A Study on the Transient Performance of Compensated PLL-Type Estimators for Sensorless IPMSMs

Abstract

The transient performance of sensorless control for interior permanent magnet synchronous motors (IPMSMs), based on back-electromotive force (back-EMF) estimation, is a critical factor in ensuring the high reliability of motor drive systems. Although rotor speed and position can be accurately estimated under steady-state conditions, estimation errors tend to increase during transient states such as acceleration, deceleration, and load torque variations. The enhancement of transient stability is closely related to the overshoot in the estimated position and speed errors. In this paper, the maximum overshoot of the estimated position and speed errors during transient operation is analyzed. Furthermore, compensation strategies are proposed to reduce the magnitude of these overshoots. The effectiveness of the proposed sensorless control method is validated through comparative analysis with existing approaches.

1. Introduction

Over the past two decades, permanent magnet synchronous motors (PMSMs) have been widely utilized in various industrial applications, including electrified vehicles (xEVs) such as battery electric vehicles (BEVs), hybrid electric vehicles (HEVs), and plug-in hybrid electric vehicles (PHEVs), due to their high torque density and efficiency. Recently, PMSMs have received particular attention as powertrain systems in automotive applications, owing to their simple structure and wide high-speed operating range.

Numerous studies have been conducted on sensorless control of PMSMs, which eliminates the need for position and speed sensors. This approach offers advantages such as compact system size, reduced cost, and improved reliability [1,2,3,4]. Rotor position estimation is essential for implementing sensorless control. Estimation methods are generally classified into two categories. The first category utilizes back electromotive force (back-EMF) to estimate rotor position and speed over a wide speed range, as the back-EMF magnitude is proportional to rotor speed [5]. The second category involves high frequency signal injection, which is effective at low speeds and standstill conditions [6].

Based on these methods, stable operation of interior PMSMs (IPMSMs) can be achieved by appropriately switching between signal injection and back-EMF estimation techniques, enabling reliable performance across the entire speed range [7,8]. In many industrial applications, including traction motor control in EVs and HEVs, sensorless control is frequently employed alongside sensored control to improve overall system reliability. To monitor the health status of position and speed sensors, residual based methods defined as the deviation between estimated and measured speed are commonly employed.

For effective fault detection and control algorithm switching, minimizing the residual is crucial. However, the residual cannot be reduced unless the peak deviation between the measured and actual values is sufficiently low. Therefore, transient-state stability analysis must consider both the residual and the overshoot in the estimated values obtained from back-EMF and position/speed estimators. If the maximum overshoot in both steady and transient states remains low without degrading overall performance, the robustness and dynamic response of sensorless control can be significantly improved [9,10,11,12,13,14,15,16,17,18,19,20,21,22]. In [23], the flux sliding mode observer was introduced for sensorless control of PMSMs, successfully mitigating the issue of flux saturation. However, its ability to suppress harmonic components remains limited. Many existing flux or back-EMF estimation techniques adopt a cascaded structure combining an observer with a phase-locked loop (PLL) [24]. Although this configuration enhances synchronization, it inevitably increases the system’s order. The addition of filters to attenuate high-frequency harmonics further complicates the system, leading to slower dynamic response and potential stability degradation. In [25], an adaptive high-gain observer presents that effectively reduces position estimation errors during sudden load torque changes and remains robust against parameter mismatches. However, its implementation involves complex mathematical modeling and leads to high computational demands.

The main contribution of this paper is the design of a phase locked loop (PLL)-based estimator for IPMSMs, aimed at enhancing sensorless control performance under transient conditions. The proposed design incorporates compensation strategies to reduce overshoot in estimated position and speed errors. Compensation terms for speed and angle are derived from the proposed equations, particularly under rapid changes in motor speed and load torque. The effectiveness and stability of the proposed method are verified through simulation and experimental results under transient operating conditions.

2. Extended EMF Model of IPMSMs

Figure 1 defines a vector diagram applied in the sensorless control of PMSMs. The relationship between the three frames is shown. The α-β axis means the stationary reference frame and d-q axis represents the rotor reference frame. Also, the γ-δ axis shows a virtual frame applied in sensorless control. The $∆\widehat{\theta }$ is the estimated angle of difference between d-q axis and γ-δ axis.

