Performance of a Direct Torque Controlled IPMDrive System in the Low Speed Region

Saeid Haghbin, Sonja Lundmark and Ola CarlsonElectric Power Engineering Division, Energy and Environment Department, Chalmers University of Technology, Sweden

Emails: saeid.haghbin@chalmers.se, sonja.lundmark@chalmers.se and ola.carlson@chalmers.se

Abstract—High power density, high speed operation, highefficiency and wide speed range made the interior permanentmagnet (IPM) synchronous motors an interesting choice for acdrive systems. Different control strategies have been proposedto reach a high performance drive system. Direct torque control(DTC) is one of these widely used methods which has fast torquedynamic and a simple structure. Different motor, inverter andcontroller parameters affect the drive system performance inthis scheme. The drive system performance is investigated forfour possible inverter switching patterns in terms of the torqueripple, stator current ripple, flux ripple and inverter switchingfrequency in the low speed region. The results show that theswitching pattern in which zero voltage is applied to reduce thetorque has better performance compared to the other switchingpatterns. The analytic solution is provided to quantify the effectsof the inverter zero voltage vector on the flux and torque of themachine and how they change when the speed varies.

I. INTRODUCTION

The overall design process of a modern high performancecost effective drive system is still a complex task. Motor,inverter and controller are the main important parts of adrive system. They should be considered as one packagewhen the whole system is designed. The IPM synchronousmotors have recently gained attention by researches for theirspecial features like high power to volume density, widespeed operation range, high speed operation range (robustmechanical structure due to buried magnets inside the rotor)and high efficiency (ideally there is no rotor losses in thismachine) [1-3]. Different control methods have been proposedfor IPM machines to achieve a high performance adjustable-speed drive system [4]. Sensorless drive systems have becomemore and more popular as a consequence of sensors price anddifficulties [5].

Direct torque control method first introduced by Takahashi[6] and Depenbrock [7] gained a lot of attentions thanks to itssimple structure and fast torque dynamics. It was consideredas an alternative to the field oriented control (FOC) methodby ABB [8]. In the DTC method, six non-zero and two zerovoltage vectors generated by the inverter are selected to keepthe motor flux and torque within the limits of two hysteresisbands [9]. The DTC method has been widely studied forinduction machines. For the induction machines there are fourdifferent switching patterns for the selection of the invertervoltage vector [10]. Each switching method affects the drivesystem performance [11]. The same concept is applied tothe IPM synchronous machine [5], so most of the methods

developed for DTC based induction motor drive systems canbe applied for the DTC based IPM synchronous motor drives.

At low speeds, the stator voltage drop can’t be neglectedcompared to the back-emf so the copper losses will be highcompared to the air-gap power (back-emf multiplied by thecurrent) so the system performance will be low.

The purpose of this paper is to investigate the impact ofdifferent inverter switching patterns on the performance of aDTC based IPM drive system in terms of the torque ripple,flux ripple, current ripple and inverter switching frequency atlow speeds. Applying a zero voltage vector by the inverterhas an important role on the overall drive system performancethat will be addressed in the sequel. The machine equationsare solved for the inverter zero voltage and an analyticalsolution for the flux trajectory and torque is provided toquantify the machine behavior. The drive system is simulatedby the use of Matlab/Simulink package. The simulation resultshave been used to compare the flux ripple, stator currentripple, torque ripple and inverter switching frequency for eachinverter switching algorithm. Moreover, the flux trajectory anddeveloped torque have been presented as the simulation resultand compared with the analytical solutions. The results showthat when a zero voltage vector is applied to the machine bythe inverter, the motor torque will be reduced regardless of thespeed.

II. DTC OF THE IPM SYNCHRONOUS MOTOR

A. Dynamic model of an IPM

The well-known d-q model of AC machines (in the rotorreference frame) is widely used for simulation purposes. Thestator voltage equations for the d and q components are:

ud = Rid + Ld

d

dtid − ωrψq (1)

uq = Riq + Lq

d

dtiq + ωrψd (2)

where R is the stator winding resistance,Ld and Lq are directand quadrature axis winding self inductances and ωr is therotor angular speed. ψd and ψq are the stator components ofthe flux. The d and q axis components of the flux can bewritten as:

ψd = ψpm + Ldid (3)

ψq = Lqiq (4)

978-1-4244-6392-3/10/$26.00 ©2010 IEEE 1420

where ψpm is the permanent magnet flux. The developedelectromagnetic torque can be expressed as:

Te =3

2P [ψpmiq + (Ld − Lq)idiq] (5)

where P is the number of pole pairs of the machine.

