Dual Quasi-Resonant Controller Position Observer Based on High Frequency Pulse Voltage Injection Method

This paper proposed a dual quasi-resonant controller position observer for conventional pulsating high frequency voltage injection method. The proposed position observer can not only improve the dynamic performance of the sensorless control, but can also compensate the position error fluctuation caused by the dead-time effect. To improve the dynamic performance, the digital bandpass filter in the traditional position observer used to extract high frequency current response is replaced by a quasi-resonant controller firstly. Moreover, an improved Luenberger observer without lowpass filter, which is usually used in traditional position observer to filter the noise in speed information, is adopted in the new position observer. Therefore, dynamic performances can be improved. Then, to reduce the sixth harmonic in the magnitude of position error and speed error caused by the dead-time effect, a frequency adaptive quasi-resonant controller is connected in parallel with the proportional-integral controller in the Luenberger observer. The experiment results verify that the proposed observer can reduce the position estimation error not only in steady state operation conditions, but in variable speed and variable load conditions, and the speed variation range can be widened as well.


I. INTRODUCTION
Permanent magnet synchronous machines (PMSMs) are widely applied due to their high torque density, power density, and efficiency. To improve the output performance, a high-resolution vector control strategy that depends on the accuracy of the rotor position information should be implemented on the controller of a PMSM. Normally, the rotor position is obtained by mechanical sensors which will bring several disadvantages such as extra cost, axial size, low reliability, and noise immunity. Therefore, a large number of publications have been focused on the rotor position estimation to achieve sensorless control [1]- [5].
The associate editor coordinating the review of this manuscript and approving it for publication was Shihong Ding .
The position sensorless estimation techniques for PMSM drives can be divided into two main categories: the methods based on back electromotive force (EMF) and high-frequency signal injection [6]- [8]. For medium-to high-speed operations, back EMF-based methods have been well founded and exhibited satisfying performance. However, the rotor position cannot be extracted effectively in zero-to low-speed operations since the amplitude of back EMF signal becomes too weak to be detected precisely [9], [10].
To realize sensorless control in low speed range of PMSM, high-frequency signal injection (HFI) method is commonly used. There are two more commonly used HFI methods: rotating signal injection and pulsating signal injection [11], [12]. The former was proposed first, but it only works for machines with salient pole, means that it cannot be used for SPMSM. As a result, the latter was proposed. It constructs a saliency for SPMSM based on the principle that the d-axis inductance will be reduced when the magnetic circuit in the d-axis is saturated. Two forms of injection scheme are compared in [13], [14]. Compared with the rotating signal injection method, the pulsating signal injection method is less sensitive to the parameter variation, more accurate and more adaptive for various kinds of machine [13], thus is more widely used.
Though the pulsating signal injection method possesses its own advantages, it is now subjected to two problems. The first is that the HF current is extracted by a constant bandwidth BPF, and a LPF is necessary to filter the noise in speed information, which will cause phase lag and decrease the dynamic performance, and this is not the best for wider speed range. [15], [16]. Mostly, the dead-time effect, which will cause the position error fluctuation in both the position error and the speed error [17], [18] is tackled as a problem in pulsating signal injection method as well.
To overcome the first drawback, an improved position observer based on square-wave voltage injection which omits the low-pass filter used to filter the noise in the speed estimation process was introduced in [19]. In [20], a simple algorithm based on square-wave voltage injection was proposed. It utilized the difference and average value of two consecutive sampling currents to figure out the high frequency current and fundamental current, so that the LPF and BPF can be removed. To cope with the position error fluctuation, numerical studies have been done. In [21], the second harmonic position error caused by parameter asymmetry in rotating or pulsating signal injection was analyzed, and a novel online strategy with dual-frequency injection was proposed to compensate the harmonic position error caused by inductance asymmetry. In [22], the position error caused by crosssaturation, which can result in sixth harmonic component in the inductances in field oriented coordinates, was analyzed. However, for the pulsating HF voltage injection method, few of them take the harmonic in the position error fluctuation caused by dead-time effect and dynamic performance into consideration at the same time.
Consequently, to compensate the position error caused by dead-time effect and improve the dynamic performance of the conventional pulsating HF voltage injection method, an improved position observer for conventional pulsating HF voltage injection method with position error compensation and fewer digital filters is designed in this paper.
In this paper, the harmonic component in position error caused by dead-time effect is analyzed firstly. Then, based on the theoretical analysis on quasi-resonant controller, an improved position observer for HF pulse voltage signal injection is proposed. In this observer, the digital BPF used to extract high frequency current response is replaced by a quasi-resonant controller, and an improved Luenberger observer is introduced to omitted the LPF used to filter the noise in speed information by using the predictive speed for the speed loop feedback. Therefore, the dynamic performance can be improved because of the reduction in the number of digital filters. Besides, to reduce the sixth harmonic in the magnitude of position error and speed error caused by dead-time effect, a PIR controller which connects a frequency adaptive quasi-resonant controller in parallel with the PI regulator in the Luenberger observer, is adopted. Finally, the effectiveness of the improved position observer is verified by experimental results on a 1.5-kW IPMSM drive platform.

