## 2 Controller Design
### Sliding Perturbation Observer (SPO)
When \(\left|\widetilde{x}_{1j}\right|<\varepsilon_{0j}\), the saturation function \(sat(\widetilde{x}_{1j})\) is reduced to \(\frac{\widetilde{x}_{1j}}{\varepsilon_{0j}}\) as shown in Equation (10). The dynamics of the error state estimation shown in matrix form is:  
\[\left[ \begin{array}{c}\dot{\widetilde{x}}_{1j}\\ \dot{\widetilde{x}}_{2j}\\ \dot{\widetilde{x}}_{3j}\end{array} \right]=\left[\begin{array}{ccc}-\frac{k _{1j}}{\varepsilon  
It shows that SPO is similar to a conventional Luenberger observer. The pole placement method is used to satisfy the stability and design the convergence time. The associated characteristic equation is:  
\[[\lambda^{3}+(\frac{k_{1j}}{\varepsilon_{0j}})\lambda^{2}+(\frac{k_{2j}}{ \varepsilon_{0j}})\lambda+\alpha^{2}{}_{3j}(\frac{k_{2j}}{\varepsilon_{0j}})]=0. \tag{14}\]  
Let \(p(\lambda_{d})=(\lambda+\lambda_{d})^{3}\) be the desired characteristic polynomial; this brings out the following design solutions:  
\[ \begin{array}{c}\frac{k_{1j}}{\varepsilon_{0j}-3\lambda_{d}}\\ \frac{k_{2j}}{k_{1j}=\lambda_{d}}\\ \alpha_{3j}=\sqrt{\frac{\lambda_{d}}{3}}.\end{array}  \tag{15}\]  
The transfer function of Equation (11) turns out to be as follows:  
\[ \begin{array}{c}\widetilde{\Psi}_{j}(p)=\frac{p\left[p^{2}+3\lambda_{d}p+3 \lambda_{d}^{2}\right]}{\left(p+\lambda_{d}\right)^{3}}\left(-\Psi_{j}(p) \right)\\ \frac{\Psi_{j}}{\Psi_{j}}=1+\frac{\widetilde{\Psi}_{j}}{\Psi_{j}}=\frac{ \lambda^{3}{}_{d}}{(p+\lambda_{d})^{3}}.\end{array}  \tag{16}\]  
**Remark 2**. : _It shows the \(\lambda_{d}\) is a design parameter, which determines the bandwidth of the high filter between the perturbation estimation error and real perturbation while a low pass filter relation is between estimated perturbation and real perturbation. The precision of estimated perturbation is restrained to only the slow motion mode._  
An example of the effect of \(\lambda_{d}\) on Equation (16) is shown by the bode diagram in Figure 2 where we give a different value to \(\lambda_{d}\). The red dash line presents \(\lambda_{d}=10\), and the solid blue line is \(\lambda_{d}=20\). The results of Figure 2 show that when \(\lambda_{d}\) increased, the estimated perturbation was much closer to the real perturbation. Although in terms of reducing errors and observing accuracy, a large gain \(\lambda_{d}\) is required. The breaking point of the sliding observer dynamics inside a manifold cannot exceed \(\frac{1}{5\tau^{\text{thr}}}\) as shown by [PERSON] et al. [21]. Where \(\tau^{\text{thr}}\) is considered the dominant time delay depending on sampling time in the hardware. A similar proposition is also presented by [PERSON] and [PERSON] [16]. When reaching the sliding phase, the dynamics of the observer becomes \(\dot{\widetilde{x}}_{1j}=\widetilde{x}_{2j}-\frac{k_{1j}}{\varepsilon_{0j}} \widetilde{x}_{1j}\). The breaking point from the dynamics of the observer becomes \(\frac{k_{1j}}{\varepsilon_{0j}}\). Therefore, the best selection of \(\lambda_{d}\) in Equation (16) is  
\[\lambda_{d}=\frac{1}{15\tau^{\text{thr}}}. \tag{17}\]  
This simple entity defines the necessary SPO parameters based on the limitation of the hardware. **Assumption 2**. : \(R_{max}\) _represents the maximum ratio between perturbation \(\Psi_{j}(p)\) and \(\widetilde{x}_{2j}\) on some frequency as:_  
\[R_{max\;j}=MAX(\frac{p(p+3\lambda_{d})}{p^{3}+3\lambda_{d}p^{2}+3\lambda_{d}^{ 2}p+3\lambda_{d}^{~{}2}\alpha_{3j}{}^{2}}). \tag{18}\]