Rotor Position Estimation for Permanent Magnet Synchronous Machines using Electromotive Force and Anisotropy Using Extended-Kalman-Filter

The use of highly efficient field-oriented controlled permanent magnet synchronous machines (PMSM) usually requires a position sensor in addition to the necessary power electronics. This position sensor and necessary signal cables increase the total drive system costs. Furthermore, most position sensors are susceptible to mechanical impacts and can therefore quickly wear out. To reduce the costs and increase the robustness of field-oriented controlled PMSM, the rotor position can be determined from the electrical characteristics of the machine. Known methods for position sensorless control of PMSM can be traced back to two rotor position-dependent machine effects.

First, the rotor position dependency on the electromotive force (EMF) is evaluated. The rotor flux caused by the permanent magnets on the rotor induce the EMF. Therefore the EMF is directly depending on the rotor position. However, the EMF is also linearly dependent on the rotor speed. Therefore there is no EMF voltage when the machine is at standstill. In practice, it has also been shown that at low speed the signal-to-noise-ratio (SNR) of the measured variables is too low for evaluation. Therefore, EMF-based rotor position estimation methods require a minimum speed [1]. Most EMF-based rotor position estimators consist of a linear EMF observer [2, 3] and an angle-Tracker [4]. Such methods have a time-delaying transfer behaviour, which reduces the phase margin of closed control loops. Alternatively, the angle information from the EMF can be directly evaluated using the Extended-Kalman-Filter (EKF). In this case, no additional angle tracker is necessary [5, 6]. An EKF also enables online disturbance and parameters estimation.

Due to the limited operating range of EMF-based rotor position estimation methods, anisotropy-based rotor position estimation methods are used alternatively or additionally. In the context of position sensorless control, the magnetic anisotropy of the machine means rotor position-dependent inductance. By injecting comparatively high-frequency carrier signals [7‐9] the inductances can be identified online, also at standstill. However, these carrier signals reduce the voltage bandwidth at high speed. The rotor angle can be estimated by evaluating the anisotropic current sequence, caused by the carrier signals, using adapted angle trackers [10]. Therefore, more or less complex signal filters with a fixed cut-off frequency, e.g. Bandpass or bandstop filter, are required to separate this anisotropic current sequence [11, 12]. The necessary signal filters also reduces the phase margin of closed control loops. Alternatively EKF can be used to separate fundamental wave and carrier signals [13]. There are many different variations of anisotropy-based estimation methods, but the main disadvantages are basically the same. The estimation accuracy and robustness of anisotropy-based methods are strongly dependent on the PMSM design and decrease with increasing PMSM speed.

It has been shown that anisotropy-based estimation methods at standstill or very low speed and EMF-based methods at medium and high speed have the optimal operating range. Therefore, a common approach for position sensorless control of PMSM in the entire speed range is to combine an EMF and an anisotropy-based estimator [14, 15]. In these methods the estimators are switched based on the speed of the PMSM. The different estimators must be synchronized when switched. Otherwise the estimated values may strongly oscillate. To not excite these oscillation, the switch-over range can be passed through at a limited speed dynamic.

It can be said that with conventional methods for a position sensorless control of a PMSM in the entire speed range, compromises must be made between a robust estimation and an optimized control behaviour. For an optimized operation of PMSM without position sensors, an EKF-based rotor position observer which evaluates EMF and anisotropy-based angle information is presented in this paper. For this purpose, the EKF uses a fundamental wave- and a HF-state space model. The main difference to conventional methods and the greatest advantage of this method is, that the angular speed and rotor position can be directly estimated at the same time by one estimator. Therefore, there is no need for additional angle trackers to evaluate the anisotropy based angle information or methods to switch or synchronize between different estimators. This reduces the number of additional transfer behaviors in the closed control circuit of the sensorless controlled PMSM. In addition, the EKF is able to separate the fundamental wave and the higher frequency component. Therefore, no further signal filters are necessary and the phase delay of the rotor position observer presented here is comparatively low. In total, this leads to optimized control behavior.

To increases the robustness of the estimator against parameter changes, the amplitude of the permanent magnetic field is estimated online by the EKF. This means that temperature and saturation dependencies can be taken into account without an analytical description of these effects. In addition to the electrical machine behaviour, the state space model includes a mechanical model of the PMSM. By the online parameter estimation of a mechanical disturbance variable, load and dynamic torques of the coupled load application along with parameter inaccuracies can be taken into account. This stabilizes the rotor position estimation especially in case of dynamic load changes.

In [16], it was already shown that the presented EKF rotor angle estimator can pick up freewheeling PMSM and transfer them in a speed controlled state. In contrast to this publication, the general accuracy of the estimator and the control behaviour of the sensorless controlled PMSM are to be examined in this paper.

