Development and New Challenges of Sensorless Control for Permanent Magnet Synchronous Motors

Abstract

Permanent magnet synchronous motors (PMSMs) are widely used in various fields due to their high efficiency and power density. To further enhance reliability and reduce cost and volume, sensorless control techniques have been extensively investigated over the past few decades. This article provides a review of major sensorless control methods, categorizing them into low-speed and high-speed methods. Virtual frequency methods, saliency-based methods, and model-based methods, along with their developments, are analyzed and compared. In addition, application-oriented analysis, implementation insights, as well as the challenges and future development trends are discussed at the end of the article.

1. Introduction

Permanent magnet synchronous motors (PMSMs) have been widely used in fields such as electric vehicles, ship propulsion, and home appliances due to their compact structure, high power density, fast dynamic response, etc. [1,2,3]. In the field-oriented control (FOC) system for PMSM drives, accurate rotor position and speed is essential for high-performance control. Thus, position sensors such as an encoder or a resolver are usually installed on the rotor shaft. However, mechanical position sensors, along with their cables and interfaces, are considered potential failure points that threaten system reliability, particularly for motors operating in extreme environments such as those with high temperatures, complex electromagnetic fields, and severe vibrations. Moreover, eliminating these sensors can significantly reduce overall cost, which has long been an industrial pursuit. To address these issues, sensorless control algorithms have been developed to replace traditional sensors by providing online estimations of speed and position. In addition, some control strategies that do not rely on position information can also be regarded as broad-sense sensorless control.

Sensorless control emerged in the 1990s and has been extensively studied over the past decades. The existing sensorless methods can be primarily classified according to their applicable speed range into high-speed methods and low-speed methods, as shown in Figure 1. It is noteworthy that the boundary between these regions is not absolute. Based on the existing research and industry practice, the low-speed region is often considered to be below 5–10% of the rated speed, while high speed covers the medium to rated speed range and above.

The low-speed sensorless methods include virtual frequency methods [4,5,6,7,8], high-frequency signal injection (HFSI) methods [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37], and fundamental PWM excitation (FPE) methods [38,39,40,41,42,43,44,45]. The virtual frequency methods are essentially open-loop control strategies. Unlike conventional closed-loop methods for speed and position estimation, the virtual frequency method directly generates voltage vectors based on an artificially given virtual position. This characteristic of being independent of position information enables it to be classified as a broad-sense sensorless strategy. HFSI methods and FPE methods achieve position estimation at low speeds by using closed-loop strategies to track rotor saliency, which originates from the d-q axis magnetic reluctance asymmetry [10]. Specifically, the HFSI method achieves saliency tracking by applying a sufficient high-frequency excitation, while the FPE method relies on measuring current responses caused by different voltage space vectors applied to the motor.

The high-speed methods, also known as the model-based methods, extract rotor position and speed by estimating the flux or back electromotive force (BEMF) related to the fundamental wave of the motor [46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66]. To achieve accurate estimation, various types of observers based on modern control theory have been studied. Many works focus on sliding-mode observers (SMOs) due to their inherent robustness and simplicity [67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84]. In addition, linear Luenberger observers are also widely used [85,86,87,88,89]. To further enhance noise immunity and estimation accuracy, the extended Kalman filter (EKF) is introduced and refined in [90,91,92,93,94,95,96,97,98,99]. In parallel, model reference adaptive systems (MRASs) offer an alternative framework for position estimation by enforcing the convergence of a reference model and an adjustable model [100,101,102,103,104,105]. However, due to factors such as inverter nonlinearity, modeling errors and signal-to-noise ratio, the model-based methods may suffer from performance degradation or even failure, particularly under zero/low speeds [106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148]. To achieve full-speed-range sensorless control, hybrid strategies incorporating both low-speed and high-speed methods are commonly adopted. Correspondingly, switching strategies between different methods in the transition region need to be investigated [149,150,151,152,153,154,155].

Today, sensorless control has approached maturity through efforts in both industry and academia. The research focus of sensorless control algorithms is shifting from principal exploration to performance optimization, aiming to enhance their applicability across various operating conditions. This article reviews existing sensorless control algorithms and their recent developments. In addition, based on the core requirements of different application scenarios, the applicability and performance tradeoffs of various methods are analyzed, aiming to provide a reference for technology selection in engineering practice. Finally, the challenges and trends in this field are discussed.

The rest of the article is organized as follows. The second and third sections introduce the low-speed and high-speed sensorless methods, respectively. The fourth section provides an application-oriented analysis and implementation insights, along with a discussion of the challenges and future development trends. Finally, the fifth section concludes the article.

2. Low-Speed Sensorless Control Methods

2.1. Virtual Frequency Methods

The virtual frequency methods do not detect the actual rotor position, which generate the voltage vector based on a virtual angle obtained by integrating a given frequency [4,5]. Figure 2 shows the control diagrams for the two common virtual frequency methods, including V/F and I/F control techniques. As an effective and easily implementable control strategy, the virtual frequency method can operate without a position sensor. Therefore, it can be regarded as a form of sensorless control.

As shown in Figure 2a, V/F control is an open-loop control for both speed and current (n* is the speed reference). Therefore, V/F control has a poor load-carrying capacity and anti-interference ability. Compared with V/F control, as can be seen from Figure 2b, I/F control achieves closed-loop current control by providing a current reference (i _d*, i _q*) with specific amplitude and frequency (${\omega }_e^v$). The motor is driven to the given speed based on the self-balancing characteristic of the power angle. However, the self-balancing characteristic limits the load-carrying capacity and is sensitive to disturbances. Thus, I/F control is unsuitable for low-speed steady-state control. Thus, it is often employed as a start-up strategy for PMSMs. Once the rotor speed is established, the system can switch to a model-based method.

When switching to the model-based method, the mismatched current vector may lead to a loss of synchronization. To address the problem, research has been carried out from different aspects. Reference [6] proposes a transition method for the switching process. Based on the error between the estimated position and reference value, an integrator is employed to adjust the reference of q-axis current, thereby achieving a smooth transition between different algorithms. In [7], a speed–position–current tracking controller is proposed to serve as the middle layer, which achieves a high acceleration rate. In [8], traditional I/F control is combined with a predictive algorithm to enhance anti-interference performance during the start-up process. However, in applications that require higher starting torque, accurate rotor position is necessary and the virtual frequency method will not be applicable.

2.2. HFSI Methods

The high-frequency signal injection method is an effective sensorless control scheme in the zero/low-speed region for PMSMs. The HFSI methods are based on the saliency characteristic of the motor and extract rotor position from the high-frequency response caused by high-frequency voltage [9,10]. For the interior PMSM (IPMSM), the asymmetry of the rotor magnetic circuit results in different d-q axis inductances, which is known as structural saliency. The d-q axis inductances of the surface-mounted PMSM (SPMSM) are the same. However, when a specific voltage is applied, it will exhibit saturation saliency due to magnetic circuit saturation. Figure 3 shows the block diagrams of the HFSI methods, where ω_r* is the speed reference. According to the applied reference frame, high-frequency injection methods can be divided into two categories: rotating signal injection in the stationary reference frame [11,12,13,14] and pulsating signal injection in the synchronous reference frame [15,16,17,18].