II. ANALYSIS OF CONVENTIONAL PULSATING SINUSOIDAL VOLTAGE INJECTION METHOD A. PRINCIPLE OF CONVENTIONAL PULSATING SINUSOIDAL VOLTAGE INJECTION METHOD
The schematic diagram of the conventional high-frequency pulsating sinusoidal voltage injection method [13] is shown in Fig. 1. Firstly, a HF pulsating sinusoidal voltage signal is injected into the estimated d axis to induce HF current. Then the position can be estimated through the minimization of the amplitude-modulated carrier current response that is measured along the axis which is orthogonal to the injection axis. The HF model of PMSM represented in the rotor dq axis can be modelled as (1), where the cross-saturation effects are neglected.
where u dh and u qh are the HF voltage of the actual dq-axis, respectively. i dh and i qh are the HF current of the actual dqaxis, L dh and L qh are the stator HF inductances, R s is the stator resistance, ω e is the electrical rotor speed, ω h is the injected HF signal, and ψ f is the magnetic linkage flux. If the motor operates at low speed and the injection frequency is set high enough, the voltage drops on the stator resistance and the items associated with the rotor speed in (1) can be neglected. Then, PMSM can be approximated to an inductive load and (1) can be simplified as If the actual rotor position is defined as θ and the estimated rotor position isθ, the difference between them is position error θ. Then, the HF-induced current in the estimated synchronous reference frame can be deduced as where L and Lare the average inductance and the difference inductance respectively, and L = (L dh +L qh )/2, L = (L dh − L qh )/2. When the HF voltage signal U h cos(ω h t) is injected into the estimated d -axis, (3) can be obtained as It can be seen from (4) that the HF current response of the estimated q-axis, which can be extracted by a BPF, contains the relevant information of the rotor position. Then multiply the HF signal with the modulation signal sin(ω h t). Then the required regulator input f ( θ) which contains position error information can be obtained by a LPF to filter the low frequency components where K = − LU h ω h L dh L qh . It can be seen from (5) that if f ( θ) is passed through a phase-locked loop (PLL) or an observer, the estimated position can converged with the actual position when the parameters are appropriate. Then, the position and speed information can be observed. In Fig. 1, a Luenberger observer is selected [15], [19].

