Research ArticleHigher-Order Sliding Mode Observer for Speed and PositionEstimation in PMSM

Suneel K Kommuri1 Kalyana C Veluvolu1 M Defoort2 and Yeng C Soh3

1 School of Electronics Engineering Kyungpook National University Daegu 702-701 Republic of Korea2 LAMIH CNRS UMR 8201 Universite Lille Nord de France and UVHC 59313 Valenciennes France3 School of Electrical and Electronics Engineering Nanyang Technological University Nanyang Avenue Singapore 639798

Correspondence should be addressed to Kalyana C Veluvolu veluvolueeknuackr

Received 17 March 2014 Accepted 26 April 2014 Published 27 May 2014

Academic Editor Her-Terng Yau

Copyright copy 2014 Suneel K Kommuri et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents a speed and position estimation method for the permanent magnet synchronous motor (PMSM) based onhigher-order sliding mode (HOSM) observer The back electromotive forces (EMFs) in the PMSM are treated as unknown inputsand are estimated with the HOSM observer without the need of low-pass filter and phase compensation modules With theestimation of back EMFs an accurate estimation of speed and rotor position can be obtained Further the proposed methodcompletely eliminates chattering Experimental results with a 26W three-phase PMSM demonstrate the effectiveness of theproposed method

1 Introduction

The permanent magnet synchronous motor (PMSM) hashigh efficiency high torque to inertia ratio and high powerdensity and hence it is popular for high performance motioncontrol applications Rotor speed estimation in a sensorlessPMSM has been extensively studied [1ndash3] and is mandatoryif speed control that is speed as a feedback is employedField oriented control has received a lot of attention [3] incontrolling the high performance PMSM drives The objec-tive is to control the stator currents represented by a vector toobtain the torque Sensorless field oriented control of PMSMrequires knowledge of the rotor position Normally the rotorposition can be measured with an encoder or hall sensorsWhen the rotor position is available it is straightforward tocalculate the speed of the PMSMby simply differentiating therotor position [4]

The presence of encoder can increase the hardware com-plexity size and cost and reduce the reliability of the driveAlso the encoder is sensitive to environmental constraintssuch as vibration and temperature [5] Its performancedegrades under uncertain conditions and may not workwell at high speeds Hence several works have focused on

replacing the hardware sensor with the software sensor thatis based on the available system measurements (voltages andcurrents) to estimate the rotor position and speed [2 3]

Several methods are available for rotor position andorspeed estimation in a sensorless PMSM drive [2 6] Themain concerns regarding speed estimation are related toaccuracy magnitude and frequency of measured electricalquantities (applied voltages and currents) dependence onmotor parameters and dynamic behavior An extendedKalman filter method and the Luenberger observer basedmethods are also available to construct full-order estimatorsbased on the machine model [7]

Sliding mode principle has been popular for estimationand control in the presence of uncertainties [3 8] Recentlysliding mode observers (SMOs) have been successful inthe estimation of unknown inputs or faults [9 10] Severalfirst-order sliding mode observers have been applied forsensorless estimation and control of industrial drives [11ndash14] Many of the existing speed estimation techniques requirelow-pass filtering and an additional position compensationfor the rotor [15 16] for instance A survey was recentlyconducted in [17] for the implementation of the sliding modecontrol In [16] the cross-coupling terms of the 119889 minus 119902 current

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 589109 12 pageshttpdxdoiorg1011552014589109

2 Mathematical Problems in Engineering

dynamics are treated as unknown disturbances Decouplingterms with improper parameters can slightly degrade thesystemrsquos performance Integral sliding mode (ISM) controllerwith a switching output was developed [18] to overcome thesedisturbances As a result ISM can guarantee the robustness ofthe system starting from the initial time instance Howeverthe speed estimation based on ISM current control requiresan additional low-pass filter which introduces the delayand which in turn reduces the systemrsquos phase margin andcan cause instability In [19] a high-speed SM observer isproposed for sensorless speed estimation in a PMSM Theselection of the boundary layer and the sliding mode gaindepends on the speed and the method is more suitable forconstant speed applications To further overcome the phase-distortion a modified SMO based is designed [12] Thismethod employs a two-stage estimation process for rotorposition estimation Both the works [12 19] employ a sigmoidfunction instead of the switching function in order to avoidchattering phenomenon

Higher-order SMOrsquos have been developed to overcomethe disadvantages of first-order SMO [20ndash22] The super-twisting algorithm (STA) (see [23]) provides finite timeand exact convergence even in the presence of boundedperturbations To analyze the robustness of the STA for awider class of disturbances strict Lyapunov functions aredeveloped in [20] This Lyapunov function makes someadditionalmodifications of the STAby including termswhichimproves its robustness and convergence properties [20]Also it can reduce the well-known chattering phenomenonTheHOSMobserver designs in [24 25] rely on STA for finite-time convergence In the above methods the performancewas only verified through simulations in the absence of noiseand experimental validation was not performed

In [20] the STA is able to converge in finite time andtolerate perturbations that have a strong influence near theorigin In the eventuality of a linearly growing perturbationthe convergence of STA fails In other words it can be saidthat the STA is unable to endure globally a linearly growingperturbation To compensate for this problem a modifiedSTA is proposed The observer gain is tuned to withstandpersistently exciting perturbation terms It is directly respon-sible for handling the linear perturbation which is boundedby a value that depends on the computed derivative ofthe sliding surface A comparative analysis of the proposedalgorithm with classical STA shows a much better reductionin estimation error with reduced chattering effect and fasterresponse Based on the above arguments it can be concludedthat the proposed modified STA offers better performance incomparison to the classical STA in the scenario when linearlygrowing perturbations are considered

The main contribution of this paper lies in applicationof a modified version of STA to design a HOSM observerThe performance of the proposed design is validated throughexperiments on a PMSM The motivation of this work isto provide the HOSM observer with the properties of finitetime convergence and low chattering effect compared to theclassical equivalent control obtained with a traditional first-order SMO that requires a low-pass filter [3] The observerenables the estimation of the rotor position and speed of

the PMSM in real time while reducing the well-knownchattering phenomenon For example in [15] cascaded SMOis proposed for the estimation of back EMFs and rotor speedThe method requires cascaded first-order SM observersand uses low-pass filtering for the unknown back EMFsestimation The requirement of low-pass filtering results ina delay in the back EMFs estimation and the correspondingspeed estimation has high chattering phenomenon In theHOSM scheme the unknown back EMFs are estimated usingthe slidingmode termsdesigned by the slidingmode observerfrom current dynamics of the PMSM With the back EMFsaccurately estimated the rotor position and speed can beobtained algebraically The proposed technique does notrequire any low-pass filtering and hence has no delay in theestimation

120582max(119860) denotes the maximum eigenvalue of a matrix119860 119860 denotes the 2-norm radic120582max(119860

119879119860) of 119860 120582min(119860)

represents its minimum singular value

2 PMSM Modeling and Problem Statement

The model of the PMSM in the stationary reference framecan be described by the following system [6]

119889119894120572

119889119905=minus119877

119871119894120572minus1

119871119890120572+1

119871119881120572

119889119894120573

119889119905=minus119877

119871119894120573minus1

119871119890120573+1

119871119881120573

(1)

with

119890120572= minus119870119864120596119904sin 120579119904

119890120573= 119870119864120596119904cos 120579119904

(2)

119889120596119904

119889119905=119875

119869120601119898(minus sin 120579

119904119894120572+ cos 120579

119904119894120573) minus

119891V

119869120596119904minus119879119897

119869 (3)

The back EMF equations involve the dynamics of speed andposition The voltages 119881

120572and 119881

120573and currents 119894

120572and 119894120573are

the known quantities The objective is to design an observerto estimate the back electromotive forces (EMFs) from (1)using the available measurements Accurate estimation ofback EMF will result in accurate estimation of speed andposition

3 High Order Sliding Mode Observer

Let us note that the extra terms (119890120572 119890120573) of the plant dynamics

in (1) act like unknown inputs The idea is to use higher-order sliding modes on both 120572 and 120573 axes to estimate theseunknown inputs

Applying the same design principles as for variablestructure control the observer trajectories are constrained toevolve after a finite time on a suitable slidingmanifoldHencethe sliding motion provides an estimate (asymptotically or infinite time) of the system states In the following a HOSMobserver will be designed to estimate the unknown inputs

Mathematical Problems in Engineering 3

For the implementation of the sliding mode observer thefollowing sliding surfaces are selected

119904120572 (119905) =

120572minus 119894120572

119904120573 (119905) =

120573minus 119894120573

(4)

where 120572and 120573are estimated currents and 119894

120572and 119894120573are actual

currentsWith the PMSMmodel defined in (1) the observer can be

designed as follows

119889120572

119889119905=minus119877

119871120572+1

119871119881120572+1

119871]1 (119905)

119889120573

119889119905=minus119877

119871120573+1

119871119881120573+1

119871]2 (119905)

(5)

The sliding mode terms are given by

]1 (119905) = minus119870

11206011(119904120572 (119905)) minus 119870

2int

119905

0

1206012(119904120572 (119905)) 119889119905

]2 (119905) = minus119870

11206011(119904120573 (119905)) minus 119870

2int

119905

0

1206012(119904120573 (119905)) 119889119905

(6)

where1206011(119904120572 (119905)) = 119904

120572 (119905) + 1198703|119904120572 (119905) |12 sign (119904

120572 (119905))

1206012(119904120572 (119905)) = 119904

120572 (119905) +1198702

4

2sign (119904

120572 (119905))

+3

21198704

1003816100381610038161003816119904120572 (119905)100381610038161003816100381612 sign (119904

120572 (119905))

(7)

where119870111987021198703 and119870

4are appropriately designed positive

constants Similarly the functions 1206011(119904120573(119905)) and 120601

2(119904120573(119905)) can

be obtained by replacing 119904120572(119905) with 119904

120573(119905) in (7)

31 Sliding Mode Stability To prove the stability of theobserver system the time derivatives of the sliding surfacesare obtained from (1) and (5) as

119904120572=minus119877

119871119904120572+1

119871119890120572+1

119871]1 (119905)

119904120573=minus119877

119871119904120573+1

119871119890120573+1

119871]2 (119905)

(8)

From (2) and (3) we can establish the boundedness ofback EMFs At least locally there are positive constants 120588

1and

1205882such that the following terms are bounded as follows

1003816100381610038161003816 1198901205721003816100381610038161003816 le 1205881

10038161003816100381610038161003816119890120573

10038161003816100381610038161003816le 1205882

(9)

for some positive constants 1205881and 120588

2 The above condition

(9) is not restrictive since 120596119904 119879119897 and 119890

120572and 119890120573 119894120572 and 119894

120573are

continuous on a compact set

Theorem 1 With the condition (9) the sliding dynamics 119904120572

and 119904120573are stabilized towards zero in finite time

Proof See Proposition A3 in the appendix

Table 1 Motor parameters

Parameters ValuesRating 26 [W]

Speed 120596 4000 [rmin]Stator resistance 119877 24 [Ω]

Stator inductance 119871 065 [mH]

Back EMF constant 119870119864

0156 [Vsrad]Inertia 119869 0004 times 10

minus4[Kgsdotm2

]

Viscous friction 119891V 0004 times 10minus4

[Nmsdotsrad]Rotor flux 120601

1198980025

Number of pole pairs 119875 4

32 Speed and Rotor Position Estimation According toTheorem 1 the origin of system (8) has a finite time stableequilibrium In the sliding mode we have 119904

120572= 119904120572= 0 and

119904120573

= 119904120573

= 0 The reduced order dynamics of system (8)becomes

0 = minus1198702int

119905

0

1206012(119904120572 (119905)) 119889119905 + 119890

120572

0 = minus1198702int

119905

0

1206012(119904120573 (119905)) 119889119905 + 119890

120573

(10)

Hence a smooth estimation of the unknown back EMFscan be obtained in finite time as follows

119890120572= 1198702int

119905

0

1206012(119904120572 (119905)) 119889119905

119890120573= 1198702int

119905

0

1206012(119904120573 (119905)) 119889119905

(11)

Using the estimated back EMF voltages the position ofthe rotor can be calculated as

120579119904= minustanminus1 (

119890120572

119890120573

) (12)

Also with estimated back EMFs using (2) the speed canbe computed algebraically as

119904=

1

119870119864

radic1198902120572+ 1198902120573 (13)

The speed estimation only uses the EMF constant119870119864and

the estimated back EMFs 119890120572and 119890

120573 The proposed HOSM

observer provides the properties of finite time convergenceand low chattering effect compared to the classical equivalentcontrol obtained using a low-pass filter [15]

4 Experimental Results

Experiments are performed with the three-phase 26WPMSM The specifications and parameters are provided inTable 1 The motor used in the experimental setup is aTBL-119894model TS4632N2050E510 3-phase PMSMThe PMSMis powered by a Fairchild FSB50325S smart power modulewhich includes 6 fast-recovery MOSFET (FRFET) invertersand 3 half-bridge high voltage integrated circuits (HVICs)

4 Mathematical Problems in Engineering

TMS320F2833335CAN

SCIPower module

JTAG emulator

PMSM

Encoder

DC link

Figure 1 Experimental setup

for FRFET gate driving Since it employs FRFET as a powerswitch it has much better robustness and larger safe opera-tion areas (SOA) than that using an IGBT-based power mod-ule or one-chip solution The experimental setup is shownin Figure 1 The space vector modulation (SVM) algorithmis used as modulation strategy and switching frequency ofthe PWM inverter is 15 kHz SMC 75 evaluation modulewith TMS320F28335 DSP controller is used It contains TexasInstruments 32-bit floating point DSP as well as analoginterfaces and JTAG emulator portThe board has analog-to-digital converter (AD) with 16 channels All the control vari-ables are monitored using graph window of Code ComposerStudio (CCS v33) after being converted to analog signalsthrough the digital-to-analog (DA) converter Real motorspeed (120596

119904) is measured using a high-resolution incremental

encoder with 2000 pulsesrotation and the estimated speed(119904) is obtained with the proposed HOSM schemeThe stator

currents of the PMSM aremeasured from the current sensorsand they are sent to TMS320F28335 viaAD converters In thesame way stator voltages are calculated using dc-bus voltagesensors and duty cycles of the inverter when the switchingfunctions are known

Three-phase currents and voltages are transformed totwo-phase stationary (120572 minus 120573) reference frame They are againtransformed to rotating (119889minus119902) reference frame for the controlPI (proportional and integral) controllers are used to regulatethe 119889 119902 synchronous frame currents 119894

119902and 119894119889 A functional

block diagram for the overall scheme is depicted in Figure 2The PMSM drive is operated in speed control mode Thespeed is also regulated using a PI controller to generate thereference current 119894ref

119902in the 119902-axis The reference current in

the 119889-axis 119894ref119889

is set to 0 For the implementation of HOSMobserver the sliding mode gains are selected as follows119870

1=

07 1198702= 60 119870

3= 35 and 119870

4= 45 The initial conditions

for the estimator are chosen as 119909(0) = [0 0 0 0]Several experiments have been performed to validate

the proposed HOSM scheme In the first part we presentthe results performance of the proposed method under no-load conditions for ramp change and step change and in thelater part similar experiments are conducted under loading

PMSM

Clarke

Position

HOSM

observer

S

S

Speed

Park120572120573

120572120573

120572120573PI PI

PIV120572

id i120572dq

dq

abc

iq

+

+

+ minus

minus

minus

ia

ib

V120573

i120573

irefd

irefq120596ref

Parkminus1

SVM3-phase

inverter

Figure 2 Functional block diagram for the overall scheme

condition For comparison the results obtained with first-order sliding mode observer are also presented

41 Under No-Load Condition In the first experiment aconstant speed reference 2000 rpm is provided for the first03 s a ramp input for the next 04 s followed by a constantspeed of 3500 rpm as shown in Figure 3(b) is employed Thereal currents of the PMSM for the first experiment are shownin Figure 3(a) The encoder speed and position are providedin Figures 3(b) and 3(c) The actual speed (120596

119904) exactly

follows the reference speed considered above Presence ofmeasurement noise can be clearly observed in (119894

120572120573) and

(120596119904 120579119904) With the proposed observer the estimated currents

and estimation error are shown in Figures 4(a) and 4(b) Realand estimated currents are very similar in both magnitudeand phase using the proposed method Figure 4(c) depictsthe estimated back EMFs obtained using (11) Despite thenoisy currents the back EMFs are relatively smooth whichconforms the theoretical claim of the proposed approachThe estimated speed computed analytically from back EMFswith (13) is shown in Figure 4(d) which exactly tracks actualspeed (120596

119904) and is shown for the comparisonThe convergence

accuracy depends on the accurate estimation of the back EMFcomponents and the back EMF constant 119870

119864 The HOSM

scheme enables a good reconstruction of the PMSM speedFigures 4(e) and 4(f) show the estimated rotor position andestimation error The estimated rotor position is robust withrespect to noise measurements and exactly matches withthe actual rotor position without any phase delay So theestimated rotor position can be used instead of the measuredone in the vector control of PMSM drive In usual practicethe values of 119877 and 119871 are not accurately known To testthe robustness the parameters (119877 and 119871) values are variedby plusmn10 and several experiments are conducted Similarperformancewas obtained in comparison to results presentedin Figure 4

For comparison the results obtained with conventionalsliding mode observer [3] are shown in Figure 5 The sliding

Mathematical Problems in Engineering 5

1

Time (s)0 01 03 05 07 09 1

minus1

0

1i 120572120573

(A)

(a)

1000

2000

3000

Time (s)0 01 03 05 07 09 1

120596s

(rpm

)

(b)

0

3

Time (s)

0 01 03 05 07 09 1

minus3

120579s

(rad

)

(c)

Figure 3 (a) Actual currents (b) Actual speed (c) Actual rotor position

mode gain is set to 50 for the observer design The presenceof noise in the estimated currents (Figure 5(a)) highly affectsthe estimation of back EMFs as shown in Figure 5(c) Theestimated back EMFs which correspond to the equivalentcontrols are obtained by filtering the switching functions ofthe observer with a 40Hz low-pass filter A proper boundarylayer is required to overcome the chattering phenomenonThe speed and position estimate in Figures 5(d)ndash5(f) exposethe problems with the conventional sliding mode observerfor back EMF estimation Also the estimated speed hasmore noise compared to the speed estimate with the HOSMobserver shown in Figure 4(d) The rotor position error inFigure 5(f) compared to Figure 4(f) clearly highlights theaccuracy obtained with proposed method Low-pass filteringclearly affects the estimation accuracyThe parameters for thefirst-order sliding mode are well-tuned to achieve the bestpossible results Errors are mainly due to filtering and the useof sigmoid function to avoid the chattering phenomenon

In the second experiment a step change in speed isprovided as reference In this experiment the speed estimateand position estimation error obtained with the proposedHOSM scheme for the step reference are shown in Figure 6Figure 6(a) shows the estimated and actual speeds obtainedusing the proposed HOSM observer and Figure 6(b) showsthe estimation error between the estimated and actual rotorpositions It can be seen from Figure 6(b) that the estimationhas no delay with the proposed approach For compari-son the results obtained with conventional first-order SMOare shown in Figure 7 Figure 7(a) shows the estimatedand actual speeds obtained using the first-order SMO andFigure 7(b) shows the estimation error between the estimatedand actual rotor positions Although the estimated speedfollows the actual speed it contains more noise compared to

the proposedHOSMobserverThese results further highlightthe robustness of the proposed method in the presenceof noise Further the chattering phenomenon is completelyeliminated and accurate position estimation can be obtainedeven in the presence of measurement noise

42 Under Loading Condition To test the performance of theproposed method a mechanical load 119869

119871= 007436 kg sdot cm2 is

connected to the motor Same set of parameters consideredfor no-load are employed for loading condition to test therobustness of the observer to parametric variations Theresults obtained with HOSM observer for a ramp changeare shown in Figure 8 Figures 8(a) and 8(b) show the esti-mated currents and their errors obtained using the proposedHOSM observer It can be observed from Figure 8(b) thatthe estimated and actual currents exactly match each otherFigure 8(c) shows the estimated unknown back EMFs whichare relatively smooth The corresponding estimated speedcalculated using (13) and the actual speed are depicted inFigure 8(d) The estimated rotor position and the estimationerror are shown in Figures 8(e) and 8(f) For comparison theestimated and actual speeds obtained with first-order SMOare shown in Figure 9(a) and the corresponding position esti-mation error is shown in Figure 9(b) In the final experimentunder loading condition step-like input reference is providedfor the system the results obtained with proposed methodand first-order SMO are shown in Figure 10 and Figure 11respectively Figures 10(a) and 10(b) show the estimatedactual speeds and position estimation error obtained withthe proposed HOSM observer while Figures 11(a) and 11(b)show the estimated actual speeds and position estimationerror obtained with the first-order SMO Under loading thenoise level is higher and the position estimation error slightly

6 Mathematical Problems in Engineering

Time (s)0 01

1

03 05 07 09 1

0

05

minus05

minus1

i 120572120573

(A)

(a)

Time (s)0 01 03 05 07 09 1

0

05

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus40

40

0

e 120572120573

(V)

(c)

1500

3000

4000

Time (s)0 01 03 05 07 09 1

EstimatedActual

120596s

(rpm

)

(d)

0

3

Time (s)0 01 03 05 07 09 1

minus3

120579s

(rad

)

(e)

0

1

2

Time (s)0 01 03 05 07 09 1

minus2

minus1120579sminus120579s (

rad)

(f)

Figure 4 Estimation using higher-order sliding mode observer under no-load (a) Estimated currents (b) Estimation current error (c)Estimated back EMFs (d) Estimated speed (e) Estimated rotor position (f) Estimation rotor position error

increasesHowever the speed andposition estimation remainrobust with the proposed observer when compared to first-order SMO Due to parametric uncertainty the speed estima-tion with proposedHOSMmethod shows a very small steadystate error at 90 of the rated speed (Figures 8(d) and 10(a))Under loading the speed and position estimation with first-order SMOare highly affected Further tuning of the observerparameters can overcome the problem

From the implementation one can conclude the follow-ing

(1) The HOSM method requires the proper selection ofsliding mode gains 119870

1 1198702 1198703 and 119870

4 The sliding

mode gains should satisfy the conditions given by(A15)-(A16) for the desired speed range If themotoroperates in wide speed range the sliding mode gainsmust be appropriately selected

(2) Since the quality of the speed estimate highly dependson the estimated back EMFs it deteriorates when

more noisy back EMFs (obtained due to high gains)are used in the calculation Compared to existingmethods the proposed scheme provides good rotorspeed and position estimation without phase delay inthe presence of noise Further one should note thatsensorless speed estimation methods based on backEMFs fail at very low speeds and standstill

(3) It should be pointed out that a chattering phe-nomenon occurs using the conventional SM observer[15]Therefore in first-order SMobserver the signumfunction is used as switching function The speedestimate is approximated by low-pass filtering thediscontinuous switching functions This delay shouldbe compensated with an additional phase compensa-tion loop As low-pass filtering is eliminated with theproposed method an additional phase compensationloop is not required

Mathematical Problems in Engineering 7

1

0

05

minus05

minus1

i 120572120573

(A)

Time (s)0 01 03 05 07 09 1

(a)

Time (s)0 01 03 05 07 09 1

0

05

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus40

40

0

e 120572120573

(V)

(c)

Time (s)0 01 03 05 07 09 1

1500

3000

4000

120596s

(rpm

)

EstimatedActual

(d)

Time (s)0 01 03 05 07 09 1

0

3

minus3

120579s

(rad

)

(e)

Time (s)0 01 03 05 07 09 1

0

1

2

minus2

minus1120579sminus120579s (

rad)

(f)

Figure 5 Estimation using conventional first-order sliding mode observer under no-load (a) Estimated currents (b) Estimation currenterror (c) Estimated back EMFs (d) Estimated speed (e) Estimated rotor position (f) Estimation rotor position error

1500

3000

4000

EstimatedActual

120596s120596

s(r

pm)

Time (s)0 02 04 05 06 08 1

(a)

120579sminus120579s

(rad

)

0

1

2

minus1

minus2

Time (s)0 02 04 05 06 08 1

(b)

Figure 6 With proposed HOSMmethod under no-load (a) Actual and estimated speed (b) Rotor position estimation error

8 Mathematical Problems in Engineering

1500

3000

4000

Time (s)0 02 04 05 06 08 1

EstimatedActual

120596s120596

s(r

pm)

(a)

Time (s)

0 02 04 05 06 08 1

120579sminus120579s

(rad

)

0

1

2

minus1

minus2

(b)

Figure 7 With conventional first-order SMO under no-load (a) Estimated speed (b) Rotor position estimation error

Time (s)0 01 03 05 07 09 1

0

06

minus06

i 120572120573

(A)

(a)

Time (s)0 01 03 05 07 09 1

0

05

12

minus12

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus60

60

0

e 120572120573

(V)

(c)

1500

3000

4000

Time (s)0 01 03 05 07 09 1

EstimatedActual

120596s

(rpm

)

(d)

0

3

Time (s)0 01 03 05 07 09 1

minus3

120579s

(rad

)

(e)

0

1

2

Time (s)0 01 03 05 07 09 1

minus2

minus1120579sminus

120579s (

rad)

(f)

Figure 8 With proposed HOSM method under load (a) Estimated currents (b) Estimation current errors (c) Estimated back EMFs (d)Estimated speed (e) Estimated rotor position (f) Rotor position error

Mathematical Problems in Engineering 9

Time (s)0 01 03 05 07 09 1

1500

3000

4000

120596s120596

s(r

pm)

EstimatedActual

(a)

Time (s)0 01 03 05 07 09 1

0

1

2

minus2

minus1

120579sminus120579s (

rad)

(b)

Figure 9 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

1500

3000

4000

0 02 04 05 06 08 1

Time (s)EstimatedActual

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1Time (s)

0

1

2

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 10 With proposed HOSMmethod under load (a) Estimated speed (b) Rotor position estimation error

(4) Furthermore compared to the classical first-orderSM technique no cutoff frequency has to be tunedInstead a simple integration is realized It enablesto reduce the time delay for the estimation (whichdepends on the sampling period) One should alsohighlight that the discontinuous part of 120601

2(depend-

ing on1198704) is usually low compared to the continuous

part of 1206012and this enables to reduce the chattering

phenomenon(5) Moreover from the experiments the proposed

method is robust to the parameter variations and themeasurement noise compared to the traditional SMobserver

(6) It is worth to point out that the proposed method iscomputationally complex compared to the traditionalSM observer However if properly tuned it has moreadvantages than the traditional SM observer Theexperiments conducted in this paper validate theadvantages of this method

(7) For the same set of parameters the speed and positionestimation remained accurate for both no-loadingand loading conditions This further highlights therobustness of the proposedmethod to parameter vari-ations that occur with loading and other conditions

5 Conclusion

This paper has presented a sensorless speed estimationmethod for the PMSM driveThe HOSMmethod is based ona modified version of super-twisting algorithmThe observerdynamics consist of sliding mode terms which are used toreconstruct the unknown back EMFs The speed is thenanalytically computed from back EMFs Experimental resultsvalidate the feasibility and effectiveness of the proposedHOSM for estimating the rotor position and speed of thePMSM Compared with the traditional SMO the proposedhigher-order SMO provides better estimation performance

Appendix

Finite-Time Stability

For any vector 119911 = [1199111 119911

119902]119879isin 119877119902 and any scalar 120572 isin 119877

we denote the following

sign (119911) = [sign (1199111) sign (119911

119902)]119879

|119911|120572= diag (10038161003816100381610038161199111

1003816100381610038161003816120572

10038161003816100381610038161003816119911119902

10038161003816100381610038161003816

120572

)

lceil119911rfloor120572= |119911|120572 sign (119911)

(A1)

10 Mathematical Problems in Engineering

1500

3000

4000

EstimatedActual

0 02 04 05 06 08 1

Time (s)

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1

0

1

2

Time (s)

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 11 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

For ease of exposition consider the following system

119904 (119905) = minus 119886119904 (119905) + ] (119905) + 119890 (119904 119905)

119904 (1199050) = 1199040

(A2)

where 119904 isin R and 119886 is a known positive constant and 119890(119904 119905) isthe unknown inputperturbation and

] (119905) = minus11987011206011 (119904 (119905)) minus 119870

2int

119905

0

1206012 (119904 (119905)) 119889119905 (A3)

where 1206011(119904(119905)) and 120601

2(119904(119905)) are defined in (7) and119870

111987021198703

and1198704are appropriately designed positive constants

Assumption A1 The time derivative of the unknowninputperturbation is upper bounded as follows

| 119890 (119904 119905)| le 120588 (A4)

for a positive constant 120588

Remark A2 The sliding dynamics 119904120572or 119904120573in (8) can be

directly expressed in the form of (A2) Further the condition(9) is similar to Assumption A1

