## 3 Control Design
### Stability Analysis of the Closed-Loop Control System
\begin{cases}-\xi\sqrt{\frac{\gamma_{1}}{2}}&\text{if } \alpha_{k}>\alpha_{m}\\ \eta&\text{if }\alpha_{k}<\alpha_{m}\end{cases}  \tag{47}\]  
Meanwhile, picking up \(\varepsilon=\frac{\xi_{2}}{2\xi_{1}}\sqrt{\frac{\gamma_{2}}{\gamma_{1}}}\) in (14) and substituting it into the time derivative of \(\beta_{k}=2\varepsilon\alpha_{k}\) yields  
\[\dot{\beta}_{k}=\left\{ \begin{array}{ll}-\xi_{2}\sqrt{\frac{\gamma_{2}}{2} }&\text{if }\alpha_{k}>\alpha_{m}\\ \frac{\xi_{2}}{\xi_{1}}\sqrt{\frac{\gamma_{2}}{\gamma_{1}}}\eta&\text{if } \alpha_{k}<\alpha_{m}\end{array} \right. \tag{48}\]  
Consequently, (42) becomes  
\[\dot{V}\leq\left\{ \begin{array}{ll}-\eta_{0}V^{1/2}+2|\varepsilon_{\alpha}| \frac{\xi_{1}}{\sqrt{2\gamma_{1}}}+2|\varepsilon_{\beta}|\frac{\xi_{2}}{ \sqrt{2\gamma_{2}}}&\text{if }\alpha_{k}>\alpha_{m}\\ -\eta_{0}V^{1/2}-|\varepsilon_{\alpha}|(\frac{\eta}{\gamma_{1}}-\frac{\xi_{1 }}{\sqrt{2\gamma_{1}}})-|\varepsilon_{\beta}|(\frac{\xi_{2}}{\xi_{1}}\frac{ \eta}{\sqrt{\gamma_{1}\gamma_{2}}}-\frac{\xi_{2}}{\sqrt{2\gamma_{2}}})&\text{if } \alpha_{k}<\alpha_{m}\end{array} \right. \tag{49}\]  
Equation (49) indicates that the sign of the time derivative of \(V\) is indefinite so that \(\|\mathbf{s}_{ik}\|_{2}\) may become larger than \(\mu\) with the decrease of \(\alpha_{k}\) and \(\beta_{k}\). Once \(\|\mathbf{s}_{ik}\|_{2}\) becomes greater than \(\mu\), the condition defined in Case 1 will be immediately triggered. In (46), the time derivative of \(V\) is negative so that the closed-looped control system possesses the inherent stability and \(\mathbf{s}_{ik}\) will enter the domain \(\|\mathbf{s}_{ik}\|_{2}<\mu\) again in finite time. This process continues back and forth until the control system becomes convergent. In this process, \(\mathbf{s}_{ik}\) may deviate from the domain for a finite time but there always exists another domain in the real sliding modes of \(\mathbf{s}_{ik}\). \[\|\mathbf{s}_{ik}\|_{2}\leq\eta_{1}\ (\eta_{1}>\mu) \tag{50}\]  
Inside the domain \(\|\mathbf{s}_{ik}\|_{2}<\mu\), the value of \(\|\mathbf{s}_{ik}\|_{2}\) can estimated from (19) and (24). \[\|\mathbf{s}_{ik}\|_{2}\leq[(1-\gamma_{1})\alpha_{k}(t_{1})+\delta_{1}]\mu^{ 1/2}+[\varepsilon(1-\gamma_{1})\alpha_{k}(t_{1})+\delta_{4}](t_{2}-t_{1})= \overline{\eta}_{2} \tag{51}\]  
Here, \(t_{1}\) is the time instant when \(\mathbf{s}_{ik}\) enters the domain \(\|\mathbf{s}_{ik}\|_{2}\leq\mu\) and \(t_{2}\) is the moment when \(\mathbf{s}_{ik}\) leaves the domain. Once \(\|\mathbf{s}_{ik}\|_{2}\) becomes \(\mu<\|\mathbf{s}_{ik}\|_{2}<\eta_{1}\), we have  
\[\|\mathbf{s}_{ik}\|_{2}\leq(1+\gamma_{1})\Big{(}\eta_{1}^{1/2}+\varepsilon \Big{)}\bigg{(}\alpha_{k}(t_{2})+\xi_{1}\sqrt{\frac{\eta_{1}\gamma_{1}}{2}}+ \varepsilon\bigg{)}(t_{3}-t_{2})+\delta_{1}\eta_{1}^{1/2}+\delta_{4}(t_{3}-t_{ 2})=\overline{\eta}_{2} \tag{52}\]  
Here, \(t_{2}\) is the time instant when \(\mathbf{s}_{ik}\) leaves the domain \(\|\mathbf{s}_{ik}\|_{2}<\mu\) and \(t_{3}\) is the moment when \(\mathbf{s}_{ik}\) enter the domain \(\|\mathbf{s}_{ik}\|_{2}<\mu\). Subsequently, (53) can be drawn from (51) and (52). \[\|\mathbf{s}_{ik}\|_{2}\leq\max\Bigl{(}\overline{\eta}_{2},\ \overline{\eta}_{2}\Bigr{)}=\eta_{2} \tag{53}\]  
From (50) and (52), there exist the real sliding modes, described by  
\[\Omega=\left\{ \begin{array}{ll}\mathbf{s}_{ik},\dot{\mathbf{s}}_{ik}:&\| \mathbf{s}_{ik}\|_{2}\leq\eta_{1}&\|\dot{\mathbf{s}}_{ik}\|_{2}\leq\eta_{2}& \eta_{1}>\mu\end{array} \right\} \tag{54}\]  
The existences of the sliding modes in (52) can be presented in theory, but \(\eta_{1}\) and \(\eta_{2}\) cannot be obtained in advance before a real control process is carried out. Further, a solution of (24) in the domain \(\mu<\|\mathbf{s}_{ik}\|_{2}\leq\eta_{1}\) can be gotten as  
\[\alpha_{k}=\alpha_{k}(0)+\xi_{1}t\sqrt{\frac{\gamma_{1}}{2}}\ 0\leq t\leq t_{F} \tag{55}\]  
Equation (54) indicates that the adaptive gain \(\alpha_{k}\) is bounded so that the gain \(\beta_{k}\) is bounded on account of \(\beta_{k}=2\varepsilon\alpha_{k}\).