A Rotor Speed and Load Torque Observer for PMSMBased on Extended Kalman Filter

Zheng Zedong 1,2, Li Yongdong2, Maurice Fadel ', Xiao Xi21. Laboratoire d'Electrotechnique et d'Electronique Industrielle, LEEI UMR-CNRS 5828, Toulouse, France

2. Department of Electrical Engineering, Tsinghua University, Beijing, ChinaE-mail: zhenggleei.enseeiht.fr

Abstract- This paper proposed a state observer based on rotor position, speed and load torque as the state variables isextended kalman filter (EKF) for the rotor speed and load torque considered to be a good solution for these problems.observation of permanent magnet synchronous motor (PMSM). Some state observers have been proposed in reference [4]This observer can be used to estimate the precise rotor position

and since thers he en isopsed direcel[4]and rotor speed in the servo systems with only optical encoder of and [5]. Since the output of the encoder is used directly, thelimited accuracy. By using the observed rotor speed and position low speed performance is limited. The observer in [6] needs toinstead of the values got directly from the encoder, the speed and measure rotor speed accurately and quickly. Observers basedposition control performances can be greatly improved. The on Kalman filter are proposed in [7] and [8]. The observer in [7]observer can also give the load torque which can be used as thefeedforward compensation of the speed PI controller. The hsolrge amount ocaulti tbesus in practice,compensation can greatly improve the speed control performance Reference [8] doesn't give the results of load torqueduring load torque change. The extended Kalman filter is used to observation and feedforward compensation. A newly observersolve the nonlinear equation of the observer, considering the based on extended Kalman filter (EKF) is proposed in thisinfluence of the parameter errors and the noise in measurement. paper. Compared with the Kalman filter and state observer, theSo it is very robust to the parameter errors.

EKF can solve nonlinear equation directly. The Kalman filterKey words- Load torque, state observer, extended Kalman considers the influence of the errors of the system parameters

filter, optical encoder, PMSM and the disturbance in the measurements of currents and rotorposition, so the EKF observer is robust to the system

I. INTRODUCTION parameters' errors, and can eliminate the noises and errorscaused by the quantization of the rotor position in the optical

The permanent magnet synchronous motor (PMSM) is encoder. The feedback gain matrix in EKF is selected based onbecoming more and more popular in servo systems because of the optimal control theory, so it can make the observer bothits high power density, large torque to inertia ratio and high stable and convergent quickly.efficiency. In order to obtain high speed and position control Some simulation and experiment results ofthe EKF observerperformance, a high accuracy position sensor is usually needed. are given in this paper, which show that the observer can getBut in many fields, this kind of rotor position sensor is too more precise rotor position and speed than the conventionalexpensive to use. Some control systems with low-precision method, also the compensation using the observed load toquepositon sensors have been proposed in reference [1] and [2], can improve the speed performance greatly. It is proved thatbut the performances cannot meet the necessary of the servo the control system with the EKF observer is very suitable andsystem. Usually an optical encoder is used to measure the rotor effective for the servo system.position by giving a fixed number of pulses in each rotation.The rotor position can be obtained by counting the number of II. MODEL OF THE OBSERVERthe encoder pulses during the detection time, then the speedcan be got by the differential of the position. Since the output The mecanical equation ofthe PMSM is as follows:of the encoder is quantization pulse, in the low speed region J-d -f_Q-Tthe pulse numbers got from the encoder in one detection period - L

become few, the accuracy of the speed estimation is limited, dOsometimes very large errors and differential noises are dt (1)generated. Some references have proposed some filters to get Where 2 is the mechanical angle speed, f is the coefficient ofoff the noise with little phase lag, but it is too complex to use friction, J is the inertia, Tern is the electromagnetic torque, TL iS

inpatc.[3]in practice. load torque, 0 is the mechanical rotor position.In the servo systems, the variation of the load torque can In theqnon-salientePMSM,aincthrotor fluioiete

' I~~~~~~~~~~nthei- nnn-,salient PMSM in thei- rotor fluix orientedcreate some ripples in the rotor speed. In order to make the coriae (d, axs,teeetoantctru'sgvnbspeed stable during load torque changes, the information of the euto 2load torque is needed to compensate the reference torque. As eqain()thoqesensor is too exesv,a state observer with the Tem =PuI 2

