Sensorless control of PMSM for low speed range

Sensorless control of PMSM for low speed range Konrad Urbański Institute of Control and Information Engineering Poznan University of Technology Poznan, Poland PREPRINT

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Structured Abstract: Purpose of this paper The purpose of this work is to develop the PMSM drive which has possibility to work in sensorless mode at low speed based on back EMFs estimation. Design/methodology/approach Estimation uses modified Luenberger observer and the preprocessing of the EMFs before calculating the speed, using derivative and the Kalman filter to obtain smooth waveform of the estimated speed. This modification is needed because of the nonlinear change of the estimated back EMFs amplitude as a function of speed, in low speed range. Findings How to use back EMF observer to estimate a speed in the low speed range was found in the course of the work. Simple and effective algorithm uses Kalman filter and can work even with a relatively big deformation of the estimated back EMF. Practical implications Such sensorless drive may be used in low-cost constant or variable speed drive in domestic use or industrial application. Such drive may work properly where there is no initial load torque and the sign of the speed does not change. Originality/value Presented results challenge the view that at low speed range (not the standstill), the back EMF based method of position estimation is very difficult or impossible. However, the problem lays in proper speed estimation, not the position estimation. Keywords: PMSM, Drive Control, Sensorless Control, Estimation, Observer, Kalman Filter

Article Classification: Research paper

I. Introduction Permanent Magnet Synchronous Motors (PMSM) are widely used in industrial and domestic drives because of theirs advantages: high power density, high torque to inertia ratio and small torque ripple. To achieve a good dynamic, effective performance especially during transients, a vector control should be used. However, a motor shaft position sensor is required to enable the effective vector control of PMSM. In typical realization, the sensorless operation in speed control of PMSM drive may be achieved in field oriented control (FOC) just by elimination of the position sensor, and use of a position estimator, even in more complex drives, e.g. fed by sinusoidal filter (Urbanski, 2013), or use the open loop mode for simple drives (Brock and Pajchrowski, 2013), or use the direct torque control (DTC) method (Ahmed Adam and Gulez, 2009). However, the speed calculation directly from estimated position value is not so trivial in the case of low speed range of operation (Raute et al., 2007)(Gu et al., 2009)(Raute et al., 2011), or in a case of high dynamic drive (Hunter, 2011), because it is assumed that the back EMF estimation does not have sufficient accuracy at lower speed range, and there is a possibility of the excitation oscillations by the estimator. Usually, as a low speed in sensorless mode is considered a range of several revolutions per second. Below that speed a “near zero” is used or “standstill region” name. Used in the paper back EMF based estimation procedure places the method as a “low speed”. Near zero or standstill speed range is typically occupied by introduction of the high frequency additional signal (Corley and Lorenz, 1998)(Consoli et al., 2001)(Schrodl and Simetzberger, 2008)(Qi et al., 2009). However, that paper proves the proper performance of the position estimator at speed range about 1-2 revolutions per second for back EMF based position observer. The influence of the nonlinear phenomena results the variability of torque generated even for constant reference current, which is visible in the measured speed variation presented on Figure 1 (no speed control, only constant reference currents: iq-ref=0.35 A, id-ref=0 A). One can see the speed distortion is related to the shaft position (which is correlated with estimated back EMFs αβ).

Fig.1. Sensorless control: constant reference current iq=0.35 A, CH1: estimated back EMF – Eα, CH2: estimated back EMF – Eβ, CH3: measured speed (average value 5 rad/s), CH4: reference current iq

Those waveforms were obtained for traditional FOC method of the PMSM, with cascade control structure using closed speed control loop, with inner current control loops contained separate controllers in d and q axis (Fig. 2). The overall structure of control, utilizes sine and cosine of the estimated position and calculated speed value instead of measured values. To obtain smooth output values of the observer, the reference voltage is used instead of measured ones. Usage of the reference voltage value in the estimation algorithm significantly facilitates the calculations (there is no need to filter the inverter's output voltage, which results in additional time delay) and affects the quality of the values obtained during shaft position calculations. The quality was defined as the low value and the small oscillations of the position error.

