Position Estimation at Zero Speed for PMSM Using Probabilistic Neural Network

Konrad Urbanski Institute of Control and Information Engineering

Poznan University of Technology Poznan, Poland

konrad.urbanski@put.poznan.pl

Abstract—The paper presents a method for estimating the shaft position of a synchronous motor with permanent magnets (PMSM) for the zero and very low speed range. The method is based on the analysis of the high frequency currents, which are induced by the additional test voltage in a stationary coordinate system associated with the stator. Although this method involves the identification of currents hodograph, the method does not need to calculate the current ellipse position. Presented method involves a comparison of obtained shape to the reference pattern using probabilistic neural network (PNN). The method can achieve satisfactory accuracy in a case the high asymmetry of the inductance, as well as in the case of small values of the inductance asymmetry ratio, also in the case of a high level of noise.

Keywords—permanent magnets synchronous motor; sensorless control; zero speed; estimation; artificial neural network; probabilistic neural network

I. INTRODUCTION

Permanent magnet synchronous motors (PMSM) are widely used in domestic appliances and industrial drive systems, both those used in drives of the highest precision machine tool, and used in drives with smaller requirements [1]–[3]. The main advantages of PMSM drive this high dynamics and energy efficiency. These features result from a small moment of inertia, high factor of the overload torque, and negligible electrical losses in the rotor. The disadvantage of traditional drives with PMSM is a need for shaft position sensor used in vector control signals to convert between different coordinate systems, calculate the speed and other advanced algorithms. Eliminating the mechanical sensor which is used to measure and calculate the shaft position and its speed, is the research subject, which is still developed in various research centers [4]–[9]. The purpose is to enable vector control without the need of the machine shaft encoder. It will allow increasing reliability and decreasing costs for the drive, which should increase the scope of its uses, as a simple drive with variable speed. Unfortunately, in order to take advantage of working range at very low and zero speed, you cannot use techniques based on back EMF estimation, due to the absence or small value of the induced back EMF [10], [11], and weak for this range of speed, ratio of the value of the measurement signals to the level of interference. For this speed range may be helpful the physical methods, using the non-linearity the engine, for example, based on an analysis of inductance variation as a function of shaft position [12].

Typically, the position estimation is carried out for this speed range, by generating additional in the stator high frequency (or suitably shaped pulses) voltage test signal. This causes high frequency currents flow, whose actual value depends on the actual inductance. Analyzing the waveforms of the current, you can get information on the current position of the shaft. Most systems in laboratory and industrial use two methods of measuring inductance: periodically are adding a test signal (INFORM method) [13], [14], and with continuous high-frequency signal. Method INFORM (Indirect Flux detection by On-Line Reactance Measurement) is a physical method, based on continuous (on-line) measured of reactance in phases. This method uses the asymmetry caused by the phenomenon of magnetic saturation or motor geometry . Another solution using high frequency excitation voltage is used in case of PMSM with cylindrical rotor or in the case with permanent magnets mounted on the surface [15]. Inductance measurement is accomplished by adding a test signal to the phase currents. This method works best at low speeds, because already for medium and high speeds, the imposition of a test signal on the phase currents causes its deformation, and thus deterioration the quality of the torque control. In the present study is used a method of continuous high-frequency signal. To estimate the position, the analysis of the currents hodograph is used (Fig. 1). It uses a phenomenon, that at the standstill, if the stator is energized by the symmetrical three-phase sinusoidal voltage, the steady-state currents hodograph will have the shape of an

Fig. 1. The idea of determination the position of the motor shaft based on stator current hodograph

978-1-4799-8322-3/15/$31.00 ©2015 IEEE

ellipse (or approximately an ellipse for the PWM inverter). Based on the actual stator phase currents, can be plotted currents hodograph. Location of the major axis of the ellipse defines the position of the shaft:

γ+Θ′=Θ (1)

where Θ – shaft position Θ’ – major axis position

γ – position adjustment depended of the stator reactance

The structure of the method using the injecting a high-frequency carrier-signal excitation is shown in Fig. 2. This is the control system of currents in the α−β stationary coordinate system. The output signals of the current controllers (block PI), are the reference voltages, to whose were added the high-frequency sinusoidal voltages. Motor currents after the transformations to stationary coordinate system (for the purpose of clarity, there are not shown in the drawing the transformation blocks) are filtered. Band stop filter (BSF) separates the first harmonic of the high frequency currents, whose are then used to currents control in the coordinate system α−β, and band pass filter (BPF) isolates a high frequency carrier signal, which is used to create the hodograph. Filtration quality affects the maximum rotor speed and dynamics, for which the algorithms can still work, with a satisfactory accuracy of estimation. The main advantage of physical methods of position estimation is the possibility of estimation for standstill and the opportunity to work with low speed. The disadvantages are the necessity to extension of control structure by the additional algorithms for current or voltage high frequency signals, and the complexity of supporting measurement systems, which require the use of complex methods of analysis of waveforms obtained in α−β axes. The additional supply voltage should have the enough high frequency and enough low value, to provide the minimization the value of additional moment which is generated. This method is used to estimate the rotor position for a standstill or rotation at low speed, and is used mostly for motors with magnets placed inside the rotor (IPMSM - Interior magnets PMSM), due to the high differences in inductance in axis d-q , what facilitates to determine of the axis of the currents ellipse. Exemplary, "ideal" hodograph for this case is presented in Fig. 3. The major and the minor ellipse diameter is here equal 1.0 and 0.6, respectively. A set of frames, for