Proposition A3 Under Assumption A1 the origin of sys-tem (A2) is a finite time stable equilibrium point Fur-ther the finite-time smooth estimation of the unknowninputperturbation 119890(119904 119905) is given by 119870

2int119905

01206012(119904(119905))119889119905

Proof Proof follows the work given in [20] Since |1206012(119904)| ge

1198702

42 one gets

| 119890 (119904 119905)| le10038161003816100381610038161206012 (119904)

1003816100381610038161003816 (A5)

if

1198704ge radic2120588 (A6)

Let us select a Hurwitz matrix 1198600

1198600= [

minus (1198701+ 119886) 1

minus1198702

0] (A7)

where1198701gt 0 and119870

2gt 0

The system (A2) (A3) can be equivalently representedby the system of two first-order equations

1199041= 1199042minus (1198701+ 119886) (119904

1+ 1198704lceil 1199041rfloor12

)

1199042= minus 119870

2(1199041+1198702

4

2sign (119904

1) +

3

21198704lceil 1199041rfloor12

) + 119890

(A8)

with 1199041= 119904 119904

2= 119890 minus 119870

2int119905

01206012(1199041) 119889119905 and

1198704=

11987011198703

1198701+ 119886

(A9)

The solutions of the discontinuous differential equations andinclusions are understood in the sense of Filippov

Let us consider the new state vector

120585 = [1205851

1205852

] = [1199041+ 1198704lceil 1199041rfloor12

1199042

] (A10)

The stability analysis of system (A8) is performed usingthe following candidate Lyapunov function [20]

119881 (120585) = 120585119879119875120585 (A11)

with 119875 = 119875119879

= [ 120582+41205982minus2120598

minus2120598 1] 120582 gt 0 and 120598 gt 0 It is worth

noting that the matrix 119875 is positive definite if 120582 and 120598 are anyreal number

Using the differential equations inclusion theory its timederivative along the solutions of the system is given by

= (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

119890]

(A12)

Mathematical Problems in Engineering 11

It can be shown that

le (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

)(120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

1205851

])

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879119876120585

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120582min (119876)100381710038171003817100381712058510038171003817100381710038172

(A13)with

119876 = [11987611198762

11987621198763

]

1198761= 2 (119870

1+ 119886) (120582 + 4120598

2) minus 4120598 (119870

2minus 1)

1198762= minus 2120598 (119870

1+ 119886) + (119870

2+ 1) minus (120582 + 4120598

2)

1198763= 4120598

(A14)

In order to guarantee the positive definiteness of matrix 119876one chooses

1198702= 120582 + 4120598

2+ 2120598 (119870

1+ 119886) (A15)

The matrix 119876 is positive definite if

1198701gt minus119886 +

4120598 + 2120598120582 + 81205983

120582+

1

4120598120582 (A16)

From (A10) one can deduce that100381710038171003817100381712058510038171003817100381710038172= 1205852

1+ 1205852

2

= 1199042

1+ 21198704

10038161003816100381610038161199041100381610038161003816100381632

+ 1198702

4

100381610038161003816100381611990411003816100381610038161003816 + 1205852

2

ge 1198702

4

100381610038161003816100381611990411003816100381610038161003816

(A17)

Since1198704gt 0

minus1198704

10038171003817100381710038171205851003817100381710038171003817

ge minus10038161003816100381610038161199041

1003816100381610038161003816minus12

(A18)

It implies that

le minus120582min (119876)

12058212

max (119875)

1198702

4

211988112

minus120582min (119876)

120582max (119875)119881 (A19)

The closed-loop system (A8) is stabilized in finite timeSince 120585 converges to zero in finite time 119904

1and 1199042converge

to 0 Therefore the term1198702int119905

01206012(119904(119905))119889119905 gives in finite time a

smooth estimation of the unknown perturbation 119890(119904 119905)

Nomenclature

120596119904 Rotor electrical speed

119894120572 119894120573 Currents in stationary reference frame

119881120572 119881120573 Voltages in stationary reference frame

119890120572 119890120573 EMFs in stationary reference frame

119877 Stator resistance119871 Synchronous inductance119870119864 EMF constant

120579119904 Rotor position angle

119879119897 Load torque

Conflict of Interests

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

Acknowledgment

This research was supported by the Basic Science ResearchProgram through theNational Research Foundation of Korea(NRF) funded by the Ministry of Education Science andTechnology (Grant no 2011-0023999)

References

[1] R Wu and G R Slemon ldquoA permanent magnet motor drivewithout a shaft sensorrdquo IEEE Transactions on Industry Applica-tions vol 27 no 5 pp 1005ndash1011 1991

[2] P Tomei and C M Verrelli ldquoObserver-based speed trackingcontrol for sensorless permanent magnet synchronous motorswith unknown load torquerdquo IEEE Transactions on AutomaticControl vol 56 no 6 pp 1484ndash1488 2011

[3] V Utkin J G Guldner and J Shi Sliding Mode Control onElectromechanical Systems Taylor and Francis New York NYUSA 1st edition 1999

[4] S Chai L Wang and R Rogers ldquoModel predictive controlof a permanent magnet synchronous motor with experimentalvalidationrdquoControl Engineering Practice vol 21 no 11 pp 1584ndash1593 2013

[5] T Orlowska-Kowalska and M Dybkowski ldquoStator-current-based MRAS estimator for a wide range speed-sensorlessinduction-motor driverdquo IEEE Transactions on Industrial Elec-tronics vol 57 no 4 pp 1296ndash1308 2010

[6] M L Corradini G Ippoliti S Longhi and G Orlando ldquoAquasi-sliding mode approach for robust control and speedestimation of PM synchronous motorsrdquo IEEE Transactions onIndustrial Electronics vol 59 no 2 pp 1096ndash1104 2012

[7] B K Bose Modern Power Electronics and AC Drives Prentice-Hall Upper Saddle River NJ USA 2002

[8] K C Veluvolu and Y C Soh ldquoMultiple sliding mode observersand unknown input estimations for Lipschitz nonlinear sys-temsrdquo International Journal of Robust and Nonlinear Controlvol 21 no 11 pp 1322ndash1340 2011

[9] K C Veluvolu and D Lee ldquoSliding mode high-gain observersfor a class of uncertain nonlinear systemsrdquoAppliedMathematicsLetters vol 24 no 3 pp 329ndash334 2011

[10] K C Veluvolu and Y C Soh ldquoFault reconstruction and stateestimationwith slidingmode observers for Lipschitz non-linearsystemsrdquo IET Control Theory amp Applications vol 5 no 11 pp1255ndash1263 2011

[11] M Comanescu and L Xu ldquoSliding-mode MRAS speed estima-tors for sensorless vector control of induction machinerdquo IEEETransactions on Industrial Electronics vol 53 no 1 pp 146ndash1532006

[12] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[13] K C Veluvolu M Y Kim and D Lee ldquoNonlinear sliding modehigh-gain observers for fault estimationrdquo International Journalof Systems Science Principles and Applications of Systems andIntegration vol 42 no 7 pp 1065ndash1074 2011

12 Mathematical Problems in Engineering

[14] K C Veluvolu M Defoort and Y C Soh ldquoHigh-gain observerwith sliding mode for nonlinear state estimation and faultreconstructionrdquo Journal of the Franklin Institute Engineeringand Applied Mathematics vol 351 no 4 pp 1995ndash2014 2014

[15] M Comanescu ldquoCascaded EMF and speed sliding modeobserver for the nonsalient PMSMrdquo in Proceedings of the 36thAnnual Conference of the IEEE Industrial Electronics Society(IECON rsquo10) pp 792ndash797 Glendale Ariz November 2010

[16] M Comanescu ldquoAn induction-motor speed estimator based onintegral sliding-mode current controlrdquo IEEE Transactions onIndustrial Electronics vol 56 no 9 pp 3414ndash3423 2009

[17] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[18] M Comanescu L Xu and T D Batzel ldquoDecoupled currentcontrol of sensorless induction-motor drives by integral slidingmoderdquo IEEE Transactions on Industrial Electronics vol 55 no11 pp 3836ndash3845 2008

[19] H Kim J Son and J Lee ldquoA high-speed sliding-mode observerfor the sensorless speed control of a PMSMrdquo IEEE Transactionson Industrial Electronics vol 58 no 9 pp 4069ndash4077 2011

[20] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

[21] T Floquet and J P Barbot ldquoSuper twisting algorithm-basedstep-by-step sliding mode observers for nonlinear systemswith unknown inputsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 38no 10 pp 803ndash815 2007

[22] J J Rath K C Veluvolu M Defoort and Y C Soh ldquoHigher-order sliding mode observer for estimation of tyre frictionin ground vehiclesrdquo IET Proceedings on Control Theory andApplications vol 8 no 6 pp 399ndash408 2014

[23] L Fridman and A Levant ldquoHigher order sliding modesSliding mode control in engineeringrdquo in Sliding Mode Controlin Engineering J P Barbot and W Perruquetti Eds MarcelDekker New York NY USA 2002

[24] M Ezzat J De Leon N Gonzalez and A GlumineauldquoObserver-controller scheme using high order sliding modetechniques for sensorless speed control of permanent magnetsynchronous motorrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 4012ndash4017 December2010

[25] D Zaltni and M N Abdelkrim ldquoRobust speed and positionobserver using HOSM for sensor-less SPMSM controlrdquoin Proceedings of the 7th International Multi-Conference onSystems Signals and Devices (SSD rsquo10) pp 1ndash6 June 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

dynamics are treated as unknown disturbances Decouplingterms with improper parameters can slightly degrade thesystemrsquos performance Integral sliding mode (ISM) controllerwith a switching output was developed [18] to overcome thesedisturbances As a result ISM can guarantee the robustness ofthe system starting from the initial time instance Howeverthe speed estimation based on ISM current control requiresan additional low-pass filter which introduces the delayand which in turn reduces the systemrsquos phase margin andcan cause instability In [19] a high-speed SM observer isproposed for sensorless speed estimation in a PMSM Theselection of the boundary layer and the sliding mode gaindepends on the speed and the method is more suitable forconstant speed applications To further overcome the phase-distortion a modified SMO based is designed [12] Thismethod employs a two-stage estimation process for rotorposition estimation Both the works [12 19] employ a sigmoidfunction instead of the switching function in order to avoidchattering phenomenon

Higher-order SMOrsquos have been developed to overcomethe disadvantages of first-order SMO [20ndash22] The super-twisting algorithm (STA) (see [23]) provides finite timeand exact convergence even in the presence of boundedperturbations To analyze the robustness of the STA for awider class of disturbances strict Lyapunov functions aredeveloped in [20] This Lyapunov function makes someadditionalmodifications of the STAby including termswhichimproves its robustness and convergence properties [20]Also it can reduce the well-known chattering phenomenonTheHOSMobserver designs in [24 25] rely on STA for finite-time convergence In the above methods the performancewas only verified through simulations in the absence of noiseand experimental validation was not performed

In [20] the STA is able to converge in finite time andtolerate perturbations that have a strong influence near theorigin In the eventuality of a linearly growing perturbationthe convergence of STA fails In other words it can be saidthat the STA is unable to endure globally a linearly growingperturbation To compensate for this problem a modifiedSTA is proposed The observer gain is tuned to withstandpersistently exciting perturbation terms It is directly respon-sible for handling the linear perturbation which is boundedby a value that depends on the computed derivative ofthe sliding surface A comparative analysis of the proposedalgorithm with classical STA shows a much better reductionin estimation error with reduced chattering effect and fasterresponse Based on the above arguments it can be concludedthat the proposed modified STA offers better performance incomparison to the classical STA in the scenario when linearlygrowing perturbations are considered

The main contribution of this paper lies in applicationof a modified version of STA to design a HOSM observerThe performance of the proposed design is validated throughexperiments on a PMSM The motivation of this work isto provide the HOSM observer with the properties of finitetime convergence and low chattering effect compared to theclassical equivalent control obtained with a traditional first-order SMO that requires a low-pass filter [3] The observerenables the estimation of the rotor position and speed of

the PMSM in real time while reducing the well-knownchattering phenomenon For example in [15] cascaded SMOis proposed for the estimation of back EMFs and rotor speedThe method requires cascaded first-order SM observersand uses low-pass filtering for the unknown back EMFsestimation The requirement of low-pass filtering results ina delay in the back EMFs estimation and the correspondingspeed estimation has high chattering phenomenon In theHOSM scheme the unknown back EMFs are estimated usingthe slidingmode termsdesigned by the slidingmode observerfrom current dynamics of the PMSM With the back EMFsaccurately estimated the rotor position and speed can beobtained algebraically The proposed technique does notrequire any low-pass filtering and hence has no delay in theestimation

120582max(119860) denotes the maximum eigenvalue of a matrix119860 119860 denotes the 2-norm radic120582max(119860

119879119860) of 119860 120582min(119860)

represents its minimum singular value

2 PMSM Modeling and Problem Statement

The model of the PMSM in the stationary reference framecan be described by the following system [6]

119889119894120572

119889119905=minus119877

119871119894120572minus1

119871119890120572+1

119871119881120572

119889119894120573

119889119905=minus119877

119871119894120573minus1

119871119890120573+1

119871119881120573

(1)

with

119890120572= minus119870119864120596119904sin 120579119904

119890120573= 119870119864120596119904cos 120579119904

(2)

119889120596119904

119889119905=119875

119869120601119898(minus sin 120579

119904119894120572+ cos 120579

119904119894120573) minus

119891V

119869120596119904minus119879119897

119869 (3)

The back EMF equations involve the dynamics of speed andposition The voltages 119881

120572and 119881

120573and currents 119894

120572and 119894120573are

the known quantities The objective is to design an observerto estimate the back electromotive forces (EMFs) from (1)using the available measurements Accurate estimation ofback EMF will result in accurate estimation of speed andposition

3 High Order Sliding Mode Observer

Let us note that the extra terms (119890120572 119890120573) of the plant dynamics

in (1) act like unknown inputs The idea is to use higher-order sliding modes on both 120572 and 120573 axes to estimate theseunknown inputs

Applying the same design principles as for variablestructure control the observer trajectories are constrained toevolve after a finite time on a suitable slidingmanifoldHencethe sliding motion provides an estimate (asymptotically or infinite time) of the system states In the following a HOSMobserver will be designed to estimate the unknown inputs

Mathematical Problems in Engineering 3

For the implementation of the sliding mode observer thefollowing sliding surfaces are selected

119904120572 (119905) =

120572minus 119894120572

119904120573 (119905) =

120573minus 119894120573

(4)

where 120572and 120573are estimated currents and 119894

120572and 119894120573are actual

currentsWith the PMSMmodel defined in (1) the observer can be

designed as follows

119889120572

119889119905=minus119877

119871120572+1

119871119881120572+1

119871]1 (119905)

119889120573

119889119905=minus119877

119871120573+1

119871119881120573+1

119871]2 (119905)

(5)

The sliding mode terms are given by

]1 (119905) = minus119870

11206011(119904120572 (119905)) minus 119870

2int

119905

0

1206012(119904120572 (119905)) 119889119905

]2 (119905) = minus119870

11206011(119904120573 (119905)) minus 119870

2int

119905

0

1206012(119904120573 (119905)) 119889119905

(6)

where1206011(119904120572 (119905)) = 119904

120572 (119905) + 1198703|119904120572 (119905) |12 sign (119904

120572 (119905))

1206012(119904120572 (119905)) = 119904

120572 (119905) +1198702

4

2sign (119904

120572 (119905))

+3

21198704

1003816100381610038161003816119904120572 (119905)100381610038161003816100381612 sign (119904

120572 (119905))

(7)

where119870111987021198703 and119870

4are appropriately designed positive

constants Similarly the functions 1206011(119904120573(119905)) and 120601

2(119904120573(119905)) can

be obtained by replacing 119904120572(119905) with 119904

120573(119905) in (7)

31 Sliding Mode Stability To prove the stability of theobserver system the time derivatives of the sliding surfacesare obtained from (1) and (5) as

119904120572=minus119877

119871119904120572+1

119871119890120572+1

119871]1 (119905)

119904120573=minus119877

119871119904120573+1

119871119890120573+1

119871]2 (119905)

(8)

From (2) and (3) we can establish the boundedness ofback EMFs At least locally there are positive constants 120588

1and

1205882such that the following terms are bounded as follows

1003816100381610038161003816 1198901205721003816100381610038161003816 le 1205881

10038161003816100381610038161003816119890120573

10038161003816100381610038161003816le 1205882

(9)

for some positive constants 1205881and 120588

2 The above condition

(9) is not restrictive since 120596119904 119879119897 and 119890

120572and 119890120573 119894120572 and 119894

120573are

continuous on a compact set

Theorem 1 With the condition (9) the sliding dynamics 119904120572

and 119904120573are stabilized towards zero in finite time

Proof See Proposition A3 in the appendix

Table 1 Motor parameters

Parameters ValuesRating 26 [W]

Speed 120596 4000 [rmin]Stator resistance 119877 24 [Ω]

Stator inductance 119871 065 [mH]

Back EMF constant 119870119864

0156 [Vsrad]Inertia 119869 0004 times 10

minus4[Kgsdotm2

]

Viscous friction 119891V 0004 times 10minus4

[Nmsdotsrad]Rotor flux 120601

1198980025

Number of pole pairs 119875 4

32 Speed and Rotor Position Estimation According toTheorem 1 the origin of system (8) has a finite time stableequilibrium In the sliding mode we have 119904

120572= 119904120572= 0 and

119904120573

= 119904120573

= 0 The reduced order dynamics of system (8)becomes

0 = minus1198702int

119905

0

1206012(119904120572 (119905)) 119889119905 + 119890

120572

0 = minus1198702int

119905

0

1206012(119904120573 (119905)) 119889119905 + 119890

120573

(10)

Hence a smooth estimation of the unknown back EMFscan be obtained in finite time as follows

119890120572= 1198702int

119905

0

1206012(119904120572 (119905)) 119889119905

119890120573= 1198702int

119905

0

1206012(119904120573 (119905)) 119889119905

(11)

Using the estimated back EMF voltages the position ofthe rotor can be calculated as

120579119904= minustanminus1 (

119890120572

119890120573

) (12)

Also with estimated back EMFs using (2) the speed canbe computed algebraically as

119904=

1

119870119864

radic1198902120572+ 1198902120573 (13)

The speed estimation only uses the EMF constant119870119864and

the estimated back EMFs 119890120572and 119890

120573 The proposed HOSM

observer provides the properties of finite time convergenceand low chattering effect compared to the classical equivalentcontrol obtained using a low-pass filter [15]

4 Experimental Results

Experiments are performed with the three-phase 26WPMSM The specifications and parameters are provided inTable 1 The motor used in the experimental setup is aTBL-119894model TS4632N2050E510 3-phase PMSMThe PMSMis powered by a Fairchild FSB50325S smart power modulewhich includes 6 fast-recovery MOSFET (FRFET) invertersand 3 half-bridge high voltage integrated circuits (HVICs)

4 Mathematical Problems in Engineering

TMS320F2833335CAN

SCIPower module

JTAG emulator

PMSM

Encoder

DC link

Figure 1 Experimental setup

for FRFET gate driving Since it employs FRFET as a powerswitch it has much better robustness and larger safe opera-tion areas (SOA) than that using an IGBT-based power mod-ule or one-chip solution The experimental setup is shownin Figure 1 The space vector modulation (SVM) algorithmis used as modulation strategy and switching frequency ofthe PWM inverter is 15 kHz SMC 75 evaluation modulewith TMS320F28335 DSP controller is used It contains TexasInstruments 32-bit floating point DSP as well as analoginterfaces and JTAG emulator portThe board has analog-to-digital converter (AD) with 16 channels All the control vari-ables are monitored using graph window of Code ComposerStudio (CCS v33) after being converted to analog signalsthrough the digital-to-analog (DA) converter Real motorspeed (120596

119904) is measured using a high-resolution incremental

encoder with 2000 pulsesrotation and the estimated speed(119904) is obtained with the proposed HOSM schemeThe stator

currents of the PMSM aremeasured from the current sensorsand they are sent to TMS320F28335 viaAD converters In thesame way stator voltages are calculated using dc-bus voltagesensors and duty cycles of the inverter when the switchingfunctions are known

Three-phase currents and voltages are transformed totwo-phase stationary (120572 minus 120573) reference frame They are againtransformed to rotating (119889minus119902) reference frame for the controlPI (proportional and integral) controllers are used to regulatethe 119889 119902 synchronous frame currents 119894

119902and 119894119889 A functional

block diagram for the overall scheme is depicted in Figure 2The PMSM drive is operated in speed control mode Thespeed is also regulated using a PI controller to generate thereference current 119894ref

119902in the 119902-axis The reference current in

the 119889-axis 119894ref119889

is set to 0 For the implementation of HOSMobserver the sliding mode gains are selected as follows119870

1=

07 1198702= 60 119870

3= 35 and 119870

4= 45 The initial conditions

for the estimator are chosen as 119909(0) = [0 0 0 0]Several experiments have been performed to validate

the proposed HOSM scheme In the first part we presentthe results performance of the proposed method under no-load conditions for ramp change and step change and in thelater part similar experiments are conducted under loading

PMSM

Clarke

Position

HOSM

observer

S

S

Speed

Park120572120573

120572120573

120572120573PI PI

PIV120572

id i120572dq

dq

abc

iq

+

+

+ minus

minus

minus

ia

ib

V120573

i120573

irefd

irefq120596ref

Parkminus1

SVM3-phase

inverter

Figure 2 Functional block diagram for the overall scheme

condition For comparison the results obtained with first-order sliding mode observer are also presented

41 Under No-Load Condition In the first experiment aconstant speed reference 2000 rpm is provided for the first03 s a ramp input for the next 04 s followed by a constantspeed of 3500 rpm as shown in Figure 3(b) is employed Thereal currents of the PMSM for the first experiment are shownin Figure 3(a) The encoder speed and position are providedin Figures 3(b) and 3(c) The actual speed (120596

119904) exactly

follows the reference speed considered above Presence ofmeasurement noise can be clearly observed in (119894

120572120573) and

(120596119904 120579119904) With the proposed observer the estimated currents

and estimation error are shown in Figures 4(a) and 4(b) Realand estimated currents are very similar in both magnitudeand phase using the proposed method Figure 4(c) depictsthe estimated back EMFs obtained using (11) Despite thenoisy currents the back EMFs are relatively smooth whichconforms the theoretical claim of the proposed approachThe estimated speed computed analytically from back EMFswith (13) is shown in Figure 4(d) which exactly tracks actualspeed (120596

119904) and is shown for the comparisonThe convergence

accuracy depends on the accurate estimation of the back EMFcomponents and the back EMF constant 119870

119864 The HOSM

scheme enables a good reconstruction of the PMSM speedFigures 4(e) and 4(f) show the estimated rotor position andestimation error The estimated rotor position is robust withrespect to noise measurements and exactly matches withthe actual rotor position without any phase delay So theestimated rotor position can be used instead of the measuredone in the vector control of PMSM drive In usual practicethe values of 119877 and 119871 are not accurately known To testthe robustness the parameters (119877 and 119871) values are variedby plusmn10 and several experiments are conducted Similarperformancewas obtained in comparison to results presentedin Figure 4

For comparison the results obtained with conventionalsliding mode observer [3] are shown in Figure 5 The sliding

Mathematical Problems in Engineering 5

1

Time (s)0 01 03 05 07 09 1

minus1

0

1i 120572120573

(A)

(a)

1000

2000

3000

Time (s)0 01 03 05 07 09 1

120596s

(rpm

)

(b)

0

3

Time (s)

0 01 03 05 07 09 1

minus3

120579s

(rad

)

(c)

Figure 3 (a) Actual currents (b) Actual speed (c) Actual rotor position

mode gain is set to 50 for the observer design The presenceof noise in the estimated currents (Figure 5(a)) highly affectsthe estimation of back EMFs as shown in Figure 5(c) Theestimated back EMFs which correspond to the equivalentcontrols are obtained by filtering the switching functions ofthe observer with a 40Hz low-pass filter A proper boundarylayer is required to overcome the chattering phenomenonThe speed and position estimate in Figures 5(d)ndash5(f) exposethe problems with the conventional sliding mode observerfor back EMF estimation Also the estimated speed hasmore noise compared to the speed estimate with the HOSMobserver shown in Figure 4(d) The rotor position error inFigure 5(f) compared to Figure 4(f) clearly highlights theaccuracy obtained with proposed method Low-pass filteringclearly affects the estimation accuracyThe parameters for thefirst-order sliding mode are well-tuned to achieve the bestpossible results Errors are mainly due to filtering and the useof sigmoid function to avoid the chattering phenomenon

In the second experiment a step change in speed isprovided as reference In this experiment the speed estimateand position estimation error obtained with the proposedHOSM scheme for the step reference are shown in Figure 6Figure 6(a) shows the estimated and actual speeds obtainedusing the proposed HOSM observer and Figure 6(b) showsthe estimation error between the estimated and actual rotorpositions It can be seen from Figure 6(b) that the estimationhas no delay with the proposed approach For compari-son the results obtained with conventional first-order SMOare shown in Figure 7 Figure 7(a) shows the estimatedand actual speeds obtained using the first-order SMO andFigure 7(b) shows the estimation error between the estimatedand actual rotor positions Although the estimated speedfollows the actual speed it contains more noise compared to

the proposedHOSMobserverThese results further highlightthe robustness of the proposed method in the presenceof noise Further the chattering phenomenon is completelyeliminated and accurate position estimation can be obtainedeven in the presence of measurement noise

42 Under Loading Condition To test the performance of theproposed method a mechanical load 119869

119871= 007436 kg sdot cm2 is

connected to the motor Same set of parameters consideredfor no-load are employed for loading condition to test therobustness of the observer to parametric variations Theresults obtained with HOSM observer for a ramp changeare shown in Figure 8 Figures 8(a) and 8(b) show the esti-mated currents and their errors obtained using the proposedHOSM observer It can be observed from Figure 8(b) thatthe estimated and actual currents exactly match each otherFigure 8(c) shows the estimated unknown back EMFs whichare relatively smooth The corresponding estimated speedcalculated using (13) and the actual speed are depicted inFigure 8(d) The estimated rotor position and the estimationerror are shown in Figures 8(e) and 8(f) For comparison theestimated and actual speeds obtained with first-order SMOare shown in Figure 9(a) and the corresponding position esti-mation error is shown in Figure 9(b) In the final experimentunder loading condition step-like input reference is providedfor the system the results obtained with proposed methodand first-order SMO are shown in Figure 10 and Figure 11respectively Figures 10(a) and 10(b) show the estimatedactual speeds and position estimation error obtained withthe proposed HOSM observer while Figures 11(a) and 11(b)show the estimated actual speeds and position estimationerror obtained with the first-order SMO Under loading thenoise level is higher and the position estimation error slightly

6 Mathematical Problems in Engineering

Time (s)0 01

1

03 05 07 09 1

0

05

minus05

minus1

i 120572120573

(A)

(a)

Time (s)0 01 03 05 07 09 1

0

05

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus40

40

0

e 120572120573

(V)

(c)

1500

3000

4000

Time (s)0 01 03 05 07 09 1

EstimatedActual

120596s

(rpm

)

(d)

0

3

Time (s)0 01 03 05 07 09 1

minus3

120579s

(rad

)

(e)

0

1

2

Time (s)0 01 03 05 07 09 1

minus2

minus1120579sminus120579s (

rad)

(f)

Figure 4 Estimation using higher-order sliding mode observer under no-load (a) Estimated currents (b) Estimation current error (c)Estimated back EMFs (d) Estimated speed (e) Estimated rotor position (f) Estimation rotor position error

increasesHowever the speed andposition estimation remainrobust with the proposed observer when compared to first-order SMO Due to parametric uncertainty the speed estima-tion with proposedHOSMmethod shows a very small steadystate error at 90 of the rated speed (Figures 8(d) and 10(a))Under loading the speed and position estimation with first-order SMOare highly affected Further tuning of the observerparameters can overcome the problem

From the implementation one can conclude the follow-ing

(1) The HOSM method requires the proper selection ofsliding mode gains 119870

1 1198702 1198703 and 119870

4 The sliding

mode gains should satisfy the conditions given by(A15)-(A16) for the desired speed range If themotoroperates in wide speed range the sliding mode gainsmust be appropriately selected

