## 1 Introduction
### Sensorless Control of BLDCMs  
Sensorless control methods for BLDCMs require position estimation techniques [4]. Indirect methods estimate rotor position indirectly by leveraging position-related quantities, such as the back-electromotive force (back-EMF) components or the third harmonic of the back-EMF signal. The most widespread indirect position estimation methods rely on the detection of the zero-crossing points of the back-EMF waveform to generate the virtual Hall signals [10]. Alternatively, the third harmonic component of the back-EMF can be used, which has a constant phase relationship with the rotor flux for any machine operating condition [11]. However, these methods become less accurate at low speeds and standstill conditions [12], which, in fact, also hinder the control during motor starting, requiringopen-loop starting procedures. This limitation is due to the reduced amplitude of the back-EMF at low speeds, which leads to an excessively low signal-to-noise ratio of the position-related system states and, consequently, delays detection [13]. To address the limitations of indirect position estimation methods at low speeds, model-based approaches are widely employed for rotor position and speed estimation, especially for medium- and high-speed applications. Among model-based approaches, closed-loop observers are the most prominent [14], designed to drive the observer output to converge with the controlled output, ensuring that the tracking error is brought to zero. Thus, the estimated values of the states are forced to converge with their actual values. Indeed, closed-loop observers can be assumed as adaptive filters. Due to their disturbance rejection capabilities and fair robustness to variations in machine parameters and noises present in current or voltage measurements. In the literature, a wide range of observers have been reported. In the following, the current literature trends in observer techniques are presented, focusing on their mechanisms, challenges, and recent advancements. Nonlinear-model-based observers, such as back-EMF observers, are among the simplest and most effective methods at medium to high speeds [7]. They estimate rotor position by leveraging the back-electromotive force (EMF) induced in the stator windings, which directly correlates with rotor flux. These observers are computationally efficient but fail at low speeds due to the diminishing back-EMF amplitude, which compromises observability. Open-loop I-f control is typically used for startup, transitioning seamlessly to sensorless methods [15]. Extended back-EMF models incorporating resistance and inductance variations have also been reported to address the issues, improving low-speed performance and position estimation accuracy [16]. Observer bandwidth is another critical factor; reducing bandwidth enhances disturbance rejection but may impair accuracy. For instance, extended EMF observers in the stator reference frame have demonstrated low position errors (<1\({}^{\circ}\)) and consistent speed estimation accuracy across wide speed ranges. A stator-frame flux observer achieved effective speed reversal in sensorless mode, while its modification enabled zero-speed starting at rated torque and rapid acceleration from zero to rated speed in just 0.1 s [17]. Sliding mode observers (SMOs) are widely used for estimating rotor speed and position due to their robustness against model uncertainties and disturbances [18]. However, traditional SMOs suffer from chattering, which induces high-frequency oscillations and high-order harmonics, adversely affecting the dynamic performance of the system. To mitigate this issue, adaptive filters or phase-locked loops (PLLs) are often integrated [19; 20; 21]. While conventional SMOs employ smoothing functions to reduce chattering, the delays introduced by low-pass filters (LPFs) in first-order SMOs necessitate the adoption of high-order SMOs [22; 23].