The voltage equation of IPMSMs in the estimated rotating reference frame (γ-δ axis) is derived as follows [5]:

$$\left[\begin{array}{c}{V}_{\gamma }\\ V_{\delta }\end{array}\right]=\left[\begin{array}{cc}R+p{L}_d& -{\omega }_r{L}_q\\ {\omega }_r{L}_d& R+p{L}_q\end{array}\right]\cdot \left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\left[\begin{array}{c}{\epsilon }_{\gamma }\\ {\epsilon }_{\delta }\end{array}\right]$$

(1)

$$\begin{array}{l}\left[\begin{array}{c}{\epsilon }_{\gamma }\\ {\epsilon }_{\delta }\end{array}\right]={\omega }_r{\varphi }_f\left[\begin{array}{c}-\mathit{sin}\Delta \theta \\ \mathit{cos}\Delta \theta \end{array}\right]+L_{1}p\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+{\omega }_r{L}_2\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\left({\widehat{\omega }}_r-{\omega }_r\right)L_3\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]\\ L_1=\left[\begin{array}{cc}-\left(L_d-L_q\right){\mathit{sin}}^2∆\theta & \left(L_d-L_q\right)\mathit{sin}∆\theta \cdot \mathit{cos}∆\theta \\ \left(L_d-L_q\right)\mathit{sin}∆\theta \cdot \mathit{cos}∆\theta & \left(L_d-L_q\right){\mathit{sin}}^2∆\theta \end{array}\right]\\ L_2=\left[\begin{array}{cc}-\left(L_d-L_q\right)\mathit{sin}∆\theta \cdot \mathit{cos}∆\theta & -\left(L_d-L_q\right){\mathit{sin}}^2∆\theta \\ -\left(L_d-L_q\right){\mathit{sin}}^2∆\theta & \left(L_d-L_q\right)\mathit{sin}∆\theta \cdot \mathit{cos}∆\theta \end{array}\right]\\ L_3=\left[\begin{array}{cc}\left(L_d-L_q\right)\mathit{sin}∆\theta \cdot \mathit{cos}∆\theta & -L_q{\mathit{cos}}^2∆\theta -L_q{\mathit{sin}}^2∆\theta \\ L_d{\mathit{sin}}^2∆\theta +L_q{\mathit{cos}}^2∆\theta & -\left(L_d-L_q\right)\mathit{sin}∆\theta \cdot \mathit{cos}∆\theta \end{array}\right]\end{array}$$

(2)

In Equation (2), the voltage equation in γ-δ axis is simple in a non-salient pole motor. However, in salient pole motors such as IPMSMs, they are very complex equations. To solve this problem, an extended EMF method is proposed as below.

In Equation (1), the voltage equation of IPMSMs in the d-q axis can be derived as follows:

$$\left[\begin{array}{c}{V}_d\\ V_q\end{array}\right]=\left[\begin{array}{cc}R+p{L}_d& -{\omega }_r{L}_q\\ {\omega }_r{L}_q& R+p{L}_d\end{array}\right]\cdot \left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{c}0\\ E_{ex}\end{array}\right]$$

(3)

$$E_{ex}={\omega }_r\left[(L_d-L_q)i_d+\psi \right]-(L_d-L_q)\left(p{i}_q\right)$$

(4)

where E_ex is the extended EMF voltage and p is d/dt. The voltage equation with respect to the γ-δ axis is given as below:

$$\left[\begin{array}{c}{V}_{\gamma }\\ V_{\delta }\end{array}\right]=\left[\begin{array}{cc}R+p{L}_d& -{\omega }_r{L}_q\\ {\omega }_r{L}_q& R+p{L}_d\end{array}\right]\cdot \left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\left[\begin{array}{c}{e}_{\gamma }\\ e_{\delta }\end{array}\right]$$

(5)

$$\left[\begin{array}{c}{e}_{\gamma }\\ e_{\delta }\end{array}\right]=E_{ex}\left[\begin{array}{c}-\mathit{sin}\Delta \theta \\ \mathit{cos}\Delta \theta \end{array}\right]+({\omega }_r-{\widehat{\omega }}_r)L_d\left[\begin{array}{c}{i}_{\delta }\\ -i_{\gamma }\end{array}\right]$$

(6)

The last term of Equation (6) has no significant impact because the speed error could be a sufficiently low value in the steady-state condition. So, Equation (5) can be rewritten as Equation (7):

$$\left[\begin{array}{c}{V}_{\gamma }\\ V_{\delta }\end{array}\right]=\left[\begin{array}{cc}R+p{L}_d& -{\omega }_r{L}_q\\ {\omega }_r{L}_q& R+p{L}_d\end{array}\right]\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+E_{ex}\left[\begin{array}{c}-\mathit{sin}\Delta \theta \\ \mathit{cos}\Delta \theta \end{array}\right]$$

(7)

From the estimated E_ex with respect to the γ-δ frame, the estimated position error $\mathsf{\Delta }\widehat{\theta }$ is finally derived as Equation (8):

$$\Delta \widehat{\theta }={\mathit{tan}}^{-1}\left({\displaystyle \frac{-E_{ex}\cdot \mathit{sin}\Delta \theta }{E_{ex}\cdot \mathit{cos}\Delta \theta }}\right)=-{\mathit{tan}}^{-1}\left({\displaystyle \frac{{\widehat{e}}_{\gamma }}{{\widehat{e}}_{\delta }}}\right)$$

(8)

where V_dq and i_dq are stator voltages and currents in the d−q axis, V_γδ is stator voltage in the γ−δ axis, e_γδ is back-EMF in the γ−δ axis, R is stator resistance, L_dq is d−q axis inductance, Ψ is permanent magnet flux linkage, θ is rotor position in the d−q axis, ω_r is rotor angular speed in the d−q axis, ^⋀ is estimated signal value, and $∆\widehat{\theta }$ is estimated position error.

