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Identifying dynamic model parameters of a BLDC motor
Simulation Modelling Practice and Theory 16 (2008) 1254–1265
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Identifying dynamic model parameters of a BLDC motor A. Kapun *, M. Cˇurkovicˇ, A. Hace, K. Jezernik Faculty of Electrical Engineering and Computer Sciences, University of Maribor, Smetanova 17, SI-2000, Slovenia
a r t i c l e
i n f o
Article history: Received 20 March 2008 Received in revised form 5 June 2008 Accepted 11 June 2008 Available online 20 June 2008
Keywords: Permanent magnet motors simulation Parameter estimation Least-squares methods Cogging-torque
a b s t r a c t An off-line identification method founded on the least-squares approximation technique and a closed-loop disturbance observer is applied for identifying the parameters of a BLDC motor model. No special configuration of the motor is required besides the availability of experimental data for back-EMF, phase currents, rotor position, and rotor speed. This method is used to identify the back-EMF harmonics and mechanical parameters, where the mechanical parameters refer to cogging-torque, viscous friction coefficient, and Coulomb friction coefficient. The proposed identification method is theoretically investigated, and the method’s effectiveness is proved by experimental results performed on a low-power BLDC motor. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Modern cars inevitably rely on several diverse electric motors. In addition to those for maintaining basic car functionality such as fuel/water pumps, power-assisted steering, and engine cooling fans, electric motors are also included in new electrical systems for improving passengers comfort and safety [4]. Traditionally brushed DC motors have been used in automotive applications, although presently permanent magnet brushless DC (BLDC) motors are gaining popularity. The benefits of the BLDC motor when compared to the DC motor concern higher reliability, longevity, lower maintenance costs, and lower electromagnetic interference (EMI). DC motor replacement by the BLDC motor, on the other hand, results in more complex control algorithms. There has been extensive research dealing with BLDC motor control, proposing diverse control algorithms for torque, speed, and position control. Intensive research work has been carried out in the field of sensorless-control algorithms for low cost applications [20,3]. Throughout the control design process there is a need for algorithm verification. Thus authentic simulation models of the BLDC motor are crucial. In [8], a BLDC motor model for MATLAB/Simulink environment simulation was developed, suitable for the analysis of BLDC motor drives and control algorithm design. However if correlation between simulation and experimental results is either over- or under-estimated, the simulation results value is questionable. Nevertheless, the system dynamic model structure is one part of the simulation environment, the second important issue is the model’s parameters. A comprehensive measurement procedure for determining the permanent magnet synchronous machine (PMSM) model and machine operational performance parameters is thoroughly described in [13]. Although the measurement procedure is very effective, its implementation requires special measurement equipment such as torque-sensor, and a high-quality servo motor in the case of cogging-torque measurement. In [14], an identification algorithm derived from least-squares techniques is used to estimate the rotor resistance, and the rotor and the stator leakage inductance of a three-phase induction machine. In [21], an auto regressive moving average (ARMA) model is used to identify the DC motor transfer function, where a recursive least square algorithm is applied to estimate the ARMA parameters. In
* Corresponding author. Tel.: +386 2 220 7176. E-mail address: [email protected] (A. Kapun). 1569-190X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2008.06.003
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[15], the particle-swarm algorithm is applied to PMSM in order to identify its stator resistance and load torque disturbances. The stator resistance, stator inductance, torque constant, inertia constant, viscous friction coefficient and Coulomb friction coefficient of the PM stepper motor are identified in [1], using an on-line batch least-squares algorithm. The real time estimation method derived from the motor inverse model is discussed in [19] to estimated the BLDC motor resistance and torque constant on-line. With adaptive controller structure the proposed estimation algorithm can also be extended for compensation of the motor parameters variation on control performances. This paper presents research work, where the main focus was on identifying those BLDC motor model parameters which significantly contribute to motor torque-ripple. The torque-ripple parameters identification is important not only for authentic simulations but also for implementation of different torque-ripple minimization control techniques, like those described in[9,16]. Minimization of torque-ripple reduce speed oscillations and consequently deteriorations in the motor performance. There are two main sources of torque-ripple, cogging-torque and non-sinusoidal stator flux linkage distribution [7]. The motor model where these two torque-ripple sources are included is derived in [12]. Based on this motor model, an off-line method was developed for identifying motor parameters. The paper is organized as follows. Firstly, the basics of the motor modelling are summarized and the three-phase BLDC motor model is presented. In Section 3, the parameters identification method is theoretically investigated. This identification method is founded on the least-squares approximation method, and a closed-loop disturbance observer, therefore, these two techniques are also briefly described. In order to verify the proposed method’s effectiveness, experiments were performed on a low-power BLDC motor. The experimental results are shown and discussed in Section 4. Finally, the conclusions are summarized is Section 5. 2. Motor model Assuming the variations of the stator self inductance with rotor position and the mutual inductance between the stator windings are negligible; the electrical dynamics of a BLDC motor may be modelled in an electrically balanced system as [17]:
disi usi V 0 ¼ Rs isi þ Ls þ ei dt X isi ¼ 0
i ¼ 1; 2; 3
ð1Þ ð2Þ
where Rs and Ls are the stator resistance and inductance, usi is the motor terminal voltages, isi is the phase current and ei is the back-EMF associated with the ith phase. The potential of the motor neutral terminal in wye-connected windings is denoted as V 0 . The back-EMF inducted in each phase by the rotor permanent magnet, can be expressed as:
dwir owir dh ow ¼ x ir ¼ dt oh dt oh
ei ¼
ð3Þ
where wir is the mutual magnetic flux between the permanent magnet and the stator windings in the ith phase, h is the rotor position, and x is the rotor speed. Since a linear magnetic model without saturation effects is assumed, the mutual magnetic flux is modelled by fictitious rotor winding with mutual inductance Lir , and fictitious constant rotor current ir ¼ 1A as:
wir ¼ Lir ir
ð4Þ
In general, the mutual magnetic flux in the air-gap is non-sinusoidal, therefore, the higher harmonics in mutual magnetic flux are introduced as higher harmonic components in fictitious rotor winding mutual inductance. The mutual inductance Lir is expressed, using the terms of the trigonometric Fourier series, as:
Lir ¼
K X 2p 2p Lirak cos k ph ði 1Þ þ Lirbk sin k ph ði 1Þ 3 3 k¼1
ð5Þ
From (3) and (4), the back-EMF can be expressed as:
ei ¼ ir x
oLir oh
ð6Þ
where the partial derivatives of the mutual inductance over the rotor position, are known as the back-EMF coefficients [5]. From the virtual work principle, the electromagnetic motor torque can be derived [12], as:
T¼
3 X i¼1
T i þ T cogg ¼ ir
3 X i¼1
oLir isi oh
!
1 2 oLrr þ ir 2 oh
ð7Þ
The first three terms in Eq. (7) are mutual torques caused by interaction between the permanent magnet field and the phase currents. The last term known as cogging-torque, is due to the attraction of the permanent to the salient portions of the stator iron. Therefore, the cogging-torque is always present, even in the absence of the phase currents. Usually, the
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cogging-torque harmonics for three-phase motors are multiples of six times the fundamental commutation frequency [9]. The cogging-torque harmonics are modelled with higher harmonic components in the fictitious rotor winding self inductance Lrr .
Lrr ¼
K0 X
ðLrrak cos kðphÞ þ Lrrbk sin kðphÞÞ
ð8Þ
k¼6
The motor mechanical dynamics is defined as:
h_ ¼ x;
1 J
x_ ¼ ðT T l Bx CsignðxÞÞ
ð9Þ
where T l is the load torque, J is the rotor inertia, B is the viscous friction coefficient, and C is the Coloumb friction coefficient. 3. Parameter estimation method The parameters of the motor model given in the previous section are Rs , Ls , J, B, C, Lirab ¼ fLira1 . . . LiraK ; Lirb1 LirbK g, and Lrrab ¼ fLrra1 LrraK , Lrrb1 LrrbK g. This section presents an off-line method for estimating the parameters Lirab , Lrrab , B, and C in Eqs. (6), (7), and (9). It is assumed that all the other parameters are known, since they are normally provided by the motor manufacturer. The estimation method is based on given experimental data for phase currents, rotor position, and rotor speed. In practice, an experimental test bench, like the one shown in Fig. 1, is used to obtain these signals. The estimation method can be divided into three steps. Firstly, the least-squares approximation method is used to estimate the parameters Lirab from the experimental data for back-EMF’s, rotor position, and rotor speed. In the second step, the motor is driven using the DSP controller and a three-phase bridge (Motor Driving Unit), while the phase currents and rotor position are measured simultaneously (Measuring Unit). The total mutual torque T 1 þ T 2 þ T 3 is calculated from the phase currents, the rotor posib signðxÞ is b dist ¼ T b cogg þ B bx þ C tion, and the parameters b L irab estimated in the first step. Then the torque disturbance signal T estimated using a closed-loop observer. Both the calculated mutual torque and measured motor speed are required for the observer’s implementation. In the last step, the least-square approximation method is again applied in order to estimate the b form the signal T b and C b dist . Each step is described in detail in the following subsections. The least-squares parameters b L rrab , B, approximation method and the disturbance observer are used during the estimation process, therefore, these two techniques are briefly described first. 3.1. Least-squares approximation method The method implementation is easy, therefore, it is widely used in practice. The method objectivity is to minimize the squared difference between the measured signal x½n and the assumed signal s½n, [10]. When a vector problem is considered, the signal s½n is determined by a signal model which depends upon the unknown parameter vector a. Signal x½n is a perturbed version of the signal s½n, due to noise and model inaccuracies. The least-squares estimator (LSE) is that value of a, which minimizes the squares error criterion, given as:
JðaÞ ¼
N X ðx½n s½nÞ2
ð10Þ
n¼1
Data Analysis
Data Flow
θ
Measuring Unit
uS1 uS2 iS1 iS2
Motor Driving Unit
Incremental Encoder
BLDC motor
Fig. 1. Test bench configuration.
