## 3 Controller Design and Stability Analysis
### Stability Analysis
** (_Young's Inequality_) _Set the real-valued \(p>1,q>1\), and \(\frac{1}{p}+\frac{1}{q}=1\), for arbitrary non-negative constant \(a\) and \(b\), there exists a relationship_  
\[V=V_{1}+V_{2}+V_{3} \tag{36}\]5  
The equality holds if and only if \(a^{p}=b^{q}\). According Lemma 1 and combining (36)-(46), \(\dot{V}\) can be calculated as follows:\[ \begin{array}{ll}\dot{V}&=\dot{V}_{1}+\dot{V}_{2}+\dot{V}_{3}\\ &\leq\mathrm{B}_{2}|y_{2}|+\mathrm{B}_{3}|y_{3}|-\frac{\dot{y}_{2}^{2}}{\tau_{2} }-\frac{\dot{y}_{3}^{2}}{\tau_{3}}+h_{1}\xi_{1}(\xi_{2}+y_{2}+\sigma_{1}^{*})- \iota_{1}\xi_{1}^{2}+\zeta_{2}(\zeta_{3}+y_{3}+\sigma_{2}^{*})-\varsigma_{2} \xi_{2}^{2}+h_{3}\zeta_{3}\sigma_{3}^{*}-\varsigma_{3}\xi_{3}^{2}\\ &-(h_{1}\widetilde{w}_{1}^{T}\widetilde{w}_{1}^{T}\widetilde{w}_{1}^{\bot}+ \widetilde{w}_{2}^{T}\widetilde{w}_{2}^{\bot}+h_{3}\widetilde{w}_{1}^{T} \widetilde{w}_{3}^{\bot})+\frac{3\kappa}{2}\\ &\leq\mathrm{B}_{2}|y_{2}|+\mathrm{B}_{3}|y_{3}|-\frac{\dot{y}_{2}^{2}}{\tau_ {3}}-\frac{\dot{y}_{3}^{2}}{\tau_{3}}-\frac{3}{\tau_{3}}\sum\limits_{i=1}^{3} \varsigma_{i}\xi_{i}^{2}+h_{1}(|\zeta_{1}||\zeta_{2}|+|\zeta_{1}||y_{2}|+| \zeta_{1}||\sigma_{1}^{*}|)+(|\zeta_{2}||\zeta_{3}|+|\zeta_{2}||y_{3}|+|\zeta_ {2}||\sigma_{2}^{*}|)+h_{3}|\xi_{3}||\sigma_{3}^{*}|\\ &-\frac{h_{1}\eta_{1}}{2}(\left\|\widetilde{w}_{1}\right\|^{2}-\|w_{1}\|^{2}) -\frac{\eta_{2}}{2}(\left\|\widetilde{w}_{2}\right\|^{2}-\|w_{2}\|^{2})-\frac {h_{2}\eta_{3}}{2}(\left\|\widetilde{w}_{3}\right\|^{2}-\|w_{3}\|^{2})+\frac{3 \kappa}{2}\end{array}  \tag{48}\]  
Allowing for the fact that  
\[\mathrm{B}_{i}|y_{i}|\leq\frac{\mathrm{B}_{i}^{2}y_{i}^{2}}{2\varepsilon}+ \frac{\varepsilon}{2}),i=2,3 \tag{49}\]  
With (47), (48) can be rewritten as follows:  
