## 3 Methodology
### Adaptive Controller Design with Input Saturation  
An adaptive controller using backstepping technology is designed for under-actuated AUVs in this section. For simplicity, a dynamic system for (9) is represented by:  
\[ \begin{cases}\dot{y}=f(y,z)\\ \dot{z}=g(y,z)\end{cases} , \tag{25}\]  
where \(y=[\gamma_{c};u_{c}]^{\mathrm{T}}\) and \(z=[v;r]^{\mathrm{T}}\). \(f(\cdot)\) and \(g(\cdot)\) are corresponding differentiable functions. Let \(z_{s}(y)=(v_{s};r_{s})\) represent the designed space configuration. \(z_{c}(y)=(v_{c};r_{c})\) represents the shift from \(z_{s}\) to \(z\), and \(z_{c}=z-z_{s}\). Following that, system (9) is shown as follows:  
\[ \begin{cases}\dot{y}=f(y,z_{s})+[f(y,z)-f(y,z_{s})]\\ \dot{z}_{c}=g(y,z_{s}+z_{c})-\dot{z}_{s}=g(y,z_{s}+z_{c})-\frac{\partial z_{s} (y)}{\partial y}\dot{y}-\frac{\partial z_{s}(y)}{\partial t}\end{cases} . \tag{26}\]  
In (26), there are two lower-order subsystems:  
\[\dot{y}=f(y,z_{s}), \tag{27}\]  
\[\dot{z}_{c}=g(y,z_{s}+z_{c}). \tag{28}\]  
Additionally, \(f(y,z)-f(y,z_{s})\) and \(\dot{z}_{s}\) are regarded as the interconnections of two subsystems. Part 1: Controller design for the subsystem (27)  
In practice, considering the input saturation, the system could be described as follows:  
\[ \begin{cases}\dot{\gamma}_{c}=r+\dot{\beta}-\dot{\gamma}_{d}\\ \dot{u}_{c}=f_{u}+M_{1}u_{1}+M_{1}d_{1}-\dot{u}_{d}\\ \dot{u}_{1}=-c_{1}u_{1}+\mathrm{sat}(\tau_{1})\end{cases} , \tag{29}\]  
where \(f_{u}=M_{1}(X_{u}u+a_{23}vr)\), \(u_{1}=T\). To determine \(u_{1}\), it is designed as follows:  
\[r_{s}=-k_{r}\gamma_{c}-\dot{\beta}+\dot{\gamma}_{d}, \tag{30}\]  
where \(k_{r}>0\). Then, the tracking error of the control system is introduced as follows:  
\[u_{e}=u-u_{d}, \tag{31}\]  
\[z_{1}=u_{1}-\alpha_{1}^{c}, \tag{32}\]  
where \(u_{d}\) is the desired surge speed, and \(\alpha_{1}^{c}\) is filtered signal of the virtual control quantity \(\alpha_{1}^{co}\) for \(u_{1}\). The compensation signal of the filter error is defined as:  
\[v_{1}=u_{e}-\xi_{1}, \tag{33}\]  
\[v_{2}=z_{1}-\xi_{2}. \tag{34}\]  
The residual compensations of the filtering error and the saturation control quantity are constructed as:  
\[\dot{\xi}_{1}=-k_{1}\xi_{1}+M_{1}(\alpha_{1}^{c}-\alpha_{1}^{co})+M_{1}\xi_{2}, \tag{35}\]\[\dot{\xi}_{2}=-k_{2}\xi_{2}+D_{1}, \tag{36}\]  
where \(k_{1}>0\), \(k_{2}>0\);\(D_{1}=\text{sat}(\tau_{1})-\tau_{1}+M_{1}\widetilde{d}_{1}\), \(\xi_{1}(0)=0\), \(\xi_{2}(0)=0\), and \(\widetilde{d}_{1}=d_{1}-\hat{d}_{1}\) represent observation error. The control signals could be designed as follows:  
\[\alpha_{1}^{co}=(-k_{1}u_{e}-f_{u}-M_{1}\hat{d}_{1}+\dot{u}_{d})/M_{1}, \tag{37}\]  
\[\tau_{1}=c_{1}u_{1}-k_{2}z_{1}+\dot{\alpha}_{1}^{c}-M_{1}v_{1}+(M_{1}-1)D_{1}. \tag{38}\]  
**Theorem 2**. : _Considering the subsystem (27), select the filter as shown in (23), construct a compensation system as shown in (35) and (36), and the control signals are designed as (37) and (38), thus the filter compensation errors (33) and (34) are asymptotically stable, and the tracking errors (31) and (32) are bound and convergent._  
**Proof. ** Using (30), (31), and (32), system (29) is put into  
\[ \begin{cases}\dot{\gamma}_{e}=-k_{r}\gamma_{e}\\ \dot{u}_{e}=-k_{1}u_{e}+M_{1}[(\alpha_{1}^{c}-\alpha_{1}^{co})+z_{1}]\\ \dot{z}_{1}=-k_{2}z_{1}+M_{1}D_{1}-M_{1}v_{1}\end{cases} . \tag{39}\]  
Step 1: According to (33) and (34), a Lyapunov function is designed:  
\[V_{1}=\frac{1}{2}(v_{1}^{2}+v_{2}^{2}). \tag{40}\]  
The derivative of \(V_{1}\) is  
\[ \begin{array}{l}\dot{V}_{1}=v_{1}(-k_{1}v_{1}+M_{1}v_{2})+v_{2}(-k_{2}v_{2}- M_{1}v_{1})\\ \leq-k_{1}v_{1}^{2}-k_{2}v_{2}^{2}\leq-n_{1}V_{1}\end{array} , \tag{41}\]  
where \(n_{1}=2\text{min}\{k_{1},k_{2}\}\),\(k_{1}>0\),\(k_{2}>0\). Multiply both sides of (41) by \(e^{n_{1}t}\), and then integrate both sides to obtain the following:  
\[V_{1}(t)\leq V_{1}(0)e^{-n_{1}t}\leq V_{1}(0). \tag{42}\]  
Additionally, we can determine that \(|v_{i}|\leq B_{1}\), \(i=1,2\), \(B_{1}=\sqrt{2V_{1}(0)}\).