## 1 Introduction
However, this method does not account for transient behavior, which has led to the development of several improved approaches aimed at enhancing performance under dynamic operating conditions [14; 15]. For example, [PERSON] [14] analyzed an equivalent steady-state circuit of the induction motor, as illustrated in Figure 4, and derived the following expression:  
\[\rho\triangleq\frac{i_{ds}^{p}}{i_{qs}^{p}}=\frac{\beta_{2}(S\omega_{e})^{2}+ \beta_{1}S\omega_{e}+\beta_{0}}{\alpha_{2}(S\omega_{e})^{2}+\alpha_{0}}. \tag{3}\]  
In this expression, \(\alpha_{2}=\omega_{e}L_{lr}(L_{m}^{2}-L_{ls}L_{lr})\), \(\alpha_{0}=-\omega_{e}r_{r}^{2}L_{ls}\), \(\beta_{2}=r_{s}L_{lr}^{2}\), \(\beta_{1}=\omega_{e}r_{r}L_{m}^{2}\), \(\beta_{0}=r_{s}r_{r}^{2}\), and the superscript \(p\) indicate that the d-axis of the reference frame is aligned with the stator voltage vector. Based on this formulation, the slip frequency is expressed as follows:  
\[S\omega_{e}=\frac{\beta_{1}\pm\sqrt{\beta_{1}^{2}-4(\rho\alpha_{0}-\beta_{0}) (\rho\alpha_{2}-\beta_{2})}}{2(\rho\alpha_{2}-\beta_{2})}. \tag{4}\]  
Furthermore, using the definition of \(\rho\), along with measured values of the stator current and voltage, the following relationship be derived:  
\[\rho=\frac{3i_{ds}^{s}\sT_{\varphi ds}^{s}-i_{0s}v_{0s}}{\sqrt{3}\mathbb{I}^{ T}\left(\sT_{qds}^{s}\sT_{qds}^{s}\right)}. \tag{5}\]  
Here, \(i_{0s}\) and \(v_{0s}\) represent the current and voltage at the neutral point, respectively. Equation (3) indicates that \(\rho\) is a function of \(S\omega_{e}\), and the power factor reaches its optimal value when the condition  
\[\left.\frac{d\rho}{d(S\omega_{e})}\right|_{S=S_{e}}=0\Rightarrow\rho_{\max}= \left.f(S\omega_{e})\right|_{S=S_{e}}\]  
Figure 1.0 The equivalent circuit of induction motors under steady state. Figure 1.1 (**a**) The common parts of all indirect FOC structures. (**b**) The most conventional indirect FOC method of estimating the slip frequency. is satisfied, where \(S_{\epsilon}\) represents the critical slip. Accordingly, this criterion can be used to determine the valid solution of (4). Another approach, presented in [15] and illustrated by the structure in Figure 5, addresses this issue by feeding the stator current \(\left(\left|\dot{i}_{qds}(k)\right|\right)\) into a fuzzy logic controller. The fuzzy control rules are designed based on the principles depicted in Figure 1.2 Once the motor reaches steady-state operation, the variation in rotor speed (\(\left|\Delta\omega_{r}(k)\right|\)) remains below a predefined threshold (\(\epsilon\)). The most recent output of the fuzzy controller is then added to the slip frequency estimated by the conventional indirect FOC scheme. The physical interpretation of Figure 6 is as follows: under the condition that the electrical rotor speed (\(\omega_{r}\)) remains constant at its steady-state value (\(\omega_{d}^{*}\)), the output parameter \(\Delta(S\omega_{\epsilon}^{*})\) is adjusted to modify \(\delta(S\omega_{\epsilon}^{*})\) such that the resulting stator current \(\left(\left|\dot{i}_{qds}(k)\right|\right)\) is minimized. This condition yields the optimal estimation of the slip frequency (\(S\omega_{\epsilon}^{*}\)). However, during this process, the deviation between \(\omega_{r}\) and \(\omega_{d}^{*}\) may increase as \(\delta(S\omega_{\epsilon}^{*})\) grows. In such cases, the system departs from steady-state operation. To restore stability, the variable \(\delta(S\omega_{\epsilon}^{*})\) must be decreased, regardless of the variation in \(\left|\dot{i}_{qds}(k)\right|\), which implies that \(\Delta(S\omega_{\epsilon}^{*})\) becomes negative. In summary, the optimal field-oriented control (FOC) under steady-state conditions corresponds to the configuration that minimizes the stator current (\(\left|\dot{i}_{qds}\right|\)).