3. Extended EMF Estimation

The block diagram for the extended EMF estimation using disturbance observer is depicted in Figure 2. The differential operator is included in the disturbance observer to determine the reverse model of the system. Therefore, a low-pass and a high-pass filter for minimizing the negative effects of the differential operation should be included in the disturbance observer as shown Equation (9). Accordingly, appropriate gain selection for the observer is vital to increase the estimator stability [5].

$${\widehat{\mathit{E}}}_{\gamma \delta }={\displaystyle \frac{g_{ob}}{s+g_{ob}}}\left({\mathit{V}}_{\gamma \delta }^{*}+j{\widehat{\omega }}_r{\overline{L}}_q\cdot {\mathit{I}}_{\gamma \delta }-\overline{R}\cdot {\mathit{I}}_{\gamma \delta }\right)-{\displaystyle \frac{s}{s+g_{ob}}}\left({\overline{L}}_d\cdot g_{ob}\cdot {\mathit{I}}_{\gamma \delta }\right)$$

(9)

where $\overline{R}$, ${\overline{L}}_d$, and ${\overline{L}}_q$ are nominal motor parameters and ${\widehat{E}}_{\gamma \delta }$ is back-EMF vector in the γ-δ axis. $V_{\gamma \delta }^{*}$ and $V_{\gamma \delta }$ are reference voltage and stator voltage vectors in the γ-δ axis, respectively. $I_{\gamma \delta }$ is stator current vector in the γ-δ axis and j is the imaginary unit. Therefore, ${\widehat{E}}_{\gamma \delta }={\widehat{e}}_{\gamma }+j{\widehat{e}}_{\delta }$.

The observer gain g_ob should be sufficiently larger than the angular speed ω_r. Generally, the g_ob can be selected as two times of ω_r. Also, the minimum value of g_ob should be considered. Therefore, the stable range of g_ob can be defined as Equation (10) [9,15,18,19].

$$\left|{\omega }_r\right|\cdot n\le g_{ob}<{\alpha }_c,\left(n=k_e/\sqrt{m_{ob}^2-(L_d-L_q{)}^2{\left|i\right|}_{\mathit{max}}^2}\right)$$

(10)

where k_e is a back-EMF constant and α_c is a current controller bandwidth. Also, ${\left|i\right|}_{\max }$ is the maximum value of stator current and m_ob is a tuning value for an appropriate back-EMF estimation. In this paper, m_ob can be selected as a minimum value to determine minimum g_ob considering maximum rotor speed and maximum n value with respect to positive denominator [9].

4. Speed and Position Estimation

4.1. Analysis of the PLL-Type Estimator

The rotor position and speed estimation with respect to the output of the disturbance observer can be derived by using the PLL-type estimator. The $\mathsf{\Delta }\widehat{\mathrm{\theta }}$ can be determined from Figure 3 and Figure 4 when the difference between actual position error and estimated position error is very small such as Equation (11). If the actual position error is very small, Equation (12) can be derived from Equation (11) and Figure 4 [19].

$$\Delta \widehat{\theta }\approx \Delta \theta =\theta -\widehat{\theta }$$

(11)

$$\widehat{\theta }={\displaystyle \frac{K_{ep}\cdot s+K_{ei}}{s^2+K_{ep}\cdot s+K_{ei}}}\cdot \theta$$

(12)

where K_ep and K_ei are PI gain for the PLL-type estimator and s is the complex frequency variable. The estimated rotor angular speed ${\widehat{\mathrm{\omega }}}_r$ can be formed from Equation (12) with Figure 4.

$${\widehat{\omega }}_r=\left({\displaystyle \frac{g_{ob}}{s+g_{ob}}}\right){\displaystyle \frac{K_{ei}}{s}}\Delta \widehat{\theta }\approx \left({\displaystyle \frac{g_{ob}}{s+g_{ob}}}\right){\displaystyle \frac{K_{ei}}{s}}\left(\theta -\widehat{\theta }\right)$$

(13)

If $\widehat{\mathrm{\theta }}$ in Equation (12) is inserted into Equation (13) and the minimum value of g_ob is selected as five times value of the PLL-type estimator bandwidth ρ to neglect g_ob bandwidth influence, the ${\widehat{\mathrm{\omega }}}_r$ is redefined as Equation (14):

$${\widehat{\omega }}_r\approx \left({\displaystyle \frac{K_{ei}s}{s^2+K_{ep}s+K_{ei}}}\right)\theta =\left({\displaystyle \frac{K_{ei}}{s^2+K_{ep}s+K_{ei}}}\right){\omega }_r$$

(14)

The standard form of the 3rd order characteristic polynomial is determined such as Equation (15).

$$c\left(s\right)=(s+{\omega }_{spd})(s^2+K_{ep}\cdot s+K_{ei})=(s+{\omega }_{spd})(s^2+2\varsigma {\omega }_{n}s+{\omega }_n^2)$$

(15)

$$\therefore K_{ep}=2\varsigma {\omega }_n,K_{ei}={\omega }_n^2$$

(16)

where ζ is the damping ratio and ω_n is the natural frequency. In order to increase the tracking performance and stability, the selection of ω_n and ζ should be considered. If the ζ is set to 1, the stable system is given by critical damping characteristic because two poles are located at same point -ρ. So, the dynamic response and stability can be determined by selecting only ω_n. In the following, the numerical equation is studied how these parameters should be chosen to obtain a stable bandwidth.

