Experimental results of EKF

To implement the EKF algorithm, the TI DSP TMS320LF2407A has been used at its maximum speed, 40MHz operating frequency. It has a fast D/A conversion time up to 500ns and thus helps reduce sampling time for the motor phase currents. For monitoring those variables in the software, an RS232 communication is established. The sampling rate for those variables is nearly 1 KHz and they can be presented in wave forms as shown in Fig.

6.20.

Fig. 6.19 Host communication software.

360∘electrical

Fig. 6.20 Actual rotor position.

360∘electrical

Fig. 6.21 Estimated rotor position.

Fig. 6.21 shows the actual rotor position at quadrant current command (i_qref) 0.4A. The EKF subroutine is executed every 200us to estimate rotor positions and speeds. The motor still uses the encoder feedback instead because we need to tune the covariant matrices P, Q and R. After the estimated parameters matches the actual ones then we can use the estimated

θ

e and

ω

_e instead of the encoder feedback.

-360^。electrical

Fig. 6.22 Error between actual rotor position and estimated one.

Tuning those covariance matrices is important and not a easy work for EKF in this experiment. It affects estimated motor speed and rotor position greatly. It needs a trial-and-error procedure to get a tradeoff between stability and convergence time. After the estimated rotor positions have been derived, it will be used to calculate the value of sine and cosine for coordinate transformations as shown in Fig. 6.26. Fig. 6.26 shows the error between actual sin(

θ

) and estimated one. The maximum error is about 0.2 due to the estimated error of rotor position. In the trial-and-error procedure, the covariant matrix P0

seems not to affect the estimated parameters too much but the covariance matrices Q and R do.

Fig. 6.27 and 6.28 show the different responses due to different Q (3, 3). In the experiment, the estimated

ω

_e sometimes has the opposite direction compared to the actual one. This is due to the inherent ambiguity of the non-linear model. -

ω

_e and +

ω

_e give the same results.

Fig. 6.23 Enlarged wave form of Fig. 6.22.

10^。

_electrical

-20^。

_electrical

Actual sine value 1 Estimated sine value

Fig. 6.24 Actual sin(

θ

). Fig. 6.25 Estimated sin(

θ

). 1

-1 -1

0.2 RPM

1000

-1000

Fig. 6.27 Estimated speed

ω

_e=550 RPM.

Fig. 6.26 Error between actual sin(

θ

)

and estimated one. Q (3, 3) =0.06.

Fig. 6.28 Estimated speed

ω

_e=550 RPM Q (3, 3) =2.

RPM

1000

-1000

0.4A 0.4A 0.2A 0.2A

-0.2A -0.2A

-0.4A -0.4A

0.3A RMS 0.3A RMS

Fig. 6.30 Estimated

I

_α. Fig. 6.29 Actual

I

_α.

180^。electrical

0.2 A

-0.2 A

-180^。electrical

Fig. 6.32 Theta error after

ω

_e direction has been corrected.

Fig. 6.31 Actual

I

_α −estimatedone

.

To overcome the wrong estimated speed direction, a strategy has been adopted as shown in (6.1).

If (

_|−1

−

_−1|−2

) ×

_|−1

< 0

k k k

k k

k e e

e

θ ω

θ

, then _|−1

= −

_|−1

k k k

k e

e

ω

ω

and

θ

_|−1

= θ

_|−1

− π

k k k

k e

e . (6.1)

Fig.6.32 shows the result using (6.1), the estimated rotor position is always positive and negative 180 degrees alternately shifted from the actual one. This problem is still under researching. Currently, the EKF is executed every 200us and the current loop is 100us.

According to the covariance matrices of the simulation results, Q12 format is chosen for real

time implement using a fixed-point DSP TMS320LF2407A. The sampling time interval is set to be 0.0002 second and that is nearly the resolution for Q12 format. The covariance matrices of the DSP software is

⎥⎥

Q15 format is preferred to increase the calculation precision, and care must be taken for the dynamic range of the covariance matrices and saturation problems.

Chapter 7

Conclusions

In this research, we find it is feasible to implement real-time EKF estimation of the rotor position and speed of a sinusoidal PMSM by using a low-cost single-chip DSP controller. The DSP can afford enough power for the matrices calculation of the EKF; it takes less than 100us cycle time by the software. The drive does not use the voltage transducers for the motor phase voltages and thus simplifies the drive and reduces the cost, so the drive is the same with the ordinary servo drives in the market unlike the other methods need the voltage transducers. This means using EKF is competitive in the cost point of view.