1-4244-0726-5/06/$20.OO '2006 IEEE 233

Where p is the number of the poles pairs, Td is the flux of d- III. SIMULATION RESULTSaxis, Iq is the current in q-axis. Since we know nothing about In order to test the effects of the proposed observer, somethe variation of the load torque, we can not get the equation of simulations have been done. In the simulation, the controlthe load. But we can consider that the load torque changes very period is 50jts, the parameters ofthe motor are as follows:slowly compared with the rotor speed and position, that is to Rotor resistance R=0.0176Q, L=0.000684H, the backassume: electromotive force coefficient is 0.0539766 v/(rad/s), inertia is

dTL Idt = 0 (3) 0.0025kg*m2, poles pairs p=4, the friction coeffiecient f isBased on the former analyses, we can get the state observer 0.0609N*m/(rad/s). The encoder gives 2500 pulses per

for the rotor speed and load torque.['] rotation.x = x + B u + w The rotor position got from the encoder and the real position

(4) when the rotor speed is 0.521 rad/s are shown in Fig. 2. Fromy=C*x+v the results we can see that the number of pulses in one

-f l detection period is few in low speed range, so we can not get-- 0 -- PWdthe precious speed from the optical encoder in every period. If

F= 1 0 0 B 0 we calculate the rotor speed every 10 periods by the,

0 0 0differential of the rotor position got from the encoder, the

0 0 0° j L ° 2 estimated speed and real speed are shown in Fig.3. The EKF(5) observer can observe the rotor speed in every sampling period.

C=[o 1 0] The observed speed is shown in Fig. 4.Q'Q Comparing Fig.3 with Fig.4, we can see that the speed

obtained by the differential of the rotor position contains a lotX = 0 ,u= Iq of noise, if we use a low-pass filter to eliminate the noise, a

VTL, "phase lag will be produced. But the speed obtained by the EKFHere w and v are the covariance matrixes of the random observer can match the real speed very well.

disturbances. In fact w is the process noise which stands for theerror of the parameters; v is the measurement noise which position got by encoder lstands for the errors in the measurement and sample. 0.185 - the real position L-The noise covariance matrixes are defined as follows: 0<

Q=cov(w)= E{ww } (6.a) .5 FT 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~R = cov(v) = E{vv } (6.b) - 0.175 - - -

The system proposed with EKF observer is shown in Fig. 1where the observed rotor speed, position are used instead of the 0 - -values got directly from optical encoder, and the observed load I

0.165 - - - - ---

torque is used to compensate the reference electromagnetictorque at the output of the speed PI regulator. Some 0 --_simulations and experiments based on this system have been 0.16 0.165 0.17 0.175done. time(s)

Fig.2. Position got from encoder and real positonDC Bus

Te i =|(Te*,) C 100dCurrent ~~~~~~~~~~~~~~~~~~~~~I I I I IP.inverter I0-

. TS S 0 ~~~~~~~~~~~~~~~~~~~~~~~0.050.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

|~~~~~~Control |||10tm(

+ | H_ ^ T \ / X E o < . L L LI =I I

+ 'q f2 ~~~~~~~~~~~~~~~~60---

d- 40(i- oLt LLLL therealspeed -

)po 2 L L L L L L L LL0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Speed time(s)

Fig. 1. The flow-chart ofthe system with state observer Fig.3. Speed got by differential and real speed

234

100 I II I I I I I

z 3 load torque estimated by EKFI

80 - - - - - - - - - - - - - - - - _78 I I I3

a i I I 0~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~2---IIr6 60 speed got by kalman filter 0 I ° I I I XaD I- I- I

40 - I I I

2 20 4-4 . .-+I- -4 - I I L- --I ---I

0 I I I I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80 time(s)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

time(s) 410 r real load torque r_80 --- I- -

.2

60 - Ithe real speed]- i_ i

D I I I -1---I---6 20 44-- -I- - - -4- - -1-_-_-_-_-- - I 1 1

II I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80 4 I l time(s)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

time(s) Fig.6. Observed load torque during load torque impact

Fig.4. Speed got by observer and real speed the rotor speed during load torque impact