4.2. Selection of Speed and Position Estimator Bandwidth ρ

To define the PLL-type estimator bandwidth ρ, it is assumed that the acceleration is constant and ω_r is rampwise changed during a short time interval. Therefore, the relationship between estimated speed and estimated position can be defined as Equation (17) [7,8,11,14].

$$\Delta {{\omega }^{\prime }}_r=\Delta {\widehat{\theta }}^{\prime }=0,\Delta {\omega }_r^{*}={\displaystyle \frac{2{{\omega }^{\prime }}_r}{\rho }},\Delta {\widehat{\theta }}^{*}={\mathit{sin}}^{-1}{\displaystyle \frac{{{\omega }^{\prime }}_r}{{\rho }^2}}$$

(17)

where $\Delta {\omega }_r^{*}$ and $\Delta {\theta }_r^{*}$ are the stable equilibrium points considering the error dynamics by Lyapunov law. The symbol ‘ ʹ ’ in superscript means the variable represented differential term. From Equation (17), ρ is given by Equation (18) [7,8].

$$\rho =\sqrt{{\displaystyle \frac{{\left|{{\omega }^{\prime }}_r\right|}_{max}}{{\mathit{sin}\left|\Delta \widehat{\theta }\right|}_{max}}}}$$

(18)

where $|\mathsf{\Delta }\hat{\mathsf{\theta }}{|}_{max}$  is the maximum error angle of allowed transient state. ${\left|{\omega }_r^{\prime }\right|}_{max}$   is the maximum acceleration at same condition. If the high ρ is selected, the tracking errors of the PLL-type estimator will be reduced. However, the noise signal effect is enlarged. Hence, the maximum error angle is increased at low speed in accordance with the low back-EMF.

5. Improvement of Speed Response in a Transient State

5.1. Compensation Design of Estimated Position Error

The $\Delta \widehat{\theta }$ is derived as Equation (8) on the assumption that the speed error is sufficiently low in a steady state. However, if the estimated speed error is high, the γ-δ axis currents can be determined by using d-q axis currents and $\Delta \widehat{\theta }$ from Figure 1 [22].

$$i_{\delta }=i_d\mathit{sin}\Delta \widehat{\theta }+i_q\mathit{cos}\Delta \widehat{\theta }=\sqrt{i_d^2+i_q^2}\cdot \mathit{cos}\left(\Delta \widehat{\theta }-{\mathit{tan}}^{-1}{\displaystyle \frac{i_d}{i_q}}\right)$$

(19)

$$i_{\gamma }=-\left(i_q\mathit{sin}\Delta \widehat{\theta }-i_d\mathit{cos}\Delta \widehat{\theta }\right)=-\sqrt{i_d^2+i_q^2}\cdot \mathit{sin}\left(\Delta \widehat{\theta }-{\mathit{tan}}^{-1}{\displaystyle \frac{i_d}{i_q}}\right)$$

(20)

Comparing (19) and (20) with (6), the back-EMF in the γ-δ axis can be obtained as Equation (21).

$${\displaystyle \frac{e_{\gamma }}{e_{\delta }}}={\displaystyle \frac{E_{ex}\cdot (-\mathit{sin}\Delta \theta )+\Delta {\omega }_r{L}_d{i}_{\delta }}{E_{ex}\cdot \mathit{cos}\Delta \theta -\Delta {\omega }_r{L}_d{i}_{\gamma }}}=-\mathit{tan}\left(\Delta \theta +{\mathit{tan}}^{-1}\left({\displaystyle \frac{\Delta {\omega }_r{L}_d{i}_q}{E_{ex}+\Delta {\omega }_r{L}_d{i}_d}}\right)\right)$$

(21)

where $\Delta {\omega }_r={\omega }_r-{\widehat{\omega }}_r$. Therefore, $\Delta \widehat{\theta }$ can be expressed by Equation (22)

$$\begin{array}{l}{\mathit{tan}}^{-1}\left(-{\displaystyle \frac{e_{\gamma }}{e_{\delta }}}\right)=\Delta \widehat{\theta }+{\mathit{tan}}^{-1}\left({\displaystyle \frac{\Delta {\omega }_r{L}_d{i}_q}{E_{ex}+\Delta {\omega }_r{L}_d{i}_d}}\right)=\Delta \widehat{\theta }+{\theta }_{SC}\\ where-{\displaystyle \frac{\pi }{2}}<\Delta \theta +{\theta }_{SC}<{\displaystyle \frac{\pi }{2}}\end{array}$$

(22)

where θ_SC is a compensation angle for a fast alignment of the γ-δ axis and the d-q axis in transient state. And ∆ω_r can be estimated by Equation (30) explained from the next section.

