## 3 SMO Design  
The proposed SMO is expressed as follows:  
\[\left\{\begin{array}{l}\dot{\mathbf{s}}_{1}=\mathbf{s}_{2}+\mathbf{v}\\ \dot{\mathbf{s}}_{2}=\mathbf{f}(\mathbf{x}_{1},\mathbf{s}_{2})+\mathbf{g}(\mathbf{x}_{1})\mathbf{u}+ \omega\mathbf{v}\end{array}\right. \tag{4}\]  
where \(\mathbf{s}_{1}\) and \(\mathbf{s}_{2}\) are the states of the observer, and \(\omega\) is the observer gain. The symbol \(v\) denotes a switching function of the observation error, which is defined by  
\[\mathbf{v}=\left\{\begin{array}{l}\frac{\eta(\mathbf{x}_{1}-\mathbf{s}_{1})}{\|\mathbf{x}_{ 1}-\mathbf{s}_{1}\|}\qquad\text{if}\|\mathbf{x}_{1}-\mathbf{s}_{1}\|\
eq 0\\ 0\qquad\text{otherwise}\end{array}\right. \tag{5}\]  
where \(\eta\) is a positive parameter.  
**Theorem 1**.: _Based on Assumptions 1 and 2, the proposed SMO (4) ensures arbitrarily bounded estimation performance for the system (2) if the observer parameters are selected with large enough values._  
Proof of Theorem 1.: Denote the error term \(\widetilde{\mathbf{s}}=\mathbf{x}-\mathbf{s}\). Subtracting both sides of (4) from those of (2), the error dynamics are expressed as follows:  
\[\begin{array}{l}\ddot{\widetilde{\mathbf{s}}}_{1}=\widetilde{\mathbf{s}}_{2}-\mathbf{v} \\ \ddot{\widetilde{\mathbf{s}}}_{2}=\mathbf{\tilde{f}}+\mathbf{d}-\omega\mathbf{v}\end{array} \tag{6}\]  
Define a Lyapunov function \(V_{s1}=1/2\widetilde{s}_{1}^{T}\widetilde{\mathbf{s}}_{1}\). Differentiating it and substituting (5) into, it becomes  
\[\dot{V}_{s1}=\widetilde{\mathbf{s}}_{1}(\widetilde{\mathbf{s}}_{2}-\mathbf{v})\leq-\| \widetilde{\mathbf{s}}_{1}\|(\eta-\|\widetilde{\mathbf{s}}_{2}\|) \tag{7}\]  
where the observer parameter \(\eta\) is selected with a large enough value so that there exists a positive constant \(\eta_{0}\) satisfying \(\|\widetilde{\mathbf{s}}_{2}\|\leq\eta-\eta_{0}\). Therefore, one obtains  
\[\dot{V}_{s1}\leq-\sqrt{2}\eta_{0}\sqrt{V_{s1}} \tag{8}\]  
Based on this, \(\mathbf{s}_{1}\) converges to zero in finite time. After that, from the first equation of (6), it is observed that the switching function \(\mathbf{v}\) is equivalent to the error \(\widetilde{\mathbf{s}}_{2}\). Substituting it into the second equation of (6), the equivalent error dynamics are given by  
\[\dot{\widetilde{\mathbf{s}}}_{2}=\widetilde{\mathbf{f}}+\mathbf{d}-\omega\widetilde{\mathbf{s }}_{2} \tag{9}\]Another Lyapunov function is designed as \(V_{s2}=1/2\widetilde{s}_{2}T^{\widetilde{s}}\widetilde{s}_{2}\). The derivative of it is derived by  
\[\begin{array}{l}\dot{V}_{s2}=\widetilde{s}_{2}^{T}\big{(}\tilde{f}+\mathbf{d}- \omega\widetilde{s}_{2}\big{)}\\ \leq-\omega\widetilde{s}_{2}T^{\widetilde{s}}\widetilde{s}_{2}+\widetilde{s}_ {2}T^{\widetilde{s}}\widetilde{s}_{2}+\frac{1}{2}\tilde{f}\tilde{f}+\frac{1}{ 2}d^{T}d\end{array} \tag{10}\]  
Combining with Assumptions 1 and 2, the following inequality holds  
\[\begin{array}{l}\dot{V}_{s2}\leq-\Big{(}\omega-\frac{1}{2}\alpha^{\prime}-1 \Big{)}\widetilde{s}_{2}T^{\widetilde{s}}\widetilde{s}_{2}+\frac{1}{2}\Delta_ {d}{}^{2}\\ \leq-\lambda_{s2}V_{s2}+\Delta_{s2}\end{array} \tag{11}\]  
where \(\lambda_{s2}=(2\omega-\alpha_{f}{}^{2}-2)\) and \(\Delta_{s2}=0.5\Delta_{d}{}^{2}\).  
From this, the proof of Theorem 1 is completed. \(\square\)  
The joint velocity and disturbance estimations are expressed as follows:  
\[\begin{array}{l}\dot{\mathbf{x}}_{2}=\mathbf{s}_{2}+\mathbf{v}\\ \dot{\mathbf{d}}=\omega_{1}\mathbf{v}\end{array} \tag{12}\]  
The first estimation is derived since \(\dot{\mathbf{x}}_{2}=\dot{\mathbf{s}}_{1}\) converges to \(\mathbf{x}_{2}\) after the convergence time of \(\widetilde{\mathbf{s}}_{1}\). Similarly, \(\mathbf{d}\) can be estimated by \(\dot{\mathbf{d}}\) when comparing (2) and (4).  
**Remark 3**.: _A modified switching function is designed to attenuate the chattering effects on the estimated values in (12) as_  
\[v_{\rm mod}=\frac{\eta(\mathbf{x}_{1}-\mathbf{s}_{1})}{\|\mathbf{x}_{1}-\mathbf{s}_{1}\|+ \varepsilon_{v}} \tag{13}\]  
_where \(\varepsilon_{v}\)is a positive constant._