## 4. Stability Analysis  
**Theorem 7**. : _For an underactuated autonomous underwater vehicle system satisfying Assumptions 1-3, consider (40), if \(\mathbf{v}_{e}=0\) and \(\boldsymbol{\alpha}\) are of the form (43), \(\boldsymbol{z}_{1}\) will converge to a small neighborhood of zero in a predefined-time._  
**Proof of Theorem 7**. : When \(\mathbf{v}_{e}=0\), and considering (30), we have \(\boldsymbol{z}_{1}^{T}\boldsymbol{\omega}z_{1}=0\). Substituting (43) into (42):  
\[\dot{V}_{1}=\boldsymbol{z}_{1}^{T}\Big{(}-C_{1}K_{1}diag\Big{(}|\boldsymbol{z }_{1}|^{1+\beta}\Big{)}\text{sgn}(z_{1})-C_{2}K_{2}diag\Big{(}|\boldsymbol{z} _{1}|^{1-\beta}\Big{)}\text{sgn}(\boldsymbol{z}_{1})\Big{)} \tag{56}\]  
By Lemma 2, it follows that  
\[ \begin{split}&-C_{1}\boldsymbol{z}_{1}^{T}K_{1}diag\Big{(}| \boldsymbol{z}_{1}|^{1+\beta}\Big{)}\text{sgn}(\boldsymbol{z}_{1})\leq-C_{1} k_{1}\boldsymbol{z}_{1}^{T}diag\Big{(}|\boldsymbol{z}_{1}|^{1+\beta}\Big{)} \text{sgn}(\boldsymbol{z}_{1})\\ &\leq-C_{1}2^{1+\beta}3\ell k_{1}\Big{(}\frac{1}{2}\boldsymbol{ z}_{1}^{T}\boldsymbol{z}_{1}\Big{)}^{1+\frac{\beta}{2}}\leq-\frac{\pi}{\beta T_{ c}}V_{1}^{1+\frac{\beta}{2}}\\ &-C_{2}\boldsymbol{z}_{1}^{T}K_{2}diag\Big{(}|\boldsymbol{z}_{1 }|^{1-\beta}\Big{)}\text{sgn}(\boldsymbol{z}_{1})\leq-C_{2}k_{2}\boldsymbol{ z}_{1}^{T}diag\Big{(}|\boldsymbol{z}_{1}|^{1-\beta}\Big{)}\text{sgn}( \boldsymbol{z}_{1})\\ &\leq-C_{2}2^{1-\beta}k_{2}\Big{(}\frac{1}{2}\boldsymbol{z}_{1}^ {T}\boldsymbol{z}_{1}\Big{)}^{1-\frac{\beta}{2}}\leq-\frac{\pi}{\beta T_{c}} V_{1}^{1-\frac{\beta}{2}}\end{split}  \tag{57}\]  
Combining (57), we obtain  
\[\dot{V}_{1}\leq-\frac{\pi}{\beta T_{c}}V_{1}^{1+\frac{\beta}{2}}-\frac{\pi}{ \beta T_{c}}V_{1}^{1-\frac{\beta}{2}} \tag{58}\]  
**Proof end \(\square\)**  
Figure 2. Controller structure. \[ \begin{split}\bar{l}=\bar{l}-l\text{, is the adaptive law estimation error, and the derivative of }V_{2}=\frac{1}{2m}\mathbf{z}_{2}^{T}M_{a}\mathbf{z}_{2},\\ V_{4}=\frac{1}{2}\sum\limits_{l=0}^{2}\frac{1}{\gamma_{l}}\widehat{l }_{i}^{2},V_{1}\text{ has been discussed in the previous section, so the derivative with respect to time is taken for }V_{2},V_{4}\text{ here:}\\ \dot{V}_{2}+\dot{V}_{4}=\mathbf{z}_{2}^{T}\Big{(}\mathbf{F}-\sigma \text{tanh}(\mathbf{z}_{2})-C_{3}K_{3}|\mathbf{z}_{2}|^{1-\beta}\text{sgn}( \mathbf{z}_{2})-C_{4}\mathbf{K}_{4}|\mathbf{z}_{2}|^{1+\beta}\text{sgn}(\mathbf{z }_{2})\Big{)}+\sum\limits_{i=0}^{2}\frac{1}{\gamma_{i}}(\dot{l}_{i}-l_{i}) \dot{\bar{l}}_{i}\\ =\mathbf{z}_{2}^{T}\Big{(}-C_{3}K_{3}|\mathbf{z}_{2}|^{1-\beta} \text{sgn}(\mathbf{z}_{2})-C_{4}\mathbf{K}_{4}|\mathbf{z}_{2}|^{1+\beta}\text{sgn} (\mathbf{z}_{2})\Big{)}-\|\mathbf{z}_{2}\|\sum\limits_{i=0}^{2}\dot{l}_{i}\ \|\ \mathbf{v}\ \|^{i}+\sum\limits_{i=0}^{2}\frac{1}{\gamma_{i}}(\dot{l}_{i}-l_{i}) \dot{\bar{l}}_{i}+3\varepsilon\\ \leq\mathbf{z}_{2}^{T}\Big{(}-C_{3}k_{3\text{min}}\mathbf{z}_{2}^ {T}|\mathbf{z}_{2}|^{1-\beta}\text{sgn}(\mathbf{z}_{2})-C_{4}k_{\text{min}} \mathbf{z}_{2}^{T}|\mathbf{z}_{2}|^{1+\beta}\text{sgn}(\mathbf{z}_{2})\Big{)} -\|\mathbf{z}_{2}\|\sum\limits_{i=0}^{2}\dot{l}_{i}\ \|\ \mathbf{v}\ \|^{i}+\sum\limits_{i=0}^{2}\frac{1}{\gamma_{i}}(\dot{l}_{i}-l_{i}) \dot{\bar{l}}_{i}+3\varepsilon\end{split}  \tag{65}\]  
Substituting (48) into (65) yields by Lemma 2 the following:  
\[