## 3 Designing Adaptive-Robust Controller
### Designing Adaptive-Robust Controller in the Presence of Model Uncertainties
Finally, the bounded controller to keep formation is proposed as follows for the \(i\)-th ASV:  
\[\overline{\tau}_{i\ a}=-K_{pi}\mbox{tanh}(e_{i})-K_{ip}\mbox{tanh}(r_{i})+u_{ Ri} \tag{44}\]where \(k_{pi}\geq 1\), \(K_{pi}=k_{pi}I_{2}\) and \(K_{ip}\in R^{2\times 1}\) is a positive definite matrix. In addition, \(u_{Ri}\) is obtained as follows:  
\[u_{Ri}=-G_{i}\ \hat{\partial}_{i}\text{tanh}\left(\frac{G_{i}\hat{\partial}_{i}r_ {i}}{\gamma_{d}}\right). \tag{45}\]  
The parameters \(\hat{\partial}_{i}\in R^{4\times 4}\) in (45) are estimated by the following adaptation rule. \[\hat{\partial}_{i}=\Gamma_{\theta}G_{i}^{T}r_{i}-\Gamma_{\theta}\delta_{ \theta}\left(\hat{\partial}_{i}-\theta_{0}\right) \tag{46}\]  
where \(\Gamma_{\theta}\in R^{4\times 4}\) is the controller gain, \(\delta_{\theta}\in R^{+}\) and \(\theta_{0}\in R^{4\times 1}\) are the initial estimation of the system parameters. To prove the stability of the closed-loop error dynamics, will require the following features:  
Property 2. The following properties can be proven for model in (39). \[x_{1i}^{T}\left(\dot{M}_{i}(e_{i})-2C_{i}\Big{(}e_{i},\dot{e}_{i}\Big{)} \right)x_{1i}=0 \tag{47}\]  
\[M_{i}(e_{i})\ddot{e}_{i}+C_{i}\Big{(}e_{i},\dot{e}_{i}\Big{)}\dot{e}_{i}+\dot {\xi}_{i}=-\overline{\tau}_{ai}(t) \tag{39}\]0  
\[M_{i}(e_{i})\ddot{e}_{i}+C_{i}\Big{(}e_{i},\dot{e}_{i}\Big{)}\dot{e}_{i}+\dot {\xi}_{i}=-\overline{\tau}_{ai}(t) \tag{39}\]1  
\[M_{i}(e_{i})\ddot{e}_{i}+C_{i}\Big{(}e_{i},\dot{e}_{i}\Big{)}\dot{e}_{i}+\dot {\xi}_{i}=-\overline{\tau}_{ai}(t) \tag{39}\]2  
where \(\lambda_{\text{ci}}\geq 0\) is a constant number. **Theorem 2**. : _By applying the bounded adaptive-robust controller in Equation (44), the formation errors in the leader-follower formation model of ASVs in Equation (26) converge into a small ball centered on the zero. Also. the estimated parameters in Equation (46) remain bounded._  
**Proof. ** Substituting Equation (43), differentiating and increasing and decreasing the similar terms in Equation (39) results in Equation (51). \[ \begin{array}{l}-\overline{\tau}_{iu}(t)=M_{i}(e_{i})\dot{r}_{i}+C_{i} \Big{(}e_{i},\dot{e}_{i}\Big{)}r_{i}-M_{i}(e_{i})\Lambda_{pi}\;sech^{2}(e_{i} )\;\dot{e}_{i}\\ -C_{i}\Big{(}e_{i},\dot{e}_{i}\Big{)}\Lambda_{pi}\;tanh(e_{i})+\dot{\xi}_{i} \end{array}  \tag{51}\]  
Now, by using Equations (48) and (50) and increasing and decreasing the similar terms in (51), the error dynamics in the closed-loop system is obtained as below:  
\[ \begin{array}{l}-\overline{\tau}_{iu}(t)=M_{i}(e_{i})\dot{r}_{i}+C_{i}(e_{i},r_{i})r_{i}-C_{i}(e_{i},r_{i})\Lambda_{pi}tanh(e_{i})\\ -M_{i}(e_{i})\Lambda_{pi}\;sech^{2}(e_{i})\dot{e}_{i}-C_{i}\Big{(}e_{i},\Lambda _{pi}tanh(e_{i})\Big{)}r_{i}\\ +C_{i}\Big{(}e_{i},\Lambda_{pi}tanh(e_{i})\Big{)}\;\Lambda_{pi}\;tanh(e_{i})+ \dot{\xi}_{i}\end{array}  \tag{52}\]  
By substituting the controller from Equation (44), we have as follows:  
\[M_{i}(e_{i})\dot{r}_{i}=-C_{i}(e_{i},r_{i})r_{i}-K_{pi}tanh(e_{i})-K_{ip}tanh( r_{i})+u_{iR}-\dot{\xi}_{i}+\chi_{i} \tag{53}\]  
where:  
\[ \begin{array}{l}\chi_{i}=M_{i}(e_{i})\ \Lambda_{pi}Sech^{2}(e_{i})\dot{e}_{i}-C_{i }\Big{(}e_{i},\Lambda_{pi}tanh(e_{i})\Big{)}\Lambda_{pi}\;tanh(e_{i})\\ +C_{i}\Big{(}e_{i},\Lambda_{pi}tanh(e_{i})\Big{)}\;r_{i}+C_{i}(e_{i},r_{i}) \Lambda_{pi}tanh(e_{i})\end{array}  \tag{54}\]  
Using property 2, \(\chi_{i}\) will be bounded as follows:  
\[\|x_{i}\|\leq\beta_{1}\|x_{i}\|+\beta_{2}\|x_{i}\|^{2}, \tag{55}\]  
and \(x_{i}=\left[tanh^{T}(e_{i}),r_{i}^{T}\right]^{T}\) where \(\beta_{1}\) and \(\beta_{2}\) are unknown positive constants.