## 1 Introduction
assumption of (1)--that the back electromotive force (EMF) dominates the stator voltage--becomes invalid due to the reduced influence of EMF at low speeds. A convergence analysis of the extended Kalman filter (EKF) applied to sensorless control of induction motors was presented in [6]. Although EKF-based methods are capable of estimating rotor flux and speed over a wide speed range, the time-varying nature of the linearized system model introduces additional complexity and may compromise estimation stability. In another study, [PERSON] et al. [5] employed an adaptive full-order observer for rotor flux and speed estimation in sensorless induction motor drives. However, this approach exhibits significant sensitivity to parameter variations, particularly under low-speed regenerative operating conditions. In [7], [PERSON] developed two discrete-time models for induction motors using variational integrators, and validated the models through both simulation and experimental results. Subsequently, a complete system framework incorporating both a controller and an observer was constructed based on a sliding mode control strategy, as described in [8]. Although the effectiveness of the proposed system was demonstrated, thisstudy did not address the method's computational efficiency or associated implementation costs. In the context of direct field-oriented control (FOC), [PERSON] et al. [9] proposed incorporating a phase-locked loop (PLL) into the voltage model to estimate the rotor speed. Based on the estimated speed, the cutoff frequency of the low-pass filter (LPF) is adaptively adjusted to enable accurate tracking of the rotor flux angle at medium and high speeds. In [10], [PERSON] argued that compared to the previously proposed low-pass filter (LPF), the second-order generalized integrator-based frequency-locked loop (SOGI-FLL), derived from the phase-locked loop (PLL), can improve estimation accuracy, reduce system complexity, and attenuate high-order harmonics in the back electromotive force (EMF). However, its performance degrades significantly during back-EMF transients, resulting in inaccurate frequency estimation and the occurrence of second harmonic oscillations. To address this issue, a dual-axis SOGI-FLL (DSOGI-FLL), constructed using the \(\alpha\) and \(\beta\) components, was proposed. This approach not only suppresses the second harmonic oscillations but also enhances estimation accuracy. Nevertheless, similar to the conventional SOGI-FLL, the DSOGI-FLL exhibits poor performance under low-speed conditions. In indirect field-oriented control (IFOC), the slip frequency (\(S\omega_{e}\)) is first estimated, and the rotor flux angle is then derived using  
\[\theta_{rf}^{s}=\int_{0}^{t}S\omega_{e}(\tau)+\omega_{r}(\tau)dt \tag{2}\]  
where \(\omega_{r}=\omega_{m}\frac{P}{2}\) represents the product of the actual rotor speed and the number of pole pairs. Strictly speaking, (2) is valid only when the slip frequency is defined as the difference between the rotor flux angular velocity and the rotor mechanical angular velocity (\(S\omega_{e}\equiv\omega_{rf}-\omega_{r}\)) [11]. Under steady-state conditions, this value approximates the difference between the synchronous speed and the rotor electrical speed (\(\omega_{e}-\omega_{r}\)). Figure 3a illustrates the representative structure of a typical indirect field-oriented control (IFOC) scheme [3]. In [12], the concept of multiple three-phase systems was further extended to enable the control of multiphase induction motors. Moreover, the concept of slip frequency has been applied to widely used and cost-effective induction machines to improve the inefficiencies associated with conventional control strategies [13]. In traditional approaches, the slip frequency is estimated by determining the d-axis and q-axis current references (\(\hat{t}_{ds}^{rf*}\) and \(\hat{t}_{qs}^{rf*}\)) based on the torque and flux commands, respectively. An estimate of the slip frequency can then be obtained, as illustrated in Figure 3b.