(2) Since the quality of the speed estimate highly dependson the estimated back EMFs it deteriorates when

more noisy back EMFs (obtained due to high gains)are used in the calculation Compared to existingmethods the proposed scheme provides good rotorspeed and position estimation without phase delay inthe presence of noise Further one should note thatsensorless speed estimation methods based on backEMFs fail at very low speeds and standstill

(3) It should be pointed out that a chattering phe-nomenon occurs using the conventional SM observer[15]Therefore in first-order SMobserver the signumfunction is used as switching function The speedestimate is approximated by low-pass filtering thediscontinuous switching functions This delay shouldbe compensated with an additional phase compensa-tion loop As low-pass filtering is eliminated with theproposed method an additional phase compensationloop is not required

Mathematical Problems in Engineering 7

1

0

05

minus05

minus1

i 120572120573

(A)

Time (s)0 01 03 05 07 09 1

(a)

Time (s)0 01 03 05 07 09 1

0

05

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus40

40

0

e 120572120573

(V)

(c)

Time (s)0 01 03 05 07 09 1

1500

3000

4000

120596s

(rpm

)

EstimatedActual

(d)

Time (s)0 01 03 05 07 09 1

0

3

minus3

120579s

(rad

)

(e)

Time (s)0 01 03 05 07 09 1

0

1

2

minus2

minus1120579sminus120579s (

rad)

(f)

Figure 5 Estimation using conventional first-order sliding mode observer under no-load (a) Estimated currents (b) Estimation currenterror (c) Estimated back EMFs (d) Estimated speed (e) Estimated rotor position (f) Estimation rotor position error

1500

3000

4000

EstimatedActual

120596s120596

s(r

pm)

Time (s)0 02 04 05 06 08 1

(a)

120579sminus120579s

(rad

)

0

1

2

minus1

minus2

Time (s)0 02 04 05 06 08 1

(b)

Figure 6 With proposed HOSMmethod under no-load (a) Actual and estimated speed (b) Rotor position estimation error

8 Mathematical Problems in Engineering

1500

3000

4000

Time (s)0 02 04 05 06 08 1

EstimatedActual

120596s120596

s(r

pm)

(a)

Time (s)

0 02 04 05 06 08 1

120579sminus120579s

(rad

)

0

1

2

minus1

minus2

(b)

Figure 7 With conventional first-order SMO under no-load (a) Estimated speed (b) Rotor position estimation error

Time (s)0 01 03 05 07 09 1

0

06

minus06

i 120572120573

(A)

(a)

Time (s)0 01 03 05 07 09 1

0

05

12

minus12

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus60

60

0

e 120572120573

(V)

(c)

1500

3000

4000

Time (s)0 01 03 05 07 09 1

EstimatedActual

120596s

(rpm

)

(d)

0

3

Time (s)0 01 03 05 07 09 1

minus3

120579s

(rad

)

(e)

0

1

2

Time (s)0 01 03 05 07 09 1

minus2

minus1120579sminus

120579s (

rad)

(f)

Figure 8 With proposed HOSM method under load (a) Estimated currents (b) Estimation current errors (c) Estimated back EMFs (d)Estimated speed (e) Estimated rotor position (f) Rotor position error

Mathematical Problems in Engineering 9

Time (s)0 01 03 05 07 09 1

1500

3000

4000

120596s120596

s(r

pm)

EstimatedActual

(a)

Time (s)0 01 03 05 07 09 1

0

1

2

minus2

minus1

120579sminus120579s (

rad)

(b)

Figure 9 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

1500

3000

4000

0 02 04 05 06 08 1

Time (s)EstimatedActual

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1Time (s)

0

1

2

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 10 With proposed HOSMmethod under load (a) Estimated speed (b) Rotor position estimation error

(4) Furthermore compared to the classical first-orderSM technique no cutoff frequency has to be tunedInstead a simple integration is realized It enablesto reduce the time delay for the estimation (whichdepends on the sampling period) One should alsohighlight that the discontinuous part of 120601

2(depend-

ing on1198704) is usually low compared to the continuous

part of 1206012and this enables to reduce the chattering

phenomenon(5) Moreover from the experiments the proposed

method is robust to the parameter variations and themeasurement noise compared to the traditional SMobserver

(6) It is worth to point out that the proposed method iscomputationally complex compared to the traditionalSM observer However if properly tuned it has moreadvantages than the traditional SM observer Theexperiments conducted in this paper validate theadvantages of this method

(7) For the same set of parameters the speed and positionestimation remained accurate for both no-loadingand loading conditions This further highlights therobustness of the proposedmethod to parameter vari-ations that occur with loading and other conditions

5 Conclusion

This paper has presented a sensorless speed estimationmethod for the PMSM driveThe HOSMmethod is based ona modified version of super-twisting algorithmThe observerdynamics consist of sliding mode terms which are used toreconstruct the unknown back EMFs The speed is thenanalytically computed from back EMFs Experimental resultsvalidate the feasibility and effectiveness of the proposedHOSM for estimating the rotor position and speed of thePMSM Compared with the traditional SMO the proposedhigher-order SMO provides better estimation performance

Appendix

Finite-Time Stability

For any vector 119911 = [1199111 119911

119902]119879isin 119877119902 and any scalar 120572 isin 119877

we denote the following

sign (119911) = [sign (1199111) sign (119911

119902)]119879

|119911|120572= diag (10038161003816100381610038161199111

1003816100381610038161003816120572

10038161003816100381610038161003816119911119902

10038161003816100381610038161003816

120572

)

lceil119911rfloor120572= |119911|120572 sign (119911)

(A1)

10 Mathematical Problems in Engineering

1500

3000

4000

EstimatedActual

0 02 04 05 06 08 1

Time (s)

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1

0

1

2

Time (s)

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 11 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

For ease of exposition consider the following system

119904 (119905) = minus 119886119904 (119905) + ] (119905) + 119890 (119904 119905)

119904 (1199050) = 1199040

(A2)

where 119904 isin R and 119886 is a known positive constant and 119890(119904 119905) isthe unknown inputperturbation and

] (119905) = minus11987011206011 (119904 (119905)) minus 119870

2int

119905

0

1206012 (119904 (119905)) 119889119905 (A3)

where 1206011(119904(119905)) and 120601

2(119904(119905)) are defined in (7) and119870

111987021198703

and1198704are appropriately designed positive constants

Assumption A1 The time derivative of the unknowninputperturbation is upper bounded as follows

| 119890 (119904 119905)| le 120588 (A4)

for a positive constant 120588

Remark A2 The sliding dynamics 119904120572or 119904120573in (8) can be

directly expressed in the form of (A2) Further the condition(9) is similar to Assumption A1

Proposition A3 Under Assumption A1 the origin of sys-tem (A2) is a finite time stable equilibrium point Fur-ther the finite-time smooth estimation of the unknowninputperturbation 119890(119904 119905) is given by 119870

2int119905

01206012(119904(119905))119889119905

Proof Proof follows the work given in [20] Since |1206012(119904)| ge

1198702

42 one gets

| 119890 (119904 119905)| le10038161003816100381610038161206012 (119904)

1003816100381610038161003816 (A5)

if

1198704ge radic2120588 (A6)

Let us select a Hurwitz matrix 1198600

1198600= [

minus (1198701+ 119886) 1

minus1198702

0] (A7)

where1198701gt 0 and119870

2gt 0

The system (A2) (A3) can be equivalently representedby the system of two first-order equations

1199041= 1199042minus (1198701+ 119886) (119904

1+ 1198704lceil 1199041rfloor12

)

1199042= minus 119870

2(1199041+1198702

4

2sign (119904

1) +

3

21198704lceil 1199041rfloor12

) + 119890

(A8)

with 1199041= 119904 119904

2= 119890 minus 119870

2int119905

01206012(1199041) 119889119905 and

1198704=

11987011198703

1198701+ 119886

(A9)

The solutions of the discontinuous differential equations andinclusions are understood in the sense of Filippov

Let us consider the new state vector

120585 = [1205851

1205852

] = [1199041+ 1198704lceil 1199041rfloor12

1199042

] (A10)

The stability analysis of system (A8) is performed usingthe following candidate Lyapunov function [20]

119881 (120585) = 120585119879119875120585 (A11)

with 119875 = 119875119879

= [ 120582+41205982minus2120598

minus2120598 1] 120582 gt 0 and 120598 gt 0 It is worth

noting that the matrix 119875 is positive definite if 120582 and 120598 are anyreal number

Using the differential equations inclusion theory its timederivative along the solutions of the system is given by

= (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

119890]

(A12)

Mathematical Problems in Engineering 11

It can be shown that

le (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

)(120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

1205851

])

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879119876120585

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120582min (119876)100381710038171003817100381712058510038171003817100381710038172

(A13)with

119876 = [11987611198762

11987621198763

]

1198761= 2 (119870

1+ 119886) (120582 + 4120598

2) minus 4120598 (119870

2minus 1)

1198762= minus 2120598 (119870

1+ 119886) + (119870

2+ 1) minus (120582 + 4120598

2)

1198763= 4120598

(A14)

In order to guarantee the positive definiteness of matrix 119876one chooses

1198702= 120582 + 4120598

2+ 2120598 (119870

1+ 119886) (A15)

The matrix 119876 is positive definite if

1198701gt minus119886 +

4120598 + 2120598120582 + 81205983

120582+

1

4120598120582 (A16)

From (A10) one can deduce that100381710038171003817100381712058510038171003817100381710038172= 1205852

1+ 1205852

2

= 1199042

1+ 21198704

10038161003816100381610038161199041100381610038161003816100381632

+ 1198702

4

100381610038161003816100381611990411003816100381610038161003816 + 1205852

2

ge 1198702

4

100381610038161003816100381611990411003816100381610038161003816

(A17)

Since1198704gt 0

minus1198704

10038171003817100381710038171205851003817100381710038171003817

ge minus10038161003816100381610038161199041

1003816100381610038161003816minus12

(A18)

It implies that

le minus120582min (119876)

12058212

max (119875)

1198702

4

211988112

minus120582min (119876)

120582max (119875)119881 (A19)

The closed-loop system (A8) is stabilized in finite timeSince 120585 converges to zero in finite time 119904

1and 1199042converge

to 0 Therefore the term1198702int119905

01206012(119904(119905))119889119905 gives in finite time a

smooth estimation of the unknown perturbation 119890(119904 119905)

Nomenclature

120596119904 Rotor electrical speed

119894120572 119894120573 Currents in stationary reference frame

119881120572 119881120573 Voltages in stationary reference frame

119890120572 119890120573 EMFs in stationary reference frame

119877 Stator resistance119871 Synchronous inductance119870119864 EMF constant

120579119904 Rotor position angle

119879119897 Load torque

Conflict of Interests

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

Acknowledgment

This research was supported by the Basic Science ResearchProgram through theNational Research Foundation of Korea(NRF) funded by the Ministry of Education Science andTechnology (Grant no 2011-0023999)

References

[1] R Wu and G R Slemon ldquoA permanent magnet motor drivewithout a shaft sensorrdquo IEEE Transactions on Industry Applica-tions vol 27 no 5 pp 1005ndash1011 1991

[2] P Tomei and C M Verrelli ldquoObserver-based speed trackingcontrol for sensorless permanent magnet synchronous motorswith unknown load torquerdquo IEEE Transactions on AutomaticControl vol 56 no 6 pp 1484ndash1488 2011

[3] V Utkin J G Guldner and J Shi Sliding Mode Control onElectromechanical Systems Taylor and Francis New York NYUSA 1st edition 1999

[4] S Chai L Wang and R Rogers ldquoModel predictive controlof a permanent magnet synchronous motor with experimentalvalidationrdquoControl Engineering Practice vol 21 no 11 pp 1584ndash1593 2013

[5] T Orlowska-Kowalska and M Dybkowski ldquoStator-current-based MRAS estimator for a wide range speed-sensorlessinduction-motor driverdquo IEEE Transactions on Industrial Elec-tronics vol 57 no 4 pp 1296ndash1308 2010

[6] M L Corradini G Ippoliti S Longhi and G Orlando ldquoAquasi-sliding mode approach for robust control and speedestimation of PM synchronous motorsrdquo IEEE Transactions onIndustrial Electronics vol 59 no 2 pp 1096ndash1104 2012

[7] B K Bose Modern Power Electronics and AC Drives Prentice-Hall Upper Saddle River NJ USA 2002

[8] K C Veluvolu and Y C Soh ldquoMultiple sliding mode observersand unknown input estimations for Lipschitz nonlinear sys-temsrdquo International Journal of Robust and Nonlinear Controlvol 21 no 11 pp 1322ndash1340 2011

[9] K C Veluvolu and D Lee ldquoSliding mode high-gain observersfor a class of uncertain nonlinear systemsrdquoAppliedMathematicsLetters vol 24 no 3 pp 329ndash334 2011

[10] K C Veluvolu and Y C Soh ldquoFault reconstruction and stateestimationwith slidingmode observers for Lipschitz non-linearsystemsrdquo IET Control Theory amp Applications vol 5 no 11 pp1255ndash1263 2011

[11] M Comanescu and L Xu ldquoSliding-mode MRAS speed estima-tors for sensorless vector control of induction machinerdquo IEEETransactions on Industrial Electronics vol 53 no 1 pp 146ndash1532006

[12] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[13] K C Veluvolu M Y Kim and D Lee ldquoNonlinear sliding modehigh-gain observers for fault estimationrdquo International Journalof Systems Science Principles and Applications of Systems andIntegration vol 42 no 7 pp 1065ndash1074 2011

12 Mathematical Problems in Engineering

[14] K C Veluvolu M Defoort and Y C Soh ldquoHigh-gain observerwith sliding mode for nonlinear state estimation and faultreconstructionrdquo Journal of the Franklin Institute Engineeringand Applied Mathematics vol 351 no 4 pp 1995ndash2014 2014

[15] M Comanescu ldquoCascaded EMF and speed sliding modeobserver for the nonsalient PMSMrdquo in Proceedings of the 36thAnnual Conference of the IEEE Industrial Electronics Society(IECON rsquo10) pp 792ndash797 Glendale Ariz November 2010

[16] M Comanescu ldquoAn induction-motor speed estimator based onintegral sliding-mode current controlrdquo IEEE Transactions onIndustrial Electronics vol 56 no 9 pp 3414ndash3423 2009

[17] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[18] M Comanescu L Xu and T D Batzel ldquoDecoupled currentcontrol of sensorless induction-motor drives by integral slidingmoderdquo IEEE Transactions on Industrial Electronics vol 55 no11 pp 3836ndash3845 2008

[19] H Kim J Son and J Lee ldquoA high-speed sliding-mode observerfor the sensorless speed control of a PMSMrdquo IEEE Transactionson Industrial Electronics vol 58 no 9 pp 4069ndash4077 2011

[20] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

[21] T Floquet and J P Barbot ldquoSuper twisting algorithm-basedstep-by-step sliding mode observers for nonlinear systemswith unknown inputsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 38no 10 pp 803ndash815 2007

[22] J J Rath K C Veluvolu M Defoort and Y C Soh ldquoHigher-order sliding mode observer for estimation of tyre frictionin ground vehiclesrdquo IET Proceedings on Control Theory andApplications vol 8 no 6 pp 399ndash408 2014

[23] L Fridman and A Levant ldquoHigher order sliding modesSliding mode control in engineeringrdquo in Sliding Mode Controlin Engineering J P Barbot and W Perruquetti Eds MarcelDekker New York NY USA 2002

[24] M Ezzat J De Leon N Gonzalez and A GlumineauldquoObserver-controller scheme using high order sliding modetechniques for sensorless speed control of permanent magnetsynchronous motorrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 4012ndash4017 December2010

[25] D Zaltni and M N Abdelkrim ldquoRobust speed and positionobserver using HOSM for sensor-less SPMSM controlrdquoin Proceedings of the 7th International Multi-Conference onSystems Signals and Devices (SSD rsquo10) pp 1ndash6 June 2010

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

For the implementation of the sliding mode observer thefollowing sliding surfaces are selected

119904120572 (119905) =

120572minus 119894120572

119904120573 (119905) =

120573minus 119894120573

(4)

where 120572and 120573are estimated currents and 119894

120572and 119894120573are actual

currentsWith the PMSMmodel defined in (1) the observer can be

designed as follows

119889120572

119889119905=minus119877

119871120572+1

119871119881120572+1

119871]1 (119905)

119889120573

119889119905=minus119877

119871120573+1

119871119881120573+1

119871]2 (119905)

(5)

The sliding mode terms are given by

]1 (119905) = minus119870

11206011(119904120572 (119905)) minus 119870

2int

119905

0

1206012(119904120572 (119905)) 119889119905

]2 (119905) = minus119870

11206011(119904120573 (119905)) minus 119870

2int

119905

0

1206012(119904120573 (119905)) 119889119905

(6)

where1206011(119904120572 (119905)) = 119904

120572 (119905) + 1198703|119904120572 (119905) |12 sign (119904

120572 (119905))

1206012(119904120572 (119905)) = 119904

120572 (119905) +1198702

4

2sign (119904

120572 (119905))

+3

21198704

1003816100381610038161003816119904120572 (119905)100381610038161003816100381612 sign (119904

120572 (119905))

(7)

where119870111987021198703 and119870

4are appropriately designed positive

constants Similarly the functions 1206011(119904120573(119905)) and 120601

2(119904120573(119905)) can

be obtained by replacing 119904120572(119905) with 119904

120573(119905) in (7)

31 Sliding Mode Stability To prove the stability of theobserver system the time derivatives of the sliding surfacesare obtained from (1) and (5) as

119904120572=minus119877

119871119904120572+1

119871119890120572+1

119871]1 (119905)

119904120573=minus119877

119871119904120573+1

119871119890120573+1

119871]2 (119905)

(8)

From (2) and (3) we can establish the boundedness ofback EMFs At least locally there are positive constants 120588

1and

1205882such that the following terms are bounded as follows

1003816100381610038161003816 1198901205721003816100381610038161003816 le 1205881

10038161003816100381610038161003816119890120573

10038161003816100381610038161003816le 1205882

(9)

for some positive constants 1205881and 120588

2 The above condition

(9) is not restrictive since 120596119904 119879119897 and 119890

120572and 119890120573 119894120572 and 119894

120573are

continuous on a compact set

Theorem 1 With the condition (9) the sliding dynamics 119904120572

and 119904120573are stabilized towards zero in finite time

Proof See Proposition A3 in the appendix

Table 1 Motor parameters

Parameters ValuesRating 26 [W]

Speed 120596 4000 [rmin]Stator resistance 119877 24 [Ω]

Stator inductance 119871 065 [mH]

Back EMF constant 119870119864

0156 [Vsrad]Inertia 119869 0004 times 10

minus4[Kgsdotm2

]

Viscous friction 119891V 0004 times 10minus4

[Nmsdotsrad]Rotor flux 120601

1198980025

Number of pole pairs 119875 4

32 Speed and Rotor Position Estimation According toTheorem 1 the origin of system (8) has a finite time stableequilibrium In the sliding mode we have 119904

120572= 119904120572= 0 and

119904120573

= 119904120573

= 0 The reduced order dynamics of system (8)becomes

0 = minus1198702int

119905

0

1206012(119904120572 (119905)) 119889119905 + 119890

120572

0 = minus1198702int

119905

0

1206012(119904120573 (119905)) 119889119905 + 119890

120573

(10)

Hence a smooth estimation of the unknown back EMFscan be obtained in finite time as follows

119890120572= 1198702int

119905

0

1206012(119904120572 (119905)) 119889119905

119890120573= 1198702int

119905

0

1206012(119904120573 (119905)) 119889119905

(11)

Using the estimated back EMF voltages the position ofthe rotor can be calculated as

120579119904= minustanminus1 (

119890120572

119890120573

) (12)

Also with estimated back EMFs using (2) the speed canbe computed algebraically as

119904=

1

119870119864

radic1198902120572+ 1198902120573 (13)

The speed estimation only uses the EMF constant119870119864and

the estimated back EMFs 119890120572and 119890

120573 The proposed HOSM

observer provides the properties of finite time convergenceand low chattering effect compared to the classical equivalentcontrol obtained using a low-pass filter [15]

4 Experimental Results

Experiments are performed with the three-phase 26WPMSM The specifications and parameters are provided inTable 1 The motor used in the experimental setup is aTBL-119894model TS4632N2050E510 3-phase PMSMThe PMSMis powered by a Fairchild FSB50325S smart power modulewhich includes 6 fast-recovery MOSFET (FRFET) invertersand 3 half-bridge high voltage integrated circuits (HVICs)

4 Mathematical Problems in Engineering

TMS320F2833335CAN

SCIPower module

JTAG emulator

PMSM

Encoder

DC link

Figure 1 Experimental setup

for FRFET gate driving Since it employs FRFET as a powerswitch it has much better robustness and larger safe opera-tion areas (SOA) than that using an IGBT-based power mod-ule or one-chip solution The experimental setup is shownin Figure 1 The space vector modulation (SVM) algorithmis used as modulation strategy and switching frequency ofthe PWM inverter is 15 kHz SMC 75 evaluation modulewith TMS320F28335 DSP controller is used It contains TexasInstruments 32-bit floating point DSP as well as analoginterfaces and JTAG emulator portThe board has analog-to-digital converter (AD) with 16 channels All the control vari-ables are monitored using graph window of Code ComposerStudio (CCS v33) after being converted to analog signalsthrough the digital-to-analog (DA) converter Real motorspeed (120596

119904) is measured using a high-resolution incremental

encoder with 2000 pulsesrotation and the estimated speed(119904) is obtained with the proposed HOSM schemeThe stator

currents of the PMSM aremeasured from the current sensorsand they are sent to TMS320F28335 viaAD converters In thesame way stator voltages are calculated using dc-bus voltagesensors and duty cycles of the inverter when the switchingfunctions are known

Three-phase currents and voltages are transformed totwo-phase stationary (120572 minus 120573) reference frame They are againtransformed to rotating (119889minus119902) reference frame for the controlPI (proportional and integral) controllers are used to regulatethe 119889 119902 synchronous frame currents 119894

119902and 119894119889 A functional

block diagram for the overall scheme is depicted in Figure 2The PMSM drive is operated in speed control mode Thespeed is also regulated using a PI controller to generate thereference current 119894ref

119902in the 119902-axis The reference current in

the 119889-axis 119894ref119889

is set to 0 For the implementation of HOSMobserver the sliding mode gains are selected as follows119870

1=

07 1198702= 60 119870

3= 35 and 119870

4= 45 The initial conditions

for the estimator are chosen as 119909(0) = [0 0 0 0]Several experiments have been performed to validate

the proposed HOSM scheme In the first part we presentthe results performance of the proposed method under no-load conditions for ramp change and step change and in thelater part similar experiments are conducted under loading

PMSM

Clarke

Position

HOSM

observer

S

S

Speed

Park120572120573

120572120573

120572120573PI PI

PIV120572

id i120572dq

dq

abc

iq

+

+

+ minus

minus

minus

ia

ib

V120573

i120573

irefd

irefq120596ref

Parkminus1

SVM3-phase

inverter

Figure 2 Functional block diagram for the overall scheme

condition For comparison the results obtained with first-order sliding mode observer are also presented

41 Under No-Load Condition In the first experiment aconstant speed reference 2000 rpm is provided for the first03 s a ramp input for the next 04 s followed by a constantspeed of 3500 rpm as shown in Figure 3(b) is employed Thereal currents of the PMSM for the first experiment are shownin Figure 3(a) The encoder speed and position are providedin Figures 3(b) and 3(c) The actual speed (120596

119904) exactly

follows the reference speed considered above Presence ofmeasurement noise can be clearly observed in (119894

120572120573) and

(120596119904 120579119904) With the proposed observer the estimated currents

and estimation error are shown in Figures 4(a) and 4(b) Realand estimated currents are very similar in both magnitudeand phase using the proposed method Figure 4(c) depictsthe estimated back EMFs obtained using (11) Despite thenoisy currents the back EMFs are relatively smooth whichconforms the theoretical claim of the proposed approachThe estimated speed computed analytically from back EMFswith (13) is shown in Figure 4(d) which exactly tracks actualspeed (120596

119904) and is shown for the comparisonThe convergence

accuracy depends on the accurate estimation of the back EMFcomponents and the back EMF constant 119870

119864 The HOSM

scheme enables a good reconstruction of the PMSM speedFigures 4(e) and 4(f) show the estimated rotor position andestimation error The estimated rotor position is robust withrespect to noise measurements and exactly matches withthe actual rotor position without any phase delay So theestimated rotor position can be used instead of the measuredone in the vector control of PMSM drive In usual practicethe values of 119877 and 119871 are not accurately known To testthe robustness the parameters (119877 and 119871) values are variedby plusmn10 and several experiments are conducted Similarperformancewas obtained in comparison to results presentedin Figure 4

For comparison the results obtained with conventionalsliding mode observer [3] are shown in Figure 5 The sliding

Mathematical Problems in Engineering 5

1

Time (s)0 01 03 05 07 09 1

minus1

0

1i 120572120573

(A)

(a)

1000

2000

3000

Time (s)0 01 03 05 07 09 1

120596s

(rpm

)

(b)

0

3

Time (s)

0 01 03 05 07 09 1

minus3

120579s

(rad

)

(c)

Figure 3 (a) Actual currents (b) Actual speed (c) Actual rotor position

mode gain is set to 50 for the observer design The presenceof noise in the estimated currents (Figure 5(a)) highly affectsthe estimation of back EMFs as shown in Figure 5(c) Theestimated back EMFs which correspond to the equivalentcontrols are obtained by filtering the switching functions ofthe observer with a 40Hz low-pass filter A proper boundarylayer is required to overcome the chattering phenomenonThe speed and position estimate in Figures 5(d)ndash5(f) exposethe problems with the conventional sliding mode observerfor back EMF estimation Also the estimated speed hasmore noise compared to the speed estimate with the HOSMobserver shown in Figure 4(d) The rotor position error inFigure 5(f) compared to Figure 4(f) clearly highlights theaccuracy obtained with proposed method Low-pass filteringclearly affects the estimation accuracyThe parameters for thefirst-order sliding mode are well-tuned to achieve the bestpossible results Errors are mainly due to filtering and the useof sigmoid function to avoid the chattering phenomenon

In the second experiment a step change in speed isprovided as reference In this experiment the speed estimateand position estimation error obtained with the proposedHOSM scheme for the step reference are shown in Figure 6Figure 6(a) shows the estimated and actual speeds obtainedusing the proposed HOSM observer and Figure 6(b) showsthe estimation error between the estimated and actual rotorpositions It can be seen from Figure 6(b) that the estimationhas no delay with the proposed approach For compari-son the results obtained with conventional first-order SMOare shown in Figure 7 Figure 7(a) shows the estimatedand actual speeds obtained using the first-order SMO andFigure 7(b) shows the estimation error between the estimatedand actual rotor positions Although the estimated speedfollows the actual speed it contains more noise compared to

the proposedHOSMobserverThese results further highlightthe robustness of the proposed method in the presenceof noise Further the chattering phenomenon is completelyeliminated and accurate position estimation can be obtainedeven in the presence of measurement noise

42 Under Loading Condition To test the performance of theproposed method a mechanical load 119869

119871= 007436 kg sdot cm2 is

connected to the motor Same set of parameters consideredfor no-load are employed for loading condition to test therobustness of the observer to parametric variations Theresults obtained with HOSM observer for a ramp changeare shown in Figure 8 Figures 8(a) and 8(b) show the esti-mated currents and their errors obtained using the proposedHOSM observer It can be observed from Figure 8(b) thatthe estimated and actual currents exactly match each otherFigure 8(c) shows the estimated unknown back EMFs whichare relatively smooth The corresponding estimated speedcalculated using (13) and the actual speed are depicted inFigure 8(d) The estimated rotor position and the estimationerror are shown in Figures 8(e) and 8(f) For comparison theestimated and actual speeds obtained with first-order SMOare shown in Figure 9(a) and the corresponding position esti-mation error is shown in Figure 9(b) In the final experimentunder loading condition step-like input reference is providedfor the system the results obtained with proposed methodand first-order SMO are shown in Figure 10 and Figure 11respectively Figures 10(a) and 10(b) show the estimatedactual speeds and position estimation error obtained withthe proposed HOSM observer while Figures 11(a) and 11(b)show the estimated actual speeds and position estimationerror obtained with the first-order SMO Under loading thenoise level is higher and the position estimation error slightly

6 Mathematical Problems in Engineering

Time (s)0 01

1

03 05 07 09 1

0

05

minus05

minus1

i 120572120573

(A)

(a)

