (MPM) This section briefly recallsthe principle ofMPMwhich is a linear predictionmethod tai-lored to the parameter estimation of the dampedundampedexponential model Starting from the signal model given in(1) MPM chooses a free parameter 119871 known as the pencilparameter such as 119889 le 119871 le 119873 minus 119889 The proper choice of 119871results in significant robustness against noise The next stepis to construct a Hankel data matrix

H =

[[[[

[

119894 (1) 119894 (2) sdot sdot sdot 119894 (119871 + 1)

119894 (2) 119894 (3) sdot sdot sdot 119894 (119871 + 2)

d

119894 (119873 minus 119871) 119894 (119873 minus 119871 + 1) sdot sdot sdot 119894 (119873)

]]]]

]

(6)

Two matrices are then obtained by removing the last andfirst columns of H In MATLAB notation they are given asfollows

Hrarr= H ( 1 119871)

Hlarr= H ( 2 119871 + 1)

(7)

The matrix pencil for the two matrices Hrarr

andHlarris defined as

their linear combination Hlarrminus 120582Hrarr with 120582 a scalar parameter

In the absence of noise and owing to the assumed signalmodel it is easily verified that H

rarrand Hlarradmit the following

Vandermonde decomposition

Hrarr= Z1RZ2

Hlarr= Z1RZ0Z2

(8)

Applied Computational Intelligence and Soft Computing 3

where

Z1 =[[[[

[

1199111 1199112 sdot sdot sdot 11991111988911991121

11991122

sdot sdot sdot 1199112119889

d

119911119873minus1198711

119911119873minus1198712

sdot sdot sdot 119911119873minus119871119889

]]]]

]

Z2 =[[[[

[

1 1199111 sdot sdot sdot 119911119871minus1

1


2

d

1 119911119889 sdot sdot sdot 119911

119871minus1

119889

]]]]

]

Z0 = diag 1199111 1199112 119911119889

R = diag 1199031 1199032 119903119889

(9)

revealing the fundamental shift-invariance property in thecolumn and row spacesThematrix pencil can then bewrittenas

Hlarrminus 120582Hrarr= Z1R [Z0 minus 120582I]Z2 (10)

where I is the identitymatrix Hence each value of 120582 = 119911119898 is arank-reducing number of the pencil The estimates of 119911119898 aretherefore the generalized eigenvalues (GEVs) of the matrixpair [Hlarr H997888rarr]

Once the complex poles 119911119898119889

119898=1are determined the

complex residues can be estimated using a least squares fithaving the following solution

r = (A119867A)minus1A119867i (11)

For noisy data total least squares matrix pencil method(TLSMPM) is usually preferred in which the singular valuedecomposition is used to prefilter the complex signals andthen conventional procedures follow For more details thereader can refer to [8]

3 Validation on Synthetic Data

31 Linear Loads To validate MPM as a feature extractionmethod we shall first compare its poles and residues withthose obtained from the theoretical expressions of the follow-ing linear elementary loads series RC series RL parallel RLand series RLCThe RC and RL circuits lead to first order dif-ferential equations in time whereas the RLC circuit leads to asecond-order differential equation Using Eulerrsquos formula andrearranging allow rewriting the current expression obtainedfrom the solution of the differential equation characterizingthe load in the form of (1) The poles and residues of eachelementary load can then be readily identified Tables 1 2 3and 4 give the residues attenuation factors and frequenciesof the four studied elementary loads As can be seen fromthese tables first-order circuits (RL andRC) are characterizedby two pure imaginary conjugate poles representing theirforced response and one real pole representing their natu-ral response whereas the second-order circuit (RLC) hasbesides the two pure imaginary conjugate poles of its forced

Table 1 The residues attenuation factors and frequencies of theseries RC load

119898 119903119898 120572119898 119891119898 (Hz)

1 119881radic2

2119895119877cos(120601)119890minus119895120601 0 +50

2 minus119881radic2

2119895119877cos(120601)119890119895120601 0 minus50

3 minus1

119877(V1198620 minus 119881

radic2 sin(120601) cos(1205961199050 minus 120601)) 1198901199050120591 minus

1

1205910

Table 2 The residues attenuation factors and frequencies of theseries RL load

119898 119903119898 120572119898 119891119898 (Hz)