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The observation interval is assumed to be n ¼ 1; . . . ; N, and the dependence of J on a is via s½n. This method is valid for Gaussian and non-Gaussian noise, however the performance of the LSE will undoubtedly depend on the properties of the corrupting noise, as well as any modelling errors, and quantization. When the linear least-squares problem is considered, the signal model is linear with respect to the unknown parameter, therefore, the signal s ¼ ½s½1 . . . s½NT may be written as:
s ¼ Ha
ð11Þ
where H is the known NxrðN > rÞ observation matrix of full rank r and the a is rx1 unknown parameter vector. The least square estimator and minimum least square error for (11) are [10]:
a^ ¼ ðHT HÞ1 HT x ^ Þ ¼ ðx Ha ^ ÞT ðx Ha ^Þ J min ¼ Jða
ð12Þ ð13Þ
3.2. Disturbance observer The disturbance observer, shown in Fig. 2, can be used to estimate the disturbance signal udist , that may be applied to improve control performances. If inaccuracies in the plant model are negligible and the compensator is able to ensure conver^ dist . The observer can be gence of the signal e towards zero, then the compensator output is the estimated disturbance signal u used in a similar way to estimate non-modelled dynamics in the case of an incomplete plant model, under the assumption that exogenous disturbances are negligible. From Fig. 2, the relationship between the estimated and applied disturbance can be evaluated as:
^ dist ðsÞ u HP ðsÞHC ðsÞ ¼ udist ðsÞ 1 þ HP ðsÞHC ðsÞ
ð14Þ
If such an HC ðsÞ is chosen, so that the denominator of the polynomial in (14) is a Hurwitz polynomial, then the system ^ dist ðtÞ will be observing udist ðtÞ from Fig. 2 is asymptotically stable. Hence, the output of the feedback compensator u asymptotically. 3.3. Back-EMF coefficients estimation It is evident from (5)-(6) that the parameters Lirab appear as higher order harmonics back-EMF coefficients due to the non-sinusoidal flux distribution phenomena. If the motor is wye-connected and the neutral point is accessible, then the easiest way to determine the back-EMF is to measure the phase to neutral voltage at open-circuit phase winding, while the rotor is rotating because of externally applied torque. If this is not the case, then the back-EMF is determined from measured phase to phase voltage but, in this case, the harmonics divisible by three can not be identified. Usually Fourier analysis is used to extract the back-EMF coefficients. Because Fourier analysis is valid for periodical functions, exactly one period of back-EMF needs to be introduced into the analysis in order to obtain accurate results. An additional restriction is in the rotor speed which should be constant during the experiment. With the proposed least-squares approximation method, the result’s accuracy is independent of the rotor rotation, since the rotor position and rotor speed are already incorporated into the estimation process. When the phase to neutral voltage is measured, the back-EMF in the first phase equals:
e1 ¼ ir xp
K X ðkL1rak sin kðphÞ þ kL1rbk cos kðphÞÞ
ð15Þ
k¼1
Eq. (15) can be written in linear form (11), if observation matrix H and parameter vector a are defined as:
udis t
u
+
y Plant Plant Model
+
HP(s)
y
+
- udist Compensator
HC(s) Fig. 2. Disturbance observer principle.
ε
0
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c
H ¼ ir p½hij ; hij ; s hij c hij
ð16Þ
¼ jx½i sin jðph½iÞ i ¼ 1 : N; ¼ jx½i cos jðph½iÞ i ¼ 1 : N;
a¼
LTirab
j¼1:K j¼1:K
¼ fLira1 . . . LiraK ; Lirb1 . . . LirbK gT
ð17Þ
The parameters Lirab are estimated by the least-squares estimator (12). The observation matrix is formed from experimental data for rotor position and rotor speed. If constant rotor speed can be accomplished, then the observation matrix can be simplified and no measurements of rotor position and speed are needed. In this case, the rotor speed is determined from the back-EMF waveform period, and the rotor position is obtained by integrating the speed. Normally, constant speed can only be accomplished during high-speed operation, where the cogging-torque’s influence could be neglected [1]. 3.4. Cogging, viscous friction, and Coulomb friction torque estimation Estimation of cogging, viscous friction, and Coulomb friction torque is realized based on the previously discussed disturbance observer, as shown in Fig. 3. The total mutual torque is calculated (7) from the simultaneously measured phase currents and rotor position, and previously identified back-EMF coefficients. The model plant is obtained from the mechanical dynamics (9), as shown in Fig. 3, where J Tot is the total inertia (motor, rotary encoder, coupling). A PI controller is used as the ^ towards the measured speed x, it can be compensator. If the controller ensures the convergence of the estimated speed x assumed that the controller output equals the estimated non-modelled dynamics signal. If the load torque is zero and the exogenous disturbances are negligible, then rotor acceleration can be written from (7) and (9) as:
x_ ¼
1 ðT 1 þ T 2 þ T 3 þ T cogg Bx CsignðxÞÞ J Tot
ð18Þ
b dist is, therefore, given as: The estimated disturbance signal T
b signðx bx ^ þC ^Þ Tb dist ¼ Tb cogg þ B
ð19Þ
3.5. Cogging-torque harmonics, viscous friction coefficient, and Coulomb friction coefficient estimation The higher harmonics in the fictitious rotor winding self inductance Lrr , which corresponds to the cogging-torque harmonics, coefficient of viscous friction, and coefficient of Coulomb friction are identified from the estimated disturbance signal:
p 2 Tb dist ¼ ir 2
K0 X b signðxÞ bx þ C ðkb L rrak sinðkphÞ kb L rrbk cosðkphÞÞ þ B
ð20Þ
k¼4
Eq. (20) can be written in the linear form of (11), if the observation matrix H and the parameter vector a are defined as: s
c
B
C
H ¼ ½hij ; hij ; hij ; hij ; ; p 2 s hij ¼ j ir sin jðph½iÞ i ¼ 1 : N; j ¼ 1 : K 2 p 2 c hij ¼ j ir cos jðph½iÞ i ¼ 1 : N; j ¼ 1 : K 2 B hij ¼ x½i i ¼ 1 : N; j ¼ 1 C
hij ¼ signðx½iÞ i ¼ 1 : N;
ð21Þ
j¼1
b gT b C a^ ¼ fbL rra1 . . . bL rraK ; bL rrb1 . . . bL rrbK ; B;
ð22Þ
^ is identified by the least-squares estimator (12). The parameter vector a Experimental Data
ω θ i1 i3 -
Electromagnetic T + T + T 2 1 3 Torque + Calculation -
Plant Model 1 Jtot T
ω+
- ω
dist
Compensator PI Controller
Fig. 3. Cogging, viscous friction, and Coulomb friction torque estimation.
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4. Experimental results 4.1. Experimental setup The block diagram of the experimental setup is shown in Fig. 1. The following lines give a brief description for each of the subsystems. 1. Motor/encoder: The tested motor was a three-phase six-pole/nine-slot BLDC motor with external rotor and maximum electrical power of 40 W. A Heidenhain ROD1080 rotary encoder with 3600 line counts, was used to measure the rotor position. The encoder interface provided sinusoidal voltage signals with an amplitude of 1 V PP . The interface signals could be highly interpolated, thus the encoder resolution could be increased. Some motor parameters and inertia of the rotary encoder and coupling are listed in Table 1. 2. Motor driving unit: Motor driving unit consisted of a DSP-based motor controller and a three-phase bridge. The so-called DSP-2 controller developed at the Faculty of Electrical Engineering and Computer Science (FERI), University of Maribor, is based on Texas Instrument TMS320C32 floating point processor and Xilinx FPGA of the Spartan family. The pulse width modulator along with the peripheral interfaces were implemented in FPGA. In addition, the DSP-2 controller included all the necessary peripheral for AC motor control [6]. The three-phase bridge with on board current sensors was used as a power stage, with the capability of delivering ±3 A at DC link of 20 V. 3. Measuring unit: Measuring unit is basically the DSP-2 controller with an additional measuring board. In this case the DSP2 controller worked as data logger. The measurement sampling time could be set from 10 ls up to 250 ls and the memory amount was available for recording 105 000 32-bits values. Data transfer between the measuring unit and a personal computer was via a USB interface with a connection speed up to 921 kBD. The measuring board contained four 12-bit A/D converters. One pair was used for the motor back EMF’s or motor phase current measurement, while the second pair was used to interface the rotary encoder. 4. Data analysis: The entire parameter estimation method was implemented off-line in MATLAB/Simulink environment.