Time (s)0 01 03 05 07 09 1

0

05

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus40

40

0

e 120572120573

(V)

(c)

1500

3000

4000

Time (s)0 01 03 05 07 09 1

EstimatedActual

120596s

(rpm

)

(d)

0

3

Time (s)0 01 03 05 07 09 1

minus3

120579s

(rad

)

(e)

0

1

2

Time (s)0 01 03 05 07 09 1

minus2

minus1120579sminus120579s (

rad)

(f)

Figure 4 Estimation using higher-order sliding mode observer under no-load (a) Estimated currents (b) Estimation current error (c)Estimated back EMFs (d) Estimated speed (e) Estimated rotor position (f) Estimation rotor position error

increasesHowever the speed andposition estimation remainrobust with the proposed observer when compared to first-order SMO Due to parametric uncertainty the speed estima-tion with proposedHOSMmethod shows a very small steadystate error at 90 of the rated speed (Figures 8(d) and 10(a))Under loading the speed and position estimation with first-order SMOare highly affected Further tuning of the observerparameters can overcome the problem

From the implementation one can conclude the follow-ing

(1) The HOSM method requires the proper selection ofsliding mode gains 119870

1 1198702 1198703 and 119870

4 The sliding

mode gains should satisfy the conditions given by(A15)-(A16) for the desired speed range If themotoroperates in wide speed range the sliding mode gainsmust be appropriately selected

(2) Since the quality of the speed estimate highly dependson the estimated back EMFs it deteriorates when

more noisy back EMFs (obtained due to high gains)are used in the calculation Compared to existingmethods the proposed scheme provides good rotorspeed and position estimation without phase delay inthe presence of noise Further one should note thatsensorless speed estimation methods based on backEMFs fail at very low speeds and standstill

(3) It should be pointed out that a chattering phe-nomenon occurs using the conventional SM observer[15]Therefore in first-order SMobserver the signumfunction is used as switching function The speedestimate is approximated by low-pass filtering thediscontinuous switching functions This delay shouldbe compensated with an additional phase compensa-tion loop As low-pass filtering is eliminated with theproposed method an additional phase compensationloop is not required

Mathematical Problems in Engineering 7

1

0

05

minus05

minus1

i 120572120573

(A)

Time (s)0 01 03 05 07 09 1

(a)

Time (s)0 01 03 05 07 09 1

0

05

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus40

40

0

e 120572120573

(V)

(c)

Time (s)0 01 03 05 07 09 1

1500

3000

4000

120596s

(rpm

)

EstimatedActual

(d)

Time (s)0 01 03 05 07 09 1

0

3

minus3

120579s

(rad

)

(e)

Time (s)0 01 03 05 07 09 1

0

1

2

minus2

minus1120579sminus120579s (

rad)

(f)

Figure 5 Estimation using conventional first-order sliding mode observer under no-load (a) Estimated currents (b) Estimation currenterror (c) Estimated back EMFs (d) Estimated speed (e) Estimated rotor position (f) Estimation rotor position error

1500

3000

4000

EstimatedActual

120596s120596

s(r

pm)

Time (s)0 02 04 05 06 08 1

(a)

120579sminus120579s

(rad

)

0

1

2

minus1

minus2

Time (s)0 02 04 05 06 08 1

(b)

Figure 6 With proposed HOSMmethod under no-load (a) Actual and estimated speed (b) Rotor position estimation error

8 Mathematical Problems in Engineering

1500

3000

4000

Time (s)0 02 04 05 06 08 1

EstimatedActual

120596s120596

s(r

pm)

(a)

Time (s)

0 02 04 05 06 08 1

120579sminus120579s

(rad

)

0

1

2

minus1

minus2

(b)

Figure 7 With conventional first-order SMO under no-load (a) Estimated speed (b) Rotor position estimation error

Time (s)0 01 03 05 07 09 1

0

06

minus06

i 120572120573

(A)

(a)

Time (s)0 01 03 05 07 09 1

0

05

12

minus12

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus60

60

0

e 120572120573

(V)

(c)

1500

3000

4000

Time (s)0 01 03 05 07 09 1

EstimatedActual

120596s

(rpm

)

(d)

0

3

Time (s)0 01 03 05 07 09 1

minus3

120579s

(rad

)

(e)

0

1

2

Time (s)0 01 03 05 07 09 1

minus2

minus1120579sminus

120579s (

rad)

(f)

Figure 8 With proposed HOSM method under load (a) Estimated currents (b) Estimation current errors (c) Estimated back EMFs (d)Estimated speed (e) Estimated rotor position (f) Rotor position error

Mathematical Problems in Engineering 9

Time (s)0 01 03 05 07 09 1

1500

3000

4000

120596s120596

s(r

pm)

EstimatedActual

(a)

Time (s)0 01 03 05 07 09 1

0

1

2

minus2

minus1

120579sminus120579s (

rad)

(b)

Figure 9 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

1500

3000

4000

0 02 04 05 06 08 1

Time (s)EstimatedActual

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1Time (s)

0

1

2

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 10 With proposed HOSMmethod under load (a) Estimated speed (b) Rotor position estimation error

(4) Furthermore compared to the classical first-orderSM technique no cutoff frequency has to be tunedInstead a simple integration is realized It enablesto reduce the time delay for the estimation (whichdepends on the sampling period) One should alsohighlight that the discontinuous part of 120601

2(depend-

ing on1198704) is usually low compared to the continuous

part of 1206012and this enables to reduce the chattering

phenomenon(5) Moreover from the experiments the proposed

method is robust to the parameter variations and themeasurement noise compared to the traditional SMobserver

(6) It is worth to point out that the proposed method iscomputationally complex compared to the traditionalSM observer However if properly tuned it has moreadvantages than the traditional SM observer Theexperiments conducted in this paper validate theadvantages of this method

(7) For the same set of parameters the speed and positionestimation remained accurate for both no-loadingand loading conditions This further highlights therobustness of the proposedmethod to parameter vari-ations that occur with loading and other conditions

5 Conclusion

This paper has presented a sensorless speed estimationmethod for the PMSM driveThe HOSMmethod is based ona modified version of super-twisting algorithmThe observerdynamics consist of sliding mode terms which are used toreconstruct the unknown back EMFs The speed is thenanalytically computed from back EMFs Experimental resultsvalidate the feasibility and effectiveness of the proposedHOSM for estimating the rotor position and speed of thePMSM Compared with the traditional SMO the proposedhigher-order SMO provides better estimation performance

Appendix

Finite-Time Stability

For any vector 119911 = [1199111 119911

119902]119879isin 119877119902 and any scalar 120572 isin 119877

we denote the following

sign (119911) = [sign (1199111) sign (119911

119902)]119879

|119911|120572= diag (10038161003816100381610038161199111

1003816100381610038161003816120572

10038161003816100381610038161003816119911119902

10038161003816100381610038161003816

120572

)

lceil119911rfloor120572= |119911|120572 sign (119911)

(A1)

10 Mathematical Problems in Engineering

1500

3000

4000

EstimatedActual

0 02 04 05 06 08 1

Time (s)

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1

0

1

2

Time (s)

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 11 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

For ease of exposition consider the following system

119904 (119905) = minus 119886119904 (119905) + ] (119905) + 119890 (119904 119905)

119904 (1199050) = 1199040

(A2)

where 119904 isin R and 119886 is a known positive constant and 119890(119904 119905) isthe unknown inputperturbation and

] (119905) = minus11987011206011 (119904 (119905)) minus 119870

2int

119905

0

1206012 (119904 (119905)) 119889119905 (A3)

where 1206011(119904(119905)) and 120601

2(119904(119905)) are defined in (7) and119870

111987021198703

and1198704are appropriately designed positive constants

Assumption A1 The time derivative of the unknowninputperturbation is upper bounded as follows

| 119890 (119904 119905)| le 120588 (A4)

for a positive constant 120588

Remark A2 The sliding dynamics 119904120572or 119904120573in (8) can be

directly expressed in the form of (A2) Further the condition(9) is similar to Assumption A1

Proposition A3 Under Assumption A1 the origin of sys-tem (A2) is a finite time stable equilibrium point Fur-ther the finite-time smooth estimation of the unknowninputperturbation 119890(119904 119905) is given by 119870

2int119905

01206012(119904(119905))119889119905

Proof Proof follows the work given in [20] Since |1206012(119904)| ge

1198702

42 one gets

| 119890 (119904 119905)| le10038161003816100381610038161206012 (119904)

1003816100381610038161003816 (A5)

if

1198704ge radic2120588 (A6)

Let us select a Hurwitz matrix 1198600

1198600= [

minus (1198701+ 119886) 1

minus1198702

0] (A7)

where1198701gt 0 and119870

2gt 0

The system (A2) (A3) can be equivalently representedby the system of two first-order equations

1199041= 1199042minus (1198701+ 119886) (119904

1+ 1198704lceil 1199041rfloor12

)

1199042= minus 119870

2(1199041+1198702

4

2sign (119904

1) +

3

21198704lceil 1199041rfloor12

) + 119890

(A8)

with 1199041= 119904 119904

2= 119890 minus 119870

2int119905

01206012(1199041) 119889119905 and

1198704=

11987011198703

1198701+ 119886

(A9)

The solutions of the discontinuous differential equations andinclusions are understood in the sense of Filippov

Let us consider the new state vector

120585 = [1205851

1205852

] = [1199041+ 1198704lceil 1199041rfloor12

1199042

] (A10)

The stability analysis of system (A8) is performed usingthe following candidate Lyapunov function [20]

119881 (120585) = 120585119879119875120585 (A11)

with 119875 = 119875119879

= [ 120582+41205982minus2120598

minus2120598 1] 120582 gt 0 and 120598 gt 0 It is worth

noting that the matrix 119875 is positive definite if 120582 and 120598 are anyreal number

Using the differential equations inclusion theory its timederivative along the solutions of the system is given by

= (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

119890]

(A12)

Mathematical Problems in Engineering 11

It can be shown that

le (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

)(120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

1205851

])

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879119876120585

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120582min (119876)100381710038171003817100381712058510038171003817100381710038172

(A13)with

119876 = [11987611198762

11987621198763

]

1198761= 2 (119870

1+ 119886) (120582 + 4120598

2) minus 4120598 (119870

2minus 1)

1198762= minus 2120598 (119870

1+ 119886) + (119870

2+ 1) minus (120582 + 4120598

2)

1198763= 4120598

(A14)

In order to guarantee the positive definiteness of matrix 119876one chooses

1198702= 120582 + 4120598

2+ 2120598 (119870

1+ 119886) (A15)

The matrix 119876 is positive definite if

1198701gt minus119886 +

4120598 + 2120598120582 + 81205983

120582+

1

4120598120582 (A16)

From (A10) one can deduce that100381710038171003817100381712058510038171003817100381710038172= 1205852

1+ 1205852

2

= 1199042

1+ 21198704

10038161003816100381610038161199041100381610038161003816100381632

+ 1198702

4

100381610038161003816100381611990411003816100381610038161003816 + 1205852

2

ge 1198702

4

100381610038161003816100381611990411003816100381610038161003816

(A17)

Since1198704gt 0

minus1198704

10038171003817100381710038171205851003817100381710038171003817

ge minus10038161003816100381610038161199041

1003816100381610038161003816minus12

(A18)

It implies that

le minus120582min (119876)

12058212

max (119875)

1198702

4

211988112

minus120582min (119876)

120582max (119875)119881 (A19)

The closed-loop system (A8) is stabilized in finite timeSince 120585 converges to zero in finite time 119904

1and 1199042converge

to 0 Therefore the term1198702int119905

01206012(119904(119905))119889119905 gives in finite time a

smooth estimation of the unknown perturbation 119890(119904 119905)

Nomenclature

120596119904 Rotor electrical speed

119894120572 119894120573 Currents in stationary reference frame

119881120572 119881120573 Voltages in stationary reference frame

119890120572 119890120573 EMFs in stationary reference frame

119877 Stator resistance119871 Synchronous inductance119870119864 EMF constant

120579119904 Rotor position angle

119879119897 Load torque

Conflict of Interests

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

Acknowledgment

This research was supported by the Basic Science ResearchProgram through theNational Research Foundation of Korea(NRF) funded by the Ministry of Education Science andTechnology (Grant no 2011-0023999)

References

[1] R Wu and G R Slemon ldquoA permanent magnet motor drivewithout a shaft sensorrdquo IEEE Transactions on Industry Applica-tions vol 27 no 5 pp 1005ndash1011 1991

[2] P Tomei and C M Verrelli ldquoObserver-based speed trackingcontrol for sensorless permanent magnet synchronous motorswith unknown load torquerdquo IEEE Transactions on AutomaticControl vol 56 no 6 pp 1484ndash1488 2011

[3] V Utkin J G Guldner and J Shi Sliding Mode Control onElectromechanical Systems Taylor and Francis New York NYUSA 1st edition 1999

[4] S Chai L Wang and R Rogers ldquoModel predictive controlof a permanent magnet synchronous motor with experimentalvalidationrdquoControl Engineering Practice vol 21 no 11 pp 1584ndash1593 2013

[5] T Orlowska-Kowalska and M Dybkowski ldquoStator-current-based MRAS estimator for a wide range speed-sensorlessinduction-motor driverdquo IEEE Transactions on Industrial Elec-tronics vol 57 no 4 pp 1296ndash1308 2010

[6] M L Corradini G Ippoliti S Longhi and G Orlando ldquoAquasi-sliding mode approach for robust control and speedestimation of PM synchronous motorsrdquo IEEE Transactions onIndustrial Electronics vol 59 no 2 pp 1096ndash1104 2012

[7] B K Bose Modern Power Electronics and AC Drives Prentice-Hall Upper Saddle River NJ USA 2002

[8] K C Veluvolu and Y C Soh ldquoMultiple sliding mode observersand unknown input estimations for Lipschitz nonlinear sys-temsrdquo International Journal of Robust and Nonlinear Controlvol 21 no 11 pp 1322ndash1340 2011

[9] K C Veluvolu and D Lee ldquoSliding mode high-gain observersfor a class of uncertain nonlinear systemsrdquoAppliedMathematicsLetters vol 24 no 3 pp 329ndash334 2011

[10] K C Veluvolu and Y C Soh ldquoFault reconstruction and stateestimationwith slidingmode observers for Lipschitz non-linearsystemsrdquo IET Control Theory amp Applications vol 5 no 11 pp1255ndash1263 2011

[11] M Comanescu and L Xu ldquoSliding-mode MRAS speed estima-tors for sensorless vector control of induction machinerdquo IEEETransactions on Industrial Electronics vol 53 no 1 pp 146ndash1532006

[12] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[13] K C Veluvolu M Y Kim and D Lee ldquoNonlinear sliding modehigh-gain observers for fault estimationrdquo International Journalof Systems Science Principles and Applications of Systems andIntegration vol 42 no 7 pp 1065ndash1074 2011

12 Mathematical Problems in Engineering

[14] K C Veluvolu M Defoort and Y C Soh ldquoHigh-gain observerwith sliding mode for nonlinear state estimation and faultreconstructionrdquo Journal of the Franklin Institute Engineeringand Applied Mathematics vol 351 no 4 pp 1995ndash2014 2014

[15] M Comanescu ldquoCascaded EMF and speed sliding modeobserver for the nonsalient PMSMrdquo in Proceedings of the 36thAnnual Conference of the IEEE Industrial Electronics Society(IECON rsquo10) pp 792ndash797 Glendale Ariz November 2010

[16] M Comanescu ldquoAn induction-motor speed estimator based onintegral sliding-mode current controlrdquo IEEE Transactions onIndustrial Electronics vol 56 no 9 pp 3414ndash3423 2009

[17] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[18] M Comanescu L Xu and T D Batzel ldquoDecoupled currentcontrol of sensorless induction-motor drives by integral slidingmoderdquo IEEE Transactions on Industrial Electronics vol 55 no11 pp 3836ndash3845 2008

[19] H Kim J Son and J Lee ldquoA high-speed sliding-mode observerfor the sensorless speed control of a PMSMrdquo IEEE Transactionson Industrial Electronics vol 58 no 9 pp 4069ndash4077 2011

[20] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

[21] T Floquet and J P Barbot ldquoSuper twisting algorithm-basedstep-by-step sliding mode observers for nonlinear systemswith unknown inputsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 38no 10 pp 803ndash815 2007

[22] J J Rath K C Veluvolu M Defoort and Y C Soh ldquoHigher-order sliding mode observer for estimation of tyre frictionin ground vehiclesrdquo IET Proceedings on Control Theory andApplications vol 8 no 6 pp 399ndash408 2014

[23] L Fridman and A Levant ldquoHigher order sliding modesSliding mode control in engineeringrdquo in Sliding Mode Controlin Engineering J P Barbot and W Perruquetti Eds MarcelDekker New York NY USA 2002

[24] M Ezzat J De Leon N Gonzalez and A GlumineauldquoObserver-controller scheme using high order sliding modetechniques for sensorless speed control of permanent magnetsynchronous motorrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 4012ndash4017 December2010

[25] D Zaltni and M N Abdelkrim ldquoRobust speed and positionobserver using HOSM for sensor-less SPMSM controlrdquoin Proceedings of the 7th International Multi-Conference onSystems Signals and Devices (SSD rsquo10) pp 1ndash6 June 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

TMS320F2833335CAN

SCIPower module

JTAG emulator

PMSM

Encoder

DC link

Figure 1 Experimental setup

for FRFET gate driving Since it employs FRFET as a powerswitch it has much better robustness and larger safe opera-tion areas (SOA) than that using an IGBT-based power mod-ule or one-chip solution The experimental setup is shownin Figure 1 The space vector modulation (SVM) algorithmis used as modulation strategy and switching frequency ofthe PWM inverter is 15 kHz SMC 75 evaluation modulewith TMS320F28335 DSP controller is used It contains TexasInstruments 32-bit floating point DSP as well as analoginterfaces and JTAG emulator portThe board has analog-to-digital converter (AD) with 16 channels All the control vari-ables are monitored using graph window of Code ComposerStudio (CCS v33) after being converted to analog signalsthrough the digital-to-analog (DA) converter Real motorspeed (120596

119904) is measured using a high-resolution incremental

encoder with 2000 pulsesrotation and the estimated speed(119904) is obtained with the proposed HOSM schemeThe stator

currents of the PMSM aremeasured from the current sensorsand they are sent to TMS320F28335 viaAD converters In thesame way stator voltages are calculated using dc-bus voltagesensors and duty cycles of the inverter when the switchingfunctions are known

Three-phase currents and voltages are transformed totwo-phase stationary (120572 minus 120573) reference frame They are againtransformed to rotating (119889minus119902) reference frame for the controlPI (proportional and integral) controllers are used to regulatethe 119889 119902 synchronous frame currents 119894

119902and 119894119889 A functional

block diagram for the overall scheme is depicted in Figure 2The PMSM drive is operated in speed control mode Thespeed is also regulated using a PI controller to generate thereference current 119894ref

119902in the 119902-axis The reference current in

the 119889-axis 119894ref119889

is set to 0 For the implementation of HOSMobserver the sliding mode gains are selected as follows119870

1=

07 1198702= 60 119870

3= 35 and 119870

4= 45 The initial conditions

for the estimator are chosen as 119909(0) = [0 0 0 0]Several experiments have been performed to validate

the proposed HOSM scheme In the first part we presentthe results performance of the proposed method under no-load conditions for ramp change and step change and in thelater part similar experiments are conducted under loading

PMSM

Clarke

Position

HOSM

observer

S

S

Speed

Park120572120573

120572120573

120572120573PI PI

PIV120572

id i120572dq

dq

abc

iq

+

+

+ minus

minus

minus

ia

ib

V120573

i120573

irefd

irefq120596ref

Parkminus1

SVM3-phase

inverter

Figure 2 Functional block diagram for the overall scheme

condition For comparison the results obtained with first-order sliding mode observer are also presented

41 Under No-Load Condition In the first experiment aconstant speed reference 2000 rpm is provided for the first03 s a ramp input for the next 04 s followed by a constantspeed of 3500 rpm as shown in Figure 3(b) is employed Thereal currents of the PMSM for the first experiment are shownin Figure 3(a) The encoder speed and position are providedin Figures 3(b) and 3(c) The actual speed (120596

119904) exactly

follows the reference speed considered above Presence ofmeasurement noise can be clearly observed in (119894

120572120573) and

(120596119904 120579119904) With the proposed observer the estimated currents

and estimation error are shown in Figures 4(a) and 4(b) Realand estimated currents are very similar in both magnitudeand phase using the proposed method Figure 4(c) depictsthe estimated back EMFs obtained using (11) Despite thenoisy currents the back EMFs are relatively smooth whichconforms the theoretical claim of the proposed approachThe estimated speed computed analytically from back EMFswith (13) is shown in Figure 4(d) which exactly tracks actualspeed (120596

119904) and is shown for the comparisonThe convergence

accuracy depends on the accurate estimation of the back EMFcomponents and the back EMF constant 119870

119864 The HOSM

scheme enables a good reconstruction of the PMSM speedFigures 4(e) and 4(f) show the estimated rotor position andestimation error The estimated rotor position is robust withrespect to noise measurements and exactly matches withthe actual rotor position without any phase delay So theestimated rotor position can be used instead of the measuredone in the vector control of PMSM drive In usual practicethe values of 119877 and 119871 are not accurately known To testthe robustness the parameters (119877 and 119871) values are variedby plusmn10 and several experiments are conducted Similarperformancewas obtained in comparison to results presentedin Figure 4

For comparison the results obtained with conventionalsliding mode observer [3] are shown in Figure 5 The sliding

Mathematical Problems in Engineering 5

1

Time (s)0 01 03 05 07 09 1

minus1

0

1i 120572120573

(A)

(a)

1000

2000

3000

Time (s)0 01 03 05 07 09 1

120596s

(rpm

)

(b)

0

3

Time (s)

0 01 03 05 07 09 1

minus3

120579s

(rad

)

(c)

Figure 3 (a) Actual currents (b) Actual speed (c) Actual rotor position

mode gain is set to 50 for the observer design The presenceof noise in the estimated currents (Figure 5(a)) highly affectsthe estimation of back EMFs as shown in Figure 5(c) Theestimated back EMFs which correspond to the equivalentcontrols are obtained by filtering the switching functions ofthe observer with a 40Hz low-pass filter A proper boundarylayer is required to overcome the chattering phenomenonThe speed and position estimate in Figures 5(d)ndash5(f) exposethe problems with the conventional sliding mode observerfor back EMF estimation Also the estimated speed hasmore noise compared to the speed estimate with the HOSMobserver shown in Figure 4(d) The rotor position error inFigure 5(f) compared to Figure 4(f) clearly highlights theaccuracy obtained with proposed method Low-pass filteringclearly affects the estimation accuracyThe parameters for thefirst-order sliding mode are well-tuned to achieve the bestpossible results Errors are mainly due to filtering and the useof sigmoid function to avoid the chattering phenomenon

In the second experiment a step change in speed isprovided as reference In this experiment the speed estimateand position estimation error obtained with the proposedHOSM scheme for the step reference are shown in Figure 6Figure 6(a) shows the estimated and actual speeds obtainedusing the proposed HOSM observer and Figure 6(b) showsthe estimation error between the estimated and actual rotorpositions It can be seen from Figure 6(b) that the estimationhas no delay with the proposed approach For compari-son the results obtained with conventional first-order SMOare shown in Figure 7 Figure 7(a) shows the estimatedand actual speeds obtained using the first-order SMO andFigure 7(b) shows the estimation error between the estimatedand actual rotor positions Although the estimated speedfollows the actual speed it contains more noise compared to

the proposedHOSMobserverThese results further highlightthe robustness of the proposed method in the presenceof noise Further the chattering phenomenon is completelyeliminated and accurate position estimation can be obtainedeven in the presence of measurement noise

42 Under Loading Condition To test the performance of theproposed method a mechanical load 119869

119871= 007436 kg sdot cm2 is

connected to the motor Same set of parameters consideredfor no-load are employed for loading condition to test therobustness of the observer to parametric variations Theresults obtained with HOSM observer for a ramp changeare shown in Figure 8 Figures 8(a) and 8(b) show the esti-mated currents and their errors obtained using the proposedHOSM observer It can be observed from Figure 8(b) thatthe estimated and actual currents exactly match each otherFigure 8(c) shows the estimated unknown back EMFs whichare relatively smooth The corresponding estimated speedcalculated using (13) and the actual speed are depicted inFigure 8(d) The estimated rotor position and the estimationerror are shown in Figures 8(e) and 8(f) For comparison theestimated and actual speeds obtained with first-order SMOare shown in Figure 9(a) and the corresponding position esti-mation error is shown in Figure 9(b) In the final experimentunder loading condition step-like input reference is providedfor the system the results obtained with proposed methodand first-order SMO are shown in Figure 10 and Figure 11respectively Figures 10(a) and 10(b) show the estimatedactual speeds and position estimation error obtained withthe proposed HOSM observer while Figures 11(a) and 11(b)show the estimated actual speeds and position estimationerror obtained with the first-order SMO Under loading thenoise level is higher and the position estimation error slightly

6 Mathematical Problems in Engineering

Time (s)0 01

1

03 05 07 09 1

0

05

minus05

minus1

i 120572120573

(A)

(a)

Time (s)0 01 03 05 07 09 1

0

05

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus40

40

0

e 120572120573

(V)

(c)

1500

3000

4000

Time (s)0 01 03 05 07 09 1

EstimatedActual

120596s

(rpm

)

(d)

0

3

Time (s)0 01 03 05 07 09 1

minus3

120579s

(rad

)

(e)

0

1

2

Time (s)0 01 03 05 07 09 1

minus2

minus1120579sminus120579s (

rad)

(f)

Figure 4 Estimation using higher-order sliding mode observer under no-load (a) Estimated currents (b) Estimation current error (c)Estimated back EMFs (d) Estimated speed (e) Estimated rotor position (f) Estimation rotor position error

increasesHowever the speed andposition estimation remainrobust with the proposed observer when compared to first-order SMO Due to parametric uncertainty the speed estima-tion with proposedHOSMmethod shows a very small steadystate error at 90 of the rated speed (Figures 8(d) and 10(a))Under loading the speed and position estimation with first-order SMOare highly affected Further tuning of the observerparameters can overcome the problem

From the implementation one can conclude the follow-ing

(1) The HOSM method requires the proper selection ofsliding mode gains 119870

1 1198702 1198703 and 119870

4 The sliding

mode gains should satisfy the conditions given by(A15)-(A16) for the desired speed range If themotoroperates in wide speed range the sliding mode gainsmust be appropriately selected

(2) Since the quality of the speed estimate highly dependson the estimated back EMFs it deteriorates when

more noisy back EMFs (obtained due to high gains)are used in the calculation Compared to existingmethods the proposed scheme provides good rotorspeed and position estimation without phase delay inthe presence of noise Further one should note thatsensorless speed estimation methods based on backEMFs fail at very low speeds and standstill

(3) It should be pointed out that a chattering phe-nomenon occurs using the conventional SM observer[15]Therefore in first-order SMobserver the signumfunction is used as switching function The speedestimate is approximated by low-pass filtering thediscontinuous switching functions This delay shouldbe compensated with an additional phase compensa-tion loop As low-pass filtering is eliminated with theproposed method an additional phase compensationloop is not required

Mathematical Problems in Engineering 7

1

0

05

minus05

minus1

i 120572120573

(A)

Time (s)0 01 03 05 07 09 1

(a)

Time (s)0 01 03 05 07 09 1

0

05

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus40

40

0

e 120572120573

(V)

(c)

Time (s)0 01 03 05 07 09 1

1500

3000

4000

120596s

(rpm

)

EstimatedActual

(d)

Time (s)0 01 03 05 07 09 1

0

3

minus3

120579s

(rad

)

(e)

Time (s)0 01 03 05 07 09 1

0

1

2

minus2

minus1120579sminus120579s (

rad)

(f)

Figure 5 Estimation using conventional first-order sliding mode observer under no-load (a) Estimated currents (b) Estimation currenterror (c) Estimated back EMFs (d) Estimated speed (e) Estimated rotor position (f) Estimation rotor position error

1500

3000

4000

EstimatedActual

120596s120596

s(r

pm)

Time (s)0 02 04 05 06 08 1

(a)

120579sminus120579s

(rad

)

0

1

2

minus1

minus2

Time (s)0 02 04 05 06 08 1

(b)

Figure 6 With proposed HOSMmethod under no-load (a) Actual and estimated speed (b) Rotor position estimation error

8 Mathematical Problems in Engineering

1500

3000

4000

Time (s)0 02 04 05 06 08 1

EstimatedActual

120596s120596

s(r

pm)

