IET Electric Power Applications

Research Article

Torque ripple reduction of brushless DCmotor with harmonic current injection basedon integral terminal sliding mode control

ISSN 1751-8660Received on 4th February 2017Revised 31st May 2017Accepted on 21st July 2017doi: 10.1049/iet-epa.2017.0070www.ietdl.org

Mojtaba Shirvani Boroujeni1, Gholamreza Arab Markadeh2 , Jafar Soltani3, Frede Blaabjerg4

1Department of Engineering, Shahrekord University, Shahrekord, Iran2Shahrekord University, Engineering Department and Center of Excellence for Mathematics, Shahrekord, Iran3Islamic Azad University, Khomeini Shahr Branch, Isfahan, Iran4Department of Energy Technology, Aalborg University, Aalborg, Denmark

E-mail: arab-gh@eng.sku.ac.ir

Abstract: Brushless DC motors have been used in many industrial applications and torque ripple reduction of these motors isan important subject. Harmonic current injection to the stator windings is one of the most effective methods based on feedingcurrent improvement. Due to multi-harmonic contents of the stator currents, the conventional methods based on rotationalreference frame cannot be used to calculate the voltage references for voltage source inverter (VSI). Sliding mode control(SMC), which has high dynamic response to track a time varying command, can be used to force the arbitrary reference currentto the stator windings without transferring the motor currents to the rotational reference frame. However, the main disadvantageof SMC is that the system states cannot reach the equilibrium point in infinite time as well as has a major chattering problem. Inthis study, a new control method called integral terminal SMC (ITSMC) is used to overcome these drawbacks. In order to injectthe reference currents to the motor windings, the ITSMC method is proposed, which generates the reference voltages for three-phase VSI. In order to show the robustness and performance of the proposed method, this method is compared with a SMC bysome simulation and experimental tests. It is concluded that the dynamic response and robustness of the proposed ITSMCmethod is higher than SMC and ITSMC is an appropriate method to inject the arbitrary reference current to the motor windings.

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1 IntroductionBrushless DC (BLDC) motors are used in many applicationsbecause of its advantages like simple construction, high powerdensity, high efficiency, long life time and easy to control [1]. Themain disadvantage of these motors is higher torque ripplegeneration. Some of the torque ripple reduction methods are likethe phase current perfectly match the back-EMF (electromotiveforce) [2], lead angle injection in respect to back-EMF zerocrossing [3], current controlled modulation technique [4], pulse-width modulation (PWM) control [5], direct torque and indirectflux control [6] and also harmonic current injection [7, 8].

In most of the harmonic current injection methods, theamplitudes of reference current harmonics are precisely calculatedto completely eliminate the selected harmonics of torque ripples[9], or by using an optimisation method, the reference currentharmonics are calculated to minimise the torque ripple [10].

In order to inject the calculated harmonic currents to the statorwindings, the conventional vector control method cannot be used.Since the harmonic contents of the stator currents in the rotationalreference frame will oscillate with sixth multiplies of thefundamental frequency, so the multi-reference frame method [11]or modified vector control method [7] may be used, which howeverare complicated and time-consuming methods.

Vector control in a multi-reference frame is proposed in [11] inorder to do harmonic injection into the stator windings. In thatmethod, the amplitude of the desired q-axis current, in the eachreference frame is assumed as a constant coefficient of the squarewave harmonics and the d-axis reference current is set to zero andthen the stator current components in the stationary reference frameat each frequency is obtained by Park transformation. Afterwards,the stator reference current components in the stationary referenceframe are calculated by the summation of each harmonic current.Then the reference of the stator voltage is calculated by using twoproportional–integral (PI) current controllers. In [12], a multiplereference frame synchronous estimator is proposed. The estimation

of the stator current harmonic amplitudes using transformationmatrices gives a heavy computational burden.

In [13], an adaptive notch filter is used to estimate the statorcurrent harmonics to be implemented in a multi-reference frame,and then, by using PI regulators the stator reference voltagecomponents in each reference frame can be derived. After that, bytransformation of these reference voltages to the stationary frameand adding them, the stator voltage references are calculated.

So, with the purpose of harmonic injection in the statorwinding, while the machine back-EMF is non-ideal, the controlmethods which are employed in the stationary reference frame arepreferred.

In order to achieve quick dynamic response of the currentcontrol loop with sinusoidal and harmonic input references, themethods such as current controlled PWM inverter [14] and PIresonance controller [15] were proposed earlier.

