## Appendix E Proof of Proposition 4  
The derivative of \(\hat{z}\) satisifies  
\[J\hat{z}=[S(J(z+\hat{\omega}_{d}))+V_{a}D](z+\hat{\omega}_{d})+V_{a}^{2}B[u-u _{\text{trim}}]+\Delta(v_{r},t)-J\hat{\omega}_{d}. \tag{10}\]  
In closed loop with the control law (45), we get  
\[J\hat{z}=\delta B\big{[}-k_{1}e_{\eta}-K_{2}\hat{z}-\tilde{\Delta}\big{]}+[V_ {a}D+S(\hat{\omega}_{d})J]\hat{z}+S(J(\hat{z}+\hat{\omega}_{d}))\hat{z}+d_{1}, \tag{11}\]  
where  
\[d_{1}=E\Delta(v_{r},t)+[J-E\hat{J}]\hat{\omega}_{d}-V_{a}[\tilde{D}-E\hat{D}] \hat{\omega}_{d}+S(\hat{J}\hat{\omega}_{d})\hat{\omega}_{d}-ES(\hat{J}\hat{ \omega}_{d})\hat{\omega}_{d}-\delta B\hat{J}(\hat{\omega}_{d}-\hat{\omega}_{d }^{{}^{\prime}}).\]  
The time derivative of \(V_{1}=k_{1}\Psi(\eta,\eta_{d})\) now becomes  
\[\dot{V}_{1}=k_{1}\Big{(}\omega-\omega_{d}^{\parallel}-\Pi_{\eta }^{\perp}\omega_{d}^{\perp}\Big{)}^{\top}e_{\eta} =k_{1}\Big{(}\hat{z}-\kappa e_{\eta}-\Pi_{\eta}^{\perp}\omega_{d} ^{\perp}\Big{)}^{\top}e_{\eta}\] \[=-\kappa k_{1}\|e_{\eta}\|^{2}+k_{1}\hat{z}^{\top}e_{\eta}-k_{1}e _{\eta}^{\top}\Pi_{\eta}^{\perp}\omega_{d}^{\perp}.\]  
Let a [PERSON] function candidate be given by \(\dot{V}_{2}=V_{1}+\hat{z}^{\top}j\hat{z}/2\), whose time derivative satisfies  
\[\dot{V}_{2}=\dot{V}_{1}+\hat{z}^{\top}j\hat{z} =-\kappa k_{1}\|e_{\eta}\|^{2}-k_{1}e_{\eta}^{\top}\Pi_{\eta}^{ \perp}\omega_{d}^{\perp}+\hat{z}^{\top}\big{[}k_{1}e_{\eta}+J\hat{z}\big{]}\] \[=-\kappa k_{1}\|e_{\eta}\|^{2}-\hat{z}^{\top}\big{[}\delta BK_{2} +S(\hat{\omega}_{d})J]\hat{z}+\hat{z}^{\top}d_{2},\]  
where \(d_{2}=d_{1}-\delta B\tilde{\Delta}+k_{1}Ee_{\eta}-k_{1}e_{\eta}^{\top}\Pi_{ \eta}^{\perp}\omega_{d}^{\perp}\). The time derivative of \(\hat{\omega}_{d}\) is  
\[\dot{\hat{\omega}}_{d}=\dot{\omega}_{d}^{\parallel}-\kappa\hat{e}_{\eta},\]  
where \(\hat{e}_{\eta}\) can be bounded using  
\[\|\hat{e}_{\eta}\|\leq\|\omega_{d}^{\perp}\|+\|e_{\omega}\|\leq 2\|\omega_{d}^{ \perp}\|+\|\hat{z}\|+\kappa\|e_{\eta}\|.\]  
From Assumption 5 we get  
\[\|\dot{\hat{\omega}}_{d}\| \leq b_{\dot{\omega}_{d}^{\parallel}}\|\omega\|+c_{\dot{\omega} _{d}^{\perp}}+2\kappa\|\omega_{d}^{\perp}\|+\kappa\|\hat{z}\|+\kappa^{2}\|e_{ \eta}\|\] \[J\hat{z}=\delta B\big{[}-k_{1}e_{\eta}-K_{2}\hat{z}-\tilde{\Delta}\big{]}+[V_ {a}D+S(\hat{\omega}_{d})J]\hat{z}+S(J(\hat{z}+\hat{\omega}_{d}))\hat{z}+d_{1}, \tag{11}\]0  
where we have used \(\|\omega\|\leq\|\hat{z}\|+\|\omega_{d}^{\parallel}\|+\kappa\|e_{\eta}\|\), from the definition of \(\hat{z}\). Now, it is not difficult to show that \(d_{2}\) satisfies  
\[J\hat{z}=\delta B\big{[}-k_{1}e_{\eta}-K_{2}\hat{z}-\tilde{\Delta}\big{]}+[V_ {a}D+S(\hat{\omega}_{d})J]\hat{z}+S(J(\hat{z}+\hat{\omega}_{d}))\hat{z}+d_{1}, \tag{11}\]1  
where  
\[J\hat{z}=\delta B\big{[}-k_{1}e_{\eta}-K_{2}\hat{z}-\tilde{\Delta}\big{]}+[V_ {a}D+S(\hat{\omega}_{d})J]\hat{z}+S(J(\hat{z}+\hat{\omega}_{d}))\hat{z}+d_{1}, \tag{11}\]2 \[J\hat{z}=\delta B\big{[}-k_{1}e_{\eta}-K_{2}\hat{z}-\tilde{\Delta}\big{]}+[V_ {a}D+S(\hat{\omega}_{d})J]\hat{z}+S(J(\hat{z}+\hat{\omega}_{d}))\hat{z}+d_{1}, \tag{11}\]3 (112) \[J\hat{z}=\delta B\big{[}-k_{1}e_{\eta}-K_{2}\hat{z}-\tilde{\Delta}\big{]}+[V_ {a}D+S(\hat{\omega}_{d})J]\hat{z}+S(J(\hat{z}+\hat{\omega}_{d}))\hat{z}+d_{1}, \tag{11}\]4 \[J\hat{z}=\delta B\big{[}-k_{1}e_{\eta}-K_{2}\hat{z}-\tilde{\Delta}\big{]}+[V_ {a}D+S(\hat{\omega}_{d})J]\hat{z}+S(J(\hat{z}+\hat{\omega}_{d}))\hat{z}+d_{1}, \tag{11}\]5  
Inserting this into the expression for \(\hat{V}_{2}\) gives  
\[J\hat{z}=\delta B\big{[}-k_{1}e_{\eta}-K_{2}\hat{z}-\tilde{\Delta}\big{]}+[V_ {a}D+S(\hat{\omega}_{d})J]\hat{z}+S(J(\hat{z}+\hat{\omega}_{d}))\hat{z}+d_{1}, \tag{11}\]6 \[J\hat{z}=\delta B\big{[}-k_{1}e_{\eta}-K_{2}\hat{z}-\tilde{\Delta}\big{]}+[V_ {a}D+S(\hat{\omega}_{d})J]\hat{z}+S(J(\hat{z}+\hat{\omega}_{d}))\hat{z}+d_{1}, \tag{11}\]7  
Let \(\gamma=\lambda_{\min}^{\text{sym}(\delta Bk_{2})}-\lambda_{\max}^{I}(c_{ \omega_{d}^{\perp}}+\kappa)-b\).