4.2. Back-EMF coefficients estimation A current-regulated DC motor with belt transmission was used to drive the non-excited BLDC motor during the test. The rotary encoder was mounted onto the BLDC motor shaft using a metal bellows coupling. The encoder outputs and phase back-EMF’s were recorded simultaneously, during the experiment, with sampling time T s ¼ 100 ls and data length of 30 000 points for each signal. The rotor position and rotor speed were then calculated off-line. Fig. 4 shows the rotor speed over ten turns of the motor, where the cogging-torque effect is noticeable as high-frequency oscillations. Fig. 5a shows the back-EMF for the first phase, measured from line to neutral. Fig. 6 shows the minimum leastsquares error J min versus the number of used harmonics k in the linear model (16) and (17), when measured back-EMF volt-
Table 1 Motor/mechanical parameters Value
Unit
Rs Ls JM JRE JC JTot
5 1 97.0 5.0 2.0 104.0
X mH g cm2 g cm2 g cm2 g cm2
Rotor Speed [rad/s]
Parameter
60 40 20 0 0
720
1440 2160 2880 Rotor Position [deg] Fig. 4. Rotor speed.
3600
1260
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1.5 Back EMF [V]
1
a)
0.5 0 -0.5
b)
-1 -1.5
0
60
120 180 240 Rotor Position [deg]
300
360
Fig. 5. Back-EMF in first phase e1 for one turn (a), and difference e1 ^ e1 (b).
70 60
J
min
50 40 30 20 10 0
1
2
3 4 5 6 7 Number of harmonics - K
8
9
Fig. 6. Minimum least-squares error for ^ e1 .
age is approximated by LSE. In practice, this kind of diagram is very useful for determining the minimal number of relevant harmonics, which gives the simplest signal model, which adequately describes the given data signal. In this determination scheme the number of harmonics is increased until the minimum least-squares error decreased only slightly with further increase in the number of harmonics. It is evident from Fig. 6, that the first, third and fifth harmonics are the only relevant ones. This result is in accordance with the fact that back-EMF waveforms usually posses half-wave symmetry, that is f ðxÞ ¼ f ðx þ T=2Þ, where T is the waveform period. It is known from Fourier analysis [2] that, for half-wave symmetry functions, the even harmonics equal zero. The estimated back-EMF coefficients for each phase, scaled by 106 , are listed in Table 2. The estimated parameters are given with their mean values and standard deviations, which provides information about reproducibility of the results. The standard deviation r of data vector x is defined as:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u m u 1 X r¼t ðxj xÞ2 m 1 j¼1
ð23Þ
where x is the mean value of the elements in vector x and m is the number of elements. In the estimation process 10 repetitions are included, therefore m ¼ 10. The parameters Lirbk ; where i ¼ 1 : 3 and k ¼ 1 : 5; are equal to zero. It is clear from the results for the individual phase, that phase non-symmetry is practically negligible. Fig. 5b shows the difference between measured back-EMF and the approximated version of back-EMF with LSE when the first, third and, fifth harmonics are taken into account.
Table 2 Estimated back EMF coefficients Parameter
b L ira1 b L ira2 b L ira3 b L ira4 b L ira5 Jimin
106 106 106 106 106
Phase 1
Phase 2
Phase 3
x
r
x
r
x
r
5817 3 57 0 28 1.96
0.7 0.5 0.0 0.0 0.0 0.94
5802 3 57 0 28 1.97
4.9 0.3 0.5 0.0 0.0 0.99
5835 2 56 0 28 2.05
0. 5 0.4 0.0 0.0 0.0 0.84
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Motor manufactures usually refer to the voltage constant K e , when describing their motors back-EMF’s. The voltage constant specify the back-EMF peak or RMS (Root Mean Square) value per unit rotor speed. The back-EMF in the first phase for the BLDC motor used in the experiment equals:
^e1 ¼ ir xpðb L 1ra1 sinðphÞ 3b L 1ra3 sinð3phÞ 5b L 1ra5 sinð5phÞÞ From the back-EMF peak value
^ep1
¼^ e1 ðph ¼
pÞ 2
¼
K pe
x, the peak voltage constant
p 3p 5p s ¼ 35:1 mV 3b L 3ra1 sin K pe ¼ ir pb L 1ra1 sin 5b L 5ra1 sin rad 2 2 2
ð24Þ K pe
can be calculated as:
ð25Þ
The back-EMF RMS value is defined as:
^erms 1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 T 2 ^e dt ¼ K rms ¼ e x T 0 1
ð26Þ
p being the period of back-EMF waveform. From (24) and (26), the RMS where K rms is the RMS voltage constant and T ¼ p2x e voltage constant is calculated as:
ir p K rms ¼ pffiffiffi e 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s b L 21ra1 þ 25b L 21ra5 ¼ 24:7 mV L 21ra1 þ 9b rad
ð27Þ
4.3. Torque disturbance estimation A field-oriented control scheme with a current-regulated pulse width modulated (PWM) inverter was applied to control the motor torque T m , as suggested in [18]. The stator current is entirely q-axis current and is equivalent to a torque comref b mand, where isq ¼ 2T ref m =ð3p L 1ra1 Þ. In the experiment, which was repeated 40 times, the control algorithm was executed with sampling time of T s ¼ 100 ls, and the reference torque was set to 4.3 mNm. Using the measurement unit, the phase currents and the rotary encoder outputs were recorded simultaneously, with the same sampling time and data length of 25 000 points for each signal. From the experimental data, the torque disturbance signal was estimated in off-line mode, as suggested in Section 3.4. Although in the estimation algorithm all data points were taken into account, not all data points are plotted in order to improve the figures readings. The calculated motor torque and measured rotor speed are shown on Figs. 7 and 8. The cogging-torque effect is noticeable as speed oscillation with frequency equal to 6p ¼ 36 per shaft turn, Fig. 8b. It is clear from the rotor speed diagram shown on Fig. 8a, that the motor and/or the whole experimental set-up system are imperfectly balanced, therefore additional speed oscillations appear with frequencies equal to 1 per shaft turn. It is evident form Fig. 9 that the disturbance observer compen^ j < 2 103 rad/s). Therefore the compensator output sator ensures convergence of the speed error towards to zero (jx x dist b shown of Fig. 10. equals to the estimated torque disturbance T 4.4. Cogging-torque harmonics, viscous friction coefficient, and Coulomb friction coefficient estimation b dist is In order to incorporate the unbalanced mechanical system effect into the estimation process, the estimated signal T redefined as:
p 2 Tb dist ¼ ir 2
K0 X
b signðx bx ^ þC ^ Þ þ Tb a1 cosð^hÞ þ Tb b1 sinð^hÞ ðkb L rrak sinðkp^hÞ kb L rrbk cosðkp^hÞÞ þ B
ð28Þ
k¼6
Calculated Torque [mNm]
b a1 and T b b1 are the Fourier coefficients of the exogenous mechanical disturbances. According to the redefinition of the where T dist b signal T , the observation matrix and parameter vector in (21) and (22) are also suitably redefined.
6 4 2 0
0
720
1440 2160 2880 Rotor Position [deg]
Fig. 7. Calculated mutual torque.
3600
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Rotor Speed [rad/s]
a 60 40 20 0
0
720
b
1440 2160 2880 Rotor Position [deg]
3600
62 60 58 2880
3000
3120
3240
Fig. 8. Rotor speed (a), and rotor speed oscillations in detail (b).
3
Speed Error [rad/s]
2
x 10
1 0 1 2
0
60
120 180 240 Rotor Position [deg]
300
360
Estimated Disturbance [mNm]
^. Fig. 9. Compensator input: speed difference x x
15 10 5 0 5
0
60
120 180 240 300 Rotor Position [deg]
360
Fig. 10. Compensator output: estimated disturbance signal T^ dist .
All 25 000 experimental data points were included in the least-squares approximation method. Fig. 11a shows the minb dist is imum least-squares error versus the number of cogging-torque harmonics, when the estimated disturbance signal T approximated using LSE. It is clear from the diagram, that the sixth cogging-torque harmonic is highly dominant, just as predicted in Section 2. A small amount of third harmonics is also present besides the sixth harmonic. Fig. 12 shows the estimated cogging-torque over one motor rotation. The estimated mechanical parameters scaled by 106 , are listed in Table 3, in the order of their mean values and standard deviations, by considering all 40 repetitions of the experiment. The high standard deviation indicates a high variation in the estimated Coulomb friction coefficient results. Therefore, the estimated Coulomb friction coefficient result value is questionable, and should be understood as a rough estimation. Friction is hard
a
0.6
Jmin
A. Kapun et al. / Simulation Modelling Practice and Theory 16 (2008) 1254–1265
0.4
1263
0.2
b
Estimated Disturbance [mNm]
0
1
2
3
4 5 6 7 8 9 10 11 12 Number of harmonics - K
15 10 5 0 -5
2900
2950
3000
3050
Rotor Position [deg]
Cogging Torque [mNm]
b dist (a), estimated disturbance signal T b dist (solid line), and his approximated version using LSE (dotted line) in Fig. 11. Minimum least-squares error for T detail (b).
