## 3 Design of Fractional Backstepping Sliding Mode Controller
### Fractional Backstepping Sliding Mode Controller  
In this section, a fractional backstepping sliding mode controller is proposed for APF in two steps. Firstly, a virtual control function is designed based on a Lyapunov function \(V_{1}\). Secondly, a real controller is proposed. In the following, we will give the two design steps of the procedure. Step 1: Assume the reference trajectory is \(x_{d}\), and \(x_{d}\) has continuous second order derivatives. The position's tracking error can be defined as  
\[e_{1}=x_{1}-x_{d} \tag{15}\]  
Then  
\[\dot{e}_{1}=\dot{x}_{1}-\dot{x}_{d}=x_{2}-\dot{x}_{d} \tag{16}\]The virtual control is designed as  
\[\alpha_{1}=-c_{1}e_{1}+\dot{x}_{d} \tag{17}\]  
where \(c_{1}\) is a positive constant. The error is defined as  
\[e_{2}=x_{2}-\alpha_{1} \tag{18}\]  
The first Lyapunov function is chosen as  
\[V_{1}=\frac{1}{2}e_{1}^{T}e_{1} \tag{19}\]  
Then one can obtain  
\[ \begin{array}{l}\dot{V}_{1}=e_{1}^{T}e_{1}=e_{1}^{T}(x_{2}-\dot{x}_{d})\\ =e_{1}^{T}(e_{2}+\alpha_{1}-\dot{x}_{d})\\ =e_{1}^{T}(e_{2}-c_{1}e_{1}+\dot{x}_{d}-\dot{x}_{d})\\ =-c_{1}e_{1}^{T}e_{1}+e_{1}^{T}e_{2}\end{array}  \tag{20}\]  
If \(e_{2}=0\), then \(\dot{V}_{1}=-c_{1}e_{1}^{T}e_{1}\leq 0\). Step 2: The derivative of Equation (18) is  
\[ \begin{array}{l}\dot{e}_{2}=\dot{x}_{2}-\dot{\alpha}_{1}\\ =f(x_{1})+bu-\dot{\alpha}_{1}\end{array}  \tag{21}\]  
The sliding surface is defined as  
\[s=\lambda_{1}e_{1}+\lambda_{2}D^{\alpha-1}e_{1}+\lambda_{3}e_{2} \tag{22}\]  
where \(\lambda_{1},\lambda_{2},\lambda_{3}\) are positive constants and \(\alpha-1\) is the fractional order in the fractional derivate operation, \(s=\left[ \begin{array}{cc}s_{1}&s_{2}&s_{3}\end{array} \right]^{T}\). The second Lyapunov function \(V_{2}=\left[ \begin{array}{cc}V_{21}&V_{22}&V_{23}\end{array} \right]^{T}\) is defined as  
\[V_{2k}=V_{1k}+\frac{1}{2}s_{k}^{2} \tag{23}\]  
**Remark 1**. : \(V_{2}=\left[ \begin{array}{cc}V_{21}&V_{22}&V_{23}\end{array} \right]^{T}\)_is a vector representing three scalar values \(V_{21}\), \(V_{22}\), \(V_{23}\) such as \(V_{21}=V_{11}+\frac{1}{2}s_{1}^{2}\), \(V_{22}=V_{12}+\frac{1}{2}s_{2}^{2}\), \(V_{23}=V_{13}+\frac{1}{2}s_{3}^{2}\). For the convenience of annotation, we use vector \(V_{2}=\left[ \begin{array}{cc}V_{21}&V_{22}&V_{23}\end{array} \right]^{T}\) to represent three Lyapunov functions that are scalar values._  
Differentiating Equation (23) with respect to time yields  
\[ \begin{array}{rcl}\dot{V}_{2k}&=&\dot{V}_{1k}+s_{k}\dot{s}_{k}\\ &=&-c_{1}e_{1k}^{2}+e_{1k}e_{2k}+s_{k}(\lambda_{1}\dot{e}_{1k}+\lambda_{2}D^{ \alpha}e_{1k}+\lambda_{3}\dot{e}_{2k})\end{array}  \tag{24}\]  
where \(k=1,2,3\), \(e_{2k}=\frac{s_{k}-\lambda_{1}e_{1k}-\lambda_{2}D^{\alpha-1}e_{1k}}{\lambda_{3}}\). Then, \(\dot{V}_{2k}\) can be further proposed as  
\[ \begin{array}{l}\dot{V}_{2k}=c_{1}e_{1k}^{2}+e_{1k}e_{2k}+s_{k}(\lambda_{1}\dot {e}_{1k}+\lambda_{2}D^{a}e_{1k}+\lambda_{3}\dot{e}_{2k})\\ =-c_{1}e_{1k}^{2}+\frac{c_{1k}}{\lambda_{3}}(s_{k}-\lambda_{1}e_{1k}-\lambda_{2 }D^{a-1}e_{1k})+s_{k}[\lambda_{1}\dot{e}_{1k}+\lambda_{2}D^{a}e_{1k}+\lambda_{ 3}(f(x_{1k})+bu_{k}-\dot{a}_{1k})]\\ =-c_{1}e_{1k}^{2}-\frac{\lambda_{1}}{\lambda_{2}}e_{1k}^{2}+\frac{s_{k}e_{1k}}{ \lambda_{3}}-\frac{3e_{2}e_{1k}D^{a-1}e_{1k}}{\lambda_{3}}+s_{k}[\lambda_{1} \dot{e}_{1k}+\lambda_{2}D^{a}e_{1k}+\lambda_{3}(f(x_{1k})+bu_{k}-\dot{a}_{1k})] \\ =-c_{1}e_{1k}^{2}-\frac{\lambda_{1}}{\lambda_{3}}e_{1k}^{2}+s_{k}[\frac{c_{1k }}{\lambda_{3}}-\frac{1}{s_{k}\lambda_{3}}\lambda_{2}e_{1k}D^{a-1}e_{1k}+ \lambda_{1}\dot{e}_{1k}+\lambda_{2}D^{a}e_{1k}+\lambda_{3}(f(x_{1k})+bu_{k}- \dot{a}_{1k})]\end{array}  \tag{25}\]  
With the aim of achieving \(\dot{V}_{2k}\leq 0\), the backstepping sliding mode controller \(U=\left( \begin{array}{ccc}U_{1}&U_{2}&U_{3}\end{array}