B. The drive system diagram

Fig. 1 shows the block diagram of a IPM synchronous motordrive system based on the DTC method. During each sampleinterval the stator currents, iA and iB , are measured alongwith the dc bus voltage Vdc. Using the inverter switching states(SASBSC), the stator voltage and current vector componentsin the stationary reference frame can be calculated as [9]:

uα =2

3Vdc(SA − SA + SB

2) (6)

uβ =1√3Vdc(SB − SC) (7)

iα = iA (8)

iβ =iA + 2iB√

3(9)

where uα, uβ, iα and iβ are α and β components of the statorvoltage and current in the stationary reference frame.

The α and β components of the stator flux, ψα and ψβ , canbe obtained by the integration of the stator voltage minus thevoltage drop in the stator resistance as:

ψα =

∫ t

0

(uα −Riα)dt+ ψα|t=0 (10)

ψβ =

∫ t

0

(uβ −Riβ)dt+ ψβ|t=0. (11)

The electromagnetic torque, Te, can be written in terms ofquantities in the stationary reference frame as:

Te =3

2P (ψαiβ − ψβiα). (12)

This equation is used in the drive system to estimate thedeveloped electromagnetic torque [12]. The stator flux vectormagnitude and phase are given by:

|ψs| =√

ψ2α + ψ2

β (13)

∠ψs = arctan(ψβ

ψα

). (14)

As shown in Fig. 1, estimated values of the stator fluxvector magnitude, |ψs|, and the electromagnetic torque, Te,are compared with their reference values. Afterward, the errorsare provided to the flux and torque hysteresis controllers. Thegoal of the control system is to limit the flux and torque withinthe hysteresis bands around their reference values. By usingthe torque error, flux error and stator flux position the controlcan be done by applying a proper inverter voltage.

For a three phase inverter, there are 6 power switches.It is not possible to turn on the upper and lower switchesin a leg simultaneously. So there are 8 possible switching

Rectifier

3 Phase

SupplyVdc

+

-

Inverter

IPM

Stator Voltage

Vector Calculation

SA

SB

SC

Switching

Table

Stator Current

Vector Calculation

iBiA

Stator Flux and Torque Calculation

u u i i

+

+

-

-

Te

*

*

Te

s

ss

Fig. 1: Block diagram of the direct torque control of IPMsynchronous motor

Axis

Axis

K=1

K=5

K=4

K=3 K=2

K=6

u1=(100)

u2=(110)u3=(010)

u4=(011)

u6=(101)u5=(001)

Fig. 2: Inverter voltage space vectors

configurations where each state defines a voltage space vector.Fig. 2 shows six non-zero inverter voltage space vectors (thereare two zero voltage vectors, u7 and u8, that are not shownin this figure). Moreover the αβ plane can be divided into 6sectors (k=1, 2, 3, 4, 5 and 6) in which the controller needsto know in what sector the stator flux is located. If two levelhysteresis controllers are used for the flux and torque control,there will be four switching strategies for the selection ofthe appropriate stator voltage vector (these possible switchingstrategies are proposed for the DTC of induction motorsoriginally). Assume that the stator flux vector is located insector k, then these four switching strategies are listed in TableI [13]. Effects of the applied voltage vector on the motor fluxand torque are summarized in table I as well. For the DTCsystem based on the IPM synchronous motor mainly solutionA and D are used [14].

Assume that the flux is located in sector k; then, to increasethe torque the voltage vectors uk+1 or uk+2 will be applied(depending on if the flux increases or decreases, one of thetwo voltage vectors will be selected). Different voltage vectorscan be applied to decrease the torque in different switchingpossibilities. uk, uk−1, uk−2 , uk+3, u7 and u8 can beapplied according to Table I. As is seen in Table I, different

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switching patterns are only different in the torque decrementcase, regardless of the flux increase or decrease demand in themotor. To decrease the torque, the simplest way is applyinga zero voltage (solution A). The main difference betweenswitching algorithms is in applying the zero vector or non-zero vector to decrease the torque. The motor current ripple,torque ripple and inverter switching frequency will vary foreach switching strategy. This will affect the whole drive systemperformance for each switching method.