(a)

Time (s)

0 02 04 05 06 08 1

120579sminus120579s

(rad

)

0

1

2

minus1

minus2

(b)

Figure 7 With conventional first-order SMO under no-load (a) Estimated speed (b) Rotor position estimation error

Time (s)0 01 03 05 07 09 1

0

06

minus06

i 120572120573

(A)

(a)

Time (s)0 01 03 05 07 09 1

0

05

12

minus12

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus60

60

0

e 120572120573

(V)

(c)

1500

3000

4000

Time (s)0 01 03 05 07 09 1

EstimatedActual

120596s

(rpm

)

(d)

0

3

Time (s)0 01 03 05 07 09 1

minus3

120579s

(rad

)

(e)

0

1

2

Time (s)0 01 03 05 07 09 1

minus2

minus1120579sminus

120579s (

rad)

(f)

Figure 8 With proposed HOSM method under load (a) Estimated currents (b) Estimation current errors (c) Estimated back EMFs (d)Estimated speed (e) Estimated rotor position (f) Rotor position error

Mathematical Problems in Engineering 9

Time (s)0 01 03 05 07 09 1

1500

3000

4000

120596s120596

s(r

pm)

EstimatedActual

(a)

Time (s)0 01 03 05 07 09 1

0

1

2

minus2

minus1

120579sminus120579s (

rad)

(b)

Figure 9 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

1500

3000

4000

0 02 04 05 06 08 1

Time (s)EstimatedActual

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1Time (s)

0

1

2

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 10 With proposed HOSMmethod under load (a) Estimated speed (b) Rotor position estimation error

(4) Furthermore compared to the classical first-orderSM technique no cutoff frequency has to be tunedInstead a simple integration is realized It enablesto reduce the time delay for the estimation (whichdepends on the sampling period) One should alsohighlight that the discontinuous part of 120601

2(depend-

ing on1198704) is usually low compared to the continuous

part of 1206012and this enables to reduce the chattering

phenomenon(5) Moreover from the experiments the proposed

method is robust to the parameter variations and themeasurement noise compared to the traditional SMobserver

(6) It is worth to point out that the proposed method iscomputationally complex compared to the traditionalSM observer However if properly tuned it has moreadvantages than the traditional SM observer Theexperiments conducted in this paper validate theadvantages of this method

(7) For the same set of parameters the speed and positionestimation remained accurate for both no-loadingand loading conditions This further highlights therobustness of the proposedmethod to parameter vari-ations that occur with loading and other conditions

5 Conclusion

This paper has presented a sensorless speed estimationmethod for the PMSM driveThe HOSMmethod is based ona modified version of super-twisting algorithmThe observerdynamics consist of sliding mode terms which are used toreconstruct the unknown back EMFs The speed is thenanalytically computed from back EMFs Experimental resultsvalidate the feasibility and effectiveness of the proposedHOSM for estimating the rotor position and speed of thePMSM Compared with the traditional SMO the proposedhigher-order SMO provides better estimation performance

Appendix

Finite-Time Stability

For any vector 119911 = [1199111 119911

119902]119879isin 119877119902 and any scalar 120572 isin 119877

we denote the following

sign (119911) = [sign (1199111) sign (119911

119902)]119879

|119911|120572= diag (10038161003816100381610038161199111

1003816100381610038161003816120572

10038161003816100381610038161003816119911119902

10038161003816100381610038161003816

120572

)

lceil119911rfloor120572= |119911|120572 sign (119911)

(A1)

10 Mathematical Problems in Engineering

1500

3000

4000

EstimatedActual

0 02 04 05 06 08 1

Time (s)

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1

0

1

2

Time (s)

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 11 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

For ease of exposition consider the following system

119904 (119905) = minus 119886119904 (119905) + ] (119905) + 119890 (119904 119905)

119904 (1199050) = 1199040

(A2)

where 119904 isin R and 119886 is a known positive constant and 119890(119904 119905) isthe unknown inputperturbation and

] (119905) = minus11987011206011 (119904 (119905)) minus 119870

2int

119905

0

1206012 (119904 (119905)) 119889119905 (A3)

where 1206011(119904(119905)) and 120601

2(119904(119905)) are defined in (7) and119870

111987021198703

and1198704are appropriately designed positive constants

Assumption A1 The time derivative of the unknowninputperturbation is upper bounded as follows

| 119890 (119904 119905)| le 120588 (A4)

for a positive constant 120588

Remark A2 The sliding dynamics 119904120572or 119904120573in (8) can be

directly expressed in the form of (A2) Further the condition(9) is similar to Assumption A1

Proposition A3 Under Assumption A1 the origin of sys-tem (A2) is a finite time stable equilibrium point Fur-ther the finite-time smooth estimation of the unknowninputperturbation 119890(119904 119905) is given by 119870

2int119905

01206012(119904(119905))119889119905

Proof Proof follows the work given in [20] Since |1206012(119904)| ge

1198702

42 one gets

| 119890 (119904 119905)| le10038161003816100381610038161206012 (119904)

1003816100381610038161003816 (A5)

if

1198704ge radic2120588 (A6)

Let us select a Hurwitz matrix 1198600

1198600= [

minus (1198701+ 119886) 1

minus1198702

0] (A7)

where1198701gt 0 and119870

2gt 0

The system (A2) (A3) can be equivalently representedby the system of two first-order equations

1199041= 1199042minus (1198701+ 119886) (119904

1+ 1198704lceil 1199041rfloor12

)

1199042= minus 119870

2(1199041+1198702

4

2sign (119904

1) +

3

21198704lceil 1199041rfloor12

) + 119890

(A8)

with 1199041= 119904 119904

2= 119890 minus 119870

2int119905

01206012(1199041) 119889119905 and

1198704=

11987011198703

1198701+ 119886

(A9)

The solutions of the discontinuous differential equations andinclusions are understood in the sense of Filippov

Let us consider the new state vector

120585 = [1205851

1205852

] = [1199041+ 1198704lceil 1199041rfloor12

1199042

] (A10)

The stability analysis of system (A8) is performed usingthe following candidate Lyapunov function [20]

119881 (120585) = 120585119879119875120585 (A11)

with 119875 = 119875119879

= [ 120582+41205982minus2120598

minus2120598 1] 120582 gt 0 and 120598 gt 0 It is worth

noting that the matrix 119875 is positive definite if 120582 and 120598 are anyreal number

Using the differential equations inclusion theory its timederivative along the solutions of the system is given by

= (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

119890]

(A12)

Mathematical Problems in Engineering 11

It can be shown that

le (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

)(120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

1205851

])

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879119876120585

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120582min (119876)100381710038171003817100381712058510038171003817100381710038172

(A13)with

119876 = [11987611198762

11987621198763

]

1198761= 2 (119870

1+ 119886) (120582 + 4120598

2) minus 4120598 (119870

2minus 1)

1198762= minus 2120598 (119870

1+ 119886) + (119870

2+ 1) minus (120582 + 4120598

2)

1198763= 4120598

(A14)

In order to guarantee the positive definiteness of matrix 119876one chooses

1198702= 120582 + 4120598

2+ 2120598 (119870

1+ 119886) (A15)

The matrix 119876 is positive definite if

1198701gt minus119886 +

4120598 + 2120598120582 + 81205983

120582+

1

4120598120582 (A16)

From (A10) one can deduce that100381710038171003817100381712058510038171003817100381710038172= 1205852

1+ 1205852

2

= 1199042

1+ 21198704

10038161003816100381610038161199041100381610038161003816100381632

+ 1198702

4

100381610038161003816100381611990411003816100381610038161003816 + 1205852

2

ge 1198702

4

100381610038161003816100381611990411003816100381610038161003816

(A17)

Since1198704gt 0

minus1198704

10038171003817100381710038171205851003817100381710038171003817

ge minus10038161003816100381610038161199041

1003816100381610038161003816minus12

(A18)

It implies that

le minus120582min (119876)

12058212

max (119875)

1198702

4

211988112

minus120582min (119876)

120582max (119875)119881 (A19)

The closed-loop system (A8) is stabilized in finite timeSince 120585 converges to zero in finite time 119904

1and 1199042converge

to 0 Therefore the term1198702int119905

01206012(119904(119905))119889119905 gives in finite time a

smooth estimation of the unknown perturbation 119890(119904 119905)

Nomenclature

120596119904 Rotor electrical speed

119894120572 119894120573 Currents in stationary reference frame

119881120572 119881120573 Voltages in stationary reference frame

119890120572 119890120573 EMFs in stationary reference frame

119877 Stator resistance119871 Synchronous inductance119870119864 EMF constant

120579119904 Rotor position angle

119879119897 Load torque

Conflict of Interests

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

Acknowledgment

This research was supported by the Basic Science ResearchProgram through theNational Research Foundation of Korea(NRF) funded by the Ministry of Education Science andTechnology (Grant no 2011-0023999)

References

[1] R Wu and G R Slemon ldquoA permanent magnet motor drivewithout a shaft sensorrdquo IEEE Transactions on Industry Applica-tions vol 27 no 5 pp 1005ndash1011 1991

[2] P Tomei and C M Verrelli ldquoObserver-based speed trackingcontrol for sensorless permanent magnet synchronous motorswith unknown load torquerdquo IEEE Transactions on AutomaticControl vol 56 no 6 pp 1484ndash1488 2011

[3] V Utkin J G Guldner and J Shi Sliding Mode Control onElectromechanical Systems Taylor and Francis New York NYUSA 1st edition 1999

[4] S Chai L Wang and R Rogers ldquoModel predictive controlof a permanent magnet synchronous motor with experimentalvalidationrdquoControl Engineering Practice vol 21 no 11 pp 1584ndash1593 2013

[5] T Orlowska-Kowalska and M Dybkowski ldquoStator-current-based MRAS estimator for a wide range speed-sensorlessinduction-motor driverdquo IEEE Transactions on Industrial Elec-tronics vol 57 no 4 pp 1296ndash1308 2010

[6] M L Corradini G Ippoliti S Longhi and G Orlando ldquoAquasi-sliding mode approach for robust control and speedestimation of PM synchronous motorsrdquo IEEE Transactions onIndustrial Electronics vol 59 no 2 pp 1096ndash1104 2012

[7] B K Bose Modern Power Electronics and AC Drives Prentice-Hall Upper Saddle River NJ USA 2002

[8] K C Veluvolu and Y C Soh ldquoMultiple sliding mode observersand unknown input estimations for Lipschitz nonlinear sys-temsrdquo International Journal of Robust and Nonlinear Controlvol 21 no 11 pp 1322ndash1340 2011

[9] K C Veluvolu and D Lee ldquoSliding mode high-gain observersfor a class of uncertain nonlinear systemsrdquoAppliedMathematicsLetters vol 24 no 3 pp 329ndash334 2011

[10] K C Veluvolu and Y C Soh ldquoFault reconstruction and stateestimationwith slidingmode observers for Lipschitz non-linearsystemsrdquo IET Control Theory amp Applications vol 5 no 11 pp1255ndash1263 2011

[11] M Comanescu and L Xu ldquoSliding-mode MRAS speed estima-tors for sensorless vector control of induction machinerdquo IEEETransactions on Industrial Electronics vol 53 no 1 pp 146ndash1532006

[12] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[13] K C Veluvolu M Y Kim and D Lee ldquoNonlinear sliding modehigh-gain observers for fault estimationrdquo International Journalof Systems Science Principles and Applications of Systems andIntegration vol 42 no 7 pp 1065ndash1074 2011

12 Mathematical Problems in Engineering

[14] K C Veluvolu M Defoort and Y C Soh ldquoHigh-gain observerwith sliding mode for nonlinear state estimation and faultreconstructionrdquo Journal of the Franklin Institute Engineeringand Applied Mathematics vol 351 no 4 pp 1995ndash2014 2014

[15] M Comanescu ldquoCascaded EMF and speed sliding modeobserver for the nonsalient PMSMrdquo in Proceedings of the 36thAnnual Conference of the IEEE Industrial Electronics Society(IECON rsquo10) pp 792ndash797 Glendale Ariz November 2010

[16] M Comanescu ldquoAn induction-motor speed estimator based onintegral sliding-mode current controlrdquo IEEE Transactions onIndustrial Electronics vol 56 no 9 pp 3414ndash3423 2009

[17] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[18] M Comanescu L Xu and T D Batzel ldquoDecoupled currentcontrol of sensorless induction-motor drives by integral slidingmoderdquo IEEE Transactions on Industrial Electronics vol 55 no11 pp 3836ndash3845 2008

[19] H Kim J Son and J Lee ldquoA high-speed sliding-mode observerfor the sensorless speed control of a PMSMrdquo IEEE Transactionson Industrial Electronics vol 58 no 9 pp 4069ndash4077 2011

[20] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

[21] T Floquet and J P Barbot ldquoSuper twisting algorithm-basedstep-by-step sliding mode observers for nonlinear systemswith unknown inputsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 38no 10 pp 803ndash815 2007

[22] J J Rath K C Veluvolu M Defoort and Y C Soh ldquoHigher-order sliding mode observer for estimation of tyre frictionin ground vehiclesrdquo IET Proceedings on Control Theory andApplications vol 8 no 6 pp 399ndash408 2014

[23] L Fridman and A Levant ldquoHigher order sliding modesSliding mode control in engineeringrdquo in Sliding Mode Controlin Engineering J P Barbot and W Perruquetti Eds MarcelDekker New York NY USA 2002

[24] M Ezzat J De Leon N Gonzalez and A GlumineauldquoObserver-controller scheme using high order sliding modetechniques for sensorless speed control of permanent magnetsynchronous motorrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 4012ndash4017 December2010

[25] D Zaltni and M N Abdelkrim ldquoRobust speed and positionobserver using HOSM for sensor-less SPMSM controlrdquoin Proceedings of the 7th International Multi-Conference onSystems Signals and Devices (SSD rsquo10) pp 1ndash6 June 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

1

Time (s)0 01 03 05 07 09 1

minus1

0

1i 120572120573

(A)

(a)

1000

2000

3000

Time (s)0 01 03 05 07 09 1

120596s

(rpm

)

(b)

0

3

Time (s)

0 01 03 05 07 09 1

minus3

120579s

(rad

)

(c)

Figure 3 (a) Actual currents (b) Actual speed (c) Actual rotor position

mode gain is set to 50 for the observer design The presenceof noise in the estimated currents (Figure 5(a)) highly affectsthe estimation of back EMFs as shown in Figure 5(c) Theestimated back EMFs which correspond to the equivalentcontrols are obtained by filtering the switching functions ofthe observer with a 40Hz low-pass filter A proper boundarylayer is required to overcome the chattering phenomenonThe speed and position estimate in Figures 5(d)ndash5(f) exposethe problems with the conventional sliding mode observerfor back EMF estimation Also the estimated speed hasmore noise compared to the speed estimate with the HOSMobserver shown in Figure 4(d) The rotor position error inFigure 5(f) compared to Figure 4(f) clearly highlights theaccuracy obtained with proposed method Low-pass filteringclearly affects the estimation accuracyThe parameters for thefirst-order sliding mode are well-tuned to achieve the bestpossible results Errors are mainly due to filtering and the useof sigmoid function to avoid the chattering phenomenon

In the second experiment a step change in speed isprovided as reference In this experiment the speed estimateand position estimation error obtained with the proposedHOSM scheme for the step reference are shown in Figure 6Figure 6(a) shows the estimated and actual speeds obtainedusing the proposed HOSM observer and Figure 6(b) showsthe estimation error between the estimated and actual rotorpositions It can be seen from Figure 6(b) that the estimationhas no delay with the proposed approach For compari-son the results obtained with conventional first-order SMOare shown in Figure 7 Figure 7(a) shows the estimatedand actual speeds obtained using the first-order SMO andFigure 7(b) shows the estimation error between the estimatedand actual rotor positions Although the estimated speedfollows the actual speed it contains more noise compared to

the proposedHOSMobserverThese results further highlightthe robustness of the proposed method in the presenceof noise Further the chattering phenomenon is completelyeliminated and accurate position estimation can be obtainedeven in the presence of measurement noise

42 Under Loading Condition To test the performance of theproposed method a mechanical load 119869

119871= 007436 kg sdot cm2 is

connected to the motor Same set of parameters consideredfor no-load are employed for loading condition to test therobustness of the observer to parametric variations Theresults obtained with HOSM observer for a ramp changeare shown in Figure 8 Figures 8(a) and 8(b) show the esti-mated currents and their errors obtained using the proposedHOSM observer It can be observed from Figure 8(b) thatthe estimated and actual currents exactly match each otherFigure 8(c) shows the estimated unknown back EMFs whichare relatively smooth The corresponding estimated speedcalculated using (13) and the actual speed are depicted inFigure 8(d) The estimated rotor position and the estimationerror are shown in Figures 8(e) and 8(f) For comparison theestimated and actual speeds obtained with first-order SMOare shown in Figure 9(a) and the corresponding position esti-mation error is shown in Figure 9(b) In the final experimentunder loading condition step-like input reference is providedfor the system the results obtained with proposed methodand first-order SMO are shown in Figure 10 and Figure 11respectively Figures 10(a) and 10(b) show the estimatedactual speeds and position estimation error obtained withthe proposed HOSM observer while Figures 11(a) and 11(b)show the estimated actual speeds and position estimationerror obtained with the first-order SMO Under loading thenoise level is higher and the position estimation error slightly

6 Mathematical Problems in Engineering

Time (s)0 01

1

03 05 07 09 1

0

05

minus05

minus1

i 120572120573

(A)

(a)

Time (s)0 01 03 05 07 09 1

0

05

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus40

40

0

e 120572120573

(V)

(c)

1500

3000

4000

Time (s)0 01 03 05 07 09 1

EstimatedActual

120596s

(rpm

)

(d)

0

3

Time (s)0 01 03 05 07 09 1

minus3

120579s

(rad

)

(e)

0

1

2

Time (s)0 01 03 05 07 09 1

minus2

minus1120579sminus120579s (

rad)

(f)

Figure 4 Estimation using higher-order sliding mode observer under no-load (a) Estimated currents (b) Estimation current error (c)Estimated back EMFs (d) Estimated speed (e) Estimated rotor position (f) Estimation rotor position error

increasesHowever the speed andposition estimation remainrobust with the proposed observer when compared to first-order SMO Due to parametric uncertainty the speed estima-tion with proposedHOSMmethod shows a very small steadystate error at 90 of the rated speed (Figures 8(d) and 10(a))Under loading the speed and position estimation with first-order SMOare highly affected Further tuning of the observerparameters can overcome the problem

From the implementation one can conclude the follow-ing

(1) The HOSM method requires the proper selection ofsliding mode gains 119870

1 1198702 1198703 and 119870

4 The sliding

mode gains should satisfy the conditions given by(A15)-(A16) for the desired speed range If themotoroperates in wide speed range the sliding mode gainsmust be appropriately selected

(2) Since the quality of the speed estimate highly dependson the estimated back EMFs it deteriorates when

more noisy back EMFs (obtained due to high gains)are used in the calculation Compared to existingmethods the proposed scheme provides good rotorspeed and position estimation without phase delay inthe presence of noise Further one should note thatsensorless speed estimation methods based on backEMFs fail at very low speeds and standstill

(3) It should be pointed out that a chattering phe-nomenon occurs using the conventional SM observer[15]Therefore in first-order SMobserver the signumfunction is used as switching function The speedestimate is approximated by low-pass filtering thediscontinuous switching functions This delay shouldbe compensated with an additional phase compensa-tion loop As low-pass filtering is eliminated with theproposed method an additional phase compensationloop is not required

Mathematical Problems in Engineering 7

1

0

05

minus05

minus1

i 120572120573

(A)

Time (s)0 01 03 05 07 09 1

(a)

Time (s)0 01 03 05 07 09 1

0

05

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus40

40

0

e 120572120573

(V)

(c)

Time (s)0 01 03 05 07 09 1

1500

3000

4000

120596s

(rpm

)

EstimatedActual

(d)

Time (s)0 01 03 05 07 09 1

0

3

minus3

120579s

(rad

)

(e)

Time (s)0 01 03 05 07 09 1

0

1

2

minus2

minus1120579sminus120579s (

rad)

(f)

Figure 5 Estimation using conventional first-order sliding mode observer under no-load (a) Estimated currents (b) Estimation currenterror (c) Estimated back EMFs (d) Estimated speed (e) Estimated rotor position (f) Estimation rotor position error

1500

3000

4000

EstimatedActual

120596s120596

s(r

pm)

Time (s)0 02 04 05 06 08 1

(a)

120579sminus120579s

(rad

)

0

1

2

minus1

minus2

Time (s)0 02 04 05 06 08 1

(b)

Figure 6 With proposed HOSMmethod under no-load (a) Actual and estimated speed (b) Rotor position estimation error

8 Mathematical Problems in Engineering

1500

3000

4000

Time (s)0 02 04 05 06 08 1

EstimatedActual

120596s120596

s(r

pm)

(a)

Time (s)

0 02 04 05 06 08 1

120579sminus120579s

(rad

)

0

1

2

minus1

minus2

(b)

Figure 7 With conventional first-order SMO under no-load (a) Estimated speed (b) Rotor position estimation error

Time (s)0 01 03 05 07 09 1

0

06

minus06

i 120572120573

(A)

(a)

Time (s)0 01 03 05 07 09 1

0

05

12

minus12

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus60

60

0

e 120572120573

(V)

(c)

1500

3000

4000

Time (s)0 01 03 05 07 09 1

EstimatedActual

120596s

(rpm

)

(d)

0

3

Time (s)0 01 03 05 07 09 1

minus3

120579s

(rad

)

(e)

0

1

2

Time (s)0 01 03 05 07 09 1

minus2

minus1120579sminus

120579s (

rad)

(f)

Figure 8 With proposed HOSM method under load (a) Estimated currents (b) Estimation current errors (c) Estimated back EMFs (d)Estimated speed (e) Estimated rotor position (f) Rotor position error

Mathematical Problems in Engineering 9

Time (s)0 01 03 05 07 09 1

1500

3000

4000

120596s120596

s(r

pm)

EstimatedActual

(a)

Time (s)0 01 03 05 07 09 1

0

1

2

minus2

minus1

120579sminus120579s (

rad)

(b)

Figure 9 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

1500

3000

4000

0 02 04 05 06 08 1

Time (s)EstimatedActual

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1Time (s)

0

1

2

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 10 With proposed HOSMmethod under load (a) Estimated speed (b) Rotor position estimation error

(4) Furthermore compared to the classical first-orderSM technique no cutoff frequency has to be tunedInstead a simple integration is realized It enablesto reduce the time delay for the estimation (whichdepends on the sampling period) One should alsohighlight that the discontinuous part of 120601

2(depend-

ing on1198704) is usually low compared to the continuous

part of 1206012and this enables to reduce the chattering

phenomenon(5) Moreover from the experiments the proposed

method is robust to the parameter variations and themeasurement noise compared to the traditional SMobserver

(6) It is worth to point out that the proposed method iscomputationally complex compared to the traditionalSM observer However if properly tuned it has moreadvantages than the traditional SM observer Theexperiments conducted in this paper validate theadvantages of this method

(7) For the same set of parameters the speed and positionestimation remained accurate for both no-loadingand loading conditions This further highlights therobustness of the proposedmethod to parameter vari-ations that occur with loading and other conditions

5 Conclusion

This paper has presented a sensorless speed estimationmethod for the PMSM driveThe HOSMmethod is based ona modified version of super-twisting algorithmThe observerdynamics consist of sliding mode terms which are used toreconstruct the unknown back EMFs The speed is thenanalytically computed from back EMFs Experimental resultsvalidate the feasibility and effectiveness of the proposedHOSM for estimating the rotor position and speed of thePMSM Compared with the traditional SMO the proposedhigher-order SMO provides better estimation performance

Appendix

Finite-Time Stability

For any vector 119911 = [1199111 119911

119902]119879isin 119877119902 and any scalar 120572 isin 119877

we denote the following

sign (119911) = [sign (1199111) sign (119911

119902)]119879

|119911|120572= diag (10038161003816100381610038161199111

1003816100381610038161003816120572

10038161003816100381610038161003816119911119902

10038161003816100381610038161003816

120572

)

lceil119911rfloor120572= |119911|120572 sign (119911)

(A1)

10 Mathematical Problems in Engineering

1500

3000

4000

EstimatedActual

0 02 04 05 06 08 1

Time (s)

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1

0

1

2

Time (s)

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 11 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

For ease of exposition consider the following system

119904 (119905) = minus 119886119904 (119905) + ] (119905) + 119890 (119904 119905)

119904 (1199050) = 1199040

(A2)

where 119904 isin R and 119886 is a known positive constant and 119890(119904 119905) isthe unknown inputperturbation and

] (119905) = minus11987011206011 (119904 (119905)) minus 119870

2int

119905

0

1206012 (119904 (119905)) 119889119905 (A3)

where 1206011(119904(119905)) and 120601

2(119904(119905)) are defined in (7) and119870

111987021198703

and1198704are appropriately designed positive constants

Assumption A1 The time derivative of the unknowninputperturbation is upper bounded as follows

| 119890 (119904 119905)| le 120588 (A4)

for a positive constant 120588

Remark A2 The sliding dynamics 119904120572or 119904120573in (8) can be

directly expressed in the form of (A2) Further the condition(9) is similar to Assumption A1

Proposition A3 Under Assumption A1 the origin of sys-tem (A2) is a finite time stable equilibrium point Fur-ther the finite-time smooth estimation of the unknowninputperturbation 119890(119904 119905) is given by 119870

2int119905

01206012(119904(119905))119889119905

Proof Proof follows the work given in [20] Since |1206012(119904)| ge

1198702

42 one gets

| 119890 (119904 119905)| le10038161003816100381610038161206012 (119904)

1003816100381610038161003816 (A5)

if

1198704ge radic2120588 (A6)

Let us select a Hurwitz matrix 1198600

1198600= [

minus (1198701+ 119886) 1

minus1198702

0] (A7)

where1198701gt 0 and119870

2gt 0

The system (A2) (A3) can be equivalently representedby the system of two first-order equations

1199041= 1199042minus (1198701+ 119886) (119904

1+ 1198704lceil 1199041rfloor12

)

1199042= minus 119870

2(1199041+1198702

4

2sign (119904

1) +

3

21198704lceil 1199041rfloor12

) + 119890

(A8)

with 1199041= 119904 119904

2= 119890 minus 119870

2int119905

01206012(1199041) 119889119905 and

1198704=

11987011198703

1198701+ 119886

(A9)

The solutions of the discontinuous differential equations andinclusions are understood in the sense of Filippov

Let us consider the new state vector

120585 = [1205851

1205852

] = [1199041+ 1198704lceil 1199041rfloor12

1199042

] (A10)

The stability analysis of system (A8) is performed usingthe following candidate Lyapunov function [20]

119881 (120585) = 120585119879119875120585 (A11)

with 119875 = 119875119879

= [ 120582+41205982minus2120598

minus2120598 1] 120582 gt 0 and 120598 gt 0 It is worth

noting that the matrix 119875 is positive definite if 120582 and 120598 are anyreal number

Using the differential equations inclusion theory its timederivative along the solutions of the system is given by

= (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

119890]

(A12)

Mathematical Problems in Engineering 11

It can be shown that

le (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

)(120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

1205851

])

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879119876120585

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120582min (119876)100381710038171003817100381712058510038171003817100381710038172

(A13)with

119876 = [11987611198762

11987621198763

]

1198761= 2 (119870

1+ 119886) (120582 + 4120598

2) minus 4120598 (119870

2minus 1)

1198762= minus 2120598 (119870

1+ 119886) + (119870

2+ 1) minus (120582 + 4120598

2)

1198763= 4120598

(A14)

In order to guarantee the positive definiteness of matrix 119876one chooses

1198702= 120582 + 4120598

2+ 2120598 (119870

1+ 119886) (A15)

The matrix 119876 is positive definite if

1198701gt minus119886 +

4120598 + 2120598120582 + 81205983

120582+

1

4120598120582 (A16)

From (A10) one can deduce that100381710038171003817100381712058510038171003817100381710038172= 1205852

1+ 1205852

2

= 1199042

1+ 21198704

10038161003816100381610038161199041100381610038161003816100381632

+ 1198702

4

100381610038161003816100381611990411003816100381610038161003816 + 1205852

2

ge 1198702

4

100381610038161003816100381611990411003816100381610038161003816

(A17)

Since1198704gt 0

minus1198704

10038171003817100381710038171205851003817100381710038171003817

ge minus10038161003816100381610038161199041

1003816100381610038161003816minus12

(A18)