different, equidistant shaft position angle is shown. In order to ensure image clarity, only first six frames are drawn. It is visible, that hodograph for every angle differs significantly, which allows to use relatively simple the identification algorithm of the shaft position. It is worth noting, that the identification of major axis should be supplemented by determine the actual angle range: 0-π or π-2π. The raw data can be shown as the 3D hodograph. In Fig. 5 there is shown the shape, which was created using all frames for complete shaft rotation. It contains also the frames presented in Fig. 3. In a case of the surface placed magnets rotor (SPMSM - Surface magnets PMSM), the impact of permanent magnets on the inductance change as a function of the rotor position is negligible, because the relative permeability coefficient of permanent magnets is similar to that for the air. For this reason, due to the symmetry of the rotor is achieved Lq≈Ld . In that case, the proper position identification is possible, if the

Fig. 2. The general view on part of the drive control system to the injecting carrier-signal excitation

Fig. 3. Simulating results - the hodograph in a case of IPMSM: first six frames equidistant at 18 degrees; current range in both axes αβ ±0.6 A

Fig. 4. Measuring results - the hodograph in a case of SPMSM: first six frames equidistant at 18 degrees; current range in both axes αβ ±0.6 A

current hodograph has a different shape for different position (even in a case of small shape differences between positions). This assumption may be correct in a case of real motor. The following waveforms were obtained using recorded data on the laboratory stand (currents α−β), and then calculations were realized offline using Matlab ®. The hodograph prepared on the basis of the recorded data is presented in Fig. 4. In order to ensure image clarity, only first six frames are drawn. In Fig. 6 is shown a 3D hodograph for the recorded data of the laboratory SPMSM. It has been created using all frames for complete shaft rotation. This hodograph is characterised generally by unsmooth changes, and these changes are very little. However, even such changes of this cross sections allows the identification of individual frames by PNN.

II. PROBABILISTIC NEURAL NETWORK

Artificial neural network (ANN) is a structure consisting of information processing elements (neurons) connected to the

network. The structure of connections and their parameters determine the operation of the network. To use ANN to classify patterns, is needed the learning procedure, which usually is based on previously prepared exemplary patterns. Used the learning procedures, according to the network structure, requires usually a lot of time to complete learning. A modified structure of ANN were introduced by D.F. Specht in late ‘80s [16]. The probabilistic neural networks (PNN) training is extremely fast procedure (compared to training time for feed-forward networks), and gives a better classification accuracy [17]. PNN are used in the area of electrical drive to improve the performance of the drive system [18], [19], and mainly in fault diagnosis [20]–[23]. The general structure of the network, which was used in investigations, is presented in Fig. 7. The network is prepared to classify "k" patterns (outputs), each pattern consist of "p" elements (inputs). Each pattern may be classified as a member of category A or category B (summation units). The PNN in presented paper is used as a position estimator. As an input is given an actual current hodograph (input vector X), and an output contains, in the ideal case, a set of low logic states (category B) and a single the high logic state (category A). The position of the output in the state A is related to the estimated shaft position. So, the number of network outputs determines the positioning resolution.

III. POSITION RECONSTRUCTION

In studies is used the additional voltage, which is sinusoidal in a shape, in both axes, with frequency equal 500 Hz. The carrier frequency of the inverter as well as the frequency of measurements is equal 10 kHz. Single frame of the hodograph contains 20 samples (in two channels, for current α and current β), so that every 2 ms is provided a new set of data. The current hodograph is interpreted just as two-dimensional shape. In the shape visible on the scope screen, it is hard to find the beginning or the end (Fig. 8). In a case of use the 2D classifier, it does not matter. However, in presented estimation strategy, used PNN has only one-dimensional the input. The measurements should be therefore preprocessed. First, both sets of samples are "normalized". It is realized by circular shift of both vectors of α and β axis (the units number is the same for α and β vector) to get the biggest value in α axis at the

Fig. 7. General view on PNN structure

Fig. 5. The 3D hodograph for data prepared in a case of IPMSM. Full record of data, as is presented partially in Fig. 3

Fig. 6. The 3D hodograph for data recorded in a case of SPMSM. Full record of data, as is presented partially in Fig. 4