8 4 0 -4 -8
0
60
120 180 240 Rotor Position [deg]
300
360
Fig. 12. Estimated cogging-torque.
to determine due to the many considerations involved, such as temperature, materials of the bodies, lubrication, manufacture precision, coupling, etc. An additional difficulty when determining the Coulomb friction is the large correlation between Coulomb friction coefficient and viscous friction coefficient, which is also reported in [1]. Perhaps improvement in Coulomb friction identification could have been achieved if the torque ramp had been used as a reference, as proposed in [11]. However, further analysis was not pursued, since the main research focus was on identifying the cogging-torque and viscous friction. Usually, information about the maximum cogging-torque can be found in motor technical data. The cogging-torque for the BLDC motor under the test is estimated as:
p 2 Tb cogg ¼ ir ð3b L rra3 sinð3p^hÞ þ 3b L rrb3 cosð3p^hÞ 6b L rra3 sinð6p^hÞ þ 6b L rrb3 cosð6p^hÞÞ 2
ð29Þ
An infinite number of minimums and maximums exist, because the function (29) is a periodical one. However, using the Matlab numerical toolbox, the first maximum for absolute value of (29) at the interval ð0; 1Þ is discovered at h ¼ 0:1483 rad. The absolute value of (29) at maximum is 7.01 mNm. According to the BLDC motor manufacture the maximum cogging-torque is 6.5 mNm, which means that the cogging-torque is overestimated by approximately 8%. 4.5. Results verification A simulation of the experiment was performed in order to verify the results of the proposed parameters identification method, where the experimental data for phase currents was used as motor excitation. The parameters from Tables 2 and 3 were used as simulation parameters, with the exception of the Coulomb friction coefficient which was reduced by 20%.
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Table 3 Estimation results for the mechanical parameters Parameter
x
r
b L rra3 106 b L rrb3 106 b L rra6 106 b L rrb6 106 b a1 106 T b b1 106 T b 106 B b 106 C
94.3 52.7 270.6 214.0 741.1 23.4 52.5 1320.0 26.8
2.3 3.0 9.4 11.9 17.5 8.5 10.6 343.8 5.5
Jmin 103
60 40 20
Rotor Speed [rad/s]
0
0
0.5
1 1.5 Time [s]
b 62
60
58
1.05
1.1
Time [s]
1.15
Rotor Speed [rad/s]
Rotor Speed [rad/s]
a
2
2.5
c
62
60 58 2880
3000
3120
3240
Rotor Position [deg]
Fig. 13. Measured (black-line) and estimated (grey-line) rotor speeds (a), rotor speed oscillation detail plotted against time (b) and rotor speed oscillation detail plotted against measured and estimated rotor position (c).
A comparison between measured and estimated rotor speeds was formed in order to highlight the verification result, see Fig. 13. The term ‘‘estimated” refers to rotor speed obtained from the simulation using the estimated parameters. It is evident from Fig. 13a, that the simulation data are in good accordance with the experimental data, especially within the steady state region. The experimental and estimated cogging-torque effects, noticeable as speed oscillation with frequency of 36 per shaft turn, are shown in Figs. 13b and 13c. A phase displacement of approximately half of the period between experimental and estimated cogging-torque is evident from Fig. 13b, where speed oscillations are plotted against time. This however should not be misunderstood, since any speed difference between measured and estimated rotor speed accumulates in its rotor position difference. The cogging-torque is a function of rotor position, therefore, the comparison between experimental and estimated cogging shown in Fig. 13c, is more pertinent. Matching of the simulated and experimental data is good enough to justify the applicability of the proposed identification method in the BLDC simulation process. 5. Conclusion This paper presents an off-line method for identifying some of the BLDC motor’s dynamic model parameters. The method is founded on the least-squares approximation method and a closed-loop disturbance observer, and is conditional on the availability of experimental data for back-EMF, phase currents, rotor position, and rotor speed. Using the proposed identification method, it is possible to identify back-EMF harmonic coefficients and mechanical parameters, where the mechanical parameters refer to cogging-torque harmonics and viscous friction coefficient. Normally the FFT analysis is performed on no-load back-EMF to find the dominant harmonics. The FFT analysis, however, can be performed only if rotor speed is constant during the period of back-EMF. Constant rotor speed is difficult to ensure, if the BLDC motor possesses high cogging-torque. In the proposed identification method, experimental data for rotor position and rotor speed are incorporated into the identification process, therefore, non-constant rotor speed has no influence on back-EMF coefficient identification performances. Using the torque-sensor seems to be the logical choice for determining
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mechanical parameters, but the torque-sensor’s application results in extra costs. Furthermore, high torque-sensor mounting demands, especially for sensors with low rated torque (<100 mNm), could be the source of additional difficulties in a nonlaboratory environment. Thus this identification method’s ability when determining mechanical parameters without torquesensor is of additional benefit. Of course, the reliability of those mechanical parameters obtained using torque-sensor is higher. However, the experimental result shows, that the proposed method sufficiently identifies the mechanical parameters. Therefore, this method enables an improvement in the field of dynamic simulations for systems using the BLDC motor. References [1] A.J. Blauch, M. Bodson, J. Chiasson, High-speed parameter estimation of stepper motors, IEEE Transactions on Control System Technology 1 (4) (1993) 270–279. [2] I.N. Bronstein, K.A. Semendjajew, G. Musiol, H. Muhlig, Matematicˇni Prirocˇnik (Tehniška zalozˇba Slovenije, Slovenia, 1997) (in Slovene). [3] K.Y. Cheng, Y.Y. Tzou, Design of a sensorless commutation IC for BLDC Motors, IEEE Transactions on Power Electronics 18 (6) (2003) 1365–1375. [4] J.H. Choi, S.H. You, J. Hur, H.G. Sung, The design and fabrication of BLDC motor and drive for 42V automotive applications, in: IEEE International Symposium on Industrial Electronics 2007 (ISIE’07), 2007, pp. 1086–1091. [5] D. Dolinar, G. Štumberger, Modeliranje in Vodenje Elektromehanskih Sistemov (Univerzitetna knjizˇnica Maribor, Slovenia, 2002) (in Slovene). [6] D. Hercog, K. Jezernik, Rapid control prototyping using MATLAB/Simulink and a DSP-based motor controller, International Journal of Engineering Education 21 (4) (2005) 596–605. [7] J. Holtz, L. Springob, Identification and compensation of torque ripple in high-precision permanent magnet motor drives, IEEE Transaction on Industrial Electronics 43 (2) (1996) 309–320. [8] W. Hong, W. Lee, B.K. Lee, Dynamic simulation of brushless DC motor drives considering phase commutation for automotive applications, in: IEEE International Electric Machines & Drives Conference (IEMDC’07), 2007, pp.1377–1383. [9] J.Y. Hung, Z. Ding, Design of currents to reduce torque ripple in brushless permanent magnet motors, IEE Proceedings-B 140 (4) (1993) 260–266. [10] S.M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice-Hall International Inc., 1993. [11] R. Kelly, J. Llamas, R. Campa, A measurement procedure for viscous and Coulomb friction, IEEE Transaction on Instrumentation and Measurement 49 (4) (2000) 857–861. [12] F. Khorrami, P. Krishnamurthy, H. Melkote, Modeling and Adaptive Nonlinear Control of Electric Motors, Springer-Verlag, New York, 2003. [13] D. Iles-Klumpner, I. Serban, M. Risticevic, I. Boldea, Comprehensive experimental analysis of the IPMSM for automotive applications, in: International Power Electronics and Motion Control Conference 2006 (EPE-PEMC’06), 2006, pp.1776–1783. [14] Y. Koubaa, Asynchronous machine parameters estimation using recursive method, Simulation Modelling Practice and Theory 14 (2006) 1010–1021. [15] L. Liu, D.A. Cartes, W. Liu, Particleswarm optimization based parameter identification applied to PMSM, in: Proceedings of the 2007 American Control Conference, 2007, pp. 2955–2960. [16] H. Lu, L. Zhang, W. Qu, A new torque control method for torque ripple minimization of BLDC motors with un-ideal back EMF, IEEE Transactions on Power Electronics 23 (2) (2008) 950–958. [17] H. Melkote, F. Khorrami, S. Jainm, M.S. Mattice, Robust adaptive control of variable reluctance stepper motors, IEEE Transaction on Control Systems Technology 7 (2) (1999) 415–425. [18] D.W. Novotny, T.A. Lipo, Vector Control and Dynamics of AC Drives, Clarendon Press, Oxford, 1996. [19] Ravindra Pantankar, Liangtao Zhu, Real-time multiple parameter estimation for voltage controlled brushless DC motor actuators, in: Proceedings of the 2004 American Control Conference, vol. 4, 2004, pp. 3851–3856. [20] J. Shao, An improved microcontroller-based sensorless brushless DC (BLDC) motor drive for automotive applications, IEEE Transactions on Industry Applications 42 (5) (2006) 1216–1221. [21] T. Tutunji, M. Molhim, E. Turki, Mechatronic system identification using an impulse response recursive algorithm, Simulation Modelling Practice and Theory 15 (2007) 970–988.
