## 3 Controller Design and Stability Analysis
### Stability Analysis  
In Section 3.1, error transformation functions (4), (12) and (20) are applied to convert the virtual errors to each surface error \(\zeta_{i}\). In this section, the Lyapunov stability theory is used to prove that the error signal is bounded by semi-global stability. Considering the introduction of the first-order filter, which is used to eliminate the \"explosion of complexity\", the error generated by filter is defined as follows:  
\[y_{i}=x_{id}-\overline{x_{i}},i=2,3 \tag{27}\]  
Invoking (11), (19) and (27) yields  
\[\dot{x_{id}}=-\frac{x_{id}-\overline{x_{i}}}{\tau_{i}}=-\frac{y_{i}}{\tau_{i }},i=2,3 \tag{28}\]  
Define the estimation error of the RBF weight  
\[\widetilde{w}_{i}=\dot{\overline{w}_{i}}-w_{i},i=1,2,3 \tag{29}\]  
Then, the differentiation of each surface error is obtained. By substituting (6)-(9) and (12), (27), (29) into (5), (5) can be rewritten as  
\[ \begin{array}{ll}\dot{\zeta}_{1}&=h_{1}(w_{1}^{*T}\zeta_{1}(x_{1})+\sigma_{1}^{* }+\zeta_{2}+y_{2}+\overline{x}_{2}+(\Delta_{1}(x,t)-\dot{x}_{1d})/h_{1})\\ &=h_{1}(w_{1}^{T}\mu_{1}+\sigma_{1}^{*}+\zeta_{2}+y_{2}-{(\stackrel{{ \wedge}}{{w_{1}}})}^{T}\mu_{1})+\Delta_{1}-c_{1}\xi_{1}-\frac{\rho_{1}^{2} \zeta_{1}}{2\varepsilon}\\ &=h_{1}(\zeta_{2}+y_{2}+\sigma_{1}^{*}-{(\stackrel{{ \sim}}{{w_{1}}})}^{T}\mu_{1})-c_{1}\zeta_{1}-\frac{\rho_{1}^{2}\zeta_{1}}{2 \varepsilon}+\Delta_{1}\end{array}  \tag{30}\]  
By substituting (14)-(17) and (20), (27), (29) into (13), (13) can be rewritten as  
\[ \begin{array}{ll}\dot{\zeta}_{2}&=w_{2}^{*T}\xi_{2}(x_{1},x_{2})+\sigma_{2}^ {*}+\zeta_{3}+y_{3}+\overline{x_{3}}+\Delta_{2}(x,t)-\dot{x}_{2d}\\ &=w_{2}^{T}\mu_{2}+\sigma_{2}^{*}+\zeta_{3}+y_{3}-{(\stackrel{{ \wedge}}{{w_{2}}})}^{T}\mu_{2}+\Delta_{2}-c_{2}\xi_{2}-\frac{\rho_{2}^{2} \zeta_{2}}{2\varepsilon}\\ &=\zeta_{3}+y_{3}+\sigma_{2}^{*}-{(\stackrel{{\sim}}{{w_{2}}})}^ {T}\mu_{2}-c_{2}\xi_{2}-\frac{\rho_{2}^{2}\zeta_{2}}{2\varepsilon}+\Delta_{2 }\end{array}  \tag{31}\]  
By substituting (22)-(25) and (27), (29) into (21), (21) can be rewritten as  
\[ \begin{array}{ll}\dot{\zeta}_{3}&=h_{3}[w_{3}^{*T}\xi_{3}(x_{2},x_{3})+ \sigma_{3}^{*}+u+(\Delta_{3}-\dot{x}_{3d})/h_{3}]\\ &=h_{3}[w_{3}^{T}\mu_{3}+\sigma_{3}^{*}-{(\stackrel{{\wedge}}{{w_ {3}}})}^{T}\mu_{3}+(\Delta_{3}-c_{3}\zeta_{3}-\frac{\rho_{2}^{2}\zeta_{3}}{2 \varepsilon})/h_{3}]\\ &=h_{3}(-{(\stackrel{{\sim}}{{w_{3}}})}^{T}\mu_{3}+\sigma_{3}^{ *})-c_{3}\zeta_{3}-\frac{\rho_{2}^{2}\zeta_{3}}{2\varepsilon}+\Delta_{3}\end{array}  \tag{32}\]  
Combining (9), (17), (27) and (28), the differentiation of filter error \(y_{i}\) stratifies  
\[\left\{ \begin{array}{ll}\dot{y}_{2}+\frac{y_{2}}{\overline{\
u}_{2}}={( \stackrel{{\wedge}}{{w_{1}}})}^{T}\mu_{1}+{(\stackrel{{ \wedge}}{{w_{1}}})}^{T}\dot{\mu}_{1}\\ \dot{y}_{3}+\frac{y_{3}}{\overline{\
u}_{3}}=-\dot{x}_{2d}+c_{2}\dot{\zeta}_{ 2}+\frac{\rho_{2}^{2}\dot{\zeta}_{2}}{2\varepsilon}+{(\stackrel{{ \wedge}}{{w_{2}}})}^{T}\mu_{2}+{(\stackrel{{\wedge}}{{w_{2}}})}^ {T}\dot{\mu}_{2})\end{array} \right. \tag{33}\]  
Reconsidering (27) and combining adaptive law (10), (18), (26) and (27)-(31), it can be seen that there are continuous non-negative functions \(\mathrm{B}_{2},\mathrm{B}_{3}\) that satisfy  
\[\left\{ \begin{array}{ll}\left|\dot{y}_{2}+\frac{y_{2}}{\overline{\
u}_{2}} \right|&\leq\mathrm{B}_{2}(\zeta_{1},\zeta_{2},y_{2},\widetilde{w}_{1},x_{1d},\dot{x}_{1d},\dot{x}_{1d})\\ \left|\dot{y}_{3}+\frac{y_{3}}{\overline{\
u}_{3}}\right|&\leq\mathrm{B}_{3}( \zeta_{1},\zeta_{2},\zeta_{3},y_{2},y_{3},\widetilde{w}_{1},\widetilde{w}_{2},x_{1d},\dot{x}_{1d},\dot{x}_{1d})\end{array} \right.