5.2. Compensation Design of Estimated Speed Error

An input error signal $\Delta \widehat{\theta }$ of PLL-type estimator using an extended EMF estimation in Figure 4 can be defined as Equation (23) by assuming no parameter errors of motor (see Appendix A), exact feedforward control and sufficiently large bandwidth of current controller.

$$\begin{array}{l}{\sigma }_d=(V_d-{\widehat{V}}_d)=V_d^{*}-\widehat{R}\cdot i_d^{*}+{\widehat{\omega }}_r\cdot {\widehat{L}}_q\cdot i_q^{*}\\ {\sigma }_q=(V_q-{\widehat{V}}_q)=V_q^{*}-\widehat{R}\cdot i_q^{*}-{\widehat{\omega }}_r\cdot {\widehat{L}}_d\cdot i_d^{*}\end{array}$$

(23)

where σ_d and σ_q. are error signals in the γ-δ axis. Also, the reference current, $i_d^{*}$, $i_q^{*}$ and reference voltage $V_d^{*}$, $V_q^{*}$ are used for finding the estimated voltage ${\widehat{V}}_d$, ${\widehat{V}}_q$ and the actual voltage, V_d, V_q  in order to decrease noise signal effect [8]. And, an error signal vector σ in the synchronous reference frame can be derived by Equation (24) [7,8].

$$\begin{array}{l}\mathit{\sigma }=\widehat{\mathit{v}}-\widehat{\mathit{Z}}\cdot \widehat{\mathit{i}}\\ \widehat{\mathit{v}}=e^{\mathit{J}\Delta \theta }\left(\mathit{Z}+p\mathit{L}\right)e^{-\mathit{J}\Delta \theta }\cdot \widehat{\mathit{i}}+e^{\mathit{J}\Delta \theta }{\omega }_r\psi \\ \widehat{\mathit{v}}=\left[\begin{array}{c}{\widehat{V}}_d\\ {\widehat{V}}_q\end{array}\right],\widehat{\mathit{i}}=\left[\begin{array}{c}{\widehat{i}}_d\\ {\widehat{i}}_q\end{array}\right],\mathit{J}=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],\mathit{L}=\left[\begin{array}{cc}{L}_d& 0\\ 0& L_q\end{array}\right]\\ \widehat{\mathit{Z}}=\left[\begin{array}{cc}\widehat{R}& -{\widehat{\omega }}_r{\widehat{L}}_q\\ {\widehat{\omega }}_r{\widehat{L}}_d& \widehat{R}\end{array}\right],e^{\mathit{J}\Delta \theta }=\left[\begin{array}{cc}\mathit{cos}\Delta \theta & -\mathit{sin}\Delta \theta \\ \mathit{sin}\Delta \theta & \mathit{cos}\Delta \theta \end{array}\right]\end{array}$$

(24)

where $\widehat{\mathit{v}}$ is the output voltage vector of inverter and $\widehat{\mathit{i}}$ is the output current vector of inverter. If ρ is set to be ten times lower than α_c, the effect of current dynamics can be neglected. From the assumption of accurate current control, $\widehat{\mathit{v}}$ and $\widehat{\mathit{i}}$ can be defined as $\widehat{\mathit{v}}={\left[\begin{array}{cc}{V}_d^{*}& V_q^{*}\end{array}\right]}^T$ and $\widehat{\mathit{i}}={\left[\begin{array}{cc}{i}_d^{*}& i_q^{*}\end{array}\right]}^T$. From Equation (24), the error signal σ_d can be described by

$$\begin{array}{l}{\sigma }_d=-\psi {\omega }_r\mathit{sin}\Delta \theta -\left(\left({\omega }_r-\Delta {\omega }_r\right)\Delta L_q+\Delta {\omega }_r\Delta L\right)i_q\\ +\Delta R{i}_d+\left({\omega }_r+\Delta {\omega }_r\right)\Delta L\mathit{sin}\Delta \theta \left(i_d\mathit{cos}\Delta \theta +i_q\mathit{sin}\Delta \theta \right)\end{array}$$

(25)

where $\Delta L_q=L_q-{\widehat{L}}_q$ and $\Delta L=L_q-L_d$. If the motor parameter errors are ignored and motor is rotating in steady state, Equation (25) can be simplified to:

$${\sigma }_d=-\psi \cdot {\omega }_r\cdot \mathit{sin}\Delta \theta +{\omega }_r\cdot i_q\cdot \Delta L{\mathit{sin}}^2\Delta \theta$$