On the other hand, according to the non-linear nature of theBLDC motor, non-linear control methods are preferred for a wideoperation range of BLDC motors [16–20]. One of the non-linearmethods is sliding mode control (SMC). This method has a fastdynamic response, a simple structure and it is robust againstparameter variations. The SMC design consists of reaching andsliding phases. The main drawback of SMC, which is mentionedby many researchers, is the chattering phenomena. This weaknesscan be improved with selection of lower discontinuous gain in thecontrol law. In this situation, the dynamic response of controllerwill be degraded. Also, in the conventional SMC method, theconvergence time of the error to zero is infinite, which decreasesthe dynamic response speed. Therefore, to obtain an infiniteconvergence time of the SMC, the integral terminal SMC (ITSMC)method has been developed [21]. The ITSMC compared with theSMC needs a lower sliding gain and thereby it has less chatteringphenomena [21].

In [22], Feng et al. have proposed a terminal sliding modeobserver for estimating the no measurable mechanical parametersof the permanent magnet synchronous motors. The observer canfollow the system states in finite time. Also an ITSMC strategy is

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designed to guarantee a global finite-time stability of the observer.However, the introduced ITSMC structure is a proportional–differential (PD) type and the mentioned structure is highlysensitive to measurement noise, since the derivative of the motorspeed is used in the control law.

In [23], a PD type TSM technique is used to generate the q-axis reference current of PMSM by using the motor speed error.The introduced method needs the derivative of the motor speed,

as well as this method cannot be used to track the harmonicallyreference current, since the vector control method with harmoniccurrent needs to be of multi-reference frame.

A position sensorless control based on SMC and terminalsliding mode observer are proposed in [24] to improve the controlperformance of BLDC. The convergence speed of the system stateis enhanced, but this method suffers from the PD type TSMCmethod weaknesses, as mentioned earlier. Most of the harmonicinjection methods for torque ripple reduction of BLDC motors arebased on hysteresis or PWM controlled current source inverter orneeds a multi-reference frame transformation with additional PIcontroller to generate the reference voltage of the voltage sourceinverter (VSI). Also all of control methods based on TSMC are ofPD type, which needs the motor speed (or stator currents) timederivatives and they are highly sensitive measurement noise [25,26].

In this paper, the stator reference currents are proposed to becalculated based on Lagrange multiplier optimisation method [27].Then by using two integral type TSMCs, the inverter referencevoltage components in the stationary reference frame are calculatedto force the calculated harmonic currents in to the stator winding.Finally, in order to evaluate the dynamic response and effectivenessof the proposed method, the results of this method are comparedwith the SMC method both in simulations and by experimentaltests.

2 Modelling and control of BLDC motor2.1 Mathematical model of the BLDC motor in a stationaryreference frame

The equivalent circuit of BLDC motor is shown in Fig. 1. Thestator voltage equations can be written as follows:

Vas = Rsias + Lsdiasdt + ea

Vbs = Rsibs + Lsdibsdt + eb

Vcs = Rsics + Lsdicsdt + ec

(1)

where Ls = Lls + 3/2Lm, Vas, Vbs and Vcs are the terminal phasevoltages, Rs is the stator resistance, ias, ibs and ics are phasecurrents, Lls and Lm are the stator leakage and magnetisinginductances, respectively, and ea, eb and ec are the back-EMFs ofthe BLDC motor.

The three-phase model of BLDC motor can be written asfollows:

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diasdt = 1

Ls−Rsias − ea + Vas

dibsdt = 1

Ls−Rsibs − eb + Vbs

dicsdt = 1

Ls−Rsics − ec + Vcs

(2)

The dynamic equations of the BLDC motor in the α–β system are

diαsdt = 1

Ls−Rsiαs − eα + Vαs

diβsdt = 1

Ls−Rsiβs − eβ + Vβs

(3)

where Vαs, Vβs and eα, eβ are two axis components of the statorvoltages and back-EMFs in the stationary reference frame,respectively.

If the iαs∗ and iβs

∗ are the reference values of currents, thedynamic errors for the system can be written as follows:

dEαdt = 1

Ls−Rsiαs − eα + Vαs − diαs

∗

dt

dEβdt = 1

Ls−Rsiβs − eβ + Vβs − diβs

∗

dt

(4)

where Eα = iαs − iαs∗ and Eβ = iβs − iβs

∗ .The electromagnetic generated torque can be expressed as

Te = 1ωm

eaiαs + ebibs + ecics (5)

where ωm is the mechanical rotor speed. The dynamic equation ofmotor speed is

dωmdt = 1

J Te − TL − Bωm (6)

where J, B and TL are the moment of inertia, friction constant andload torque, respectively.

