## 3 Controller Design
### Stability Analysis
Then, we can obtain Therefore, we have the following inequality:  
\[-a_{1}\Big{(}{s_{1}}^{2}\Big{)}^{\frac{{{\tau_{1}}+1}}{2}}-c_{1}\Big{(}{s_{1}}^{2} \Big{)}^{\frac{{{\tau_{1}}+1}}{2}}+\frac{1}{2}{s_{1}}^{2}\leq-\bigg{(}a_{1}- \frac{1}{2}\bigg{)}\Big{(}{s_{1}}^{2}\Big{)}^{\frac{{{\tau_{1}}+1}}{2}}-\bigg{(} c_{1}-\frac{1}{2}\bigg{)}\Big{(}{s_{1}}^{2}\Big{)}^{\frac{{{\tau_{2}}+1}}{2}} \tag{53}\]  
\[-\frac{1}{\tau}\Big{(}{y_{1}}^{2}\Big{)}^{\frac{{{\tau_{1}}+1}}{2}}-\frac{1}{ \tau}\Big{(}{y_{1}}^{2}\Big{)}^{\frac{{{\tau_{2}}+1}}{2}}+{y_{1}}^{2}\leq- \bigg{(}\frac{1}{\tau}-1\bigg{)}\Big{(}{y_{1}}^{2}\Big{)}^{\frac{{{\tau_{1}}+1 }}{2}}-\bigg{(}\frac{1}{\tau}-1\bigg{)}\Big{(}{y_{1}}^{2}\Big{)}^{\frac{{{\tau _{2}}+1}}{2}}, \tag{54}\]  
Substituting Equations (53) and (54) into Equation (50), we obtain  
\[ \begin{array}{l}\dot{V}_{1}\leq-\Big{(}a_{1}-\frac{1}{2}\Big{)}\big{(}{s_{1 }}^{2}\Big{)}^{\frac{{{\tau_{1}}+1}}{2}}-\Big{(}c_{1}-\frac{1}{2}\Big{)}\big{(} {s_{1}}^{2}\Big{)}^{\frac{{{\tau_{2}}+1}}{2}}-a_{2}\big{(}{s_{2}}^{2}\Big{)}^ {\frac{{{\tau_{1}}+1}}{2}}-c_{2}\big{(}{s_{2}}^{2}\big{)}^{\frac{{{\tau_{2}}+1 }}{2}}\\ -\Big{(}\frac{1}{\tau}-1\Big{)}\big{(}{y_{1}}^{2}\Big{)}^{\frac{{{\tau_{1}}+1 }}{2}}-\Big{(}\frac{1}{\tau}-1\Big{)}\big{(}{y_{1}}^{2}\big{)}^{\frac{{{\tau_{ 2}}+1}}{2}}+\frac{1}{2}{\xi_{{\omega_{z}}}}^{2}\\ \leq-M\bigg{[}\big{(}{s_{1}}^{2}\big{)}^{\frac{{{\tau_{1}}+1}}{2}}+\big{(}{s_{ 2}}^{2}\big{)}^{\frac{{{\tau_{1}}+1}}{2}}-N\bigg{[}\big{(}{s_{1}}^{2}\big{)}^{ \frac{{{\tau_{2}}+1}}{2}}+\big{(}{s_{2}}^{2}\big{)}^{\frac{{{\tau_{2}}+1}}{2} }\bigg{]}\\ -K\big{(}{y_{1}}^{2}\big{)}^{\frac{{{\tau_{1}}+1}}{2}}-K\big{(}{y_{1}}^{2} \big{)}^{\frac{{{\tau_{2}}+1}}{2}}+\frac{1}{2}{\xi_{{\omega_{z}}}}^{2}\end{array}  \tag{55}\]  