Fig.2. Structure of the presented sensorless drive

II. Position estimation The position estimation concept is based on estimation the back EMF induced in stator windings. An observer which is used in presented paper, it is just a motor model, expanded with corrective feedback. The observer estimates the back EMFs and currents. Estimated currents and measured currents are used in the corrective feedback to enable the proper estimation performance. Used in the control chain the observer is based on modified (Urbanski, 2013) structure of the Luenberger observer (Luenberger, 1971). Observer is prepared as an αβ model. It is based on a motor model described only by the two first electrical equations in stationary coordinates system αβ, with state variables iα , iβ , input variables vα , vβ (reference voltage) and the back EMF eα , eβ may be considered as disturbances (Parasiliti et al., 1997). One can write the observer equations set as in the following form:

dˆi α R 1 1 = − ˆi α − eˆ α + v α + Fiα (ˆi α − i α ) dt L L L dˆiβ R 1 1 = − ˆiβ − eˆ β + v β + Fiβ (ˆiβ − i β ) dt L L L deˆ α = Feα (ˆi α − i α ) dt deˆ β = Feβ (ˆi β − i β ) dt

(1)

where Fiα and Fiβ are the current correction function, Feα and Feβ are the back EMF correction function, for α and β axis of coordinate system, respectively. The observer output signals are estimated currents iˆα , iˆβ and the estimated back EMF components eˆα , eˆβ . The form of the correction function Fxy (x means i/e , y means α/β) in all four cases is the same, however, their factors may differ. Because of assumption of symmetry in axis α and β, correction function Fiα has the same factors as Fiβ and correction function Feα has the same factors as Feβ . Correction function for use at low speed range may be a proportional, so the corrector is just a set of gains. Such form of the corrector is used in basic structure of the Luenberger observer. In order to increase the accuracy of observation, especially in dynamic states, the higher speed range, or in a case of big-step calculations, a structure of the corrector may be expanded by adding another integrator, to give correction function „PI”, as is presented in matrix form:

Fxy [∆i ] = K p x [∆i ] + K i x ∫ [∆i ] dt

(2 )

where Kp and Ki are arrays of correction factors, respectively for proportional and integral component of the observer’s corrector. The factors are prepared using the random optimization algorithm (Urbanski and Zawirski, 2004). The quality function used in presented investigations has a form shown below: t1 +τ

Q = K1

∫ (ε (t ) + ε (t ))dt + K 2 sin

2 cos

2

⋅ ∆ε Θ (τ )

(3)

t1

where εsin and εcos are the estimation errors of sine and cosine of the shaft position, ∆εΘ is the range of those errors changes in a time range τ. Symbols K1 and K2 are the weight factors for the components of quality function. By proper set of those weights, one can combine the task of minimization the position estimation errors, and penalize its oscillation. To obtain position information and a rotor speed values, some calculations should be made. Estimation the sine and cosine of the rotor position is enough to use a vector control:

()

ˆ =− sin Θ

eˆα eˆ

()

ˆ = cos Θ

eˆβ eˆ

(4)

The quality of the estimated position at low speed is visible on the Figure 3, where estimated and measured sine of the shaft position waveforms are shown. It is noticeable, that measured and estimated sine of the shaft position may be very close, even in the case of a step change of reference speed. Estimated position is sufficiently smooth (Fig. 3, channel 2), even of estimating it from rough estimated back EMF (Fig. 3, channel 4).

ˆ ), CH3: measured speed, Fig.3. Sensor control: step change of reference speed ωref=0.8→1.6 rad/s, CH1: measured sin(Θ) CH2: estimated sin( Θ CH4: estimated back EMF– Eα

III. Speed calculation The back EMFs estimated at low speed is not noisy or somehow damaged (Fig. 3, channel 4), however some defects do not allow simply calculation of speed: first – there are irregularities in the back EMFs (related to the shaft position), and second – there is no noticeable amplitude change (Fig. 4, channel 1 & 2). The almost constant estimated amplitude is visible at experimental test where reference speed was changed in sequence 0-0.4-0.8-1.6 rad/s however, the speed information is still inside covert.