2.2 SMC design

SMC is an efficient non-linear control method, which has beenwidely used for non-linear and uncertain systems [28]. The SMCdesign involves two steps:

a. Selecting a stable sliding surface in state space on which thestate trajectory must finally lie in (sliding phase).

b. Designing a suitable control law that makes this slidingsurface attractive for the state trajectory to reach it in finitetime (reaching phase).

Thus, the first step in SMC is to select the sliding surface S(t). S(t)is chosen to represent a desired performance for instance stability.Slotine [28] proposed a form of general equation to determine thesliding surface, which ensures the convergence of a state variabletowards its reference value as (7) [28].

S = ddt + k

n − 1e (7)

where n is the system order, e is the tracking error signal and k is apositive constant that determines the bandwidth of the system.After selecting the sliding surface, the next step is to find thecontrol law (u) that forces the state trajectory to reach the slidingsurface. Finally, the control law should be designed in such a waythat the following condition is met (reaching condition):

SS < 0 (8)

Fig. 1  Equivalent circuit for BLDC motor

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In this section, the non-linear SMC design is performed in thestationary reference frame. The sliding surfaces are selectedaccording to system errors as follows:

Sα = iαs − iαs∗

Sβ = iβs − iβs∗ (9)

and their time derivatives are as follows:

dSαdt = 1

Ls−Rsiαs − eα + Vαs − diαs

∗

dt

dSβdt = 1

Ls−Rsiβs − eβ + Vβs − diβs

∗

dt

(10)

To facilitate the derivative of the sliding surfaces to be equal tozero, the control efforts of this system, Vαs, Vβs can be selected asfollows:

Vαs = Rsiαs + eα + Lsdiαs

∗

dt

Vβs = Rsiβs + eβ + Lsdiβs

∗

dt

(11)

As shown in (11), the obtained control efforts need to the actualvalues of motor parameters such as stator resistance andinductance. Since these parameters may vary with temperature andcore saturation of the BLDC motor, the controllers must be robustagainst parametric uncertainties. So the control efforts of thissystem can be obtained as follows:

Vαs = V^αs − kssign Sα − kαSα

Vβs = V^βs − kssign Sβ − kβSβ

(12)

where kα and kβ are positive Lyapunov constants, ks is a positiveconstant known as a discontinuous term gain, Sα and Sβ are thesliding surfaces, and V^

αs and V^βs are as follows:

V^αs = R^

siαs + eα + L^sdiαs

∗

dt

V^βs = R^siβs + eβ + L^

sdiβs

∗

dt

(13)

where R^s and L^

σs are the nominal values of the stator resistance andinductance, respectively.

Also the Signum function is defined as

Sign s = 1 if S > 0−1 if S < 0 (14)

Since the Signum function caused to chattering phenomena, thisfunction is replaced with saturation function as

Vαs = V^αs − kssat Sα

∅ − kαSα

Vβs = V^βs − kssat Sβ

∅ − kβSβ

(15)

The saturation function is defined as

Sat S∅ =

−1, S < − ∅S/∅, − ∅ ≤ S ≤ + ∅+1, S > + ∅

(16)

where ∅ is a positive constant.

2.3 ITSMC design

In this section, the control objective is to drive the BLDC motorwith the desired reference currents iαs

∗ and iβs∗ in the stationary

reference frame. Define the tracking errors as Eα = iαs − iαs∗ and

Eβ = iβs − iβs∗ and the integral terminal sliding functions as

Sα t = Eα t + λEIα

Sβ t = Eβ t + λEIβ(17)

where

EIα = ∫0

tEα

qp τ d(τ)

EIβ = ∫0

tEβ

qp τ d τ

(18)

and

dEIαdt = Eα

q/ p t

dEIβdt = Eβ

q/ p t(19)

where λ > 0 is a constant, p and q are odd integers which satisfythe condition

p > q > 0 (20)

when the state trajectories reach to the sliding surfacesSα t = 0, Sβ t = 0, then,

Eα t = − λEIα

Eβ t = − λEIβ(21)

and

dEIαdt = − λq/ pEIα

q/ p t

dEIβdt = − λq/ pEIβ

q/ p t(22)

and

dEαdt = − λEα

q/ p t

dEβdt = − λEβ

q/ p t(23)

The convergence of (22) and (23) is proved by taking theLyapunov functions VI and V as the following, respectively

VI = 12 EIα

2 + EIβ2 (24)

V = 12 Eα

2 + Eβ2 (25)

