## 3 Design Process of the PACFTB Controller
 \tag{26}\]  
where \(f_{1}\left(X\right)=\frac{K_{T}}{M}i_{qs}+\Phi v-i_{qs}\), and \(X=\left[v,\ i_{qs}\right]^{T}\) is defined the input of the FLS. According to the Lemma 1, we have the following FLS:  
\[f_{1}\left(X\right)=W_{1}^{T}B_{1}\left(X\right)+\xi_{1}\left(X\right) \tag{27}\]  
where \(\xi_{1}\left(X\right)\) denotes the approximation error and satisfies the condition \(\left\|\xi_{1}\left(X\right)\right\|\leq\xi_{1}\). Referring to (32) and [PERSON]'s inequality, the following inequality can be obtained:  
\[ \begin{split}\tilde{\epsilon}_{1}f_{1}\left(X\right)& =\tilde{\epsilon}_{1}W_{1}^{T}B_{1}\left(X\right)+\tilde{\epsilon}_{1}\xi_{1 }\left(X\right)\\ &\leq\tilde{\epsilon}_{1}W_{1}^{T}B_{1}\left(X\right)+\frac{1}{2} \tilde{\epsilon}_{1}^{2}+\frac{1}{2}\xi_{1}^{2}\end{split}  \tag{28}\]Then, the \(\dot{V}_{1}\) has the following unequal relation:  
\[\dot{V}_{1}\leq-k_{1}\bar{e}_{1}^{2}+\frac{1}{2}\bar{e}_{1}^{2}+\bar{e}_{1}\left( e_{2}-\bar{v}^{d}+\bar{i}_{qs}^{d}+W_{1}^{T}B_{1}(X)+F+k_{1}e_{1}+\frac{1}{2} \bar{e}_{1}\right) \tag{29}\]  
Therefore, the virtual controller \(\dot{i}_{qs}^{d}\) can be chosen as:  
\[\dot{i}_{qs}^{d}=\bar{v}^{d}-\hat{W_{1}}^{T}B_{1}\left(X\right)-k_{1}e_{1}- \frac{1}{2}\bar{e}_{1}-\hat{F}-e_{2} \tag{30}\]  
where \(\hat{F}\) means the estimation value of \(F\) that is used to deal with the problem caused by load disturbance, \(\hat{W}_{1}\) the estimation value of weight vector \(W_{1}\), then the following inequations can be obtained as:  
\[\dot{V}_{1}\leq-k_{1}\bar{e}_{1}^{2}+\frac{1}{2}\bar{e}_{1}^{2}+\bar{e}_{1} \big{(}W_{1}-\hat{W_{1}}\big{)}^{T}B_{1}\left(X\right)-\bar{e}_{1}\hat{F} \tag{31}\]  
where \(\hat{F}=\hat{F}-F\) denotes the adaptive estimation error. **Step 2:** In order to obtain the control law in \(q-axis\), the Lyapunov function \(V_{2}\) is selected as:  
\[V_{2}=V_{1}+\frac{1}{2}\bar{S}_{q}^{2} \tag{32}\]  
Then the derivative of \(V_{2}\) can be figured by:  
\[\dot{V}_{2}=\dot{V}_{1}+S_{q}\dot{S}_{q} \tag{33}\]  
From (6) and (23), \(S_{q}\dot{S}_{q}\) is calculated as:  
\[ \begin{split} S_{q}\dot{S}_{q}&=S_{q}\left(-\omega _{e}\left[i_{ds}+\frac{L_{m}\left(1-f(Q)\right)}{L(Q)\left(L_{r}-L_{m}f(Q) \right)}\psi_{dr}\right]-\frac{R_{s}}{L(Q)}i_{qs}+\frac{U_{qs}}{L(Q)}-\dot{i}_ {qs}^{c}+k_{q}{}^{p_{2}/q_{2}}e_{2}I_{q}^{p_{2}/q_{2}-1}\right)\\ &=S_{q}\left(f_{2}\left(X\right)+U_{qs}-\dot{i}_{qs}^{c}+k_{q}{} ^{p_{2}/q_{2}}e_{2}I_{q}^{p_{2}/q_{2}-1}\right)\end{split}  \tag{34}\]  
where  
\[ \begin{split} f_{2}\left(X\right)&=-\omega_{e} \left[i_{ds}+\frac{L_{m}\left(1-f(Q)\right)}{L(Q)\left(L_{r}-L_{m}f(Q)\right)} \psi_{dr}\right]-\frac{R_{s}}{L(Q)}i_{qs}+\frac{U_{qs}}{L(Q)}-U_{qs}\\ &=W_{2}^{T}B_{2}\left(X\right)+\xi_{2}\left(X\right)\end{split}  \tag{35}\]  
with \(\left\|\xi_{2}\left(X\right)\right\|\leq\xi_{2}\). \(X=\left[i_{ds},\ i_{qs}\right]^{T}\) is defined the input of the FLS. Similarly, for given \(\xi_{2}>0\), the following inequality can be confirmed:  
\[S_{q}f_{2}\left(X\right)\leq S_{q}W_{2}^{T}B_{2}\left(X\right)+\frac{1}{2}S_{ q}^{2}+\frac{1}{2}\bar{c}_{2}^{2} \tag{36}\]  
Substituting (34) and (36) into (33), then gets  
\[ \begin{split}\dot{V}_{2}\leq&-k_{1}\bar{e}_{1}^{2}+ \frac{1}{2}\bar{c}_{1}^{2}+\frac{1}{2}\bar{c}_{2}^{2}+\bar{e}_{1}\big{(}W_{1}- \hat{W_{1}}\big{)}^{T}B_{1}\left(X\right)-\bar{e}_{1}\hat{F}\\ &+S_{q}\left(W_{2}^{T}B_{2}\left(X\right)+U_{qs}-\dot{i}_{qs}^{c} +\frac{1}{2}S_{q}+k_{q}{}^{p_{2}/q_{2}}e_{2}I_{q}^{p_{2}/q_{2}-1}\right)\end{split}  \tag{37}\]  
To meet the conditions \(S_{q}\dot{S}_{q}<0\), the exponential convergence law is adopted as the following:  
\[\dot{S}_{q}=-k_{2}S_{q}-k_{q}\text{sign}\left(S_{q}\right) \tag{38}\]where \(k_{2}>0\) and \(k_{q}>0\) are the designed positive constant, function sign \(\left(\cdot\right)\) represents the symbolic function which defined as:  
\[\text{sign}\left(\cdot\right)=\left\{ \begin{array}{ll}\tilde{x}/\left|\tilde{x }\right|,&\text{if }\tilde{x}\
eq 0\\ 0,&\text{if }\tilde{x}=0\end{array} \right.