2.2.1. PMSM HF Model

Considering that the injection frequency ω_h is much higher than the fundamental frequency ω_e of the motor, the voltage drops on resistance, and variables associated with ω_e can be neglected. The high-frequency model of the PMSM in the d-q frame is derived as

$$\left[\begin{array}{c}{u}_{dh}\\ u_{qh}\end{array}\right]=\left[\begin{array}{cc}{L}_d& 0\\ 0& L_q\end{array}\right]p\left[\begin{array}{c}{i}_{dh}\\ i_{qh}\end{array}\right]$$

(1)

where u_dh, u_qh, i_dh and i_qh are high-frequency voltages and currents in the d-q frame; L_d and L_q are d-q axis inductances; and p is the differential operator.

Transforming (1) to the α-β frame yields

$$\left[\begin{array}{l}{u}_{\alpha h}\hfill \\ u_{\beta h}\hfill \end{array}\right]=\left[\begin{array}{cc}\Sigma L+\Delta L\cos 2{\theta }_e& \Delta L\sin 2{\theta }_e\\ \Delta L\sin 2{\theta }_e& \Sigma L-\Delta L\cos 2{\theta }_e\end{array}\right]p\left[\begin{array}{c}{i}_{\alpha h}\\ i_{\beta h}\end{array}\right]$$

(2)

where u_αh, u_βh, i_αh and i_βh are high-frequency voltages and currents in the α-β frame; θ_e is the roto position; ∑L = (L_d + L_q)/2 and ΔL = (L_d − L_q)/2.

2.2.2. Rotating Signal Injection

As shown in Figure 3a, a rotating voltage vector (3) is injected into the voltage reference in the α-β frame.

$$\left[\begin{array}{c}{u}_{\alpha h}\\ u_{\beta h}\end{array}\right]=V_{inj}\left[\begin{array}{c}\cos ({\omega }_{h}t)\\ \sin ({\omega }_{h}t)\end{array}\right]$$

(3)

where V_inj is the magnitude of the injected voltage vector.

The HF current response in the α-β frame can be derived as

$$\left[\begin{array}{c}{i}_{\alpha h}\\ i_{\beta h}\end{array}\right]=\left[\begin{array}{c}{I}_{sp}\sin ({\omega }_{h}t)+I_{sn}\sin (-{\omega }_{h}t+2{\theta }_e)\\ -I_{sp}\cos ({\omega }_{h}t)-I_{sn}\cos (-{\omega }_{h}t+2{\theta }_e)\end{array}\right]$$

(4)

where I_sp and I_sn are the magnitudes of the positive-sequence and negative-sequence components, respectively.

It can be seen that the negative-sequence component includes the rotor position. In the signal process, the HF current is obtained through a band-pass filter and transformed to negative-sequence rotating reference frame, yielding

$${\left[\begin{array}{c}{i}_{dh}\\ i_{qh}\end{array}\right]}_n={\left[\begin{array}{c}{I}_{sp}\sin (2{\omega }_{h}t)+I_{sn}\sin (2{\theta }_e)\\ -I_{sp}\cos (2{\omega }_{h}t)-I_{sn}\cos (2{\theta }_e)\end{array}\right]}_n$$

(5)

After low-pass filtering, the rotor position can be obtained through an arctangent function as

$${\theta }_e={\displaystyle \frac{1}{2}}\arctan \left[-{\displaystyle \frac{LPF(i_{dh})}{LPF(i_{qh})}}\right]$$

(6)

For the rotating signal injection, the application in the α-β frame allows direct rotor position estimation in theory, eliminating the need for an additional position tracker. However, the use of filters inevitably introduces errors and delays. In addition, the frequency of the injected signal is usually less than one tenth of the pulse width modulation (PWM) frequency, ensuring that the high-frequency current response can be sampled to obtain a complete cycle.

2.2.3. Pulsating Signal Injection

Figure 3b shows the block diagram of a pulsating signal injection. The HF signal is injected into the estimated d-axis. Thus, the initial rotor position is needed and position tracking is achieved by eliminating the error between the actual d-axis and the estimated one. According to the form of the injected signal, the pulsating signal injection can be further divided into a pulsating sinusoidal injection (7) and pulsating square-wave injection (8).

$$\left[\begin{array}{c}{u}_{dh}\\ u_{qh}\end{array}\right]=V_{inj}\left[\begin{array}{c}\cos ({\omega }_{h}t)\\ 0\end{array}\right]$$

(7)

$$\left[\begin{array}{c}{u}_{dh}\\ u_{qh}\end{array}\right]=\left[\begin{array}{c}\pm V_{inj}\\ 0\end{array}\right]$$

(8)

The HF current response can be derived as

$$\left[\begin{array}{c}{i}_{dh}\\ i_{qh}\end{array}\right]=K\left[\begin{array}{c}\Sigma L-\Delta L\cos 2{\tilde{\theta }}_e\\ -\Delta L\sin 2{\tilde{\theta }}_e\end{array}\right]{\displaystyle \int u_{inj}dt}$$

(9)

where K is the amplitude coefficient and ${\tilde{\theta }}_e={\theta }_e-{\widehat{\theta }}_e$, ${\widehat{\theta }}_e$ is the estimated rotor position.

According to (9), the error of the rotor position can be obtained through extracting the envelope of the HF current. By inputting the error into a position observer, the estimated position and speed can be further obtained.

In practical applications, the frequency of the injected signal is typically set between several hundred Hz and several kHz [17], which fundamentally limits the bandwidth of the position observation [18]. Furthermore, signal processing components such as LPFs, BPFs, and demodulators introduce additional delay and phase lag. To enhance the dynamic response, a pulsating square-wave injection is preferred because square-wave signals are easy to generate and their frequency can reach up to the PWM switching frequency [19]. Additionally, solutions such as filterless carrier separation [20], fast time-domain demodulator [21] and PID-type Luenberger position observer [22] have also been proposed. However, considerable work remains for the HFSI methods to achieve dynamic performance comparable to that of mechanical sensors.

Another critical issue in HFSI applications is the acoustic noise caused by high-frequency signals, which is unacceptable for noise-sensitive applications such as household motors and underwater propulsion. Common methods for noise reduction include increasing the injection frequency [23], reducing the amplitude of the injected signal [24], and adopting random injection [25,26,27,28]. Reference [25] modifies the injected voltage signal from a fixed frequency to a varying frequency, spreading the acoustic energy on a frequency band, thereby reducing the peak level of noise. Figure 4 shows the HF voltage and corresponding current response waveforms with a random-frequency injection method, where the injected signal randomly switches between two specific frequencies [26]. According to power spectral density (PSD) analysis in [27], the randomized high-frequency injection method achieves a smoother distribution of the current PSD, thereby effectively mitigating high-frequency noise [28].

