## Appendix A Proof of Theorem 1
The time derivative of \(V_{oi}\) is given as  
\[\dot{V}_{oi}\!=\!2\gamma_{s}s_{is}s_{is}\!-\!2\hat{W}_{i}^{T}F_{\,i}^{-1}\hat{ W}_{i}\!+\!\hat{W}_{i}^{T}F_{\,i}^{-1}\hat{W}_{i}\!+\!\frac{2\gamma_{f}s_{if} ds_{if}}{dt_{\epsilon}}.\] (A3)  
Noting the result of the projection operation, if \(\hat{W}_{i}(0)\in\Omega_{\epsilon_{ui}}\), then the parameter learning law ensures \(\hat{W}_{i}(t)\in\Omega_{\epsilon_{ui}}\) and  
\[\hat{W}_{i}^{T}F_{\,i}^{-1}\hat{W}_{i}\!\geq\!\hat{W}_{i}^{T}F_{\,i}^{-1} \Bigg{(}\frac{\xi_{i}}{\xi_{fi}^{2}}b_{i}F_{\,i}\Phi_{f}\vec{\psi}_{fi}\!+\! \gamma_{s}s_{is}F_{\,i}\Phi s_{is}\!-\!\gamma_{f}K_{0}F_{\,i}\Phi s_{if}\Bigg{)}.\] (A4)  
Applying the foregoing inequality to (A3), one obtains  
\[\dot{V}_{oi}\leq-2\frac{\xi_{i}}{\xi_{fi}^{2}}b_{i}\hat{W}_{i}^{T }\[PERSON]_{f}\vec{\psi}_{fi}\!-\!2\gamma_{s}g_{is}F_{\,i}\tilde{W}_{i}^{T}F_{\,i} ^{-1}\[PERSON] s_{is}+2\gamma_{f}K_{0}F\,\tilde{W}_{i}^{T}\dot{F}_{\,i}^{-1}\[PERSON] s _{if}\] \[+2\gamma_{s}s_{is}\dot{s}_{is}\!+\!\hat{W}_{i}^{T}F_{\,i}^{-1}\hat {W}_{i}\!+\!\frac{2\gamma_{f}s_{if}ds_{if}}{dt_{\epsilon}}.\] (A5)  
Case 1a: When \(|\vec{\psi}_{fi}|>\overline{\xi}_{fi}\) & \(\[PERSON]_{f}^{T}F_{\,i}\[PERSON]_{f}>\overline{\xi}_{fi}\) hold, and \(||\hat{W}_{i}||<c_{xi}\) or \(||\hat{W}_{i}||=c_{xi}\) & \(\hat{W}_{i}^{T}\bullet\leq 0\) hold, that is, \(b_{i}=1\) and \(b_{p}=1\) in this case. Combining (A5) and (39a), one obtains\[\dot{V}_{oi}\leq -\frac{\xi_{i}}{\xi_{fi}^{2}}\tilde{W}_{i}^{T}\[PERSON]_{f}(\tilde{W}_{i}^ {T}\[PERSON]_{f}+\xi_{fi})-\frac{\xi_{i}}{\xi_{fi}^{2}}(\tilde{\psi}_{fi}-\zeta_{fi}) \tilde{\psi}_{fi}\] \[-2\gamma_{s}g_{is}F_{i}\dot{W}_{i}^{T}F_{i}^{-1}\[PERSON]_{is}+2\gamma_ {f}K_{0}F_{i}\dot{W}_{i}^{T}F_{i}^{-1}\[PERSON]_{is}-L_{i}\tilde{W}_{i}^{T}F_{i}^{- 1}\tilde{F}_{i}\dot{F}_{i}^{-1}\tilde{W}_{i}\] \[+j_{i}\tilde{W}_{i}^{T}F_{i}^{-1}(F_{i}\[PERSON]_{f}\[PERSON]_{f}^{T}F_{i}- F_{i}\[PERSON]_{f}\[PERSON]_{f}^{T}F_{i})F_{i}^{-1}\tilde{W}_{i}\] \[+2\gamma_{s}s_{is}g_{is}\left(-k_{1i}^{s}s_{is}-k_{2i}^{s}\text{ sig}(s_{is})^{e_{s}}-k_{3i}^{s}\tanh(\frac{s_{is}}{\epsilon_{s}})+\tilde{W}_{i}^{T} \Phi+\zeta_{i}\right)\] \[V_{oi}=\gamma_{s}s_{is}^{2}+\hat{W}_{i}^{T}F_{\,i}^{-1}\hat{W}_{i}+\epsilon \gamma_{f}s_{if}^{2},\]0 \[V_{oi}=\gamma_{s}s_{is}^{2}+\hat{W}_{i}^{T}F_{\,i}^{-1}\hat{W}_{i}+\epsilon \gamma_{f}s_{if}^{2},\]1 \[V_{oi}=\gamma_{s}s_{is}^{2}+\hat{W}_{i}^{T}F_{\,i}^{-1}\hat{W}_{i}+\epsilon \gamma_{f}s_{if}^{2},\]2 \[V_{oi}=\gamma_{s}s_{is}^{2}+\hat{W}_{i}^{T}F_{\,i}^{-1}\hat{W}_{i}+\epsilon \gamma_{f}s_{if}^{2},\]3  
Using Lemma 2, the above equation can be further rewritten as:  
\[V_{oi}=\gamma_{s}s_{is}^{2}+\hat{W}_{i}^{T}F_{\,i}^{-1}\hat{W}_{i}+\epsilon \gamma_{f}s_{if}^{2},\]4 \[V_{oi}=\gamma_{s}s_{is}^{2}+\hat{W}_{i}^{T}F_{\,i}^{-1}\hat{W}_{i}+\epsilon \gamma_{f}s_{if}^{2},\]5 \[V_{oi}=\gamma_{s}s_{is}^{2}+\hat{W}_{i}^{T}F_{\,i}^{-1}\hat{W}_{i}+\epsilon \gamma_{f}s_{if}^{2},\]6 \[V_{oi}=\gamma_{s}s_{is}^{2}+\hat{W}_{i}^{T}F_{\,i}^{-1}\hat{W}_{i}+\epsilon \gamma_{f}s_{if}^{2},\]7 \[V_{oi}=\gamma_{s}s_{is}^{2}+\hat{W}_{i}^{T}F_{\,i}^{-1}\hat{W}_{i}+\epsilon \gamma_{f}s_{if}^{2},\]8  
\[V_{oi}=\gamma_{s}s_{is}^{2}+\hat{W}_{i}^{T}F_{\,i}^{-1}\hat{W}_{i}+\epsilon \gamma_{f}s_{if}^{2},\]9 From \(0<\xi_{i}<1\) and \(\phi_{f}^{T}F_{i}\phi_{f}>\xi_{fi}^{2}\), one obtains \((1-\varsigma_{i})\xi_{fi}^{2}+\varsigma_{i}\phi_{f}^{T}F_{i}\phi_{f}>\xi_{fi}^{2}\). Noting \(0<L_{i0}<1\), it can be obtained \((\varsigma_{i}/\xi_{fi}^{2})-j_{i}>0\). It is clear that \(V_{oi}<0\) is true as long as \(V>\frac{\xi_{ni}}{\alpha_{v1i}}\). Case 1b: When \(|\tilde{\psi}_{fi}|>\overline{\zeta}_{fi}\) & \(\Phi_{f}^{T}F_{i}\Phi_{f}>\overline{\xi}_{fi}^{2}\) hold, and \(||\tilde{W}_{i}||>c_{wi}\) or \(||\tilde{W}_{i}||=c_{wi}\) & \(\tilde{W}_{i}^{T}\bullet>0\) hold, that is, \(b_{i}=1\) and \(b_{p}=0\) in this case, which implies \(\dot{F}_{i}=0\).