1 119881radic2

2119895119877cos(120601)119890minus119895120601 0 +50

2 minus119881radic2

2119895119877cos(120601)119890119895120601 0 minus50

3 1198941198710 minus119881radic2

119877cos(120601) sin(1205961199050 minus 120601)119890

1199050120591 minus1

1205910

Table 3 The residues attenuation factors and frequencies of theparallel RL load

119898 119903119898 120572119898 119891119898 (Hz)

1 119881radic2

2119895119877 cos(120601)119890minus119895120601 0 +50

2 minus119881radic2

2119895119877 cos(120601)119890119895120601 0 minus50

3 1198941198710minus

119881radic2

119877 cos(120601)sin(120596119905

0minus 120601) 0 0

Table 4 The residues attenuation factors and frequencies of theseries RLC load

119898 119903119898 120572119898 119891119898 (Hz)

1 119881radic2

2119895119877cos(120601)119890minus119895120601 0 +50

2 minus119881radic2

2119895119877cos(120601)119890119895120601 0 minus50

3 119860119890minus119896112059601199050 11989611205960 0

4 119861119890minus119896212059601199050 11989621205960 0

response two conjugate complex poles related to its naturalresponse The expressions of the dependent parameters aregiven in the appendix

32 Nonlinear Loads A nonlinear load is one for which therelationship between the current through the load and thevoltage across the load is a nonlinear function A simple viewof the nature of nonlinear loads can be presented usingOhmrsquosLaw which states that the voltage is the product of the loadresistance and the current (119881 = 119877119868) For a linear load theresistance (119877) is a constant for a nonlinear load the resistancevaries When AC power is supplied to a nonlinear load


Table 5 Current composition of a nonlinear load

1198681

1198685

1198687

11986811

11986813

100 189 11 59 48

the result is the creation of currents that do not oscillate atthe supply frequency These currents are called harmonicsHarmonics occur at multiples of the supply (fundamental)frequency For instance if the fundamental frequency is50Hz the so-called second harmonic is 100Hz the thirdharmonic is 150Hz and so on Any number of harmonicscan be created by a particular piece of equipment dependingon that equipmentrsquos electrical characteristics Therefore thecurrent drawn by nonlinear loads can still be representedby (1) where harmonics appear in the form of pole-residuecouples at frequency multiples of 50Hz

33 Results Assuming zero initial conditions (1198941198710 = 0 andorV1198620 = 0) the following numerical values were used todetermine the electric current data sequence from whichMPM extracted poles and residues 119877 = 100Ω 119862 = 01mFfor the series RC circuit 119877 = 10Ω 119871 = 100mH for boththe series and parallel RL circuits and 119877 = 1Ω 119871 = 20mH119862 = 60mF for the series RLC circuit A duration of tenperiods or 02 seconds was chosen for the current which at119905119904 = 625 times 10

minus4 is equivalent to 320 samples and MPM wasapplied at each period Figures 1 2 3 and 4 show the currentobtained from the analytic expression of poles and residuesin the tables above and its reconstruction obtained from thepoles and residues extracted by MPM An almost perfectagreement can be seen between the two curves indicating theaccuracy of the characteristic complex numbers extracted byMPM In addition the figures show the forced and naturalresponses of each of the four elementary circuits

To evaluate the performance ofMPM on nonlinear loadswe considered the current shown in Table 5 It consists ofa fundamental and four harmonics and hence can be rep-resented by ten pairwise complex conjugate pole-residuecouples We then used MPM to extract these ten coupleswhich served to reconstruct the current as shown in Figure 5As can be seen MPM is successful in estimating the pole-residue couples of the load

4 Validation on Real Data

41 Reconstruction Results In this section the validation ofMPM is carried out on currents of three representative loadsa television set a vacuum cleaner and an economy lampAs for the case of synthetic data MPM was applied at eachperiod Figures 6 7 and 8 show the current drawn by theappliances and its reconstruction based on the pole-residueestimates of MPMThe close agreement shown in the figuresindicate that the exponential model of (1) and its parametersestimated by the MPM accurately predict the response ofthe actual loads It is worth mentioning that the number ofpole-residue couples 119889 increases with the nonlinearity of theload For instance the current of the vacuum cleaner couldbe accurately reconstructed from four pole-residue couples