The derivatives of Lyapunov functions (24) and (25) are asfollows:

V I = EIαdEIαdt + EIβ

dEIβdt = − EIαλq/ pEIα

q/ p − EIβλq/ pEIβq/ p =

− λq/ p EIαp + q/ p + EIβ

p + q/ p < 0(26)

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V I = EαdEαdt + Eβ

dEβdt = − Eαλq/ pEα

q/ p − Eβλq/ pEβq/ p = − λq/ p

Eαp + q/ p + Eβ

p + q/ p < 0(27)

Furthermore, from solving the error dynamic equations (22) and(23), the convergence time of EIα, EIβ, Eα and Eβ are, respectively,obtained as

tIα = EIα(0) 1 − q/ p

λq/ p 1 − q/ p

tIβ = EIβ(0) 1 − q/ p

λq/ p 1 − q/ p

(28)

tα = Eα(0) 1 − q/ p

λ 1 − q/ p

tβ = Eβ(0) 1 − q/ p

λ 1 − q/ p

(29)

where EIα 0 , EIβ 0 , Eα(0) and Eβ(0) are the initial values of theerrors. It is clear that if the state trajectory reach to the slidingsurface S t = 0, the errors EIα, EIβ, Eα and Eβ will converge tozero in finite time.

The control laws of the ITSMC are proposed to force the errorssystem to the surface Sα t = 0 and Sβ t = 0. Therefore, we takethe time derivative on the error functions Sα t = 0 and Sα t = 0 asthe following: (see (30)) Therefore, the control efforts in thestationary reference frame are obtained as

Vαs = Rsiαs + eα + Lsdiαs

∗

dt − λLsEαq/ p

Vβs = Rsiβs + eβ + Lsdiβs

∗

dt − λLsEβq/ p

(31)

As (31) shows, the obtained control efforts need to the actualvalues of motor parameters such as stator resistance andinductance. Since these parameters may vary with temperature and

core saturation of the BLDC motor, the controllers must berobust against parametric uncertainties. In Appendix, it is provedthat the ITSMC method is robust against uncertainties. So thereference voltage values of the inverter can be obtained as follows:

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Vαs = V^αs − kssign Sα − kαSα

Vβs = V^βs − kssign Sβ − kβSβ

(32)

where

V^αs = R^

siαs + eα + L^sdiαs

∗

dt − λL^sEα

q/ p

V^βs = R^

siβs + eβ + L^sdiβs

∗

dt − λL^sEβ

q/ p(33)

where R^s and L^

s are the nominal values of the stator resistance andinductance, respectively.

3 Reference current production based onharmonic injection blockSince the torque harmonics are a function of the back-EMF and thestator current harmonics, the injection of some specific harmoniccurrents in the stator windings, some of the torque harmonics canbe eliminated or minimised so the torque ripple can be decreased.The amplitude of the stator reference current harmonics, whichshould be injected to the stator windings, is calculated by‘reference current production based on harmonic injection block’in Fig. 2. It should be noted that, in order to minimise the motorlosses and maximise the torque per ampere ratio, each phase statorreference current should be in-phase with the corresponding phaseback-EMF [29]. As well as, by multiplying a constant coefficient inthe amplitude of the stator current harmonics, the motor averagetorque can be controlled. Therefore, one of the inputs of this blockis the magnitude of this coefficient, which can be obtained from aPI speed controller, and the other input is the rotor position.

In general, the major components of the torque profile are T6and T12 which are functions of low-order harmonics of the back-EMF and the stator currents [9, 30]. If the harmonic order of thestator current increased, the effective resistance of the motorwindings will be increased, so the copper loss will be increased. Aswell as, in higher harmonic order of the stator current the iron losswill be enlarged incredible. So, by assuming the stator harmoniccurrents higher than seventh order to be zero, the problem will besummarised as follows:

T0 = 32ωm

E1i1 + E5i5 + E7i7 (34)

dSαdt = dEα

dt + λdEIαdt = diαs

dt − diαs∗

dt + λEαq/ p = 1

Ls−Rsiαs − eα + Vαs − diαs

∗

dt + λEαq/ p

= 0

dSβdt = dEβ

dt + λdEIβdt = diβs

dt − diβs∗

dt + λEβq/ p = 1

Ls−Rsiβs − eβ + Vβs − diβs

∗

dt + λEβq/ p = 0

(30)

Fig. 2  Block diagram of the proposed control method

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T6 = 32ωm

E7 − E5 i1 − E1i5 + E1i7 (35)

