## 1 Introduction  
Induction motors are widely employed in industrial applications due to their inherent ruggedness, high reliability, and cost-effectiveness. When precise and efficient rotor speed control is required, field-oriented control (FOC), also known as vector control, is commonly adopted. With advancements in power electronics and the widespread availability of high-speed digital signal processors (DSPs), model-based control strategies have become the mainstream approach in motor drive systems. In addition, comprehensive training platforms for implementing FOC have also been developed [1, 2]. The core concept of FOC lies in transforming the motor dynamic equations into a rotating reference frame in which the d-axis is aligned with the rotor flux vector [3]. Under this condition, rotor dynamics can be described by Equation \(L_{r}\frac{d}{d}\lambda_{dr}^{rf}+r_{r}\lambda_{dr}^{rf}=r_{r}L_{m}\tau_{ds}^{ rf}\), \(T_{e}=\frac{3 PL_{m}}{4L_{r}}\lambda_{dr}^{rf}\iota_{ds}^{rf}\). Accurate control of the current components \(\iota_{ds}^{rf}\) and \(\iota_{ds}^{rf}\) in this frame enables the generation of the desired electromagnetic torque \(T_{e}\). Most FOC-based speed control schemes for induction motors share a similar structural configuration, as illustrated in Figure 1. It is important to emphasize that an induction machine's rotor magnetic pole \(\lambda_{\mathit{old}}^{s}\) arises from the relative motion between the stator's rotating magnetic field \(\lambda_{\mathit{ols}}^{s}\) and the physical rotor. Perfect alignment is not always guaranteed even under steady-state conditions, where \(\lambda_{\mathit{old}}^{s}\) and \(\lambda_{\mathit{ols}}^{s}\) rotate synchronously. Since the magnitude (\(\lambda_{rf}\)) and direction (\(\theta_{rf}^{s}\)) of the rotor flux cannot be directly measured, it is necessary to estimate the rotor flux angle (\(\theta_{rf}^{s}\)) prior to applying the control strategy shown in Figure 1. Consequently, various rotor flux angle estimation techniques have been developed, which are generally categorized into two main approaches: direct field-oriented control (DFOC) and indirect field-oriented control (IFOC). In direct field-oriented control (DFOC), the rotor flux angle (\(\theta_{rf}^{s}\)) is directly estimated. As shown in Figure 2a,b, two representative DFOC structures are commonly adopted, depending on whether a gap-flux sensor is employed [3]. The gap-fluxsensor-based method, depicted in Figure 2a, presents several limitations. First, the sensing region is highly localized and cannot accurately represent the total rotor flux. Second, the sensor structure lacks mechanical stability and is highly susceptible to noise interference. Moreover, the inclusion of a gap-flux sensor increases the overall system cost, making this method less desirable for practical applications. The method illustrated in Figure 2b, along with the approach proposed by [PERSON] [4], employs a sensorless structure that does not require a gap-flux sensor. This design leverages the fact that, under medium- to high-speed conditions, the back electromotive force (EMF) dominates the stator voltage, as described by  
\[\underline{v}_{\text{qds}}^{s}\approx\underline{v}_{\text{qs}}^{e}\approx-j \frac{L_{\text{m}}}{L_{r}}\omega_{r}\underline{\lambda}_{\text{dr}}^{rf}. \tag{1}\]  
[PERSON] [4] estimated the rotor flux angle (\(\theta_{rf}^{s}\)) based on the stator voltage (\(\underline{v}_{\text{qds}}^{s}\)). Under steady-state conditions, the resulting rotor flux angle (\(\underline{\lambda}_{rf}\)) typically lags by a fixed phase angle. However, both the structure shown in Figure 2b and the algorithm proposed in [5] exhibit significantly degraded performance at low speeds. Specifically, the integration-based estimation in Figure 2b may lead to numerical drift or overflow, while the underlying  
Figure 1: The common structure of most FOC mechanisms. Figure 2: (**a**) The direct FOC structure using gap-flux sensors. (**b**) The sensorless direct FOC structure.