Fig.4. Sensor control: step change of reference speed ωref=0→0.4→0.8→1.6 rad/s, CH1: estimated back EMF – Eα, CH2: estimated back EMF – Eβ, CH3: measured speed, CH4: reference current iq

In such case, a typical method of the speed calculation based on the dependency (5) (

eˆ is a length of the back EMF vector

and ke is scaling factor) cannot be used, because of strongly non-linear relationship of the back EMF amplitude and the speed

value. More importantly, difficulty is rugged in the shape of the back EMF signal, which generates oscillations in speed value due to calculations the length of the back EMF vector.

ωˆ =

eˆ ke

(5)

Finally, for tested range of the speed, the speed controller parameters, chosen for higher speed range, make it not fast enough (Fig. 5, channel 3). Presented speed waveforms for sensor control at low speed range are not acceptable. Increase of the speed controller coefficients, does increase the smoothness of the speed waveforms (Fig. 6, channel 3). However, the sharper dynamics decreased definitely a possibility to achieve stable sensorless performance of the drive.

ˆ ), Fig.5. Sensor control – slow speed controller: step change of reference speed ωref=1.6→0.8→0.4 rad/s, CH1: measured sin(Θ) CH2: estimated sin( Θ CH3: measured speed, CH4: estimated back EMF– Eα

ˆ ), Fig.6. Sensor control – fast speed controller: step change of reference speed ωref=1.6→0.8→0.4 rad/s, CH1: measured sin(Θ) CH2: estimated sin( Θ CH3: measured speed, CH4: estimated back EMF– Eα

To solve those problems, a proposed estimation system was developed (Fig. 7). In typical sensorless realization, the Kalman filter (KF) (Kalman, 1960) is used as an estimator, e.g. the shaft position or state variables (Janiszewski and Muszynski, 2007)(Szabat et al., 2006). In presented paper, KF is used now as an easy implementation of the discrete filter. Estimated speed extraction procedure is as follows: the position estimator produces back EMFs in stationary reference frameαβ: eˆα , eˆ β . The estimated position

ˆ is calculated using (9), and then is used to calculate modified values of the back EMFs eˆα 2 , eˆ β 2 . Θ

After adding correction value depending on the shaft position, by the use of derivative, the variable amplitude signal is now obtained e&ˆ α 2 , e&ˆ β 2 . Then, after filtration using KF, the shaft speed is calculated using relationship (8).

Fig.7. Estimated speed extraction procedure

Finally, estimated speed may be used to control the drive. The filtering structure which is used in the algorithm is one of the simplest structure in the KF family, the discrete form of the Kalman filter. The equations of KF are fully described (Bishop and Welch, 2001)(Grewal and Andrews, 2008), therefore in this paper the part of computer program for calculating the KF is shown instead (Fig. 8). The input values are "meas1" and "meas2", and the output values "xhatOut1" and "xhatOut2" are the smoothed values of derivative of the modified back EMFs. The input vector "u" is zero.

Fig.8. The Kalman filter calculation procedure presented as a Matlab code

The values of the process noise covariance Q and measurement noise covariance R are chosen experimentally:

10 −7 0  Q= −7   0 10 

10 −1 0  R= −1   0 10 

(6)

The initial state of error covariance matrix P does not have much impact on filter operation. The process arrays and initial state estimate are presented below:

1 0 A=  0 1 

0  B=  0 

1 0 C=  0 1 

0  xˆ 0|0 =   0 

(7)

After calculations, the filtered by KF the instant values from the back EMF derivatives are used to calculate the speed. It is still used method based on length of the EMF vector however, that signals were pre-processed using Kalman filter: eˆ 2 KF 2 ˆ2 eˆ 2 KF = eˆα2 ωˆ = (8) KF + eβ2 KF k e2 eˆ 2 KF

(

)

where eˆ 2 KF is a length of the KF output vector (corresponds to the modified value of back EMF vector) and ke2 is scaling factor. That factor, depend on used calculation method for speed estimation, may be a constant for proposed method, or may be used as a function that depends on the obtained vector length eˆ 2 KF , for wider speed range, for which it may be use. The other signals produced by observer are sine and cosine of motor shaft position. Using calculated estimated position (9) and previously identified back EMF deviation from pure sinusoid, a more pure sinusoidal in shape back EMFs are obtained in block “modifier”, which acts as a look-up table.