0 002 004 006 008 01 012 014 016 018 02

40

30

20

10

0

minus10

minus20

minus30

minus40

Time (s)

Curr

ent (

A)

Analytic currentReconstructed current

Forced responseNatural response

Figure 1 The analytic and reconstructed currents of the series RCcircuit along with its forced and natural responses

0 002 004 006 008 01 012 014 016 018 02

150

100

50

0

minus50

minus100

Time (s)

Curr

ent (

A)



Figure 2 The analytic and reconstructed currents of the series RLcircuit along with its forced and natural responses

0 002 004 006 008 01 012 014 016 018 02

500

400

300

200

100

0

minus100

minus200

minus300

minus400

Time (s)

Curr

ent (

A)



Figure 3The analytic and reconstructed currents of the parallel RLcircuit along with its forced and natural responses


0 002 004 006 008 01 012 014 016 018 02

Time (s)

Curr

ent (

A)

100

80

60

40

20

0

minus20

minus40

minus60



Figure 4The analytic and reconstructed currents of the series RLCcircuit along with its forced and natural responses

0 002 004 006 008 01 012 014 016 018 02

minus15

minus1

minus05

0

05

1

15

Time (s)

Curr

ent (

A)


Figure 5 The analytic and reconstructed currents of a nonlinearload The inset zooms in on a half period of the drawn current inorder to show the accuracy of reconstruction

whereas that of the economy lamp needed up to twelvecouples

42 Feature Space The feature space contains 900 signaturesuniformly distributed among the following nine appliancesincandescent lamp halogen lamp economy lamp waterheater electric convector oven two-burner hot plate tele-vision set and computer As shown in Figure 9 each sig-nature (represented by a point in the the three-dimensionalfeature space) is characterized by three pole-residue productscorresponding to the maxima of the fundamental third andfifth harmonic currents The restriction to three frequencieshas the sole aim of representing the feature space graphicallyFrom the feature space ten clusters representing the nineappliances can be clearly distinguishedThe additional clusteris due to the two-burner hot plate which is represented by twoclusters one for each burner It can hence be concluded that

0 002 004 006 008 01 012 014 016 018 02

Time (s)

Curr

ent (

A)

Measured currentReconstructed current

minus15

minus10

minus5

0

5

10

15

Figure 6Themeasured and reconstructed currents of the televisionset

0 002 004 006 008 01 012 014 016 018 02

Time (s)


40

30

20

10

0

minus10

minus20

minus30

minus40

Curr

ent (

A)

Figure 7 The measured and reconstructed currents of the vacuumcleaner

the studied appliances can be fairly distinguished using thefundamental and higher harmonics

5 Conclusion

This paper presented a novel feature extraction method fornon intrusive appliance load monitoring First the polesand residues estimated by the matrix pencil method wereshown to enable accurate reconstruction of synthetic and realcurrent signals Second these complex numbers were usedto determine a three-dimensional feature space with reducedintercluster overlap Future research will make use of theextracted features for the classification phase

Appendix

The dependent parameters of first-order circuits are given inTable 6 where 120591 is the time constant and 120601 is the phase angle


0 002 004 006 008 01 012 014 016 018 02

Time (s)

Curr

ent (

A)


minus10

minus8

minus6

minus4

minus2

0

2

4

Figure 8Themeasured and reconstructed currents of the economylamp

05

1015

20

0

Fundamental (A)

06

04

02

0

minus02

06

04

02

minus02

Third harmonic (A)

TelevisionEconomy lampComputerIncandescent lampHalogen lamp

Hotplate 1 burnerElectric convectorOvenHotplate 2 burnersWater heater

Fifth

har

mon

ic (A

)

Figure 9The feature space showing the disaggregated contributionof the fundamental and harmonic currents to the maximum of thetotal current for several appliances

In addition to the phase angle 120601 the dependent param-eters of the series RLC circuit include the resonance angularfrequency 1205960 and the damping factor 120585

120601 = arctan [ 1119877(119871120596 minus

1

119862120596)] [rad]

1205960 =1

radic119871119862[rads]

120585 =119877

2radic119862

119871[Np]

(A1)