C. The drive system simulation

The whole drive system has been simulated by the use ofMatlab/Simulink to study the drive system performance. Themotor parameters were selected according to the motor modelin [15] and are shown in Table II. The controller parameterslike sampling frequency, torque reference value, flux referencevalue and hysteresis band values for the flux and torquehave considerable effects on the drive system performance.To investigate the impact of the inverter switching algorithmon the drive system performance, the controller parameters arekept the same for all switching patterns according to Table III.

III. IMPACT OF SWITCHING PATTERN ON THE DRIVESYSTEM PERFORMANCE

To evaluate the drive system performance at low speed(100 rpm in this case according to table III) for differentinverter switching algorithms according to Table I, the motortorque ripple, stator current ripple, stator flux ripple andinverter switching frequency have been considered. Using thesame motor, controller and load parameters, simulations havebeen conducted for different inverter switching patterns. Thenormalized torque ripple, normalized stator current ripple,normalized stator flux ripple and average inverter switchingfrequency have been determined. thus, after removing theaverage part of the signals (torque, magnitude of the statorcurrent and flux vectors), the root mean square (rms) values arecalculated. Moreover, the values are normalized by dividingwith the related average values. The results are presented inTable IV.

As is presented in Table IV, the torque ripple and averageinverter switching frequency are lower in solution A comparedto the other switching patterns. The reason for this is explainedlater in this section. For solution D, the inverter switchingfrequency is the highest, making inverter loss higher thanthose of the other switching algorithms. Thus, high valuesof the torque ripple and inverter switching frequency makethis solution (solution D) an unfavorable choice for the drivesystem at low speeds. The drive system with the switchingpattern A has better performance compared to the othermethods.

To show high frequency effects, the frequency spectrum ofstator current ripple is shown in Fig. 3 for the four differentswitching patterns. As can be seen in Fig. 3, the waveformcorresponding to solution D shows the lowest harmonics. Thisis related to the high switching frequency of the inverter

TABLE I: SWITCHING STRATEGIES FOR DTC SYSTEM

Te ↑ |ψs| ↑ Te ↑ |ψs| ↓ Te ↓ |ψs| ↑ Te ↓ |ψs| ↓

Solution A uk+1 uk+2 u7,u8 u7,u8

Solution B uk+1 uk+2 uk u7,u8

Solution C uk+1 uk+2 uk uk+3

Solution D uk+1 uk+2 uk−1 uk−2

TABLE II: PARAMETERS OF SYNCHRONOUS RELUCTANCEMOTOR

Rated power (kW) 2.2Rated line voltage (V) 380

Rated current (A) 4.1Rated speed(rev/min) 1750

No of poles 6Permanent magnet flux (Wb) 0.48

Stator resistance (ohm) 3.3d axis inductance (mH) 42q axis inductance (mH) 57

Inertia (Kg.m2) 0.01Viscous friction coefficient (Nms/rad) 0.002

TABLE III: CONTROLLER PARAMETERS OF DRIVE SYSTEM

Reference torque (N-m) 13Reference flux (Wb) 0.5

Torque hysteresis upper band (N-m) 14Torque hysteresis lower band (N-m) 12

Flux hysteresis upper band (Wb) 0.55Flux hysteresis lower band (Wb) 0.45

DC Link voltage (V) 510Motor steady state speed (rev/min) 100

Sampling frequency (KHz) 20

producing a more symmetric waveform, especially comparedto solution A that applies zero voltage vector in both fluxincrease and decrease cases.

To explain the situation it is useful to approach the torquecontrol process in terms of rotor reference frame quantities.The developed electromagnetic torque in an IPM synchronousmotor in terms of stator and rotor fluxes in the rotor referenceframe can be expressed as [14]:

Te =3P |ψs|8LdLq

[2ψpmLq sin δ − |ψs|(Lq − Ld) sin 2δ] (15)

where δ is the load angle, the angle between the stator androtor flux linkage vectors (Fig. 4). The torque is controlledby regulating the amplitude of the stator flux and its anglewith respect to the rotor flux. As expressed in equation (15)the torque is sensitive to the angle variations. So it is possibleto rapidly change the torque by changing the angle δ evenwith fixed magnitude of the stator flux vector. As mentioned

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101 102 103 104−120

−100

−80

−60

−40

−20

0

Frequency (Hz)(a)

Spe

ctru

m o

f Sta

tor

Cur

rent

(dB

)

101 102 103 104−120

−100

−80

−60

−40

−20

0

Frequency (Hz)(b)

Spe

ctru

m o

f Sta

tor

Cur

rent

(dB

)