\[ \begin{array}{ll}\dot{V}&\leq\sum\limits_{i=2}^{3}\left(\frac{\mathrm{B}_{i }^{2}y_{i}^{2}}{2\varepsilon}+\frac{\xi}{2}-\frac{\dot{y}_{i}^{2}}{\tau_{3}} \right)-\sum\limits_{i=1}^{3}c_{i}\zeta_{i}^{2}+\frac{h_{1}}{2}\left[(\zeta_{ 1}^{2}+\zeta_{2}^{2})+(\zeta_{1}^{2}+y_{2}^{2})+(\zeta_{1}^{2}+\sigma_{1}^{* 2})\right]\\ &+\frac{1}{2}\left[(\zeta_{2}^{2}+\zeta_{3}^{2})+(\zeta_{2}^{2}+y_{3}^{2})+( \zeta_{2}^{2}+\sigma_{2}^{*2})\right]+\frac{h_{2}}{2}(\zeta_{3}^{2}+\sigma_{ 3}^{*2})\\ &-\frac{h_{1}\eta_{1}}{2}(\left\|\widetilde{w}_{1}\right\|^{2}-\|w_{1}\|^{2}) -\frac{\eta_{2}}{2}(\left\|\widetilde{w}_{2}\right\|^{2}-\|w_{2}\|^{2})-\frac {h_{2}\eta_{3}}{2}(\left\|\widetilde{w}_{3}\right\|^{2}-\|w_{3}\|^{2})+\frac{3 \kappa}{2}\\ &\leq(\frac{\mathrm{B}_{2}^{2}}{2\varepsilon}+\frac{h_{2}}{2}-\frac{1}{2}) \vartheta_{2}^{*}+(\frac{\mathrm{B}_{2}^{2}}{2\varepsilon}+\frac{1}{2}-\frac{ 1}{2})\vartheta_{3}^{*}+(\frac{3h_{1}}{2}-\iota_{1})\zeta_{1}^{2}+(\frac{h_{ 2}}{2}+\frac{3}{2}-\iota_{2})\zeta_{2}^{2}+(\frac{h_{2}}{2}+\frac{1}{2}- \varsigma_{3})\zeta_{3}^{2}+\frac{h_{1}}{2}\sigma_{1}^{*2}+\frac{1}{2}\sigma_ {2}^{*2}+\frac{h_{2}}{2}\sigma_{3}^{*2}\\ &-\frac{h_{1}\eta_{1}}{2\lambda_{\mathrm{max}}(\widetilde{\Gamma}_{1}^{-1})} \widetilde{w}_{1}^{T}\Gamma_{1}^{-1}\widetilde{w}_{1}^{\widetilde{w}_{1}}- \frac{\eta_{2}}{2\lambda_{\mathrm{max}}(\widetilde{\Gamma}_{1}^{-1})}\widetilde {w}_{2}^{T}\Gamma_{2}^{-1}\widetilde{w}_{2}-\frac{h_{2}\eta_{1}}{2\lambda_{ \mathrm{max}}(\widetilde{\Gamma}_{1}^{-1})}\widetilde{w}_{3}^{\widetilde{T}} \Gamma_{3}^{-1}\widetilde{w}_{3}^{\widetilde{T}}\Gamma_{3}^{-1}\widetilde{w}_{ 3}^{\widetilde{\eta}}+\frac{h_{1}\eta_{1}}{2}\|w_{1}\|^{2}+\frac{\eta_{2}}{2}\|w_ {2}\|^{2}+\frac{h_{2}\eta_{1}}{2}\|w_{3}\|^{2}+\frac{\kappa}{2}\end{array}  \tag{50}\]  
Based on the above analysis, the conditions that the design control parameters should meet are as follows:  
\[\left\{ \begin{array}{ll}\frac{1}{\tau_{2}}\geq\frac{M_{2}^{2}}{2\varepsilon}+ \frac{h_{1M}}{2}+\vartheta\\ \frac{1}{\tau_{3}}\geq\frac{M_{3}^{2}}{2\varepsilon}+\frac{1}{2}+\vartheta\\ \end{array} \right.\quad\left\{ \begin{array}{ll}c_{1}\geq\frac{3h_{1M}}{2 }+\vartheta\\ c_{2}\geq\frac{h_{1M}}{2}+\frac{3}{2}+\vartheta\\ c_{3}\geq\frac{h_{3M}}{2}+\frac{1}{2}+\vartheta\\ \end{array} \right.\quad\left\{