## 3 Methodology
### NESO Design  
As is well known, ESO can estimate lumped disturbances effectively. However, ESO can achieve fast convergence by selecting high gain, which may also cause differential peak problems. To overcome the mentioned restriction, a NESO combines the advantages of the hyperbolic tangent function and high-gain structure to estimate the unknown disturbances \(d_{1}\) and \(d_{2}\) in the system (9). The upper bound of the estimation error depends on the high gain value, and the hyperbolic tangent function is employed to reduce the effect of the differential peak problem. Let \(x_{11}=u_{e}\), \(x_{12}=d_{1}\), \(x_{21}=r\), \(x_{22}=d_{2}\), \(\omega_{1}=\dot{d}_{1}\), and \(\omega_{2}=\dot{d}_{2}\), then there are two subsystems such that:  
Subsys1:  
\[ \begin{cases}\dot{x}_{11}=M_{1}x_{12}+M_{1}(X_{u}u+a_{23}vr+T)-\dot{u}_{d}\\ \dot{x}_{12}=\omega_{1}\end{cases} . \tag{10}\]  
Subsys2:  
\[ \begin{cases}\dot{x}_{21}=M_{3}N_{r}x_{21}+M_{3}x_{22}+M_{3}(a_{12}uv+\tau_{r} )\\ \dot{x}_{22}=\omega_{2}\end{cases} . \tag{11}\]  
Additionally, the NESOs are designed as follows:  
NESO1:  
\[ \begin{cases}\dot{x}_{11}=\dot{x}_{12}-gk_{11}^{\text{seq}}\tilde{x}_{11}+M_{1} (X_{u}u+a_{23}vr+T)-\dot{u}_{d}\\ \dot{x}_{12}=-gk_{12}^{\text{seq}}\text{tanh}(gk_{th}\tilde{x}_{11})\end{cases} . \tag{12}\]  
NESO2:  
\[ \begin{cases}\dot{x}_{21}=M_{3}N_{r}\hat{x}_{21}+M_{3}\hat{x}_{22}+M_{3}(a_{12 }uv+\tau_{r})-gk_{21}^{\text{seq}}\tilde{x}_{21}\\ \dot{x}_{22}=-gk_{22}^{\text{seq}}\text{tanh}(gk_{th}\tilde{x}_{21})\end{cases} , \tag{13}\]  
where \(\hat{x}_{ij}\) is the estimated value of \(x_{ij}\), \(\tilde{x}_{ij}=\hat{x}_{ij}-x_{ij}\) is the estimation error, \(k_{ij}^{\text{seq}}>0\) is the observer gain coefficient, \(i=1,2\), \(j=1,2\), \(g>1\) is the high gain coefficient, and \(k_{th}>0\) is the hyperbolic tangent function \(\text{tanh}(x)=(e^{x}-e^{-x})/(e^{x}+e^{-x})\). Then, the estimated value of the disturbance could be obtained, \(\hat{d}_{i}=\hat{x}_{2}\), \(i=1,2\). **Theorem 1**. : _Under the condition of satisfying Assumption 1, the NESOs (12) and (13) designed for subsystems (10) and (11) can ensure the consistent final bound convergence of the estimation errors, and the upper bound of the errors depends on the high gain coefficient \(g\)._  
Proof. : Since the two NESOs have the same structure, only the NESO1 convergence analysis is provided below. Let \(z_{1j}(t)=g^{2-j}\tilde{x}_{1j}(\epsilon t)\), \(j=1,2\), and \(\xi_{1}=z_{11}\), \(\xi_{2}=z_{12}-k_{11}^{\text{seq}}z_{11}\). Then, we can obtain the following from (10) and (12):  
\[ \begin{cases}\dot{\xi}_{1}=\xi_{2}\\ \dot{\xi}_{2}=-k_{12}^{\text{seq}}\text{tanh}(k_{th}\xi_{1})-k_{11}^{\text{seq }}\xi_{2}-g^{-1}\omega_{1}\end{cases} . \tag{14}\]  
On the basis of this observation, the following Lyapunov function is considered. \[V_{\xi 1}(\xi_{1},\xi_{2})=\int_{0}^{\xi_{1}(t)}\text{tanh}(k_{th}\xi_{1}(\tau)) d\xi_{1}(\tau)+\frac{1}{2k_{12}^{\text{seq}}\xi_{2}^{2}}. \tag{15}\]  
The derivative of \(V_{\xi 1}(\xi_{1},\xi_{2})\) along the system (15) is:  
\[\dot{V}_{\xi 1}\leq-\frac{k_{11}^{\text{seq}}}{k_{12}^{\text{seq}}}\left[\xi_{2}+ \frac{\omega_{1}}{2 gk_{11}^{\text{seq}}}\right]^{2}+\frac{\overline{\omega}_{1 }^{2}}{4 gk_{11}^{\text{seq}}k_{12}^{\text{seq}}}. \tag{16}\]From (16), it follows that \(\xi_{1}\) and \(\xi_{2}\) are bound. Additionally, according to (14), when the system reaches a steady state, it can be obtained that  
\[\left\{ \begin{aligned} &|\xi_{1}|\leq\frac{1}{k_{th}}{\tanh^{-1} \left(\frac{22\mathcal{D}_{1}}{8k_{12}^{2\mathcal{D}}}\right)}\\ &|\xi_{2}|\leq\frac{\xi_{1}}{8k_{11}^{2\mathcal{D}}}\end{aligned}  \right.. \tag{17}\]  
Additionally,  
\[\left\{ \begin{aligned} &|\widetilde{x}_{11}|=\frac{|\xi_{1}|}{g}\leq \frac{1}{8k_{th}}{\tanh^{-1}\left(\frac{2\mathcal{D}_{1}}{k_{12}^{2\mathcal{D} }}\right)}\\ &|\widetilde{x}_{12}|=|z_{12}|\leq|\xi_{2}|+k_{11}^{eso}|\xi_{1}| \leq\frac{\mathcal{D}_{1}}{8k_{11}^{eso}}+\frac{k_{11}^{eso}}{8k_{th}}{\tanh^{ -1}\left(\frac{2\mathcal{D}_{1}}{k_{12}^{eso}}\right)}\end{aligned} \right.