## 1 Introduction
Although the procedures described above yield an accurate or optimal slip frequency (\(S\omega_{\epsilon}^{*}\)) under steady-state conditions, the estimation of \(\theta_{rf}^{*\ast}\) remains affected by accumulated error during transients due to the integration process in (2). As a result, the slip frequency derived purely under steady-state assumptions cannot ensure accurate estimation of the rotor flux angle (\(\theta_{rf}^{s}\)). Therefore, unless the slip frequency (\(S\omega_{\epsilon}^{*}\)) can be precisely estimated during transients, the resulting field-oriented control will be inaccurate, limiting the effectiveness of these conventional approaches. Figure 1.3 The estimation method used in [15]. Figure 1.4 The fuzzy rules used in [4]. Recently, [PERSON] et al. [16] proposed a rotor speed sensing technique based on the observation that discontinuities in the magnetic path, caused by rotating rotor slots, induce high-frequency harmonics in the stator phase currents. These harmonics are utilized to infer the rotor speed. However, the stator current is also affected by magnetic and electrical interferences, necessitating the use of filters to extract the true speed signal. Since the machine operates over a wide speed range, the filter's cutoff frequency must adapt to the varying electrical frequency. To achieve this, the authors incorporated complex algorithms such as phase-locked loops (PLLs). Nevertheless, skewed slot designs--commonly adopted in motor manufacturing to reduce cogging torque--conflict with the assumptions required by this approach. As a result, the practical viability of this technique remains under investigation. A combined FOC and Direct Torque Control (DTC) approach was proposed to enhance the performance of three-phase induction motors across varying torque loads. This method utilizes a dynamic reconfiguration between control algorithms to optimize performance during different operating conditions [17]. A sensorless speed control method employing a Model Reference Adaptive System (MRAS) and a deadbeat regulator was introduced to improve affordability and efficiency by eliminating the need for physical speed sensors [18]. A compensated direct field-oriented sensorless control using a Luenberger observer was developed for induction motors connected via long cables and filters, addressing challenges in submarine applications where sensor deployment is impractical [19]. A hybrid sensorless speed estimator based on flux error was presented for electric vehicle applications, aiming to improve speed tracking accuracy in indirect FOC systems [20]. A nonlinear field-oriented control technique utilizing tensor product representations and linear matrix inequalities was proposed to enhance control performance in induction machines [21]. The technique proposed in this paper does not fall under conventional indirect field-oriented control (IFOC) or direct field-oriented control (DFOC). Instead, it integrates the advantages of both approaches while avoiding their respective limitations. As shown in Figure 3b, conventional IFOC estimates the rotor flux angle (\(S\omega_{e}^{*}\)) based on command values (\(i_{qs}^{ff*}\) and \(\lambda_{dr}^{ff*}\)), which are then integrated and treated as the actual rotor flux angle to be fed back into the control system. Clearly, such an approach may deviate from real-world behavior. For example, although [PERSON] [22] incorporated a phase-locked loop (PLL) algorithm along with an adaptive model to mitigate issues associated with pure integration, the model still relies on the structural framework depicted in Figure 3b. Similarly, the sensorless DFOC scheme shown in Figure 2b continues to depend on integration to derive \(\underline{\lambda}_{qds}^{s}\), which is then fed back to compute \(\underline{\lambda}_{qdr}^{s}\) and eventually obtain \(\theta_{rf}^{s*}\).