In addition, the influence of non-ideal factors such as inverter nonlinearity, cross-coupling saturation, and multiple saliency effect on the HFSI should be taken into consideration to achieve more accurate position estimations [29,30,31].

2.2.4. Magnetic Polarity Identification

For saliency-based methods, the position information varies periodically at twice the fundamental frequency. Therefore, the obtained rotor position does not contain polarity information, which may result in the estimation error of π, as shown in Figure 5a. Figure 5b shows the nonlinear saturation characteristic of the stator core, which can be utilized for polarity identification [32]. Positive d-axis current increases the saturation level of the stator core, while negative current reduces saturation. By injecting different test signals to the d-axis, the polarity can be identified through the response of the injected signal, such as the dual pulse injection method [33]. As reported in [34], by comparing the amplitude of the current responses of two opposite voltage pulses, the pulse with a higher amplitude represents the actual d-axis direction. In addition, the second harmonic-based methods are studied in [35,36,37]. The 2nd harmonics in the HF current response or HF zero-sequence voltage response can be utilized to determine polarity.

2.3. Fundamental PWM Excitation Methods

Compared with the HFSI methods, the FPE methods directly use the PWM signal as the excitation source, without the need for additional signal injection. Thus, the FPE methods will not cause additional high-frequency noise. The rotor position is obtained by sampling the current change rate generated by the voltage vectors. Existing FPE methods include indirect flux detection by online reactance measurement (INFORM) [38,39,40], the zero-sequence current derivative (ZSCD) measurements method [41,42,43] and the zero voltage vector injection method (ZVVI) [44,45]. Due to the fact that the FPE method detects the current change rate multiple times within one PWM cycle, traditional continuous sampling cannot meet the requirements of high-frequency current measurements. Therefore, complex high-speed and high-precision current sampling circuits and specially designed modulation signals are necessary, which are unfriendly to low-cost control systems. The characteristics of the major low-speed sensorless methods are summarized in

Table 1. Summary and comparison of major low-speed sensorless methods.

Strategy	Control Structure	Minimum Operating Speed	Dynamic Response	Additional Losses and Noise	Requirements for Motors	Implementation Complexity
V/F or I/F	Open-loop	Near zero	Low	None	None	Low
HFSI methods	Close loop	Zero	Medium	High	Saliency	High (HF signal generation and demodulation)
FPE methods	Close loop	Zero	Medium	Ultra-low	Saliency	High (High-precision current sampling)

.

3. High-Speed Sensorless Control Methods

The low-speed methods discussed in the previous section have been proven effective and applied in the industry. However, due to adverse effects such as torque ripple, limited dynamic response, and acoustic noise, these methods are recommended to be used only in the zero/low-speed range. With the increase in speed, switching to model-based methods is a commonly adopted strategy. Since the rotor position information is included in the flux or back EMF, the model-based method generally consists of two stages: flux or back-EMF estimation, position and speed extraction, as shown in Figure 6. In the first stage, both flux and back EMF can be obtained through open-loop calculation and close-loop observation. The former is directly calculated from the model equation, and the latter estimates the state variables through an observer. In the second stage, the position and speed can be obtained through an open-loop arctangent function and differential operation or close-loop phase-lock loop (PLL) and Luenberger observer [46,47,48,49,50,51].

3.1. Flux or Back-EMF Estimation

3.1.1. Flux-Based Methods

Sensorless control based on flux linkage estimation has been widely used [52,53,54,55,56]. It operates by estimating the flux linkage produced by the permanent magnets and then extracting the rotor position from this estimated flux.

In the stationary frame, the stator flux linkage of SPMSM can be expressed as

$$\left[\begin{array}{c}{\psi }_{\alpha }\\ {\psi }_{\beta }\end{array}\right]=\left[\begin{array}{cc}{L}_{\alpha }& 0\\ 0& L_{\beta }\end{array}\right]\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]+\left[\begin{array}{c}{\psi }_{m\alpha }\\ {\psi }_{m\beta }\end{array}\right]$$

(10)

$$\left[\begin{array}{c}{\psi }_{m\alpha }\\ {\psi }_{m\beta }\end{array}\right]={\psi }_f\left[\begin{array}{c}\cos {\theta }_e\\ \sin {\theta }_e\end{array}\right]$$

(11)

where ψ_α and ψ_β are stator flux linkage, ψ_mα and ψ_mβ are PM flux linkages, ψ_f is the magnitude of the PM flux linkage and L_α = L_β = L_s is the phase inductance of the SPMSM. The phasor diagram of the flux linkage is shown in Figure 7, where ψ_s and ψ_m represent synthetic stator flux and PM flux, respectively.

Generally, the stator flux linkage is the integral of the back EMF, as

$$\left\{\begin{array}{c}{\psi }_{\alpha }={\displaystyle \int \left(u_{\alpha }-R_s{i}_{\alpha }\right)} dt\\ {\psi }_{\beta }={\displaystyle \int \left(u_{\beta }-R_s{i}_{\beta }\right)} dt\end{array}\right.$$

(12)

Then, the rotor position can be obtained as

$${\theta }_e=\arctan {\displaystyle \frac{{\psi }_{m\beta }}{{\psi }_{m\alpha }}}=\arctan {\displaystyle \frac{{\psi }_{\beta }-L_s{i}_{\beta }}{{\psi }_{\alpha }-L_s{i}_{\alpha }}}$$

(13)

The block diagram of flux-based sensorless control is presented in Figure 8.

For an IPMSM, its stator flux satisfies

$$\left[\begin{array}{c}{\psi }_{\alpha }\\ {\psi }_{\beta }\end{array}\right]=\left[\begin{array}{cc}\Sigma L+\Delta L\cos 2{\theta }_e& \Delta L\sin 2{\theta }_e\\ \Delta L\sin 2{\theta }_e& \Sigma L-\Delta L\cos 2{\theta }_e\end{array}\right]\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]+{\psi }_f\left[\begin{array}{c}\cos {\theta }_e\\ \sin {\theta }_e\end{array}\right]$$

(14)

The inductances in the α-β frame are coupled with the rotor position, which increases the difficulty of estimating the PM flux linkage. To address this problem, [57] proposed the concept of an active flux, which unifies a salient-pole motor as a virtual non-salient machine.

The active flux is defined as

$${\psi }_a={\psi }_f+\left(L_d-L_q\right)i_d$$

(15)

The stator flux of IPMSM can be rewritten as

$$\left[\begin{array}{c}{\psi }_{\alpha }\\ {\psi }_{\beta }\end{array}\right]=\left[\begin{array}{cc}{L}_q& 0\\ 0& L_q\end{array}\right]\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]+{\psi }_a\left[\begin{array}{c}\cos {\theta }_e\\ \sin {\theta }_e\end{array}\right]$$

(16)

Similarly, the rotor position can be obtained as

$${\theta }_e=\arctan {\displaystyle \frac{{\psi }_{\beta }-L_q{i}_{\beta }}{{\psi }_{\alpha }-L_q{i}_{\alpha }}}$$

(17)

However, calculating flux through an integrator is highly susceptible to non-ideal factors such as core saturation, temperature drift, and initial value mismatch. This causes errors to accumulate progressively, culminating in system instability. Hence, the use of a pure integrator is typically avoided.

