## 5 Conclusions  
It is worth emphasizing that the proposed framework is neither a conventional direct field-oriented control (DFOC) nor an indirect field-oriented control (IFOC) scheme. Instead, it combines the advantages of both approaches while avoiding their respective drawbacks. For example, as shown in Figure 2(b), the conventional IFOC method calculates the slip frequency (\(S\omega_{e}^{*}\)) from command values and integrates it to obtain the rotor flux angle, which is then fed back into the control loop. This approach can easily diverge from actual system behavior. Similarly, the sensorless DFOC structure in Figure 2(b) continues to rely on the integration of measured signals to compute \(\theta_{rf}^{*s}\). However, continuously integrating inevitably accumulates errors and lacks a correction mechanism. In contrast, the proposed method introduces a dynamic switching mechanism between transient and steady-state modes, with a phase-locked loop (PLL) gradually aligning the estimated rotor flux angle with its true value during steady-state operation. This hybrid approach improves estimation accuracy while maintaining system stability. Moreover, since the proposed control strategy achieves convergence within a few seconds, it is particularly well suited for applications that operate primarily at constant speeds. These include air conditioning systems, household appliances, electric vehicles, and other energy-efficient industrial drives.In the context of increasing emphasis on energy efficiency and environmental sustainability, control strategies that reduce power consumption without requiring hardware modifications are of significant value. Compared to the conventional indirect FOC method, the proposed approach demonstrates this potential by achieving higher steady-state efficiency. One potential limitation of the proposed method lies in its sensitivity to variations in motor parameters. However, this challenge can be addressed through well-established system identification techniques [23, 24, 25]. For example, in [26], an adaptive control framework was developed by combining observers for unmeasured states, such as rotor flux and electromagnetic torque, with estimators for time-varying parameters, including rotor resistance and load torque. Rather than sacrificing system performance to mitigate parameter sensitivity, online identification methods can be employed to ensure accurate estimation. At the very least, the proposed method provides a clear framework and tunable structure for rotor flux angle estimation. Simulation results confirm that, when accurate system parameters, such as those obtained via online identification, are available, the proposed approach remains both feasible and effective. \begin{table}
\begin{tabular}{p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline \hline
**Feature** & **Proposed Method (Hybrid PLL-Based FOC)** & **Extended Kalman Filter (EKF)** & **Sliding Mode Observer (SMO)** & **Conventional Indirect FOC** \\ \hline \multirow{2}{*}{Sensitivity to Noise capability} & Moderate (PLL filtering capability) & Moderate-High (Kalman filter requires tuning) & Moderate (observer robust but chattering issue) & Moderate-Low (integration issues) \\ \hline Practical Implementation and Validation & Validated experimentally and in simulation & Validated mostly in simulation & Experimental validation common & Established practical use \\ \hline Efficiency Improvement (Stator Current Reduction) & Significant improvement & Moderate (dependent on accurate estimation) & Moderate (efficiency not primary focus) & Moderate (standard approach) \\ \hline \multirow{2}{*}{Error Accumulation} & Minimal (PLL continuously corrects angle) & Moderate (cumulative estimation errors) & Moderate (observer inaccuracies) & High (integration error accumulates) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._  
**Author Contributions:** Conceptualization, C.-W.D. and P.-C.T. ; Methodology, C.-W.D. and P.-C.T. ; Software, C.-W.D. ; Validation, P.-C.T. ; Formal analysis, C.-W.D.