101 102 103 104−120

−100

−80

−60

−40

−20

0

Frequency (Hz)(c)

Spe

ctru

m o

f Sta

tor

Cur

rent

(dB

)

101 102 103 104−120

−100

−80

−60

−40

−20

0

Frequency (Hz)(d)

Spe

ctru

m o

f Sta

tor

Cur

rent

(dB

)

Fig. 3: Stator current ripple frequency spectrum for differentswitching patterns: (a) solution A, (b): solution B, (c): solutionC and (d): solution D

TABLE IV: IMPACT OF SWITCHING ALGORITHM ON THEDRIVE SYSTEM PERFORMANCE

InverterSwitching Normalized Normalized Normalized switchingalgorithm torque ripple stator current stator flux frequency

(%) ripple (%) ripple (%) [kHz]

Solution A 5.31 6.81 7.54 0.35

Solution B 11.15 10.36 7.50 0.27

Solution C 10.11 7.81 8.91 1.46

Solution D 8.84 7.48 7.57 2.47

above when a zero voltage is applied the torque has lowerripple compared to the torque ripple when a non-zero voltagevector is applied by the inverter. In other words, the torquehas a lower slope when a zero voltage vector is applied to themotor compared to the case when a non-zero voltage vectoris applied at low speeds. The reason is that when a non-zerovoltage is applied, the flux will change more rapidly and theload angle will change rapidly compared to the situation whena zero voltage vector is applied [16]. Fig. 5 shows the torquewaveform for the inverter switching pattern A and D. For atorque decrement in solution D, by applying a non-zero vectorthe torque sharply decreases. In solution A, by applying a zerovector, the torque decreases smoothly which means that theswitching frequency will be lower in solution A compared tosolution D.

IV. IMPACT OF THE ZERO VOLTAGE VECTOR ON THE IPMMOTOR FLUX AND TORQUE:ANALYTICAL SOLUTION

Simulation results show that applying a zero-voltage vectorwill reduce the torque (solution A) and the torque will havelower ripple compared to other switching algorithms. Moredetail analysis is provided in this section to quantify machineflux trajectory and torque by solving the machine equations.During each sampling period the speed of the machine isassumed to be constant. The machine state-space equationsin the rotor reference frame with zero voltage can be writtenas:

[ψ̇d

ψ̇q

] = [− R

Ldωr

−ωr − RLq

][ψd

ψq] + [

RLd

0]ψpm. (16)

If we assume that the speed is constant then this system willbe a time-invariant linear system. The state solutions are [17]:

[ψd

ψq] = eAt[

ψd0

ψq0] + eAt

∫ t

0

e−Aσ[RLd

0]ψpmdσ (17)

where A = [− R

Ldωr

−ωr − RLq

].

ψd0 and ψq0 are the rotor flux components in d and q axes atthe initial time instant (the time that the zero voltage vector is

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d Axis

Axis

q Axis

Axi

s

d

sy

rq pmy

di

qisidd

iL

qqiL

Fig. 4: Vector diagram of IPM synchronous motor

1 1.0005 1.001 1.0015 1.002 1.0025 1.003 1.0035 1.004 1.0045 1.00510

10.5

11

11.5

12

12.5

13

13.5

14

14.5

15

time (Sec)

Torq

ue(N

m)

Solution ASolution D

Fig. 5: Torque waveform for switching algorithm A and D

applied) which here is assumed to be zero for simplicity. Thestates equilibrium points are points where the time derivativesare zero. These state equilibrium points, ψe

d and ψeq , can be

calculated as [18]:

ψed =

R2

R2 + ω2rLdLq

ψpm (18)

ψeq = − RLqωr

R2 + ω2rLdLq

ψpm. (19)

In this case the parametric calculation of the exponentialmatrix function is difficult but it is possible to describe thesystem behavior by looking at the eigenvalues of matrix A.The characteristic function of matrix A is [18]:

λ2 +R(1

Ld

+1

Lq

)λ+R2

LdLq

+ ω2r = 0. (20)

This equation can be solved to obtain eigenvalues λ. Depend-ing on the speed, the system has two real negative eigenvaluesor two complex conjugate eigenvalues with negative real parts.These eigenvalues, λ1 and λ2, are:

λ1,2 = −1

2R(

1

Ld

+1

Lq

)

± 1

2

√

R2(1

Ld

+1

Lq

)2 − 4(ω2r +

R2

LdLq

)

= −1

2R(

1

Ld

+1

Lq

) ± 1

2

√

R2(1

Ld

− 1

Lq

)2 − 4ω2r . (21)

If ωrc is defined as ωrc = 1

2R( 1

Ld− 1

Lq), then for ωr <

ωrc there are two negative real eigenvalues and for ωr > ωrc

there are two complex conjugate eigenvalues with negative realparts. For the machine used in this paper ωrc=10.34 rad/secwhich is equivalent to the mechanical speed of 32.9 rpm.