Table 6 Dependent parameters of first-order circuits

Load 120591 [s] 120601 [rad]Series RC 119877119862 arctan( minus1

119877119862120596)

Series RL 119871

119877arctan(119871120596

119877)

Parallel RL mdash arctan( 119877119871120596)

The roots of the characteristic equation of the second-orderdifferential equation 1198961 and 1198962 are expressed in terms of thethe damping factor 120585 as follows

1198961 = minus120585 minus radic1205852 minus 1

1198962 = minus120585 + radic1205852 minus 1

(A2)

Finally 119860 and 119861 shown in Table 4 are expressed as follows

119860 =1198961

1198961 minus 1198962(1198602 minus 11989621198601)

119861 =1198962

1198962 minus 1198961(1198602 minus 11989611198601)

(A3)

where

1198601 = V1198880119862

119871+119881radic2

119877cos (120601) cos (1205961199050 minus 120601)

1205960

120596

1198602 = 1198941198710 minus119881radic2

119877cos (120601) sin (1205961199050 minus 120601)

(A4)

Acknowledgment

This work was supported in part by Landis+Gyr

References

[1] GWHart ldquoNonintrusive appliance loadmonitoringrdquo Proceed-ings of the IEEE vol 80 no 12 pp 1870ndash1891 1992

[2] C Laughman K Lee R Cox et al ldquoPower signature analysisrdquoIEEE Power and Energy Magazine vol 1 no 2 pp 56ndash63 2003

[3] Y Du L Du B Lu RHarley and THabetler ldquoA review of iden-tification and monitoring methods for electric loads in com-mercial and residential buildingsrdquo in Proceedings of the 2ndIEEEEnergyConversionCongress andExposition (ECCE rsquo10) pp4527ndash4533 Atlanta Ga USA September 2010

[4] M Zeifman and K Roth ldquoNonintrusive appliance load mon-itoring review and outlookrdquo IEEE Transactions on ConsumerElectronics vol 57 no 1 pp 76ndash84 2011

[5] H Najmeddine K E K Drissi C Pasquier et al ldquoSmartMeter-ing by using matrix pencilrdquo in Proceedings of the 9th Interna-tional Conference on Environment and Electrical Engineering(EEEIC rsquo10) pp 238ndash241 Prague Czech Republic May 2010

[6] H Najmeddine K El Khamlichi Drissi A Diop and T Jouan-net ldquoMethod and device for the non-intrusive determination ofthe electrical power consumed by an installation by analysingload transientsrdquo French Patent FR 0856717 October 2008


[7] K Chahine K El Khamlichi Drissi C Pasquier et al ldquoElectricload disaggregation in smart metering using a novel featureextraction method and supervised classificationrdquo Energy Pro-cedia vol 6 pp 627ndash632 2011

[8] Y Hua and T K Sarkar ldquoMatrix pencil method for estimatingparameters of exponentially dampedundamped sinusoids innoiserdquo IEEE Transactions on Acoustics Speech and Signal Pro-cessing vol 38 no 5 pp 814ndash824 1990

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and off manually to learn their signatures the latter sets itselfup using prior information about potential appliances AS-NIALMhence extracts the signatures and labels themwithoutany sort ofmanual interventionwhichwould greatly facilitatemass installation of smart meters To the authorsrsquo knowledgeno AS-NIALM system has hitherto been implemented It ishence the main goal of this work to pave the way for such asolution

In this paper the matrix pencil method a well-knownparametric estimation technique is applied to the electriccurrent drawn by some elementary linear and nonlinear elec-tric loads driven by a sinusoidal voltage source as well as realloadsThe result is a compact representation of the current interms of complex numbers referred to as poles and residues[5 6]These complex numbers are shown to be characteristicof the considered load and thus can serve as features for thesubsequent classification phase [7] For both synthetic andreal data results indicate that poles and residues extracted bythe MPM allow an almost perfect reconstruction of drawnelectric currents Results obtained from a database of ahousehold indicate that the extracted features succeed inreducing the intercluster overlap of different appliances

The objectives of this paper are summarized in the follow-ing two points

(1) show that the reduced number of poles and residuesestimated byMPM enable an accurate reconstructionof synthetic and real signals

(2) show that the fundamental and higher harmoniccurrents determined from poles and residues yield afeature space with reduced intercluster overlap