(26)

From the same method, ${\sigma }_q$ is derived as:

$${\sigma }_q=\psi \cdot {\omega }_r\cdot \mathit{cos}\Delta \theta -{\omega }_r\cdot i_q\cdot \Delta L\mathit{sin}\Delta \theta \cdot \mathit{cos}\Delta \theta$$

(27)

From Equations (26) and (27), the speed information can be determined by the absolute value of error signal.

$$\left|\sigma \right|=\sqrt{\left({\sigma }_d^2+{\sigma }_q^2\right)}=\left|{\omega }_r\right|\cdot (\psi -\Delta L\cdot i_q\cdot \mathit{sin}\Delta \theta )$$

(28)

$$\left|{\omega }_r\right|={\displaystyle \frac{\left|\sigma \right|}{\psi -\Delta L\cdot i_q\cdot \mathit{sin}\Delta \theta }}$$

(29)

The estimated speed error $\Delta {\widehat{\omega }}_r$ can be obtained by Equation (29) because the sign of rotor speed is already given in the equation:

$$\Delta {\widehat{\omega }}_r={\displaystyle \frac{\left|\sigma \right|}{\psi -\Delta L\cdot i_q\cdot \mathit{sin}\Delta \theta }}\cdot sign\left({\widehat{\omega }}_r\right)-{\widehat{\omega }}_r$$

(30)

$$\Delta {\widehat{\omega }}_c=m_{sc}\cdot \Delta {\widehat{\omega }}_r$$

(31)

The absolute value $\left|\sigma \right|$ of error signal can be calculated in Equation (23). Therefore, m_sc is a manual tuning value with respect to speed variation and $\Delta {\widehat{\omega }}_c$ can be used as the compensation term of a speed difference error. The block diagram for the compensation of estimated speed error and angle error is depicted as shown in Figure 5. $\Delta {\widehat{\omega }}_c$ using Equation (31) and θ_SC using Equation (22) have a compensation value with respect to speed variation such as acceleration and deceleration.

6. Improvement of Torque Response in a Transient State

6.1. Compensation Design Using Current Feedback Control

The overshoot of estimated position error has to be decreased to improve the transient stability. If the reference torque is decreased, the q-axis current is decreased. And the q-axis inductance is increased instantaneously by the magnetic saturation. The increased q-axis inductance causes the overshoot of estimated position error. The high estimated position error can increase the possibility of control angle slip. This angle slip can result in the increased instability of the sensorless control system. Therefore, the low overshoot of $\Delta \widehat{\theta }$ is necessary. The overshoot of $\Delta \widehat{\theta }$ occurs when the speed is changed during a short time from Equations (2) and (6). If it is assumed that the estimated speed error is not small, Equation (6) can be redefined in Equation (32) as mentioned in previous section and Equation (22).

$${\mathit{tan}}^{-1}\left(-{\displaystyle \frac{e_{\gamma }}{e_{\delta }}}\right)=\Delta \theta +{\mathit{tan}}^{-1}\left({\displaystyle \frac{\Delta {\omega }_r{L}_d{i}_q}{E_{ex}+\Delta {\omega }_r{L}_d{i}_d}}\right)=\Delta \theta +{\theta }_{FC}$$

(32)

where θ_FC is a compensated angle for the alignment between d-q axis and γ-δ axis in transient state. E_ex and ∆ω_r are dominant terms related to q-axis current. And, the estimation error of q-axis current is fed to the PI controller to find the speed estimation value [21]. Therefore, the θ_FC can be compensated by current feedback control as in Equation (33):

$${\displaystyle \frac{d{\theta }_{FC}}{dt}}=m_{ac}\times \left(k_p\cdot (i_q^{*}-i_{\delta })+k_i\int (i_q^{*}-i_{\delta })\cdot dt\right)$$

(33)

where m_ac is a manual tuning value to make zero level the estimated position error in transient state and k_p and k_i are PI gain for current feedback controller. Typically, the reference value of i_q is changed according to the input torque variation in transient state. This rapid change occurs with the inductance variation of the motor during a short time and the error $\mathsf{\Delta }\widehat{\mathrm{\theta }}$  also occurs. Variations in the input reference torque lead to changes in both iq and Lq, which in turn result in significant transient errors in the estimated angle ∆θ, indicating the occurrence of the control angle slip.

To mitigate the overshoot in ∆θ, an appropriate selection of mac gain is required following the determination of a stable PI gain in the proposed current feedback control scheme. A higher mac value effectively reduces the overshoot of ∆θ; however, it also increases the sensitivity of the control system to noise signals. Therefore, in this study, mac is manually tuned based on the reference torque to balance transient performance and noise robustness. Therefore, the compensation method using the difference value between q-axis reference current and δ-axis current is effective. The result of the block diagram for angle compensation is depicted in Figure 6.

