## 4 Design of Fractional Backstepping Sliding Mode Adaptive Fuzzy Controller  
By designing the fractional backstepping sliding mode controller, the system is proved to be stable based on Lyapunov theory. However, considering the unknown function \(f(x_{1})\), the control law (26) cannot be implemented directly. Due to the fuzzy system's advantage of approximating an arbitrary nonlinear function to accuracy, an adaptive fuzzy system \(\hat{f}(x_{1})\) is constructed to estimate the unknown function \(f(x_{1})\). In the following, the comprehensive analysis of designing a fractional backstepping sliding mode adaptive fuzzy controller will be given, as shown in Figure 2.  
Applying the strategy of singleton fuzzification, product inference and center average defuzzification, the output of adaptive fuzzy system can be expressed as  
\[\hat{f}(x_{k}|\theta_{f})=\theta_{fk}^{T}\xi(x_{k})^{T} \tag{28}\]  
Figure 2: Block diagram of fractional-order backstepping sliding adaptive fuzzy controller. where \(\xi_{k}(x)=\frac{\prod\limits_{i=1}^{n}\mu_{k}(x_{i})}{\sum\limits_{i=1}^{M} \left(\prod\limits_{i=1}^{n}\mu_{k}(x_{i})\right)}\) is fuzzy basis function and \(\theta_{fk}{}^{T}\) is an adjustable parameter which can be updated by the adaptive law  
\[\dot{\theta}_{fk}=r_{k}\xi(x_{k})^{T} \tag{29}\]  
where \(r\) is a positive constant. Then we can get the improved control law by replacing \(f_{k}\) with \(\hat{f_{k}}\)  
\[u_{k}=\frac{1}{b}(-e_{1k}+\frac{1}{s_{k}}\lambda_{2}D^{\alpha-1}e_{1k}-\lambda _{1}\dot{e}_{1k}-\hat{f_{k}}+\dot{\alpha}_{1k}-\lambda_{2}D^{\alpha}e_{1k}) \tag{30}\]  
**Proof. ** Define the optimal parameter as  
\[\theta_{f}^{*}=\arg\min_{\theta_{f}\in\Omega_{f}}[\sup\lvert\hat{f}\underbrace {x\rvert\theta_{f}}_{x\in R^{n}}-f(x)\rvert] \tag{31}\]  
where \(\Omega_{f}\) is the aggregation of \(\theta_{f}\). The minimum approximation error is defined as  
\[\omega=f(x)-\hat{f}(x\rvert\theta_{f}^{*}) \tag{32}\]  
where \(\omega\) is bounded by a positive constant \(\omega_{\max}\) as  
\[\lvert\omega\rvert\leq\omega_{\max} \tag{33}\]  
A Lyapunov function is defined as  
\[V_{3k}=V_{2k}+\frac{1}{2r}\varphi_{fk}^{T}\varphi_{fk} \tag{34}\]  
where \(\varphi_{fk}=\theta_{fk}^{*}-\theta_{fk}\). \(\Box\)  
Then the derivative of \(V_{3k}\) becomes  
\[ \begin{array}{l}\dot{V}_{3k}=\dot{V}_{2k}+\frac{1}{r}\varphi_{fk}^{T}\dot{ \varphi}_{fk}\\ =-c_{1}e_{1k}^{2}-\frac{\lambda_{1}}{\lambda_{3}}e_{1k}^{2}+s_{k}(f_{k}-f_{k}) +\frac{1}{r}\varphi_{fk}^{T}\dot{\varphi}_{f}\\ =-c_{1}e_{1k}^{2}-\frac{\lambda_{1}}{\lambda_{3}}e_{1k}^{2}+s_{k}[\dot{f}(x_{ k}\rvert\theta_{fk}^{*})-\dot{f}(x_{k})+\omega_{k}]+\frac{1}{r}\varphi_{fk}^{T} \dot{\varphi}_{fk}\\ =-c_{1}e_{1k}^{2}-\frac{\lambda_{1}}{\lambda_{3}}e_{1k}^{2}+s_{k}[\varphi_{fk }^{T}\dot{\varepsilon}(x_{k})^{T}+\omega_{k}]+\frac{1}{r}\varphi_{fk}^{T} \dot{\varphi}_{fk}\\ =-c_{1}e_{1k}^{2}-\frac{\lambda_{1}}{\lambda_{3}}e_{1k}^{2}+\frac{1}{r} \varphi_{fk}^{T}[r_{k}\dot{\varepsilon}(x_{k})^{T}+\dot{\varphi}_{fk}]+s_{k} \omega_{k}\end{array}  \tag{35}\]  
where \(\dot{\varphi}_{fk}=-\dot{\theta}_{fk}\). Applying the adaptive law (Equation (29)) into Equation (35) yields  
\[\dot{V}_{3k}=-c_{1}e_{1k}^{2}-\frac{\lambda_{1}}{\lambda_{3}}e_{1k}^{2}+s_{k} \omega_{k} \tag{36}\]  
According to the fuzzy approximation theory, adaptive fuzzy systems can achieve the results that the approximation error \(\omega_{k}\) would be a tiny number which can be ignored, then \(\dot{V}_{3k}\leq 0\) is semi-negative definite. From Barbalart lemma, we can prove that \(e_{1k},e_{2k}\), and \(s_{k}\) will converge to zero. As a result, the proposed controller can ensure that the system is stable. Furthermore, the controlled system will exhibit a certain degree of robustness to external disturbances. **Remark 2**. : _The sliding surface designed in this paper is an ordinary sliding surface with an integrated fractional term which is not a terminal sliding surface. The reaching time obtained in this sliding mode control is same as the that of an ordinary sliding surface.