## 4 Controller Design
\begin{array}{ll}z_{1}(\theta_{1}^{*T}\varphi_{1}(x_{1})-\theta_{1}^{*T} \varphi_{1}(\widetilde{x}_{1})-w_{1}^{*T}\varphi_{1}(\xi,\widetilde{x}_{1})) \leq\frac{3}{2}z_{1}^{2}+\|\theta_{1}^{*}\|^{2}+\frac{1}{2}\|w_{1}^{*}\|^{2} \\ \\ (z_{1}-\hat{z}_{1})\widetilde{\theta}_{1}^{T}\varphi_{1}(\widetilde{x}_{1}) +(z_{1}-\hat{z}_{1})\widetilde{w}_{1}^{T}\varphi_{1}(\xi,\widetilde{x}_{1})\\ \\ \leq(\widetilde{w}_{1}^{T}\varphi_{1}(\xi,\widetilde{x}_{1})-w_{1}^{*T}\varphi _{1}(\xi,\widetilde{x}_{1})-\epsilon_{10})(\widetilde{w}_{1}^{T}+\widetilde{ \theta}_{1}^{T})\varphi_{1}(\xi,\widetilde{x}_{1})\\ \\ =(w_{1}^{*T}\varphi_{1}(\xi,\widetilde{x}_{1})-\widetilde{w}_{1}^{T}\varphi _{1}(\xi,\widetilde{x}_{1})-w_{1}^{*T}\varphi_{1}(\xi,\widetilde{x}_{1})- \varepsilon_{10})(\widetilde{w}_{1}^{T}+\widetilde{\theta}_{1}^{T})\varphi_{1 }(\xi,\widetilde{x}_{1})\\ \\ \leq(2\left\|w_{1}^{*}\right\|+\left\|\widetilde{w}_{1}\right\|-\varepsilon_{10 })(\left\|\widetilde{w}_{1}\right\|+\left\|\widetilde{\theta}_{1}^{*}\right\| \right)\\ \\ \leq\left\|w_{1}^{*}\right\|^{2}+\left\|\widetilde{w}_{1}\right\|^{2}+2\left\| \widetilde{\theta}_{1}\right\|^{2}+\left\|\varepsilon_{10}^{*}\right\|^{2} \end{array}  \tag{34}\]  
Substituting (30)-(32) into (12) yields  
\[ \begin{array}{ll}V_{1}&\leq z_{1}(3z_{1}+\theta_{1}^{T}\varphi_{1}( \widetilde{x}_{1})+\widetilde{w}_{1}^{T}\varphi_{1}(\xi,\widetilde{x}_{1})+ \beta_{1})+\frac{1}{2}z_{2}^{2}+\frac{1}{2}\xi_{2}^{2}\\ &+2\left\|\widetilde{\theta}_{1}\right\|^{2}+\left\|\widetilde{w}_{1}\right\|^ {2}+\frac{1}{p_{1}}\widetilde{\theta}_{1}^{T}(p_{1}\hat{z}_{1}\varphi_{1}( \widetilde{x}^{{}^{\prime}}_{1})-\dot{\theta}_{1})+\left\|\theta_{1}^{*}\right\| ^{2}\\ &+\frac{1}{q_{1}}\widetilde{w}_{1}^{T}(q_{1}\hat{z}_{1}\varphi_{1}(\xi, \widetilde{x}_{1})-\dot{w}_{1})+\frac{3}{2}\|w_{1}^{*}\|^{2}+\frac{1}{2}\| \varepsilon_{1}^{*}\|^{2}+\left\|\varepsilon_{10}^{*}\right\|^{2}\end{array}  \tag{35}\]  
Substituting (14)-(16) into (35) yields  
\[ \begin{array}{ll}V_{1}&\leq z_{1}(-c_{1}\hat{z}_{1}+3z_{1}-3\hat{z}_{1})+ \frac{1}{2}z_{2}^{2}+\frac{1}{2}\xi_{2}^{2}+2\left\|\widetilde{\theta}_{1} \right\|^{2}+\left\|\widetilde{w}_{1}\right\|^{2}\\ &+\frac{c_{1}}{p_{1}}\widetilde{\theta}_{1}^{T}\hat{\theta}_{1}+\frac{\pi_{1}} {q_{1}}\widetilde{w}_{1}^{T}\hat{w}_{1}+\left\|\theta_{1}^{*}\right\|^{2}+ \frac{3}{2}\left\|w_{1}^{*}\right\|^{2}+\frac{1}{2}\left\|\varepsilon_{1}^{*} \right\|^{2}+\left\|\varepsilon_{10}^{*}\right\|^{2}\end{array}  \tag{36}\]  
According to (45) and (46), we have  
\[ \begin{array}{ll}\dot{V}_{s}&\leq-\sum\limits_{k=1}^{s}{(c_{k}-3)z_{k}^{2}+ \frac{1}{2}z_{s+1}^{2}}-\sum\limits_{k=2}^{s}{(\frac{1}{\tau_{k}}-N_{k}^{2}- \frac{1}{2})\xi_{k}^{2}}+\sum\limits_{k=1}^{s}{(\frac{\sigma_{k}}{p_{k}}\widetilde {\theta}_{k}^{\top}\delta_{k}+\frac{\overline{\sigma}_{k}}{q_{k}}\widetilde{w}_ {k}^{\top}\vartheta_{k})}\\ &+2s\left\|\widetilde{\theta}_{1}\right\|^{2}+\sum\limits_{k=1}^{s}{(\frac{9+ 4c_{k}^{2}}{2}\left\|\widetilde{w}_{1}\right\|^{2}+\left\|\widetilde{w}_{k} \right\|^{2})}+D_{s}\\ \end{array}  \tag{47}\]  
where \(D_{s}=D_{s-1}+\left\|\theta_{s}^{*}\right\|^{2}+(\frac{39}{2}+2\sigma_{s}^{2} )\left\|w_{s}^{*}\right\|^{2}+\frac{1}{2}\left\|e_{s}^{*}\right\|^{2}+6\left\| e_{s0}^{*}\right\|^{2}\). According to (1), (3), (9) and (10), we can obtain  
\[