This paper is divided into the following sections: Section 2 describes the fundamental wave-, higher frequency- and mechanical-model of the PMSM as well as the resulting EKF. Section 3 describes how the rotor position observer is integrated into the cascade control of the PMSM. In Sect. 4, the resulting EKF-based observer is tried and tested on a test bench and evaluated using measurements. In the last Sect. 5 the results and possible subsequent work are discussed.

In this Section, the fundamental wave model and the model for high frequency carrier signals (HF-model) are described. In addition, a mechanical model of the PMSM is considered. To clearly differentiate between the associated variables and parameters of the two electrical models, they are marked with \(f\) for the fundamental wave model or with \(c\) for the HF-model (carrier signal). Both models are represented in the stator-fixed \(\alpha\beta\)-coordinate system.

The fundamental wave voltages \(u_{\mathrm{f,\alpha}}\) and \(u_{\mathrm{f,\beta}}\) and the carrier signals \(u_{\mathrm{c,\alpha}}\) and \(u_{\mathrm{c,\beta}}\) are separated input variables of the EKF (see Fig. 1). Therefore, the measured control voltage cannot be used as an input, since the fundamental wave component and the HF component are always superimposed there. Therefore the outputs of the PI current regulators are used as fundamental wave input voltages. The HF voltages are defined by the amplitude \(U_{\mathrm{c}}\) and the frequency \(f_{\mathrm{c}}\) and can therefore be set separately as an input. The supper position from fundamental waves and HF component is then modulated with a PWM frequency of \(16\,\mathrm{kHz}\) and provided by the power electronics. The currents \(i_{\mathrm{u}}\), \(i_{\mathrm{v}}\) and \(i_{\mathrm{w}}\), which are measured anyhow, are exploited in the EKF. The estimated fundamental wave currents \(\hat{i}_{\mathrm{f,d}}\) and \(\hat{i}_{\mathrm{f,q}}\) are fed back for current control. The estimated electrical rotor angle \(\hat{\varphi}_{\mathrm{el}}\) is used to commute the voltages from the rotor-fixed to a stator-fixed coordinate system. The observed mechanical rotor speed \(\hat{\omega}_{\mathrm{mech}}\) is fed back to the speed control loop. This is calculated from the electrical angular speed \(\hat{\omega}_{\mathrm{el}}\) divided by the pole number \(p\). Additionally, the estimated mechanical disturbance \(\hat{M}_{\mathrm{load}}\) is used as a compensating feedforward control term.

As already indicated in Sect. 2.2, the amplitude of the HF carrier signals depends on the electrical speed of the PMSM. Therefore the amplitude of the Signal is not constant. The amplitude at standstill and at a low speed must be high enough to excite the anisotropic current sequence with sufficient strength. If the speed is high enough to robustly identify the rotor position from the EMF, the carrier signals reduce the voltage bandwidth unnecessarily. Therefore the amplitude is reduced linear from a maximum of \(U_{\mathrm{c,max}}=30\,\mathrm{V}\) at standstill to \(U_{\mathrm{c,min}}=0\,\mathrm{V}\) at \(26.3\%\) of the rated speed. At higher speeds, the amplitude is kept constant at \(U_{\mathrm{c,min}}=0\,\mathrm{V}\). The frequency of the carrier signals is constant at \(f_{\mathrm{c}}=1000\,\mathrm{Hz}\).

To evaluate the EKF-based rotor position estimator and the superimposed position sensorless speed control, different measurements on the test bench are shown in this section and the results are discussed. For the tests, a PMSM was coupled with a load machine and controlled via a frequency converter from the company LENZE (i950 4kW). The presented position sensorless control concept was implemented on a real-time target (RTT), which transmits the voltages generated by the current control to the converter and receives the measured current values. The connected motor is also a PMSM from LENZE with the type designation MDSKSAG056-23. For the separate evaluation of fundamental waves and high-frequency signals, these can be determined by offline filtering from the total currents or voltages measured. In this paper, however, only the total currents and voltages are shown.

To evaluate the angle estimation error and the speed estimation error of the EKF over the entire speed operating range, a speed ramp is traversed from standstill to the nominal speed. The machine is idle during this experiment. Figure 2 shows high estimation quality over the entire speed range, both for the rotor position and the speed estimation. It turns out that at the transition between the anisotropy and the EMF range, at approx. \(100\) to \(500\,\mathrm{rpm}\), there are no negative effects on the speed control. The transition takes place very slowly, but the results are accordant even in the event of a jump speed setting in a later experiment.

In Fig. 2 it can be seen that over the entire speed range the maximal estimation error of the electrical rotor position \(\Delta\varphi_{\mathrm{el}}\) is \(4^{\circ}\). The maximal estimation error of the mechanical speed \(\Delta n_{\mathrm{mech}}\) is \(11\,\mathrm{rpm}\). In this experiment, however, the estimation accuracies are not considered in the steady state. Therefore, it cannot be ruled out that dynamic errors influence the results. Therefore, in a second experiment, the PMSM is driven to different stationary speed operating points and the estimation errors of the rotor position and speed are statistically evaluated.