The rest of the paper is organized as follows Section 2presents the signal model and the principle of the MPMSections 3 and 4 show the validation on simulated and realdata respectively Finally Section 5 provides the summaryand conclusion

2 Feature Extraction

21 Signal Model For a sinusoidal driving voltage of theform V(119905) = 119881radic2 sin(120596119905) the drawn electric current canbe modeled as a linear combination of 119889 cisoids (complex-valued sinusoidal signals) weighted by complex residuesaccording to the following signal model

119894 (119905) asymp

119889

sum119898=1

119903119898 exp (120572119898 + 1198952120587119891119898) 119905 + 119887 (119905) (1)

where 119903119898 is the residue of the119898th cisoid 120572119898 is its attenuationfactor 119891119898 is its frequency and 119887(119905) is additive white Gaussiannoise After sampling the time variable 119905 is replaced by 119905119896 =119896119905119904 where 119905119904 = 625times10

minus4 is the chosen sampling periodThediscrete current signal becomes

119894 (119896) asymp

119889

sum119898=1

119903119898119911119896

119898+ 119887 (119896) 119896 = 1 2 119873 (2)

where

119911119898 = exp (120572119898 + 1198952120587119891119898) 119905119904 119898 = 1 2 119889 (3)

is the119898th complex pole Undermatrix form the signalmodelis expressed by

i = Ar + b (4)

with the following notational definitions

i = [119894 (1) 119894 (2) 119894 (119873)]119879

A = [a1 a2 a119889]

a119898 = [119911119898 1199112

119898 119911119873

119898]119879

r = [1199031 1199032 119903119889]119879

b = [119887(1) 119887(2) 119887(119873)]119879

(5)

The superscript 119879 denotes the transpose operatorThe feature extraction problem can now be stated as fol-

lows Given the electric current data sequence 119894(119896)119873119896=1

usea feature extraction method to extract the complex poles119911119898119889

119898=1and residues 119903119898

119889

119898=1of the load

22Matrix Pencil Method (MPM) This section briefly recallsthe principle ofMPMwhich is a linear predictionmethod tai-lored to the parameter estimation of the dampedundampedexponential model Starting from the signal model given in(1) MPM chooses a free parameter 119871 known as the pencilparameter such as 119889 le 119871 le 119873 minus 119889 The proper choice of 119871results in significant robustness against noise The next stepis to construct a Hankel data matrix

H =

[[[[

[

119894 (1) 119894 (2) sdot sdot sdot 119894 (119871 + 1)

119894 (2) 119894 (3) sdot sdot sdot 119894 (119871 + 2)

d

119894 (119873 minus 119871) 119894 (119873 minus 119871 + 1) sdot sdot sdot 119894 (119873)

]]]]

]

(6)

Two matrices are then obtained by removing the last andfirst columns of H In MATLAB notation they are given asfollows

Hrarr= H ( 1 119871)

Hlarr= H ( 2 119871 + 1)

(7)

The matrix pencil for the two matrices Hrarr

andHlarris defined as

their linear combination Hlarrminus 120582Hrarr with 120582 a scalar parameter

In the absence of noise and owing to the assumed signalmodel it is easily verified that H

rarrand Hlarradmit the following

Vandermonde decomposition

Hrarr= Z1RZ2

Hlarr= Z1RZ0Z2

(8)


where

Z1 =[[[[

[

1199111 1199112 sdot sdot sdot 11991111988911991121

11991122


d

119911119873minus1198711

119911119873minus1198712


]]]]

]

Z2 =[[[[

[


1


2

d

1 119911119889 sdot sdot sdot 119911

119871minus1

119889

]]]]

]

Z0 = diag 1199111 1199112 119911119889

R = diag 1199031 1199032 119903119889

(9)







r = (A119867A)minus1A119867i (11)





119898 119903119898 120572119898 119891119898 (Hz)

1 119881radic2

2119895119877cos(120601)119890minus119895120601 0 +50

2 minus119881radic2

2119895119877cos(120601)119890119895120601 0 minus50

3 minus1

119877(V1198620 minus 119881


1

1205910


119898 119903119898 120572119898 119891119898 (Hz)