For ωr � ωrc the eigenvalues can be approximated as:

λ1,2 = −1

2R(

1

Ld

+1

Lq

) ± 1

2R(

1

Ld

− 1

Lq

) (22)

and for ωr � ωrc the eigenvalues can be approximated as:

λ1,2 = −1

2R(

1

Ld

+1

Lq

) ± jωr. (23)

For all speeds the real part of eigenvalues are negative soexponential stability of the system (time-invariant system) isguaranteed [18]. Moreover as time increases the trajectoryof states will move towards the equilibrium points that arecalculated in (18) and (19). The speed is assumed to beconstant for a long time to be able to identify the statestrajectory treatment. Figure 6 shows the stator flux trajectoryfor different speeds and how the equilibrium points changesas a function of speed. The time is not shown in this graphso in a real system the trajectory will be different (after a fewswitching samples a non-zero voltage vector will be appliedto increase the torque and this will change the trajectory).

If both eigenvalues are real and negative, say λ1 and λ2,then in the state response the terms eλ1t and eλ2t tend tozero after a while (λ1 and λ2 are negative values)[18]. Thestate trajectory in ψd and ψq plane directly moves towards theequilibrium point (that is a function of speed). If eigenvaluesare complex conjugate (ωr > ωrc) then the state responsesincludes eαtcosωrt which means that there will be someoscillations in the time response (α is the real part of theeigenvalue λ and is negative also). The oscillation frequencyis equivalent to the imaginary part of eigenvalues. The higherthe speed, the higher the time domain oscillations and that willcause circulation in the state plane trajectory (Fig. 6) [18].

In this simulations (when the zero voltage vector is applied)it has been assumed that the load angle is positive, because thetorque should be decreased by applying zero voltage. Howeverthis assumption does not affect the results. The magnitude ofthe stator flux and torque are shown in Fig. 7 and 8. As wasexpected, both are decreasing when a zero voltage vector isapplied to the machine. The decrement of flux magnitude isnot sensitive to the speed for a short time (10 sample timesin this case) but the torque decrement is very sensitive to thespeed. The reason for the fast torque decrement is that thetorque is very sensitive to the flux angle (load angle) as canbe seen from (15).

When the zero voltage vector is applied, in the low speedregion the torque decrement is smoother than that in the highspeed region. As a result, for high speed applications, applyinga zero voltage vector to decrease the torque will create moretorque ripple compared to the low speed region.

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−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

d component (Wb)

q co

mpo

nent

(Wb)

Sator flux dq compentns trajectory

ωr= 1 rad/sec

ωr= 70 rad/sec

ωr= 150 rad/sec

Trajectory ofEquilibrium Pointwhen ωr increases

ωr= 400 rad/sec

Fig. 6: Stator flux trajectories for different speeds and inverterzero-voltage vector (constant speed assumption)

0 20 40 60 80 100 120 140 160 180 2000.5484

0.5486

0.5488

0.549

0.5492

0.5494

0.5496

0.5498

0.55

0.5502

time (µ sec)

Mag

nitu

de o

f the

Sta

tor

Flux

(Wb)

ωr= 400 rad/sec

ωr= 1, 70 and 150 rad/sec

Fig. 7: Magnitude of the stator flux for different speeds as afunction of time with the inverter zero-voltage vector (constantspeed assumption)

0 20 40 60 80 100 120 140 160 180 20011.5

12

12.5

13

13.5

14

14.5

time (µ sec)

Torq

ue(N

m)

ωr= 400 rad/sec

ωr= 150 rad/sec

ωr= 70 rad/sec

ωr= 1 rad/sec

Fig. 8: The developed motor torque for different speeds withthe inverter zero-voltage vector (constant speed assumption)

V. CONCLUSION

For a DTC based IPM synchronous motor drive system, theeffects of the inverter switching pattern on the drive systemperformance in terms of the torque ripple, stator current ripple,stator flux magnitude ripple and inverter switching frequencywas investigated at low speeds. It is shown that the inverterswitching pattern A which employs the zero voltage vectorto reduce the torque has better performance compared to theother switching methods at low speeds. The analytical analysisof the system was made for the inverter zero-voltage vector

to quantify the machine torque and flux. The flux trajectoryand torque as a function of time has been presented fordifferent speeds. The analytical results show that applyinga zero voltage vector reduces the motor torque for differentspeeds.