In [58], the pure integrator is replaced with LPF to enhance the robustness against DC offset. Due to the phase lag introduced by the LPF, a flux observer with delay compensation is employed in [59]. Further, a second-order generalized integrator (SOGI)-based flux observer is introduced in [60], featuring improved performance in both harmonic suppression and system robustness. In [61], a nonlinear flux observer is proposed, which employs the magnitude of the PM flux for feedback correction. The practicality has been demonstrated. Based on this, [62] has improved the mechanism of error correction for the nonlinear flux observer. Dynamic response and estimation accuracy are further improved. In [63], a high-order nonlinear adaptive observer is constructed. It features a nonlinear update law for parameters and employs higher-order sliding-mode terms to correct estimation errors, achieving an improved dynamic response.

3.1.2. Back-EMF-Based Methods

In addition, back-EMF-based sensorless control algorithms have also been extensively studied [64,65,66]. As the rotor rotates, the back EMF is generated in the stator winding and is proportional to the speed. The back EMF of SPMSM can be expressed as

$$\left[\begin{array}{c}{e}_{\alpha }\\ e_{\beta }\end{array}\right]=\left[\begin{array}{c}{u}_{\alpha }\\ u_{\beta }\end{array}\right]-\left[\begin{array}{cc}{R}_s+p{L}_s& 0\\ 0& R_s+p{L}_s\end{array}\right]\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]={\omega }_e{\psi }_f\left[\begin{array}{c}-\sin {\theta }_e\\ \cos {\theta }_e\end{array}\right]$$

(18)

The rotor position can be obtained as

$${\theta }_e=\arctan \left(-{\displaystyle \frac{e_{\alpha }}{e_{\beta }}}\right)$$

(19)

For IPMSMs, the concept of an extended back EMF was proposed in [65]. The extended back EMF of an IPMSM is defined as

$$\left[\begin{array}{c}{e}_{\alpha }\\ e_{\beta }\end{array}\right]=\left[\begin{array}{c}{u}_{\alpha }\\ u_{\beta }\end{array}\right]-\left[\begin{array}{cc}{R}_s+p{L}_d& {\omega }_e\left(L_d-L_q\right)\\ -{\omega }_e\left(L_d-L_q\right)& R_s+p{L}_d\end{array}\right]\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]=E_{ex}\left[\begin{array}{c}-\sin {\theta }_e\\ \cos {\theta }_e\end{array}\right]$$

(20)

$$E_{ex}=\left(L_d-L_q\right)\left({\omega }_e{i}_d-p{i}_q\right)+{\omega }_e{\psi }_f$$

(21)

During motor operation, the back EMF is difficult to measure directly and is therefore estimated from the measurable stator voltage and current. The estimated back EMF can be directly calculated based on (20), but closed-loop observers are more favored due to their high accuracy and robustness. Figure 9 illustrates the general block diagram of an observer [66]. Assuming the controlled object is given by

$$\left\{\begin{array}{c}px=Ax+Bu\\ y=Cx\end{array}\right.$$

(22)

where x is the state variable, u is the excitation, and y is the measured response. The estimation model can be described as

$$\left\{\begin{array}{c}p\widehat{x}=A\widehat{x}+Bu+G\left(y-C\widehat{x}\right)\\ \widehat{y}=C\widehat{x}\end{array}\right.$$

(23)

where the variable with the symbol “^” represents the estimated value. G is the feedback gain matrix, which has various forms.

For the back-EMF observers, the estimation model is generally established based on the fundamental model of the controlled PMSM. The current error between the estimated state and sampling value is used for model correction under a specific control law. When the error converges to zero, the estimated state can reflect the system state. To this end, various control laws and state variable selections create a variety of observers, including the sliding-mode observer, full-order/reduced-order state observer, extended state observer (ESO), model reference adaptive system, extended Kalman filter and so on.

(1)

Sliding-Mode Observer

The SMO has been widely studied due to its simple structure and strong robustness [67,68,69,70,71,72]. The key to the SMO lies in the variable structure control. As a nonlinear observer, it typically introduces a signum function as a correction term. The discontinuous gain switching depends entirely on the sign of the estimation error. When the error reaches zero, the back EMF, which is treated as an unknown disturbance, can be fully compensated by the output of the signum function. The LPF value of this output thereby provides an estimate of the back EMF. The block diagram of SMO is present as Figure 10a.

However, the high-frequency switching of the signum function also leads to the undesirable chattering problem, manifesting as high-frequency noise in the estimated back EMF. As the chattering originates from the inherent discreteness of the signum function, the smoother saturation function and sigmoid function are used to weaken chattering in [73,74,75,76,77]. Inside the boundary layer, the gain continuously changes with error, which can effectively suppress chattering. A comparative study in [78] indicates that the sigmoid function offers a better performance for chattering suppression compared with two alternative switching functions. Nevertheless, the selection of the boundary layer involves a critical tradeoff between the estimation performance and the level of chattering [79]. In [80], an adaptive gain control law is proposed. The SMO gain varies with the speed range, and the chattering is suppressed. Furthermore, the high-order SMO is also designed for chattering suppression [81]. By selecting both the back EMF and the current as state variables, the analysis in [48] shows that the required sliding-mode gain only needs to be greater than the maximum estimation error of the back EMF. This allows for a lower gain setting, which effectively mitigates the chattering phenomenon.

Another type of higher-order SMO, known as the super-twisting SMO [82,83], is illustrated in Figure 10b. The super-twisting SMO is based on the super-twisting algorithm (STA), and the control law is defined as

$$\left\{\begin{array}{l}{Z}_{STA}=-k_{STA1}{\left|{\tilde{i}}_{\alpha \beta }\right|}^{\frac{1}{2}}\mathrm{sign}\left({\tilde{i}}_{\alpha \beta }\right)+{\xi }_{STA}\\ {\displaystyle \frac{d}{dt}}{\xi }_{STA}=-k_{STA2}\mathrm{sign}\left({\tilde{i}}_{\alpha \beta }\right)\end{array}\right.$$

(24)

where Z_STA is the output of control law. k_STA1 and k_STA2 are the STA gains. The integral part in (24) has a suppressing effect on chattering, but the performance of STA-SMO is affected by STA gains. The relevant parameter design methods can be found in [84]. However, due to the involvement of discontinuous dynamics, it is difficult to achieve ideal parameter selection.

(2)

Luenberger Observer

The Luenberger observer, functioning as a linear observer, achieves state tracking through an error feedback coefficient matrix. This matrix is designed through pole placement or optimization methods [85,86,87,88,89].

