## 3 Sliding Mode Control with Sliding Perturbation Observer (SMCSPO)
The desired \(\dot{s}_{j}\) dynamics is represented as  
\[\dot{\hat{s}}_{j}=-K_{j}sat(\hat{s}_{j}) \tag{16}\]  
Thus, \(\dot{\hat{s}}_{j}\) can be calculated as  
\[\dot{\hat{s}}_{j}=\alpha_{3j}\overline{u}_{j}-\Big{[}k_{2j}/\varepsilon_{0j}+c _{j1}\Big{(}k_{1j}/\varepsilon_{0j}\Big{)}-\Big{(}k_{1j}/\varepsilon_{0j} \Big{)}^{2}\Big{]}\overline{x}_{1j}-\Big{(}k_{1j}/\varepsilon_{0j}\Big{)} \overline{x}_{2j}-\ddot{x}_{1 jd}+c_{j1}\Big{(}\hat{x}_{2j}-\dot{x}_{1 jd}\Big{)} +\dot{\psi}_{j} \tag{17}\]  
The control law is defined as  
\[\overline{u}_{j}=\frac{1}{\alpha_{3j}}\{-K_{j}sat(\hat{s}_{j})+\big{[}\frac{ k_{2j}}{\varepsilon_{0j}}+c_{j1}\frac{k_{1j}}{\varepsilon_{0j}}-\big{(}\frac{k_{1j}} {\varepsilon_{0j}}\big{)}^{2}\big{]}\overline{x}_{1j}+\ddot{x}_{1 jd}-c_{j1}( \hat{x}_{2j}-\dot{x}_{1 jd})-\dot{\psi}_{j}\} \tag{18}\]  
In the above equation, \(\overline{u}_{j}\) is a control input of SMCSPO. The actual \(s_{j}\) dynamics within the boundary layer \(\big{|}\dot{s}_{j}\big{|}\leq\varepsilon_{0j}\) becomes  
\[\dot{s}_{j}+\frac{K_{j}}{\varepsilon_{0j}}s_{j}=\bigg{[}\frac{k_{2j}}{ \varepsilon_{0j}}-\bigg{(}\frac{k_{1j}}{\varepsilon_{0j}}-\frac{K_{j}}{ \varepsilon_{0j}}\bigg{)}\bigg{(}c_{j1}-\frac{k_{1j}}{\varepsilon_{0j}}\bigg{)} \bigg{]}\overline{x}_{1j}-\bigg{(}c_{j1}+\frac{K_{j}}{\varepsilon_{0j}}\bigg{)} \widetilde{x}_{2j}-\dot{\overline{\psi}}_{j} \tag{19}\]  
The designed controller (SMCSPO) reduces the error between the real and the desired trajectories. The mechanical hardware limitations restrict the design procedure as described in [PERSON] et al. [40]. The instant at which \(\left|\hat{s}_{j}\right|\leq\varepsilon_{0j}\), the observer design, and the \(s_{j}\) dynamics can be represented in mathematical form as shown below:  
\[\left[ \begin{array}{c}\dot{\widetilde{x}}_{1j}\\ \dot{\widetilde{x}}_{2j}\\ \dot{\widetilde{x}}_{3j}\\ \dot{\hat{s}}_{j}\end{array} \right]=\left[ \begin{array}{cccc}-k_{1j}/ \varepsilon_{0j}&1&0&0\\ -\frac{k_{2j}}{\varepsilon_{0j}}&\alpha^{2}{}_{3j}&-\alpha_{3j}&0\\ 0&\alpha^{3}{}_{3j}&-\alpha^{2}{}_{3j}&0\\ k_{2j}/\varepsilon_{0j}-\left(c-k_{1j}/\varepsilon_{0j}\right)^{2}&-\left(2c+ \alpha^{2}{}_{3j}\right)&\alpha_{3j}&-c\end{array} \right]\cdot\left[ \begin{array} []{c}\widetilde{x}_{1j}\\ \widetilde{x}_{2j}\\ \widetilde{x}_{3j}\\ s_{j}\end{array} \right]+\left[ \begin{array}{c}0\\ 0\\ 1\\ 0\end{array} \right]\dot{\psi}_{j}/\alpha_{3j} \tag{20}\]  
where a square matrix of order 4 in Equation (20) represents the state matrix. Further supposing that \(\lambda\) represents the eigenvalues of the state matrix A, then its characteristic equation \(\mathrm{det}|\lambda I-A|=0\) can be expressed as,  
\[\left[\lambda+c_{j1}\right]\left[\lambda^{3}+\left(k_{1j}/\varepsilon_{0j} \right)\lambda^{2}+\left(k_{2j}/\varepsilon_{0j}\right)\lambda+\alpha^{2}{}_{ 3j}\left(k_{2j}/\varepsilon_{0j}\right)\right]=0 \tag{21}\]  
By implementing the pole-placement method, let us introduce a desired characteristic polynomial \(p(\lambda_{d})=\left(\lambda+\lambda_{d}\right)^{4}\) which leads to a design solution  
\[k_{1j}/\varepsilon_{0j}=3\lambda_{d}k_{2j}/k_{1j}=\lambda_{d}\alpha_{3j}= \sqrt{\lambda_{d}/3}c=K_{j}/\varepsilon_{0j}=\lambda_{d} \tag{22}\]  
It is clear from the equation that higher values of gain \(\lambda_{d}\) guarantee higher accuracy. In [41], the authors showed the limitations of sliding function dynamics. They suggested that \(\lambda_{d}\) should be less than \(1/5\tau^{\mathit{thr}}\), where \(w\) is the frequency and \(h\) is a positive number. These results were also validated in another study [42]. In this work, we selected the value of optical gain by using following formula:  
\[\lambda_{d}=\frac{1}{15\tau^{\mathit{thr}}} \tag{23}\]