Besides the rotor speed, the EKF observer can also observe 80 -- _-7-- V--V--,--the load torque which will have a very great influence to the 70 -- - - --L ttspeed control performance. When the load torque is zero, the 60 -

observed results of the load torque during the start-up of the 50 _ _ _ _ _ _ _ _ _ _motor are shown in Fig.5. At 0.5s, there is a load torque 4L-change, the observed load torque is shown in Fig. 6. We can m Isee that the observer can observe the load torque very precisely. 2 30 - rEven in the quick dynamic process, the observed torque can

track the real torque well. 10Once the load torque is observed by the EKF, a feed- o0-- -- - Vr--r- T T

forward compensation can be built. The observed load toque is -10 0 2 0 0 0 0 0 0 00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1used to compensate the reference torque at the output of the time(s)

speed PI regulator. This compensation can greatly improve the Fig.7. Speed during load sudden change without feedforward compensationspeed control performance, especially when the load torque the rotor speed during load torque impactvaries very quickly. 90l

If there is a load torque sudden change, the motor speeds 80 ----________ ________with and without the load torque feedforward compensation are 70 fshown in Fig. 7 and Fig. 8.When the load torque compensation is added, the speed drop

is less than the system without load torque compensation. a0Because of the lag between the observed toque and real one, /Dthe compensation cannot remove all the speed drop during the 2 lload change. 30 T- -- - T - - - -- - -

1 ~~~~~~~~~~~~~~~~~~~~20|------ ----- - -- - ------ --- -- --

E 0 5 - torque estimated by EKF 10

.2 C--o I I ll 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9- time(s)

I-0S r X E XI IFig.8. Speed during load sudden change with feedforward compensation

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4time(s) In order to test the robust characters of the EKF observer to

realtorquethe parameters errors, some simulations have been done when

+E 0.5 --- t - - - e - - - - - - - - - - - - | - 1 real torque1the friction coefficient and inertia are not precise. When thereoo is some errors in the coefficient of friction, the part of the

I0.5I___|___l____________l___l___ _ Ifriction torque caused by the parameter error will be included< ~~~~~~~~~~~~~inthe observed load torque.

01 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Fig. 9 shows the observed load torque when the frictiontime(s) coefficient used in the EKF observer is only lO0% of the real

Fig.5.Observed load torque during start-up one. At 0.5s there is a load torque impact of 3Nm.

235

If we consider the friction torque as a kind of load torque, the rotor speed during load torque impactthe load torque is as follows: 90

T=f Q+TL (7.a) 80

The estimated load torque is: 70 ______

T =f Q+TL (7.b) 60 ____ __ I__ 11 -_I___ _

Then the observed load toque is the same with the real one _ 50-even if the friction coefficient is not precise. The simulation a Iresults are shown in Fig. 10. The speed control performance is I I I IV I I Ishown in Fig. 11 with the feedforward compensation.It can be 2 30 - --------- ------ --------- ---

seen that when the friction coefficient is not correct, the 2 | I- I- I I-20observed load torque contains parts of the friction torque,

which cause some overshoot in the speed control in dynamicprocess.When the inertia parameter is 50°0 of the actual one, the time(s)

observed load torque is shown in Fig. 12. It can be seen that the Fig.1 1. Speed control performance when the friction coefficient is 10% of theerror of the inertia parameter makes the observed load torque actual onecontain some errors in the dynamic process, but in the steadytime, the observed load torque agrees with the real one very the inertia parameter is onle 50% of the actual valuewell. The speed during load torque impact using the observed 3.5

torque as the feedforward compensation is shown in Fig. 13. torque estimated by EKF~~~~~~~~~~~~~~~~~~~~~~~~~~~3- -e..real load torque

The error of the inertia make the dynamic performance worse, 3 Ilbut it has no influence in the steady state. 2.5 -

I 'I I '|~~I I I I

4.5 | l e 05

-______________ 0IF tque estimated EK1 k I II I I II I I

3t. 5-I I- I I-

E~~ 3.- I -I t

2.5 -_ -t - i- i- t0 -1 -

0.5 I I I 0.5 time(s)

7 0 0. -.-.-.7.-.-.-.T. h oo pe uigla oqeipc

ED1.5 4- 4 - - ----7C -

2__W_4__A___-----W_;____ __0_-_-------------e---e--- t - - - t - - - t - ---__

t )I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.5 4 -4-----1 . time(s)

0 -- -Fig.12. Observed load toque when the inertia parameter is 5-0 of the actualI I I III I I I ~~~~~~~~~~~~~~one