The state equation of the Luenberger observer is expressed as

$$\left\{\begin{array}{l}s{\widehat{\mathit{i}}}_{\alpha \beta }=\left(-{\displaystyle \frac{R_s}{L_d}}+j{\displaystyle \frac{{\widehat{\omega }}_e\left(L_d-L_q\right)}{L_d}}\right){\widehat{\mathit{i}}}_{\alpha \beta }+{\displaystyle \frac{1}{L_d}}\left({\mathit{u}}_{\alpha \beta }-{\widehat{\mathit{e}}}_{\alpha \beta }\right)+k_1\left({\widehat{\mathit{i}}}_{\alpha \beta }-{\mathit{i}}_{\alpha \beta }\right)\\ s{\widehat{\mathit{e}}}_{\alpha \beta }=j{\widehat{\omega }}_e{\widehat{\mathit{e}}}_{\alpha \beta }+k_2\left({\widehat{\mathit{i}}}_{\alpha \beta }-{\mathit{i}}_{\alpha \beta }\right)\end{array}\right.$$

(25)

where k₁ and k₂ are gains of feedback. The gain design can be found in [86], and the block diagram of the Luenburger observer is present as Figure 11.

The transfer function between the estimated and actual back EMF is

$$G_c\left(s\right)={\displaystyle \frac{{\widehat{e}}_{\alpha \beta }\left(s\right)}{e_{\alpha \beta }\left(s\right)}}=={\displaystyle \frac{k_2}{\left(s-j{\widehat{\omega }}_e\right)\left(s{L}_d+R_s-j{\widehat{\omega }}_e\left(L_d-L_q\right)-k_1{L}_d\right)+k_2}}$$

(26)

According to (26), the observer exhibits a second-order low-pass characteristic, which consequently provides a continuous and smooth output. This avoids the chattering issue of sliding-mode-based observers. Reference [88] systematically compares reduced-order and full-order back-EMF observers for sensorless drives. The reduced-order observer is easier to tune, while the full-order observer offers superior robustness against parameter variation and noise.

(3)

Extended Kalman Filter

The extended Kalman filter is a recursive computational method for nonlinear estimation [90,91,92]. The EKF algorithm consists of two parts: prediction and correction, as illustrated in Figure 12. It primarily predicts the system states by utilizing the system model and reference voltage information, then corrects these predictions using the measured current signal. Applications of the EKF for PMSM sensorless control are documented in [93,94,95,96,97,98,99]. Due to its prediction–correction mechanism, the EKF can effectively suppress the influence of system noise and disturbances, exhibiting strong robustness [96].

In [97], the full-order form of the EKF is presented. The selection of the EKF covariance matrices can refer to [93]. It is observed that the recursive algorithm involves operations with fourth-order matrices, which imposes high demands on the computational capability of processors. Considering that some states can be directly obtained from measurements, the reduced-order EKF has been investigated in [98,99]. This reduces the model complexity and further decreases the computational load of the system.

(4)

Model Reference Adaptive System

In the MRAS algorithms, the physical motor with known parameters serves as the reference model, while a parametric motor model constitutes the adjustable model [100,101,102,103,104,105]. The current error between these two models is fed back to the adjustable model through a specific adaptive law. The output of the adaptive law is typically the rotor position or speed. Although the back EMF is not directly observed, it is implicitly reconstructed during the speed identification process. The block diagram of the MRAS method is shown in Figure 13.

Based on the conventional MRAS, ref. [102] introduces an improved adaptive law that enables simultaneous estimation of both speed and load torque, thereby improving disturbance rejection and dynamic response. To address parameter variations, the MRAS algorithm in [103] incorporates online estimations of both resistance and flux. Additionally, the grey wolf optimizer algorithm is utilized to adjust the parameters of the adaptive law. In [104], an MRAS algorithm based on a finite position set is proposed. It utilizes a model predictive algorithm to replace the PI regulator in the adaptive law, thus offering a novel design approach. Reference [105] considers the impact of high-speed operation on the MRAS algorithm. A discrete-time MRAS algorithm based on the second-order Taylor expansion is proposed, which improves the discretization accuracy and stability of the algorithm.

The characteristics of major model-based sensorless algorithms are presented in

Table 2. Summary and comparison of major model-based sensorless control algorithms.

Estimation Strategy	Observer Type	Convergence Rate	Parameter Robustness	Computational Complexity	Advantages	Disadvantages
Open-Loop Calculation	None	Low	Low	Low (Direct calculation based on model function)	Simple Easy calculation	No error correction Low performance
Close-Loop Observer	Nonlinear Flux Observer [61,62,63]	High	Medium	Medium (Involves integration and amplitude correction)	Low-speed performance	PM flux parameter dependence
	SMO [67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84]	High	High	Low to Medium (Switching function as the core)	Robustness Fast dynamic response	Chattering problem Phase lag
	Luenberger Observer [85,86,87,88,89,107]	High	Medium	Medium (State matrix operation for pole placement)	High accuracy Systematized design	Accurate linearization model dependency
	Extended Kalman Filter [93,94,95,96,97,98,99]	High	High	High (Real-time high-order matrix multiplication, inversion)	Parameter robustness Anti-measurement noise	Large computational burden Difficult parameter tuning
	MRAS [100,101,102,103,104,105]	High	Medium	Medium (PI-type adaptive law for online parameter tuning)	Flexible design Easy implementation	Complex adaptive law design Motor parameter dependence

. In addition, these observers can be constructed either in the stationary frame (α-β) or in the estimated frame (γ-δ) [88,106,107]. The relationship of these frames is shown in Figure 14. In the α-β frame, the stator voltage, current and back EMF are orthogonal. The rotor position and speed are included in the phase and amplitude of the back EMF. The γ-δ frame is aligned with the estimated rotor position. The estimated back EMF reflects the position error between the estimated γ-δ frame and the actual d-q frame. Forcing the error to converge to zero is the key of estimation in the synchronous frame.

3.2. Position and Speed Extraction

After obtaining the rotor flux or back EMF, the rotor position and speed need to be extracted to achieve sensorless control. As shown in Figure 15, the rotor position can be directly obtained using the arctangent function, and the speed is the derivative of the position. This method is simple and fast, but the calculation process is sensitive to disturbances, and differential operations are easily affected by noise [46,108]. Currently, the phase-locked loop is the mainstream method for position and speed extraction. Figure 16 shows the block diagrams of PLL in both the α-β frame and the γ-δ frame. A PI regulator is commonly used to eliminate the angular error for position tracking. The low-pass characteristic of the PLL can filter out unwanted harmonics and improve estimation accuracy but at the expense of degraded dynamic performance [47,48,49,50,51]. Compared with the PLL, the Luenberger observer has faster dynamic response, but the moment of inertia is necessary [22,109]. The block diagram of the Luenburger position observer is presented in Figure 17.

3.3. Model-Based Observers Under Non-Ideal Conditions

3.3.1. Parameter Mismatch

Due to the model dependency of the aforementioned methods, motor parameters have a significant impact on the sensorless control. Usually, the nominal parameters of the motor are used in the flux or back-EMF observer, which are provided by the motor manufacturer. However, actual motor parameters vary with operating conditions [110,111]. For instance, winding resistance is sensitive to temperature and the PM flux decays with temperature rise [112,113]. The key parameters of the motor are also influenced by variations in load conditions [114]. These variations lead to observer model mismatch, and there is a deviation between the estimated state and the actual state. This will further lead to errors in position and speed estimation and degrades the performance of sensorless control [115,116].

