## 3 Controller Design
### Stability Analysis
\begin{split}\hat{V}_{c}&=\hat{W}_{c}^{\top}\bar{ \theta}\Big{[}-\theta^{\top}\hat{W}_{c}-\lambda\hat{W}_{a}^{\top}\
abla\sigma_{ s}B_{1}\tanh\Big{(}\hat{M}_{a}^{*}\Big{)}-\epsilon_{H}-\lambda B_{1}^{\top} \
abla\sigma_{s}^{\top}\hat{W}_{a}\text{sign}\left(\hat{M}_{a}^{*}\right)\\ +&\lambda B_{1}^{\top}\
abla\sigma_{s}^{\top}W^{*} \text{sign}\Big{(}\hat{M}_{o}^{*}\Big{)}+\lambda^{2}\bar{\omega}(\epsilon_{a} -\epsilon_{a^{*}})\Big{]}+p_{c}\hat{W}_{c}^{\top}\left(\hat{W}_{c}-\hat{W}_{a} \right)\\ &=\hat{W}_{c}^{\top}\bar{\theta}\Big{[}-\epsilon_{H}+\lambda^{2} \bar{\omega}(\epsilon_{a}-\epsilon_{a^{*}})+\lambda B_{1}^{\top}\
abla\sigma_{ s}^{\top}W^{*}\text{sign}\Big{(}\hat{M}_{o}^{*}\Big{)}\Big{]}\\ &-\hat{W}_{c}^{\top}\bar{\theta}\bigg{\{}\lambda\hat{W}_{a}^{ \top}\
abla\sigma_{s}B_{1}\tanh\left(\hat{M}_{a}^{*}\right)+\lambda\hat{W}_{a} ^{\top}\
abla\sigma_{s}B_{1}\Bigg{[}\tanh\left(\frac{\hat{M}_{a}^{*}}{\kappa} \right)+\kappa_{1}\Bigg{]}\bigg{\}}\\ &-\hat{W}_{c}^{\top}\bar{\theta}\theta^{\top}\hat{W}_{c}+p_{c} \hat{W}_{c}^{\top}\left(\hat{W}_{c}-\hat{W}_{a}\right)\end{split}  \tag{44}\]  
where \(||\epsilon_{a}||\), \(||\epsilon_{a^{*}}||\) and \(|\kappa_{1}|\) are bounded. For ease of notation, two bounded vectors \(\aleph_{1}\) and \(\aleph_{2}\) are defined as:  
\[ \begin{split}\aleph_{1}&=\Bigg{\{}\lambda\
abla \sigma_{s}B_{1}\tanh\left(\frac{\hat{M}_{a}^{*}}{\kappa}\right)-\lambda\
abla \sigma_{s}B_{1}\tanh\left(\hat{M}_{a}^{*}\right)\Bigg{\}}\bar{\theta}^{\top}W^ {*},\\ \aleph_{2}&=\lambda W^{*\top}\
abla\sigma_{s}B_{1} \text{sign}\Big{(}\hat{W}_{o}^{*}\Big{)}-\lambda W^{*\top}\
abla\sigma_{s}B_{ 1}\Bigg{[}\tanh\left(\frac{\hat{M}_{a}^{*}}{\kappa}\right)+\kappa_{1}\Bigg{]} \\ &-\epsilon_{H}+\lambda^{2}\bar{\omega}(\epsilon_{a}-\epsilon_{a^{ *}}),\end{split}  \tag{45}\]  
then (44) can be further sorted into the following structure  
\[ \begin{split}\hat{V}_{c}&=-\hat{W}_{c}^{\top}\bar{ \theta}\theta^{\top}\hat{W}_{c}+p_{c}\hat{W}_{c}^{\top}\big{(}\hat{W}_{c}-\hat{ W}_{a}\big{)}+\hat{W}_{c}\bar{\theta}\aleph_{2}+\hat{W}_{a}^{\top}\aleph_{1}+ \lambda\kappa_{1}\hat{W}_{c}^{\top}\bar{\theta}\hat{W}_{a}^{\top}\
abla\sigma _{s}B_{1}\\ &+\lambda\hat{W}_{c}^{\top}\bar{\theta}\hat{W}_{a}^{\top}\
abla \sigma_{s}B_{1}\boldsymbol{\
abla}.\end{split}  \tag{46}\]  
Substituting (33), (41), and (46) into (35) yields  
\[ \begin{split}\hat{V}&\leq-V_{s}-\lambda\hat{W}_{a}^{ \top}\
abla\sigma_{s}B_{1}\tanh\left(\hat{M}_{a}^{*}\right)+\epsilon_{-}\hat{W }_{c}^{\top}\bar{\theta}\theta^{\top}\hat{W}_{c}+p_{c}\hat{W}_{c}^{\top}\big{(} \hat{W}_{c}-\hat{W}_{a}\big{)}\\ &+\hat{W}_{c}\bar{\theta}\aleph_{2}+\hat{W}_{a}^{\top}\aleph_{1}+ \lambda\kappa_{1}\hat{W}_{c}^{\top}\bar{\theta}\hat{W}_{a}^{\top}\
abla \sigma_{s}B_{1}-\lambda\hat{W}_{c}^{\top}\bar{\theta}\hat{W}_{a}^{\top}\
abla \sigma_{s}B_{1}\\ &+\lambda\hat{W}_{c}^{\top}\bar{\theta}\hat{W}_{a}^{\top}\
abla \sigma_{s}B_{1}\boldsymbol{\
abla}+\hat{W}_{a}^{\top}\Big{[}p_{c}\big{(}\hat{W }_{a}-\hat{W}_{c}\big{)}-\lambda\
abla\sigma_{s}B_{1}\boldsymbol{\
abla}\bar{ \theta}^{\top}\hat{W}_{c}\Big{]}\\ &\leq-V_{s}-\lambda\hat{W}_{a}^{\top}\
abla\sigma_{s}B_{1} \tanh\left(\hat{M}_{a}^{*}\right)+\epsilon_{v}-\hat{W}_{c}^{\top}\bar{\theta} \theta^{\top}\hat{W}_{c}+p_{c}\big{(}\hat{W}_{a}-\hat{W}_{c}\big{)}^{\top} \big{(}\hat{W}_{a}-\hat{W}_{c}\big{)}\\ &+\hat{W}_{c}\bar{\theta}\aleph_{2}+\hat{W}_{a}^{\top}\aleph_{1}+ \lambda\kappa_{1}\hat{W}_{c}^{\top}\bar{\theta}\hat{W}_{a}^{\top}\
abla\sigma_{s }B_{1}\\ &\leq-V_{s}-\hat{W}_{c}^{\top}\bar{\theta}\theta^{\top}\hat{W}_{c}+ \epsilon_{v1}+\hat{W}_{c}^{\top}\bar{\theta}\aleph_{3}\end{split}  \tag{47}\]  
where \(\epsilon_{v1}=\epsilon_{v}-\lambda\hat{W}_{a}^{\top}\
abla\sigma_{s}B_{1} \tanh\left(\hat{M}_{a}^{*}\right)+\hat{W}_{a}^{\top}\aleph_{1}\) and \(\aleph_{3}=\aleph_{2}+\lambda\kappa_{1}\hat{W}_{a}^{\top}\
abla\sigma_{s}B_{1}\).