## 3 A Model-Based TSMC Approach Using PPCA
### Design of Model-Based TSMC Method Using PPCA  
Substituting Equation (13) into Equation (15) yields:  
\[ \begin{array}{l}-(e_{2i}-\wp_{i})=\mathcal{K}_{1}\text{Si}\mathrm{g}^{\xi_{ 1}}(E_{1i})+\mathcal{K}_{2}\text{Si}\mathrm{g}^{\xi_{2}}(E_{1i})-S_{i}\\ -e_{2i}=\mathcal{K}_{1}\text{Si}\mathrm{g}^{\xi_{1}}(E_{1i})+\mathcal{K}_{2} \text{Si}\mathrm{g}^{\xi_{2}}(E_{1i})-S_{i}-\wp_{i}\\ (\alpha_{2i}-\alpha_{2 di})=\mathcal{K}_{1}\text{Si}\mathrm{g}^{\xi_{1}}(E_{1i})+ \mathcal{K}_{2}\text{Si}\mathrm{g}^{\xi_{2}}(E_{1i})-S_{i}-\wp_{i}\\ \alpha_{2i}=\mathcal{K}_{1}\text{Si}\mathrm{g}^{\xi_{1}}(E_{1i})+\mathcal{K}_{2 }\text{Si}\mathrm{g}^{\xi_{2}}(E_{1i})-S_{i}-\wp_{i}+\alpha_{2 di}\end{array} . \tag{17}\]We define \(\mathcal{V}=\left[ \begin{array}{cccc}\mathcal{V}_{1}&\ldots&\mathcal{V}_{n}\\ \end{array} \right]^{T}\) with \(\mathcal{V}_{i}=\mathcal{K}_{1}\text{Sig}^{\xi_{1}}(E_{1i})+\mathcal{K}_{2} \text{Sig}^{\xi_{2}}(E_{1i})-\wp_{i}+\alpha_{2 di}\), for \(i=1,\ldots,n\). Therefore, Equation (17) can be written as \(\alpha_{2i}=\mathcal{V}_{i}-S_{i}\), and its vector form is as follows:  
\[\alpha_{2}=\mathcal{V}-S\;. \tag{18}\]  
With respect to time, taking the derivative of Equation (15) yields the following results:  
\[\dot{S}_{i}=\dot{E}_{2i}+\mathcal{K}_{1}\xi_{1}|E_{1i}|^{\xi_{1}-1}\dot{E}_{1i }+\mathcal{K}_{2}\xi_{2}|E_{1i}|^{\xi_{2}-1}\dot{E}_{1i}\;. \tag{19}\]  
Substituting Equation (14) into Equation (19) yields:  
\[ \begin{array}{l}\dot{S}_{i}=(\dot{e}_{2i}-\mathcal{H}_{i})+\mathcal{K}_{1} \xi_{1}|E_{1i}|^{\xi_{1}-1}\dot{E}_{1i}+\mathcal{K}_{2}\xi_{2}|E_{1i}|^{\xi_{ 2}-1}\dot{E}_{1i}\\ \dot{S}_{i}=\dot{e}_{2i}+\mathcal{K}_{1}\xi_{1}|E_{1i}|^{\xi_{1}-1}\dot{E}_{1i }+\mathcal{K}_{2}\xi_{2}|E_{1i}|^{\xi_{2}-1}\dot{E}_{1i}-\mathcal{H}_{i}\\ \dot{S}_{i}=(\dot{\alpha}_{2 di}-\dot{\alpha}_{2i})+\mathcal{K}_{1}\xi_{1}|E_{1 i}|^{\xi_{1}-1}\dot{E}_{1i}+\mathcal{K}_{2}\xi_{2}|E_{1i}|^{\xi_{2}-1}\dot{E}_{1i}- \mathcal{H}_{i}\\ \dot{S}_{i}=\dot{\alpha}_{2 di}+\mathcal{K}_{1}\xi_{1}|E_{1i}|^{\xi_{1}-1}\dot{ E}_{1i}+\mathcal{K}_{2}\xi_{2}|E_{1i}|^{\xi_{2}-1}\dot{E}_{1i}-\mathcal{H}_{i}- \dot{\alpha}_{2i}\end{array} \;. \tag{20}\]  
By defining \(\mathcal{Y}=\left[ \begin{array}{cccc}\mathcal{Y}_{1}&\ldots&\mathcal{Y}_{n }\\ \end{array} \right]^{T}\) with \(\mathcal{Y}_{i}=\dot{\alpha}_{2 di}+\mathcal{K}_{1}\xi_{1}|E_{1i}|^{\xi_{1}-1 }\dot{E}_{1i}+\mathcal{K}_{2}\xi_{2}|E_{1i}|^{\xi_{2}-1}\dot{E}_{1i}\)\(-\mathcal{H}_{i}\), for \(i=1,\ldots,n\), Equation (20) can be written as \(\dot{S}_{i}=\mathcal{Y}_{i}-\dot{\alpha}_{2i}\), and its vector form is as follows:  
\[\dot{S}=\mathcal{Y}-\dot{\alpha}_{2}\;. \tag{21}\]  
By multiplying both sides of Equation (21) with \(\bar{M}(\alpha_{1})\) and applying Equations (4) and (18), the following expression is derived:  
\[ \begin{array}{l}\bar{M}(\alpha_{1})\dot{S}=\bar{M}(\alpha_{1})(\mathcal{Y}- \dot{\alpha}_{2})\\ =\bar{M}(\alpha_{1})\mathcal{Y}-\bar{M}(\alpha_{1})\dot{\alpha}_{2}\\ =\bar{M}(\alpha_{1})\mathcal{Y}-\bar{M}(\alpha_{1})\big{(}\bar{M}^{-1}(\alpha_ {1})(-\bar{C}(\alpha_{1},\alpha_{2})\alpha_{2}-\bar{G}(\alpha_{1})+\mathcal{F }+\tau)\big{)}\\ =\bar{M}(\alpha_{1})\mathcal{Y}+\bar{C}(\alpha_{1},\alpha_{2})\alpha_{2}+\bar {G}(\alpha_{1})-\mathcal{F}-\tau\\ =\bar{M}(\alpha_{1})\mathcal{Y}+\bar{C}(\alpha_{1},\alpha_{2})(\mathcal{V}-S)+ \bar{G}(\alpha_{1})-\mathcal{F}-\tau\\ =\bar{M}(\alpha_{1})\mathcal{Y}+\bar{C}(\alpha_{1},\alpha_{2})\mathcal{V}-\bar {C}(\alpha_{1},\alpha_{2})S+\bar{G}(\alpha_{1})-\mathcal{F}-\tau\end{array} \;. \tag{22}\]  
A Lyapunov function is considered as follows:  
\[V_{0}=\frac{1}{2}S^{T}\bar{M}(\alpha_{1})S\;. \tag{23}\]  
By differentiating \(V_{0}\) in Equation (23) with respect to time, we obtain the following expression:  
\[