 ˆ = arctan eˆα Θ  eˆ  β

   

(9)

The block “modifier”, depending on the estimated position, adds a stored in memory value to estimated back EMF, to get smoother estimated speed value. The added value is independent for an α axis and β axis. The modifier and position calculator (9) works using raw back EMFs, the KF uses as an input the modified back EMFs. Effects of such an approach are visible at figure 9, at the waveforms of filtered measured speed and estimated speed. Such waveforms were obtained for observer input signals recorded at the laboratory stand (there were reference voltages and measured currents), in the sensor mode operation and then the calculations are processed off-line. It proves that the idea of speed value processing is correct. At figure 10 estimated and corrected waveforms of the back EMF are visible. Using the correction value depending on the shaft position, a smoothed back EMF signals were obtained. This correction signal is added using stored values with interpolation mechanism. Such obtained smooth waveforms allow calculation of derivative of those signals. Such way of generating correction values, only in function of the position, is possible due to almost no change of the amplitude of the estimated back EMF in that speed range. Discrete derivative of the corrected estimation of the back EMF (with step equal 100 µs and output limited in this case to the range ±300) is furthermore filtered using KF. That structure is chosen to get smoother waveforms at shorter processing angle (processing time is not as important, in opposed to processing angle, in sense of the motor shaft position in variable speed mode). The output of the KF is shown at figure 11. It is noticeable, that obtained waveforms are quite smooth, despite the implementation of the derivative operation. The effect of rapid speed change is clearly visible at time 5 s and 8 s (Fig. 10, 11). This is noticeable at Fig. 10 as peaks and as a quick change of the angular frequency of those waveforms, and at Fig. 11 as a change of the amplitude of those waveforms. The time delay has acceptable value for low-dynamic drives (Fig. 9). The phase shift of the estimated back EMF signal after filtering does not matter, because that procedure is used only to calculate vector length, not to the position of the back EMF vector.

1.6

speed [rad/s]

1.4

measured speed

1.2 1 0.8 0.6 0.4

estimated speed

0.2 0

1

2

3

4

5 Time [s]

6

7

8

9

10

Fig.9. Open loop mode (recorded experimental data) – speed comparison: estimated speed is calculated using presented in Fig. 7 algorithm for data as in Fig. 4

Fig.10. Open loop mode (recorded experimental data): estimated and corrected back EMFs using presented in Fig. 7 algorithm for data as in Fig. 4

Fig.11. Open loop mode (recorded experimental data): Kalman filter output signals using presented in Fig. 7 algorithm for data as in Fig. 4

IV. Conclusions Even in the low speed range (excluding standstill) the estimated back EMF values still hold the information useful to the position calculation with acceptable accuracy. However, the speed calculation using only estimated by observer the back EMFs is problematic. Presented speed extraction procedure is simple and useful and it was tested on signals recorded on laboratory stand using 1.23 kW PMSM. For that speed range, below single revolutions per second, is hard to use typical sensorless structure, with the observer used instead position measurement, because of the phenomena which force the “fast” speed controller settings. Therefore, the estimation of the speed must not introduce additional oscillations. Such pre-processing of the signal gives possibility to meet this requirement. Filtering must be strong enough to neutralize rough EMF signal derivative. From the other hand, usage of the filter forces the delay the estimated speed value. Reduction of this delay is the next task, which is necessary to achieve the better dynamics.

Additional tests demonstrated (not presented here), if working online with drive, the performance of the extraction speed algorithm should be more improved by modification of the correction values on working drive. The reason for that is the interaction between correction values and position estimation error, which dynamically forces the modification of the correction values. The next approach is to prepare modifier values online, using random method to tuning correction values online, during normal working of the drive. Appendix The parameters of the motor used in experiments are: • rated power - 1.23 kW • nominal speed - 3000 rpm • rated torque - 3.9 N·m • number of poles - 6 • measured resistance - 2 Ω • measured inductance - 5.7 mH • total moment of inertia - 24.96 kg·cm2

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To cite this document:

Konrad Urbanski , (2015), "Sensorless control of PMSM for low speed range", COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 34 Issue 3, pp. 754 – 765 Permanent link to the final version: http://dx.doi.org/10.1108/COMPEL-10-2014-0275