## 2 Design of a New Adaptive Fuzzy Controller
### Robustness of Input Controls
It is remarked that coefficients of the conventional sliding surface are positive to guarantee the stability of control system. Then, the derivative of Equation (5) is obtained as follows. \[\dot{\sigma}_{s\
abla}=\ddot{\varphi}+c_{s\
abla}\dot{\varphi}=M_{1\
abla}+M_ {2\
abla}+M_{3\
abla}\big{(}f_{\
abla}(x_{\
abla})+g_{\
abla}(x_{\
abla})u_{ \
abla}(t)+d_{\
abla}(t)-\tilde{x}_{d\
abla}\big{)}+c_{s\
abla}\dot{\varphi} \tag{6}\]  
In the above, \(\dot{\varphi}=\frac{1}{2}\bigg{[}\frac{\dot{\lambda}+\dot{\varpi}(t)}{\lambda+ \varpi(t)}-\frac{\dot{\lambda}-\dot{\varpi}(t)}{\lambda-\varpi(t)}\bigg{]}\), \(\ddot{\varphi}=M_{1\
abla}+M_{2\
abla}+M_{3\
abla}\tilde{e}_{\
abla}(t)\), \(M_{1\
abla}=\frac{\ddot{\lambda}(\lambda+e_{\
abla}(t))-\left(\dot{\lambda}+ \dot{e}_{\
abla}(t)\right)^{2}}{2(\lambda+e_{\
abla}(t))^{2}}\), \(M_{2\
abla}=-\frac{\ddot{\lambda}(\lambda-e_{\
abla}(t))-\left(\dot{\lambda}- \dot{e}_{\
abla}(t)\right)^{2}}{2(\lambda-e_{\
abla}(t))^{2}}\), \(M_{3\
abla}=\left(\frac{\lambda+e_{\
abla}(t)}{2(\lambda+e_{\
abla}(t))^{2}}+ \frac{\lambda-e_{\
abla}(t)}{2(\lambda-e_{\
abla}(t))^{2}}\right)\)  
The lumped uncertainty of system is then defined by:  
\[w_{\
abla}=M_{3\
abla}\widetilde{\gamma}_{f\
abla}\xi_{f\
abla}+M_{3\
abla} \widetilde{\gamma}_{g\
abla}\xi_{g\
abla}u_{\
abla}+M_{3\
abla}d_{\
abla}(t) \tag{7}\]  
where, \(\gamma_{f\
abla}=f_{\
abla}(x_{\
abla})-f_{\
abla}^{*}(x_{\
abla}),\gamma_{g \
abla}=g_{\
abla}(x_{\
abla})-g_{\
abla}^{*}(x_{\
abla})\); \(\xi_{f\
abla},\xi_{g\
abla}\) are two consequent membership values of functions \(f_{\
abla}(x_{\
abla}),g_{\
abla}(x_{\
abla})\), respectively. Using Equations (6) and (7), the time derivative of Equation (6) is rewritten as follows:  
\[\dot{\sigma}_{s\
abla}=M_{1\
abla}+M_{2\
abla}+M_{3\
abla}f_{\
abla}^{*}(x_{ \
abla})+M_{3\
abla}g_{\
abla}^{*}(x_{\
abla})u_{\
abla}(t)-M_{3\
abla}\tilde {x}_{d\
abla}+c_{s\
abla}\dot{\varphi}+w_{\
abla} \tag{8}\]  
The relationship between Equation (8) and IT2 FNN model is expressed as follows:  
\[\dot{\sigma}_{s\
abla}=M_{1\
abla}+M_{2\
abla}+M_{3\
abla}\theta_{f\
abla}^{* }\xi_{f\
abla}+M_{3\
abla}\theta_{g\
abla}^{*}\xi_{g\
abla}u_{\
abla}-M_{3 \
abla}\tilde{x}_{d\
abla}+c_{s\
abla}\dot{\varphi}+w_{\
abla} \tag{9}\]  
where,  
\[\theta_{f\
abla}^{*}=arg\ min_{\theta_{g\
abla}\in\Delta\theta_{f\
abla}} \Big{[}sup_{x_{\
abla}\in\Delta x_{\
abla}}\big{|}f_{\
abla}(x_{\
abla})-f_{ \
abla}^{*}(x_{\
abla})\big{|}\Big{]},\]  
\[ \begin{array}{l}\theta_{g\
abla}^{*}=arg\ min_{\theta_{g\
abla}\in\Delta \theta_{g\
abla}}\Big{[}sup_{x_{\
abla}\in\Delta x_{\
abla}}\big{|}g_{\
abla}( x_{\
abla})-g_{\
abla}^{*}(x_{\
abla})\big{|}\Big{]},\ \Delta\theta_{f\
abla}=\Big{\{}\theta_{f\
abla}\in R^{n},\|\theta_{f\
abla}\| \leq\Theta_{f\
abla}\Big{\}},\\ \Delta\theta_{g\
abla}=\big{\{}\theta_{g\
abla}\in R^{n},\|\theta_{g\
abla}\| \leq\Theta_{g\
abla}\big{\}},\ \Delta x_{\
abla}=\{x_{\
abla}\in R^{n},\|x_{\
abla}\|\leq\Theta_{x\
abla}\}. \end{array} \]  
In the above definitions, \(\mathbf{\Theta}_{f\
abla}\), \(\mathbf{\Theta}_{g\
abla}\), \(\mathbf{\Theta}_{x\
abla}\) are constant boundaries. Now, an equivalent controller is determined from Equation (9) based on the assumption \(\dot{\sigma}_{s\
abla}\approx 0\):  
\[u_{1}=\frac{1}{M_{3\
abla}\dot{\theta}_{g\
abla}\xi_{g\
abla}}\Big{(}-M_{1 \
abla}-M_{2\
abla}-M_{3\
abla}\dot{\theta}_{f\
abla}\tilde{\xi}_{f\
abla}+M_{ 3\
abla}\tilde{x}_{d\
abla}-c_{s}\
abla\dot{\varphi}\Big{)} \tag{10}\]  
The equivalent control term \(u_{1}\) cannot stabilize the system because it cannot compensate the error caused from the fuzzy approximation. In fact, the input control \(u_{1}\) can be considered as the internal energy of the system. In the physical model, related to the potential energy, the derivative (9) must less than or equal zero. When this situation is obtained, the system will be stable. Thus, it can be assumed that the derivative \(\dot{\sigma}_{s\
abla}\) is approximated to be zero to find the original control energy (10).