## 3 Main Result  
### Command-Filter Based Controller Design with Prescribed Performance  
In this section, a command-filter based backstepping controller with prescribed performance is designed. Firstly, some new error variables are defined as follows,  
\[z_{1}=x_{1}-x_{1}^{c} \tag{19}\]  
\[z_{2}=x_{2}-x_{2}^{c} \tag{20}\]  
where the definitions of \(x_{1}^{c}\) and \(x_{2}^{c}\) are along with the classic command-filter backstepping controller design in [22]. In addition, \(x_{1}^{c}=x_{1}^{d}\) and \(x_{1}^{d}\) is the desired rolling angle signal. \(x_{2}^{c}\) is the output of the command-filter. The compensated tracking error \(\bar{z}_{1}\) is defined as  
\[\bar{z}_{1}=z_{1}-\zeta_{1} \tag{21}\]  
and the \(\zeta_{1}\) signal is chosen as  
\[\zeta_{1}=-k_{1}\zeta_{1}+(x_{2}^{c}-x_{2}^{d}) \tag{22}\]  
where \(x_{2}^{d}\) is the desired value of \(x_{2}\), which can be utilized to stabilize the outer loop. In order to obtain the prescribed performance, the stability of the transformation error \(\varepsilon(t)\) should be validated. By taking the derivation of \(\varepsilon(t)\) with respect to time, it yields  
\[\dot{\varepsilon} =\frac{\partial\mathcal{L}^{-1}}{\partial(z_{1}/\rho)}\frac{1}{ \rho}\left(\dot{x}_{1}-\dot{x}_{1}^{c}-\dot{\zeta}_{1}-\frac{\partial\bar{z}_ {1}}{\rho}\right) \tag{23}\] \[=r(cx_{2}-\dot{x}_{1}^{c}-\dot{\zeta}_{1}+\
u)\]  
where \(r=\frac{\partial\mathcal{L}^{-1}}{\partial(z_{1}/\rho)}\frac{1}{\rho}\), \(\
u=-\frac{\partial\bar{z}_{1}}{\rho}\). And for the inner loop, an integral sliding mode surface is given as follows  
\[s=z_{2}+a\int_{0}^{t}z_{2}^{m/n}dt \tag{24}\]  
where \(a\) is a positive scalar, \(m\) and \(n\) are add integers satisfying \(n>m>0\). The reaching law for the sliding mode surface is selected as  
\[\dot{s}=-k_{3}s-\tau sign(s) \tag{25}\]  
where \(k_{3}>0\) and \(\tau>0\) are two positive real scalars. Before presenting the first main result of this study, the desired virtual controller \(x_{2}^{d}\) is selected as  
\[x_{2}^{d}=-k_{1}\zeta_{1}+\dot{x}_{1}^{c}-\
u-k_{2}\varepsilon \tag{26}\]  
and the controller \(u^{d}\) is chosen as  
\[u=b^{-1}\left(-f(x)+\dot{x}_{2}^{c}-az_{2}^{m/n}-k_{3}s-\tau sign(s)-\bar{F} sign(s)\right). \tag{27}\]  
**Theorem 1**. : _For the rolling angle dynamic system (7) and (8), suppose that Assumptions 1 and 2 hold, the transformation error \(\varepsilon(t)\) and \(z_{2}\) are asymptotically stable by the proposed virtual controller (26) and the sliding mode controller (27) while \(z_{1}\) converges to the neighbor domain of \(0\) with the prescribed performance._  
Proof of Theorem 1.: In this position, the Lyapunov function is selected as  
\[z_{2}=x_{2}-x_{2}^{c} \tag{20}\]0Taking the derivation of \(V_{1}\) with respect to time, it yields  
\[ \begin{split}\dot{V}_{1}&=\varepsilon\cdot\left(x_{2}- \dot{x}_{1}^{c}-\dot{\xi}_{1}+\
u\right)+s\left(z_{2}+az_{2}^{m/n}\right)\\ &=\varepsilon\cdot\left(x_{2}-\dot{x}_{1}^{c}+k_{1}\xi_{1}-x_{2} ^{c}+x_{2}^{d}+\
u\right)+s\left(\dot{z}_{2}+az_{2}^{m/n}\right).\end{split}  \tag{29}\]  
Substituting (26) into (29), it can be obtained that  
\[\dot{V}_{1}=-k_{2}\cdot\varepsilon^{2}+\varepsilon z_{2}+s\left(f(x)+bu+F- \dot{x}_{2}^{c}+az_{2}^{m/n}\right), \tag{30}\]  
and the following relationship holds as  
\[\varepsilon\cdot z_{2}\leq\frac{1}{2}\varepsilon^{2}+\frac{1}{2}z_{2}^{2}\leq \frac{1}{2}\varepsilon^{2}+\frac{1}{2}s^{2}.\]  
Substituting the sliding mode controller (27) into (30),  
\[ \begin{split}\dot{V}_{1}&\leq-\left(k_{2}-\frac{1 }{2}\right)\cdot\varepsilon^{2}+\frac{1}{2}s^{2}+s\left(f(x)+bu+F-\dot{x}_{2} ^{c}+az_{2}^{m/n}\right)\\ &\leq-\left(k_{2}-\frac{1}{2}\right)\cdot\varepsilon^{2}-\left( k_{3}-\frac{1}{2}\right)s^{2}+s\left(F-\bar{F}sign(s)\right)-\tau|s|\\ &\leq-\left(k_{2}-\frac{1}{2}\right)\cdot\varepsilon^{2}-\left( k_{3}-\frac{1}{2}\right)s^{2}\leq 0.\end{split}  \tag{31}\]  
This implies that, with the implement of the proposed virtual controller, the transformation errors \(\varepsilon\) and \(z_{2}\) are asymptotically stable.