It implies that

le minus120582min (119876)

12058212

max (119875)

1198702

4

211988112

minus120582min (119876)

120582max (119875)119881 (A19)

The closed-loop system (A8) is stabilized in finite timeSince 120585 converges to zero in finite time 119904

1and 1199042converge

to 0 Therefore the term1198702int119905

01206012(119904(119905))119889119905 gives in finite time a

smooth estimation of the unknown perturbation 119890(119904 119905)

Nomenclature

120596119904 Rotor electrical speed

119894120572 119894120573 Currents in stationary reference frame

119881120572 119881120573 Voltages in stationary reference frame

119890120572 119890120573 EMFs in stationary reference frame

119877 Stator resistance119871 Synchronous inductance119870119864 EMF constant

120579119904 Rotor position angle

119879119897 Load torque

Conflict of Interests

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

Acknowledgment

This research was supported by the Basic Science ResearchProgram through theNational Research Foundation of Korea(NRF) funded by the Ministry of Education Science andTechnology (Grant no 2011-0023999)

References

[1] R Wu and G R Slemon ldquoA permanent magnet motor drivewithout a shaft sensorrdquo IEEE Transactions on Industry Applica-tions vol 27 no 5 pp 1005ndash1011 1991

[2] P Tomei and C M Verrelli ldquoObserver-based speed trackingcontrol for sensorless permanent magnet synchronous motorswith unknown load torquerdquo IEEE Transactions on AutomaticControl vol 56 no 6 pp 1484ndash1488 2011

[3] V Utkin J G Guldner and J Shi Sliding Mode Control onElectromechanical Systems Taylor and Francis New York NYUSA 1st edition 1999

[4] S Chai L Wang and R Rogers ldquoModel predictive controlof a permanent magnet synchronous motor with experimentalvalidationrdquoControl Engineering Practice vol 21 no 11 pp 1584ndash1593 2013

[5] T Orlowska-Kowalska and M Dybkowski ldquoStator-current-based MRAS estimator for a wide range speed-sensorlessinduction-motor driverdquo IEEE Transactions on Industrial Elec-tronics vol 57 no 4 pp 1296ndash1308 2010

[6] M L Corradini G Ippoliti S Longhi and G Orlando ldquoAquasi-sliding mode approach for robust control and speedestimation of PM synchronous motorsrdquo IEEE Transactions onIndustrial Electronics vol 59 no 2 pp 1096ndash1104 2012

[7] B K Bose Modern Power Electronics and AC Drives Prentice-Hall Upper Saddle River NJ USA 2002

[8] K C Veluvolu and Y C Soh ldquoMultiple sliding mode observersand unknown input estimations for Lipschitz nonlinear sys-temsrdquo International Journal of Robust and Nonlinear Controlvol 21 no 11 pp 1322ndash1340 2011

[9] K C Veluvolu and D Lee ldquoSliding mode high-gain observersfor a class of uncertain nonlinear systemsrdquoAppliedMathematicsLetters vol 24 no 3 pp 329ndash334 2011

[10] K C Veluvolu and Y C Soh ldquoFault reconstruction and stateestimationwith slidingmode observers for Lipschitz non-linearsystemsrdquo IET Control Theory amp Applications vol 5 no 11 pp1255ndash1263 2011

[11] M Comanescu and L Xu ldquoSliding-mode MRAS speed estima-tors for sensorless vector control of induction machinerdquo IEEETransactions on Industrial Electronics vol 53 no 1 pp 146ndash1532006

[12] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[13] K C Veluvolu M Y Kim and D Lee ldquoNonlinear sliding modehigh-gain observers for fault estimationrdquo International Journalof Systems Science Principles and Applications of Systems andIntegration vol 42 no 7 pp 1065ndash1074 2011

12 Mathematical Problems in Engineering

[14] K C Veluvolu M Defoort and Y C Soh ldquoHigh-gain observerwith sliding mode for nonlinear state estimation and faultreconstructionrdquo Journal of the Franklin Institute Engineeringand Applied Mathematics vol 351 no 4 pp 1995ndash2014 2014

[15] M Comanescu ldquoCascaded EMF and speed sliding modeobserver for the nonsalient PMSMrdquo in Proceedings of the 36thAnnual Conference of the IEEE Industrial Electronics Society(IECON rsquo10) pp 792ndash797 Glendale Ariz November 2010

[16] M Comanescu ldquoAn induction-motor speed estimator based onintegral sliding-mode current controlrdquo IEEE Transactions onIndustrial Electronics vol 56 no 9 pp 3414ndash3423 2009

[17] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[18] M Comanescu L Xu and T D Batzel ldquoDecoupled currentcontrol of sensorless induction-motor drives by integral slidingmoderdquo IEEE Transactions on Industrial Electronics vol 55 no11 pp 3836ndash3845 2008

[19] H Kim J Son and J Lee ldquoA high-speed sliding-mode observerfor the sensorless speed control of a PMSMrdquo IEEE Transactionson Industrial Electronics vol 58 no 9 pp 4069ndash4077 2011

[20] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

[21] T Floquet and J P Barbot ldquoSuper twisting algorithm-basedstep-by-step sliding mode observers for nonlinear systemswith unknown inputsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 38no 10 pp 803ndash815 2007

[22] J J Rath K C Veluvolu M Defoort and Y C Soh ldquoHigher-order sliding mode observer for estimation of tyre frictionin ground vehiclesrdquo IET Proceedings on Control Theory andApplications vol 8 no 6 pp 399ndash408 2014

[23] L Fridman and A Levant ldquoHigher order sliding modesSliding mode control in engineeringrdquo in Sliding Mode Controlin Engineering J P Barbot and W Perruquetti Eds MarcelDekker New York NY USA 2002

[24] M Ezzat J De Leon N Gonzalez and A GlumineauldquoObserver-controller scheme using high order sliding modetechniques for sensorless speed control of permanent magnetsynchronous motorrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 4012ndash4017 December2010

[25] D Zaltni and M N Abdelkrim ldquoRobust speed and positionobserver using HOSM for sensor-less SPMSM controlrdquoin Proceedings of the 7th International Multi-Conference onSystems Signals and Devices (SSD rsquo10) pp 1ndash6 June 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Time (s)0 01

1

03 05 07 09 1

0

05

minus05

minus1

i 120572120573

(A)

(a)

Time (s)0 01 03 05 07 09 1

0

05

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus40

40

0

e 120572120573

(V)

(c)

1500

3000

4000

Time (s)0 01 03 05 07 09 1

EstimatedActual

120596s

(rpm

)

(d)

0

3

Time (s)0 01 03 05 07 09 1

minus3

120579s

(rad

)

(e)

0

1

2

Time (s)0 01 03 05 07 09 1

minus2

minus1120579sminus120579s (

rad)

(f)

Figure 4 Estimation using higher-order sliding mode observer under no-load (a) Estimated currents (b) Estimation current error (c)Estimated back EMFs (d) Estimated speed (e) Estimated rotor position (f) Estimation rotor position error

increasesHowever the speed andposition estimation remainrobust with the proposed observer when compared to first-order SMO Due to parametric uncertainty the speed estima-tion with proposedHOSMmethod shows a very small steadystate error at 90 of the rated speed (Figures 8(d) and 10(a))Under loading the speed and position estimation with first-order SMOare highly affected Further tuning of the observerparameters can overcome the problem

From the implementation one can conclude the follow-ing

(1) The HOSM method requires the proper selection ofsliding mode gains 119870

1 1198702 1198703 and 119870

4 The sliding

mode gains should satisfy the conditions given by(A15)-(A16) for the desired speed range If themotoroperates in wide speed range the sliding mode gainsmust be appropriately selected

(2) Since the quality of the speed estimate highly dependson the estimated back EMFs it deteriorates when

more noisy back EMFs (obtained due to high gains)are used in the calculation Compared to existingmethods the proposed scheme provides good rotorspeed and position estimation without phase delay inthe presence of noise Further one should note thatsensorless speed estimation methods based on backEMFs fail at very low speeds and standstill

(3) It should be pointed out that a chattering phe-nomenon occurs using the conventional SM observer[15]Therefore in first-order SMobserver the signumfunction is used as switching function The speedestimate is approximated by low-pass filtering thediscontinuous switching functions This delay shouldbe compensated with an additional phase compensa-tion loop As low-pass filtering is eliminated with theproposed method an additional phase compensationloop is not required

Mathematical Problems in Engineering 7

1

0

05

minus05

minus1

i 120572120573

(A)

Time (s)0 01 03 05 07 09 1

(a)

Time (s)0 01 03 05 07 09 1

0

05

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus40

40

0

e 120572120573

(V)

(c)

Time (s)0 01 03 05 07 09 1

1500

3000

4000

120596s

(rpm

)

EstimatedActual

(d)

Time (s)0 01 03 05 07 09 1

0

3

minus3

120579s

(rad

)

(e)

Time (s)0 01 03 05 07 09 1

0

1

2

minus2

minus1120579sminus120579s (

rad)

(f)

Figure 5 Estimation using conventional first-order sliding mode observer under no-load (a) Estimated currents (b) Estimation currenterror (c) Estimated back EMFs (d) Estimated speed (e) Estimated rotor position (f) Estimation rotor position error

1500

3000

4000

EstimatedActual

120596s120596

s(r

pm)

Time (s)0 02 04 05 06 08 1

(a)

120579sminus120579s

(rad

)

0

1

2

minus1

minus2

Time (s)0 02 04 05 06 08 1

(b)

Figure 6 With proposed HOSMmethod under no-load (a) Actual and estimated speed (b) Rotor position estimation error

8 Mathematical Problems in Engineering

1500

3000

4000

Time (s)0 02 04 05 06 08 1

EstimatedActual

120596s120596

s(r

pm)

(a)

Time (s)

0 02 04 05 06 08 1

120579sminus120579s

(rad

)

0

1

2

minus1

minus2

(b)

Figure 7 With conventional first-order SMO under no-load (a) Estimated speed (b) Rotor position estimation error

Time (s)0 01 03 05 07 09 1

0

06

minus06

i 120572120573

(A)

(a)

Time (s)0 01 03 05 07 09 1

0

05

12

minus12

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus60

60

0

e 120572120573

(V)

(c)

1500

3000

4000

Time (s)0 01 03 05 07 09 1

EstimatedActual

120596s

(rpm

)

(d)

0

3

Time (s)0 01 03 05 07 09 1

minus3

120579s

(rad

)

(e)

0

1

2

Time (s)0 01 03 05 07 09 1

minus2

minus1120579sminus

120579s (

rad)

(f)

Figure 8 With proposed HOSM method under load (a) Estimated currents (b) Estimation current errors (c) Estimated back EMFs (d)Estimated speed (e) Estimated rotor position (f) Rotor position error

Mathematical Problems in Engineering 9

Time (s)0 01 03 05 07 09 1

1500

3000

4000

120596s120596

s(r

pm)

EstimatedActual

(a)

Time (s)0 01 03 05 07 09 1

0

1

2

minus2

minus1

120579sminus120579s (

rad)

(b)

Figure 9 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

1500

3000

4000

0 02 04 05 06 08 1

Time (s)EstimatedActual

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1Time (s)

0

1

2

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 10 With proposed HOSMmethod under load (a) Estimated speed (b) Rotor position estimation error

(4) Furthermore compared to the classical first-orderSM technique no cutoff frequency has to be tunedInstead a simple integration is realized It enablesto reduce the time delay for the estimation (whichdepends on the sampling period) One should alsohighlight that the discontinuous part of 120601

2(depend-

ing on1198704) is usually low compared to the continuous

part of 1206012and this enables to reduce the chattering

phenomenon(5) Moreover from the experiments the proposed

method is robust to the parameter variations and themeasurement noise compared to the traditional SMobserver

(6) It is worth to point out that the proposed method iscomputationally complex compared to the traditionalSM observer However if properly tuned it has moreadvantages than the traditional SM observer Theexperiments conducted in this paper validate theadvantages of this method

(7) For the same set of parameters the speed and positionestimation remained accurate for both no-loadingand loading conditions This further highlights therobustness of the proposedmethod to parameter vari-ations that occur with loading and other conditions

5 Conclusion

This paper has presented a sensorless speed estimationmethod for the PMSM driveThe HOSMmethod is based ona modified version of super-twisting algorithmThe observerdynamics consist of sliding mode terms which are used toreconstruct the unknown back EMFs The speed is thenanalytically computed from back EMFs Experimental resultsvalidate the feasibility and effectiveness of the proposedHOSM for estimating the rotor position and speed of thePMSM Compared with the traditional SMO the proposedhigher-order SMO provides better estimation performance

Appendix

Finite-Time Stability

For any vector 119911 = [1199111 119911

119902]119879isin 119877119902 and any scalar 120572 isin 119877

we denote the following

sign (119911) = [sign (1199111) sign (119911

119902)]119879

|119911|120572= diag (10038161003816100381610038161199111

1003816100381610038161003816120572

10038161003816100381610038161003816119911119902

10038161003816100381610038161003816

120572

)

lceil119911rfloor120572= |119911|120572 sign (119911)

(A1)

10 Mathematical Problems in Engineering

1500

3000

4000

EstimatedActual

0 02 04 05 06 08 1

Time (s)

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1

0

1

2

Time (s)

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 11 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

For ease of exposition consider the following system

119904 (119905) = minus 119886119904 (119905) + ] (119905) + 119890 (119904 119905)

119904 (1199050) = 1199040

(A2)

where 119904 isin R and 119886 is a known positive constant and 119890(119904 119905) isthe unknown inputperturbation and

] (119905) = minus11987011206011 (119904 (119905)) minus 119870

2int

119905

0

1206012 (119904 (119905)) 119889119905 (A3)

where 1206011(119904(119905)) and 120601

2(119904(119905)) are defined in (7) and119870

111987021198703

and1198704are appropriately designed positive constants

Assumption A1 The time derivative of the unknowninputperturbation is upper bounded as follows

| 119890 (119904 119905)| le 120588 (A4)

for a positive constant 120588

Remark A2 The sliding dynamics 119904120572or 119904120573in (8) can be

directly expressed in the form of (A2) Further the condition(9) is similar to Assumption A1

Proposition A3 Under Assumption A1 the origin of sys-tem (A2) is a finite time stable equilibrium point Fur-ther the finite-time smooth estimation of the unknowninputperturbation 119890(119904 119905) is given by 119870

2int119905

01206012(119904(119905))119889119905

Proof Proof follows the work given in [20] Since |1206012(119904)| ge

1198702

42 one gets

| 119890 (119904 119905)| le10038161003816100381610038161206012 (119904)

1003816100381610038161003816 (A5)

if

1198704ge radic2120588 (A6)

Let us select a Hurwitz matrix 1198600

1198600= [

minus (1198701+ 119886) 1

minus1198702

0] (A7)

where1198701gt 0 and119870

2gt 0

The system (A2) (A3) can be equivalently representedby the system of two first-order equations

1199041= 1199042minus (1198701+ 119886) (119904

1+ 1198704lceil 1199041rfloor12

)

1199042= minus 119870

2(1199041+1198702

4

2sign (119904

1) +

3

21198704lceil 1199041rfloor12

) + 119890

(A8)

with 1199041= 119904 119904

2= 119890 minus 119870

2int119905

01206012(1199041) 119889119905 and

1198704=

11987011198703

1198701+ 119886

(A9)

The solutions of the discontinuous differential equations andinclusions are understood in the sense of Filippov

Let us consider the new state vector

120585 = [1205851

1205852

] = [1199041+ 1198704lceil 1199041rfloor12

1199042

] (A10)

The stability analysis of system (A8) is performed usingthe following candidate Lyapunov function [20]

119881 (120585) = 120585119879119875120585 (A11)

with 119875 = 119875119879

= [ 120582+41205982minus2120598

minus2120598 1] 120582 gt 0 and 120598 gt 0 It is worth

noting that the matrix 119875 is positive definite if 120582 and 120598 are anyreal number

Using the differential equations inclusion theory its timederivative along the solutions of the system is given by

= (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

119890]

(A12)

Mathematical Problems in Engineering 11

It can be shown that

le (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

)(120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

1205851

])

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879119876120585

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120582min (119876)100381710038171003817100381712058510038171003817100381710038172

(A13)with

119876 = [11987611198762

11987621198763

]

1198761= 2 (119870

1+ 119886) (120582 + 4120598

2) minus 4120598 (119870

2minus 1)

1198762= minus 2120598 (119870

1+ 119886) + (119870

2+ 1) minus (120582 + 4120598

2)

1198763= 4120598

(A14)

In order to guarantee the positive definiteness of matrix 119876one chooses

1198702= 120582 + 4120598

2+ 2120598 (119870

1+ 119886) (A15)

The matrix 119876 is positive definite if

1198701gt minus119886 +

4120598 + 2120598120582 + 81205983

120582+

1

4120598120582 (A16)

From (A10) one can deduce that100381710038171003817100381712058510038171003817100381710038172= 1205852

1+ 1205852

2

= 1199042

1+ 21198704

10038161003816100381610038161199041100381610038161003816100381632

+ 1198702

4

100381610038161003816100381611990411003816100381610038161003816 + 1205852

2

ge 1198702

4

100381610038161003816100381611990411003816100381610038161003816

(A17)

Since1198704gt 0

minus1198704

10038171003817100381710038171205851003817100381710038171003817

ge minus10038161003816100381610038161199041

1003816100381610038161003816minus12

(A18)

It implies that

le minus120582min (119876)

12058212

max (119875)

1198702

4

211988112

minus120582min (119876)

120582max (119875)119881 (A19)

The closed-loop system (A8) is stabilized in finite timeSince 120585 converges to zero in finite time 119904

1and 1199042converge

to 0 Therefore the term1198702int119905

01206012(119904(119905))119889119905 gives in finite time a

smooth estimation of the unknown perturbation 119890(119904 119905)

Nomenclature

120596119904 Rotor electrical speed

119894120572 119894120573 Currents in stationary reference frame

119881120572 119881120573 Voltages in stationary reference frame

119890120572 119890120573 EMFs in stationary reference frame

119877 Stator resistance119871 Synchronous inductance119870119864 EMF constant

120579119904 Rotor position angle

119879119897 Load torque

Conflict of Interests

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

Acknowledgment

This research was supported by the Basic Science ResearchProgram through theNational Research Foundation of Korea(NRF) funded by the Ministry of Education Science andTechnology (Grant no 2011-0023999)

References

[1] R Wu and G R Slemon ldquoA permanent magnet motor drivewithout a shaft sensorrdquo IEEE Transactions on Industry Applica-tions vol 27 no 5 pp 1005ndash1011 1991

[2] P Tomei and C M Verrelli ldquoObserver-based speed trackingcontrol for sensorless permanent magnet synchronous motorswith unknown load torquerdquo IEEE Transactions on AutomaticControl vol 56 no 6 pp 1484ndash1488 2011

[3] V Utkin J G Guldner and J Shi Sliding Mode Control onElectromechanical Systems Taylor and Francis New York NYUSA 1st edition 1999

[4] S Chai L Wang and R Rogers ldquoModel predictive controlof a permanent magnet synchronous motor with experimentalvalidationrdquoControl Engineering Practice vol 21 no 11 pp 1584ndash1593 2013

[5] T Orlowska-Kowalska and M Dybkowski ldquoStator-current-based MRAS estimator for a wide range speed-sensorlessinduction-motor driverdquo IEEE Transactions on Industrial Elec-tronics vol 57 no 4 pp 1296ndash1308 2010

[6] M L Corradini G Ippoliti S Longhi and G Orlando ldquoAquasi-sliding mode approach for robust control and speedestimation of PM synchronous motorsrdquo IEEE Transactions onIndustrial Electronics vol 59 no 2 pp 1096ndash1104 2012

[7] B K Bose Modern Power Electronics and AC Drives Prentice-Hall Upper Saddle River NJ USA 2002

[8] K C Veluvolu and Y C Soh ldquoMultiple sliding mode observersand unknown input estimations for Lipschitz nonlinear sys-temsrdquo International Journal of Robust and Nonlinear Controlvol 21 no 11 pp 1322ndash1340 2011

[9] K C Veluvolu and D Lee ldquoSliding mode high-gain observersfor a class of uncertain nonlinear systemsrdquoAppliedMathematicsLetters vol 24 no 3 pp 329ndash334 2011

[10] K C Veluvolu and Y C Soh ldquoFault reconstruction and stateestimationwith slidingmode observers for Lipschitz non-linearsystemsrdquo IET Control Theory amp Applications vol 5 no 11 pp1255ndash1263 2011

[11] M Comanescu and L Xu ldquoSliding-mode MRAS speed estima-tors for sensorless vector control of induction machinerdquo IEEETransactions on Industrial Electronics vol 53 no 1 pp 146ndash1532006

[12] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[13] K C Veluvolu M Y Kim and D Lee ldquoNonlinear sliding modehigh-gain observers for fault estimationrdquo International Journalof Systems Science Principles and Applications of Systems andIntegration vol 42 no 7 pp 1065ndash1074 2011

12 Mathematical Problems in Engineering

[14] K C Veluvolu M Defoort and Y C Soh ldquoHigh-gain observerwith sliding mode for nonlinear state estimation and faultreconstructionrdquo Journal of the Franklin Institute Engineeringand Applied Mathematics vol 351 no 4 pp 1995ndash2014 2014

[15] M Comanescu ldquoCascaded EMF and speed sliding modeobserver for the nonsalient PMSMrdquo in Proceedings of the 36thAnnual Conference of the IEEE Industrial Electronics Society(IECON rsquo10) pp 792ndash797 Glendale Ariz November 2010

[16] M Comanescu ldquoAn induction-motor speed estimator based onintegral sliding-mode current controlrdquo IEEE Transactions onIndustrial Electronics vol 56 no 9 pp 3414ndash3423 2009

[17] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[18] M Comanescu L Xu and T D Batzel ldquoDecoupled currentcontrol of sensorless induction-motor drives by integral slidingmoderdquo IEEE Transactions on Industrial Electronics vol 55 no11 pp 3836ndash3845 2008

[19] H Kim J Son and J Lee ldquoA high-speed sliding-mode observerfor the sensorless speed control of a PMSMrdquo IEEE Transactionson Industrial Electronics vol 58 no 9 pp 4069ndash4077 2011

[20] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

[21] T Floquet and J P Barbot ldquoSuper twisting algorithm-basedstep-by-step sliding mode observers for nonlinear systemswith unknown inputsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 38no 10 pp 803ndash815 2007

[22] J J Rath K C Veluvolu M Defoort and Y C Soh ldquoHigher-order sliding mode observer for estimation of tyre frictionin ground vehiclesrdquo IET Proceedings on Control Theory andApplications vol 8 no 6 pp 399ndash408 2014

[23] L Fridman and A Levant ldquoHigher order sliding modesSliding mode control in engineeringrdquo in Sliding Mode Controlin Engineering J P Barbot and W Perruquetti Eds MarcelDekker New York NY USA 2002

[24] M Ezzat J De Leon N Gonzalez and A GlumineauldquoObserver-controller scheme using high order sliding modetechniques for sensorless speed control of permanent magnetsynchronous motorrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 4012ndash4017 December2010

[25] D Zaltni and M N Abdelkrim ldquoRobust speed and positionobserver using HOSM for sensor-less SPMSM controlrdquoin Proceedings of the 7th International Multi-Conference onSystems Signals and Devices (SSD rsquo10) pp 1ndash6 June 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

1

0

05

minus05

minus1

i 120572120573

(A)

Time (s)0 01 03 05 07 09 1

(a)

Time (s)0 01 03 05 07 09 1

0

05

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus40

40

0

e 120572120573

(V)

(c)

Time (s)0 01 03 05 07 09 1

1500

3000

4000

120596s

(rpm

)

EstimatedActual

(d)

Time (s)0 01 03 05 07 09 1

0

3

minus3

120579s

(rad

)

(e)

Time (s)0 01 03 05 07 09 1

0

1

2

minus2

minus1120579sminus120579s (

rad)

(f)

Figure 5 Estimation using conventional first-order sliding mode observer under no-load (a) Estimated currents (b) Estimation currenterror (c) Estimated back EMFs (d) Estimated speed (e) Estimated rotor position (f) Estimation rotor position error

1500

3000

4000

EstimatedActual

120596s120596

s(r

pm)

Time (s)0 02 04 05 06 08 1

(a)

120579sminus120579s

(rad

)

0

1

2

minus1

minus2

Time (s)0 02 04 05 06 08 1

(b)

Figure 6 With proposed HOSMmethod under no-load (a) Actual and estimated speed (b) Rotor position estimation error

8 Mathematical Problems in Engineering

1500

3000

4000

Time (s)0 02 04 05 06 08 1

EstimatedActual

120596s120596

s(r

pm)

(a)

Time (s)

0 02 04 05 06 08 1

120579sminus120579s

(rad

)

0

1

2

minus1

minus2

(b)

Figure 7 With conventional first-order SMO under no-load (a) Estimated speed (b) Rotor position estimation error

Time (s)0 01 03 05 07 09 1

0

06

minus06

i 120572120573

(A)

(a)

Time (s)0 01 03 05 07 09 1

0

05

12

minus12

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus60

60

0

e 120572120573

(V)

(c)

1500

3000

4000

Time (s)0 01 03 05 07 09 1

EstimatedActual

120596s

(rpm

)

(d)

0

3

Time (s)0 01 03 05 07 09 1

minus3

120579s

(rad

)

(e)

0

1

2

Time (s)0 01 03 05 07 09 1

minus2

minus1120579sminus

120579s (

rad)

(f)

Figure 8 With proposed HOSM method under load (a) Estimated currents (b) Estimation current errors (c) Estimated back EMFs (d)Estimated speed (e) Estimated rotor position (f) Rotor position error

Mathematical Problems in Engineering 9

Time (s)0 01 03 05 07 09 1

1500

3000

4000

120596s120596

s(r

pm)

EstimatedActual

(a)

Time (s)0 01 03 05 07 09 1

0

1

2

minus2

minus1

120579sminus120579s (

rad)

(b)

Figure 9 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

1500

3000

4000

0 02 04 05 06 08 1

Time (s)EstimatedActual

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1Time (s)

0

1

2

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 10 With proposed HOSMmethod under load (a) Estimated speed (b) Rotor position estimation error

(4) Furthermore compared to the classical first-orderSM technique no cutoff frequency has to be tunedInstead a simple integration is realized It enablesto reduce the time delay for the estimation (whichdepends on the sampling period) One should alsohighlight that the discontinuous part of 120601

2(depend-

ing on1198704) is usually low compared to the continuous

part of 1206012and this enables to reduce the chattering

phenomenon(5) Moreover from the experiments the proposed

method is robust to the parameter variations and themeasurement noise compared to the traditional SMobserver

(6) It is worth to point out that the proposed method iscomputationally complex compared to the traditionalSM observer However if properly tuned it has moreadvantages than the traditional SM observer Theexperiments conducted in this paper validate theadvantages of this method

(7) For the same set of parameters the speed and positionestimation remained accurate for both no-loadingand loading conditions This further highlights therobustness of the proposedmethod to parameter vari-ations that occur with loading and other conditions

5 Conclusion

This paper has presented a sensorless speed estimationmethod for the PMSM driveThe HOSMmethod is based ona modified version of super-twisting algorithmThe observerdynamics consist of sliding mode terms which are used toreconstruct the unknown back EMFs The speed is thenanalytically computed from back EMFs Experimental resultsvalidate the feasibility and effectiveness of the proposedHOSM for estimating the rotor position and speed of thePMSM Compared with the traditional SMO the proposedhigher-order SMO provides better estimation performance

Appendix

Finite-Time Stability

For any vector 119911 = [1199111 119911

119902]119879isin 119877119902 and any scalar 120572 isin 119877

we denote the following

sign (119911) = [sign (1199111) sign (119911

119902)]119879

|119911|120572= diag (10038161003816100381610038161199111

1003816100381610038161003816120572

10038161003816100381610038161003816119911119902

10038161003816100381610038161003816

120572

)

lceil119911rfloor120572= |119911|120572 sign (119911)

(A1)

10 Mathematical Problems in Engineering

1500

3000

4000

EstimatedActual

0 02 04 05 06 08 1

Time (s)

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1

0

1

2

Time (s)

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 11 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

For ease of exposition consider the following system

119904 (119905) = minus 119886119904 (119905) + ] (119905) + 119890 (119904 119905)

119904 (1199050) = 1199040

(A2)

where 119904 isin R and 119886 is a known positive constant and 119890(119904 119905) isthe unknown inputperturbation and

