## 3 Adaptive Sliding Mode Controller Design via Backstepping
### The Design Steps for the Controller
\begin{array}{ll}\dot{V}_{n}=\dot{V}_{n-1}+\mathbf{S}^{T}\dot{\mathbf{S}}+\widetilde {\mathbf{F}}_{n}^{T}\Gamma_{n}^{-1}\dot{\widetilde{F}}_{n}\end{array}  \tag{49}\]  
Substituting Equation (47) into Equation (49), we get:  
\[ \begin{array}{ll}\dot{V}_{n}&=\dot{V}_{n-1}+\mathbf{S}^{T}[c_{1}\dot{\mathbf{z}}_{1}+ c_{2}\dot{\mathbf{z}}_{2}+\cdots+c_{n-1}\dot{\mathbf{z}}_{n-1}\\ &+\mathbf{f}(\mathbf{x},t)+\mathbf{G}(\mathbf{x},t)\mathbf{u}+\mathbf{F}_{n}-\dot{\mathbf{\alpha}}_{n}]+ \widetilde{\mathbf{F}}_{n}^{T}\Gamma_{n}^{-1}\widetilde{\mathbf{F}}_{n}\end{array}  \tag{50}\]  
Substituting Equation (15) into Equation (50), we have:  
\[ \begin{array}{ll}\dot{V}_{n}=\dot{V}_{n-1}+\mathbf{S}^{T}[c_{1}\dot{\mathbf{z}}_{1}+ c_{2}\dot{\mathbf{z}}_{2}+\cdots+c_{n-1}\dot{\mathbf{z}}_{n-1}\\ +\mathbf{f}(\mathbf{x},t)+\mathbf{G}(\mathbf{x},t)\mathbf{u}+\widehat{\mathbf{F}}_{n}-\dot{\mathbf{\alpha}}_{ n})]+\mathbf{S}^{T}\widetilde{\mathbf{F}}_{n}+\widetilde{\mathbf{F}}_{n}^{T}\Gamma_{n}^{-1} \biggl{(}\dot{\mathbf{F}}_{n}-\dot{\widehat{F}}_{n}\biggr{)}\end{array}  \tag{51}\]Assuming that the parameter uncertainty, \(\mathbf{F}_{n}\), changes slowly, we take \(\dot{\mathbf{F}}_{n}=0\); then, Equation (51) can be rewritten as:  
\[ \begin{array}{ll}\dot{V}_{n}&=\dot{V}_{n-1}+\mathbf{S}^{T}[c_{1}\dot{\mathbf{z}}_{1}+c _{2}\dot{\mathbf{z}}_{2}+\cdots+c_{n-1}\dot{\mathbf{z}}_{n-1}\\ &+\mathbf{f}(\mathbf{x},t)+\mathbf{G}(\mathbf{x},t)\mathbf{u}+\widehat{\mathbf{F}}_{n}-\dot{\mathbf{\dot{u} }}_{n})]+\widetilde{\mathbf{F}}_{n}^{T}(\mathbf{S}-\Gamma_{n}^{-1}\widehat{\mathbf{F}}_{n}) \end{array}  \tag{52}\]  
Substituting Equation (43) into Equation (52), we have:  
\[ \begin{array}{ll}\dot{V}_{n}&=\sum\limits_{j=1}^{n-\mathbf{1}}[-\mathbf{k}_{j}\mathbf{z}_ {j}^{T}\mathbf{z}_{j}+\widetilde{\mathbf{F}}_{j}^{T}(\mathbf{z}_{j}-\Gamma_{j}^{-1}\dot{ \widehat{\mathbf{F}}}_{j})]+\mathbf{z}_{n-1}^{T}\mathbf{z}_{n}+\mathbf{S}^{T}[c_{1}\dot{\mathbf{z} }_{1}+c_{2}\dot{\mathbf{z}}_{2}\\ &+\cdots+c_{n-1}\dot{\mathbf{z}}_{n-1}+\mathbf{f}(\mathbf{x},t)+\mathbf{G}(\mathbf{x},t)\mathbf{u}+ \widehat{\mathbf{F}}_{n}-\dot{\mathbf{\dot{u}}}_{n})]+\widetilde{\mathbf{F}}_{n}^{T}(\mathbf{S }-\Gamma_{n}^{-1}\dot{\widehat{\mathbf{F}}}_{n})\end{array}  \tag{53}\]  
According to Equation (45), we have \(\mathbf{z}_{n}=\mathbf{S}-c_{1}\mathbf{z}_{1}-c_{2}\mathbf{z}_{2}-\cdots-c_{n-1}\mathbf{z}_{n-1}\); thus Equation (53) can be rewritten as  
\[ \begin{array}{ll}\dot{V}_{n}&=\sum\limits_{j=1}^{n-\mathbf{1}}[-\mathbf{k}_{j}\mathbf{z }_{j}^{T}\mathbf{z}_{j}+\widetilde{\mathbf{F}}_{j}^{T}(\mathbf{z}_{j}-\Gamma_{j}^{-1}\dot{ \widehat{\mathbf{F}}}_{j})]+\mathbf{z}_{n-1}^{T}(\mathbf{S}-\mathbf{c}_{1}\mathbf{z}_{1}-c_{2}\mathbf{ z}_{2}-\cdots-\mathbf{c}_{n-1}\mathbf{z}_{n-1})\\ &+\mathbf{S}^{T}\Big{[}c_{1}\dot{\mathbf{z}}_{1}+c_{2}\dot{\mathbf{z}}_{2}+\cdots+c_{n-1} \dot{\mathbf{z}}_{n-1}+\mathbf{f}(\mathbf{x},t)+\mathbf{G}(\mathbf{x},t)u+\widehat{\mathbf{F}}_{n}- \dot{\mathbf{\dot{u}}}_{n}\Big{)}\big{]}\\ &+\widetilde{\mathbf{F}}_{n}^{T}(\mathbf{S}-\Gamma_{n}^{-1}\dot{\widehat{\mathbf{F}}}_{n} )\\ &=\sum\limits_{j=1}^{n-\mathbf{1}}[-\mathbf{k}_{j}\mathbf{z}_{j}^{T}\mathbf{z}_{j}+\widetilde{ \mathbf{F}}_{j}^{T}(\mathbf{z}_{j}-\Gamma_{j}^{-1}\dot{\widehat{\mathbf{F}}}_{j})]-\mathbf{z}_ {n-1}^{T}(c_{1}\mathbf{z}_{1}+c_{2}\mathbf{z}_{2}+\cdots+c_{n-1}\mathbf{z}_{n-1})\\ &+\mathbf{S}^{T}\Big{[}\mathbf{z}_{n-1}+c_{1}\dot{\mathbf{z}}_{1}+c_{2}\dot{\mathbf{z}}_{2}+ \cdots+c_{n-1}\dot{\mathbf{z}}_{n-1}+\mathbf{f}(\mathbf{x},t)+\mathbf{G}(\mathbf{x},t)u+\widehat{ \mathbf{F}}_{n}-\dot{\mathbf{\dot{u}}}_{n}\Big{)}]\\ &+\widetilde{\mathbf{F}}_{n}^{T}(\mathbf{S}-\Gamma_{n}^{-1}\dot{\widehat{\mathbf{F}}}_{n} )\end{array}