T12 = 32ωm

−E7i5 − E5i7 (36)

where Ej is the amplitude of jth harmonic order of the BLDC motorback-EMF, Ij is the amplitude of jth harmonic order of the statorcurrent and T0 is the average demanded torque. Calculation of (34)may be time consuming and highly dependent to back-EMFharmonic contents of motor. So, for simplicity, if the amplitude ofthe harmonic currents is selected such as

I5

I7= E7

E5(37)

The amplitude of generated torque harmonics will be obtained asfollows:

T0 = 32ωm

E1i1 + 2E5i5 (38)

T6 = 0 (39)

T12 = 3ωm

−E7i5 (40)

4 Block diagram of the proposed methodA block diagram of the proposed control method is shown inFig. 2. The reference harmonic currents are produced in thereference current production based on harmonic injection block.Also, the ITSMC method is used to generate the required referencevoltage for the VSI. This method has an external speed controlloop, which produces the magnitude of the reference statorcurrents. The parameters of BLDC motor and ITSMC controller

are listed in Table 1. Also, the harmonic content of the motor back-EMF and the stator current are listed in Tables 2 and 3,respectively.

5 Simulation resultsIn this section, the performance of the proposed ITSMC method iscompared with the conventional SMC method by simulations.Simulations are established in C++ software for a three-phase,four-pole BLDC motor. The switching frequency and thesimulation step time interval are selected as 10 kHz and 1 μs,respectively. The parameters used in the speed controller in bothITSMC and SMC methods are listed in Table 4.

In order to demonstrate the four quadrant operation of theproposed controller, the load torque is considered as a multiple ofmotor speed. This multiple is obtained from the ratio of thenominal motor load and speed. The nominal of motor load andspeed are 1 Nm and 90 rad/s, respectively.

In order to present the effectiveness of the proposed ITSMCmethod, the simulations of the ITSMC and SMC methods areperformed and they are presented in Fig. 3. In order to show thedynamic response of the proposed ITSMC method, a fourquadrature speed reference is considered and the motor speed andtorque in TSMC and SMC methods are presented in Figs. 3a and b,respectively. In these figures, the reference speed (yellow), theBLDC motor speeds (cyan) and the BLDC motor torque (pink) areshown. Also the enlarged figures of Figs. 3a and b are presented inFigs. 3c and d, respectively. In these figures, the reference speed(yellow), the BLDC motor speeds (cyan), the BLDC motor torque(pink) and the BLDC motor stator current (green) in the ITSMCand SMC methods are depicted. As shown in this figure, the riseand fall times of the actual motor speed in the ITSMC method arelower than the SMC method and the results are listed in Table 5.

In order to show the robustness of the proposed ITSMCmethod, an initial error for the stator resistance is assumed in thecontroller block (Rs is assumed to be 0.45 Ω at t = 12 s while itstrue value is 0.15 Ω). The results of this scenario are shown inFigs. 4a and b, where the motor speed (cyan), the motor torque(yellow) and Rs variation (pink) in the ITSMC and SMC methodsare shown. It can be seen that, the robustness of the proposedITSMC method (Fig. 4a) is better than the SMC method (Fig. 4b)and the motor torque ripple in ITSMC is less than SMC.

Also, in order to demonstrate the robustness of the proposedITSMC method against the stator inductance variation, an initialerror for the stator inductance is assumed in the controller block (Lsis assumed to be 0.5 mH at t = 12 s, while its true value is 0.25 mH). The simulation results of this case are shown in Figs. 5a andb. In these figures, the motor speed (cyan), the motor torque(yellow) and the stator inductance variation (pink) are presented.As it can be seen in these figures, the robustness of the proposedmethod against the variation of Ls is better than SMC method.

In order to evaluate the proposed method with other controlmethods, experimental tests are performed and their results arepresented in the next section.

6 Experimental resultsFor experimental evaluation of the actual system, a DSP-basedprototype is built and tested. The practical setup with respect to theoverall system block diagram shown in Fig. 6 consists of thefollowing parts: a 100 W BLDC motor with non-sinusoidal back-EMF, VSI and its driver board, sensor board and a TMS320F28335discrete signal processor board.

To measure the stator phase currents, two Hall-effect currentsensors (LEM LTS-6-NP) are used and the line-to-line voltage iscalculated by voltage sensors (LEM LV-25-P). All measured statorcurrent and voltage signals are filtered by analogue second-orderlow-pass filters with cut-off frequency of about 2.6 kHz andconverted to digital by a 12-bit on-chip A/D converter. Rotorposition is detected by means of an incremental encoder with 1024pulses per round mounted on the DC generator shaft.

