## 2 Controller Design
### Sliding Perturbation Observer (SPO)  
The state space with a new state variable \(x_{3j}\) can be represented as:  
\[ \begin{array}{c}\dot{x}_{1j}=x_{2j}\\ \dot{x}_{2j}=a_{3j}\overline{u}_{j}+\Psi_{j}\\ \dot{x}_{3j}=\alpha_{3j}\dot{x}_{2j}-\frac{\dot{\Psi}_{j}}{\alpha_{3j}}\\ y_{j}=x_{1j}.\end{array}  \tag{4}\]  
The new state variable \(x_{3j}\) is defined as:  
\[x_{3j}=\alpha_{3j}x_{2j}-\frac{\Psi_{j}}{\alpha_{3j}}. \tag{5}\]  
The sliding perturbation observer (SPO) equations for a \(j\)-th link can be described as follows:  
\[\begin{array}{c}\dot{\dot{x}}_{1j}=\dot{x}_{2j}-k_{1j}sat(\overline{x}_{1j} ),\\ \dot{\dot{x}}_{2j}=\alpha_{3}\overline{u}_{j}-k_{2j}sat(\overline{x}_{1  
The existence condition of the sliding mode to be satisfied is derived as:  
\[ \begin{array}{l}\widetilde{x}_{2j}\leq k_{1j}(\text{ if }\overline{x}_{1j}> \varepsilon_{0j})\\ \widetilde{x}_{2j}\geq-k_{1j}(\text{ if }\overline{x}_{1j}<-\varepsilon_{0j}). \end{array}  \tag{8}\]  
where \(k_{1j}\) is a constant and larger than the maximum of \(\widetilde{x}_{2j}\), i.e., \(k_{1j}>\left|max(\widetilde{x}2_{j})\right|\). After the observer sliding mode begins (\(\widetilde{x}_{1j}=0\)), \(\widetilde{x}_{2j}\) dynamics becomes:  
\[\dot{\widetilde{x}}_{2j}+(\frac{k_{2j}}{k_{1j}})\widetilde{x}_{2j}=\widetilde {\Psi}_{j}. \tag{9}\]  
It is desirable to place the breakpoint \(\frac{k_{2j}}{k_{1j}}\) as high as possible in order to increase the attenuation from \(\widetilde{\Psi}_{j}\) to \(\widetilde{x}_{2j}\), and improve the estimation accuracy of \(x_{2j}\) at the same time. If we select that \(k_{2j}\) is larger than the maximum of \(\widetilde{\Psi}_{j}\), i.e., \(k_{2j}\geq\left|max(\widetilde{\Psi}_{j})\right|\), it can assure the steady state \(\left|\widetilde{x}_{2j}\right|\leq k_{1j}\) from Equation (9), will force the states to stay in the sliding region. When \(\left|\widetilde{x}_{1j}\right|\leq\varepsilon_{0j}\), the error estimation dynamics becomes:  
\[ \begin{array}{l}\dot{\widetilde{x}}_{1j}=\widetilde{x}_{2j}-k_{1j}\frac{ \widetilde{x}_{ij}}{\varepsilon_{0j}}\\ \dot{\widetilde{x}}_{2j}=-k_{2j}\frac{\widetilde{x}_{1j}}{\varepsilon_{0j}}+ \widetilde{\Psi}_{j}.\end{array}  \tag{10}\]  
**Remark 1**. : _If the gain \(\alpha_{3j}\) is selected high enough, the term \(\frac{\dot{\Psi}_{j}}{\alpha_{3j}}\) gives very little influence on the changes of \(\widetilde{x}_{3j}\) in Equation (7). \(\widetilde{x}_{3j}\) also converges to 0 after \(\widetilde{x}_{2j}\) reaches its equilibrium point (i.e., \(\widetilde{x}_{2j}=0\)) when \(\frac{\dot{\Psi}_{j}}{\alpha_{3j}}\) is small and can be neglected with a large selected value of \(\alpha_{3j}\). This can be proven using Lyapunov stability theory when both sides of Equation (7) are multiplied with \(\widetilde{x}_{3j}\)._  
Using Equations (4) and (5), the relation between \(\widetilde{\Psi}_{j}\) and \(\Psi_{j}\) in the Laplace domain is (see the Appendix A):  
\[\widetilde{\Psi}_{j}(p)=\frac{p[p^{2}+(k_{1j}/\varepsilon_{0j})p+k_{2j}/ \varepsilon_{0j}]}{p^{3}+(k_{1j}/\varepsilon_{0j})p^{2}+(k_{2j}/\varepsilon _{0j})p+\alpha_{3j}^{2}(k_{2j}/\varepsilon_{0j})}(-\Psi_{j}(p)). \tag{11}\]  
This relation in Equation (11), between the real perturbation and the error of perturbation estimation, is a high pass filter. It means if the perturbation is in the low-frequency area the observer will estimate accurately. Using Equations (10) and (11), a relation between \(\widetilde{x}_{2j}\) and \(\Psi_{j}\) in the Laplace domain is derived as:  
\[\widetilde{x}_{2j}=\frac{p(p+\frac{k_{1j}}{\varepsilon_{0j}})}{p^{3}+(\frac{k _{1j}}{\varepsilon_{0j}})p^{2}+(\frac{k_{2j}}{\varepsilon_{0j}})p+\alpha_{3j }2(\frac{k_{2j}}{\varepsilon_{0j}})}\Big{(}-\Psi_{j}(p)\Big{)}. \tag{12}\]  
Only utilizing the partial state feedback, the estimated perturbation by the SPO can be used to reduce the effect of real perturbation for a more accurate estimation [21]. Moreover, the more accurate estimation can be used in the sliding mode control to follow the desired state.