B. ANALYSIS OF POSITION ERROR CAUSED BY DEAD-TIME EFFECT
In a pulse width modulation (PWM) inverter-fed PMSM drive, a dead time should be inserted in the IGBT drive signal to prevent short circuit in DC link caused by the simultaneous conduction of both switches in one inverter leg. However, such a blanking time causes serious distortion in output voltages, and then, the harmonics will occur in phase current Assuming t on is the turn-on time delay of the switching device, t off is the turn-off time delay of the switching device, t d is the dead time. Then the inverter time error T err can be expressed as Subsequently, the phase voltage error can be given as where T is the PWM carrier period, sign(i p ) is a sign function of phase current, U dc is the DC-link voltage. In (7), the phase voltage error is a constant value. Therefore, the harmonic content in phase current will increase further when the motor is operating at low speed and light load. The main harmonic component superposed in the fundamental amplitude of phase current caused by dead-time is sixth harmonic [23]. Hence, the phase current can be expressed as where x is three phase a, b and c, respectively. i corresponds to 0, −1 and 1, respectively. ω e is the electrical rotor speed. I 1 and ϕ 1 are fundamental wave amplitude and initial phase, respectively. I 6 and ϕ 6 are the amplitude and initial phase of sixth harmonic which is superposed in the fundamental amplitude of phase current. By further calculating (8) with trigonometric function formula, the fifth and seventh harmonics of fundamental component in phase current can be obtained as Transform (9) to SRF, therefore, the current can be expressed as i d = I 1 sin ϕ 1 − I 6 sin(6ω e t + ϕ 6 ) sin ϕ 1 i q = −I 1 cos ϕ 1 + I 6 sin(6ω e t + ϕ 6 ) cos ϕ 1 (10) It can be simplified as where I d = I 1 sin ϕ 1 , I q = −I 1 cos ϕ 1 , I d6 = −I 6 sin ϕ 1 , It can be seen from (11) that dead-time effect results in sixth harmonics in SRF. Therefore, dead-time effect will cause the sixth harmonic to be superimposed on the fundamental amplitude of the HF current in SRF [24].
where I xh and ϕ xh are the amplitude and initial phase of the high-frequency current, respectively. I xh6 and ϕ xh6 are the amplitude and initial phase of the sixth harmonic superimposed on the amplitude of the HF current, respectively.
Transforming the high frequency current response of the actual dq axis in (12) to the estimated dq axis.
Subsequently, the estimated q axis HF current response with the sixth harmonic component superimposed on the amplitude of the current is finally demodulated by the signal to obtain f ( θ) It can be seen from (14) that in f ( θ), besides the DC component, the sixth harmonic component of the fundamental wave is superimposed. If the estimated rotor position converges to the actual rotor position in this case, θ can be expressed as If the sixth harmonic signal in f ( θ) is not processed, the sixth harmonic component of the fundamental wave will be present in the final rotor position error.

III. REVIEW OF THE QUASI-RESONANT CONTROLLER
In recent years, the Quasi-Resonant Controller has been widely used to supress multiple harmonics at resonant frequencies because of its flexibility, computation efficiency, and infinite gain of resonant controllers at resonant frequencies [25]- [27]. The transfer function of ideal resonant controller [25] can be expressed as where k ir is the resonant coefficient, ω 0 is the resonant frequency.
For the ideal resonant controller, the gain at the resonant frequency is expressed in (17). It can be seen that the gain of G R (s) at the resonant frequency ω 0 is infinite, therefore, the steady tracking error of the frequency ω 0 in the AC reference signal can be eliminated, and the influences of the external disturbance can be completely resisted.
However, since the gain of the resonant controller is infinite only at the resonant frequency, and the gain outside the resonant frequency decreases dramatically, which will cause the ideal resonant controller to be sensitive to the changes in the reference signal frequency, resulting in the system to fluctuate easily. Moreover, the ideal resonant controller is not conducive in practice due to its narrow bandwidth and the limitations in the precisions of the digital system. Hence, this paper adopts the quasi-resonant controller [23], [24], the transfer function can be expressed as where ω c is the cutoff frequency of the quasi-resonant controller. Fig. 3 represents the Bode diagram which shows the effect of parameters ω c and k ir on the performance of the quasi-resonant controller. Fig. 3(a) is the Bode diagram with k ir constant at 1, ω c changes when ω 0 = 1000π(f = 500HZ). Fig. 3(b) is the Bode diagram with ω c constant at 250 π, k ir changes when ω 0 = 1000π(f = 500HZ). Fig. 3(a) exhibits that the gain and phase of the controller at the resonant frequency point do not change, only the bandwidth of the controller changes with the change of ω c . The bandwidth of the controller increases with the increase of ω c , nevertheless, the excessive bandwidth will affect the frequency selection characteristics of the controller, and the speed of the system response will be affected when the bandwidth is too small. Fig. 3(b) shows that the gain of the quasi-resonant controller is proportional to k ir , and large k ir leads to large peak gain, but the phase and bandwidth is fixed. It also can be seen that G R (s) provides a high open-loop gain at ω 0 . Therefore, the quasi-resonant controller can suppress harmonics and has an excellent tracking for the input when used in closed loop. Subsequently, the quasi-resonant controller is used to extract high frequency current response and reduce the sixth harmonic in the magnitude of position error and speed error caused by dead-time effect.
Subsequently, the equivalent principle block diagram of the quasi-resonant controller can be obtained as shown in Fig. 2.

