## 4 Control System Design
### Surge Speed Controller
\[V_{ra}=V_{\psi}+\frac{m_{33}\tilde{r}^{2}}{2}+\frac{m_{33}\hat{\Theta}_{\tau}^ {2}}{2}+\frac{m_{33}\hat{\zeta}_{\tau}^{2}}{2} \tag{68}\]4 \[V_{ra}=V_{\psi}+\frac{m_{33}\tilde{r}^{2}}{2}+\frac{m_{33}\hat{\Theta}_{\tau}^ {2}}{2}+\frac{m_{33}\hat{\zeta}_{\tau}^{2}}{2} \tag{68}\]5Define  
\[V_{ra}=V_{\psi}+\frac{m_{33}\tilde{r}^{2}}{2}+\frac{m_{33}\hat{\Theta}_{\tau}^ {2}}{2}+\frac{m_{33}\hat{\zeta}_{\tau}^{2}}{2} \tag{68}\]6  
The time derivative of (79) is  
\[V_{ra}=V_{\psi}+\frac{m_{33}\tilde{r}^{2}}{2}+\frac{m_{33}\hat{\Theta}_{\tau}^ {2}}{2}+\frac{m_{33}\hat{\zeta}_{\tau}^{2}}{2} \tag{68}\]7 \[V_{ra}=V_{\psi}+\frac{m_{33}\tilde{r}^{2}}{2}+\frac{m_{33}\hat{\Theta}_{\tau}^ {2}}{2}+\frac{m_{33}\hat{\zeta}_{\tau}^{2}}{2} \tag{68}\]8 \[V_{ra}=V_{\psi}+\frac{m_{33}\tilde{r}^{2}}{2}+\frac{m_{33}\hat{\Theta}_{\tau}^ {2}}{2}+\frac{m_{33}\hat{\zeta}_{\tau}^{2}}{2} \tag{68}\]9  
Substituting (61) into (80),  
\[\dot{V}_{ra} =\ddot{\psi}\dot{\psi}+m_{33}\dot{\psi}\dot{\psi}-m_{33}\dot{ \Theta},\dot{\Theta}_{r}+m_{33}\zeta_{r}\dot{\zeta}_{r}\]0 \[\dot{V}_{ra} =\ddot{\psi}\dot{\psi}+m_{33}\dot{\psi}\dot{\psi}-m_{33}\dot{ \Theta},\dot{\Theta}_{r}+m_{33}\zeta_{r}\dot{\zeta}_{r}\]1  
In light of [PERSON]'s inequality,  
\[\dot{V}_{ra} =\ddot{\psi}\dot{\psi}+m_{33}\dot{\psi}\dot{\psi}-m_{33}\dot{ \Theta},\dot{\Theta}_{r}+m_{33}\zeta_{r}\dot{\zeta}_{r}\]2  
\[\dot{V}_{ra} =\ddot{\psi}\dot{\psi}+m_{33}\dot{\psi}\dot{\psi}-m_{33}\dot{ \Theta},\dot{\Theta}_{r}+m_{33}\zeta_{r}\dot{\zeta}_{r}\]3  
\[\dot{V}_{ra} =\ddot{\psi}\dot{\psi}+m_{33}\dot{\psi}\dot{\psi}-m_{33}\dot{ \Theta},\dot{\Theta}_{r}+m_{33}\zeta_{r}\dot{\zeta}_{r}\]4  
\[\dot{V}_{ra} =\ddot{\psi}\dot{\psi}+m_{33}\dot{\psi}\dot{\psi}-m_{33}\dot{ \Theta},\dot{\Theta}_{r}+m_{33}\zeta_{r}\dot{\zeta}_{r}\]5  
\[\dot{V}_{ra} =\ddot{\psi}\dot{\psi}+m_{33}\dot{\psi}\dot{\psi}-m_{33}\dot{ \Theta},\dot{\Theta}_{r}+m_{33}\zeta_{r}\dot{\zeta}_{r}\]6  
Substituting (82)-(86) into (87), the following equation is obtained:  
\[\dot{V}_{ra} =\ddot{\psi}\dot{\psi}+m_{33}\dot{\psi}\dot{\psi}-m_{33}\dot{ \Theta},\dot{\Theta}_{r}+m_{33}\zeta_{r}\dot{\zeta}_{r}\]7 \[\dot{V}_{ra} =\ddot{\psi}\dot{\psi}+m_{33}\dot{\psi}\dot{\psi}-m_{33}\dot{ \Theta},\dot{\Theta}_{r}+m_{33}\zeta_{r}\dot{\zeta}_{r}\]8  
Then, \(\tilde{d}_{u}\) represents the upper limit of surge speed disturbance estimation. \[\dot{V}_{ra} =\ddot{\psi}\dot{\psi}+m_{33}\dot{\psi}\dot{\psi}-m_{33}\dot{ \Theta},\dot{\Theta}_{r}+m_{33}\zeta_{r}\dot{\zeta}_{r}\]9 \[=-k_{\psi}\dot{\psi}^{2}-U\phi y_{r}\dot{\psi}+\ddot{r}(m_{33} \Delta f_{r}-k_{r}\ddot{r}-m_{33}c_{r}\dot{\Theta}_{r}\hat{\phi}^{2}(r)\ddot{ r}+m_{33}k_{\zeta r}\dot{\zeta}_{r}+\tilde{d}_{r})\]0  
Define  
\[=-k_{\psi}\dot{\psi}^{2}-U\phi y_{r}\dot{\psi}+\ddot{r}(m_{33} \Delta f_{r}-k_{r}\ddot{r}-m_{33}c_{r}\dot{\Theta}_{r}\hat{\phi}^{2}(r)\ddot{ r}+m_{33}k_{\zeta r}\dot{\zeta}_{r}+\tilde{d}_{r})\]1  
\[=-k_{\psi}\dot{\psi}^{2}-U\phi y_{r}\dot{\psi}+\ddot{r}(m_{33} \Delta f_{r}-k_{r}\ddot{r}-m_{33}c_{r}\dot{\Theta}_{r}\hat{\phi}^{2}(r)\ddot{ r}+m_{33}k_{\zeta r}\dot{\zeta}_{r}+\tilde{d}_{r})\]2 The results of the straight-line path following under weak interference: (**a**) Path following result. (**b**) Along-track and cross-track error. (**c**) Profile of the heading and surge velocity tracking. (**d**) Estimations of disturbance. (**e**) Profile of the control inputs. (**f**) Estimations \(\hat{\Phi}_{r}\) and \(\hat{\Phi}_{r}\). (**g**) Estimations of sideslip angle. \begin{table}
\begin{tabular}{c c c} \hline
**Performance Indicator** & **AFTC** & **ANNC** \\ \hline \(\text{IEA}(x_{e})\) & 0.36 & 0.848 \\ \(\text{IEA}(y_{e})\) & 8.74 & 10.2 \\ \hline \end{tabular}
\end{table} 
Table 1: Performance indicator straight-line path following under weak interference.