## 3 Designing Adaptive-Robust Controller
### Designing Adaptive-Robust Controller in the Presence of Model Uncertainties
To prove the stability of tracking error, the Lyapunov function defined as a function of the following variables:  
\[V=\sum_{i=1}^{n}V_{i}\Big{(}e_{i},r_{i},\widetilde{\theta}_{i}\Big{)},i=1,2, \ldots,N \tag{56}\]  
where for the \(i\)-th ASV, the Lyapunov function defined as below:  
\[V_{i}=k_{pi}\ln\cosh(e_{i})+\frac{1}{2}r_{i}^{T}M_{i}r_{i}+\frac{1}{2} \widetilde{\theta}_{i}^{T}\Gamma_{\theta}^{-1}\widetilde{\theta}_{i} \tag{57}\]  
where \(\widetilde{\theta}_{i}=\theta_{i}-\hat{\theta}_{i}\). Differentiating of (57) results in (58) and (59). \[\dot{V}_{i}=tanh^{T}(e_{i})\ K_{pi}\dot{e}_{i}+r_{i}^{T}M_{i}\dot{r}_{i}+0.5r_{ i}^{T}\dot{M}_{i}r_{i}+\widetilde{\theta}_{i}^{T}\Gamma_{\theta}^{-1}\dot{ \widetilde{\theta}}_{i} \tag{58}\]  
By substituting Equations (43) and (53) into Equation (58), we have as follows:  
\[\dot{V}_{i}=\frac{1}{2}\ r_{i}^{T}\Big{(}\dot{M}_{i}-2C_{i}\Big{)}r_{i}-tanh ^{T}(e_{i})K_{pi}\Lambda_{pi}\Lambda_{pi}(e_{i}) \tag{59}\] \[-r_{i}^{T}K_{xi}tanh(r_{i})+r_{i}^{T}u_{iR}-r_{i}^{T}\xi_{i}+r_{i }^{T}\dot{x}_{i}+\widetilde{\theta}_{i}^{T}\Gamma_{\theta}^{-1}\dot{ \widetilde{\theta}}_{i}\]  
Using Equation (47), the first term in the derivative of the Lyapunov function equals zero. Thus, we have as follows:  
\[\dot{V}_{i}=-tanh^{T}(e_{i})\ K_{pi}\Lambda_{pi}tanh(e_{i})-r_{i}^{T}K_{pi} tanh(r_{i}) \tag{60}\] \[+r_{i}^{T}u_{iR}-r_{i}^{T}\xi_{i}+r_{i}^{T}\dot{x}_{i}+\widetilde{ \theta}_{i}^{T}\ \Gamma_{\theta}^{-1}\dot{\widetilde{\theta}}_{i}\]  
Now using (46) that is the adaptation rule and the relations \(\xi_{i}\leq G_{i}\theta_{i}\), \(a^{T}b\leq\|a\|\|b\|\) and \(\lambda_{\min}(M_{i})\|x_{i}\|^{2}\leq x_{i}^{T}M_{i}x_{i}\leq\lambda_{\max}(M _{i})\|x_{i}\|^{2}\), one can easily find a new upper bound for the first term of \(\dot{V}_{i}\). Thus, we have as follows:  
\[\dot{V}_{i}\leq-\lambda_{min}\big{[}K_{pi}\Lambda_{pi}\big{]}\big{\|}tanh(e_{ i})\big{\|}^{2}-\lambda_{di}\|r_{i}\|^{2}-r_{i}^{T}K_{iw}\ tanh(r_{i})+r_{i}^{T}u_{iR} \tag{61}\] \[+\big{\|}r_{i}^{T}\big{\|}G_{i}\theta_{i}+\big{\|}r_{i}^{T}\big{\|} \|\chi_{i}\|-\widetilde{\theta}_{i}^{T}G_{i}^{T}\|r_{i}\|+\widetilde{\theta}_ {i}^{T}\delta_{\theta}\Big{(}\hat{\theta}_{i}-\theta_{0}\Big{)}\]  
Using property \(\|x_{i}\|\leq\beta_{1}\|x_{i}\|+\beta_{2}\|x_{i}\|^{2}\) and assuming \(-r_{i}^{T}K_{iw}tanh(r_{i})\leq 0\) and substituting \(\big{\|}r_{i}^{T}\big{\|}G_{i}\hat{\theta}_{i}-r_{i}^{T}G_{i}\hat{\theta}_{i} \text{tanh}\Big{(}\frac{G_{i}\widetilde{\theta}_{i}}{\gamma_{d}}\Big{)}\leq n \gamma_{di}\) instead of \(r_{i}^{T}u_{iR}-\big{\|}r_{i}^{T}\big{\|}G_{i}\theta_{i}\), according to [33] and property \(ab\leq\frac{\big{(}a^{2}+b^{2}\big{)}}{2}\), we can easily write Equation (61) as below:  
\[V=\sum_{i=1}^{n}V_{i}\Big{(}e_{i},r_{i},\widetilde{\theta}_{i}\Big{)},i=1,2, \ldots,N \tag{56}\]0 \[V=\sum_{i=1}^{n}V_{i}\Big{(}e_{i},r_{i},\widetilde{\theta}_{i}\Big{)},i=1,2, \ldots,N \tag{56}\]1  
where:  
\[V=\sum_{i=1}^{n}V_{i}\Big{(}e_{i},r_{i},\widetilde{\theta}_{i}\Big{)},i=1,2, \ldots,N \tag{56}\]2  
\[V=\sum_{i=1}^{n}V_{i}\Big{(}e_{i},r_{i},\widetilde{\theta}_{i}\Big{)},i=1,2, \ldots,N \tag{56}\]3  
According to \(\lambda_{di}>0.5\beta_{1}+0.5\beta_{2}\) and \(\beta_{m}=\min\Big{\{}\lambda_{min}\Big{\{}K_{pi}\Lambda_{pi}\Big{\}},\{\lambda _{di}-0.5\beta_{1}+0.5\beta_{2}\big{\}}\), we have as follows:  
\[V=\sum_{i=1}^{n}V_{i}\Big{(}e_{i},r_{i},\widetilde{\theta}_{i}\Big{)},i=1,2, \ldots,N \tag{56}\]4  
Now, assuming \(\beta_{m}>0.5\beta_{1}+0.5\beta_{2}\|x_{i}\|^{2}\) it can be written as below:  
\[V=\sum_{i=1}^{n}V_{i}\Big{(}e_{i},r_{i},\widetilde{\theta}_{i}\Big{)},i=1,2, \ldots,N \tag{56}\]5\[ \begin{split}&\sum\limits_{i=1}^{n}\dot{V}_{i}=\dot{V}_{1}+\dot{V}_{2}+ \ldots+\dot{V}_{n}\leq-\min((c_{m1}\|x_{1}\|^{2}-c_{0_{1}}\|\widetilde{\theta}_ {1}\|^{2}\\ +\gamma_{1}),\left(c_{m2}\|x_{2}\|^{2}-c_{0_{2}}\|\widetilde{\theta }_{2}\|^{2}+\gamma_{2}\right)+\ldots+\left(c_{mm}\|x_{n}\|^{2}-c_{0_{n}}\| \widetilde{\theta}_{n}\|^{2}+\gamma_{n}\right)\end{split}