IV. DESIGN AND ANALYSIS OF IMPROVED ROTOR POSITION OBSERVER BASED ON QUASI-RESONANT CONTROLLER
Due to the effects of digital filter and dead-time effect, conventional pulsating sinusoidal voltage injection method suffers from chattering. Therefore, in this paper, an improved rotor position observer based on quasi-resonant controller is proposed to improve the sensorless performance. The structure of the improved position observer is shown in Fig. 4. It includes HF current extraction module, Luenberger based position observer without LPF module and position error compensation module.

A. HF CURRENT EXTRACTION MODULE DESIGN
BPF is used to extract high frequency current response in conventional pulsating sinusoidal voltage injection method. For digital control systems, the BPF is generally implemented in the form of a digital filter. However, the tuning of the parameters is cumbersome, and the introduction of BPF will limit the bandwidth of position observers. Additionally, if the low-order BPF is selected, the required high-frequency current component cannot be effectively extracted, the signalto-noise ratio (SNR) of the high-frequency signal is reduced, and the rotor position estimation will fluctuate greatly. While high-order BPF will introduce serious delay problem, which will affect the dynamic performance of sensorless control.
As shown in the ''HF current extraction module'' in Fig. 4, the BPF is replaced by a quasi-resonant controller. When the k ir is constant to 1, the resonant frequency is set as the frequency of the injected signal, therefore, the adjustment of the filter characteristic can be realized by adjusting ω c . This improvement can reduce the workload in the debugging process and the phase delay caused by the digital filter.
In addition, since the conventional BPF does not have inner loop adjustment, the HF disturbance generated by the step input cannot be effectively suppressed. Nevertheless, for the quasi-resonant controller, an inner-loop regulation exists. When the step signal is input, the output signal is quickly tracked by the signal y 1 obtained by integral feedback, thereby reducing the influence of the step input on y 2 .

B. LUENBERGER BASED POSITION OBSERVER WITHOUT LPF MODULE DESIGN
For the traditional PI observer based on PLL or Luenberger observer with electromagnetic torque feedforward, the observed speed information will contain certain noise. To make the observed speed can be used for normal speed closed-loop, it is necessary to add a LPF after the observed speed to filter out the noise. However, the introduction of the LPF will greatly reduce the bandwidth of the speed loop and affect the dynamic performance of the system.
As shown in the ''Luenberger based position observer without LPF module'' in Fig. 4, a modified Luenberger observer is used to observe the speed. The predictive speed is used to replace the speed after the LPF. So that the LPF used to filter out the noise can be omitted, which will improve the dynamic performance of the system.
When the viscous friction is neglected, the mechanical model of the motor can be expressed as [28] p ω e θ e = 0 0 1 0 where p is the differential operator, ω e and θ e are the rotor electrical speed and electrical angle, T e and T l are the electromagnetic torque and load torque, J is the moment of inertia, n p is the pole pair number. In (19), the electromagnetic torque can be expressed as n p ψ f i q (20) where ψ f is the permanent magnet flux linkage. Then the estimated states can be expressed as Through (20) (21), by selecting the observer gains K pi as proportional and integral (PI) gains and K d as a proportional gain, the ''Luenberger based position observer without LPF module'' in Fig. 4 can be built.
The closed-loop transfer function of Luenberger observer can be expressed aŝ where J andĴ are the moment of inertia and its observation, respectively, k p , k i and k d are the proportional, integral, differential gain of the Luenberger observer, respectively. Finally, the modified Luenberger observer can not only accurately identify the rotor position without phase lag, but also omits the speed LPF and improves the dynamic performance of the system.