The inverter has been designed using low-loss IGBT moduleSKM40GD124D (with 40 A, 1200 V ratings) and intelligent IGBT

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Table 1 BLDC motor and controllers parametersParameters Amountnumber of pole pairs: P 2moment of inertia: J 0.0003 N ms2

stator resistance: Rs 0.15 Ωequivalent inductance of phase windings: Ls − M 0.25 mHDC voltage VDC 90 VLyapunov constant kα, kβ 100discontinuous term gain ks 3positive constant λ 1boundary layer ∅ 0.1

Table 2 Harmonic contents of motor back-EMFHarmonic order Per unit amount1 15 −0.2507 −0.236

Table 3 Harmonic contents of the stator reference currentHarmonic order Per unit amount1 15 −0.2367 −0.250

Table 4 Parameters of speed controller for BLDC motorControl method KP KISMC and proposed ITSMC 20 143

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drivers, HCPL-316J, which guarantee electrical separation betweenthe power and control systems. The inverter switching frequency is10 kHz.

Experiments are performed to evaluate the ITSMC using theSMC method. Some distinct results are presented in the following.The back-EMF of the BLDC motor in the 2200 RPM is presented

in Fig. 7. Q2

In order to show the dynamic response of the proposed ITSMCand the SMC methods, four quadrant operation of BLDC motor isconsidered and the reference speed is changed from −90 RPM to + 90 RPM and vice versa. The reference and actual speed and alsotorque motor in the ITSMC and the SMC are presented in Figs. 8aand b, respectively. In this figure, the reference speed (yellow), themotor speed (cyan) and the generated torque (pink) are depicted.

It must be noted, in order to show the calculated torque on theoscilloscope, we use the PWM-DAC module in the DSP chip andconvert the calculated torque to an analogue voltage which isproportional to the main torque superimposed to a 1.75 V offsetvalue. So, if the generated torque is −1 N m, the analogue voltagewill be 1.3 V, and if the value of torque is +1 N m, the analoguevoltage will be 2.2 V on oscilloscope screen.

As shown in this figure, the rise and fall times of the actualspeed in the ITSMC method are less than in the conventional SMCmethod and this is also listed in Table 5. Figs. 8c and d arepresenting the reference speed (yellow), the motor speed (cyan),

Fig. 3  Simulation results(a) ITSMC method, (b) SMC method (speed reference: yellow, BLDC motor speed: cyan, BLDC motor torque: pink), (c) ITSMC method, (d) SMC method (speed reference: yellow,BLDC motor speed: cyan, BLDC motor torque: pink, BLDC motor stator current: green)

Table 5 Rise and fall times in the proposed ITSMC andSMC methods in simulations and experimentsControlmethod

Simulation results Experimental resultsRise time,

sFall time, s Rise time, s Fall time, s

SMC 1.2 1.06 1.6 2.2proposedITSMC

1.0 0.75 1.2 1.3

Fig. 4  Simulation results: controller robustness against Rs variation(a) ITSMC method, (b) SMC method (BLDC motor stator resistance: pink, BLDC motor speed: cyan, BLDC motor torque: yellow)

Fig. 5  Simulation results: controller robustness against Ls variation(a) ITSMC method, (b) SMC method (BLDC motor stator resistance: pink, BLDC motor speed: cyan, BLDC motor torque: yellow)

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the generated torque (pink) and the stator current (green) of theBLDC motor. Here the fast Fourier transform (FFT) of the motortorque in the ITSMC and SMC methods are demonstrated inFigs. 8e and f, respectively, and it is shown that the amplitude ofsixth harmonic oscillation, which is seen on the motor torque in theproposed method, is less than in the SMC method. The injectedstator current in ITSMC and SMC methods is shown in Figs. 9aand b, respectively. Also, the FFTs of the injected stator current inthe ITSMC and SMC methods are depicted in Figs. 9c and d,respectively, and it is shown that the harmonic content and rippleof the injected stator current in the proposed method is less than inthe SMC method.

For detailed comparison between ITSMC and SMC methodseffects on the generated torque ripple, the steady-state operations ofthese methods are analysed in Figs. 8e, f and 9c, d. The THD of the

stator current and ripple factor of generated torque are calculatedas follows:

THDi = I52 + I7

2 + I112 + I13

2

I1

RFT = T62 + T12

2

T0

(41)

THDi and RFT are calculated and listed in Table 6. As can be seen,the RFT index in the ITSMC is less than SMC. As well as, theTHDi of the proposed method is better than SMC.

