## 3 Design Process of the PACFTB Controller
\tag{39}\]  
Then, the controller \(U_{qs}\) can be chosen as:  
\[U_{qs}=-\hat{W_{2}}^{T}B_{2}\left(X\right)+i_{qs}^{c}-\frac{1}{2}S_{q}-k_{2}S_ {q}-k_{q}\text{sign}\left(S_{q}\right)-k_{q}{}^{p_{2}/q_{2}}e_{2}l_{q}^{p_{2} /q_{2}-1} \tag{40}\]  
Afterwards, the derivative of \(V_{2}\) can be calculated as:  
\[ \begin{split}\dot{V}_{2}&\leq-k_{1}\varepsilon_{1}^{2 }-k_{2}S_{q}^{2}-k_{q}\left|S_{q}\right|+\frac{1}{2}\varepsilon_{1}^{2}+\frac {1}{2}\varepsilon_{2}^{2}-\varepsilon_{1}\tilde{F}+\varepsilon_{1}\left(W_{ 1}-\hat{W_{1}}\right)^{T}B_{1}\left(X\right)\\ &\quad+S_{q}\left(W_{2}-\hat{W_{2}}\right)^{T}B_{2}\left(X\right) \end{split}  \tag{41}\]  
**Step 3:** In order to design the control law of d-axis, the Lyapunov function \(V_{3}\) is chosen as:  
\[V_{3}=V_{2}+\frac{1}{2}S_{d}^{2} \tag{42}\]  
Then we can get the derivative of \(V_{3}\) as:  
\[ \begin{split}\dot{V}_{3}&=\dot{V}_{2}+S_{d}\left(- \frac{R_{s}}{L(Q)}i_{ds}+\frac{U_{ds}}{L(Q)}+\omega_{c}i_{qs}-i_{ds}^{d}+k_{d} {}^{p_{1}/q_{1}}e_{3}l_{d}^{p_{1}/q_{1}-1}\right)\\ &=\dot{V}_{2}+S_{d}\left(U_{ds}+f_{3}\left(X\right)-i_{ds}^{d}+k_ {d}{}^{p_{1}/q_{1}}e_{3}l_{d}^{p_{1}/q_{1}-1}\right)\end{split}  \tag{43}\]  
where  
\[ \begin{split} f_{3}\left(X\right)&=-\frac{R_{s}}{L (Q)}i_{ds}+\frac{U_{ds}}{L(Q)}+\omega_{c}i_{qs}-U_{ds}\\ &=W_{3}^{T}B_{3}\left(X\right)+\xi_{3}\left(X\right)\end{split}  \tag{44}\]  
with \(\left\|\xi_{3}\left(X\right)\right\|\leq\xi_{3}\). \(X=\left[i_{ds},\,i_{qs}\right]^{T}\) is defined as the input of the FLS. Similarly to the front, through Lemma 1, (46) and Young's inequality, the following inequality can be obtained:  
\[S_{d}f_{3}\left(X\right)\leq S_{d}W_{3}^{T}B_{3}\left(X\right)+\frac{1}{2}S_{ d}^{2}+\frac{1}{2}\varepsilon_{3}^{2} \tag{45}\]  
where \(\xi_{3}>0\) is any given constant. Substituting (45) into (43), one obtain:  
\[ \begin{split}\dot{V}_{3}&\leq\dot{V}_{2}+\frac{1}{2 }\varepsilon_{3}^{2}+S_{d}\bigg{(}U_{ds}+W_{3}^{T}B_{3}\left(X\right)\\ &\quad+\frac{1}{2}S_{d}-i_{ds}^{d}+k_{d}{}^{p_{1}/q_{1}}e_{3}l_{ d}^{p_{1}/q_{1}-1}\bigg{)}\end{split}  \tag{46}\]  
Similarly, the exponential convergence law is adopted as follows:  
\[\dot{S}_{d}=-k_{3}S_{d}-k_{d}\text{sign}\left(S_{d}\right) \tag{47}\]  
where \(k_{3}>0\) and \(k_{d}>0\) are the designed positive constant. Then, the real controller for \(U_{ds}\) can be chosen as:  
\[U_{ds}=-\hat{W_{3}}^{T}B_{3}\left(X\right)-\frac{1}{2}S_{d}+i_{ds}^{d}-k_{3}S_ {d}-k_{d}\text{sign}\left(S_{d}\right)-k_{d}{}^{p_{1}/q_{1}}e_{3}l_{d}^{p_{1} /q_{1}-1} \tag{48}\]Thus, we can reach \(\dot{V}_{3}\) as  
\[\dot{V}_{3} \leq-k_{1}\tilde{e}_{1}^{2}-k_{2}S_{q}^{2}-k_{3}S_{d}^{2}-\tilde{e}_ {1}\hat{F}-k_{d}\left|S_{d}\right|-k_{q}\left|S_{q}\right|+\frac{1}{2}\sum_{i=1}^ {3}\varsigma_{i}^{2}-\tilde{e}_{1}\tilde{W_{1}}^{T}B_{1}\left(X\right) \tag{49}\] \[S_{q}\hat{W_{2}}^{T}B_{2}\left(X\right)-S_{d}\hat{W_{3}}^{T}B_{3} \left(X\right)\]  
where \(\tilde{W}_{i}=\hat{W}_{i}-W_{i}\), \((i=1,2,3)\) represent the adaptive estimation errors. **Step 4:** This step is used to get the precise adaptive law for FLS, then select the following Lyapunov function \(V\) as:  
\[V=V_{3}+\sum_{i=1}^{3}\frac{\tilde{W}_{i}^{T}\tilde{W}_{i}}{2\gamma_{i}}+\frac {\tilde{F}^{2}}{2\gamma_{4}} \tag{50}\]  
where \(\gamma_{1},\ \gamma_{2},\ \gamma_{3},\ \gamma_{4}>0\) are self-defined gain constant. The derivative of \(V\) can be easily deduced by (51) which is shown as follows.