The model sensitivity of the model-based method has been discussed in [117], which highlights the effects of the q-axis inductance and resistance. This part will further analyze the impact of parameter mismatch on the accuracy of back-EMF-based methods. The parameter deviation is defined as follows.

$$\left\{\begin{array}{c}{\tilde{R}}_s=R_s-{\widehat{R}}_s\\ {\tilde{L}}_d=L_d-{\widehat{L}}_d\\ {\tilde{L}}_q=L_q-{\widehat{L}}_q\end{array}\right.$$

(27)

where variables with “~” represent the deviation values and variables with “^” represent the nominal values.

In the γ-δ frame, the position error caused by parameter mismatch can be derived as

$$\tilde{\theta }\approx \sin \tilde{\theta }={\displaystyle \frac{{\tilde{L}}_d\frac{d{\widehat{i}}_{\gamma }}{dt}+{\tilde{R}}_s{\widehat{i}}_{\gamma } - \widehat{\omega }{\tilde{L}}_q{\widehat{i}}_{\delta }+D_{\gamma }{V}_{dead}}{\omega {\psi }_f}}$$

(28)

where D_γV_dead is the influence of dead-time effect for the PWM inverter.

It can be found that the d-axis inductance influences the error in the transient state. When the motor is operating in the steady state, the current differential term can be ignored. Errors in the q-axis inductance and stator resistance result in a position estimation error, which appears as a DC offset. The impact of dead time presents as error fluctuations of six times the fundamental frequency and will increase during low-speed operation. For the SPMSM with i_d = 0 control, the error Equation (28) can be simplified as

$$\tilde{\theta }\approx \sin \tilde{\theta }={\displaystyle \frac{-\widehat{\omega }{\tilde{L}}_q{\widehat{i}}_{\delta }+D_{\gamma }{V}_{dead}}{\omega {\psi }_f}}$$

(29)

According to (29), the mismatched resistance will not cause a position error under i_d = 0 control.

To address the parameter mismatch, parameter identification is a direct solution [118,119,120,121]. However, conventional parameter identification methods with position sensors are not suitable for the sensorless system, as position error $\tilde{\theta }$ is an extra unknown state [122]. To address this issue, [123] utilizes the inherent PWM current ripple of the voltage source inverter for inductance identification. Reference [124] uses a first-order discrete-time model in the γ-δ frame to identify the inductance, which is independent of flux error, initial rotor position error, and inverter nonlinearity. In [125], a rotor position-offset injection method is proposed for parameter estimation of sensorless controlled PMSMs. The amplitude of the q-axis voltage fluctuation decreases with the increasing accuracy of parameters during the position-offset injection, and thus, stator inductance and resistance can be estimated by controlling the q-axis voltage fluctuation. Reference [126] proposes a noniterative method to simultaneously obtain the resistance and inductance of HSPMSM based on a multi-frequency disturbance injection. The voltage distortion caused by the dead-time effect (DTE) is modeled as an equivalent DTE resistance. In [127], virtual signal injection methods are proposed for full-parameter estimation. When a positive and a negative virtual signal is injected into a flux observer, the position error is only related to the ratio of the γ-δ axis voltage fluctuation. Therefore, the parameters can be independent of position error and inverter nonlinearity at different speeds and loads.

3.3.2. Low Carrier Ratio

As the power and speed increase, the PMSM drive system often operates at a low carrier ratio (f_ratio = f_PWM/f_e), with the lowest carrier ratio generally below 10 [128]. The performance degradation, discretization error and instability of sensorless control have gradually attracted attention [129,130,131].

The existing sensorless control algorithms are generally designed in the continuous-time domain, and then some approximate discretization methods are used to implement the observer in the digital controller, such as the forward Euler method, backward Euler method, Tustin method, etc. Essentially, these methods are approximations of the state-space equations discretized using the zero-order hold (ZOH). The discrete models obtained by different discretization methods (in the d-q frame, neglecting back EMF) can be uniformly expressed as

$$i_{dq}\left(k+1\right)=A_{dis}{i}_{dq}\left(k\right)+B_{dis}{u}_{dq}\left(k\right)$$

(30)

The discretization errors between different discretization methods and the ZOH method are shown in Figure 18. The backward Euler method has a smaller error than the forward Euler method, but the errors at low carrier ratio are both relatively large. Compared with using approximate methods, the observer designed by precise discretization can achieve a lower carrier ratio.

In [132], the discretization errors of forward Euler, backward Euler, Tustin, and pre-corrected Tustin methods are compared, and it points out that the discretization accuracy of the Tustin method is higher at different sampling frequencies. Reference [77] uses a precisely discretized SMO to achieve sensorless control with a carrier ratio of 20 at a switching frequency of 2 kHz. Reference [133] proposes a discrete-time adaptive synchronous-frequency observer, achieving a carrier ratio of 5. Reference [134] derives the discretization model of the IPMSM and proposes a discrete full-order state observer. The zero-pole matching method is adopted to configure the discrete-domain poles, achieving stable control with a minimum carrier ratio of 2.5.

3.3.3. Back-EMF Harmonics

Another critical issue is the harmonics in the estimation results of the model-based method, which are introduced by periodic disturbances in the motor drive system [135]. The various types of disturbance sources and back-EMF harmonic orders are summarized in [107]. In the process of motor design and manufacturing, achieving an ideal sinusoidal PM flux is challenging. The harmonics of back EMF are mainly low-order harmonics, such as the 3rd, 5th and 7th harmonics. For traditional three-phase star-winding PMSMs, the 3rd harmonics are zero-sequence components and do not cause current harmonics due to a no flow path. Therefore, the non-ideal flux will cause the −5th and +7th harmonics, which is the ±6th harmonics in the d-q frame. The effect of dead time is similar. The distortion of voltage caused by dead time will generate 6th harmonics in the estimation of rotor position and speed, which is significant at low speeds [136,137]. The three-phase asymmetries will cause negative-sequence components and cause 2nd harmonics in estimated back EMFs [15,138].

In order to suppress the harmonics in the estimated back EMF, additional filter networks are studied [139]. Frequency-adaptive band-pass filters are widely used for the characteristic of frequency selection. In [72], a frequency-adaptive complex-coefficient filter (FACCF) is proposed to suppress the low-order harmonics of the estimated back EMF and the chattering of the SMO. The structure and Bode diagram of FACCF are shown in Figure 19. In addition, adaptive notch filters (ANFs) are adopted to eliminate specific-order harmonics [140]. Compared with band-pass filters, the band-stop characteristic offers it a stronger ability to suppress the 5th and 7th harmonics. In addition to employing filters, constructing an observer with a harmonic suppression capability is also a feasible solution. In [141], a quasi-proportional-resonant controller (QPRC) is used to replace the signum function of SMO, which introduces a band-pass characteristic for the proposed observer. Reference [142] proposes a master–slave-structure observer for multi-harmonics suppression. The proposed observer does not affect the fundamental frequency signal but has a harmonic suppression effect similar to ANFs.