6.2. Stability Analysis of Current Feedback Control

For stability analysis, the error dynamics of the estimator by Equations (17) and (33) and Figure 6 are given as:

$$\begin{array}{l}\Delta {\widehat{\omega }}_r^{\prime }={\rho }^2·\left(∆\theta +{\theta }_{FC}\right)\\ {\widehat{\theta }}^{\prime }={\widehat{\omega }}_r+2\rho \cdot (\Delta \theta +{\theta }_{FC})\end{array}$$

(34)

The error dynamics can be expressed as Equation (35) in nominal and high speeds with $\mathsf{\Delta }{\mathsf{\omega }}_r^{\prime }\approx \mathsf{\Delta }{\mathsf{\omega }}_r$

$$\begin{gathered} \Delta {\widehat{\omega }}_r^{\prime }={\omega }_r^{\prime }-{\widehat{\omega }}_r^{\prime }=-{\widehat{\omega }}_r^{\prime }=-{\rho }^2·∆\theta -{\rho }^2·{\theta }_{FC} \\ ∆{\widehat{\theta }}^{\prime }={\theta }^{\prime }-{\widehat{\theta }}^{\prime }={\omega }_r-\left({\widehat{\omega }}_r+2\rho \cdot \left(\Delta \theta +{\theta }_{FC}\right)\right) \\ ={∆\omega }_r-2\rho \cdot \Delta \theta -2\rho \cdot {\theta }_{FC} \end{gathered}$$

(35)

The stability of the nonlinear system can be defined by the coefficients of the characteristic polynomial.

$$\mathit{det}(sI-(A-BK))=\mathit{det}\left(\left[\begin{array}{cc}s& 0\\ 0& s\end{array}\right]-\left(\left[\begin{array}{cc}0& -{\rho }^2\\ 1& -2\rho \end{array}\right]-\left[\begin{array}{c}-{\rho }^2\\ -2\rho \end{array}\right]\left[\begin{array}{cc}{k}_1& k_2\end{array}\right]\right)\right)$$

(36)

$$\mathit{det}(sI-(A-BK))=s^2-\left(k_1\rho +2(k_2-1)\cdot \rho \right)\cdot s+(1+k_2)\cdot {\rho }^2$$

(37)

In accordance with the stable gain selection process in [19], the PLL-type estimator bandwidth is set to 100 rad/s. Therefore, the characteristic polynomial equation is given by:

$$\begin{array}{l}\mathit{det}(sI-(A-BK))=s^2-\left(100k_1+200\left(k_2-1\right)\right)\cdot s+(1+k_2)\cdot 10000\\ =s^2+m_1\cdot s+m_2\end{array}$$

(38)

Using the Routh–Hurwitz stability criterion, if the coefficients of the characteristic polynomial are positive such as m₁ > 0 and m₂ > 0, the nonlinear system is stable. Therefore, the stability condition can be defined as:

$$\therefore -K_1>2(K_2-1),K_2>-1$$

(39)

If K₂ is set to 0.15, K₁ should set less value than 1.7. Also, the closed loop poles, damping ratio, and undamped natural frequency are defined as below:

$$s_{\mathrm{1,2}}={\displaystyle \frac{-m_1\pm \sqrt{m_1^2-4m_2}}{2}}$$

(40)

$$\zeta ={\displaystyle \frac{m_1}{2\sqrt{m_2}}},{\omega }_n=\sqrt{m_2}$$

(41)

7. Simulation and Experimental Results

The configuration of the sensorless drive system for the simulation and experiment is shown in Figure 7. The PLL-type estimator and disturbance observer were applied to obtain the estimated angle and speed. The accuracy of estimated signals can be evaluated by encoder output signals. The motor parameters listed in

Table 1. Motor parameters.

Parameters	Value
Number of poles	4
Rated speed	1500 [min−1]
Rated torque	1.8 [Nm]
Stator resistance	0.814 [Ω]
D-axis inductance	10.7 [mH]
Q-axis inductance	26.3 [mH]
Back-EMF constant	0.14693 [V·s/rad]
Rotor inertia	0.001641 [kg-m2]

can be considered in the evaluation to compare the performance of the proposed method. Also, the control variables for sensorless control can be set as below:

$$\begin{array}{l}{\alpha }_c=3140 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s},\hspace{1em}|\Delta \hat{\theta }{|}_{\max }=10 \mathrm{deg}.,\hspace{1em}{|{\stackrel{.}{\omega }}_r|}_{\max }=2073 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}\hspace{1em}{g}_{ob}=1000 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s},\hspace{1em}\\ \rho =100 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s},\hspace{1em}{m}_{ob}=0.12\hspace{1em},\hspace{1em}{m}_{ac}=0.15\hspace{1em}\end{array}$$

where α_c is current controller bandwidth.

7.1. Dynamic Performance of Speed Response

Figure 8 shows the PSIM simulation result of speed and d-q axis current response on the conventional PLL-type estimator when the reference speed is rampwise changed from 500 min⁻¹ to 1500 min⁻¹ under rated torque 1.8 Nm. ∆ω_r has a maximum overshoot value about 262 min⁻¹ in acceleration and −252 min⁻¹ in deceleration. The rising time and falling time is set to 75 ms considering motor inertia.