1 119881radic2

2119895119877cos(120601)119890minus119895120601 0 +50

2 minus119881radic2

2119895119877cos(120601)119890119895120601 0 minus50

3 1198941198710 minus119881radic2

119877cos(120601) sin(1205961199050 minus 120601)119890

1199050120591 minus1

1205910


119898 119903119898 120572119898 119891119898 (Hz)

1 119881radic2

2119895119877 cos(120601)119890minus119895120601 0 +50

2 minus119881radic2

2119895119877 cos(120601)119890119895120601 0 minus50

3 1198941198710minus

119881radic2

119877 cos(120601)sin(120596119905

0minus 120601) 0 0


119898 119903119898 120572119898 119891119898 (Hz)

1 119881radic2

2119895119877cos(120601)119890minus119895120601 0 +50

2 minus119881radic2

2119895119877cos(120601)119890119895120601 0 minus50

3 119860119890minus119896112059601199050 11989611205960 0

4 119861119890minus119896212059601199050 11989621205960 0





1198681

1198685

1198687

11986811

11986813

100 189 11 59 48







0 002 004 006 008 01 012 014 016 018 02

40

30

20

10

0

minus10

minus20

minus30

minus40

Time (s)

Curr

ent (

A)




0 002 004 006 008 01 012 014 016 018 02

150

100

50

0

minus50

minus100

Time (s)

Curr

ent (

A)




0 002 004 006 008 01 012 014 016 018 02

500

400

300

200

100

0

minus100

minus200

minus300

minus400

Time (s)

Curr

ent (

A)





0 002 004 006 008 01 012 014 016 018 02

Time (s)

Curr

ent (

A)

100

80

60

40

20

0

minus20

minus40

minus60




0 002 004 006 008 01 012 014 016 018 02

minus15

minus1

minus05

0

05

1

15

Time (s)

Curr

ent (

A)





0 002 004 006 008 01 012 014 016 018 02

Time (s)

Curr

ent (

A)


minus15

minus10

minus5

0

5

10

15


0 002 004 006 008 01 012 014 016 018 02

Time (s)


40

30

20

10

0

minus10

minus20

minus30

minus40

Curr

ent (

A)



5 Conclusion


Appendix



0 002 004 006 008 01 012 014 016 018 02

Time (s)

Curr

ent (

A)


minus10

minus8

minus6

minus4

minus2

0

2

4


05

1015

20

0

Fundamental (A)

06

04

02

0

minus02

06

04

02

minus02

Third harmonic (A)



Fifth

har

mon

ic (A

)



120601 = arctan [ 1119877(119871120596 minus

1

119862120596)] [rad]

1205960 =1

radic119871119862[rads]

120585 =119877

2radic119862

119871[Np]

(A1)



119877119862120596)

Series RL 119871

119877arctan(119871120596

119877)





(A2)


119860 =1198961

1198961 minus 1198962(1198602 minus 11989621198601)

119861 =1198962

1198962 minus 1198961(1198602 minus 11989611198601)

(A3)

where

1198601 = V1198880119862

119871+119881radic2

119877cos (120601) cos (1205961199050 minus 120601)

1205960

120596

1198602 = 1198941198710 minus119881radic2

119877cos (120601) sin (1205961199050 minus 120601)

(A4)

Acknowledgment


References

















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




where

Z1 =[[[[

[

1199111 1199112 sdot sdot sdot 11991111988911991121

11991122


d

119911119873minus1198711

119911119873minus1198712


]]]]

]

Z2 =[[[[

[


1


2

d

1 119911119889 sdot sdot sdot 119911

119871minus1

119889

]]]]

]

Z0 = diag 1199111 1199112 119911119889

R = diag 1199031 1199032 119903119889

(9)







r = (A119867A)minus1A119867i (11)





119898 119903119898 120572119898 119891119898 (Hz)

1 119881radic2

2119895119877cos(120601)119890minus119895120601 0 +50

2 minus119881radic2

2119895119877cos(120601)119890119895120601 0 minus50

3 minus1

119877(V1198620 minus 119881


1

1205910


119898 119903119898 120572119898 119891119898 (Hz)