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 the rotor speed during load torque impacttime(s) 9C l l l l l l

Fig.9. Observed load torque when the friction coefficient is 1O0o of the actualone

z 4

o o.i 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fi.3 Roo spe duin loa imac whe th ietaprmeris5%otime(s) th acua oneI I

Fig 10 Obevdad ellatorque containing byctoEKtorque r r-

=3~~~ ~ ~ ~ ~ ~ ~ ~ 3

3 -+ - -~I I I I I

1 70 ~ ~ I I I

CD ~~~~~~~~~~~~~~~~~~~0 0.1. .. . . 0. 0.1 .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fg1.Rtrseddrn odipc hnteietaprm tri Oootime(s) the--I--- acua one4 --4

Fig.10.Obsevedandreal latorque cnann rcintru

=3~~~ ~ ~ ~ ~ ~ ~ ~ 3

From the simulations with some parameter errors, it is When a load torque of 2. INm is applied on the rotor, theproved that the observer has a very good robustness to the load torque observed by the EKF observer is shown in Fig. 17.motor parameters' variation. We cannot get the curve of the real load toque because there is

no torque sensor in the platform, but we can compare theIV. EXPERIMENTS RESULTS estimated load torque with the electromagnetic toque which is

shown in Fig. 18. It can be seen that the observed load torqueTo verify the simulation results, some experiments have is almost the same with the electromagnet torque. This can

been done on a motor control platform based on DSP verify the precision of the observer.TMS320F28 12. The motor parameters are as follows: Some experiments can also been donw when there are errorsP=750w, R=3.69Q2, L=13.22mH, J= 0.00943kg*m,2 in the system parameters to detect the robustness of the

1=3 .65A, poles pairs p=4; The optical encoder generates 2500 observer. The observed load torque when the frictionpulses per rotation. The friction coefficient is unknown, but it coefficient parameter is 0.000INm/(rad/s) is shown in Fig. 19.is supposed to be 0.0009 N*m/(rad/s). And the observed load torque when the inertia parameter is

In the experiment system, there are three kinds of speeds. 0.006 kg*m2 is shown in Fig. 20.First is the one got from the differential of the rotor position From the experimental results, we can see that thewhich is the output of the encoder. But in this speed there is a observer can estimate the rotor speed and load torque exactly.lot of noise especially in the low speed region. To remove the The observed speed has a better precision than the speed gotnoise, a first order low pass filter is used. The output of the from the direct differential of encoder pulses numbers. Byfilter is the second kind of speed. The third speed is the using the observed load torque as the feedforwardobserved one by the EKF observer. The experiment results of compensation, the speed control performance can be greatlythese speeds are shown in Fig. 14. It can be seen that the speed improved during load torque impact. The observer can improvegot by the LPF filter has a large lag compared with the the speed control precision both in steady and transient state.observed one. But the observed rotor speed can match the

300-speed got by differential very well even when the latter hasa28lot of noise. 260 rotor speed~]

In experiment, we use the observed rotor speed and position 240.....................

instead of the values got from the encoder directly in the vector20control system. The control system used in the experiment is E- 2800.....the same as the flowchart of Fig. 1. 160

If there is no feedforward compensation, the motor speed1420

during load torque impact is shown in Fig. 15 when the 10

reference speed is 10OHz. The load torque after impact is 2. INm. 80In the system with the observed load torque as the 60

feedforward compensation, the speed during a same load 20torque impact is shown in Fig. 16. Because the frictioncoefficient used in the observer is not the same with the actual 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

one, tere i anoershot simlar t the imulaion rsults Fig. 15. Rotor speed control performance without feedforward compensationshown in Fig. 11. In spite of the overshoot, the speed controlperformance is still better than the situation without thecompensation.30-

1500 __roto_ spee

1400

-1350 p dtbK 71EE

CL25 16

.1300

sp--ied-o~ydferedngtb ifferentia1050

Fig14Rtorsped urig cceertio Fg. 6.potresee cotro prfomacefitffedfowadicmpnsaio

1050~~ ~ ~ ~ ~ ~ ~ ~ 3

2.50 -_ _ _ _

Observed Load Torqe2522.25 .25... -- --- -Observed Load Torque

2.00 ..2.00 -A A~

.1 .75-

.5 .51 .25 .

o 1.25 - .. .-- .C

I- -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~J 1.00 -...

