## 2 Proposition of EBSMC
Consider the system state \(e_{2}\) as an independent input, and let:  
\[e_{2}=\phi_{1}(e_{1})=-k_{1}e_{1},\;k_{1}>0. \tag{8}\]  
Select a Lyapunov function \(V_{B1}=\frac{1}{2}e_{1}^{2}\). It can then be obtained that:  
\[\dot{V}_{B1}=e_{1}\dot{e}_{1}=-k_{1}e_{1}^{2}\leq 0 \tag{9}\]  
Therefore, state \(e_{1}\) is asymptotically stable. Step 2:  
Actually, there may be differences between \(e_{2}\) and \(\phi_{1}(e_{1})\). Therefore, a new error variable \(z_{1}=e_{2}-\phi_{1}(e_{1})\) is defined, which presents the difference between the stabilizing control law \(\phi_{1}(e_{1})\) and the error state \(e_{2}\). By adding and subtracting the virtual control law \(\phi_{1}(e_{1})\) to the first equation of (7), we can get the dynamic of the subsystem \((e_{1})\) as follows:  
\[\dot{e}_{1}=\phi_{1}(e_{1})+[e_{2}-\phi_{1}(e_{1})]=\phi_{1}(e_{1})+z_{1} \tag{10}\]  
Equations (8) and (9) can be satisfied if \(z_{1}\) in (10) equals zero. Further, consider the dynamic of \(z_{1}\):  
\[\dot{z}_{1}=\dot{e}_{2}-\dot{\phi}_{1}(e_{1})=e_{3}-\dot{\phi}_{1}(e_{1}) \tag{11}\]  
In a similar manner, treat the state \(e_{3}\) as an independent input of the form \(\phi_{2}(e_{1},z_{1})\) as the following:  
\[e_{3}=\phi_{2}(e_{1},z_{1})=-e_{1}-k_{2}z_{1}+\dot{\phi}_{1}(e_{1}),\ k_{2}>0. \tag{12}\]  
Select a Lyapunov function of the subsystem \((e_{1},z_{1})\) in the form of:  
\[V_{B2}=V_{B1}+\frac{1}{2}z_{1}^{2}=\frac{1}{2}e_{1}^{2}+\frac{1}{2}z_{1}^{2} \tag{13}\]  
From (10)-(12), the derivative of (13) is obtained as:  
\[\dot{V}_{B2}=e_{1}\dot{e}_{1}+z_{1}\dot{z}_{1}=-k_{1}e_{1}^{2}-k_{2}z_{1}^{2}\leq 0 \tag{14}\]  
Thus, the subsystem \((e_{1},z_{1})\) is asymptotically stable. Step 3:  
By adding and subtracting the virtual control law \(\phi_{2}(e_{1},z_{1})\) to (11) and defining an error variable as \(z_{2}=e_{3}-\phi_{2}(e_{1},z_{1})\), (11) can then be represented as:  
\[ \begin{array}{ll}\dot{z}_{1}&=[e_{3}-\phi_{2}(e_{1},z_{1})]+\phi_{2}(e_{1}, z_{1})-\dot{\phi}_{1}(e_{1})\\ &=z_{2}+\phi_{2}(e_{1},z_{1})-\dot{\phi}_{1}(e_{1})\end{array}  \tag{15}\]  
and:  
\[ \begin{array}{ll}\dot{z}_{2}&=\dot{e}_{3}-\dot{\phi}_{2}(e_{1},z_{1})\\ &=\dot{u}_{C}-\dot{\phi}_{2}(e_{1},z_{1})+\dot{\dot{D}}\Big{(}e_{3},\ddot{x}_{ d},\ddot{x}_{d},\dot{u}_{C},\dot{F}_{B}\big{(}x,\dot{\kappa}\big{)},\dot{d} \Big{)}\end{array}  \tag{16}\]  
From Step 1 and Step 2, it is found that the desired behavior of the subsystem \((e_{1},z_{1})\) can be achieved if the condition \(z_{2}=0\) is satisfied. Therefore, the control purpose can be simplified to the regulation of the virtual state \(z_{2}\) in the presence of external disturbances. The sliding mode control can be introduced to the backstepping design as follows. Select a sliding surface of:  
\[S=z_{2} \tag{17}\]  
In the absence of \(\dot{\dot{D}}\) in (16), the corresponding equivalent control force can be obtained by \(\dot{S}=0\), that is:  
\[\dot{u}_{Ceq}=\dot{\phi}_{2}(e_{1},z_{1}) \tag{18}\]  
A switching control action is applied to enhance the system robustness, such that system states stay on the sliding surface even in the presence of disturbances:  
\[\dot{u}_{C}=\dot{u}_{Ceq}-w\mathrm{sgn}(S) \tag{19}\]where the robust gain \(w\) satisfies the condition \(w>\eta\). Choose a Lyapunov function as \(V_{S}=S^{2}/2\), and it can be obtained that:  
\[ \begin{array}{ll}\dot{V}_{S}&=S[\dot{u}_{C}-\dot{\varphi}_{2}(e_{1},z_{1})+ \ddot{\widetilde{D}}\big{(}e_{3},\ddot{x}_{d},\dddot{x}_{d},\dot{u}_{C},\dot{F}_ {B}\big{(}\kappa,\dot{\kappa}\big{)},\dot{d}\big{)}]\\ &=S[-w\text{sgn}(S)+\ddot{\widetilde{D}}\big{(}e_{3},\ddot{x}_{d},\dddot{x}_{d },\dot{u}_{C},\dot{F}_{B}\big{(}\kappa,\dot{\kappa}\big{)},\dot{d}\big{)}]\\ &\leq-|S|[w-\eta]\leq 0\end{array}  \tag{20}\]  
It shows that the approaching condition can be reached, and thus, the system is asymptotically stable.