3.3.4. Zero/Low-Speed Operation

As aforementioned, the model-based sensorless methods possibly fail during the zero/low-speed operation. The model observability analysis of PMSMs provides a theoretical explanation for the poor performance at zero/low speed [143,144,145,146]. The convergence of model-based sensorless methods relies on a sufficient condition that the state variables must be locally weakly observable at a given operating point. For the highly nonlinear motor system, this property can be verified through Lie derivative analysis. The observability criterion matrix for determining local weak observability is defined as

$$\mathit{O}={\left[\begin{array}{cccc}{\displaystyle \frac{\partial {\mathcal{L}}_f^0{h}_1}{\partial i_{\alpha }}}& {\displaystyle \frac{\partial {\mathcal{L}}_f^0{h}_1}{\partial i_{\beta }}}& {\displaystyle \frac{\partial {\mathcal{L}}_f^0{h}_1}{\partial {\theta }_e}}& {\displaystyle \frac{\partial {\mathcal{L}}_f^0{h}_1}{\partial {\omega }_e}}\\ {\displaystyle \frac{\partial {\mathcal{L}}_f^0{h}_2}{\partial i_{\alpha }}}& {\displaystyle \frac{\partial {\mathcal{L}}_f^0{h}_2}{\partial i_{\beta }}}& {\displaystyle \frac{\partial {\mathcal{L}}_f^0{h}_2}{\partial {\theta }_e}}& {\displaystyle \frac{\partial {\mathcal{L}}_f^0{h}_2}{\partial {\omega }_e}}\\ ⋮& ⋮& ⋮& ⋮\\ {\displaystyle \frac{\partial {\mathcal{L}}_f^3{h}_2}{\partial i_{\alpha }}}& {\displaystyle \frac{\partial {\mathcal{L}}_f^3{h}_2}{\partial i_{\beta }}}& {\displaystyle \frac{\partial {\mathcal{L}}_f^3{h}_2}{\partial {\theta }_e}}& {\displaystyle \frac{\partial {\mathcal{L}}_f^3{h}_2}{\partial {\omega }_e}}\end{array}\right]}_{(8\times 4)}$$

(31)

As demonstrated in [143], the determinant of the observability matrix can be derived as

$$-\delta L\left({\displaystyle \frac{d{i}_q}{dt}}\left(\delta L+{\psi }_f\right)-\delta L{\displaystyle \frac{d{i}_d}{dt}}\right)+\omega \left({\left(\delta L\right)}^2\left(i_d^2+i_q^2\right)+{\psi }_f\left({\psi }_f+2\delta L{i}_d\right)\right)\ne 0$$

(32)

where δL = L_d − L_q.

For the IPMSMs, δL ≠ 0. When the motor runs at zero speed (or near to zero speed), the observability condition can be reduced to

$$\left|i_d+{\displaystyle \frac{{\psi }_f}{\delta L}}\right|\left|C\right|\ne \left|i_q\right|$$

(33)

where C is a constant. Equation (33) indicates that the state of IPMSMs in the zero or low-speed is observable when the d-q axis currents change but are not linearly synchronized. For the SPMSMs, δL = 0. Thus, a well-known sufficient observability condition can be obtained as

$${\omega }_e\ne 0$$

(34)

However, the derivation of (32) is based on the assumption that the rotor speed varies slowly. Considering rotor acceleration, the observability criteria in [147] can be expressed as

$${\displaystyle \frac{{\psi }_f^2}{L_s^2}}\left({\displaystyle \frac{R_s}{L_s}}+{\displaystyle \frac{f_v}{J}}\right){\dot{\omega }}_e\ne 0$$

(35)

Thus, the state of an SPMSM is observable when ω_e satisfies

$${\omega }_e\ne 0 \vee {\dot{\omega }}_e\ne 0$$

(36)

According to the analysis above, the main challenges for model-based sensorless control in the zero/low-speed region can be summarized as follows: compensation for inverter nonlinearity, parameter robustness, start-up and speed reversal. When the dead-time compensation and motor parameters are accurate, the motor state is observable at low speeds. However, as the errors exist in reality, the state variables become unobservable under certain conditions, ultimately leading to system instability. Therefore, proactive identification of both inverter nonlinearity and motor parameters is necessary.

Typically, to achieve full-speed-range sensorless control, the I/F control or HFSI methods are usually adopted as the start-up strategy. However, this multi-strategy switching is complex and time-consuming. For situations with a fast start-up and frequent speed reversal, such switching is not desirable. Recently, research attention has been shifting toward sensorless methods capable of full-speed-range operation. For the SPMSMs at standstill, the sufficient condition for observability is the existence of rotor acceleration. It is necessary to apply torque to the rotor to break it from the stationary state. Subsequently, the observer promptly acquires reliable position information, which is then used for speed and torque control. Thus, fast and reliable rotor position acquisition following a brief uncontrollable interval is the critical requirement for start-up. Currently, several cases of the single sensorless control method achieving full-speed-range operation have been documented. Reference [147] proposes a full-order adaptive flux observer featuring virtual hybrid voltage injection, which achieves stable operation at zero speed at a rated load. The nonlinear flux observer in [61] and the voltage model observer in [148] perform well during start-up and speed reversal. However, load-driving capability, reliable start-up and stable rotation reversal are still common challenges for these methods.

3.4. High–Low Speed Switching Strategy

When combined strategies of high-speed and low-speed methods are adopted, a smooth and robust transition between different algorithms is the key to avoiding speed or torque ripple and maintaining system stability. Since the switching between the virtual frequency method and the model-based method has been analyzed in the previous section, this subsection mainly introduces the switching strategies between saliency-based and model-based methods.

Currently, the dominant approach is weighted switching, which can be further categorized into position/speed weighting [149,150] and error signal weighting [151,152]. The former operates dual position observers simultaneously and fuses their estimated results via weighting coefficients for output. Since two position observers are employed, the low-speed and high-speed responses can be adjusted independently, but the computational burden increases accordingly. As for the error signal weighting, the normalized position errors of both the HF model and fundamental model are combined by a weighting function and used as the input to a PLL. This approach is easier to implement and requires fewer parameters to tune. The speed-based weighting function is shown as Figure 20. The estimated result of the weighted average is

$${\widehat{\theta }}_r=(1-k)\times {\widehat{\theta }}_h+k\times {\widehat{\theta }}_f$$

(37)

where ${\widehat{\theta }}_r$ is the final result, and ${\widehat{\theta }}_h$ and ${\widehat{\theta }}_f$ are results of saliency-based and model-based methods. k is the weighting coefficient, which can be calculated as

$$k=\left\{\begin{array}{c}0\\ ({\widehat{\omega }}_e-{\omega }_{low})/({\omega }_{high}-{\omega }_{low})\\ 1\end{array}\right.\begin{array}{c}{\widehat{\omega }}_e<{\omega }_{low}\\ {\omega }_{low}<{\widehat{\omega }}_e<{\omega }_{high}\\ {\widehat{\omega }}_e>{\omega }_{high}\end{array}$$

(38)

where ω_low and ω_high are the low-speed and high-speed boundaries of the transition region, respectively. Besides linearly varying weighting coefficients, [149] adopts fuzzy weighting coefficients, which also achieves a smooth transition between algorithms.