Contents lists available at ScienceDirect
Simulation Modelling Practice and Theory journal homepage: www.elsevier.com/locate/simpat
Identifying dynamic model parameters of a BLDC motor A. Kapun *, M. Cˇurkovicˇ, A. Hace, K. Jezernik Faculty of Electrical Engineering and Computer Sciences, University of Maribor, Smetanova 17, SI-2000, Slovenia
a r t i c l e
i n f o
Article history: Received 20 March 2008 Received in revised form 5 June 2008 Accepted 11 June 2008 Available online 20 June 2008
Keywords: Permanent magnet motors simulation Parameter estimation Least-squares methods Cogging-torque
a b s t r a c t An off-line identification method founded on the least-squares approximation technique and a closed-loop disturbance observer is applied for identifying the parameters of a BLDC motor model. No special configuration of the motor is required besides the availability of experimental data for back-EMF, phase currents, rotor position, and rotor speed. This method is used to identify the back-EMF harmonics and mechanical parameters, where the mechanical parameters refer to cogging-torque, viscous friction coefficient, and Coulomb friction coefficient. The proposed identification method is theoretically investigated, and the method’s effectiveness is proved by experimental results performed on a low-power BLDC motor. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Modern cars inevitably rely on several diverse electric motors. In addition to those for maintaining basic car functionality such as fuel/water pumps, power-assisted steering, and engine cooling fans, electric motors are also included in new electrical systems for improving passengers comfort and safety [4]. Traditionally brushed DC motors have been used in automotive applications, although presently permanent magnet brushless DC (BLDC) motors are gaining popularity. The benefits of the BLDC motor when compared to the DC motor concern higher reliability, longevity, lower maintenance costs, and lower electromagnetic interference (EMI). DC motor replacement by the BLDC motor, on the other hand, results in more complex control algorithms. There has been extensive research dealing with BLDC motor control, proposing diverse control algorithms for torque, speed, and position control. Intensive research work has been carried out in the field of sensorless-control algorithms for low cost applications [20,3]. Throughout the control design process there is a need for algorithm verification. Thus authentic simulation models of the BLDC motor are crucial. In [8], a BLDC motor model for MATLAB/Simulink environment simulation was developed, suitable for the analysis of BLDC motor drives and control algorithm design. However if correlation between simulation and experimental results is either over- or under-estimated, the simulation results value is questionable. Nevertheless, the system dynamic model structure is one part of the simulation environment, the second important issue is the model’s parameters. A comprehensive measurement procedure for determining the permanent magnet synchronous machine (PMSM) model and machine operational performance parameters is thoroughly described in [13]. Although the measurement procedure is very effective, its implementation requires special measurement equipment such as torque-sensor, and a high-quality servo motor in the case of cogging-torque measurement. In [14], an identification algorithm derived from least-squares techniques is used to estimate the rotor resistance, and the rotor and the stator leakage inductance of a three-phase induction machine. In [21], an auto regressive moving average (ARMA) model is used to identify the DC motor transfer function, where a recursive least square algorithm is applied to estimate the ARMA parameters. In
* Corresponding author. Tel.: +386 2 220 7176. E-mail address: [email protected] (A. Kapun). 1569-190X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2008.06.003
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[15], the particle-swarm algorithm is applied to PMSM in order to identify its stator resistance and load torque disturbances. The stator resistance, stator inductance, torque constant, inertia constant, viscous friction coefficient and Coulomb friction coefficient of the PM stepper motor are identified in [1], using an on-line batch least-squares algorithm. The real time estimation method derived from the motor inverse model is discussed in [19] to estimated the BLDC motor resistance and torque constant on-line. With adaptive controller structure the proposed estimation algorithm can also be extended for compensation of the motor parameters variation on control performances. This paper presents research work, where the main focus was on identifying those BLDC motor model parameters which significantly contribute to motor torque-ripple. The torque-ripple parameters identification is important not only for authentic simulations but also for implementation of different torque-ripple minimization control techniques, like those described in[9,16]. Minimization of torque-ripple reduce speed oscillations and consequently deteriorations in the motor performance. There are two main sources of torque-ripple, cogging-torque and non-sinusoidal stator flux linkage distribution [7]. The motor model where these two torque-ripple sources are included is derived in [12]. Based on this motor model, an off-line method was developed for identifying motor parameters. The paper is organized as follows. Firstly, the basics of the motor modelling are summarized and the three-phase BLDC motor model is presented. In Section 3, the parameters identification method is theoretically investigated. This identification method is founded on the least-squares approximation method, and a closed-loop disturbance observer, therefore, these two techniques are also briefly described. In order to verify the proposed method’s effectiveness, experiments were performed on a low-power BLDC motor. The experimental results are shown and discussed in Section 4. Finally, the conclusions are summarized is Section 5. 2. Motor model Assuming the variations of the stator self inductance with rotor position and the mutual inductance between the stator windings are negligible; the electrical dynamics of a BLDC motor may be modelled in an electrically balanced system as [17]:
disi usi V 0 ¼ Rs isi þ Ls þ ei dt X isi ¼ 0
i ¼ 1; 2; 3
ð1Þ ð2Þ
where Rs and Ls are the stator resistance and inductance, usi is the motor terminal voltages, isi is the phase current and ei is the back-EMF associated with the ith phase. The potential of the motor neutral terminal in wye-connected windings is denoted as V 0 . The back-EMF inducted in each phase by the rotor permanent magnet, can be expressed as:
dwir owir dh ow ¼ x ir ¼ dt oh dt oh
ei ¼
ð3Þ
where wir is the mutual magnetic flux between the permanent magnet and the stator windings in the ith phase, h is the rotor position, and x is the rotor speed. Since a linear magnetic model without saturation effects is assumed, the mutual magnetic flux is modelled by fictitious rotor winding with mutual inductance Lir , and fictitious constant rotor current ir ¼ 1A as:
wir ¼ Lir ir
ð4Þ
In general, the mutual magnetic flux in the air-gap is non-sinusoidal, therefore, the higher harmonics in mutual magnetic flux are introduced as higher harmonic components in fictitious rotor winding mutual inductance. The mutual inductance Lir is expressed, using the terms of the trigonometric Fourier series, as:
Lir ¼
K X 2p 2p Lirak cos k ph ði 1Þ þ Lirbk sin k ph ði 1Þ 3 3 k¼1
ð5Þ
From (3) and (4), the back-EMF can be expressed as:
ei ¼ ir x
oLir oh
ð6Þ
where the partial derivatives of the mutual inductance over the rotor position, are known as the back-EMF coefficients [5]. From the virtual work principle, the electromagnetic motor torque can be derived [12], as:
T¼
3 X i¼1
T i þ T cogg ¼ ir
3 X i¼1
oLir isi oh
!
1 2 oLrr þ ir 2 oh
ð7Þ
The first three terms in Eq. (7) are mutual torques caused by interaction between the permanent magnet field and the phase currents. The last term known as cogging-torque, is due to the attraction of the permanent to the salient portions of the stator iron. Therefore, the cogging-torque is always present, even in the absence of the phase currents. Usually, the
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cogging-torque harmonics for three-phase motors are multiples of six times the fundamental commutation frequency [9]. The cogging-torque harmonics are modelled with higher harmonic components in the fictitious rotor winding self inductance Lrr .
Lrr ¼
K0 X
ðLrrak cos kðphÞ þ Lrrbk sin kðphÞÞ
ð8Þ
k¼6
The motor mechanical dynamics is defined as:
h_ ¼ x;
1 J
x_ ¼ ðT T l Bx CsignðxÞÞ
ð9Þ
where T l is the load torque, J is the rotor inertia, B is the viscous friction coefficient, and C is the Coloumb friction coefficient. 3. Parameter estimation method The parameters of the motor model given in the previous section are Rs , Ls , J, B, C, Lirab ¼ fLira1 . . . LiraK ; Lirb1 LirbK g, and Lrrab ¼ fLrra1 LrraK , Lrrb1 LrrbK g. This section presents an off-line method for estimating the parameters Lirab , Lrrab , B, and C in Eqs. (6), (7), and (9). It is assumed that all the other parameters are known, since they are normally provided by the motor manufacturer. The estimation method is based on given experimental data for phase currents, rotor position, and rotor speed. In practice, an experimental test bench, like the one shown in Fig. 1, is used to obtain these signals. The estimation method can be divided into three steps. Firstly, the least-squares approximation method is used to estimate the parameters Lirab from the experimental data for back-EMF’s, rotor position, and rotor speed. In the second step, the motor is driven using the DSP controller and a three-phase bridge (Motor Driving Unit), while the phase currents and rotor position are measured simultaneously (Measuring Unit). The total mutual torque T 1 þ T 2 þ T 3 is calculated from the phase currents, the rotor posib signðxÞ is b dist ¼ T b cogg þ B bx þ C tion, and the parameters b L irab estimated in the first step. Then the torque disturbance signal T estimated using a closed-loop observer. Both the calculated mutual torque and measured motor speed are required for the observer’s implementation. In the last step, the least-square approximation method is again applied in order to estimate the b form the signal T b and C b dist . Each step is described in detail in the following subsections. The least-squares parameters b L rrab , B, approximation method and the disturbance observer are used during the estimation process, therefore, these two techniques are briefly described first. 3.1. Least-squares approximation method The method implementation is easy, therefore, it is widely used in practice. The method objectivity is to minimize the squared difference between the measured signal x½n and the assumed signal s½n, [10]. When a vector problem is considered, the signal s½n is determined by a signal model which depends upon the unknown parameter vector a. Signal x½n is a perturbed version of the signal s½n, due to noise and model inaccuracies. The least-squares estimator (LSE) is that value of a, which minimizes the squares error criterion, given as:
JðaÞ ¼
N X ðx½n s½nÞ2
ð10Þ
n¼1
Data Analysis
Data Flow
θ
Measuring Unit
uS1 uS2 iS1 iS2
Motor Driving Unit
Incremental Encoder
BLDC motor
Fig. 1. Test bench configuration.