In Fig. 3 the mean value and standard deviation of the estimation errors of rotor position and speed are illustrated. The maximum angle estimation error at speed of \(n_{\mathrm{mech}}=0\,\mathrm{rpm}\) is \(\Delta\varphi_{\mathrm{el,max}}=4.17^{\circ}\mathrm{el}\) and the maximum speed estimation error at speed of \(n_{\mathrm{mech}}=100\,\mathrm{rpm}\) is \(\Delta n_{\mathrm{mech,max}}=2.9\,\mathrm{rpm}\) can be determined from Fig. 3. The evaluation shows that the maximum estimation error is present in both cases in the low speed range. In this range the anisotropy is used as the basis for the angle estimation.

In order to evaluate the dynamic behaviour of the sensorless speed controlled PMSM, a speed setpoint jump to \(1000\,\mathrm{rpm}\) is performed. The PMSM is idling during this attempt. As a reference, the same jump is made for speed control with a position sensor. Reset time and controller gain of speed and current control are identical for both experiments.

Figure 4 shows, that the EKF-based sensorless speed control behaviour is comparably good. However, the comparison of the measured currents shows that, with position sensorless control, the current in \(d\)-direction is higher and the current in the \(q\)-direction is slightly lower shortly after the jump. This is cased by temporarily higher estimation errors. Due to the reduced \(q\)-current, the internal torque of the PMSM is also lower, which explains the slightly lagging speed for sensorless control. Overall, this experiment shows that the EKF-based sensorless control has approximately the same behaviour as speed control with a position sensor.

As described in Sect. 3, the estimated fundamental wave currents are fed back as actual values for the current control. This means that no additional filtering is necessary. For proof of the current sequence separation by the EKF, the frequency spectrum of the measured total currents, the estimated fundamental wave and higher-frequency current sequences are compared next. For this purpose, the frequency spectrum of the currents in \(\alpha\beta\)-coordinates for the previous experiment with speed jump are considered.

The frequency spectrum of the \(\alpha\) current in Fig. 5 shows that the sequences of fundamental wave and carrier signals are separated effectively by the EKF. The dominant frequency of the fundamental wave at \(50\,\mathrm{Hz}\) (\(1000\,\mathrm{rpm}\cdot p/60\,\mathrm{s}=50\,\mathrm{Hz}\)) and the dominant frequency of the carrier signal at \(1000\,\mathrm{Hz}\) are only expressed in the respective estimated current sequences, which confirms the high quality of the current separation.

Finally, the robustness of the estimation and the superimposed position sensorless control with regard to load and disturbance torques should be examined. For this purpose, a load torque jump of \(1.6\,\mathrm{Nm}\) is applied with the help of the coupled load machine. This happens once when the machine is at a standstill and once at a speed of \(1000\,\mathrm{rpm}\). At standstill, only the anisotropy-based rotor position information is used for estimation, while at \(1000\,\mathrm{rpm}\) only the EMF is evaluated for rotor position estimation.

In Fig. 6 it can be seen that the disturbance jump is regulated a little bit slower with sensorless control compared to the control with a position sensor. However, the difference is insufficient small and can also be traced back to the dynamic estimation error. It also shows that the rotor position estimation error resulting from the load jump is higher at standstill than at \(1000\,\mathrm{rpm}\). The estimation based on the EMF is therefore more robust.

In this paper, a new EKF-based rotor position observer for position sensorless control of a PMSM is presented. This EKF simultaneously evaluates EMF and anisotropy for rotor position estimation. Switching between or synchronizing different estimators is not necessary. The estimation accuracy and the control quality were examined on the basis of experiments. The measurements show that the maximum stationary estimation error of the electrical rotor position in the entire speed range is below \(4.2^{\circ}\). The later comparison shows that the EKF-based position sensorless control has no significant disadvantages compared to PMSM control with position sensor. The EKF can clearly separate the fundamental wave current components from the higher-frequency current. Thus, in addition to the EKF rotor position observer, no additional angle trackers or signal filters are necessary. Therefore, the phase margin of the control loop is not unnecessarily reduced. It was also shown that disturbance torques can be estimated with a high accuracy by the EKF. This disturbance variable estimation has a high potential for an additional error detection, such as e.g. for detecting gear breakdowns or mechanical wear. The new EKF-based rotor position estimator presented in this paper enables an optimized position sensorless operation of the PMSM with regard to the control dynamics without the necessity to make compromises with regard to estimation robustness. Therefore, the presented rotor position estimator enables applications with high demands on dynamics and accuracy to be controlled without position sensors.

An analytical investigation of the transmission behaviour of the presented EKF and an optimization of the controller design are planned as the content of future work.