## 3 Control Design
### Stability Analysis of the Closed-Loop Control System  
**Theorem 1**. : _Consider the dynamics (19), where \(\mathbf{a}_{1}\), \(\mathbf{a}_{2}\) and \(\mathbf{b}_{1}\) satisfy Assumptions 1 and 2 with the unknown gains \(\delta_{1}\), \(\delta_{2}\), and \(\gamma_{1}\). The adaptive gains satisfy Assumptions 3 and 4 and they are determined by (24). Then, for any \(\mathbf{x}_{ik}\) at \(t=0\), there exist_  
* _a parameter_ \(\mu>0\) _so that_ \(\alpha_{k}\) _satisfies (_25_) if_ \(\|\mathbf{s}_{ik}\|_{2}>\mu\) _at_ \(t=0\)_;_ \[\alpha_{k}>\frac{\delta_{1}(\lambda+4\varepsilon^{2})-\varepsilon(4\delta_{4}+ 1)}{\lambda(1-\gamma_{1})}+\frac{\left[2\varepsilon\delta_{1}-2\delta_{4}- \lambda-4\varepsilon^{2}\right]^{2}}{12\varepsilon\lambda(1-\gamma_{1})}\] (25) _Here_ \(\lambda\) _is an arbitrary positive constant and_ \(\delta_{4}\) _is determined by (_23_)._
* _a finite time_ \(t_{\text{F}}>0\) _so that the sliding modes of_ \(\mathbf{s}_{ik}\) _are reached in the finite time_ \(t_{\text{F}}\) _regarding to the adaptive-gain super-twisting sliding mode control method, that is,_ \(\forall t>t_{\text{F}}\)_,_ \(\exists\|\mathbf{s}_{ik}\|_{2}\leq\eta_{1}\) _and_ \(\|\mathbf{s}_{ik}\|_{2}\leq\eta_{2}\)_. Here,_ \(\eta_{1}>\mu\) _and_ \(\eta_{2}>0\)_._
* _both_ \(\alpha_{k}\) _and_ \(\beta_{k}\) _are bounded._  
**Proof. **  
* _Preparation_. Define a_ \(4\times 1\) _augmented vector_ \(\mathbf{Z}\)__ \[\mathbf{Z}=\left[ \begin{array}{cc}\mathbf{Z}_{1}^{T}&\mathbf{Z}_{2}^{T}\\ \end{array} \right]^{T}=\left[ \begin{array}{cc}\sqrt{\|\mathbf{s}_{ik}\|_{2 }}[\text{sgn}(\mathbf{s}_{ik})]^{T}&[\omega_{k}]^{T}\\ \end{array} \right]^{T}\] (26)  
_Here,_ \(\|\mathbf{Z}_{1}\|_{\infty}=\sqrt{\|\mathbf{s}_{ik}\|_{2}}\)_,_ \(\text{sgn}(\mathbf{Z}_{1})=\text{sgn}(\mathbf{s}_{ik})\) _and_ \(\mathbf{Z}_{2}=\omega_{k}\)_. Further, the time derivative of_ \(\mathbf{Z}_{1}\) _and_ \(\mathbf{Z}_{2}\) _can be written as_  
\[ \begin{array}{l}\dot{\mathbf{Z}}_{1}=\frac{1}{2}\frac{1}{2\|\mathbf{Z}_{1}\| _{\infty}}[-\alpha_{k}\mathbf{b}_{1}\mathbf{Z}_{1}+\mathbf{Z}_{2}+\mathbf{a}_{ 1}]\\ \dot{\mathbf{Z}}_{2}=-\frac{\ddot{\mathbf{b}}_{2}}{2}\mathbf{b}_{1}\frac{1}{2\| \mathbf{Z}_{1}\|_{\infty}}+\dot{\chi}\end{array}  \tag{27}\]  
_From (_27_), we have_  
\[\left[ \begin{array}{c}\dot{\mathbf{Z}}_{1}\\ \dot{\mathbf{Z}}_{2}\end{array} \right]=\frac{1}{2\|\mathbf{Z}_{1}\|_{\infty}} \left[ \begin{array}{cc}-\alpha_{k}\mathbf{b}_{1}&\mathbf{I}_{2}\\ -\beta_{k}\mathbf{b}_{1}&\mathbf{O}_{2}\\ \end{array} \right]\left[ \begin{array}{c}\mathbf{Z}_{1}\\ \mathbf{Z}_{2}\end{array} \right]+\frac{1}{2\|\mathbf{Z}_{1}\|_{\infty}} \left[ \begin{array}{c}\mathbf{a}_{1}\\ 2\|\mathbf{Z}_{1}\|_{\infty}\dot{\chi}\\ \end{array} \right] \tag{28}\]  
_Here,_ \(\mathbf{O}_{2}\) _is a_ \(2\times 2\) _zero matrix._  
_Further, considering (_16_) and (_23_), two bounded scalar functions_ \(\rho_{1}(\mathbf{x}_{ik},t)\) _and_ \(\rho_{2}(\mathbf{x}_{ik},t)\) _can be constructed as_  
\[ \begin{array}{l}\mathbf{a}_{1}=\rho_{1}(\mathbf{x}_{ik},t)\sqrt{\|\mathbf{s}_ {ik}\|_{2}}\text{sgn}(\mathbf{s}_{ik})=\rho_{1}(\mathbf{x}_{ik},t)\mathbf{Z}_{1 }\\ \dot{\chi}=\frac{\rho_{2}(\mathbf{x}_{ik},t)}{2}\frac{\sqrt{\|\mathbf{s}_{ik}\|_{2 }}}{\sqrt{\|\mathbf{s}_{ik}\|_{2}}}\text{sgn}(\mathbf{s}_{ik})=\frac{\rho_{2}( \mathbf{x}_{ik},t)}{2}\frac{\mathbf{Z}_{1}}{\|\mathbf{Z}_{1}\|_{\infty}}\end{array}  \tag{29}\]Then, we can have  
\[0<\rho_{1}(\mathbf{x}_{ik},t)<\delta_{1} \tag{30}\] \[0<\rho_{2}(\mathbf{x}_{ik},t)<2\delta_{4}\]  
Substituting (29) into (28) yields  
\[\left[ \begin{array}{c}\dot{\mathbf{Z}}_{1}\\ \dot{\mathbf{Z}}_{2}\end{array} \right]=\overline{\mathbf{A}}\left[ \begin{array} []{c}\mathbf{Z}_{1}\\ \mathbf{Z}_{2}\end{array} \right] \tag{31}\]  
Here, \(\overline{\mathbf{A}}=\frac{1}{2\|\mathbf{Z}_{1}\|_{\infty}}\left[ \begin{array} []{cc}-\alpha_{k}\mathbf{b}_{1}+\rho_{1}(\mathbf{x}_{ik},t)\mathbf{I}_{2}& \mathbf{I}_{2}\\ -\beta_{k}\mathbf{b}_{1}+\rho_{2}(\mathbf{x}_{ik},t)\mathbf{I}_{2}&\mathbf{O}_ {2}\end{array} \right]\).