## 3 A Model-Based TSMC Approach Using PPCA
### Design of Model-Based TSMC Method Using PPCA
\begin{array}{l}\dot{V}_{0}=S^{T}\bar{M}(\alpha_{1})\dot{S}+\frac{1}{2}S^{ T}\dot{\bar{M}}(\alpha_{1})S\\ =S^{T}(\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)+\frac{ 1}{2}S^{T}\dot{\bar{M}}(\alpha_{1})S\\ =S^{T}(\bar{M}(\alpha_{1})\mathcal{Y}+\bar{C}(\alpha_{1},\alpha_{2})\mathcal{V }+\bar{G}(\alpha_{1})-\mathcal{F}-\tau)+\frac{1}{2}S^{T}\Big{(}\dot{\bar{M}}( \alpha_{1})-2\bar{C}(\alpha_{1},\alpha_{2})\Big{)}S\end{array} \;. \tag{24}\]  
Referring to Property 2, we have \(S^{T}\Big{(}\dot{\bar{M}}(\alpha_{1})-2\bar{C}(\alpha_{1},\alpha_{2})\Big{)}S=0\). Consequently, we obtain:  
\[\dot{V}_{0}=S^{T}(\bar{M}(\alpha_{1})\mathcal{Y}+\bar{C}(\alpha_{1},\alpha_{2}) \mathcal{V}+\bar{G}(\alpha_{1})-\mathcal{F}-\tau)\;. \tag{25}\]  
Based on Equation (25), the control law is formulated as follows:  
\[ \begin{cases}\tau=\tau_{eq}+\tau_{r}\\ \tau_{eq}=\bar{M}(\alpha_{1})\mathcal{Y}+\bar{C}(\alpha_{1},\alpha_{2})\mathcal{ V}+\bar{G}(\alpha_{1})-\mathcal{F}\\ \tau_{r}=\gamma\text{sign}(S)+\psi_{1}\text{Sig}^{2\alpha_{1}-1}(S)+\psi_{2} \text{Sig}^{2\alpha_{2}-1}(S)\end{cases} \;. \tag{26}\]  
In the control law, two terms are considered: \(\tau_{eq}\) and \(\tau_{r}\). The term \(\tau_{eq}\) is designed based on the mathematical model of the robotic manipulator, while the term \(\tau_{r}\) is designed as a fixed-time reaching control law. The constants \(\psi_{1}\), \(\psi_{2}\), and \(\gamma\) are positive, with \(\varpi_{1}>1\) and \(\frac{1}{2}<\varpi_{2}<1\). A summary of the control design is stated in the following theorem. Theorem 1: _Assuming that we have knowledge of the dynamic model of the robotic manipulator, including \(\bar{M}(\alpha_{1})\), \(\bar{C}(\alpha_{1},\alpha_{2})\), \(\bar{G}(\alpha_{1})\), and \(\mathcal{F}\), we can design the control law as in Equation (26) to achieve the prescribed performance for the robot in a fixed time._  
Proof of Theorem 1: By applying the control law (26) to Equation (25), we obtain:  
\[ \begin{array}{l}\dot{V}_{0}=S^{T}\Big{(}-\gamma\text{sign}(S)-\psi_{1}\text{ Sig}^{2\varpi_{1}-1}(S)-\psi_{2}\text{Sig}^{2\varpi_{2}-1}(S)\Big{)}\\ =-S^{T}\gamma\text{sign}(S)-S^{T}\psi_{1}\text{Sig}^{2\varpi_{1}-1}(S)-S^{T} \psi_{2}\text{Sig}^{2\varpi_{2}-1}(S)\\ \leq-\lambda_{\text{min}}(\gamma)\|S\|-S^{T}\psi_{1}\text{Sig}^{2\varpi_{1}-1} (S)-S^{T}\psi_{2}\text{Sig}^{2\varpi_{2}-1}(S)\\ \leq-S^{T}\psi_{1}\text{Sig}^{2\varpi_{1}-1}(S)-S^{T}\psi_{2}\text{Sig}^{2 \varpi_{2}-1}(S)\end{array} . \tag{27}\]  
Rewriting Equation (27) and using Property 1 and Lemma 3, we can obtain the following expression:  
\[ \begin{array}{l}\dot{V}_{0}\leq-\sum\limits_{i=1}^{n}\psi_{1i}|S_{i}|^{2 \varpi_{1}}-\sum\limits_{i=1}^{n}\psi_{2i}|S_{i}|^{2\varpi_{2}}\\ \leq-\lambda_{\text{min}}(\psi_{1})2^{\varpi_{1}}n^{1-\varpi_{1}}\Big{(}\frac {1}{2}\sum\limits_{i=1}^{n}S_{i}^{2}\Big{)}^{\varpi_{1}}-\lambda_{\text{min} }(\psi_{2})2^{\varpi_{2}}\Big{(}\frac{1}{2}\sum\limits_{i=1}^{n}S_{i}^{2} \Big{)}^{\varpi_{2}}\\ \leq-\lambda_{\text{min}}(\psi_{1})2^{\varpi_{1}}n^{1-\varpi_{1}}\Big{(}\frac {1}{2}\|S\|^{2}\Big{)}^{\varpi_{1}}-\lambda_{\text{min}}(\psi_{2})2^{\varpi_{ 2}}\Big{(}\frac{1}{2}\|S\|^{2}\Big{)}^{\varpi_{2}}\\ \leq-\frac{\lambda_{\text{min}}(\psi_{1})}{\lambda_{\text{max}}(M)^{\varpi_{ 1}}}2^{\varpi_{1}}n^{1-\varpi_{1}}\Big{(}\frac{1}{2}S^{T}MS\Big{)}^{\varpi_{1}} -\frac{\lambda_{\text{min}}(\psi_{2})}{\lambda_{\text{max}}(M)^{\varpi_{2}}} 2^{\varpi_{2}}\Big{(}\frac{1}{2}S^{T}MS\Big{)}^{\varpi_{2}}\\ \leq-\eta_{1}(V_{0})^{\varpi_{1}}-\eta_{2}(V_{0})^{\varpi_{2}}\end{array} \, \tag{28}\]  
where \(\eta_{1}=\frac{\lambda_{\text{min}}(\psi_{1})}{\lambda_{\text{max}}(M)^{\varpi _{1}}}2^{\varpi_{1}}n^{1-\varpi_{1}}\) and \(\eta_{2}=\frac{\lambda_{\text{min}}(\psi_{2})}{\lambda_{\text{max}}(M)^{\varpi _{2}}}2^{\varpi_{2}}\). In practical scenarios, computing the dynamic model of a robot manipulator can be a challenging task due to several factors.