Also, in order to compare the dynamic response of the proposedITSMC method with the SMC method, a step change in thereference speed from +30 to +90 rad/s is considered and results aredepicted in Figs. 10a and b. In this figure, the reference speed(yellow), the motor speed (cyan) and the generated torque (pink)are shown.

In order to demonstrate the robustness of the proposed ITSMCmethod, an initial error for the stator resistance is assumed in thecontroller block (Rs is assumed to be 0.45 Ω at t = 1.6 s while itstrue value is 0.15 Ω). The results of this scenario are shown inFigs. 11a and b, where the motor speed (cyan), the motor torque(yellow) and Rs variation (pink) are shown. It can be seen that, therobustness of the proposed ITSMC method is better than the SMCmethod and the motor torque ripple in the ITSMC method is lessthan the SMC method.

Robustness of the proposed ITSMC method against to the statorinductance variation is presented in Figs. 11c and d, where the

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Fig. 6  Experimental setup of the BLDC motor drive system

Fig. 7  Experimental results: BLDC motor back-EMF at 2200 RPM (10 v/div)

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motor speed (cyan), the motor torque (yellow) and the statorinductance variation (pink) are presented. In this test, an initialerror for the stator inductance is assumed in the controller block (Lsis assumed to be 0.5 mH at t = 0.9 s, while its true value is 0.25 mH). As it can be seen in these figures, the robustness of theproposed ITSMC method against the variation of Ls is better thanthe SMC method.

In the final stage, for evaluation of the robustness of theproposed ITSMC method, both of the Rs and Ls parameters arechanged (Rs is assumed to be 0.45 Ω at t = 1.6 s while its true valueis 0.15 Ω and Ls is assumed to be 0.5 mH at t = 0.9 s, while its truevalue is 0.25 mH). The result of this case is shown in Figs. 11e andf. In these figures, the motor speed (cyan), the motor torque

(yellow), the Rs variation: (green) and the Ls stator inductancevariation (pink) are depicted. As shown in these figures, therobustness of the proposed ITSMC method against the variationsboth of the Rs and Ls is better than the SMC method.

Finally, the robustness of the proposed method against the loadvariations is tested and shown in Fig. 12. As shown in this figure,the robustness of ITSMC is better than SMC against loadvariations. It is important to note that the load variations areperformed in two steps (at first from +1 to +0.2 N m and then from+0.2 to +0.7 N m).

Fig. 8  Experimental results(a) ITSMC method, (b) SMC method (speed reference: yellow, BLDC motor speed: cyan, BLDC motor torque: pink), (c) ITSMC method, (d) SMC method (speed reference: yellow,BLDC motor speed: cyan, BLDC motor torque: pink, BLDC stator current (2 A/div): green), (e) FFT of the motor torque in ITSMC method, (f) FFT of the motor torque in SMCmethod

8 IET Electr. Power Appl.© The Institution of Engineering and Technology 2017

7 Conclusions The aim of this paper is to develop a robust control techniquenamed ITSMC for torque ripple reduction in BLDC motor. In thiswork, the ITSMC method is proposed to generate the referencevoltages for three-phase VSI. Unlike previous research work, theVSI used in this paper is controllable with voltage and therefore thenon-linear controller can follow each arbitrary reference currents.The effectiveness of the proposed method is evaluated bysimulations and experimental tests and the dynamic response androbustness of the proposed ITSMC method is compared with theconventional SMC method. It can be concluded that the dynamicresponse of the proposed ITSMC method is higher than SMC.

Fig. 9  Experimental results(a) Injected stator current in ITSMC method, (b) Injected stator current in SMC method, (c) FFT of the injected stator current in ITSMC method, (d) FFT of the injected statorcurrent in SMC method

Table 6 THD of the stator current and ripple factor of thegenerated torque and T/I ratioMethod SMC TSMCTHDi 0.323 0.271RFT 0.18 0.10T/I 0.145 0.174

Fig. 10  Experimental results(a) ITSMC method, (b) SMC method (speed reference: yellow, BLDC motor speed: cyan, BLDC motor torque: pink)

IET Electr. Power Appl.© The Institution of Engineering and Technology 2017

9

Also, the robustness of the ITSMC method against Rs and Lsvariations is better than the SMC method and the motor torqueripple in ITSMC is less than SMC.

