## 3 Proposed Control Methodology
### Adaptive Backstepping Sliding Mode Algorithm
Despite using the new adaptive law in Equation (19), an error of estimated lumped uncertainties is still existing, but its estimated error is bounded and predetermined as \(\|\hat{\Delta}\|=\|\Delta-\hat{\Delta}\|\leq\varepsilon_{\Delta}\). Then \(\mathbf{\eta}_{1}\) is chosen to be greater than the error of estimated lumped uncertainties, i.e., \(\mathbf{\eta}_{1i}\geq\varepsilon_{\Delta i}\). This means the controller gains are much smaller instead of designing with large value enough to cover the upper bound of uncertainties in the conventional design._  
From Equation (18), the derivative of the Lyapunov function \(V_{1}\) is semi-negative if the error \(\mathbf{e}_{3}\) converges to zero. Therefore, this error should be tackled as small as possible. Step 2:  
Define the sliding surface as  
\[\mathbf{s}_{2}=\mathbf{e}_{3}\in R^{n} \tag{20}\]  
Taking the sliding surface \(\mathbf{s}_{2}\) as derivative, one obtains:  
\[\dot{\mathbf{s}}_{2}=\dot{\mathbf{e}}_{3}=\mathbf{f}(\mathbf{y}_{1},\mathbf{y }_{2})+\mathbf{g}\mathbf{u}+\mathbf{\
abla}\mathbf{C}_{\text{l}ak}-\dot{ \mathbf{y}}_{3d} \tag{21}\]  
Choosing the second Lyapunov function \(V_{2}\) as:  
\[V_{2}=V_{1}+\frac{1}{2}\mathbf{s}_{2}^{T}\mathbf{s}_{2}+\frac{1}{2}\mathbf{e} _{C_{\text{l}ak}}^{T}\mathbf{e}_{C_{\text{l}ak}} \tag{22}\]  
where \(\mathbf{e}_{C_{\text{l}ak}}=\mathbf{C}_{\text{l}ak}-\hat{\mathbf{C}}_{\text{ l}ak}\) is an estimated error between a real and estimated leakage coefficient. Taking derivative of the Lyapunov function \(V_{2}\) and substituting Equation (21) results in:  
\[\dot{V}_{2}=\mathbf{s}_{1}^{T}\bigg{(}\mathbf{J}_{a}^{T}\mathbf{e}_{3}- \mathbf{K}_{1}\mathbf{s}_{1}-\mathbf{\eta}_{1}\text{tanh}\bigg{(}\frac{ \mathbf{s}_{1}}{\mathbf{\sigma}_{1}}\bigg{)}\bigg{)}+\mathbf{s}_{2}^{T}\big{(} \mathbf{f}(\mathbf{y}_{1},\mathbf{y}_{2})+\mathbf{g}\mathbf{u}+\mathbf{\
abla }\mathbf{C}_{\text{l}ak}-\dot{\mathbf{y}}_{3d}\big{)}+\mathbf{e}_{C_{\text{l}ak }}^{T}\dot{\mathbf{e}}_{C_{\text{l}ak}} \tag{23}\]  
To guarantee the system robustness and global stability, the control input signal \(\mathbf{u}_{v}\) is designed as  
\[\mathbf{u}_{v}=\mathbf{g}^{-1}\bigg{(}-\mathbf{f}(\mathbf{y}_{1},\mathbf{y}_ {2})-\mathbf{\
abla}\hat{\mathbf{C}}_{\text{l}ak}+\dot{\mathbf{y}}_{3d}- \mathbf{K}_{2}\mathbf{s}_{2}-\mathbf{\eta}_{2}\text{tanh}\bigg{(}\frac{ \mathbf{s}_{2}}{\mathbf{\sigma}_{2}}\bigg{)}-\mathbf{J}_{a}^{T}\mathbf{s}_{1} \bigg{)} \tag{24}\]  
where \(\mathbf{K}_{2}\triangleq\text{diag}\Big{(} \begin{array}{cccccc}K_{21}&K_{22}& \ldots&K_{2i}&\ldots&K_{2n}\end{array} \Big{)}\) and \(\mathbf{\eta}_{2}\triangleq\text{diag}\Big{(} \begin{array}{cccccc}\eta_{21}& \eta_{22}&\ldots&\eta_{2i}&\ldots&\eta_{2n}\end{array} \Big{)}\) are positive definite matrices. \(\mathbf{\sigma}_{2}\) is the width vector, the term \(-\mathbf{K}_{2}\mathbf{s}_{2}\) is added to enhance the convergence rate when \(\mathbf{s}_{2}\) is large. A function \(\text{tanh}\bigg{(}\frac{\mathbf{s}_{i}}{\mathbf{\sigma}_{j}}\bigg{)}=\bigg{(} \text{tanh}\bigg{(}\frac{s_{j1}}{\mathbf{\sigma}_{j1}}\bigg{)},\text{tanh} \bigg{(}\frac{s_{2}}{\mathbf{\sigma}_{2}}\bigg{)}\ldots,\text{tanh}\bigg{(} \frac{s_{jn}}{\mathbf{\sigma}_{jn}}\bigg{)}\bigg{)}^{T}(j=1,2)\) in both Equations (16) and (24) are used to replace a conventional signum function to achieve chattering-free and facilitate control design.