on integral terminal sliding mode control motor with...
TRANSCRIPT
![Page 1: on integral terminal sliding mode control motor with ...cup-co.ir/wp-content/uploads/2019/02/EPA-2017-0070-PROOF.pdf · Torque ripple reduction of brushless DC motor with harmonic](https://reader033.vdocuments.mx/reader033/viewer/2022050221/5f66edefa012985e3b3a7dbb/html5/thumbnails/1.jpg)
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: [email protected]
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.
Q1
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
IET Electr. Power Appl.© The Institution of Engineering and Technology 2017
1
![Page 2: on integral terminal sliding mode control motor with ...cup-co.ir/wp-content/uploads/2019/02/EPA-2017-0070-PROOF.pdf · Torque ripple reduction of brushless DC motor with harmonic](https://reader033.vdocuments.mx/reader033/viewer/2022050221/5f66edefa012985e3b3a7dbb/html5/thumbnails/2.jpg)
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:
Q2
Q2
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
2 IET Electr. Power Appl.© The Institution of Engineering and Technology 2017
![Page 3: on integral terminal sliding mode control motor with ...cup-co.ir/wp-content/uploads/2019/02/EPA-2017-0070-PROOF.pdf · Torque ripple reduction of brushless DC motor with harmonic](https://reader033.vdocuments.mx/reader033/viewer/2022050221/5f66edefa012985e3b3a7dbb/html5/thumbnails/3.jpg)
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)
IET Electr. Power Appl.© The Institution of Engineering and Technology 2017
3
![Page 4: on integral terminal sliding mode control motor with ...cup-co.ir/wp-content/uploads/2019/02/EPA-2017-0070-PROOF.pdf · Torque ripple reduction of brushless DC motor with harmonic](https://reader033.vdocuments.mx/reader033/viewer/2022050221/5f66edefa012985e3b3a7dbb/html5/thumbnails/4.jpg)
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:
Q3
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
4 IET Electr. Power Appl.© The Institution of Engineering and Technology 2017
![Page 5: on integral terminal sliding mode control motor with ...cup-co.ir/wp-content/uploads/2019/02/EPA-2017-0070-PROOF.pdf · Torque ripple reduction of brushless DC motor with harmonic](https://reader033.vdocuments.mx/reader033/viewer/2022050221/5f66edefa012985e3b3a7dbb/html5/thumbnails/5.jpg)
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
Q2
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
IET Electr. Power Appl.© The Institution of Engineering and Technology 2017
5
![Page 6: on integral terminal sliding mode control motor with ...cup-co.ir/wp-content/uploads/2019/02/EPA-2017-0070-PROOF.pdf · Torque ripple reduction of brushless DC motor with harmonic](https://reader033.vdocuments.mx/reader033/viewer/2022050221/5f66edefa012985e3b3a7dbb/html5/thumbnails/6.jpg)
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)
6 IET Electr. Power Appl.© The Institution of Engineering and Technology 2017
![Page 7: on integral terminal sliding mode control motor with ...cup-co.ir/wp-content/uploads/2019/02/EPA-2017-0070-PROOF.pdf · Torque ripple reduction of brushless DC motor with harmonic](https://reader033.vdocuments.mx/reader033/viewer/2022050221/5f66edefa012985e3b3a7dbb/html5/thumbnails/7.jpg)
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
Q2
Fig. 6 Experimental setup of the BLDC motor drive system
Fig. 7 Experimental results: BLDC motor back-EMF at 2200 RPM (10 v/div)
IET Electr. Power Appl.© The Institution of Engineering and Technology 2017
7
![Page 8: on integral terminal sliding mode control motor with ...cup-co.ir/wp-content/uploads/2019/02/EPA-2017-0070-PROOF.pdf · Torque ripple reduction of brushless DC motor with harmonic](https://reader033.vdocuments.mx/reader033/viewer/2022050221/5f66edefa012985e3b3a7dbb/html5/thumbnails/8.jpg)
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
![Page 9: on integral terminal sliding mode control motor with ...cup-co.ir/wp-content/uploads/2019/02/EPA-2017-0070-PROOF.pdf · Torque ripple reduction of brushless DC motor with harmonic](https://reader033.vdocuments.mx/reader033/viewer/2022050221/5f66edefa012985e3b3a7dbb/html5/thumbnails/9.jpg)
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
![Page 10: on integral terminal sliding mode control motor with ...cup-co.ir/wp-content/uploads/2019/02/EPA-2017-0070-PROOF.pdf · Torque ripple reduction of brushless DC motor with harmonic](https://reader033.vdocuments.mx/reader033/viewer/2022050221/5f66edefa012985e3b3a7dbb/html5/thumbnails/10.jpg)
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
![Page 11: on integral terminal sliding mode control motor with ...cup-co.ir/wp-content/uploads/2019/02/EPA-2017-0070-PROOF.pdf · Torque ripple reduction of brushless DC motor with harmonic](https://reader033.vdocuments.mx/reader033/viewer/2022050221/5f66edefa012985e3b3a7dbb/html5/thumbnails/11.jpg)
8 References[1] Hendershort Jrand, J.R., Miller, T.J.E.: ‘Design of brushless permanent-
magnet machines’ (Motor Design Books, Venice, FL, 2010)[2] Park, S.J., Park, H.W., Lee, M.H., et al.: ‘A new approach for minimum-