Fig. 11  Experimental results: controller robustness against Rs variation(a) ITSMC method, (b) SMC method (BLDC motor speed: cyan, BLDC motor torque (0.3 Nm/div): yellow, Rs variation: pink) controller robustness against Ls variation, (c) ITSMCmethod, (d) SMC method (BLDC motor speed: cyan, BLDC motor torque: yellow, Ls variation: pink) controller robustness against Rs and Ls variations, (e) ITSMC method, (f) SMCmethod (BLDC motor speed: cyan, BLDC motor torque: yellow, Rs variation: green, Ls variation: pink)

10 IET Electr. Power Appl.© The Institution of Engineering and Technology 2017

8 References[1] Hendershort Jrand, J.R., Miller, T.J.E.: ‘Design of brushless permanent-

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[4] Karthikeyan, J., Sekaran, R.D.: ‘Current control of brushless dc motor basedon a common dc signal for space operated vehicles’, Electr. Power EnergySyst., 2011, 33, pp. 1721–1727

[5] Krishnan, G., Ajmal, K.T.: ‘A neoteric method based on PWM ON PWMscheme with buck converter for torque ripple minimization in BLDC Drive’.Proc. Int. Conf. on Emerging Research Area: Magnetics, Machines & Drives(AICERA-2014 iCMMD), 2014, pp. 1–6

[6] Ozturk, S.B., Alexander, W.C., Toliyat, H.A.: ‘Direct torque control of four-switch brushless DC motor with non-sinusoidal back EMF’, IEEE Trans.Power Electron., 2010, 25, (2), pp. 263–271

[7] Kshirsagar, P., Krishnan, R.: ‘High-efficiency current excitation strategy forvariable-speed nonsinusoidal back-EMF PMSM machines’, IEEE Trans. Ind.Appl., 2012, 48, (6), pp. 1875–1889

[8] Sheng, T., Wang, X., Zhang, J., et al.: ‘Torque-ripple mitigation for brushlessDC machine drive system using one-cycle average torque control’, IEEETrans. Ind. Electron., 2015, 62, (4), pp. 2114–2122

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Q4

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9 Appendix If the control laws are selected as follows:

Vαs = R^siαs + eα + L^

sdiαs

∗

dt − λL^sEα

q/ p − kαSα − kssign(Sα)

Vβs = R^siβs + eβ + L^

sdiβs

∗

dt − λL^sEβ

q/ p − kβSβ − kssign(Sβ)(42)

then the time derivatives of Sα and Sβ can be written as

Sαβ(t) = 1Ls

−Rsiαβs + R^siαβs + L^

sdiαβs

∗

dt − λL^sEαβ

q/ p − kαβSαβ − kssign(Sαβ) − diαβs∗

dt +λEαβq/ p

= −1Ls

Rs − R^s iαβs + L^

sLs

− 1 diαβs∗

dt − λL^

sLs

− 1 Eαβq/ p − kαβ

LsSαβ − ks

Lssign(Sαβ)

(43)

The sliding condition can be satisfied as

SS < − ks S (44)

so

Fig. 12  Experimental results: robustness against load variation(a) ITSMC method, (b) SMC method (BLDC motor speed: cyan, BLDC motor torque: yellow)

IET Electr. Power Appl.© The Institution of Engineering and Technology 2017

11

SαSα(t) = Sα−1Ls

Rs − R^s iαs + L^

sLs

− 1 diαs∗

dt − λL^

sLs

− 1 Eαq/ p − kα

LsSα − ks

Lssign(Sα) < − ks Sα

SβSβ(t) = Sβ−1Ls

Rs − R^s iβs + L^

sLs

− 1 diβs∗

dt − λL^

sLs

− 1 Eβq/ p − kβ

LsSβ − ks

Lssign(Sβ) < − ks Sβ

(45)

then, the condition (44) can be written as follows:

ks > −1Ls

Rs − R^s iαs + L^

sLs

− 1 diαs∗

dt − λL^

sLs

− 1 Eαq/ p

ks > −1Ls

Rs − R^s iβs + L^

sLs

− 1 diβs∗

dt − λL^

sLs

− 1 Eβq/ p

(46)

IET-EPA20170070Author Queries

Q Please make sure the supplied images are correct for both online (colour) and print (black and white). If changes arerequired please supply corrected source files along with any other corrections needed for the paper.

Q1 Please reduce the number of words in the Abstract to 200 words.Q2 Please provide expansion for 'TSM', 'PMSM', 'DSP', 'RPM', 'THD'.Q3 Please check the sentence 'As (31) shows, the obtained control efforts need....' seems to be not clear. Please rephrase.Q4 Please confirm the given year in Refs. [11, 13, 18]

12 IET Electr. Power Appl.© The Institution of Engineering and Technology 2017