## 3 Controller Design and Stability Analysis
### Stability Analysis
\tag{34}\]  
Considering (34), the following can be obtained:  
\[\dot{y}_{i}+\frac{y_{i}}{\overline{\tau}_{i}}\leq\mathrm{B}_{i}\Rightarrow \dot{y}_{i}\leq\mathrm{B}_{i}-\frac{y_{i}}{\overline{\tau}_{i}}\Rightarrow y_{ i}\dot{y}_{i}\leq\mathrm{B}_{i}|y_{i}|-\frac{y_{i}^{2}}{\overline{\tau}_{i}} \tag{35}\]  
Now, design the following candidate Lyapunov function for the controller  
\[V=V_{1}+V_{2}+V_{3} \tag{36}\]  
where \(V_{1},V_{2},V_{3}\) are defined as follows  
\[V_{1}=\frac{1}{2}(y_{2}^{2}+y_{3}^{2}),V_{2}=\frac{1}{2}(\zeta_{1}^{2}+\zeta_{2 }^{2}+\zeta_{3}^{2}),V_{3}=\frac{1}{2}(h_{1}\widetilde{w}_{1}^{T}\Gamma_{1} \widetilde{w}_{1}+\widetilde{w}_{2}^{T}\Gamma_{2}\widetilde{w}_{2}+h_{3} \widetilde{w}_{3}^{T}\Gamma_{3}\widetilde{w}_{3}) \tag{37}\]  
**Theorem 1. **_Consider the closed-loop system established by (2) and (25). If assumptions 1-3 are true and for an arbitrary positive constant \(p\), inequality \(V(0)\leq p\) holds, then there are parameters \(\tau_{2},\tau_{3}\) and \(c_{i},\eta_{i},\Gamma_{i},i=1,2,3\) that ensure all signals of the closed-loop system above are semi-global bounded, and the tracking error of the system can converge to any small residual set [29]._  
**Proof. ** Differentiating (37) \(V_{1}\) with respect to time and invoking (35), it has the following inequality:  
\[\dot{V}_{1}\leq\mathrm{B}_{2}|y_{2}|+\mathrm{B}_{3}|y_{3}|-\frac{y_{2}^{2}}{ \tau_{2}}-\frac{y_{3}^{2}}{\tau_{3}} \tag{38}\]  
Differentiating (37) \(V_{2}\) with respect to time and considering (30), (31) and (32), it has  
\[\dot{V}_{2} =\zeta_{1}\bigg{[}h_{1}(\zeta_{2}+y_{2}+\sigma_{1}^{*}-w_{1}^{ \widetilde{\top}}\mu_{1})-c_{1}\zeta_{1}-\frac{\rho_{1}^{2}\zeta_{1}}{2\varepsilon }+\Delta_{1}\bigg{]}+\zeta_{2}\bigg{[}\zeta_{3}+y_{3}+\sigma_{2}^{*}-\widetilde {w_{2}^{\widetilde{\top}}}\mu_{2}-c_{2}\zeta_{2}-\frac{\rho_{2}^{2}\zeta_{2}} {2\varepsilon}+\Delta_{2}\bigg{]} \tag{39}\] \[+\zeta_{3}\bigg{[}h_{3}(-\widetilde{w_{3}^{\widetilde{\top}}}\mu_{ 3}+\sigma_{3}^{*})-c_{3}\zeta_{3}-\frac{\rho_{3}^{2}\zeta_{3}}{2\varepsilon}+ \Delta_{3}\bigg{]}\]  
Differentiating (37) \(V_{3}\) with respect to time and considering (10), (18) and (26), it has  
\[\dot{V}_{3}=h_{1}(\widetilde{w_{1}^{\widetilde{\top}}}\mu_{1}\zeta_{1}- \widetilde{w_{1}^{\widetilde{\top}}}\eta_{1}\widetilde{w_{1}^{\widetilde{ \bot}}})+(\widetilde{w_{2}^{\widetilde{\top}}}\mu_{2}\zeta_{2}-\widetilde{w_{ 2}^{\widetilde{\top}}}\eta_{2}\widetilde{w_{2}^{\widetilde{\bot}}})+h_{3}( \widetilde{w_{3}^{\widetilde{\top}}}\mu_{3}\zeta_{3}-\widetilde{w_{3}^{ \widetilde{\top}}}\eta_{3}\widetilde{w_{3}^{\widetilde{\bot}}}) \tag{40}\]  
Assumption 3, when \(V\leq p\), defines compact set \(\Omega=\Omega_{1}\times\Omega_{2}\), where  
\[\Omega_{1}=\left\{(x_{1d},\dot{x}_{1d},\dot{x}_{1d}):x_{1d}^{2}\ +\ \dot{x_{1d}^{2}}\ +\ \dot{x_{1d}^{2}}\ \leq \Theta\right\} \tag{41}\]  
\[\Omega_{2}=\left\{\sum_{i=1}^{2}y_{i}^{2}+\sum_{i=1}^{3}\zeta_{i}^{2}+h_{1} \widetilde{w_{1}^{\widetilde{\top}}}T_{1}^{-1}\widetilde{w_{1}}+h_{3} \widetilde{w_{3}^{\widetilde{\top}}}T_{1}^{-1}\widetilde{w_{3}}\leq 2p\right\} \tag{42}\]  
\(\mathrm{B}_{2},\mathrm{B}_{3}\) have maxima \(M_{2},M_{3}\) in compact set \(\Omega\). Meanwhile, notice that there is equality  
\[-\frac{\rho_{i}^{2}\zeta_{i}^{2}}{2\varepsilon}\leq\frac{\varepsilon}{2}-\rho_ {i}|\zeta_{i}|\leq\frac{\varepsilon}{2}-\Delta_{i}\zeta_{i}\Rightarrow-\frac {\rho_{i}^{2}\zeta_{i}^{2}}{2\varepsilon}+\Delta_{i}\zeta_{i}\leq\frac{ \varepsilon}{2} \tag{43}\]  
To eliminate the common term \(\widetilde{w_{i}^{\widetilde{\top}}}\mu_{i}\zeta_{i}\), combining (39) and (40), it has  
\[V=V_{1}+V_{2}+V_{3} \tag{36}\]0 \[V=V_{1}+V_{2}+V_{3} \tag{36}\]1  
Substituting (43) into (44) yields  
\[V=V_{1}+V_{2}+V_{3} \tag{36}\]2 \[V=V_{1}+V_{2}+V_{3} \tag{36}\]3  
Considering the following inequality  
\[V=V_{1}+V_{2}+V_{3} \tag{36}\]4  
\(\square\)  
**Lemma 1.