] (119905) = minus11987011206011 (119904 (119905)) minus 119870

2int

119905

0

1206012 (119904 (119905)) 119889119905 (A3)

where 1206011(119904(119905)) and 120601

2(119904(119905)) are defined in (7) and119870

111987021198703

and1198704are appropriately designed positive constants

Assumption A1 The time derivative of the unknowninputperturbation is upper bounded as follows

| 119890 (119904 119905)| le 120588 (A4)

for a positive constant 120588

Remark A2 The sliding dynamics 119904120572or 119904120573in (8) can be

directly expressed in the form of (A2) Further the condition(9) is similar to Assumption A1

Proposition A3 Under Assumption A1 the origin of sys-tem (A2) is a finite time stable equilibrium point Fur-ther the finite-time smooth estimation of the unknowninputperturbation 119890(119904 119905) is given by 119870

2int119905

01206012(119904(119905))119889119905

Proof Proof follows the work given in [20] Since |1206012(119904)| ge

1198702

42 one gets

| 119890 (119904 119905)| le10038161003816100381610038161206012 (119904)

1003816100381610038161003816 (A5)

if

1198704ge radic2120588 (A6)

Let us select a Hurwitz matrix 1198600

1198600= [

minus (1198701+ 119886) 1

minus1198702

0] (A7)

where1198701gt 0 and119870

2gt 0

The system (A2) (A3) can be equivalently representedby the system of two first-order equations

1199041= 1199042minus (1198701+ 119886) (119904

1+ 1198704lceil 1199041rfloor12

)

1199042= minus 119870

2(1199041+1198702

4

2sign (119904

1) +

3

21198704lceil 1199041rfloor12

) + 119890

(A8)

with 1199041= 119904 119904

2= 119890 minus 119870

2int119905

01206012(1199041) 119889119905 and

1198704=

11987011198703

1198701+ 119886

(A9)

The solutions of the discontinuous differential equations andinclusions are understood in the sense of Filippov

Let us consider the new state vector

120585 = [1205851

1205852

] = [1199041+ 1198704lceil 1199041rfloor12

1199042

] (A10)

The stability analysis of system (A8) is performed usingthe following candidate Lyapunov function [20]

119881 (120585) = 120585119879119875120585 (A11)

with 119875 = 119875119879

= [ 120582+41205982minus2120598

minus2120598 1] 120582 gt 0 and 120598 gt 0 It is worth

noting that the matrix 119875 is positive definite if 120582 and 120598 are anyreal number

Using the differential equations inclusion theory its timederivative along the solutions of the system is given by

= (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

119890]

(A12)

Mathematical Problems in Engineering 11

It can be shown that

le (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

)(120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

1205851

])

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879119876120585

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120582min (119876)100381710038171003817100381712058510038171003817100381710038172

(A13)with

119876 = [11987611198762

11987621198763

]

1198761= 2 (119870

1+ 119886) (120582 + 4120598

2) minus 4120598 (119870

2minus 1)

1198762= minus 2120598 (119870

1+ 119886) + (119870

2+ 1) minus (120582 + 4120598

2)

1198763= 4120598

(A14)

In order to guarantee the positive definiteness of matrix 119876one chooses

1198702= 120582 + 4120598

2+ 2120598 (119870

1+ 119886) (A15)

The matrix 119876 is positive definite if

1198701gt minus119886 +

4120598 + 2120598120582 + 81205983

120582+

1

4120598120582 (A16)

From (A10) one can deduce that100381710038171003817100381712058510038171003817100381710038172= 1205852

1+ 1205852

2

= 1199042

1+ 21198704

10038161003816100381610038161199041100381610038161003816100381632

+ 1198702

4

100381610038161003816100381611990411003816100381610038161003816 + 1205852

2

ge 1198702

4

100381610038161003816100381611990411003816100381610038161003816

(A17)

Since1198704gt 0

minus1198704

10038171003817100381710038171205851003817100381710038171003817

ge minus10038161003816100381610038161199041

1003816100381610038161003816minus12

(A18)

It implies that

le minus120582min (119876)

12058212

max (119875)

1198702

4

211988112

minus120582min (119876)

120582max (119875)119881 (A19)

The closed-loop system (A8) is stabilized in finite timeSince 120585 converges to zero in finite time 119904

1and 1199042converge

to 0 Therefore the term1198702int119905

01206012(119904(119905))119889119905 gives in finite time a

smooth estimation of the unknown perturbation 119890(119904 119905)

Nomenclature

120596119904 Rotor electrical speed

119894120572 119894120573 Currents in stationary reference frame

119881120572 119881120573 Voltages in stationary reference frame

119890120572 119890120573 EMFs in stationary reference frame

119877 Stator resistance119871 Synchronous inductance119870119864 EMF constant

120579119904 Rotor position angle

119879119897 Load torque

Conflict of Interests

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

Acknowledgment

This research was supported by the Basic Science ResearchProgram through theNational Research Foundation of Korea(NRF) funded by the Ministry of Education Science andTechnology (Grant no 2011-0023999)

References

[1] R Wu and G R Slemon ldquoA permanent magnet motor drivewithout a shaft sensorrdquo IEEE Transactions on Industry Applica-tions vol 27 no 5 pp 1005ndash1011 1991

[2] P Tomei and C M Verrelli ldquoObserver-based speed trackingcontrol for sensorless permanent magnet synchronous motorswith unknown load torquerdquo IEEE Transactions on AutomaticControl vol 56 no 6 pp 1484ndash1488 2011

[3] V Utkin J G Guldner and J Shi Sliding Mode Control onElectromechanical Systems Taylor and Francis New York NYUSA 1st edition 1999

[4] S Chai L Wang and R Rogers ldquoModel predictive controlof a permanent magnet synchronous motor with experimentalvalidationrdquoControl Engineering Practice vol 21 no 11 pp 1584ndash1593 2013

[5] T Orlowska-Kowalska and M Dybkowski ldquoStator-current-based MRAS estimator for a wide range speed-sensorlessinduction-motor driverdquo IEEE Transactions on Industrial Elec-tronics vol 57 no 4 pp 1296ndash1308 2010

[6] M L Corradini G Ippoliti S Longhi and G Orlando ldquoAquasi-sliding mode approach for robust control and speedestimation of PM synchronous motorsrdquo IEEE Transactions onIndustrial Electronics vol 59 no 2 pp 1096ndash1104 2012

[7] B K Bose Modern Power Electronics and AC Drives Prentice-Hall Upper Saddle River NJ USA 2002

[8] K C Veluvolu and Y C Soh ldquoMultiple sliding mode observersand unknown input estimations for Lipschitz nonlinear sys-temsrdquo International Journal of Robust and Nonlinear Controlvol 21 no 11 pp 1322ndash1340 2011

[9] K C Veluvolu and D Lee ldquoSliding mode high-gain observersfor a class of uncertain nonlinear systemsrdquoAppliedMathematicsLetters vol 24 no 3 pp 329ndash334 2011

[10] K C Veluvolu and Y C Soh ldquoFault reconstruction and stateestimationwith slidingmode observers for Lipschitz non-linearsystemsrdquo IET Control Theory amp Applications vol 5 no 11 pp1255ndash1263 2011

[11] M Comanescu and L Xu ldquoSliding-mode MRAS speed estima-tors for sensorless vector control of induction machinerdquo IEEETransactions on Industrial Electronics vol 53 no 1 pp 146ndash1532006

[12] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[13] K C Veluvolu M Y Kim and D Lee ldquoNonlinear sliding modehigh-gain observers for fault estimationrdquo International Journalof Systems Science Principles and Applications of Systems andIntegration vol 42 no 7 pp 1065ndash1074 2011

12 Mathematical Problems in Engineering

[14] K C Veluvolu M Defoort and Y C Soh ldquoHigh-gain observerwith sliding mode for nonlinear state estimation and faultreconstructionrdquo Journal of the Franklin Institute Engineeringand Applied Mathematics vol 351 no 4 pp 1995ndash2014 2014

[15] M Comanescu ldquoCascaded EMF and speed sliding modeobserver for the nonsalient PMSMrdquo in Proceedings of the 36thAnnual Conference of the IEEE Industrial Electronics Society(IECON rsquo10) pp 792ndash797 Glendale Ariz November 2010

[16] M Comanescu ldquoAn induction-motor speed estimator based onintegral sliding-mode current controlrdquo IEEE Transactions onIndustrial Electronics vol 56 no 9 pp 3414ndash3423 2009

[17] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[18] M Comanescu L Xu and T D Batzel ldquoDecoupled currentcontrol of sensorless induction-motor drives by integral slidingmoderdquo IEEE Transactions on Industrial Electronics vol 55 no11 pp 3836ndash3845 2008

[19] H Kim J Son and J Lee ldquoA high-speed sliding-mode observerfor the sensorless speed control of a PMSMrdquo IEEE Transactionson Industrial Electronics vol 58 no 9 pp 4069ndash4077 2011

[20] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

[21] T Floquet and J P Barbot ldquoSuper twisting algorithm-basedstep-by-step sliding mode observers for nonlinear systemswith unknown inputsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 38no 10 pp 803ndash815 2007

[22] J J Rath K C Veluvolu M Defoort and Y C Soh ldquoHigher-order sliding mode observer for estimation of tyre frictionin ground vehiclesrdquo IET Proceedings on Control Theory andApplications vol 8 no 6 pp 399ndash408 2014

[23] L Fridman and A Levant ldquoHigher order sliding modesSliding mode control in engineeringrdquo in Sliding Mode Controlin Engineering J P Barbot and W Perruquetti Eds MarcelDekker New York NY USA 2002

[24] M Ezzat J De Leon N Gonzalez and A GlumineauldquoObserver-controller scheme using high order sliding modetechniques for sensorless speed control of permanent magnetsynchronous motorrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 4012ndash4017 December2010

[25] D Zaltni and M N Abdelkrim ldquoRobust speed and positionobserver using HOSM for sensor-less SPMSM controlrdquoin Proceedings of the 7th International Multi-Conference onSystems Signals and Devices (SSD rsquo10) pp 1ndash6 June 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

1500

3000

4000

Time (s)0 02 04 05 06 08 1

EstimatedActual

120596s120596

s(r

pm)

(a)

Time (s)

0 02 04 05 06 08 1

120579sminus120579s

(rad

)

0

1

2

minus1

minus2

(b)

Figure 7 With conventional first-order SMO under no-load (a) Estimated speed (b) Rotor position estimation error

Time (s)0 01 03 05 07 09 1

0

06

minus06

i 120572120573

(A)

(a)

Time (s)0 01 03 05 07 09 1

0

05

12

minus12

minus05

i 120572minusi 120572

i120573minusi 120573

(A)

(b)

Time (s)0 01 03 05 07 09 1

minus60

60

0

e 120572120573

(V)

(c)

1500

3000

4000

Time (s)0 01 03 05 07 09 1

EstimatedActual

120596s

(rpm

)

(d)

0

3

Time (s)0 01 03 05 07 09 1

minus3

120579s

(rad

)

(e)

0

1

2

Time (s)0 01 03 05 07 09 1

minus2

minus1120579sminus

120579s (

rad)

(f)

Figure 8 With proposed HOSM method under load (a) Estimated currents (b) Estimation current errors (c) Estimated back EMFs (d)Estimated speed (e) Estimated rotor position (f) Rotor position error

Mathematical Problems in Engineering 9

Time (s)0 01 03 05 07 09 1

1500

3000

4000

120596s120596

s(r

pm)

EstimatedActual

(a)

Time (s)0 01 03 05 07 09 1

0

1

2

minus2

minus1

120579sminus120579s (

rad)

(b)

Figure 9 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

1500

3000

4000

0 02 04 05 06 08 1

Time (s)EstimatedActual

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1Time (s)

0

1

2

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 10 With proposed HOSMmethod under load (a) Estimated speed (b) Rotor position estimation error

(4) Furthermore compared to the classical first-orderSM technique no cutoff frequency has to be tunedInstead a simple integration is realized It enablesto reduce the time delay for the estimation (whichdepends on the sampling period) One should alsohighlight that the discontinuous part of 120601

2(depend-

ing on1198704) is usually low compared to the continuous

part of 1206012and this enables to reduce the chattering

phenomenon(5) Moreover from the experiments the proposed

method is robust to the parameter variations and themeasurement noise compared to the traditional SMobserver

(6) It is worth to point out that the proposed method iscomputationally complex compared to the traditionalSM observer However if properly tuned it has moreadvantages than the traditional SM observer Theexperiments conducted in this paper validate theadvantages of this method

(7) For the same set of parameters the speed and positionestimation remained accurate for both no-loadingand loading conditions This further highlights therobustness of the proposedmethod to parameter vari-ations that occur with loading and other conditions

5 Conclusion

This paper has presented a sensorless speed estimationmethod for the PMSM driveThe HOSMmethod is based ona modified version of super-twisting algorithmThe observerdynamics consist of sliding mode terms which are used toreconstruct the unknown back EMFs The speed is thenanalytically computed from back EMFs Experimental resultsvalidate the feasibility and effectiveness of the proposedHOSM for estimating the rotor position and speed of thePMSM Compared with the traditional SMO the proposedhigher-order SMO provides better estimation performance

Appendix

Finite-Time Stability

For any vector 119911 = [1199111 119911

119902]119879isin 119877119902 and any scalar 120572 isin 119877

we denote the following

sign (119911) = [sign (1199111) sign (119911

119902)]119879

|119911|120572= diag (10038161003816100381610038161199111

1003816100381610038161003816120572

10038161003816100381610038161003816119911119902

10038161003816100381610038161003816

120572

)

lceil119911rfloor120572= |119911|120572 sign (119911)

(A1)

10 Mathematical Problems in Engineering

1500

3000

4000

EstimatedActual

0 02 04 05 06 08 1

Time (s)

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1

0

1

2

Time (s)

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 11 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

For ease of exposition consider the following system

119904 (119905) = minus 119886119904 (119905) + ] (119905) + 119890 (119904 119905)

119904 (1199050) = 1199040

(A2)

where 119904 isin R and 119886 is a known positive constant and 119890(119904 119905) isthe unknown inputperturbation and

] (119905) = minus11987011206011 (119904 (119905)) minus 119870

2int

119905

0

1206012 (119904 (119905)) 119889119905 (A3)

where 1206011(119904(119905)) and 120601

2(119904(119905)) are defined in (7) and119870

111987021198703

and1198704are appropriately designed positive constants

Assumption A1 The time derivative of the unknowninputperturbation is upper bounded as follows

| 119890 (119904 119905)| le 120588 (A4)

for a positive constant 120588

Remark A2 The sliding dynamics 119904120572or 119904120573in (8) can be

directly expressed in the form of (A2) Further the condition(9) is similar to Assumption A1

Proposition A3 Under Assumption A1 the origin of sys-tem (A2) is a finite time stable equilibrium point Fur-ther the finite-time smooth estimation of the unknowninputperturbation 119890(119904 119905) is given by 119870

2int119905

01206012(119904(119905))119889119905

Proof Proof follows the work given in [20] Since |1206012(119904)| ge

1198702

42 one gets

| 119890 (119904 119905)| le10038161003816100381610038161206012 (119904)

1003816100381610038161003816 (A5)

if

1198704ge radic2120588 (A6)

Let us select a Hurwitz matrix 1198600

1198600= [

minus (1198701+ 119886) 1

minus1198702

0] (A7)

where1198701gt 0 and119870

2gt 0

The system (A2) (A3) can be equivalently representedby the system of two first-order equations

1199041= 1199042minus (1198701+ 119886) (119904

1+ 1198704lceil 1199041rfloor12

)

1199042= minus 119870

2(1199041+1198702

4

2sign (119904

1) +

3

21198704lceil 1199041rfloor12

) + 119890

(A8)

with 1199041= 119904 119904

2= 119890 minus 119870

2int119905

01206012(1199041) 119889119905 and

1198704=

11987011198703

1198701+ 119886

(A9)

The solutions of the discontinuous differential equations andinclusions are understood in the sense of Filippov

Let us consider the new state vector

120585 = [1205851

1205852

] = [1199041+ 1198704lceil 1199041rfloor12

1199042

] (A10)

The stability analysis of system (A8) is performed usingthe following candidate Lyapunov function [20]

119881 (120585) = 120585119879119875120585 (A11)

with 119875 = 119875119879

= [ 120582+41205982minus2120598

minus2120598 1] 120582 gt 0 and 120598 gt 0 It is worth

noting that the matrix 119875 is positive definite if 120582 and 120598 are anyreal number

Using the differential equations inclusion theory its timederivative along the solutions of the system is given by

= (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

119890]

(A12)

Mathematical Problems in Engineering 11

It can be shown that

le (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

)(120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

1205851

])

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879119876120585

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120582min (119876)100381710038171003817100381712058510038171003817100381710038172

(A13)with

119876 = [11987611198762

11987621198763

]

1198761= 2 (119870

1+ 119886) (120582 + 4120598

2) minus 4120598 (119870

2minus 1)

1198762= minus 2120598 (119870

1+ 119886) + (119870

2+ 1) minus (120582 + 4120598

2)

1198763= 4120598

(A14)

In order to guarantee the positive definiteness of matrix 119876one chooses

1198702= 120582 + 4120598

2+ 2120598 (119870

1+ 119886) (A15)

The matrix 119876 is positive definite if

1198701gt minus119886 +

4120598 + 2120598120582 + 81205983

120582+

1

4120598120582 (A16)

From (A10) one can deduce that100381710038171003817100381712058510038171003817100381710038172= 1205852

1+ 1205852

2

= 1199042

1+ 21198704

10038161003816100381610038161199041100381610038161003816100381632

+ 1198702

4

100381610038161003816100381611990411003816100381610038161003816 + 1205852

2

ge 1198702

4

100381610038161003816100381611990411003816100381610038161003816

(A17)

Since1198704gt 0

minus1198704

10038171003817100381710038171205851003817100381710038171003817

ge minus10038161003816100381610038161199041

1003816100381610038161003816minus12

(A18)

It implies that

le minus120582min (119876)

12058212

max (119875)

1198702

4

211988112

minus120582min (119876)

120582max (119875)119881 (A19)

The closed-loop system (A8) is stabilized in finite timeSince 120585 converges to zero in finite time 119904

1and 1199042converge

to 0 Therefore the term1198702int119905

01206012(119904(119905))119889119905 gives in finite time a

smooth estimation of the unknown perturbation 119890(119904 119905)

Nomenclature

120596119904 Rotor electrical speed

119894120572 119894120573 Currents in stationary reference frame

119881120572 119881120573 Voltages in stationary reference frame

119890120572 119890120573 EMFs in stationary reference frame

119877 Stator resistance119871 Synchronous inductance119870119864 EMF constant

120579119904 Rotor position angle

119879119897 Load torque

Conflict of Interests

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

Acknowledgment

This research was supported by the Basic Science ResearchProgram through theNational Research Foundation of Korea(NRF) funded by the Ministry of Education Science andTechnology (Grant no 2011-0023999)

References

[1] R Wu and G R Slemon ldquoA permanent magnet motor drivewithout a shaft sensorrdquo IEEE Transactions on Industry Applica-tions vol 27 no 5 pp 1005ndash1011 1991

[2] P Tomei and C M Verrelli ldquoObserver-based speed trackingcontrol for sensorless permanent magnet synchronous motorswith unknown load torquerdquo IEEE Transactions on AutomaticControl vol 56 no 6 pp 1484ndash1488 2011

[3] V Utkin J G Guldner and J Shi Sliding Mode Control onElectromechanical Systems Taylor and Francis New York NYUSA 1st edition 1999

[4] S Chai L Wang and R Rogers ldquoModel predictive controlof a permanent magnet synchronous motor with experimentalvalidationrdquoControl Engineering Practice vol 21 no 11 pp 1584ndash1593 2013

[5] T Orlowska-Kowalska and M Dybkowski ldquoStator-current-based MRAS estimator for a wide range speed-sensorlessinduction-motor driverdquo IEEE Transactions on Industrial Elec-tronics vol 57 no 4 pp 1296ndash1308 2010

[6] M L Corradini G Ippoliti S Longhi and G Orlando ldquoAquasi-sliding mode approach for robust control and speedestimation of PM synchronous motorsrdquo IEEE Transactions onIndustrial Electronics vol 59 no 2 pp 1096ndash1104 2012

[7] B K Bose Modern Power Electronics and AC Drives Prentice-Hall Upper Saddle River NJ USA 2002

[8] K C Veluvolu and Y C Soh ldquoMultiple sliding mode observersand unknown input estimations for Lipschitz nonlinear sys-temsrdquo International Journal of Robust and Nonlinear Controlvol 21 no 11 pp 1322ndash1340 2011

[9] K C Veluvolu and D Lee ldquoSliding mode high-gain observersfor a class of uncertain nonlinear systemsrdquoAppliedMathematicsLetters vol 24 no 3 pp 329ndash334 2011

[10] K C Veluvolu and Y C Soh ldquoFault reconstruction and stateestimationwith slidingmode observers for Lipschitz non-linearsystemsrdquo IET Control Theory amp Applications vol 5 no 11 pp1255ndash1263 2011

[11] M Comanescu and L Xu ldquoSliding-mode MRAS speed estima-tors for sensorless vector control of induction machinerdquo IEEETransactions on Industrial Electronics vol 53 no 1 pp 146ndash1532006

[12] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[13] K C Veluvolu M Y Kim and D Lee ldquoNonlinear sliding modehigh-gain observers for fault estimationrdquo International Journalof Systems Science Principles and Applications of Systems andIntegration vol 42 no 7 pp 1065ndash1074 2011

12 Mathematical Problems in Engineering

[14] K C Veluvolu M Defoort and Y C Soh ldquoHigh-gain observerwith sliding mode for nonlinear state estimation and faultreconstructionrdquo Journal of the Franklin Institute Engineeringand Applied Mathematics vol 351 no 4 pp 1995ndash2014 2014

[15] M Comanescu ldquoCascaded EMF and speed sliding modeobserver for the nonsalient PMSMrdquo in Proceedings of the 36thAnnual Conference of the IEEE Industrial Electronics Society(IECON rsquo10) pp 792ndash797 Glendale Ariz November 2010

[16] M Comanescu ldquoAn induction-motor speed estimator based onintegral sliding-mode current controlrdquo IEEE Transactions onIndustrial Electronics vol 56 no 9 pp 3414ndash3423 2009

[17] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[18] M Comanescu L Xu and T D Batzel ldquoDecoupled currentcontrol of sensorless induction-motor drives by integral slidingmoderdquo IEEE Transactions on Industrial Electronics vol 55 no11 pp 3836ndash3845 2008

[19] H Kim J Son and J Lee ldquoA high-speed sliding-mode observerfor the sensorless speed control of a PMSMrdquo IEEE Transactionson Industrial Electronics vol 58 no 9 pp 4069ndash4077 2011

[20] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

[21] T Floquet and J P Barbot ldquoSuper twisting algorithm-basedstep-by-step sliding mode observers for nonlinear systemswith unknown inputsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 38no 10 pp 803ndash815 2007

[22] J J Rath K C Veluvolu M Defoort and Y C Soh ldquoHigher-order sliding mode observer for estimation of tyre frictionin ground vehiclesrdquo IET Proceedings on Control Theory andApplications vol 8 no 6 pp 399ndash408 2014

[23] L Fridman and A Levant ldquoHigher order sliding modesSliding mode control in engineeringrdquo in Sliding Mode Controlin Engineering J P Barbot and W Perruquetti Eds MarcelDekker New York NY USA 2002

[24] M Ezzat J De Leon N Gonzalez and A GlumineauldquoObserver-controller scheme using high order sliding modetechniques for sensorless speed control of permanent magnetsynchronous motorrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 4012ndash4017 December2010

[25] D Zaltni and M N Abdelkrim ldquoRobust speed and positionobserver using HOSM for sensor-less SPMSM controlrdquoin Proceedings of the 7th International Multi-Conference onSystems Signals and Devices (SSD rsquo10) pp 1ndash6 June 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Time (s)0 01 03 05 07 09 1

1500

3000

4000

120596s120596

s(r

pm)

EstimatedActual

(a)

Time (s)0 01 03 05 07 09 1

0

1

2

minus2

minus1

120579sminus120579s (

rad)

(b)

Figure 9 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

1500

3000

4000

0 02 04 05 06 08 1

Time (s)EstimatedActual

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1Time (s)

0

1

2

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 10 With proposed HOSMmethod under load (a) Estimated speed (b) Rotor position estimation error

(4) Furthermore compared to the classical first-orderSM technique no cutoff frequency has to be tunedInstead a simple integration is realized It enablesto reduce the time delay for the estimation (whichdepends on the sampling period) One should alsohighlight that the discontinuous part of 120601

2(depend-

ing on1198704) is usually low compared to the continuous

part of 1206012and this enables to reduce the chattering

phenomenon(5) Moreover from the experiments the proposed

method is robust to the parameter variations and themeasurement noise compared to the traditional SMobserver

(6) It is worth to point out that the proposed method iscomputationally complex compared to the traditionalSM observer However if properly tuned it has moreadvantages than the traditional SM observer Theexperiments conducted in this paper validate theadvantages of this method

(7) For the same set of parameters the speed and positionestimation remained accurate for both no-loadingand loading conditions This further highlights therobustness of the proposedmethod to parameter vari-ations that occur with loading and other conditions

5 Conclusion

This paper has presented a sensorless speed estimationmethod for the PMSM driveThe HOSMmethod is based ona modified version of super-twisting algorithmThe observerdynamics consist of sliding mode terms which are used toreconstruct the unknown back EMFs The speed is thenanalytically computed from back EMFs Experimental resultsvalidate the feasibility and effectiveness of the proposedHOSM for estimating the rotor position and speed of thePMSM Compared with the traditional SMO the proposedhigher-order SMO provides better estimation performance

Appendix

Finite-Time Stability

For any vector 119911 = [1199111 119911

119902]119879isin 119877119902 and any scalar 120572 isin 119877

we denote the following

sign (119911) = [sign (1199111) sign (119911

119902)]119879

|119911|120572= diag (10038161003816100381610038161199111

1003816100381610038161003816120572

10038161003816100381610038161003816119911119902

10038161003816100381610038161003816

120572

)

lceil119911rfloor120572= |119911|120572 sign (119911)

(A1)

10 Mathematical Problems in Engineering

1500

3000

4000

EstimatedActual

0 02 04 05 06 08 1

Time (s)

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1

0

1

2

Time (s)

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 11 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

For ease of exposition consider the following system

119904 (119905) = minus 119886119904 (119905) + ] (119905) + 119890 (119904 119905)

119904 (1199050) = 1199040

(A2)

where 119904 isin R and 119886 is a known positive constant and 119890(119904 119905) isthe unknown inputperturbation and

] (119905) = minus11987011206011 (119904 (119905)) minus 119870

2int

119905

0

1206012 (119904 (119905)) 119889119905 (A3)

where 1206011(119904(119905)) and 120601

2(119904(119905)) are defined in (7) and119870

111987021198703

and1198704are appropriately designed positive constants

Assumption A1 The time derivative of the unknowninputperturbation is upper bounded as follows

| 119890 (119904 119905)| le 120588 (A4)

for a positive constant 120588

Remark A2 The sliding dynamics 119904120572or 119904120573in (8) can be

directly expressed in the form of (A2) Further the condition(9) is similar to Assumption A1

Proposition A3 Under Assumption A1 the origin of sys-tem (A2) is a finite time stable equilibrium point Fur-ther the finite-time smooth estimation of the unknowninputperturbation 119890(119904 119905) is given by 119870

2int119905

01206012(119904(119905))119889119905

Proof Proof follows the work given in [20] Since |1206012(119904)| ge

1198702

42 one gets

| 119890 (119904 119905)| le10038161003816100381610038161206012 (119904)

1003816100381610038161003816 (A5)

if

1198704ge radic2120588 (A6)

Let us select a Hurwitz matrix 1198600

1198600= [

minus (1198701+ 119886) 1

minus1198702

0] (A7)

where1198701gt 0 and119870

2gt 0

The system (A2) (A3) can be equivalently representedby the system of two first-order equations

1199041= 1199042minus (1198701+ 119886) (119904

1+ 1198704lceil 1199041rfloor12

)

1199042= minus 119870

2(1199041+1198702

4

2sign (119904

1) +

3

21198704lceil 1199041rfloor12

) + 119890

(A8)

with 1199041= 119904 119904

2= 119890 minus 119870

2int119905

01206012(1199041) 119889119905 and

1198704=

11987011198703

1198701+ 119886

(A9)

The solutions of the discontinuous differential equations andinclusions are understood in the sense of Filippov