C. POSITION ERROR COMPENSATION MODULE DESIGN BASED ON FREQUENCY ADAPTIVE QUASI-RESONANT CONTROLLER
According to the analysis in the second section, the position error information f ( θ ) contains the sixth harmonic component of the fundamental wave due to the dead-time effect. Then the position information is observed from f ( θ ) through the Luenberger observer.
A Luenberger observer is composed of three parts: PI regulator, PMSM mechanical system mathematical model and error feedforward compensation. Specifically, f ( θ) outputs the signal -n p T l /J required by the later mechanical mathematical mode through PI regulator, then calculates the speed information through the mathematical model of the motor. Finally, the stability of the system will be enhanced by error feedforward compensation.
However, the traditional PI controller can only control the DC signal without static error. When the input signal f ( θ ) contains AC component, the PI controller cannot compensate it, which will eventually lead to the sixth harmonic component of the fundamental wave in the rotor position error. Therefore, for the choice of the controller, it is not only required to adjust the DC component, but also to suppress the AC disturbance component.
In this paper, a proportional-integral-resonant (PIR) controller, which connects a quasi-resonant controller in parallel with the PI regulator, is adopted. The block diagram of the PIR controller scheme is shown in Fig. 5. The transfer function of the PIR controller can be expressed as Fig. 6 represents the Bode diagram of the PIR controller when ω 0 = 300, K P = 10, K I = 1, k ir = 10, ω c = 30. It can be seen that the open-loop gain of the controller at the resonance frequency point increases sharply, which means the controller can suppress the AC signal at the resonance frequency point well.
Therefore, to suppress the sixth harmonic component in f ( θ) on the position error, the ''position error compensation module'' as shown in Fig. 4, which adopts a PIR controller, is proposed in this paper. Since the quasi-resonant controller can achieve zero static error control of the sinusoidal signal at the resonant frequency point, therefore, a frequency adaptive quasi-resonant controller with a resonant frequency of 6ω e is connected in parallel with the PI regulator of the Luenberger observer, and ω e adopts the observed speed. Thus, the resonant frequency of the quasi-resonant controller can be adaptively changed with the change of the angular frequency of the motor to achieve a more accurate rotor position estimation.
Finally, the improved position observer with a Luenberger observer and a quasi-resonant controller can be seen as a PIDR observer, which can accurately identify the rotor position without phase lag and reduce the influence of dead-time effect on position estimation simultaneously.

V. EXPERIMENTAL VALIDATION
To verify the validity of theoretical analysis of the proposed control method, the proposed algorithm adopted for a IPMSM, a dSPACE 1103 controller is used to drive the prototype. The experiment platform is shown in Fig. 7, which includes IPMSM for research in this paper, its parameters are shown in Table 1. The magnetic powder brake is used as a load. The actual rotor position is obtained by an incremental encoder with 1024 pulses/revolution for comparison with the observed rotor position. The switching frequency and  deadtime of the voltage source inverters are set at 10kHz and 2µs, the current sampling frequency is 10 kHz, the DC bus voltage is 100 V.
For the proposed position observer, to obtain the position information effectively, the magnitude and frequency of the injected carrier voltage is set as 14.5V and 500Hz. ω h of the quasi-resonant controller in the front stage is taken 1000π, because the frequency of the injected carrier voltage is 500Hz. ω c is 500π according to the bode diagram of the quasi-resonant controller. To extract the effective component and avoid introducing serious phase lag, the parameter T of the LPF used to demodulate f ( θ ) is selected to be 1/(900π). The k p , k i and k d of the Luenberger observer is 2.25, 30 and 100, and the parameterĴ is 0.0015. k ir and ω c of the quasi-resonant controller to suppress the rotor position error in the latter stage is taken as 5 and 200 according to the bode diagram of the quasi-resonant controller. The second-order Butterworth bandpass filter is selected for the traditional position observer used for comparison and the passband is set to [450HZ,550HZ]. Fig. 8 shows the comparison experimental results before and after the improved algorithm is enabled when n = 50rpm (2% rated speed) in no load condition. Fig. 8(a) shows the actual speed, estimated speed and speed error before and after the improved algorithm is enabled when the dead time is 2µs. Fig. 8(b) shows the actual rotor position, estimated rotor position and position error before and after the improved algorithm is enabled when the dead time is 2µs. Fig. 8(c) shows actual rotor position, estimated rotor position, speed error and position error before and after the improved algorithm is enabled when the dead time is 5µs.