Furthermore, an alternative strategy is to build a unified hybrid model combining the high-frequency and fundamental models. In [153], a quadratic BEMF model was applied, which integrates the position extraction of the injection method and the model-based method into a unified framework, thereby achieving seamless switching. Research following the same approach can be found in [154,155].

4. Discussion

This article reviews the state-of-the-art sensorless control techniques for PMSMs. Research on sensorless control has progressively shifted toward performance optimization to enhance its engineering applicability. This section focuses on the analysis of sensorless algorithms for specific application scenarios, serving as a reference for engineering practice. Furthermore, key challenges and emerging trends are discussed as well.

4.1. Application-Oriented Analysis and Implementation Insights

Different sensorless control techniques possess distinct strengths and limitations, and their suitability is critically determined by the particular needs and constraints of the application field. Key systematic evaluation indicators for these methods include minimum operating speed, dynamic response, parameter robustness, computational complexity, cost, and dependence on motor saliency, as partially summarized in Table 1 and Table 2. The following analysis will be conducted in combination with specific application scenarios.

Table 3. Typical application areas of sensorless control and their core requirements.

Target Application Areas	Typical Operating Conditions	Core Requirements
Industrial fans and pumps	Wide-range speed regulation, low dynamic response requirements, rare ultra-low speed	High reliability, long service life
Home appliances (air conditioner fans, refrigerator compressors)	Long-term uninterrupted operation, relatively fixed load, steady-state or slowly variable speed operation	Ultimate low system cost, high reliability, good efficiency
Electric-vehicle drive motor	Transient load torque, wide speed range (zero speed to ultra-high-speed flux-weakening region), vibration and humidity environments	Strong adaptability to motor parameter variations, absolute safety under full operating conditions

presents some typical application scenarios and their core requirements. For the numerous fans and pumps in industrial systems, their load is proportional to the square of the rotational speed. During the start-up, I/F control is sufficient. As the speed increases, a transition is made to model-based high-speed methods.

For home appliances, which have stringent requirements for cost and failure rate, reliable and easy-to-implement algorithms are the preferred choice. Since a high-frequency injection generates noise and the FPE method has high hardware requirements, I/F control can be adopted for start-up in the low-speed stage and then switched to high-speed methods. Among high-speed methods, the sliding-mode observer is widely used due to its simple implementation and strong robustness. Algorithms with high computational burden, such as EKF, are not applicable in these low-cost systems.

In recent years, electric vehicles have developed rapidly. For safety considerations, sensorless control can be employed as a backup in the event of position-sensor failure. Current main drive PM motors are mostly IPMSMs to fully utilize reluctance torque and flux-weakening speed expansion capability. The saliency-based HFSI methods have significant application value, but it is necessary to reduce injection noise to improve driving comfort. Since motor parameters vary significantly under different operating conditions, the adopted sensorless algorithm should be incorporated with online parameter identification to enhance parameter robustness.

Currently, there is no universally optimal algorithm as the selection of a technique is the result of tradeoffs under specific application constraints. Future research should focus on scenario-oriented algorithm design and the establishment of standardized testing benchmarks to accelerate technology transfer.

4.2. Challenge and Future Research Trends

4.2.1. Dynamic Response Enhancement in Full Speed Range

Achieving dynamic performance comparable to that of sensored PMSM control has always been a primary goal for sensorless techniques. The inherent tradeoff between dynamic response and estimation accuracy is a common challenge faced by both low-speed and high-speed sensorless algorithms. Consequently, enhancing dynamic response is a crucial prerequisite for realizing high-performance sensorless control and ultimately replacing mechanical position sensors.

4.2.2. Sensorless Drives Under Low Carrier Ratio

Sensorless control under low carrier ratios is another critical challenge. High-power motor systems with low switching frequencies and ultra-high-speed PMSM drives for special applications are current research focuses. For both operating conditions, sensorless techniques under low carrier ratios still face unresolved issues that require further investigation.

4.2.3. Improvement in Parameter Robustness for Sensorless Control

Parameter sensitivity remains a fundamental constraint for sensorless control algorithms, directly impacting stability and accuracy. In addition to enhancing the inherent robustness of algorithms, integrating real-time motor parameter identification is also a viable solution. Looking forward, the focus should shift toward adaptive, self-learning systems combined with intelligent algorithms to achieve universal sensorless control.

4.2.4. Zero/Low-Speed Performance Enhancement of Model-Based Algorithms

Performance degradation in the zero/low-speed region remains a persistent weakness for model-based sensorless algorithms. However, a unified full-speed-range method is the optimal engineering solution, rather than the currently adopted hybrid strategy. Overcoming difficulties of the mode-based method in start-up and speed reversal, as well as further expanding the low-speed operating range, are crucial for the practical application of sensorless algorithms.

4.2.5. AI and Data-Driven Sensorless Control

The aforementioned challenges (e.g., dynamic response, parameter robustness, low-speed performance) are fundamentally rooted in the strong dependence on accurate physical models and a priori knowledge. Recently, artificial intelligence (AI) and data-driven approaches have offered new solutions to these inherent constraints. The intrinsic advantage of such methods lies in their ability to directly learn the complex dynamic characteristics and nonlinear mapping relationships of motors from operational data, thereby reducing reliance on precise models.

Deep learning and reinforcement learning have been utilized for the online identification of key motor parameters and load disturbances [156,157]. Compared with traditional identification methods, they exhibit superior robustness to noise and model inaccuracies. AI techniques have also been incorporated into observer structures to improve the performance of position estimation [158,159]. Benefiting from the potential to learn complex nonlinear dynamics, the AI-based observers may achieve superior dynamic estimation performances across a wide speed range, particularly in the low-speed region. Moreover, using neural networks to model unknown or non-ideal factors proves to be a feasible and effective approach [160,161].

However, several obstacles must be overcome to facilitate the proliferation of AI in sensorless control, including the acquisition of sufficient and effective training data, the deployment of complex AI models on resource-constrained control chips, and concerns regarding reliability and safety.

5. Conclusions

For cost-sensitive and high-reliability applications, PMSM sensorless control is irreplaceable as it eliminates the need for traditional position sensors. This article provides a comprehensive review of existing sensorless control techniques and their latest developments. For the low-speed region, virtual frequency methods, high-frequency injection methods, and fundamental PWM excitation methods are introduced. For the medium-to-high-speed region, flux and back-EMF-based methods are reviewed, while also analyzing their performance degradation under non-ideal conditions. Finally, application-oriented analysis, implementation insights, as well as the challenges and future development trends of sensorless control are discussed. Currently, a performance gap still exists between sensorless control and sensored control. In the future, sensorless control with enhanced robustness and dynamic performance is expected to completely replace position sensors in a wider range of applications.