## 3 Predefined-Time Backstepping Control
### Control Law Design
\[\mathbf{\tau}_{a}=m\Big{(}-\Lambda^{T}\mathbf{z}_{1}-\sigma\text{tanh}(\mathbf{z}_{2})-C_{3 }\mathbf{K}_{3}|\mathbf{z}_{2}|^{1-\beta}\text{sgn}(\mathbf{z}_{2})-C_{4}\mathbf{K}_{4}|\mathbf{z}_{ 2}|^{1+\beta}\text{sgn}(\mathbf{z}_{2})-\mathbf{\hat{J}}_{NN}\Big{)} \tag{47}\]  
where \(\mathbf{z}_{2}=\mathbf{v}_{e}\), \(\mathbf{K}_{3},\mathbf{K}_{4}\), \(\mathbf{C}_{3}\), \(\mathbf{C}_{4}\) are the control gains of the actual control law; \(\mathbf{K}_{3}=\text{diag}([k_{31},k_{32},k_{33}]),\mathbf{K}_{4}=\text{diag}([k_{41},k_{ 42},k_{43}])\) is a positive definite symmetric matrix; \(k_{31},k_{32},k_{33},k_{41},k_{42},k_{43}\), is a given parameter; \(\mathbf{C}_{3}\), \(\mathbf{C}_{4}\) is a predefined time control parameter defined as follows: \(\mathbf{C}_{3}=\frac{\pi}{\beta T_{c}k_{\text{min}2^{1-\beta}}},\mathbf{C}_{4}=\frac{ \pi}{3\beta\beta T_{c}k_{\text{min}2^{1+\beta}}}\); \(\sigma=\hat{l}_{0}+\hat{l}_{1}\parallel\mathbf{v}\parallel+\hat{l}_{2}\parallel \mathbf{v}\parallel^{2}\) is an estimate of the unmodeled part of the system; and \(l_{0},\ l_{1},\ l_{2}\) is a predefined time-adaptive law that is an estimate of \(l_{0},\ l_{1},\ l_{2}\) that is updated for and follows the following law:  
\[\hat{l}_{i}=\gamma_{i}\|\mathbf{z}_{2}\|\|\mathbf{v}\|^{i}-C_{\text{min}}2\beta\gamma_{i }^{2-\beta}\hat{l}_{i}^{1-\beta}-C_{\text{min}}2\beta\gamma_{i}^{2+\beta}\hat{l}_{ i}^{1+\beta},i=1,2,3 \tag{48}\]where \(C_{\min}=\min\{C_{5},C_{6}\},C_{5}=-\frac{\pi}{T_{c}\beta}\left(\frac{1}{2}\right) ^{1-\frac{\beta}{2}},C_{6}=-\frac{\pi}{T_{c}\beta}\left(\frac{1}{2}\right)^{1+ \frac{\beta}{2}}2^{\frac{\beta}{2}}\) is the predefined-time adaptive law parameter; \(\gamma_{0},\gamma_{1},\gamma_{2}\) is the adaptive law gain; and \(\hat{\mathbf{f}}_{NN}\) is the estimated value of \(\mathbf{f}_{NN}\). The neural network weight update law \(\hat{\mathbf{W}}\) is designed as follows, as defined in (9). Define the cost function as [24]:  
\[I(t)=\int_{t}^{\infty}e^{-\frac{l-t}{\Psi}}\varphi(l)dl \tag{49}\]  
\(\psi\) is the positive constant, and \(\varphi(l)\) is the instantaneous cost function defined as  
\[\varphi(t)=(\mathbf{\eta}_{1}-\mathbf{\eta}_{1d})^{T}\mathbf{D}(\mathbf{\eta}_{1}-\mathbf{\eta}_{1 d})+\mathbf{\tau}_{a}^{T}\mathbf{R}\mathbf{\tau}_{a} \tag{50}\]  
where \(\mathbf{D},\mathbf{R}\) is a positive definite diagonal matrix used to regulate the effect of trajectory tracking errors and input moments on weight updating, and the loss function \(\vartheta(t)\) is defined according to (50) as  
\[\vartheta(t)=\varphi(t)-\frac{1}{\psi}\hat{\mathbf{f}}_{NN}+\dot{\hat{f}}_{NN} \tag{51}\]  
According to the first-order gradient descent rule, the update law of the weight \(\hat{\mathbf{W}}\) has the following form:  
\[\dot{\hat{\mathbf{W}}}=-\delta\frac{\partial E}{\partial\hat{\mathbf{W}}} \tag{52}\]  
where \(E_{c}=(1/2)\vartheta^{2}\), \(\delta>0\) is the learning rate, obtained by substituting (51) into (52):  
\[ \begin{array}{l}\dot{\hat{\mathbf{W}}}=-\delta\vartheta\frac{\partial \vartheta}{\partial\hat{\mathbf{W}}}\\ =-\delta\vartheta\frac{\partial[\varphi(t)-(1/\psi)\hat{\mathbf{f}}_{NN}+\hat{ \mathbf{f}}_{NN}]}{\partial\hat{\mathbf{W}}}\\ =-\delta\vartheta\left[-\frac{1}{\psi}\frac{\partial\hat{\mathbf{f}}_{NN}}{ \partial\hat{\mathbf{W}}}+\frac{\partial}{\partial\hat{\mathbf{W}}}\left( \frac{\partial\hat{\mathbf{f}}_{NN}}{\partial z_{1}}\right)\hat{z}_{1}+\frac{ \partial}{\partial\hat{\mathbf{W}}}\left(\frac{\partial\hat{\mathbf{f}}_{NN}}{ \partial z_{2}}\right)\hat{z}_{2}\right]\\ =-\delta\left(\varphi(t)+\hat{\mathbf{W}}\Xi\right)\Xi=-\delta\rho\left(\hat{ \mathbf{W}},z_{1},\hat{z}_{1},z_{2},\hat{z}_{2}\right)\end{array}  \tag{53}\]  
\(\Xi=-(\overline{\alpha}/\psi)+\
abla\overline{\alpha}\hat{z}_{1}+\
abla \overline{\alpha}\hat{z}_{2}\).