Sh

aft

po

siti

on

[ra

d]

Sh

aft

po

siti

on

[ra

d]

iα

iβ

iα iβ

start of data vector. It acts as a simple synchronization mechanism in a case, where high-frequency additional voltage is activated only time to time, under certain conditions. Then both vectors (2x20) are joined into single vector (1x40), which will be called in the simulations as the key frame, in opposite to calculated input vector, using interpolation - which will be called the frame. It gives as a result an input pattern, which is then classified using PNN. Thus used in research PNN has 40 inputs. At this stage of the study, were recorded 20 different shaft position (every 18 degree). So the PNN has 20 outputs. In the next step of the research, a bigger resolution will be accomplished. In Fig. 8 is visible a hodograph, which consist of 20 measured points (channel 1 and 2). The origin of the

other samples, which are visible as a noise around those "main" points, is explained by Fig. 9, which presents waveforms of the measured currents in α−β axes. The currents values are disturbed by pulses every 100 µs, which corresponds to the carrier frequency of the inverter. The tests of the estimator were realized by the following way: in time range 0-10 s, the position is changed in range 0-2π. Depending on the instantaneous position, the adequate frame is given to the PNN inputs.The tests showed that used estimator works well, even in such conditions. Moreover, in a presence of the additional noises, the PNN estimates properly the position (considering the resolution, which is used), up-to noise level equal 10% of the high frequency current amplitude (Fig. 10). So, the key frames are classified perfectly, and this figure shows the estimation accuracy for presented system. The mean value of used in this study the amplitude of current is about

Fig. 8. Exemplary single frame of the current hodograph (channel 1-2) for SPMSM. Shape is similar to a circle rather than to ellipse however, ANN has capabilities to distinguish even very similar shapes

Fig. 9. Exemplary waveforms of high frequency currents which are used to create the hodograph

Fig. 10. Real position and estimated position (sector number) in a case 10% of additional noise – 100% accuracy for assumed resolution

Fig. 11. Comparison of the exemplary hodograph from the learning pattern (red, "x"-marker) and modified by use the 10% noise (blue, "o"-marker)

0.4 A, and for that value is calculated additional distortion. The exemplary hodograph from the training pattern compared with the hodograph modified by adding 10% noise (testing pattern) is shown in Fig. 11. Interesting results are obtained during the test, where the input pattern (during the simulation) is changed smoothly, by means the interpolation between key frames. Interpolation result is visible in Fig. 6., the smoothed 3D hodograph is built from 20 key frames. In a case of network trained only for recorded key frames, the generated outputs are not satisfactory (Fig. 12). New data set for training was prepared, now using the frames based on interpolation. The results are significantly better however, they are still not satisfactory (Fig. 13). From the other hand it is noticeable, that sector number is estimated perfectly, or is estimated with an error corresponding to 180 degree. The next test of the network training, contains the same data as in case of Fig. 13 however, an additional noise (15% of current amplitude) was added to the training points. The performance of such prepared PNN is shown in Fig.14, where a perfect match is visible, without chattering of the estimated value.

The network calculations are accomplished using the Neural Network Toolbox for Matlab®.

IV. CONCLUSION

In the paper PNN is used for position estimation of the SPMSM at standstill, using high frequency currents, which are induced by the additional test voltage in a stationary coordinate system associated with the stator. Such method uses the casual asymmetry, which is a side effect of the structural inaccuracy. PNN has important advantages: extremely short time of training (compared to feed-forward networks) and promising performance in a task of hodograph classification, even in a presence of noise. However, to proper operation of the estimator, using neural networks, the number of key frames should be increased in a case of studied machine. In a case if the hodograph shape of real motor is near circle (as may be in SPMSM), it is hard to see any regularity during smooth position change, as is easy find in a case of IPMSM (Fig. 3 vs. Fig. 4). It is visible using animation of the hodograph, which is modified according to smooth change of position. This work presents an early stage of the investigation however, the results of the method used in the study are promising. The next step is to increase the number of key frames in order to increase the resolution and accuracy. Robustness on the noise, as is visible in Fig. 10., suggest the robustness on some fluctuation of the hodograph shape, which may be resulted e.g. of temperature change or machine aging.

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

Fig. 12. Real position and estimated position (sector number) using interpolated frames as an input without additional learning (the same network as in a case of Fig. 10)

Fig. 13. Real position and estimated position (sector number) using interpolated frames as an input after additional learning

Fig. 14. Real position and estimated position (sector number) using interpolated frames as an input after additional learning using the noisy data

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Urbanski K.: “Position Estimation at Zero Speed for PMSM Using Probabilistic Neural Network”, Proceedings of the 2nd IEEE International Conference on Cybernetics – CYBCONF 2015, Gdynia, Poland, 24-26 June 2015, USB stick, ISBN: 978-1-4799-8321-6