Due to the nonlinearity of the state equations, the estimated speed could be in the wrong direction and solving it is kind of tricky. Another consideration is that the covariance matrices could be in a value of big variation, which means we could not use the same Q format throughout all the calculations and this makes the software more complicated. The experiment results show that the maximum estimation error of the rotor position is -20 electrical degrees in steady state which differs from the results of the simulation. In the simulation the motor parameters are the same in the motor block and EKF, and all the calculations are based on the floating point method, but there are parameter errors between the actual motor and software, sampling errors, quantization errors and the characteristic of those covariance matrices P, Q and R are not well known in the experiment. In the future, the PWM will be used instead of the RMS value for the motor three-phase voltages and fixed-point method will be used instead of the floating-point method in the simulation to present the actual circumstances as similar as possible in order to understand the differences which can be a reference to improve the sensorless drive.

Appendix

The technical data for the motor in the experiment are as below:

Rated power: 90W/3000 RPM

Pole number: 4

Stator phase resistance: 3.4 Ω Stator inductance

L

_d: 9mH Stator inductance

L

_q: 12mH

Flux linkage: 0.11327 Wb-T

References

[1] Silverio Bolognani, Roberto Oboe, and Mauro Zigliotto, “Sensorless full-digital PMSM drive with EKF estimation of speed and rotor position,” IEEE Trans. Ind.

Electron., vol. 46, pp. 184-191. Feb. 1999.

[2] M. A. Jabbar, M. A. Hoque, and M. A. Rahman, “Sensorless permanent magnet synchronous motor drives,” IEEE CCECE’97, 1997, pp. 878-883.

[3] Kazutaka Tatematsu, Daisuke Hamada, and Kenji Uchida, “Sensorless control for permanent magnet synchronous motor with reduced order observer,” in Proc. of

The 29th IEEE Power Electronics Specialists Conf. (PESC'98), 1998, pp. 125-131.

[4] “Sensorless control with Kalman filter on TMS320 fixed-point DSP (Literature Number: BPRA057),” Texas Instruments Europe, 1997.

[5] Yuan-Rui Chen, Norbert C. Cheung, and Jie Wu, “Sensorless drive of permanent magnet linear motors using modified Kalman filter,” IEEE Powel Electronics

Specialist Conf., PESC'2001, 2001, pp.17-21.

[6] Mohand A. Ouhrouche, “EKF-Based estimation of rotor flux, speed and rotor resistance in cage induction motor sensorless drive,” Proceeding of the IASTED

International Conf. Modeling and Simulation (MS’2000), May 15-17, 2000 -

Pittsburgh, Pennsylvania, USA.

[7] Bozo Terzic and Martin Jadric, “Design and implementation of the extended Kalman filter for the speed and rotor position estimation of brushless DC motor,”

IEEE Trans. Ind. Electron., vol. 48, no. 6, pp. 1065-1073

, December 2001.

[8] A. Bado, S. Bolognani, and M. Zigliotto, “Effective estimation of speed and rotor position of a PM synchronous motor drive by a Kalman filtering technique,” Proc.

IEEE-PESC, vol. 2, pp. 951-957, 1992.

[9] Peter Vas, Sensorless Vector and Direct Torque Control, Oxford, New York, 1998.

[10] Chee-Mun Ong, Dynamic Simulation of Electric Machinery Using

Matlab/Simulink, Prentice Hall, New Jersey, 1998.

[11] Power System Blockset User’s Guide, TEQSIM International Inc., 2000.

[12] Yasuhiko Dote and Sakan Kinoshita, Brushless Servomotors Fundamentals and

Applications, Clarendon Press. Oxford, 1990.

[13] Duane C. Hanselman, Brushless Permanent-Magnet Motor Design, McGraw-Hill

Inc., 1994.

[14] Peter Vas, Parameter Estimation, Condition Monitoring, and Diagnosis of

Electrical Machines, Clarendon Press. Oxford, 1993.

Author Resume

姓名: 葛育中 出生: 1972/2/22

求學歷程: 逢甲大學 自動控制工程

台中一中

經歷: 現任職於愛德利科技股份有限公司(1996 至今)

主要作品: 電動機車無刷馬達控制器

工業用無刷馬達控制器D305 系列

92

碩 士 論 文

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