## 3 Main Results
### Stability Analysis
\begin{split}\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{p+1}{2}}-\frac{\gamma_{1}}{4\gamma_{2}}\widetilde{B}_{s}^{2}\\ &\leq(1-\overline{p})\overline{p}^{\frac{\overline{p}}{1-p}} \end{split}  \tag{32}\]  
where \(\overline{p}=\frac{p+1}{2}\). Assume that \(\Phi\) satisfies \(\Phi=\left\{\widetilde{B}_{s}\Big{|}\Big{|}\widetilde{B}_{s}\Big{|}\leq \lambda\right\}\), where \(\lambda>0\). Then, according to (32) and (33), we obtain:  
\[\left(\frac{\gamma_{1}}{4\gamma_{2}}\widetilde{B}_{s}^{2}\right)^{\frac{p+1}{ 2}}+\left(\frac{\gamma_{1}}{4\gamma_{2}}\widetilde{B}_{s}^{2}\right)^{\frac{q +1}{2}}-\frac{\gamma_{1}}{2\gamma_{2}}\widetilde{B}_{s}^{2}\leq\mu \tag{33}\]  
where \(\mu\) is defined as:  
\[\mu= \begin{cases}(1-\overline{p})\overline{p}^{\frac{\overline{p}}{1-p}},& \lambda<2\sqrt{\frac{\gamma_{2}}{\gamma_{1}}}\\ \left(\frac{\gamma_{1}}{4\gamma_{2}}\lambda^{2}\right)^{\frac{p+1}{2}}-\frac{ \gamma_{1}}{4\gamma_{2}}\lambda^{2},&\lambda\geq 2\sqrt{\frac{\gamma_{2}}{\gamma_{1}}} \end{cases}  \tag{34}\]  
Substituting (34) into (31), combined with Lemma 4, we can obtain:  
\[\dot{V}\leq-aV^{\frac{p+1}{2}}-\beta V^{\frac{q+1}{2}}+\alpha \tag{35}\]  
where \(\alpha=\min\left\{2^{\frac{p+1}{2}}k_{1},\left(\frac{\gamma_{1}}{2}\right)^{ \frac{p+1}{2}}\right\}\), \(\beta=2^{\frac{1-q}{2}}\min\left\{2^{\frac{q+1}{2}}k_{2},\left(\frac{\gamma_{1} }{2}\right)^{\frac{q+1}{2}}\right\}\), and \(\alpha=\mu+\frac{\gamma_{1}}{2\gamma_{2}}B_{s}^{2}+\frac{\sigma^{2}}{2}\). According to Lemma 1, the system tracking error can be stabilized within a fixed time \(T_{1}\) and converge to the following residual set:  
\[\Delta(\eta_{e},\xi_{e})\leq\min\left\{\left(\frac{\alpha}{\alpha(1-\varepsilon )}\right)^{\frac{2}{p+1}},\left(\frac{\alpha}{\beta(1-\varepsilon)}\right)^{ \frac{2}{q+1}}\right\} \tag{36}\]  
where \(0<\varepsilon<1\). The upper bound of convergence time satisfies the following:  
\[T_{1}\leq\frac{2}{\alpha\varepsilon(1-p)}+\frac{2}{\beta\varepsilon(q-1)} \tag{37}\]  
**Step 2:** When the system state reaches the sliding mode surface, there is the following:  
\[s=0,\dot{s}=0 \tag{38}\]  
Combining (20) and (21), the tracking error system can be written as follows:  
\[\left\{ \begin{array}{ll}\dot{\eta}_{e}=&\xi_{e}\\ \ddot{\xi}_{e}=&-m_{1}sig^{a_{1}}(\eta_{e})-m_{2}sig^{a_{2}}(\eta_{e})-m_{3} sign(\eta_{e})\\ &-n_{1}sig^{b_{1}}(\bar{\xi}_{e})-n_{2}sig^{b_{2}}(\bar{\xi}_{e})-n_{3}sign( \bar{\xi}_{e})\end{array} \right. \tag{39}\]  
According to Lemma 2, when the tracking error remains on the sliding mode surface, the tracking errors \(\eta_{e}\) and \(\bar{\xi}_{e}\) can converge to the equilibrium point within a fixed time \(T_{2}\). In summary, the designed fixed-time adaptive fault-tolerant control scheme can ensure the stability of the entire closed-loop system within a fixed time \(T=T_{1}+T_{2}\). **Remark 1**. : _In Lemma 2, although the system tracking errors can converge to the origin along the sliding mode surface within a fixed time \(T_{2}\), the upper bound of the convergence time during the sliding stage cannot be explicitly given. The detailed proof process of Lemma 2 can be found in reference [21]._