## 2 New Field-Oriented Control (FOC) Technique
By substituting the previously obtained coefficients into Equation (19), the closed-form solution is derived as follows:  
\[0=r_{r}\dot{r}^{rf}_{qr}+\Big{(}\omega_{rf}-\omega_{r}\Big{)}\lambda^{rf}_{ dr}, \tag{10}\]5  
In this expression, \(\widetilde{a_{3}}=-L_{r}\big{(}L_{m}^{2}-L_{r}L_{s}\big{)}\), \(\widetilde{a_{2}}=L_{r}(r_{s}L_{r}+r_{L}L_{s})-r_{r}\big{(}L_{m}^{2}-L_{r}L_{ s}\big{)}\), \(\widetilde{a_{1}}=r_{r}(2r_{s}L_{r}+r_{r}L_{s})-(\omega_{e}-\omega_{r})^{2}L_{ r}\big{(}L_{m}^{2}-L_{r}L_{s}\big{)}\), \(\widetilde{a_{0}}=r_{r}^{2}r_{s}+r_{s}L_{r}^{2}(\omega_{e}-\omega_{r})^{2}+ \omega_{e}r_{r}L_{m}^{2}(\omega_{e}-\omega_{r})\), \(\widetilde{b_{2}}=\omega_{e}L_{r}\big{(}L_{m}^{2}-L_{r}L_{s}\big{)}\), \(\widetilde{b_{1}}=r_{r}\big{[}\omega_{e}\big{(}L_{m}^{2}-2L_{r}L_{s}\big{)}+( \omega_{e}-\omega_{r})L_{m}^{2}\big{]}\), and \(\widetilde{b_{0}}=(\omega_{e}-\omega_{r})^{2}\omega_{e}L_{r}\)\((L_{m}^{2}-L_{r}L_{s})-r_{r}^{2}L_{s}\omega_{e}\) represent system-specific constants and intermediate variables derived from the stator and rotor parameters. Under steady-state conditions, the following relation holds: \(\omega_{rf}\rightarrow\omega_{e}\Rightarrow S\omega_{e}\rightarrow(\omega_{e }-\omega_{r})\). Furthermore, based on the definition of \(\rho\) and its geometric interpretation illustrated in Figure 7, the following expression can be derived [14]:  
\[0=r_{r}\dot{r}^{rf}_{qr}+\Big{(}\omega_{rf}-\omega_{r}\Big{)}\lambda^{rf}_{ dr}, \tag{10}\]6  
Similarly, applying the final value theorem to (20) yields the following expression:  
\(\lim\limits_{t\rightarrow\infty}\rho=\lim\limits_{t\rightarrow\infty}^{ \frac{p_{rf}^{2}}{\rho}(t)}=\frac{\lim\limits_{t\rightarrow\infty}^{\frac{p_ {rf}^{2}}{\rho}(t)}}{\lim\limits_{t\rightarrow\infty}^{\frac{p_{rf}^{2}}{ \rho}(t)}}=\frac{\lim\limits_{t\rightarrow\infty}^{\frac{p_{rf}^{2}}{\rho}(s) }}{\lim\limits_{s\rightarrow\infty}^{\frac{p_{rf}^{2}}{\rho}(s)}}=\lim \limits_{s\to 0}\frac{i_{gs}^{p}(s)}{However, due to the fact that the domain of variable \(\tan^{-1}\) is defined over \(-\frac{\pi}{2}\sim\frac{\pi}{2}\) with non-differentiable endpoints, the above computational process poses challenges for numerical implementation. This discontinuity hinders the accurate calculation of intermediate variables \(\theta_{is}^{s}\), \(\omega_{e}^{s}\), and \(\gamma(t)\) through direct analytical methods. To address this issue, the \(\theta_{is}^{s}\) and \(\omega_{e}^{s}\) computation blocks in Figure 8 are replaced by a phase-locked loop (PLL) module, as shown in \[0=r_{r}\dot{r}^{rf}_{qr}+\Big{(}\omega_{rf}-\omega_{r}\Big{)}\lambda^{rf}_{ dr}, \tag{10}\]7 This modification eliminates the dependence on the function involving \(\tan^{-1}\), thereby enabling a more robust and continuous estimation process. \[0=r_{r}\dot{r}^{rf}_{qr}+\Big{(}\omega_{rf}-\omega_{r}\Big{)}\lambda^{rf}_{ dr}, \tag{10}\]8 System structure of the proposed method in Stage I. The outputs of the \(\omega_{\epsilon}^{s}\) and \(\rho\) estimators are subsequently fed into (21) and (16) to compute \(\tau_{\gamma}y=\tan[\gamma(t)]\). This result is then input into the computational structure shown in \[0=r_{r}\dot{r}^{rf}_{qr}+\Big{(}\omega_{rf}-\omega_{r}\Big{)}\lambda^{rf}_{ dr}, \tag{10}\]9 The calculation framework in Figure 10 is based on the expression given in  
\[ \begin{split}& A\sin\left(\theta_{is}^{s}-\gamma\right)=i_{qs}^{s} \cos\gamma-i_{ds}^{s}\sin\gamma=A\sin\left(\theta_{rf}^{s}\right)\\ & A\cos\left(\theta_{is}^{s}-\gamma\right)=i_{ds}^{s}\cos\gamma+i_ {qs}^{s}\sin\gamma=A\cos\left(\theta_{rf}^{s}\right)\end{split}  \tag{24}\]  
which involves variables \(i_{qs}^{s}=A\sin\left(\theta_{is}^{s}\right)\) and \(i_{ds}^{s}=A\cos\left(\theta_{is}^{s}\right)\). Although (24) reveals information related to the rotor flux angle, it still depends on the estimation of \(\hat{\theta}_{rf}^{-s*}=\tan^{-1}\left[\frac{\sin\left(\theta_{is}^{s}-\gamma \right)}{\cos\left(\theta_{is}^{s}-\gamma\right)}\right]\). This necessitates the use of phase-locked loop (PLL) techniques once again.