## 4 A Model-Free TSMC Approach Using RBFNN and PPCA
### Design of Model-Free TSMC Method Using RBFNN and PPCA
\begin{array}{c}\dot{V}_{2}=-\mathcal{K}_{N}\sum_{i=1}^{n}\bar{\mathcal{W}} _{i}^{T}\dot{\mathcal{W}}_{i}\\ =-\sum_{i=1}^{n}\bar{\mathcal{W}}_{i}^{T}\Big{(}\mathcal{G}(\mathcal{X})S_{i}- \psi_{3}\dot{\mathcal{W}}_{i}^{2\alpha_{1}-1}-\psi_{4}\dot{\mathcal{W}}_{i}^{ 2\alpha_{2}-1}\Big{)}\\ =-\sum_{i=1}^{n}\bar{\mathcal{W}}_{i}^{T}\mathcal{G}(\mathcal{X})S_{i}+\psi_{3} \sum_{i=1}^{n}\bar{\mathcal{W}}_{i}^{T}\dot{\mathcal{W}}_{i}^{2\alpha_{1}-1}+ \psi_{4}\sum_{i=1}^{n}\bar{\mathcal{W}}_{i}^{T}\dot{\mathcal{W}}_{i}^{2\alpha _{2}-1}\end{array} \ . \tag{42}\]  
By utilizing the inequality:  
\[ \begin{cases}\bar{w}_{ij}\bar{w}_{ij}^{2\alpha_{1}-1}\leq w_{ij}^{2\alpha_{1}}- \bar{w}_{ij}^{2\alpha_{1}}\\ \bar{w}_{ij}\bar{w}_{ij}^{2\alpha_{2}-1}\leq\
u_{1}w_{ij}^{2\alpha_{2}}-\
u_{2 }\bar{w}_{ij}^{2\alpha_{2}}\,\end{cases} \, \tag{43}\]  
where \(w_{ij}\) is the \(j\)th element of the vector \(\mathcal{W}_{i}\), \(i=1,\ldots,n\), and \(j=1,\ldots,l\). Consequently, Equation (42) can be rewritten as follows:  
\[ \begin{split}\dot{V}_{2}&\leq-\sum\limits_{i=1}^{n} \bar{\mathcal{W}}_{i}^{T}\mathcal{G}(\mathcal{X})S_{i}+\psi_{3}\sum\limits_{i=1 }^{n}\sum\limits_{j=1}^{l}\left(w_{ij}^{2\omega_{1}}-\psi_{ij}^{2\omega_{1}} \right)+\psi_{4}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{l}\left(\
u_{1}w_{ij }^{2\omega_{2}}-\
u_{2}\psi_{ij}^{2\omega_{2}}\right)\\ &=-\sum\limits_{i=1}^{n}\bar{\mathcal{W}}_{i}^{T}\mathcal{G}( \mathcal{X})S_{i}-\psi_{3}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{l}\left( \bar{w}_{ij}^{2\omega_{1}}\right)-\psi_{4}\
u_{2}\sum\limits_{i=1}^{n}\sum \limits_{j=1}^{l}\left(\bar{w}_{ij}^{2\omega_{2}}\right)+\psi_{3}\sum\limits_ {i=1}^{n}\sum\limits_{i=1}^{l}\left(w_{ij}^{2\omega_{1}}\right)+\psi_{4}\
u_ {1}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{l}\left(w_{ij}^{2\omega_{2}}\right) \\ &=-\sum\limits_{i=1}^{n}\bar{\mathcal{W}}_{i}^{T}\mathcal{G}( \mathcal{X})S_{i}-\psi_{3}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{l}\left( \bar{w}_{ij}^{2\omega_{1}}\right)-\psi_{4}\
u_{2}\sum\limits_{i=1}^{n}\sum \limits_{j=1}^{l}\left(\bar{w}_{ij}^{2\omega_{2}}\right)+\hbar\\ &\leq-\sum\limits_{i=1}^{n}\bar{\mathcal{W}}_{i}^{T}\mathcal{G}( \mathcal{X})S_{i}-\psi_{3}1^{1-\omega_{1}}\sum\limits_{i=1}^{n}\left(\sum \limits_{j=1}^{l}\bar{w}_{ij}^{2}\right)^{\omega_{1}}-\psi_{4}\
u_{2}\sum \limits_{i=1}^{n}\left(\sum\limits_{j=1}^{l}\bar{\omega}_{ij}^{2}\right)^{ \omega_{2}}+\hbar\\ &=-\sum\limits_{i=1}^{n}\bar{\mathcal{W}}_{i}^{T}\mathcal{G}( \mathcal{X})S_{i}-\left(\frac{2}{\kappa_{N}}\right)^{\omega_{1}}\psi_{3}1^{1- \omega_{1}}\left(\frac{\kappa_{N}}{2}\sum\limits_{i=1}^{n}\bar{\mathcal{W}}_{ i}^{T}\dot{\mathcal{W}}_{i}\right)^{\omega_{1}}-\left(\frac{2}{\kappa_{N}} \right)^{\omega_{2}}\psi_{4}\
u_{2}\left(\frac{\kappa_{N}}{2}\sum\limits_{i=1 }^{n}\bar{\mathcal{W}}_{i}^{T}\dot{\mathcal{W}}_{i}\right)^{\omega_{2}}+\hbar\\ &\leq-\sum\limits_{i=1}^{n}\bar{\mathcal{W}}_{i}^{T}\mathcal{G}( \mathcal{X})S_{i}-\eta_{3}(V_{2})^{\omega_{1}}-\eta_{4}(V_{2})^{\omega_{2}}+ \hbar\\ \end{split} \ \, \tag{44}\]  
\[\text{where }\eta_{3}=\left(\frac{2}{\kappa_{N}}\right)^{\omega_{1}}\psi_{3}1^{ -\omega_{1}},\,\eta_{4}=\left(\frac{2}{\kappa_{N}}\right)^{\omega_{2}}\psi_{4} \
u_{2},\,\text{and }\hbar=\psi_{3}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{l}\left(w_{ij}^{2 \omega_{1}}\right)+\psi_{4}\
u_{1}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{l} \left(w_{ij}^{2\omega_{2}}\right)+\psi_{4}\
u_{1}\sum\limits_{i=1}^{n}\sum \limits_{j=1}^{l}\left(w_{ij}^{2\omega_{2}}\right)\]  
\[\left(w_{ij}^{2\omega_{2}}\right).\]  
Choose \(V\) as the Lyapunov function for the convergence and stability of the entire control system that includes the NN. Hence, we have \(V=V_{1}+V_{2}\). From the results of Equations (40) and (44), the derivative of \(V\) is given by:  
\[ \begin{split}\dot{V}&\leq-\eta_{1}(V_{1})^{\omega_{1 }}-\eta_{2}(V_{1})^{\omega_{2}}-\eta_{3}(V_{2})^{\omega_{1}}-\eta_{4}(V_{2})^{ \omega_{2}}+\hbar\\ &\leq-\theta_{1}(V)^{\omega_{1}}-\phi_{2}(V)^{\omega_{2}}+\hbar \end{split}