## 2 Materials and Methods
### The Design of NDO  
A nonlinear disturbance observer is developed to estimate the actual disturbance in the system for an unknown disturbance D, thereby strengthening the system's robustness. To address practical considerations, it is assumed that any disturbance is bounded as follows [33; 34; 35]. **Lemma 1**. : _For initial conditions that are bounded, a Liapunov function is also uniformly bounded \(x(t)\) if there exists a continuous positive definite Liapunov function \(V(x)\) satisfying the following conditions:_  
\[\delta_{1}(\|x\|)\leq V(x)\leq\delta_{2}(\|x\|),\ \dot{V}(x)\leq-\kappa V(x)+c, \tag{13}\]  
_where \(\delta_{1},\delta_{2}:\mathbb{R}^{n}\rightarrow\mathbb{R}\) is the \(V\) class function, and \(\kappa\), \(c\) all are positive constants._  
**Assumption 2**. : _Since no disturbance can be infinite in the real world, we assume that the perturbations in the WIPR system studied in this paper are all bounded, and their first-order derivatives and second-order derivatives are assumed to be bounded; thus, the following equations can be obtained._  
\[\|\dot{D}\|\leq\varepsilon_{1},\ \|\dot{D}\|\leq\varepsilon_{2},\ \varepsilon_{1}>0, \varepsilon_{2}>0\]  
\(\|\cdot\|\) represents the Euclidean norm of the vector. The NDO is designed as in Equation (14). \[ \begin{split}\hat{D}=Z_{1}+P_{1}(\xi_{2})\dot{Z}_{1}=-L_{1}(\xi_ {2})\big{[}f(\xi)+g(\xi)\tau+\hat{D}\big{]}+\hat{\tilde{D}}\\ \hat{\tilde{D}}=Z_{2}+P_{2}(\xi_{2})\dot{Z}_{2}=-L_{2}(\xi_{2}) \big{[}f(\xi)+g(\xi)\tau+\hat{D}\big{]},\end{split}  \tag{14}\]  
\(\hat{D}\) and \(\hat{\tilde{D}}\) represent the estimates of the total perturbation and its derivative, respectively, while \(Z_{1}\) and \(Z_{2}\) are intermediate variables in the observer. To meet the requirements of the system, the self-designed nonlinear functions \(P_{1}(\xi_{2})\) and \(P_{2}(\xi_{2})\) are used and must satisfy the conditions \(L_{1}(\xi_{2})=\frac{\partial P_{1}(\xi_{2})}{\partial\xi_{2}}\) and \(L_{2}(\xi_{2})=\frac{\partial P_{2}(\xi_{2})}{\partial\xi_{2}}\). **Property 1**. : _The error between the estimated and actual values of the disturbance is represented by \(\widetilde{D}=D-\hat{D}\), while\(\widetilde{D}=\dot{D}-\hat{\tilde{D}}\)represents the error between the derivative of the actual value of a perturbation and the derivative of the estimated value of the same perturbation._  
The derivation of \(\widetilde{D}\) and \(\widetilde{\tilde{D}}\) substitution of Equations (7) and (14) into the above equation leads to results \(\dot{\widetilde{D}}\) and \(\widetilde{\tilde{D}}\), which are the equations of the NDO, (15) and (16), respectively. \[ \begin{split}\dot{\widetilde{D}}=\dot{D}-\dot{\tilde{D}}=\dot{D}- \dot{Z}_{1}-\frac{\partial P(\xi_{2})}{\partial\xi_{2}}\cdot\frac{d\xi_{2}}{dt }=\dot{D}-\dot{Z}_{1}-L_{1}(\xi_{2})\dot{\xi}_{2}\\ =\dot{D}+L_{1}(\xi_{2})\big{[}f(\xi)+g(\xi)\tau+\hat{D}\big{]}- \dot{\tilde{D}}-L_{1}(\xi_{2})\dot{\xi}_{2}\\ \dot{D}+L_{1}(\xi_{2})\big{[}f(\xi)+g(\xi)\tau+\hat{D}\big{]}- \dot{\tilde{D}}-L_{1}(\xi_{2})[f(\xi)+g(\xi)\tau+D]\\ =-L_{1}(\xi_{2})\widetilde{D}+\widetilde{\dot{D}},\end{split}  \tag{15}\]\[ \begin{split}&\dot{\widetilde{D}}=\widetilde{D}-\dot{\widetilde{D}}= \dot{D}-\dot{Z}_{2}-\frac{\partial\mathcal{P}(\zeta_{2})}{\partial\zeta_{2}} \cdot\frac{d\zeta_{2}}{dt}=\widetilde{D}-\dot{Z}_{2}-L_{2}(\zeta_{2})\dot{ \zeta}_{2}\\ &=\ddot{D}+L_{2}(\zeta_{2})\big{[}f(\xi)+g(\xi)\tau+\dot{D}\big{]} -L_{2}(\xi_{2})[f(\xi)+g(\xi)\tau+D]\\ &\qquad\qquad\qquad\qquad\qquad=-L_{2}(\xi_{2})\widetilde{D}+ \ddot{D},\end{split}  \tag{16}\]  
By letting \(E=\left[\widetilde{D}^{T},\ \widetilde{D}^{T}\right]^{T}\) and substituting the appropriate Equations (15)-(17) can be obtained. \[\dot{E}=LE+\delta\widetilde{D}, \tag{17}\]  
where \(L= \begin{bmatrix}-L_{1}(\zeta_{2})&I_{3}\\ -L_{2}(\zeta_{2})&0\end{bmatrix} \), \(\delta= \begin{bmatrix}0\\ I_{3}\end{bmatrix} \). The observer's stability is examined, and a Lyapunov function is selected to make sure that it can reliably predict the system state despite any nonlinear disturbances.