1.00-J 0.75 ..........~ ~~~~~~~~~~~0.7 0.50-

.50 -0.25

.2 --0.00

0.0 -0.25~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~020.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

-0.50 '- - ri itime(s)0.00.51.0 1.5

time(s)30 35 40 45

Fig.20. Observed load torque when the parameterJis 0.006Kg*m2 (6400 of thetime(s) ~~~~~~~~~~~~~~~~actualvalue)Fig. 17. Observed load torque with precise parameters

2.50 _CONCLUSION2.5 electromagnet torqu:e:]__- - - - ~~~~~~~~~~~~~Thispaper proposed a state observer based on extended

2.00 ~~~~~~~~~~~~Kalmanfilter. The observer can eliminate the position error1.75 ~~~~~~~~~~~~causedby the quantization of the optical encoder and obtain1.50 . ~~~~~~~~~~~~~~~~theaccurate rotor speed in the system. It can overcome the$ 1.25 . I I I I ~~~~~~~~~~~~~defectsof low precision and differential noise in the general1.00 ~~~~~~~~~~~~speedcalculation method. The sate observer can also observe

0.75 . . ~~~~~~~~~theload torque and precise rotor position. By using the0 0.50 ~~~~~~~~~~~observedload toque, a feedforward compensation is built

0.25 ~~~ ~ ~ ~ ~ ~~----- which can greatly improve the speed control performance0.00 -------------- --------- when the load toque varies suddenly. Generally, the state-0.25 ~~~~~~~~~~~observerbased on extended Kalman filter proposed in this

-0.50- i' -'- r-'--- r -'-1-'- paper is very suitable for the hihperformance servosytm0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 hihlwpeis oo oi igh systems

time(s)wihlwpeiertrpsto sno.Fig.18. Electromagnetic torque of the motor

REFERENCES2.50-__ _ _

2.25 L ____ Observed Load Torque [1] Todd D. Batzel, and Kwang Y. Lee, "Commutation Torque RippleMinimization for Permanent Magnet Synchronous Machines with Hall

2.00 -----.......... .....-----Effect Position Feedback", IEEE Trans. Energy Coversion, vol. 13,No.3,EZ 1.75 ----- ----pp. 257-261, Sept. 1998.

1.50 . ~~~~~~~~~~~~[2]Shigeo Morimoto, Masayuki Sanada and Yoji Takeda, "High-2- ~~~~~~~~~~~~~~~performance current drive for PMSM and SynRM with only low-

c'1.25 - -----resolution position sensor", IEEE Trans. Industry Applications, vol. 39,co 1.00 ..............No. 3, pp. 792-801, May/June 2003.

_j ~~~~~~~~~~~~~~~~~[3]Seppo J. Ovaska, "Improving the velocity sensing resolution of pulse-~~~~ 0.75 . . ~~~~~~~~~~~~encoders by FIR prediction," IEEE Trans. Instrum. Meas., vol. 40. No. 3,0.50

- ----......--- --------- pp657-658, June 1991.0,

0 0.00 ~~~~~~~~~~~~~~~~estimationfor all-digital, ac servo drives," IEEE Trans. Industry.Applications., vol. 27, No. 4, pp70 1-705, July/August 1991.

-0.25...... ......... [5] Guchuan Zhu, Lousi-A. Dessaint, Ouassima Akhrif, and Azeddine-0.50- - T 1'T r T 1'1 Kaddouri, "Speed Tracking Control of a Permanent-magnet Synchronous

0.0 0.5 1.0 1.5 2.0 2 5 3.0 3.5 4.0 4.5 5.0 Motor with State and Load Torque Observer," IEEE Trans. Industry.time(s) Electronics., vol. 47, No. 2, pp346-355, April. 2000.

[6] L. Gasc, M. Fadel, S. Astier, and L. Calegari, "Load torque observer forFig.19. Observed load torque when the parameterf is 0.OOOlINm/(rad/s) minimising torque ripple in PMSM", in proc. ICEMS 2003, vol.2,

(11.110I% of the value used in former) pp.473-476, 2003.[6] Taeg-Joon Kweon, and Dong-Seok Hyun, "High-performance speed

control of electric machine using low-precision shaft encoder", IEEETrans. Power Elcrois,vl 14, pp. 838-849ASpt 1999.