1 119881radic2

2119895119877cos(120601)119890minus119895120601 0 +50

2 minus119881radic2

2119895119877cos(120601)119890119895120601 0 minus50

3 1198941198710 minus119881radic2

119877cos(120601) sin(1205961199050 minus 120601)119890

1199050120591 minus1

1205910


119898 119903119898 120572119898 119891119898 (Hz)

1 119881radic2

2119895119877 cos(120601)119890minus119895120601 0 +50

2 minus119881radic2

2119895119877 cos(120601)119890119895120601 0 minus50

3 1198941198710minus

119881radic2

119877 cos(120601)sin(120596119905

0minus 120601) 0 0


119898 119903119898 120572119898 119891119898 (Hz)

1 119881radic2

2119895119877cos(120601)119890minus119895120601 0 +50

2 minus119881radic2

2119895119877cos(120601)119890119895120601 0 minus50

3 119860119890minus119896112059601199050 11989611205960 0

4 119861119890minus119896212059601199050 11989621205960 0





1198681

1198685

1198687

11986811

11986813

100 189 11 59 48







0 002 004 006 008 01 012 014 016 018 02

40

30

20

10

0

minus10

minus20

minus30

minus40

Time (s)

Curr

ent (

A)




0 002 004 006 008 01 012 014 016 018 02

150

100

50

0

minus50

minus100

Time (s)

Curr

ent (

A)




0 002 004 006 008 01 012 014 016 018 02

500

400

300

200

100

0

minus100

minus200

minus300

minus400

Time (s)

Curr

ent (

A)





0 002 004 006 008 01 012 014 016 018 02

Time (s)

Curr

ent (

A)

100

80

60

40

20

0

minus20

minus40

minus60




0 002 004 006 008 01 012 014 016 018 02

minus15

minus1

minus05

0

05

1

15

Time (s)

Curr

ent (

A)





0 002 004 006 008 01 012 014 016 018 02

Time (s)

Curr

ent (

A)


minus15

minus10

minus5

0

5

10

15


0 002 004 006 008 01 012 014 016 018 02

Time (s)


40

30

20

10

0

minus10

minus20

minus30

minus40

Curr

ent (

A)



5 Conclusion


Appendix



0 002 004 006 008 01 012 014 016 018 02

Time (s)

Curr

ent (

A)


minus10

minus8

minus6

minus4

minus2

0

2

4


05

1015

20

0

Fundamental (A)

06

04

02

0

minus02

06

04

02

minus02

Third harmonic (A)



Fifth

har

mon

ic (A

)



120601 = arctan [ 1119877(119871120596 minus

1

119862120596)] [rad]

1205960 =1

radic119871119862[rads]

120585 =119877

2radic119862

119871[Np]

(A1)



119877119862120596)

Series RL 119871

119877arctan(119871120596

119877)





(A2)


119860 =1198961

1198961 minus 1198962(1198602 minus 11989621198601)

119861 =1198962

1198962 minus 1198961(1198602 minus 11989611198601)

(A3)

where

1198601 = V1198880119862

119871+119881radic2

119877cos (120601) cos (1205961199050 minus 120601)

1205960

120596

1198602 = 1198941198710 minus119881radic2

119877cos (120601) sin (1205961199050 minus 120601)

(A4)

Acknowledgment


References

















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





1198681

1198685

1198687

11986811

11986813

100 189 11 59 48







0 002 004 006 008 01 012 014 016 018 02

40

30

20

10

0

minus10

minus20

minus30

minus40

Time (s)

Curr

ent (

A)




0 002 004 006 008 01 012 014 016 018 02

150

100

50

0

minus50

minus100

Time (s)

Curr

ent (

A)




0 002 004 006 008 01 012 014 016 018 02

500

400

300

200

100

0

minus100

minus200

minus300

minus400

Time (s)

Curr

ent (

A)





0 002 004 006 008 01 012 014 016 018 02

Time (s)

Curr

ent (

A)

100

80

60

40

20

0

minus20

minus40

minus60




0 002 004 006 008 01 012 014 016 018 02

minus15

minus1

minus05

0

05

1

15

Time (s)

Curr

ent (

A)





0 002 004 006 008 01 012 014 016 018 02

Time (s)

Curr

ent (

A)


minus15

minus10

minus5

0

5

10

15


0 002 004 006 008 01 012 014 016 018 02

Time (s)


