## 3 Design of Control System
### Design of Fractional Sliding Mode Backstepping Control  
The estimated states \(\hat{x}_{1}\) and \(\hat{x}_{2}\) of the LESO are used in the design of fractional sliding-mode backstepping control. Define the tracking error of the inclination angle as  
\[e_{1}=\sigma_{d}-\hat{x}_{1}. \tag{17}\]  
Calculating the derivative of (17) yields  
\[\dot{e}_{1} =\dot{\sigma}_{d}-\hat{x}_{1} \tag{18}\] \[=\dot{\sigma}_{d}-\hat{x}_{2}.\]  
Define an auxiliary error \(e_{2}\) as  
\[e_{2}=x_{2d}-\hat{x}_{2}. \tag{19}\]  
where \(x_{2d}\) is a virtual control variable. It is easy to obtain  
\[\dot{e}_{1}=e_{2}+\dot{\sigma}_{d}-x_{2d}. \tag{20}\]  
According to the error system, let the virtual control variable satisfy  
\[x_{2d}=\dot{\sigma}_{d}+k_{1}e_{1}, \tag{21}\]  
where \(k_{1}\in R^{+}\) is the feedback gain. To avoid differential explosion, the algorithm of dynamic surface control is employed. Designing a low-pass filter and passing the virtual control variable \(x_{2d}\) through it yield  
\[T\hat{x}_{2d}+\hat{x}_{2d}=x_{2d},\ \hat{x}_{2d}(0)=x_{2d}(0), \tag{22}\]where \(T\) is the time constant of the filter and \(\hat{x}_{2d}\) is the filtered virtual control variable. Define the filter error as  
\[\hat{x}_{2d}=\hat{x}_{2d}-x_{2d}. \tag{23}\]  
Calculating the derivative of (23) along (22) yields  
\[\hat{x}_{2d} =\hat{x}_{2}d-\hat{x}_{2d} \tag{24}\] \[=-\frac{1}{T}\hat{x}_{2d}-(\ddot{\sigma}_{d}+k_{1}\hat{e}_{1})\] \[\dot{e}_{1} =\dot{\sigma}_{d}-\hat{x}_{1} \tag{18}\]0  
where \(B(\ddot{\sigma}_{d},\dot{e}_{1})\) is a function about \(\ddot{\sigma}_{d}\) and \(\dot{e}_{1}\). There exists a positive constant \(B_{M}\) such that \(|B|\leq B_{M}\)[22]. Substituting (21) into (20) yields  
\[\dot{e}_{1} =\dot{\sigma}_{d}-\hat{x}_{1} \tag{18}\]1  
Choose a Lyapunov candidate to be  
\[\dot{e}_{1} =\dot{\sigma}_{d}-\hat{x}_{1} \tag{18}\]2  
It is easy to obtain  
\[\dot{e}_{1} =\dot{\sigma}_{d}-\hat{x}_{1} \tag{18}\]3  
In this study, the methodology of the sliding surface and the backstepping control is used to ensure that \(e_{1}\) and \(e_{2}\) converge quickly. Moreover, the fractional calculus operator is employed to improve transient performance in the design of a sliding mode surface, which is denoted by  
\[\dot{e}_{1} =\dot{\sigma}_{d}-\hat{x}_{1} \tag{18}\]4  
where \(\lambda_{1}\) is the sliding surface gain and \({}_{a}D_{t}^{\alpha}\) is the calculus operator  
\[{}_{a}D_{t}^{\alpha}= \begin{cases}\dfrac{d^{\alpha}}{dt^{\alpha}}&Re(\alpha)> 0\\ 1&Re(\alpha)=0\\ \int_{a}^{t}d(\tau)^{-\alpha}&Re(\alpha)<0.\end{cases}  \tag{29}\]  
In (29), \(\alpha\) is the order of the operator and can be used to adjust transient performance of the control system. For simplicity of implementation, we choose the Caputo fractional calculus [23; 24; 25]  
\[{}_{a}D_{t}^{\alpha}f(t)=\frac{1}{\Gamma(m-\alpha)}\int_{a}^{t}\frac{f^{m}( \tau)}{(t-\tau)^{1+\alpha-m}}d\tau,m-1<\alpha<m, \tag{30}\]  
where \(\Gamma(\cdot)\) is Gamma Function, \(\Gamma(\eta)=\int_{0}^{\infty}e^{-t}t^{\eta-1}dt\) and \(m\) is the least integer that is not less than \(\alpha\). The fractional differential operation is transformed to the particular form of integral operation; therefore, the fractional calculus has the heritability and the memorability. To simplify the notation, let \(D^{\alpha}\) represent \({}_{0}D_{t}^{\alpha}\). This should not cause confusion. In order to make the system state converge to the sliding-mode surface and further weaken the chattering of the system, the fractional reaching law is designed as  
\[D^{\beta}s=-\epsilon sgn(s), \tag{31}\]where \(\epsilon\) is a positive constant.