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The observation interval is assumed to be n ¼ 1; . . . ; N, and the dependence of J on a is via s½n. This method is valid for Gaussian and non-Gaussian noise, however the performance of the LSE will undoubtedly depend on the properties of the corrupting noise, as well as any modelling errors, and quantization. When the linear least-squares problem is considered, the signal model is linear with respect to the unknown parameter, therefore, the signal s ¼ ½s½1 . . . s½NT may be written as:
s ¼ Ha
ð11Þ
where H is the known NxrðN > rÞ observation matrix of full rank r and the a is rx1 unknown parameter vector. The least square estimator and minimum least square error for (11) are [10]:
a^ ¼ ðHT HÞ1 HT x ^ Þ ¼ ðx Ha ^ ÞT ðx Ha ^Þ J min ¼ Jða
ð12Þ ð13Þ
3.2. Disturbance observer The disturbance observer, shown in Fig. 2, can be used to estimate the disturbance signal udist , that may be applied to improve control performances. If inaccuracies in the plant model are negligible and the compensator is able to ensure conver^ dist . The observer can be gence of the signal e towards zero, then the compensator output is the estimated disturbance signal u used in a similar way to estimate non-modelled dynamics in the case of an incomplete plant model, under the assumption that exogenous disturbances are negligible. From Fig. 2, the relationship between the estimated and applied disturbance can be evaluated as:
^ dist ðsÞ u HP ðsÞHC ðsÞ ¼ udist ðsÞ 1 þ HP ðsÞHC ðsÞ
ð14Þ
If such an HC ðsÞ is chosen, so that the denominator of the polynomial in (14) is a Hurwitz polynomial, then the system ^ dist ðtÞ will be observing udist ðtÞ from Fig. 2 is asymptotically stable. Hence, the output of the feedback compensator u asymptotically. 3.3. Back-EMF coefficients estimation It is evident from (5)-(6) that the parameters Lirab appear as higher order harmonics back-EMF coefficients due to the non-sinusoidal flux distribution phenomena. If the motor is wye-connected and the neutral point is accessible, then the easiest way to determine the back-EMF is to measure the phase to neutral voltage at open-circuit phase winding, while the rotor is rotating because of externally applied torque. If this is not the case, then the back-EMF is determined from measured phase to phase voltage but, in this case, the harmonics divisible by three can not be identified. Usually Fourier analysis is used to extract the back-EMF coefficients. Because Fourier analysis is valid for periodical functions, exactly one period of back-EMF needs to be introduced into the analysis in order to obtain accurate results. An additional restriction is in the rotor speed which should be constant during the experiment. With the proposed least-squares approximation method, the result’s accuracy is independent of the rotor rotation, since the rotor position and rotor speed are already incorporated into the estimation process. When the phase to neutral voltage is measured, the back-EMF in the first phase equals:
e1 ¼ ir xp
K X ðkL1rak sin kðphÞ þ kL1rbk cos kðphÞÞ
ð15Þ
k¼1
Eq. (15) can be written in linear form (11), if observation matrix H and parameter vector a are defined as:
udis t
u
+
y Plant Plant Model
+
HP(s)
y
+
- udist Compensator
HC(s) Fig. 2. Disturbance observer principle.
ε
0
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c
H ¼ ir p½hij ; hij ; s hij c hij
ð16Þ
¼ jx½i sin jðph½iÞ i ¼ 1 : N; ¼ jx½i cos jðph½iÞ i ¼ 1 : N;
a¼
LTirab
j¼1:K j¼1:K
¼ fLira1 . . . LiraK ; Lirb1 . . . LirbK gT
ð17Þ
The parameters Lirab are estimated by the least-squares estimator (12). The observation matrix is formed from experimental data for rotor position and rotor speed. If constant rotor speed can be accomplished, then the observation matrix can be simplified and no measurements of rotor position and speed are needed. In this case, the rotor speed is determined from the back-EMF waveform period, and the rotor position is obtained by integrating the speed. Normally, constant speed can only be accomplished during high-speed operation, where the cogging-torque’s influence could be neglected [1]. 3.4. Cogging, viscous friction, and Coulomb friction torque estimation Estimation of cogging, viscous friction, and Coulomb friction torque is realized based on the previously discussed disturbance observer, as shown in Fig. 3. The total mutual torque is calculated (7) from the simultaneously measured phase currents and rotor position, and previously identified back-EMF coefficients. The model plant is obtained from the mechanical dynamics (9), as shown in Fig. 3, where J Tot is the total inertia (motor, rotary encoder, coupling). A PI controller is used as the ^ towards the measured speed x, it can be compensator. If the controller ensures the convergence of the estimated speed x assumed that the controller output equals the estimated non-modelled dynamics signal. If the load torque is zero and the exogenous disturbances are negligible, then rotor acceleration can be written from (7) and (9) as:
x_ ¼
1 ðT 1 þ T 2 þ T 3 þ T cogg Bx CsignðxÞÞ J Tot
ð18Þ
b dist is, therefore, given as: The estimated disturbance signal T
b signðx bx ^ þC ^Þ Tb dist ¼ Tb cogg þ B
ð19Þ
3.5. Cogging-torque harmonics, viscous friction coefficient, and Coulomb friction coefficient estimation The higher harmonics in the fictitious rotor winding self inductance Lrr , which corresponds to the cogging-torque harmonics, coefficient of viscous friction, and coefficient of Coulomb friction are identified from the estimated disturbance signal:
p 2 Tb dist ¼ ir 2
K0 X b signðxÞ bx þ C ðkb L rrak sinðkphÞ kb L rrbk cosðkphÞÞ þ B
ð20Þ
k¼4
Eq. (20) can be written in the linear form of (11), if the observation matrix H and the parameter vector a are defined as: s
c
B
C
H ¼ ½hij ; hij ; hij ; hij ; ; p 2 s hij ¼ j ir sin jðph½iÞ i ¼ 1 : N; j ¼ 1 : K 2 p 2 c hij ¼ j ir cos jðph½iÞ i ¼ 1 : N; j ¼ 1 : K 2 B hij ¼ x½i i ¼ 1 : N; j ¼ 1 C
hij ¼ signðx½iÞ i ¼ 1 : N;
ð21Þ
j¼1
b gT b C a^ ¼ fbL rra1 . . . bL rraK ; bL rrb1 . . . bL rrbK ; B;
ð22Þ
^ is identified by the least-squares estimator (12). The parameter vector a Experimental Data
ω θ i1 i3 -
Electromagnetic T + T + T 2 1 3 Torque + Calculation -
Plant Model 1 Jtot T
ω+
- ω
dist
Compensator PI Controller
Fig. 3. Cogging, viscous friction, and Coulomb friction torque estimation.
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4. Experimental results 4.1. Experimental setup The block diagram of the experimental setup is shown in Fig. 1. The following lines give a brief description for each of the subsystems. 1. Motor/encoder: The tested motor was a three-phase six-pole/nine-slot BLDC motor with external rotor and maximum electrical power of 40 W. A Heidenhain ROD1080 rotary encoder with 3600 line counts, was used to measure the rotor position. The encoder interface provided sinusoidal voltage signals with an amplitude of 1 V PP . The interface signals could be highly interpolated, thus the encoder resolution could be increased. Some motor parameters and inertia of the rotary encoder and coupling are listed in Table 1. 2. Motor driving unit: Motor driving unit consisted of a DSP-based motor controller and a three-phase bridge. The so-called DSP-2 controller developed at the Faculty of Electrical Engineering and Computer Science (FERI), University of Maribor, is based on Texas Instrument TMS320C32 floating point processor and Xilinx FPGA of the Spartan family. The pulse width modulator along with the peripheral interfaces were implemented in FPGA. In addition, the DSP-2 controller included all the necessary peripheral for AC motor control [6]. The three-phase bridge with on board current sensors was used as a power stage, with the capability of delivering ±3 A at DC link of 20 V. 3. Measuring unit: Measuring unit is basically the DSP-2 controller with an additional measuring board. In this case the DSP2 controller worked as data logger. The measurement sampling time could be set from 10 ls up to 250 ls and the memory amount was available for recording 105 000 32-bits values. Data transfer between the measuring unit and a personal computer was via a USB interface with a connection speed up to 921 kBD. The measuring board contained four 12-bit A/D converters. One pair was used for the motor back EMF’s or motor phase current measurement, while the second pair was used to interface the rotary encoder. 4. Data analysis: The entire parameter estimation method was implemented off-line in MATLAB/Simulink environment.