40

30

20

10

0

minus10

minus20

minus30

minus40

Curr

ent (

A)



5 Conclusion


Appendix



0 002 004 006 008 01 012 014 016 018 02

Time (s)

Curr

ent (

A)


minus10

minus8

minus6

minus4

minus2

0

2

4


05

1015

20

0

Fundamental (A)

06

04

02

0

minus02

06

04

02

minus02

Third harmonic (A)



Fifth

har

mon

ic (A

)



120601 = arctan [ 1119877(119871120596 minus

1

119862120596)] [rad]

1205960 =1

radic119871119862[rads]

120585 =119877

2radic119862

119871[Np]

(A1)



119877119862120596)

Series RL 119871

119877arctan(119871120596

119877)





(A2)


119860 =1198961

1198961 minus 1198962(1198602 minus 11989621198601)

119861 =1198962

1198962 minus 1198961(1198602 minus 11989611198601)

(A3)

where

1198601 = V1198880119862

119871+119881radic2

119877cos (120601) cos (1205961199050 minus 120601)

1205960

120596

1198602 = 1198941198710 minus119881radic2

119877cos (120601) sin (1205961199050 minus 120601)

(A4)

Acknowledgment


References

















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0 002 004 006 008 01 012 014 016 018 02

Time (s)

Curr

ent (

A)

100

80

60

40

20

0

minus20

minus40

minus60




0 002 004 006 008 01 012 014 016 018 02

minus15

minus1

minus05

0

05

1

15

Time (s)

Curr

ent (

A)





0 002 004 006 008 01 012 014 016 018 02

Time (s)

Curr

ent (

A)


minus15

minus10

minus5

0

5

10

15


0 002 004 006 008 01 012 014 016 018 02

Time (s)


40

30

20

10

0

minus10

minus20

minus30

minus40

Curr

ent (

A)



5 Conclusion


Appendix



0 002 004 006 008 01 012 014 016 018 02

Time (s)

Curr

ent (

A)


minus10

minus8

minus6

minus4

minus2

0

2

4


05

1015

20

0

Fundamental (A)

06

04

02

0

minus02

06

04

02

minus02

Third harmonic (A)



Fifth

har

mon

ic (A

)



120601 = arctan [ 1119877(119871120596 minus

1

119862120596)] [rad]

1205960 =1

radic119871119862[rads]

120585 =119877

2radic119862

119871[Np]

(A1)



119877119862120596)

Series RL 119871

119877arctan(119871120596

119877)





(A2)


119860 =1198961

1198961 minus 1198962(1198602 minus 11989621198601)

119861 =1198962

1198962 minus 1198961(1198602 minus 11989611198601)

(A3)

where

1198601 = V1198880119862

119871+119881radic2

119877cos (120601) cos (1205961199050 minus 120601)

1205960

120596

1198602 = 1198941198710 minus119881radic2

119877cos (120601) sin (1205961199050 minus 120601)

(A4)

Acknowledgment


References

















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0 002 004 006 008 01 012 014 016 018 02

Time (s)

Curr

ent (

A)


minus10

minus8

minus6

minus4

minus2

0

2

4


05

1015

20

0

Fundamental (A)

06

04

02

0

minus02

06

04

02

minus02

Third harmonic (A)



Fifth

har

mon

ic (A

)



120601 = arctan [ 1119877(119871120596 minus

1

119862120596)] [rad]

1205960 =1

radic119871119862[rads]

120585 =119877

2radic119862

119871[Np]

(A1)



119877119862120596)

Series RL 119871

119877arctan(119871120596

119877)





(A2)


119860 =1198961

1198961 minus 1198962(1198602 minus 11989621198601)

119861 =1198962

1198962 minus 1198961(1198602 minus 11989611198601)

(A3)

where

1198601 = V1198880119862

119871+119881radic2

119877cos (120601) cos (1205961199050 minus 120601)

1205960

120596

1198602 = 1198941198710 minus119881radic2

119877cos (120601) sin (1205961199050 minus 120601)

(A4)

Acknowledgment


References

















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in













Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Documents

Research Article A Novel Feature Extraction Method for ...downloads.hindawi.com/journals/acisc/2013/686345.pdf · Research Article A Novel Feature Extraction Method for Nonintrusive