torque-ripple maximum-efficiency control of BLDC motor’, IEEE Trans. Ind.Appl., 2000, 47, (1), pp. 109–114
[3] Gu, B.G., Choi, J.H., Jung, I.S.: ‘Simple lead angle adjustment method forbrushless DC motors’, J. Power Electron., 2014, 14, (3), pp. 541–548
[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
[9] Hanselman, D.C.: ‘Minimum torque ripple, maximum efficiency excitation ofbrushless permanent magnet motors’, IEEE Trans. Ind. Appl., 1994, 41, (3),pp. 292–300
[10] LeHuy, H., Perret, R., Feuillet, R.: ‘Minimization of torque ripple in brushlessDC motor drive’, IEEE Trans. Ind. Appl., 1986, IA-22, (4), pp. 748–755
[11] Kim, T.S., Ahn, S.C., Hyun, D.S.: ‘A new current control algorithm for torqueripple reduction of BLDC motors’. The 27th Annual Conf. of the IEEE Proc.Industrial Electronics Society, IECON ‘01, 2001, pp. 1521–1526
[12] Chapman, P.L., Sudhoff, S.D.: ‘A multiple reference frame synchronousestimator/regulator’, IEEE Trans. Energy Convers., 2000, 15, (2), pp. 197–202
[13] Amirian, M.A., Rashidiand, A., Saghaeian Nejad, S.M.: ‘Torque ripplereduction of permanent magnet synchronous motor with non-sinusoidal backEMF using a modified and simplified current optimization technique’. Proc.6th Power Electronics, Drive Systems & Technologies Conf. (PEDSTC2015),2015, pp. 322–327
[14] Wu, G., Zhu, H.: ‘Combined hysteresis current-controlled PWM inverter androbust control for a permanent-magnet synchronous motor’. Proc. Int. Conf.on Electric Information and Control Engineering (ICEICE), 2011, pp. 5753–5757
[15] Xia, C., Ji, B., Yan, Y.: ‘Smooth speed control for low-speed high-torquepermanent-magnet synchronous motor using proportional–integral–resonantcontroller’, IEEE Trans. Ind. Electron., 2015, 62, (4), pp. 2123–2134
[16] Utkin, V.: ‘Sliding mode control design principles and application to electricdrives’, IEEE Trans. Ind. Electron., 1993, 40, (1), pp. 23–36
[17] Kim, S.K., Lee, J.S., Lee, K.B.: ‘Offset-free robust adaptive back-steppingspeed control for uncertain permanent magnet synchronous motor’, J. PowerElectron., 2016, 31, (10), pp. 7065–7076
[18] Mao, Q., Na, R., Wang, H.: ‘Direct torque control for surface permanentmagnet synchronous motor drives based on feedback linearization’. Proc. Int.Symp. on Computer, Consumer and Control (IS3C 2016), 2016, pp. 1097–1100
[19] Chai, H., Yan, P., Guo, L.: ‘Feedback linearization design for permanentmagnet synchronous motor with disturbance observer’. Proc. 32nd ChineseControl Conf. (CCC 2013), pp. 2739–2744
[20] Kim, S.K., Lee, J.S., Lee, K.B.: ‘Robust current control-based sliding modecontrol with simple uncertainties estimation in permanent magnetsynchronous motor drive systems’, IET Electr. Power Appl., 2010, 4, (6), pp.441–450
Q4
[21] Zhihong, M., Yu, X.H.: ‘Terminal sliding mode control of MIMO linearsystems’, IEEE Trans. Circuits Syst., 1997, 44, (11), pp. 1065–1070
[22] Feng, Y., Yu, X., Han, F.: ‘High-order terminal sliding mode observer forparameter estimation of a permanent-magnet synchronous motor’, IEEETrans. Ind. Electron., 2013, 10, (10), pp. 4272–4280
[23] Li, S., Zhou, M., Yu, X.: ‘Design and implementation of terminal slidingmode control method for PMSM speed regulation system’, IEEE Trans. Ind.Inf., 2013, 9, (4), pp. 1879–1891
[24] Wang, X., Fu, T., Wang, X.: ‘Position sensorless control of BLDC motorsbased on global fast terminal sliding mode observer’, J. Power Electron.,2015, 15, (6), pp. 1559–1566
[25] Xu, W., Jiang, Y., Mu, C.: ‘Nonsingular terminal sliding mode control for thespeed regulation of permanent magnet synchronous motor with parameteruncertainties’. Proc. 18th Int. Conf. on Electrical Machines and Systems(ICEMS), 2015, pp. 2034–2039
[26] Xu, W., Jiang, Y., Mu, C.: ‘Novel composite sliding mode control for PMSMdrive system based on disturbance observer’, IEEE Trans. Appl. Supercond.,2016, 26, (7), pp. 1–7
[27] Chapman, P., Sudhoff, S., Whitcomb, C.: ‘Optimal current control strategiesfor surface-mounted permanent-magnet synchronous machine drives’, IEEETrans. Energy Convers., 1999, 14, (4), pp. 1043–1050
[28] Slotine, J.J.E., Li, W.: ‘Applied nonlinear control’ (Prentice-Hall, Inc, 1991)[29] Jeon, M.H., Kim, D.H., Kim, C.E.: ‘Optimum design of BLDC motor for
cogging torque minimization using genetic algorithm and response surfacemethod’, J. Electr. Eng. Technol. (JEET), 2006, 1, (40), pp. 466–471
[30] Shirvani Boroujeni, M., Arab Markadeh, G.R., Soltani, J.: ‘Torque ripplereduction of brushless DC motor based on adaptive input-output feedbacklinearization’ (ISA Trans, 2017), in press. Available at https://doi.org/10.1016/j.isatra.2017.05.006
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
![Page 12: on integral terminal sliding mode control motor with ...cup-co.ir/wp-content/uploads/2019/02/EPA-2017-0070-PROOF.pdf · Torque ripple reduction of brushless DC motor with harmonic](https://reader033.vdocuments.mx/reader033/viewer/2022050221/5f66edefa012985e3b3a7dbb/html5/thumbnails/12.jpg)
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