where \(M=\min\Big{(}a_{2},a_{1}-\frac{1}{2}\Big{)}\), \(N=\min\Big{(}c_{2},c_{1}-\frac{1}{2}\Big{)}\), \(K=\frac{1}{\tau}-1\). **Assumption 2**. : The control parameters are selected appropriately so that \(M\), \(N\), \(K\) are positive. Since \(0<\frac{{{\tau_{1}}+1}}{2}<1\) and \(\frac{{{\tau_{2}}+1}}{2}>1\) hold, according to Lemma 4, for the first two terms in Equation (55), we have  
\[-\bigg{[}\big{(}{s_{1}}^{2}\big{)}^{\frac{{{\tau_{1}}+1}}{2}}+\big{(}{s_{2}}^{2 }\big{)}^{\frac{{{\tau_{1}}+1}}{2}}\bigg{]}\leq-\Big{(}{s_{1}}^{2}+{s_{2}}^{2 }\Big{)}^{\frac{{{\tau_{1}}+1}}{2}} \tag{56}\]  
\[-\bigg{[}\big{(}{s_{1}}^{2}\big{)}^{\frac{{{\tau_{2}}+1}}{2}}+\big{(}{s_{2}}^{ 2}\big{)}^{\frac{{{\tau_{2}}+1}}{2}}\bigg{]}\leq-2^{\frac{{{\tau_{2}}+1}}{2}} \Big{(}{s_{1}}^{2}+{s_{2}}^{2}\Big{)}^{\frac{{{\tau_{2}}+1}}{2}} \tag{57}\]  
Thus, we can obtain  
\[ \begin{array}{l}\dot{V}_{1}\leq-M\big{(}{s_{1}}^{2}+{s_{2}}^{2}\big{)}^{ \frac{{{\tau_{1}}+1}}{2}}-2^{\frac{{{\tau_{2}}}}{2}}N\big{(}{s_{1}}^{2}+{s_{2} }^{2}\big{)}^{\frac{{{\tau_{2}}+1}}{2}}-K\big{(}{y_{1}}^{2}\big{)}^{\frac{{{ \tau_{1}}+1}}{2}}-K\big{(}{y_{1}}^{2}\big{)}^{\frac{{{\tau_{2}}+1}}{2}}+\frac{1 }{2}{\xi_{{\omega_{z}}}}^{2}\\ \leq-\min(M,K\bigg{[}\big{(}{s_{1}}^{2}+{s_{2}}^{2}\big{)}^{\frac{{{\tau_{2}}+1 }}{2}}+\big{(}{y_{1}}^{2}\big{)}^{\frac{{{\tau_{1}}+1}}{2}}\bigg{]}\leq-2^{ \frac{{{\tau_{2}}+1}}{2}}{\xi_{{\omega_{z}}}}^{2}\\ \end{array}  \tag{58}\]  
For the terms with the same power in Equation (58), using Lemma 4, we have  
\[-\bigg{[}\big{(}{s_{1}}^{2}+{s_{2}}^{2}\big{)}^{\frac{{{\tau_{1}}+1}}{2}}+ \big{(}{y_{1}}^{2}\big{)}^{\frac{{{\tau_{1}}+1}}{2}}\bigg{]}\leq-\Big{(}{s_{1}}^{ 2}+{s_{2}}^{2}+{y_{1}}^{2}\Big{)}^{\frac{{{\tau_{1}}+1}}{2}} \tag{59}\]  
\[-\bigg{[}\big{(}{s_{1}}^{2}+{s_{2}}^{2}\big{)}^{\frac{{{\tau_{1}}+1}}{2}}+ \big{(}{y_{1}}^{2}\big{)}^{\frac{{{\tau_{2}}+1}}{2}}\bigg{]}\leq-2^{\frac{{{ \tau_{1}}+1}}{2}}\Big{(}{s_{1}}^{2}+{s_{2}}^{2}+{y_{1}}^{2}\Big{)}^{\frac{{{\tau_{ 2}}+1}}{2}} \tag{60}\]  
Then, we can obtain  
\[