## 4 Control System Design
### Yaw Rate Controller  
The heading tracking error \(\psi_{e}\) is defined as follows:  
\[\psi_{e}=\psi-\psi_{d} \tag{34}\]  
Hence, \(\psi_{e}=r-\psi_{d}\). Consider the following Lyapunov function candidate:  
\[V_{\psi}=\frac{1}{2}\tilde{p}^{2} \tag{35}\]  
Differentiating Equation (36),  
\[\dot{V}_{\psi}=\tilde{\psi}(r-\psi_{d}) \tag{36}\]  
Define the yaw angle rate tracking error variable  
\[\tilde{r}=r-r_{d} \tag{37}\]  
where \(r_{d}\) denotes virtual control input:  
\[r_{d}=-k_{\psi}\tilde{\psi}+\dot{\psi}_{d}-U\delta y_{e} \tag{38}\]  
where \(k_{\psi}\) is a positive control design parameter. Next,  
\[V_{\psi}=-k_{\psi}\tilde{p}^{2}+\tilde{\psi}\tilde{r}-U\psi_{e}\tilde{\psi} \tag{39}\]  
According to Remark 1 and Equation (4), differentiating Equation (38),  
\[\dot{r}=\dot{r}-\dot{r}_{d}=\tilde{r}_{r}+\frac{\tau_{r}}{m_{33}}+\frac{d_{r} }{m_{33}}-\dot{r}_{d} \tag{40}\]  
Consider the following Lyapunov function:  
\[V_{r}=V_{\psi}+\frac{1}{2}m_{33}\dot{r}^{2} \tag{41}\]  
Differentiating Equation (42),  
\[\dot{V}_{r} =\dot{V}_{\psi}+m_{33}\dot{r}\dot{r}\] \[=-k_{\psi}\tilde{p}^{2}+\tilde{\psi}\tilde{r}-U\phi y_{e}\tilde{ \psi}+\tilde{r}(m_{33}\tilde{r}_{r}+\tau_{r}+d_{r}-m_{33}\dot{r}_{d}) \tag{42}\]Based on Equation (43), the design heading path following control law is as follows:  
\[V_{\psi}=\frac{1}{2}\tilde{p}^{2} \tag{35}\]0  
where \(k_{r}>0\) denotes control design parameters. Since the model has deviation in practice, the model deviation is approximated by utilizing the single-parameter neural network [29]. \[V_{\psi}=\frac{1}{2}\tilde{p}^{2} \tag{35}\]1  
where  
\[V_{\psi}=\frac{1}{2}\tilde{p}^{2} \tag{35}\]2  
\(\Theta_{r}\) has no clear physical meaning, so it is called a virtual parameter. \[V_{\psi}=\frac{1}{2}\tilde{p}^{2} \tag{35}\]3  
The third-order TD with \(\dot{\psi}_{d}\) current as input signal is designed as follows:  
\[\left\{ \begin{array}{l}\dot{v}_{1}=v_{2}\\ \dot{v}_{2}=v_{3}\\ \dot{v}_{3}=-\dot{\iota}_{1}^{3}\bigl{(}a_{1}(v_{1}-\dot{\psi}_{d})+a_{2}v_{2 }/\iota_{1}+a_{3}v_{3}/\iota_{1}^{2}\bigr{)}\end{array} \right. \tag{47}\]  
where \(\iota_{1}\), \(a_{1}\), \(a_{2}\) and \(a_{3}\) are positive constants; \(v_{1}\), \(v_{2}\) and \(v_{3}\) denote the corresponding estimations, which are the state variables of TD; \(v_{1}=\psi_{dc}\); \(v_{2}=\dot{\psi}_{dc}\) and \(v_{3}=\ddot{\psi}_{dc}\). Define \(\vartheta=U\delta y_{e}\), the second-order TD with \(\vartheta\) current as input signal, is designed as follows:  
\[\left\{ \begin{array}{l}v_{4}=-v_{5}\\ v_{5}=-\dot{\iota}_{2}^{2}(a_{4}(v_{4}-\vartheta)+a_{5}v_{5}/\iota_{2})\end{array} \right. \tag{48}\]  
where \(\iota_{2}\), \(a_{4}\) and \(a_{5}\) denote positive design parameters; \(v_{4}\) and \(v_{5}\) denote the corresponding estimates, which are the state variables of TD; \(v_{4}=\dot{\vartheta}_{c}\) and \(v_{5}=\dot{\vartheta}_{c}\). Substituting (48) and (49) into (39), the following equation is obtained:  
\[\dot{r}_{d}=-k_{2}(r-\dot{\psi}_{dc})+\ddot{\psi}_{dc}-\dot{\vartheta}_{c} \tag{49}\]  
In light of Formula (44), the adaptive single-parameter neural network yaw rate controller can be proposed as  
\[\tau_{r}=-k_{r}\tilde{r}-m_{33}k_{\dot{\psi}}(r-\dot{\psi}_{dc})+m_{33}\ddot{ \psi}_{dc}-m_{33}\dot{\vartheta}_{c}-\dot{d}_{r}-m_{33}\ddot{f}_{r}-m_{33}c_{ r}\dot{\Theta}_{r}\dot{\phi}^{2}(r)\tilde{r}-\ddot{\psi} \tag{50}\]  
with the adaptive law  
\[\dot{\Theta}_{r}=c_{r}\dot{\phi}^{2}(r)\tilde{r}^{2}-\sigma_{r}\dot{\Theta}_{ r},\ \ \ \dot{\Theta}_{r}(0)\geq 0 \tag{51}\]  
For the sake of reducing the impact of actuator saturation on the control system [38], the following auxiliary dynamic system equation is constructed:  
\[\dot{\zeta}_{r}=-k_{\zeta r1}\zeta_{r}-k_{\zeta r0}S_{\zeta r}+\Delta\tau_{r} \tag{52}\]  
where  
\[S_{\zeta r}=\left\{ \begin{array}{ll}|\zeta_{r}|^{\frac{1}{2}}sgn(\zeta_{r} )&|\zeta_{r}|\geq\varepsilon_{r}\\ \frac{3}{2}\dot{\varepsilon}_{r}^{-\frac{1}{2}}\zeta_{r}-\frac{1}{2}\varepsilon _{r}^{-\frac{3}{2}}\zeta_{r}&|\zeta_{r}|<\varepsilon_{r}\end{array} \right.