## 3 The Design of the Adaptive Fuzzy Sliding-Mode Controller
The fuzzy switching functions, which are also the switching input control laws, are  
\[{}^{C}D_{t}^{a}s_{i}=u_{isw}=\hat{u}_{isw}(s_{i}|\theta_{iu}) \tag{12}\]  
As the sliding-mode surface and its differential are approaching zero, a global sliding-mode surface function can be obtained:  
\[{}^{C}D_{t}^{a}s_{i}={}^{C}D_{t}^{a}e_{i}+m_{i}e_{i}-{}^{C}D_{t}^{a}h_{i}(t)=0 \tag{13}\]  
Let \({}^{C}D_{t}^{a}s_{i}=0\), then one can obtain the following functions:  
\[{}^{C}D_{t}^{a}e_{i}=-m_{i}e_{i}+{}^{C}D_{t}^{a}h_{i}(t) \tag{14}\]  
Firstly, the system designed must be stable and satisfy the definition of global stability. Let the first Lyapunov function be  
\[V_{1}(t)=\sum_{i=1}^{n}\frac{1}{2}{e_{i}}^{2} \tag{15}\]  
From (14), the \(\alpha\)-th order derivative of \(V_{1}(t)\) is expressed as  
\[{}^{C}D_{t}^{a}V_{1}(t) =\sum_{i=1}^{n}e_{i}D_{t}^{a}e_{i} \tag{16}\] \[=\sum_{i=1}^{n}e_{i}(-m_{i}e_{i}+{}^{C}D_{t}^{a}h_{i}(t))\] \[=-\sum_{i=1}^{n}m_{i}e_{i}{}^{2}+\sum_{i=1}^{n}e_{i}h_{i}(0)^{a} D_{t}^{C}e^{-pt}\]It is obvious that if \(t\rightarrow\infty\), \(e^{-pt}=0\) and \({}^{\alpha}D_{t}^{C}e^{-pt}=0\); thus, the final formulation can be obtained  
\[e_{i}=x_{i}-x_{id} \tag{10}\]0  
where \(m=\min\{m_{1},m_{2},\cdots,m_{n}\}\). Based on the fractional Lyapunov stability theorem, the tracking errors of the dynamical system (9) are driven to the origin asymptotically. Based on APSMC in this paper, the designed input control laws can be obtained by the following:  
\[ \begin{array}{l}u_{i}=u_{leq}-u_{isw};\\ u_{leq}=-f_{i}(x)+{}^{C}D_{t}^{\alpha}x_{id}-m_{i}e_{i}-\hat{g}_{i}(x|\theta_{ ig})+{}^{C}D_{t}^{\alpha}h_{i}(t)-\hat{\delta}_{i};\\ u_{isw}=\hat{u}_{isw}(s_{i}|\theta_{iu}).\end{array}  \tag{18}\]  
where \(\hat{\delta}_{i}\) is the excepted value of \(\delta_{i}\), which will be mentioned in the following analysis; \(\hat{g}_{i}(x|\theta_{ig})=\theta_{ig}^{T}\hat{g}(x)\) are the fuzzy of unknown terms \(g_{i}(x,t)\); and \(\hat{u}_{isw}(s_{i}|\theta_{iu})=\theta_{iu}^{T}\phi(s_{i})\) are the fuzzy of switching functions. Adaptive control laws are  
\[{}^{C}D_{t}^{\alpha}\theta_{ig}=r_{i}s_{i}\hat{g}(x) \tag{19}\]  
\[{}^{C}D_{t}^{\alpha}\theta_{iu}=l_{i}s_{i}\phi(s_{i}) \tag{20}\]  
\[{}^{C}D_{t}^{\alpha}\hat{\delta}_{i}=\gamma_{i}s_{i} \tag{21}\]  
According to Definition 4, suppose that the optimal parameters of fuzzy approximation are as follows:  
\[\theta_{ig}^{*}=\arg\min_{\theta_{ig}\in\Omega_{ig}}[\sup|\hat{g}_{i}(x|\theta _{ig})-g_{i}(x)|] \tag{22}\]  
\[\theta_{iu}^{*}=\arg\min_{\theta_{iu}\in\Omega_{isw}}[\sup|\hat{u}_{isw}(S_{i} |\theta_{iu})-u_{isw}|] \tag{23}\]  
The expected switching functions are sign functions:  
\[\hat{u}_{isw}(s_{i}|\theta_{iu}^{*})=\eta_{i}\text{sgn}(s_{i}) \tag{24}\]  
To improve the control performance of FLSs, the minimum approximation error is as follows:  
\[w_{i}=g_{i}(x)-\hat{g}_{i}(x|\theta_{ig}^{*}) \tag{25}\]  
Designing that \(\tilde{\theta}_{ig}=\theta_{ig}-\theta_{ig}^{*}\tilde{\theta}_{iu}=\theta_{iu }-\theta_{iu}^{*}\) and \(\hat{\delta}_{i}=w_{i}+d_{i}\), one can obtain the differential forms as  
\[ \begin{array}{l}{}^{C}D_{t}^{\alpha}\tilde{\theta}_{ig}={}^{C}D_{t}^{\alpha }\theta_{ig}\\ {}^{C}D_{t}^{\alpha}\tilde{\theta}_{iu}={}^{C}D_{t}^{\alpha}\theta_{iu}\\ {}^{C}D_{t}^{\alpha}\tilde{\delta}_{i}={}^{C}D_{t}^{\alpha}\hat{\delta}_{i} \end{array}  \tag{26}\]Choose the second Lyapunov function:  
\[V_{2}(t)=\frac{1}{2}\sum_{i=1}^{n}\left(s_{i}^{2}(t)+\frac{1}{r_{i}}\bar{\theta}_ {ig}^{2}+\frac{1}{l_{i}}\bar{\theta}_{iu}^{2}+\frac{1}{\gamma_{i}}\bar{\delta}_ {i}^{2}\right) \tag{27}\]  
where \(\bar{\delta}_{i}=\delta_{i}-\hat{\delta}_{i}\).