Let us consider the new state vector

120585 = [1205851

1205852

] = [1199041+ 1198704lceil 1199041rfloor12

1199042

] (A10)

The stability analysis of system (A8) is performed usingthe following candidate Lyapunov function [20]

119881 (120585) = 120585119879119875120585 (A11)

with 119875 = 119875119879

= [ 120582+41205982minus2120598

minus2120598 1] 120582 gt 0 and 120598 gt 0 It is worth

noting that the matrix 119875 is positive definite if 120582 and 120598 are anyreal number

Using the differential equations inclusion theory its timederivative along the solutions of the system is given by

= (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

119890]

(A12)

Mathematical Problems in Engineering 11

It can be shown that

le (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

)(120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

1205851

])

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879119876120585

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120582min (119876)100381710038171003817100381712058510038171003817100381710038172

(A13)with

119876 = [11987611198762

11987621198763

]

1198761= 2 (119870

1+ 119886) (120582 + 4120598

2) minus 4120598 (119870

2minus 1)

1198762= minus 2120598 (119870

1+ 119886) + (119870

2+ 1) minus (120582 + 4120598

2)

1198763= 4120598

(A14)

In order to guarantee the positive definiteness of matrix 119876one chooses

1198702= 120582 + 4120598

2+ 2120598 (119870

1+ 119886) (A15)

The matrix 119876 is positive definite if

1198701gt minus119886 +

4120598 + 2120598120582 + 81205983

120582+

1

4120598120582 (A16)

From (A10) one can deduce that100381710038171003817100381712058510038171003817100381710038172= 1205852

1+ 1205852

2

= 1199042

1+ 21198704

10038161003816100381610038161199041100381610038161003816100381632

+ 1198702

4

100381610038161003816100381611990411003816100381610038161003816 + 1205852

2

ge 1198702

4

100381610038161003816100381611990411003816100381610038161003816

(A17)

Since1198704gt 0

minus1198704

10038171003817100381710038171205851003817100381710038171003817

ge minus10038161003816100381610038161199041

1003816100381610038161003816minus12

(A18)

It implies that

le minus120582min (119876)

12058212

max (119875)

1198702

4

211988112

minus120582min (119876)

120582max (119875)119881 (A19)

The closed-loop system (A8) is stabilized in finite timeSince 120585 converges to zero in finite time 119904

1and 1199042converge

to 0 Therefore the term1198702int119905

01206012(119904(119905))119889119905 gives in finite time a

smooth estimation of the unknown perturbation 119890(119904 119905)

Nomenclature

120596119904 Rotor electrical speed

119894120572 119894120573 Currents in stationary reference frame

119881120572 119881120573 Voltages in stationary reference frame

119890120572 119890120573 EMFs in stationary reference frame

119877 Stator resistance119871 Synchronous inductance119870119864 EMF constant

120579119904 Rotor position angle

119879119897 Load torque

Conflict of Interests

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

Acknowledgment

This research was supported by the Basic Science ResearchProgram through theNational Research Foundation of Korea(NRF) funded by the Ministry of Education Science andTechnology (Grant no 2011-0023999)

References

[1] R Wu and G R Slemon ldquoA permanent magnet motor drivewithout a shaft sensorrdquo IEEE Transactions on Industry Applica-tions vol 27 no 5 pp 1005ndash1011 1991

[2] P Tomei and C M Verrelli ldquoObserver-based speed trackingcontrol for sensorless permanent magnet synchronous motorswith unknown load torquerdquo IEEE Transactions on AutomaticControl vol 56 no 6 pp 1484ndash1488 2011

[3] V Utkin J G Guldner and J Shi Sliding Mode Control onElectromechanical Systems Taylor and Francis New York NYUSA 1st edition 1999

[4] S Chai L Wang and R Rogers ldquoModel predictive controlof a permanent magnet synchronous motor with experimentalvalidationrdquoControl Engineering Practice vol 21 no 11 pp 1584ndash1593 2013

[5] T Orlowska-Kowalska and M Dybkowski ldquoStator-current-based MRAS estimator for a wide range speed-sensorlessinduction-motor driverdquo IEEE Transactions on Industrial Elec-tronics vol 57 no 4 pp 1296ndash1308 2010

[6] M L Corradini G Ippoliti S Longhi and G Orlando ldquoAquasi-sliding mode approach for robust control and speedestimation of PM synchronous motorsrdquo IEEE Transactions onIndustrial Electronics vol 59 no 2 pp 1096ndash1104 2012

[7] B K Bose Modern Power Electronics and AC Drives Prentice-Hall Upper Saddle River NJ USA 2002

[8] K C Veluvolu and Y C Soh ldquoMultiple sliding mode observersand unknown input estimations for Lipschitz nonlinear sys-temsrdquo International Journal of Robust and Nonlinear Controlvol 21 no 11 pp 1322ndash1340 2011

[9] K C Veluvolu and D Lee ldquoSliding mode high-gain observersfor a class of uncertain nonlinear systemsrdquoAppliedMathematicsLetters vol 24 no 3 pp 329ndash334 2011

[10] K C Veluvolu and Y C Soh ldquoFault reconstruction and stateestimationwith slidingmode observers for Lipschitz non-linearsystemsrdquo IET Control Theory amp Applications vol 5 no 11 pp1255ndash1263 2011

[11] M Comanescu and L Xu ldquoSliding-mode MRAS speed estima-tors for sensorless vector control of induction machinerdquo IEEETransactions on Industrial Electronics vol 53 no 1 pp 146ndash1532006

[12] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[13] K C Veluvolu M Y Kim and D Lee ldquoNonlinear sliding modehigh-gain observers for fault estimationrdquo International Journalof Systems Science Principles and Applications of Systems andIntegration vol 42 no 7 pp 1065ndash1074 2011

12 Mathematical Problems in Engineering

[14] K C Veluvolu M Defoort and Y C Soh ldquoHigh-gain observerwith sliding mode for nonlinear state estimation and faultreconstructionrdquo Journal of the Franklin Institute Engineeringand Applied Mathematics vol 351 no 4 pp 1995ndash2014 2014

[15] M Comanescu ldquoCascaded EMF and speed sliding modeobserver for the nonsalient PMSMrdquo in Proceedings of the 36thAnnual Conference of the IEEE Industrial Electronics Society(IECON rsquo10) pp 792ndash797 Glendale Ariz November 2010

[16] M Comanescu ldquoAn induction-motor speed estimator based onintegral sliding-mode current controlrdquo IEEE Transactions onIndustrial Electronics vol 56 no 9 pp 3414ndash3423 2009

[17] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[18] M Comanescu L Xu and T D Batzel ldquoDecoupled currentcontrol of sensorless induction-motor drives by integral slidingmoderdquo IEEE Transactions on Industrial Electronics vol 55 no11 pp 3836ndash3845 2008

[19] H Kim J Son and J Lee ldquoA high-speed sliding-mode observerfor the sensorless speed control of a PMSMrdquo IEEE Transactionson Industrial Electronics vol 58 no 9 pp 4069ndash4077 2011

[20] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

[21] T Floquet and J P Barbot ldquoSuper twisting algorithm-basedstep-by-step sliding mode observers for nonlinear systemswith unknown inputsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 38no 10 pp 803ndash815 2007

[22] J J Rath K C Veluvolu M Defoort and Y C Soh ldquoHigher-order sliding mode observer for estimation of tyre frictionin ground vehiclesrdquo IET Proceedings on Control Theory andApplications vol 8 no 6 pp 399ndash408 2014

[23] L Fridman and A Levant ldquoHigher order sliding modesSliding mode control in engineeringrdquo in Sliding Mode Controlin Engineering J P Barbot and W Perruquetti Eds MarcelDekker New York NY USA 2002

[24] M Ezzat J De Leon N Gonzalez and A GlumineauldquoObserver-controller scheme using high order sliding modetechniques for sensorless speed control of permanent magnetsynchronous motorrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 4012ndash4017 December2010

[25] D Zaltni and M N Abdelkrim ldquoRobust speed and positionobserver using HOSM for sensor-less SPMSM controlrdquoin Proceedings of the 7th International Multi-Conference onSystems Signals and Devices (SSD rsquo10) pp 1ndash6 June 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

1500

3000

4000

EstimatedActual

0 02 04 05 06 08 1

Time (s)

120596s120596

s(r

pm)

(a)

0 02 04 05 06 08 1

0

1

2

Time (s)

minus1

minus2

120579sminus

120579s

(rad

)

(b)

Figure 11 With conventional first-order SMO under load (a) Estimated speed (b) Rotor position estimation error

For ease of exposition consider the following system

119904 (119905) = minus 119886119904 (119905) + ] (119905) + 119890 (119904 119905)

119904 (1199050) = 1199040

(A2)

where 119904 isin R and 119886 is a known positive constant and 119890(119904 119905) isthe unknown inputperturbation and

] (119905) = minus11987011206011 (119904 (119905)) minus 119870

2int

119905

0

1206012 (119904 (119905)) 119889119905 (A3)

where 1206011(119904(119905)) and 120601

2(119904(119905)) are defined in (7) and119870

111987021198703

and1198704are appropriately designed positive constants

Assumption A1 The time derivative of the unknowninputperturbation is upper bounded as follows

| 119890 (119904 119905)| le 120588 (A4)

for a positive constant 120588

Remark A2 The sliding dynamics 119904120572or 119904120573in (8) can be

directly expressed in the form of (A2) Further the condition(9) is similar to Assumption A1

Proposition A3 Under Assumption A1 the origin of sys-tem (A2) is a finite time stable equilibrium point Fur-ther the finite-time smooth estimation of the unknowninputperturbation 119890(119904 119905) is given by 119870

2int119905

01206012(119904(119905))119889119905

Proof Proof follows the work given in [20] Since |1206012(119904)| ge

1198702

42 one gets

| 119890 (119904 119905)| le10038161003816100381610038161206012 (119904)

1003816100381610038161003816 (A5)

if

1198704ge radic2120588 (A6)

Let us select a Hurwitz matrix 1198600

1198600= [

minus (1198701+ 119886) 1

minus1198702

0] (A7)

where1198701gt 0 and119870

2gt 0

The system (A2) (A3) can be equivalently representedby the system of two first-order equations

1199041= 1199042minus (1198701+ 119886) (119904

1+ 1198704lceil 1199041rfloor12

)

1199042= minus 119870

2(1199041+1198702

4

2sign (119904

1) +

3

21198704lceil 1199041rfloor12

) + 119890

(A8)

with 1199041= 119904 119904

2= 119890 minus 119870

2int119905

01206012(1199041) 119889119905 and

1198704=

11987011198703

1198701+ 119886

(A9)

The solutions of the discontinuous differential equations andinclusions are understood in the sense of Filippov

Let us consider the new state vector

120585 = [1205851

1205852

] = [1199041+ 1198704lceil 1199041rfloor12

1199042

] (A10)

The stability analysis of system (A8) is performed usingthe following candidate Lyapunov function [20]

119881 (120585) = 120585119879119875120585 (A11)

with 119875 = 119875119879

= [ 120582+41205982minus2120598

minus2120598 1] 120582 gt 0 and 120598 gt 0 It is worth

noting that the matrix 119875 is positive definite if 120582 and 120598 are anyreal number

Using the differential equations inclusion theory its timederivative along the solutions of the system is given by

= (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

119890]

(A12)

Mathematical Problems in Engineering 11

It can be shown that

le (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

)(120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

1205851

])

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879119876120585

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120582min (119876)100381710038171003817100381712058510038171003817100381710038172

(A13)with

119876 = [11987611198762

11987621198763

]

1198761= 2 (119870

1+ 119886) (120582 + 4120598

2) minus 4120598 (119870

2minus 1)

1198762= minus 2120598 (119870

1+ 119886) + (119870

2+ 1) minus (120582 + 4120598

2)

1198763= 4120598

(A14)

In order to guarantee the positive definiteness of matrix 119876one chooses

1198702= 120582 + 4120598

2+ 2120598 (119870

1+ 119886) (A15)

The matrix 119876 is positive definite if

1198701gt minus119886 +

4120598 + 2120598120582 + 81205983

120582+

1

4120598120582 (A16)

From (A10) one can deduce that100381710038171003817100381712058510038171003817100381710038172= 1205852

1+ 1205852

2

= 1199042

1+ 21198704

10038161003816100381610038161199041100381610038161003816100381632

+ 1198702

4

100381610038161003816100381611990411003816100381610038161003816 + 1205852

2

ge 1198702

4

100381610038161003816100381611990411003816100381610038161003816

(A17)

Since1198704gt 0

minus1198704

10038171003817100381710038171205851003817100381710038171003817

ge minus10038161003816100381610038161199041

1003816100381610038161003816minus12

(A18)

It implies that

le minus120582min (119876)

12058212

max (119875)

1198702

4

211988112

minus120582min (119876)

120582max (119875)119881 (A19)

The closed-loop system (A8) is stabilized in finite timeSince 120585 converges to zero in finite time 119904

1and 1199042converge

to 0 Therefore the term1198702int119905

01206012(119904(119905))119889119905 gives in finite time a

smooth estimation of the unknown perturbation 119890(119904 119905)

Nomenclature

120596119904 Rotor electrical speed

119894120572 119894120573 Currents in stationary reference frame

119881120572 119881120573 Voltages in stationary reference frame

119890120572 119890120573 EMFs in stationary reference frame

119877 Stator resistance119871 Synchronous inductance119870119864 EMF constant

120579119904 Rotor position angle

119879119897 Load torque

Conflict of Interests

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

Acknowledgment

This research was supported by the Basic Science ResearchProgram through theNational Research Foundation of Korea(NRF) funded by the Ministry of Education Science andTechnology (Grant no 2011-0023999)

References

[1] R Wu and G R Slemon ldquoA permanent magnet motor drivewithout a shaft sensorrdquo IEEE Transactions on Industry Applica-tions vol 27 no 5 pp 1005ndash1011 1991

[2] P Tomei and C M Verrelli ldquoObserver-based speed trackingcontrol for sensorless permanent magnet synchronous motorswith unknown load torquerdquo IEEE Transactions on AutomaticControl vol 56 no 6 pp 1484ndash1488 2011

[3] V Utkin J G Guldner and J Shi Sliding Mode Control onElectromechanical Systems Taylor and Francis New York NYUSA 1st edition 1999

[4] S Chai L Wang and R Rogers ldquoModel predictive controlof a permanent magnet synchronous motor with experimentalvalidationrdquoControl Engineering Practice vol 21 no 11 pp 1584ndash1593 2013

[5] T Orlowska-Kowalska and M Dybkowski ldquoStator-current-based MRAS estimator for a wide range speed-sensorlessinduction-motor driverdquo IEEE Transactions on Industrial Elec-tronics vol 57 no 4 pp 1296ndash1308 2010

[6] M L Corradini G Ippoliti S Longhi and G Orlando ldquoAquasi-sliding mode approach for robust control and speedestimation of PM synchronous motorsrdquo IEEE Transactions onIndustrial Electronics vol 59 no 2 pp 1096ndash1104 2012

[7] B K Bose Modern Power Electronics and AC Drives Prentice-Hall Upper Saddle River NJ USA 2002

[8] K C Veluvolu and Y C Soh ldquoMultiple sliding mode observersand unknown input estimations for Lipschitz nonlinear sys-temsrdquo International Journal of Robust and Nonlinear Controlvol 21 no 11 pp 1322ndash1340 2011

[9] K C Veluvolu and D Lee ldquoSliding mode high-gain observersfor a class of uncertain nonlinear systemsrdquoAppliedMathematicsLetters vol 24 no 3 pp 329ndash334 2011

[10] K C Veluvolu and Y C Soh ldquoFault reconstruction and stateestimationwith slidingmode observers for Lipschitz non-linearsystemsrdquo IET Control Theory amp Applications vol 5 no 11 pp1255ndash1263 2011

[11] M Comanescu and L Xu ldquoSliding-mode MRAS speed estima-tors for sensorless vector control of induction machinerdquo IEEETransactions on Industrial Electronics vol 53 no 1 pp 146ndash1532006

[12] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[13] K C Veluvolu M Y Kim and D Lee ldquoNonlinear sliding modehigh-gain observers for fault estimationrdquo International Journalof Systems Science Principles and Applications of Systems andIntegration vol 42 no 7 pp 1065ndash1074 2011

12 Mathematical Problems in Engineering

[14] K C Veluvolu M Defoort and Y C Soh ldquoHigh-gain observerwith sliding mode for nonlinear state estimation and faultreconstructionrdquo Journal of the Franklin Institute Engineeringand Applied Mathematics vol 351 no 4 pp 1995ndash2014 2014

[15] M Comanescu ldquoCascaded EMF and speed sliding modeobserver for the nonsalient PMSMrdquo in Proceedings of the 36thAnnual Conference of the IEEE Industrial Electronics Society(IECON rsquo10) pp 792ndash797 Glendale Ariz November 2010

[16] M Comanescu ldquoAn induction-motor speed estimator based onintegral sliding-mode current controlrdquo IEEE Transactions onIndustrial Electronics vol 56 no 9 pp 3414ndash3423 2009

[17] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[18] M Comanescu L Xu and T D Batzel ldquoDecoupled currentcontrol of sensorless induction-motor drives by integral slidingmoderdquo IEEE Transactions on Industrial Electronics vol 55 no11 pp 3836ndash3845 2008

[19] H Kim J Son and J Lee ldquoA high-speed sliding-mode observerfor the sensorless speed control of a PMSMrdquo IEEE Transactionson Industrial Electronics vol 58 no 9 pp 4069ndash4077 2011

[20] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

[21] T Floquet and J P Barbot ldquoSuper twisting algorithm-basedstep-by-step sliding mode observers for nonlinear systemswith unknown inputsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 38no 10 pp 803ndash815 2007

[22] J J Rath K C Veluvolu M Defoort and Y C Soh ldquoHigher-order sliding mode observer for estimation of tyre frictionin ground vehiclesrdquo IET Proceedings on Control Theory andApplications vol 8 no 6 pp 399ndash408 2014

[23] L Fridman and A Levant ldquoHigher order sliding modesSliding mode control in engineeringrdquo in Sliding Mode Controlin Engineering J P Barbot and W Perruquetti Eds MarcelDekker New York NY USA 2002

[24] M Ezzat J De Leon N Gonzalez and A GlumineauldquoObserver-controller scheme using high order sliding modetechniques for sensorless speed control of permanent magnetsynchronous motorrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 4012ndash4017 December2010

[25] D Zaltni and M N Abdelkrim ldquoRobust speed and positionobserver using HOSM for sensor-less SPMSM controlrdquoin Proceedings of the 7th International Multi-Conference onSystems Signals and Devices (SSD rsquo10) pp 1ndash6 June 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

It can be shown that

le (1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

)(120585119879(119860119879

0119875 + 119875119860

0) 120585 + 2120585

119879119875[

0

1205851

])

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120585119879119876120585

le minus(1 +1198704

2

100381610038161003816100381611990411003816100381610038161003816minus12

) 120582min (119876)100381710038171003817100381712058510038171003817100381710038172

(A13)with

119876 = [11987611198762

11987621198763

]

1198761= 2 (119870

1+ 119886) (120582 + 4120598

2) minus 4120598 (119870

2minus 1)

1198762= minus 2120598 (119870

1+ 119886) + (119870

2+ 1) minus (120582 + 4120598

2)

1198763= 4120598

(A14)

In order to guarantee the positive definiteness of matrix 119876one chooses

1198702= 120582 + 4120598

2+ 2120598 (119870

1+ 119886) (A15)

The matrix 119876 is positive definite if

1198701gt minus119886 +

4120598 + 2120598120582 + 81205983

120582+

1

4120598120582 (A16)

From (A10) one can deduce that100381710038171003817100381712058510038171003817100381710038172= 1205852

1+ 1205852

2

= 1199042

1+ 21198704

10038161003816100381610038161199041100381610038161003816100381632

+ 1198702

4

100381610038161003816100381611990411003816100381610038161003816 + 1205852

2

ge 1198702

4

100381610038161003816100381611990411003816100381610038161003816

(A17)

Since1198704gt 0

minus1198704

10038171003817100381710038171205851003817100381710038171003817

ge minus10038161003816100381610038161199041

1003816100381610038161003816minus12

(A18)

It implies that

le minus120582min (119876)

12058212

max (119875)

1198702

4

211988112

minus120582min (119876)

120582max (119875)119881 (A19)

The closed-loop system (A8) is stabilized in finite timeSince 120585 converges to zero in finite time 119904

1and 1199042converge

to 0 Therefore the term1198702int119905

01206012(119904(119905))119889119905 gives in finite time a

smooth estimation of the unknown perturbation 119890(119904 119905)

Nomenclature

120596119904 Rotor electrical speed

119894120572 119894120573 Currents in stationary reference frame

119881120572 119881120573 Voltages in stationary reference frame

119890120572 119890120573 EMFs in stationary reference frame

119877 Stator resistance119871 Synchronous inductance119870119864 EMF constant

120579119904 Rotor position angle

119879119897 Load torque

Conflict of Interests

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

Acknowledgment

This research was supported by the Basic Science ResearchProgram through theNational Research Foundation of Korea(NRF) funded by the Ministry of Education Science andTechnology (Grant no 2011-0023999)

References

[1] R Wu and G R Slemon ldquoA permanent magnet motor drivewithout a shaft sensorrdquo IEEE Transactions on Industry Applica-tions vol 27 no 5 pp 1005ndash1011 1991

[2] P Tomei and C M Verrelli ldquoObserver-based speed trackingcontrol for sensorless permanent magnet synchronous motorswith unknown load torquerdquo IEEE Transactions on AutomaticControl vol 56 no 6 pp 1484ndash1488 2011

[3] V Utkin J G Guldner and J Shi Sliding Mode Control onElectromechanical Systems Taylor and Francis New York NYUSA 1st edition 1999

[4] S Chai L Wang and R Rogers ldquoModel predictive controlof a permanent magnet synchronous motor with experimentalvalidationrdquoControl Engineering Practice vol 21 no 11 pp 1584ndash1593 2013

[5] T Orlowska-Kowalska and M Dybkowski ldquoStator-current-based MRAS estimator for a wide range speed-sensorlessinduction-motor driverdquo IEEE Transactions on Industrial Elec-tronics vol 57 no 4 pp 1296ndash1308 2010

[6] M L Corradini G Ippoliti S Longhi and G Orlando ldquoAquasi-sliding mode approach for robust control and speedestimation of PM synchronous motorsrdquo IEEE Transactions onIndustrial Electronics vol 59 no 2 pp 1096ndash1104 2012

[7] B K Bose Modern Power Electronics and AC Drives Prentice-Hall Upper Saddle River NJ USA 2002

[8] K C Veluvolu and Y C Soh ldquoMultiple sliding mode observersand unknown input estimations for Lipschitz nonlinear sys-temsrdquo International Journal of Robust and Nonlinear Controlvol 21 no 11 pp 1322ndash1340 2011

[9] K C Veluvolu and D Lee ldquoSliding mode high-gain observersfor a class of uncertain nonlinear systemsrdquoAppliedMathematicsLetters vol 24 no 3 pp 329ndash334 2011

[10] K C Veluvolu and Y C Soh ldquoFault reconstruction and stateestimationwith slidingmode observers for Lipschitz non-linearsystemsrdquo IET Control Theory amp Applications vol 5 no 11 pp1255ndash1263 2011

[11] M Comanescu and L Xu ldquoSliding-mode MRAS speed estima-tors for sensorless vector control of induction machinerdquo IEEETransactions on Industrial Electronics vol 53 no 1 pp 146ndash1532006

[12] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[13] K C Veluvolu M Y Kim and D Lee ldquoNonlinear sliding modehigh-gain observers for fault estimationrdquo International Journalof Systems Science Principles and Applications of Systems andIntegration vol 42 no 7 pp 1065ndash1074 2011

12 Mathematical Problems in Engineering

[14] K C Veluvolu M Defoort and Y C Soh ldquoHigh-gain observerwith sliding mode for nonlinear state estimation and faultreconstructionrdquo Journal of the Franklin Institute Engineeringand Applied Mathematics vol 351 no 4 pp 1995ndash2014 2014

[15] M Comanescu ldquoCascaded EMF and speed sliding modeobserver for the nonsalient PMSMrdquo in Proceedings of the 36thAnnual Conference of the IEEE Industrial Electronics Society(IECON rsquo10) pp 792ndash797 Glendale Ariz November 2010

[16] M Comanescu ldquoAn induction-motor speed estimator based onintegral sliding-mode current controlrdquo IEEE Transactions onIndustrial Electronics vol 56 no 9 pp 3414ndash3423 2009

[17] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[18] M Comanescu L Xu and T D Batzel ldquoDecoupled currentcontrol of sensorless induction-motor drives by integral slidingmoderdquo IEEE Transactions on Industrial Electronics vol 55 no11 pp 3836ndash3845 2008

[19] H Kim J Son and J Lee ldquoA high-speed sliding-mode observerfor the sensorless speed control of a PMSMrdquo IEEE Transactionson Industrial Electronics vol 58 no 9 pp 4069ndash4077 2011

[20] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

[21] T Floquet and J P Barbot ldquoSuper twisting algorithm-basedstep-by-step sliding mode observers for nonlinear systemswith unknown inputsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 38no 10 pp 803ndash815 2007

[22] J J Rath K C Veluvolu M Defoort and Y C Soh ldquoHigher-order sliding mode observer for estimation of tyre frictionin ground vehiclesrdquo IET Proceedings on Control Theory andApplications vol 8 no 6 pp 399ndash408 2014

[23] L Fridman and A Levant ldquoHigher order sliding modesSliding mode control in engineeringrdquo in Sliding Mode Controlin Engineering J P Barbot and W Perruquetti Eds MarcelDekker New York NY USA 2002

[24] M Ezzat J De Leon N Gonzalez and A GlumineauldquoObserver-controller scheme using high order sliding modetechniques for sensorless speed control of permanent magnetsynchronous motorrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 4012ndash4017 December2010

[25] D Zaltni and M N Abdelkrim ldquoRobust speed and positionobserver using HOSM for sensor-less SPMSM controlrdquoin Proceedings of the 7th International Multi-Conference onSystems Signals and Devices (SSD rsquo10) pp 1ndash6 June 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[14] K C Veluvolu M Defoort and Y C Soh ldquoHigh-gain observerwith sliding mode for nonlinear state estimation and faultreconstructionrdquo Journal of the Franklin Institute Engineeringand Applied Mathematics vol 351 no 4 pp 1995ndash2014 2014

[15] M Comanescu ldquoCascaded EMF and speed sliding modeobserver for the nonsalient PMSMrdquo in Proceedings of the 36thAnnual Conference of the IEEE Industrial Electronics Society(IECON rsquo10) pp 792ndash797 Glendale Ariz November 2010

[16] M Comanescu ldquoAn induction-motor speed estimator based onintegral sliding-mode current controlrdquo IEEE Transactions onIndustrial Electronics vol 56 no 9 pp 3414ndash3423 2009

[17] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[18] M Comanescu L Xu and T D Batzel ldquoDecoupled currentcontrol of sensorless induction-motor drives by integral slidingmoderdquo IEEE Transactions on Industrial Electronics vol 55 no11 pp 3836ndash3845 2008

[19] H Kim J Son and J Lee ldquoA high-speed sliding-mode observerfor the sensorless speed control of a PMSMrdquo IEEE Transactionson Industrial Electronics vol 58 no 9 pp 4069ndash4077 2011

[20] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

[21] T Floquet and J P Barbot ldquoSuper twisting algorithm-basedstep-by-step sliding mode observers for nonlinear systemswith unknown inputsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 38no 10 pp 803ndash815 2007

[22] J J Rath K C Veluvolu M Defoort and Y C Soh ldquoHigher-order sliding mode observer for estimation of tyre frictionin ground vehiclesrdquo IET Proceedings on Control Theory andApplications vol 8 no 6 pp 399ndash408 2014

[23] L Fridman and A Levant ldquoHigher order sliding modesSliding mode control in engineeringrdquo in Sliding Mode Controlin Engineering J P Barbot and W Perruquetti Eds MarcelDekker New York NY USA 2002

[24] M Ezzat J De Leon N Gonzalez and A GlumineauldquoObserver-controller scheme using high order sliding modetechniques for sensorless speed control of permanent magnetsynchronous motorrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 4012ndash4017 December2010

[25] D Zaltni and M N Abdelkrim ldquoRobust speed and positionobserver using HOSM for sensor-less SPMSM controlrdquoin Proceedings of the 7th International Multi-Conference onSystems Signals and Devices (SSD rsquo10) pp 1ndash6 June 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of