## 3 Adaptive Sliding Mode Controller Design via Backstepping
### The Design Steps for the Controller  
In this subsection, a backstepping-based adaptive sliding mode control is presented for a class of high-order nonlinear system with uncertainties. The considered high order nonlinear system with uncertainties is as follows:  
\[\left\{ \begin{array}{l}\dot{\mathbf{x}}_{1}=\mathbf{x}_{2}+\mathbf{F}_{1}\\ \dot{\mathbf{x}}_{2}=\mathbf{x}_{3}+\mathbf{F}_{2}\\ \vdots\\ \dot{\mathbf{x}}_{i}=\mathbf{x}_{i+1}+\mathbf{F}_{i}\\ \vdots\\ \dot{\mathbf{x}}_{n}=\mathbf{f}(\mathbf{x},t)+\mathbf{G}(\mathbf{x},t)\mathbf{u}+\mathbf{F}_{n}\\ \mathbf{y}=\mathbf{x}_{1}\end{array} \right. \tag{14}\]  
where \(\mathbf{x}_{i}=\left[x_{i,1},x_{i,2}\right]^{T}\in\mathbb{R}^{2}\), \(i=1,2\cdots n\) are states; \(\mathbf{f}(\mathbf{x},t)\) and \(\mathbf{G}(\mathbf{x},t)\) are given nonlinear functions; \(\mathbf{u}\in\mathbb{R}^{2}\) is control input; \(\mathbf{y}\in\mathbb{R}^{2}\) is the system output; \(\mathbf{F}_{i}=\left[F_{i,1},F_{i,2}\right]^{T}\in\mathbb{R}^{2}\), \(i=1,2\cdots n\) represent uncertainties. Uncertainties are defined as follows:  
\[\mathbf{F}_{i}=\widehat{\mathbf{F}}_{i}+\widetilde{\mathbf{F}}_{i} \tag{15}\]  
where \(\widehat{\mathbf{F}}_{i}\) represents the estimates of uncertainties; \(\widetilde{\mathbf{F}}_{i}\) represents the estimation errors of uncertainties. The controller is developed as follows. Step 1  
Defining the tracking error as:  
\[\mathbf{z}_{1}=\mathbf{x}_{1}-\mathbf{\alpha}_{1} \tag{16}\]  
where \(\mathbf{\alpha}_{1}=\mathbf{x}_{1\text{d}}\) is the command signal for the first subsystem, then:  
\[\dot{\mathbf{z}}_{1}=\dot{\mathbf{x}}_{1}-\dot{\mathbf{\alpha}}_{1}=\mathbf{x}_{2}+\mathbf{F}_{1} -\dot{\mathbf{\alpha}}_{1} \tag{17}\]  
Defining the tracking error as:  
\[\mathbf{z}_{2}=\mathbf{x}_{2}-\mathbf{\alpha}_{2} \tag{18}\]  
To stabilize the subsystem, a virtual control input is defined as follows:  
\[\mathbf{\alpha}_{2}=-k_{1}\mathbf{z}_{1}+\dot{\mathbf{\alpha}}_{1}-\widehat{\mathbf{F}}_{1} \tag{19}\]where \(k_{1}>0\). Substituting Equation (19) into Equation (18), we have:  
\[\mathbf{x}_{2}=-k_{1}\mathbf{z}_{1}+\dot{\mathbf{a}}_{1}+\mathbf{z}_{2}-\widehat{\mathbf{F}}_{1} \tag{20}\]  
Substituting Equation (20) into Equation (17), we have:  
\[\dot{\mathbf{z}}_{1}=-k_{1}\mathbf{z}_{1}+\mathbf{z}_{2}-\widehat{\mathbf{F}}_{1}+\mathbf{F}_{1}=-k_ {1}\mathbf{z}_{1}+\mathbf{z}_{2}+\widetilde{\mathbf{F}}_{1} \tag{21}\]  
A Lyapunov function candidate is chosen as:  
\[V_{1}=\frac{1}{2}\mathbf{z}_{1}^{T}\mathbf{z}_{1}+\frac{1}{2}\widetilde{\mathbf{F}}_{1}^{T }\Gamma_{1}^{-1}\widetilde{\mathbf{F}}_{1} \tag{22}\]  
where \(\Gamma_{1}\) is a symmetric positive definite matrix. Taking the derivative of Equation (22), we have:  
\[\dot{V}_{1}=\mathbf{z}_{1}^{T}\dot{\mathbf{z}}_{1}+\widetilde{\mathbf{F}}_{1}^{T}\Gamma_{ 1}^{-1}\dot{\widetilde{\mathbf{F}}}_{1} \tag{23}\]  
Substituting Equation (21) into Equation (23), we have:  
\[ \begin{array}{rl}\dot{V}_{1}&=\mathbf{z}_{1}^{T}\Big{(}-k_{1}\mathbf{z}_{1}+\mathbf{z}_{ 2}+\widetilde{\mathbf{F}}_{1}\Big{)}+\widetilde{\mathbf{F}}_{1}^{T}\Gamma_{1}^{-1} \bigg{(}\dot{\mathbf{F}}_{1}-\dot{\widetilde{\mathbf{F}}}_{1}\bigg{)}\\ &=-k_{1}\mathbf{z}_{1}^{T}\mathbf{z}_{1}+\mathbf{z}_{1}^{T}\mathbf{z}_{2}+\mathbf{z}_{1}^{T} \widetilde{\mathbf{F}}_{1}+\widetilde{\mathbf{F}}_{1}^{T}\Gamma_{1}^{-1}\bigg{(}\dot {\mathbf{F}}_{1}-\dot{\widetilde{\mathbf{F}}}_{1}\bigg{)}\end{array}