## 4 A Model-Free TSMC Approach Using RBFNN and PPCA
### Design of Model-Free TSMC Method Using RBFNN and PPCA  
The control law is expressed in the following manner:  
\[ \begin{cases}\tau=\hat{f}(s)+\tau_{r}\\ \hat{f}(s)=\hat{\mathcal{W}}^{T}\mathcal{G}(\mathcal{X})\\ \tau_{r}=\gamma^{*}\text{sign}(S)+\psi_{1}\text{Si}^{2\omega_{1}-1}(S)+\psi_{2 }\text{Si}^{2\omega_{2}-1}(S)\end{cases} . \tag{34}\]  
In the control law, \(\hat{f}(s)\) and \(\tau_{r}\) represent two distinct terms. \(\hat{f}(s)\) is obtained from the output of the NN, while the term \(\tau_{r}\) is designed as a fixed-time reaching control law. The constants \(\psi_{1}\) and \(\psi_{2}\) are positive, with \(\omega_{1}>1\), \(\frac{1}{2}<\omega_{2}<1\), and \(\gamma^{*}>|\iota|\). We design update laws for the NN to achieve fixed-time stability for both the tracking error and the neural network itself, as follows:  
\[\hat{\mathcal{W}}_{i}=\mathcal{K}_{N}^{-1}\Big{(}\mathcal{G}(\mathcal{X})S_{i }-\psi_{3}\hat{\mathcal{W}}_{i}^{2\omega_{1}-1}-\psi_{4}\hat{\mathcal{W}}_{i} ^{2\omega_{2}-1}\Big{)}\ . \tag{35}\]  
Here, \(\hat{\mathcal{W}}_{i}\) represents the \(i\)th element of the matrix \(\mathcal{W}\). A summary of the control design is stated in the following theorem, and the control diagram is shown in Figure 1.  
Figure 1: Graph of the control system. **Theorem 2**. : _By employing an RBFNN with update laws (35), we can estimate the unknown function \(f(s)\), which is associated with the robot's dynamics and dynamic uncertainties. Utilizing the control law defined in Equation (34), we can achieve both the desired performance for the robot's tracking errors and the convergence of the estimation errors of the NN within a fixed time._  
Proof of Theorem 2.: Substituting Equation (34) into Equation (22) yields:  
\[ \begin{array}{c}\bar{M}(\alpha_{1})\dot{S}=f(s)-\bar{C}(\alpha_{1},\alpha_{2} )S-\bar{f}(s)-\tau_{r}\\ =\bar{\mathcal{W}}^{T}\mathcal{G}(\mathcal{X})+\iota-\bar{C}(\alpha_{1},\alpha _{2})S-\tau_{r}\end{array} \ . \tag{36}\]  
The Lyapunov function is selected as follows:  
\[V_{1}=\frac{1}{2}S^{T}\bar{M}(\alpha_{1})S. \tag{37}\]  
Computing the time derivative of \(V_{1}\) using the result of Equation (36), we obtain:  
\[ \begin{array}{c}\dot{V}_{1}=S^{T}\bar{M}(\alpha_{1})\dot{S}+\frac{1}{2}S^{T }\dot{\bar{M}}(\alpha_{1})S\\ =S^{T}(\bar{\mathcal{W}}^{T}\mathcal{G}(\mathcal{X})+\iota-\bar{C}(\alpha_{1},\alpha_{2})S-\tau_{r})+\frac{1}{2}S^{T}\dot{\bar{M}}(\alpha_{1})S\\ =\big{(}S^{T}\bar{\mathcal{W}}^{T}\mathcal{G}(\mathcal{X})+S^{T}l-S^{T}\bar{C }(\alpha_{1},\alpha_{2})S-S^{T}\tau_{r}\big{)}+\frac{1}{2}S^{T}\dot{\bar{M}}( \alpha_{1})S\\ =S^{T}\bar{\mathcal{W}}^{T}\mathcal{G}(\mathcal{X})+S^{T}l-S^{T}\tau_{r}+0.5S ^{T}\Big{(}\dot{\bar{M}}(\alpha_{1})-2\bar{C}(\alpha_{1},\alpha_{2})\Big{)}S \end{array} \ . \tag{38}\]  
Since \(S^{T}\Big{(}\dot{\bar{M}}(\alpha_{1})-2\bar{C}(\alpha_{1},\alpha_{2})\Big{)}S=0\), we can subsequently obtain:  
\[ \begin{array}{c}\dot{V}_{1}=S^{T}\bar{\mathcal{W}}^{T}\mathcal{G}(\mathcal{X })+S^{T}l-S^{T}\tau_{r}\\ =S^{T}\bar{\mathcal{W}}^{T}\mathcal{G}(\mathcal{X})-S^{T}\gamma^{*}\text{sign }(S)-S^{T}\psi_{1}\text{Sig}^{2\alpha_{1}-1}(S)-S^{T}\psi_{2}\text{Sig}^{2 \alpha_{2}-1}(S)+S^{T}l\\ \leq S^{T}\bar{\mathcal{W}}^{T}\mathcal{G}(\mathcal{X})-S^{T}\psi_{1}\text{Sig }^{2\alpha_{1}-1}(S)-S^{T}\psi_{2}\text{Sig}^{2\alpha_{2}-1}(S)\end{array} \ . \tag{39}\]  
Like Equations (27) and (28), we can obtain the following equality:  
\[\dot{V}_{1}\leq S^{T}\bar{\mathcal{W}}^{T}\mathcal{G}(\mathcal{X})-\eta_{1}(V_ {1})^{\alpha_{1}}-\eta_{2}(V_{1})^{\alpha_{2}}\, \tag{40}\]  
where \(\eta_{1}=\frac{\lambda_{\min}(\phi_{1})}{\lambda_{\max}(M)^{\alpha_{1}}}2^{ \alpha_{1}}n^{1-\alpha_{1}}\) and \(\eta_{2}=\frac{\lambda_{\min}(\phi_{1})}{\lambda_{\max}(M)^{\alpha_{2}}}2^{ \alpha_{2}}\). Select a Lyapunov function for the convergence of RBFNN as follows:  
\[V_{2}=\frac{1}{2}\mathcal{K}_{N}\sum_{i=1}^{n}\bar{\mathcal{W}}_{i}^{T}\dot{ \mathcal{W}}_{i}. \tag{41}\]  
By considering update law (35), we can derive the following expression for the derivative of Equation (41):  
\[