Figure 9 shows the simulation waveforms of speed and d-q axis current on the proposed method under 1.8 Nm. The maximum overshoot of ∆ω_r in acceleration is about 155 min⁻¹ and −133 min⁻¹ in deceleration. Therefore, the proposed method has a lower overshoot value in the transient state than that of the conventional method.

Figure 10 shows the experimental setup in order to evaluate the feasibility of the proposed compensation method. The specification of the four-pole IPMSM is 1.8 Nm, 3 Arms, and 1500 min⁻¹ such as shown in Table 1.

Figure 11 shows the speed response in the conventional method when the speed is under gradual speed variation from 500 min⁻¹ to 1500 min⁻¹ at the rising and falling time of 75 ms at 1 Nm. The maximum overshoot of ∆ω_r is about 400 min⁻¹ and −370 min⁻¹ at acceleration and deceleration time, respectively, with m_sc = 1.

Figure 12 shows the transient waveforms of the difference value on the estimated speed and position error when the proposed method is applied to a PLL-type estimator under 1 Nm. The maximum overshoot of ∆ω_r in acceleration time is about 200 min⁻¹ and −190 min⁻¹ in deceleration time.

In this result, the proposed compensation strategy shows lower overshoot values than of the method using an uncompensated PLL-type estimator in accordance with experimental results. Therefore, good dynamics can be obtained by the compensated PLL-type estimator.

7.2. Dynamic Performance of Torque Response

Figure 13 shows the simulation result on the overshoot waveforms of estimated position error when the reference torque is rapidly stepwise decreased from 1.8 Nm to 0.1 Nm at 500 min⁻¹. This overshoot can be decreased by the angle compensation method considering the current feedback control to improve a stable sensorless control.

Figure 14 shows the experimental result of the comparison of maximum overshoot values during torque variation from 1.8 Nm to 0.1 Nm at 500 min⁻¹. The maximum overshoot values of estimated speed error and angle error without compensation using the current feedback control are 360 min⁻¹ and 53 degrees, respectively, whereas the overshoot values with the proposed compensation method are 120 min⁻¹ and 23.5 degrees, respectively, when the mac value was selected as 0.15 in Equation (33).

Figure 15 represents the comparison result of overshoot with respect to the proposed current feedback control and without it at 1500 min⁻¹ in the experiment. The maximum overshoot value of the estimated position error in the proposed compensation method shows a lower value than that of the uncompensated sensorless control.

7.3. Performance Comparison of Speed and Postion Estimators

Figure 16 illustrates the structural configurations of conventional speed and position estimators employed in the sensorless control of IPMSMs. The control parameter settings used to determine the bandwidths of each estimator are also indicated in the figure.

To evaluate transient response characteristics, the control schemes of the conventional estimators were individually implemented within the PLL-type estimator block in the simulation circuit shown in Figure 7.

Figure 17 presents the maximum overshoot values of the PLL-type estimator under variations in speed and torque. It is observed that rapid acceleration and deceleration in speed, as well as the step response in torque, result in significant instantaneous overshoots in the estimated speed and position.

Figure 18 shows the simulation results of the double-integrator PLL-type estimator, while Figure 19 depicts those of the LO-type estimator. Figure 20 presents the simulation results obtained using the estimator proposed in this paper.

From the simulation results, it can be concluded that the proposed estimator yields the smallest peak estimation error in speed under torque variations, demonstrating superior transient performance compared to conventional methods.

The comparison results of the speed and position estimators in a transient state are shown in

Table 2. Comparison results on speed and the position estimator using PSIM simulation.

Item	Step Torque Max. ∆ωr [min−1]	Rampwise Speed Max. ∆ωr [min−1]
PLL-type estimator	26.8	213.4
Double integral PLL-type estimator	81.4	−130.6
Luenberger Observer estimator	144	76.5
Proposed current feedback control	22.5	213

. From this result, the maximum overshoot of the estimator using the proposed current feedback method is lower than other estimator methods, although the overshoot of estimated speed error ∆ω_r with linearly increasing speed reference is higher than other methods. However, this overshoot can be decreased by the proposed method for speed response improvement. Therefore, the high performance of the PLL-type estimator on speed and torque variation can be achieved with the proposed control strategy.

8. Conclusions

In this paper, compensation methods for a PLL-type estimator are proposed to enhance dynamic performance during transient conditions, such as rapid acceleration, deceleration, and torque variations. The proposed strategies incorporate speed and angle compensation terms, which effectively reduce the maximum overshoot in estimated speed and position during transient operation. As a result, improved sensorless control performance, including a faster dynamic response, can be achieved.

Simulation and experimental results confirm that the proposed compensation method limits the maximum overshoot of estimated speed and position errors to significantly lower values compared to those observed in the uncompensated approach.