4.2. Back-EMF coefficients estimation A current-regulated DC motor with belt transmission was used to drive the non-excited BLDC motor during the test. The rotary encoder was mounted onto the BLDC motor shaft using a metal bellows coupling. The encoder outputs and phase back-EMF’s were recorded simultaneously, during the experiment, with sampling time T s ¼ 100 ls and data length of 30 000 points for each signal. The rotor position and rotor speed were then calculated off-line. Fig. 4 shows the rotor speed over ten turns of the motor, where the cogging-torque effect is noticeable as high-frequency oscillations. Fig. 5a shows the back-EMF for the first phase, measured from line to neutral. Fig. 6 shows the minimum leastsquares error J min versus the number of used harmonics k in the linear model (16) and (17), when measured back-EMF volt-
Table 1 Motor/mechanical parameters Value
Unit
Rs Ls JM JRE JC JTot
5 1 97.0 5.0 2.0 104.0
X mH g cm2 g cm2 g cm2 g cm2
Rotor Speed [rad/s]
Parameter
60 40 20 0 0
720
1440 2160 2880 Rotor Position [deg] Fig. 4. Rotor speed.
3600
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A. Kapun et al. / Simulation Modelling Practice and Theory 16 (2008) 1254–1265
1.5 Back EMF [V]
1
a)
0.5 0 -0.5
b)
-1 -1.5
0
60
120 180 240 Rotor Position [deg]
300
360
Fig. 5. Back-EMF in first phase e1 for one turn (a), and difference e1 ^ e1 (b).
70 60
J
min
50 40 30 20 10 0
1
2
3 4 5 6 7 Number of harmonics - K
8
9
Fig. 6. Minimum least-squares error for ^ e1 .
age is approximated by LSE. In practice, this kind of diagram is very useful for determining the minimal number of relevant harmonics, which gives the simplest signal model, which adequately describes the given data signal. In this determination scheme the number of harmonics is increased until the minimum least-squares error decreased only slightly with further increase in the number of harmonics. It is evident from Fig. 6, that the first, third and fifth harmonics are the only relevant ones. This result is in accordance with the fact that back-EMF waveforms usually posses half-wave symmetry, that is f ðxÞ ¼ f ðx þ T=2Þ, where T is the waveform period. It is known from Fourier analysis [2] that, for half-wave symmetry functions, the even harmonics equal zero. The estimated back-EMF coefficients for each phase, scaled by 106 , are listed in Table 2. The estimated parameters are given with their mean values and standard deviations, which provides information about reproducibility of the results. The standard deviation r of data vector x is defined as:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u m u 1 X r¼t ðxj xÞ2 m 1 j¼1
ð23Þ
where x is the mean value of the elements in vector x and m is the number of elements. In the estimation process 10 repetitions are included, therefore m ¼ 10. The parameters Lirbk ; where i ¼ 1 : 3 and k ¼ 1 : 5; are equal to zero. It is clear from the results for the individual phase, that phase non-symmetry is practically negligible. Fig. 5b shows the difference between measured back-EMF and the approximated version of back-EMF with LSE when the first, third and, fifth harmonics are taken into account.
Table 2 Estimated back EMF coefficients Parameter
b L ira1 b L ira2 b L ira3 b L ira4 b L ira5 Jimin
106 106 106 106 106
Phase 1
Phase 2
Phase 3
x
r
x
r
x
r
5817 3 57 0 28 1.96
0.7 0.5 0.0 0.0 0.0 0.94
5802 3 57 0 28 1.97
4.9 0.3 0.5 0.0 0.0 0.99
5835 2 56 0 28 2.05
0. 5 0.4 0.0 0.0 0.0 0.84
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Motor manufactures usually refer to the voltage constant K e , when describing their motors back-EMF’s. The voltage constant specify the back-EMF peak or RMS (Root Mean Square) value per unit rotor speed. The back-EMF in the first phase for the BLDC motor used in the experiment equals:
^e1 ¼ ir xpðb L 1ra1 sinðphÞ 3b L 1ra3 sinð3phÞ 5b L 1ra5 sinð5phÞÞ From the back-EMF peak value
^ep1
¼^ e1 ðph ¼
pÞ 2
¼
K pe
x, the peak voltage constant
p 3p 5p s ¼ 35:1 mV 3b L 3ra1 sin K pe ¼ ir pb L 1ra1 sin 5b L 5ra1 sin rad 2 2 2
ð24Þ K pe
can be calculated as:
ð25Þ
The back-EMF RMS value is defined as:
^erms 1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 T 2 ^e dt ¼ K rms ¼ e x T 0 1
ð26Þ
p being the period of back-EMF waveform. From (24) and (26), the RMS where K rms is the RMS voltage constant and T ¼ p2x e voltage constant is calculated as:
ir p K rms ¼ pffiffiffi e 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s b L 21ra1 þ 25b L 21ra5 ¼ 24:7 mV L 21ra1 þ 9b rad
ð27Þ
4.3. Torque disturbance estimation A field-oriented control scheme with a current-regulated pulse width modulated (PWM) inverter was applied to control the motor torque T m , as suggested in [18]. The stator current is entirely q-axis current and is equivalent to a torque comref b mand, where isq ¼ 2T ref m =ð3p L 1ra1 Þ. In the experiment, which was repeated 40 times, the control algorithm was executed with sampling time of T s ¼ 100 ls, and the reference torque was set to 4.3 mNm. Using the measurement unit, the phase currents and the rotary encoder outputs were recorded simultaneously, with the same sampling time and data length of 25 000 points for each signal. From the experimental data, the torque disturbance signal was estimated in off-line mode, as suggested in Section 3.4. Although in the estimation algorithm all data points were taken into account, not all data points are plotted in order to improve the figures readings. The calculated motor torque and measured rotor speed are shown on Figs. 7 and 8. The cogging-torque effect is noticeable as speed oscillation with frequency equal to 6p ¼ 36 per shaft turn, Fig. 8b. It is clear from the rotor speed diagram shown on Fig. 8a, that the motor and/or the whole experimental set-up system are imperfectly balanced, therefore additional speed oscillations appear with frequencies equal to 1 per shaft turn. It is evident form Fig. 9 that the disturbance observer compen^ j < 2 103 rad/s). Therefore the compensator output sator ensures convergence of the speed error towards to zero (jx x dist b shown of Fig. 10. equals to the estimated torque disturbance T 4.4. Cogging-torque harmonics, viscous friction coefficient, and Coulomb friction coefficient estimation b dist is In order to incorporate the unbalanced mechanical system effect into the estimation process, the estimated signal T redefined as:
p 2 Tb dist ¼ ir 2
K0 X
b signðx bx ^ þC ^ Þ þ Tb a1 cosð^hÞ þ Tb b1 sinð^hÞ ðkb L rrak sinðkp^hÞ kb L rrbk cosðkp^hÞÞ þ B
ð28Þ
k¼6
Calculated Torque [mNm]
b a1 and T b b1 are the Fourier coefficients of the exogenous mechanical disturbances. According to the redefinition of the where T dist b signal T , the observation matrix and parameter vector in (21) and (22) are also suitably redefined.
6 4 2 0
0
720
1440 2160 2880 Rotor Position [deg]
Fig. 7. Calculated mutual torque.
3600
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Rotor Speed [rad/s]
a 60 40 20 0
0
720
b
1440 2160 2880 Rotor Position [deg]
3600
62 60 58 2880
3000
3120
3240
Fig. 8. Rotor speed (a), and rotor speed oscillations in detail (b).
3
Speed Error [rad/s]
2
x 10
1 0 1 2
0
60
120 180 240 Rotor Position [deg]
300
360
Estimated Disturbance [mNm]
^. Fig. 9. Compensator input: speed difference x x
15 10 5 0 5
0
60
120 180 240 300 Rotor Position [deg]
360
Fig. 10. Compensator output: estimated disturbance signal T^ dist .
All 25 000 experimental data points were included in the least-squares approximation method. Fig. 11a shows the minb dist is imum least-squares error versus the number of cogging-torque harmonics, when the estimated disturbance signal T approximated using LSE. It is clear from the diagram, that the sixth cogging-torque harmonic is highly dominant, just as predicted in Section 2. A small amount of third harmonics is also present besides the sixth harmonic. Fig. 12 shows the estimated cogging-torque over one motor rotation. The estimated mechanical parameters scaled by 106 , are listed in Table 3, in the order of their mean values and standard deviations, by considering all 40 repetitions of the experiment. The high standard deviation indicates a high variation in the estimated Coulomb friction coefficient results. Therefore, the estimated Coulomb friction coefficient result value is questionable, and should be understood as a rough estimation. Friction is hard
a
0.6
Jmin
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0.4
1263
0.2
b
Estimated Disturbance [mNm]
0
1
2
3
4 5 6 7 8 9 10 11 12 Number of harmonics - K
15 10 5 0 -5
2900
2950
3000
3050
Rotor Position [deg]
Cogging Torque [mNm]
b dist (a), estimated disturbance signal T b dist (solid line), and his approximated version using LSE (dotted line) in Fig. 11. Minimum least-squares error for T detail (b).