A. EXPERIMENT RESULTS OF NO-LOAD STEADY STATE
From Fig. 8, it can be seen that under different dead times, the improved algorithm can effectively suppress the speed error and the rotor position error, and the fluctuation of the actual speed can be reduced at the same time, so that the motor can run more smoothly. Through the FFT analysis of the speed error and the rotor position error it is found that, when the dead time is 2µs and the improved algorithm is disabled, the amplitude of the harmonics in the speed error and the rotor position error is 4.02rpm, 5.7 • . When the improved algorithm is enabled, the amplitude of the harmonics in the speed error and the rotor position error is 1.57rpm, 1.49 • . Hence, compared with the traditional algorithm, the improved algorithm improves the speed estimation accuracy by 60.9% and the rotor position estimation accuracy by 74%. Moreover, when the dead time is 5µs and the improved algorithm is disabled, the amplitude of the harmonics in the speed error and the rotor position error is 4.25rpm, 6.24 • . When the improved algorithm is enabled, the amplitude of the harmonics in the speed error and the rotor position error is 1.725rpm, 1.6 • , the improved algorithm improves the speed estimation accuracy by 59.4% and the rotor position estimation accuracy by 74.3%. Therefore, the improved method has good effect in different dead time. Fig.9 shows the comparison experimental results before and after the improved algorithm is enabled when the motor  is close to standstill (n = 10rpm). It can be seen that the improved algorithm works well too.

B. COMPARISON OF ENABLING RESULTS OF IMPROVED ALGORITHM AT DIFFERENT SPEED
To further verify the effectiveness of the improved algorithm, an experimental comparison is made every 10 rpm for the speed from 10rpm to 100rpm, and then the 6 th harmonic amplitude in the error information is obtained by FFT analysis. Subsequently, the comparison chart shown in Fig. 10 is obtained. Fig. 10(a) shows the effect of speed error compensation at different speed and dead time. Fig. 10(b) shows the effect of theta error compensation at different speed and dead time.

C. ANALYSIS OF DYNAMIC TRACKING PERFORMANCE
To verify the dynamic performance of the improved algorithm, Fig. 11 shows the shifting comparison between the traditional algorithm and the improved algorithm in no-load condition. When the conventional position observer is adopted with the initial operation speed at 50rpm, the maximum reachable speed is 250rpm as shown in Fig. 11(a). To achieve a fair comparison, the experimental results using the improved position observer are shown in Fig. 11(b) as well when the motor operates at the same speed changing from 50rpm to 250rpm and then to 50rpm. In Fig. 11, the actual speed θ, estimated speedθ, speed error n, and position error θ are all presented. From Fig. 11, it can be seen that the speed overshoot and response time of the improved position observer are much smaller than those of the traditional observer during the shifting process, and the speed error and position error are reduced a lot. In the traditional position observer, the overshoot of speed error is about 25rpm, and the position error overshoot is about 28.5 • . In the improved position observer,  the overshoot of speed error is about 13rpm, and the position error overshoot is about 14.3 • . It is calculated that the speed error overshoot of the improved position observer is 48% lower than that of the conventional observer, and the overshoot of the rotor position error is 49.8%. It can be found that although the estimated rotor position cannot track the actual rotor positon without any difference for the improved position observer because the digital filter still exists in the observer. However, compared with the conventional method, the tracking performance is improved a lot.
As shown in Fig. 12, with the initial operation speed at 50rpm, the maximum reachable speed for the improved position observer is 400rpm. It can be found that the improved algorithm can still run smoothly when the motor is subjected to a wider speed range, indicating that the improved position observer has good dynamic performance and can operate well under medium speed region.

D. ANALYSIS OF ANTI-DISTURBANCE PERFORMANCE
To verify the anti-disturbance performance of the new algorithm, Fig. 13 shows the load-changing process of the traditional algorithm and the improved algorithm at 50rpm. Fig. 13(a) shows the rotor position, q-axis current and position errors when the conventional algorithm suddenly accumulates 4.4 N·m load. Fig. 13(b) shows the rotor position, q-axis current and position errors when the improved algorithm suddenly accumulates 4.4 N·m load. Fig. 13 exhibits that the improved algorithm can better track the actual rotor position during the process of variable load, and the motor runs more smoothly.

VI. CONCLUSION
In this paper, an improved position observer, which can compensate the position error caused by dead-time effect and improve the dynamic performance of the conventional high frequency signal injection method, is proposed. According to the experimental validation, it can be concluded that compared with conventional pulsating sinusoidal voltage injection method, the improved algorithm can reduce the position estimation error not only in steady state operation conditions, but in variable speed and variable load conditions. And the variable speed range can be widened as well. However, for the proposed observer, there are a lot of parameters, which will bring challenges in selection of the values. Therefore, it should be simplified in the future work.