## 1 Introduction
### Sensorless Control of BLDCMs
The super-twisting sliding mode observer (STO) is a notable advancement, effectively reducing chattering while maintaining robustness, albeit with increased complexity in gain design [24]. Flux linkage observers are another popular approach, particularly for high-speed applications [25]. These observers estimate rotor flux linkage using stator voltage and current measurements. However, they are sensitive to parameter mismatches caused by variations in temperature, saturation, and load conditions. Adaptive parameter estimation techniques [26], such as Extended Kalman Filters (EKFs) [27], and model reference adaptive systems (MRASs) [28], have also been integrated with flux linkage observers to dynamically adjust for these variations. A common challenge across observer-based position estimation methods lies in determining rotor position from the estimated quantities. This often involves using the arc-tangent function, which is prone to noise-induced spikes at low speeds. These spikes complicate speed estimation, necessitating the use of low-pass filters with adaptive cutoff frequencies and variable gains for improved accuracy. In variable-speed drives, frequency-locked loops(FLLs) and PLLs have emerged as critical tools to refine position estimation and mitigate the limitations of arc-tangent-based methods [29]. Among these, the conventional synchronous reference frame PLL is widely employed due to its simplicity [30; 31]. However, it uses a fixed cutoff frequency, making it less adaptable to changes in operating conditions. Comprehensive reviews [32] have highlighted other PLL variants, such as the moving average filter PLL (MAF-PLL), tangent-based PLL (TPLL), and quadrature-PLL (QPLL) [33]. These PLLs are extensively used for position and frequency estimation but remain sensitive to voltage variations, affecting loop gain, stability margins, and dynamic behavior. Furthermore, voltage disturbances can lead to performance inconsistencies, especially in systems with trapezoidal back-EMF or quasi-square wave currents, as in permanent magnet BLDCM drives. In such cases, extracting the fundamental positive-sequence components of voltage is essential for implementing sensorless algorithms effectively. Speed-adaptive variable-frequency filters, such as second-order generalized integrators (SOCIs), multiple SOCIs (MSOCIs), and dual SOCIs (DSOCIs), have been developed to extract these components [19]. These filters offer robust harmonic rejection and enable PLLs to accurately extract rotor position information. However, significant errors may still occur when the back-EMF approaches zero, which can be mitigated by gain-scheduling strategies to dynamically adjust system parameters. This introduces additional control complexity but ensures reliable performance. Frequency-dependent observers [14], such as adaptive bandpass position observers and harmonic elimination techniques, address challenges posed by inverter nonlinearity, harmonic distortions, and noise. Adaptive bandpass filters, like SOCIs, are widely used to isolate fundamental components while suppressing DC offsets and harmonic disturbances [34; 35]. The SOCI's ability to dynamically adapt to rotor frequency has made it a preferred choice for frequency-locked-loop designs. Advanced configurations, such as multiple harmonic elimination (MHE) observers [36] utilize bandstop filters to dynamically cancel low-order harmonics like the fifth and seventh. These methods significantly enhance robustness but at the cost of increased computational complexity and control design challenges. Machine parameter-dependent observers [14], on the other hand, focus on estimating parameters such as resistance, inductance, and flux to maintain accuracy under dynamic operating conditions. Techniques like recurrent least squares (RLS), MRAS, and adaptive PI controllers are often employed for real-time parameter adaptation.