8 4 0 -4 -8
0
60
120 180 240 Rotor Position [deg]
300
360
Fig. 12. Estimated cogging-torque.
to determine due to the many considerations involved, such as temperature, materials of the bodies, lubrication, manufacture precision, coupling, etc. An additional difficulty when determining the Coulomb friction is the large correlation between Coulomb friction coefficient and viscous friction coefficient, which is also reported in [1]. Perhaps improvement in Coulomb friction identification could have been achieved if the torque ramp had been used as a reference, as proposed in [11]. However, further analysis was not pursued, since the main research focus was on identifying the cogging-torque and viscous friction. Usually, information about the maximum cogging-torque can be found in motor technical data. The cogging-torque for the BLDC motor under the test is estimated as:
p 2 Tb cogg ¼ ir ð3b L rra3 sinð3p^hÞ þ 3b L rrb3 cosð3p^hÞ 6b L rra3 sinð6p^hÞ þ 6b L rrb3 cosð6p^hÞÞ 2
ð29Þ
An infinite number of minimums and maximums exist, because the function (29) is a periodical one. However, using the Matlab numerical toolbox, the first maximum for absolute value of (29) at the interval ð0; 1Þ is discovered at h ¼ 0:1483 rad. The absolute value of (29) at maximum is 7.01 mNm. According to the BLDC motor manufacture the maximum cogging-torque is 6.5 mNm, which means that the cogging-torque is overestimated by approximately 8%. 4.5. Results verification A simulation of the experiment was performed in order to verify the results of the proposed parameters identification method, where the experimental data for phase currents was used as motor excitation. The parameters from Tables 2 and 3 were used as simulation parameters, with the exception of the Coulomb friction coefficient which was reduced by 20%.
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Table 3 Estimation results for the mechanical parameters Parameter
x
r
b L rra3 106 b L rrb3 106 b L rra6 106 b L rrb6 106 b a1 106 T b b1 106 T b 106 B b 106 C
94.3 52.7 270.6 214.0 741.1 23.4 52.5 1320.0 26.8
2.3 3.0 9.4 11.9 17.5 8.5 10.6 343.8 5.5
Jmin 103
60 40 20
Rotor Speed [rad/s]
0
0
0.5
1 1.5 Time [s]
b 62
60
58
1.05
1.1
Time [s]
1.15
Rotor Speed [rad/s]
Rotor Speed [rad/s]
a
2
2.5
c
62
60 58 2880
3000
3120
3240
Rotor Position [deg]
Fig. 13. Measured (black-line) and estimated (grey-line) rotor speeds (a), rotor speed oscillation detail plotted against time (b) and rotor speed oscillation detail plotted against measured and estimated rotor position (c).
A comparison between measured and estimated rotor speeds was formed in order to highlight the verification result, see Fig. 13. The term ‘‘estimated” refers to rotor speed obtained from the simulation using the estimated parameters. It is evident from Fig. 13a, that the simulation data are in good accordance with the experimental data, especially within the steady state region. The experimental and estimated cogging-torque effects, noticeable as speed oscillation with frequency of 36 per shaft turn, are shown in Figs. 13b and 13c. A phase displacement of approximately half of the period between experimental and estimated cogging-torque is evident from Fig. 13b, where speed oscillations are plotted against time. This however should not be misunderstood, since any speed difference between measured and estimated rotor speed accumulates in its rotor position difference. The cogging-torque is a function of rotor position, therefore, the comparison between experimental and estimated cogging shown in Fig. 13c, is more pertinent. Matching of the simulated and experimental data is good enough to justify the applicability of the proposed identification method in the BLDC simulation process. 5. Conclusion This paper presents an off-line method for identifying some of the BLDC motor’s dynamic model parameters. The method is founded on the least-squares approximation method and a closed-loop disturbance observer, and is conditional on the availability of experimental data for back-EMF, phase currents, rotor position, and rotor speed. Using the proposed identification method, it is possible to identify back-EMF harmonic coefficients and mechanical parameters, where the mechanical parameters refer to cogging-torque harmonics and viscous friction coefficient. Normally the FFT analysis is performed on no-load back-EMF to find the dominant harmonics. The FFT analysis, however, can be performed only if rotor speed is constant during the period of back-EMF. Constant rotor speed is difficult to ensure, if the BLDC motor possesses high cogging-torque. In the proposed identification method, experimental data for rotor position and rotor speed are incorporated into the identification process, therefore, non-constant rotor speed has no influence on back-EMF coefficient identification performances. Using the torque-sensor seems to be the logical choice for determining
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mechanical parameters, but the torque-sensor’s application results in extra costs. Furthermore, high torque-sensor mounting demands, especially for sensors with low rated torque (<100 mNm), could be the source of additional difficulties in a nonlaboratory environment. Thus this identification method’s ability when determining mechanical parameters without torquesensor is of additional benefit. Of course, the reliability of those mechanical parameters obtained using torque-sensor is higher. However, the experimental result shows, that the proposed method sufficiently identifies the mechanical parameters. Therefore, this method enables an improvement in the field of dynamic simulations for systems using the BLDC motor. References [1] A.J. Blauch, M. Bodson, J. Chiasson, High-speed parameter estimation of stepper motors, IEEE Transactions on Control System Technology 1 (4) (1993) 270–279. [2] I.N. Bronstein, K.A. Semendjajew, G. Musiol, H. Muhlig, Matematicˇni Prirocˇnik (Tehniška zalozˇba Slovenije, Slovenia, 1997) (in Slovene). [3] K.Y. Cheng, Y.Y. Tzou, Design of a sensorless commutation IC for BLDC Motors, IEEE Transactions on Power Electronics 18 (6) (2003) 1365–1375. [4] J.H. Choi, S.H. You, J. Hur, H.G. Sung, The design and fabrication of BLDC motor and drive for 42V automotive applications, in: IEEE International Symposium on Industrial Electronics 2007 (ISIE’07), 2007, pp. 1086–1091. [5] D. Dolinar, G. Štumberger, Modeliranje in Vodenje Elektromehanskih Sistemov (Univerzitetna knjizˇnica Maribor, Slovenia, 2002) (in Slovene). [6] D. Hercog, K. Jezernik, Rapid control prototyping using MATLAB/Simulink and a DSP-based motor controller, International Journal of Engineering Education 21 (4) (2005) 596–605. [7] J. Holtz, L. Springob, Identification and compensation of torque ripple in high-precision permanent magnet motor drives, IEEE Transaction on Industrial Electronics 43 (2) (1996) 309–320. [8] W. Hong, W. Lee, B.K. Lee, Dynamic simulation of brushless DC motor drives considering phase commutation for automotive applications, in: IEEE International Electric Machines & Drives Conference (IEMDC’07), 2007, pp.1377–1383. [9] J.Y. Hung, Z. Ding, Design of currents to reduce torque ripple in brushless permanent magnet motors, IEE Proceedings-B 140 (4) (1993) 260–266. [10] S.M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice-Hall International Inc., 1993. [11] R. Kelly, J. Llamas, R. Campa, A measurement procedure for viscous and Coulomb friction, IEEE Transaction on Instrumentation and Measurement 49 (4) (2000) 857–861. [12] F. Khorrami, P. Krishnamurthy, H. Melkote, Modeling and Adaptive Nonlinear Control of Electric Motors, Springer-Verlag, New York, 2003. [13] D. Iles-Klumpner, I. Serban, M. Risticevic, I. Boldea, Comprehensive experimental analysis of the IPMSM for automotive applications, in: International Power Electronics and Motion Control Conference 2006 (EPE-PEMC’06), 2006, pp.1776–1783. [14] Y. Koubaa, Asynchronous machine parameters estimation using recursive method, Simulation Modelling Practice and Theory 14 (2006) 1010–1021. [15] L. Liu, D.A. Cartes, W. Liu, Particleswarm optimization based parameter identification applied to PMSM, in: Proceedings of the 2007 American Control Conference, 2007, pp. 2955–2960. [16] H. Lu, L. Zhang, W. Qu, A new torque control method for torque ripple minimization of BLDC motors with un-ideal back EMF, IEEE Transactions on Power Electronics 23 (2) (2008) 950–958. [17] H. Melkote, F. Khorrami, S. Jainm, M.S. Mattice, Robust adaptive control of variable reluctance stepper motors, IEEE Transaction on Control Systems Technology 7 (2) (1999) 415–425. [18] D.W. Novotny, T.A. Lipo, Vector Control and Dynamics of AC Drives, Clarendon Press, Oxford, 1996. [19] Ravindra Pantankar, Liangtao Zhu, Real-time multiple parameter estimation for voltage controlled brushless DC motor actuators, in: Proceedings of the 2004 American Control Conference, vol. 4, 2004, pp. 3851–3856. [20] J. Shao, An improved microcontroller-based sensorless brushless DC (BLDC) motor drive for automotive applications, IEEE Transactions on Industry Applications 42 (5) (2006) 1216–1221. [21] T. Tutunji, M. Molhim, E. Turki, Mechatronic system identification using an impulse response recursive algorithm, Simulation Modelling Practice and Theory 15 (2007) 970–988.