## 3 Main Results
### Stability Analysis  
Based on the above discussions, the following Theorem 1 can be derived:  
**Theorem 1**: _For the dynamic positioning ship fault-tolerant system (10), satisfying Assumption 1, the sliding mode surface is given by (21). Combined with the fixed-time adaptive fault-tolerant controller (25) and the adaptive law (24), all signals of the closed-loop system can be guaranteed to be stable within a fixed time in the case of thruster faults, external ocean disturbances, and model uncertainties._  
**Proof. ** The proof of Theorem 1 is completed in two steps. Firstly, it is proved that the tracking error can converge to the sliding mode surface within a fixed time, and then it is proved that the system can converge to the equilibrium point within a fixed time when it reaches the sliding mode surface. \(\square\)  
**Step 1:** Consider the following Lyapunov candidate function:  
\[V=\frac{1}{2}s^{T}s+\frac{1}{2\gamma_{2}}\widetilde{B}_{s}^{2} \tag{25}\]  
where \(\widetilde{B}_{s}=B_{s}-\hat{B}_{s}\) represents the estimation error. Combining (20) and (22), taking the derivative of (26) yields:  
\[\dot{V}=s^{T}(S(r)\xi+J(\psi)\overline{M}^{-1}(-\overline{D}(\
u)v+\theta\tau _{c}+\Omega)-\dot{\xi}_{d}+\dot{\varphi})-\frac{1}{\gamma_{2}}\widetilde{B}_ {s}\dot{\hat{B}}_{s} \tag{26}\]  
Furthermore, substituting the fixed-time fault-tolerant controller (25) and adaptive law (24) into (27) yields:  
\[ \begin{array}{ll}\dot{V}=s^{T}(-k_{1}sig^{p}(s)-k_{2}sig^{q}(s)-\frac{\hat{B }_{s}s}{2\sigma^{2}}+\Omega)-\widetilde{B}_{s}(-\frac{\gamma_{1}}{2}\widetilde {B}_{s}+\frac{\|s\|^{2}}{2\sigma^{2}})\\ =-k_{1}\|s\|^{p+1}-k_{2}\|s\|^{q+1}+s^{T}\Omega-\frac{\hat{B}_{s}\|^{2}}{2 \sigma^{2}}+\frac{\gamma_{1}}{\gamma_{2}}\widetilde{B}_{s}\hat{B}_{s}\end{array}  \tag{27}\]  
According to [PERSON]'s inequality and Assumption 1, the following is obtained:  
\[s^{T}\Omega\leq\frac{B^{2}\|s\|^{2}}{2\sigma^{2}}+\frac{\sigma^{2}}{2}=\frac{ B_{s}\|s\|^{2}}{2\sigma^{2}}+\frac{\sigma^{2}}{2} \tag{28}\]Moreover, it is not difficult to obtain:  
\[\frac{\gamma_{1}}{\gamma_{2}}\widetilde{B}_{s}\hat{B}_{s}=\frac{\gamma_{1}}{ \gamma_{2}}\widetilde{B}_{s}B_{s}-\frac{\gamma_{1}}{\gamma_{2}}\widetilde{B}_{ s}^{2}\leq\frac{\gamma_{1}}{2\gamma_{2}}B_{s}^{2}-\frac{\gamma_{1}}{2\gamma_{2}} \widetilde{B}_{s}^{2} \tag{29}\]  
Next, by substituting (29) and (30) into (28), we obtain:  
\[ \begin{split}&\dot{V}\leq-k_{1}\|s\|^{p+1}-k_{2}\|s\|^{q+1}- \frac{\gamma_{1}}{2\gamma_{2}}\widetilde{B}_{s}^{2}+\frac{\gamma_{1}}{2\gamma_ {2}}B_{s}^{2}+\frac{\sigma^{2}}{2}\\ &=-k_{1}\|s\|^{p+1}-k_{2}\|s\|^{q+1}-\left(\frac{\gamma_{1}}{ \gamma_{2}}\widetilde{B}_{s}^{2}\right)^{\frac{p+1}{2}}-\left(\frac{\gamma_{1} }{\gamma_{2}}\widetilde{B}_{s}^{2}\right)^{\frac{q+1}{2}}+\Lambda\end{split}  \tag{30}\]  
where \(\Lambda=\left(\frac{\gamma_{1}}{\gamma_{2}}\widetilde{B}_{s}^{2}\right)^{ \frac{p+1}{2}}+\left(\frac{\gamma_{1}}{\gamma_{2}}\widetilde{B}_{s}^{2}\right) ^{\frac{q+1}{2}}-\frac{\gamma_{1}}{2\gamma_{2}}\widetilde{B}_{s}^{2}+\frac{ \gamma_{1}}{2\gamma_{2}}B_{s}^{2}+\frac{\sigma^{2}}{2}\). It is worth noting that \(\Lambda\) has two cases that need to be discussed. When \(\frac{\gamma_{1}}{\gamma_{2}}\widetilde{B}_{s}^{2}\geq 1\), we obtain:  
\[\left(\frac{\gamma_{1}}{4\gamma_{2}}\widetilde{B}_{s}^{2}\right)^{\frac{p+1}{ 2}}+\left(\frac{\gamma_{1}}{4\gamma_{2}}\widetilde{B}_{s}^{2}\right)^{\frac{q +1}{2}}-\frac{\gamma_{1}}{2\gamma_{2}}\widetilde{B}_{s}^{2}\leq\left(\frac{ \gamma_{1}}{4\gamma_{2}}\widetilde{B}_{s}^{2}\right)^{\frac{q+1}{2}}-\frac{ \gamma_{1}}{4\gamma_{2}}\widetilde{B}_{s}^{2} \tag{31}\]  
When \(\frac{\gamma_{1}}{4\gamma_{2}}\widetilde{B}_{s}^{2}<1\), combined with Lemma 3, we obtain:  
\[