## 3 Adaptive Sliding Mode Controller Design via Backstepping
### The Design Steps for the Controller
Taking the derivative of Equation (38), we have:  
\[\dot{\mathbf{z}}_{2}=\dot{\mathbf{x}}_{2}-\dot{\mathbf{a}}_{2}=\mathbf{x}_{3}+\mathbf{F}_{2}-\dot{ \mathbf{a}}_{2} \tag{26}\]6  
Substituting Equation (37) to Equation (39), we have:  
\[\dot{\mathbf{z}}_{2}=\dot{\mathbf{x}}_{2}-\dot{\mathbf{a}}_{2}=\mathbf{x}_{3}+\mathbf{F}_{2}-\dot{ \mathbf{a}}_{2} \tag{26}\]7 \[\dot{\mathbf{z}}_{2}=\dot{\mathbf{x}}_{2}-\dot{\mathbf{a}}_{2}=\mathbf{x}_{3}+\mathbf{F}_{2}-\dot{ \mathbf{a}}_{2} \tag{26}\]8  
Assuming that the parameter uncertainty, \(\mathbf{F}_{i}\), changes slowly, we take \(\dot{\mathbf{F}}_{i}=0\); then, Equation (40) can be rewritten as:  
\[\dot{\mathbf{z}}_{2}=\dot{\mathbf{x}}_{2}-\dot{\mathbf{a}}_{2}=\mathbf{x}_{3}+\mathbf{F}_{2}-\dot{ \mathbf{a}}_{2} \tag{26}\]9 \[\mathbf{z}_{3}=\mathbf{x}_{3}-\mathbf{\alpha}_{3} \tag{27}\]0 \[\mathbf{z}_{3}=\mathbf{x}_{3}-\mathbf{\alpha}_{3} \tag{27}\]1 \[\mathbf{z}_{3}=\mathbf{x}_{3}-\mathbf{\alpha}_{3} \tag{27}\]2  
Step \(n-1\):  
A Lyapunov function candidate is chosen as:\[\mathbf{z}_{3}=\mathbf{x}_{3}-\mathbf{\alpha}_{3} \tag{27}\]3  
where \(\Gamma_{n-1}\) is a symmetric positive definite matrix. Then:  
\[ \begin{array}{ll}&\dot{V}_{n-1}=\dot{V}_{n-2}+z_{n-1}^{T}(-k_{n-1}z_{n-1}-z_{ n-2}+z_{n-1})+\widetilde{\mathbf{F}}_{n-1}^{T}(z_{n-1}-\Gamma_{n-1}^{-1}\dot{ \widehat{F}}_{n-1})\\ &=-k_{1}z_{1}^{T}z_{1}-k_{2}z_{2}^{T}z_{2}-\cdots-k_{n-1}z_{n-1}^{T}z_{n-1}+z_{ n-1}^{T}z_{n}+\widetilde{\mathbf{F}}_{1}^{T}(z_{1}-\Gamma_{1}^{-1}\widehat{\mathbf{F}}_{1})\\ &+\widetilde{\mathbf{F}}_{2}^{T}(z_{2}-\Gamma_{2}^{-1}\widehat{\mathbf{F}}_{2})+\cdots +\widetilde{\mathbf{F}}_{n-1}^{T}(z_{n-1}-\Gamma_{n-1}^{-1}\widehat{\mathbf{F}}_{n-1} )\\ &=\sum\limits_{j=1}^{n-1}[-k_{j}z_{j}^{T}z_{j}+\widetilde{\mathbf{F}}_{j}^{T}(z_{j }-\Gamma_{j}^{-1}\dot{\widehat{F}}_{j})]+z_{n-1}^{T}z_{n}\end{array}  \tag{43}\]  
where the tracking error and virtual control input are defined as follows:  
\[\left\{ \begin{array}{ll}z_{n}=x_{n}-\mathbf{\alpha}_{n}\\ \mathbf{\alpha}_{n}=-k_{n-1}z_{n-1}+\dot{\mathbf{\alpha}}_{n-1}-z_{n-2}-\widehat{\mathbf{ F}}_{n}\end{array} \right. \tag{44}\]  
Step \(n\)  
Designing the sliding surface as:  
\[ \begin{array}{ll}S=\begin{bmatrix}s_{1}\\ s_{2}\end{bmatrix}=c_{1}\mathbf{z}_{1}+c_{2}\mathbf{z}_{2}+\cdots c_{i}\mathbf{z}_{i}+ \cdots+c_{n-1}\mathbf{z}_{n-1}+\mathbf{z}_{n}\end{array}  \tag{45}\]  
where \(\mathbf{S}\in\mathbb{R}^{2}\), \(c_{i}>0\), \(i=1,2,\cdots n-1\). Taking the derivative of Equation (45), we have:  
\[ \begin{array}{ll}\dot{\mathbf{S}}&=c_{1}\dot{\mathbf{z}}_{1}+c_{2}\dot{\mathbf{z}}_{2}+ \cdots+c_{n-1}\dot{\mathbf{z}}_{n-1}+\dot{\mathbf{z}}_{n}\\ &=c_{1}\dot{\mathbf{z}}_{1}+c_{2}\dot{\mathbf{z}}_{2}+\cdots+c_{n-1}\dot{\mathbf{z}}_{n-1}+ \dot{\mathbf{x}}_{n}-\dot{\mathbf{x}}_{n}\end{array}  \tag{46}\]  
Substituting Equation (14) into Equation (46), we have:  
\[ \begin{array}{ll}\dot{\mathbf{S}}=c_{1}\dot{\mathbf{z}}_{1}+c_{2}\dot{\mathbf{z}}_{2}+ \cdots+c_{n-1}\dot{\mathbf{z}}_{n-1}+\mathbf{f}(x,t)+\mathbf{G}(x,t)\mathbf{u}+\mathbf{F}_{n}-\dot{ \mathbf{\alpha}}_{n}\end{array}  \tag{47}\]  
A Lyapunov function candidate is chosen as:  
\[ \begin{array}{ll}V_{n}&=V_{n-1}+\frac{1}{2}\mathbf{S}^{T}\mathbf{S}+\frac{1}{2} \widetilde{\mathbf{F}}_{n}^{T}\Gamma_{n}^{-1}\widetilde{\mathbf{F}}_{n}\\ &=\frac{1}{2}\sum\limits_{j=1}^{n-1}\Bigl{(}k_{j}z_{j}^{T}z_{j}+\widetilde{ \mathbf{F}}_{j}^{T}\Gamma_{j}^{-1}\widetilde{\mathbf{F}}_{j}\Bigr{)}+\frac{1}{2}\mathbf{S} ^{T}\mathbf{S}+\frac{1}{2}\widetilde{\mathbf{F}}_{n}^{T}\Gamma_{n}^{-1}\widetilde{\mathbf{ F}}_{n}\\ &=\frac{1}{2}\sum\limits_{j=1}^{n-1}k_{j}\mathbf{z}_{j}^{T}\mathbf{z}_{j}+\frac{1}{2} \mathbf{S}^{T}\mathbf{S}+\frac{1}{2}\sum\limits_{j=1}^{n}\widetilde{\mathbf{F}}_{j}^{T} \Gamma_{j}^{-1}\widetilde{\mathbf{F}}_{j}\end{array}  \tag{48}\]  
where \(\Gamma_{n}\) is a symmetric positive definite matrix. Taking the derivative of Equation (48), we have:  
\[