REFERENCES

[1] H. Murakami, Y. Honda, H. Kiriyama, S. Morimoto and Y.Takeda, ”The performance comparison of SPMSM, IPMSM andSynRM in use as air-conditioning compressor,” Conf. Rec. IEEE IAS,vol. 2, pp. 840-845, 1999.[2] Zeraoulia Mounir, Benbouzid Mohamed El Hachemi, DialloDemba, ”Electric motor drive selection issues for HEV propulsionsystems : A comparative study,” IEEE transactions on vehiculartechnology ISSN 0018-9545 2006, vol. 55, no6, pp. 1756-1764.[3] M. A. Rahman, ”Recent Advances of IPM Motor Drives in Powerelectronics word”, PDES 2005, pp. 24-31.[4] T. M. Jahns, G. B. Kliman, and T. W. Neuman, ”Interior PMsynchronousmotors for adjustable speed drives,” IEEE Trans. Ind.Appl., vol. IA-22, no. 4, pp. 738-747, Jul./Aug. 1986.[5] Li Yongdong and Zhu Hao, ”Sensorless Control of PermanentMagnet Synchronous Motor - A Survey,” IEEE Vehicle Power andPropulsion Conference (VPPC), September 3-5, 2008, Harbin, China.[6] I. Takahashi and T. Noguchi, ”A new quick-response and high-efficiency control strategy of an induction motor,” IEEE Trans. onIndustry Applications, vol. 22, no. 5, pp. 820-827, 1986.[7] M. Depenbrock, ”Direct self-control (DSC) of inverter-fed induc-tion machine,” IEEE Trans. on Power Electronics, vol. 3, no. 4, pp.420-429, 1988.[8] P. Tiitinen and M. Surandra, ”The next generation motor controlmethod, DTC direct torque control,” Power Electronics, Drives andEnergy Systems for Industrial Growth, 1996., Proceedings of the 1996International Conference on, Vol. 1 (1996), pp. 37-43 vol.1.[9] Peter.Vas, ”Sensorless vector and direct torque control,” OxfordPress, 1998.[10] G. Buja, Domenico Casadei and Giovanni Serra, ”Direct torquecontrol of induction motor drives,” ISIE’97, Guimaraes, Portugal,IEEE Catalogue Number: 97TH8280.[11] Y. W. Hu, C. Tian, Y. K. Gu, Z. Q. You, L. X. Tang and M.F. Rahman, ”In-depth research on direct torque control of permanentmagnet synchronous motor”, IEEE Proceedings on industrial elec-tronics society, Vol. 2, pp. 1060-1065, 2002.[12] L. Zhong, M.F.Rahman, W.Y.Hu, ”Analysis of direct torquecontrolin PMSM drives,” IEEE Trans. Power Electronics, vol. 12,no3, pp. 528-536, 1997.[13] G. Buja, Domenico Casadei and Giovanni Serra, ”Direct torquecontrol of induction motor drives,” ISIE’97, Guimaraes, Portugal,IEEE Catalogue Number: 97TH8280.[14] M. F. Rahman and L. Zhong, ”Voltage switching tables for DTCcontrolled interior permanent magnet motor,” IECON’99, 25th annualConference June 1999. p. 1445-51.[15] Frede Blaabjerg, G. D. Andreescu, C.I. Pitic and I. Boldea,”Combined Flux Observer With Signal Injection Enhancement forWide Speed Range Sensorless Direct Torque Control of IPMSMDrives,” IEEE Transactions on Energy Conversion, vol. 23, no. 2,June 2008.[16] Saeid Haghbin and Torbjorn Thiringer, ”Impact of inverterswitching pattern on the performance of a direct torque controlledsynchronous reluctance motor drive,” International Conference onPower Engineering, Energy and Electrical Drives, POWERENG 09,18-20 March 2009, Lisboa, Portugal, pp. 337-341.[17] Wilson J. Rugh, ”Linear system theory,” second edition, PrenticeHall, 1996.[18] H. Khalil, ”Nonlinear systems,” third edition, Prentice Hall,2002.

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