## 5 Numerical Simulation  
### Simulation Setup  
To verify the effectiveness of the proposed control strategy, the hydraulic servo system [38] is adopted. System parameters for the simulation are given in Table 1.  
To illustrate the superiority of the proposed method, some controllers are employed for comparison as follows:  
1. The proposed controller with the control parameters \(\lambda=1000\), \(k=150\), \(\eta_{1}=1000\), and \(\eta_{2}=4.1\times 10^{4}\). The parameters for the first Levant's differentiator are chosen as \(\varsigma_{1}=100\), \(\varsigma_{2}=50\), \(\varsigma_{3}=100\), and \(\varsigma_{4}=100\). For the second differentiator \(\varsigma_{1}=350\), \(\varsigma_{2}=1.5\times 10^{4}\), and \(\varsigma_{3}=1.5\times 10^{4}\) are selected. The RBF NNs are designed with 51 nodes in the hidden layer, and their parameters are \(\gamma_{f_{1}}=0.15\), \(\gamma_{df_{1}}=0.2\), \(\gamma_{f_{2}}=0.02\), \(\gamma_{g_{2}}=0.1\), \(\gamma_{G_{1}}=0.01\), \(\gamma_{G_{2}}=0.1\), \(\Gamma_{f_{1}}=10^{3}\), \(\Gamma_{df_{1}}=10^{-7}\), \(\Gamma_{f_{2}}=10^{3}\), \(\Gamma_{g_{2}}=10^{-7}\), \(\Gamma_{G_{1}}=2\times 10^{3}\), and \(\Gamma_{G_{2}}=10^{-7}\). The disturbance observer gains are chosen as \(\kappa_{1}=0.005\) and \(\kappa_{2}=0.005\). 2. RBF NN-based sliding mode control without disturbance observer (RBF-SMC), in which larger controller gains are designed \(\eta_{1}=1500\) and \(\eta_{2}=4.1\times 10^{4}\) to attenuate the effects of disturbances while the avoidance of the chattering phenomenon is guaranteed. The structures and parameters of RBF NNs are the same as the proposed method. 3. Extended state observer-based backstepping controller (ESO-BC) [27], with the assumption that nominal system parameters are known. The control law is designed as \[ \begin{split}& u=\frac{1}{g_{20}}\Big{(}-f_{20}-\hat{d}_{2}+\hat{u}_{ 2}-k_{3}(x_{3}-\alpha_{2})\Big{)}\\ &\alpha_{2}=-f_{10}+\hat{u}_{1}-k_{2}z_{2}-\hat{d}_{1}\\ &\alpha_{1}=\hat{x}_{1d}-k_{1}(x_{1}-x_{1d})\end{split} \] (59) where \(z_{1}=x_{1}-x_{1d}\), \(z_{2}=\hat{x}_{2}-\alpha_{1}\), \(z_{3}=x_{3}-\alpha_{2}\), and the nominal dynamic functions are \[ \begin{split}& f_{10}=-\frac{1}{J}B_{0}\hat{x}_{2}-\left( \tau_{c0}+\tau_{\text{s0}}e^{\left(-\frac{|\hat{x}_{2}|}{\omega_{s}}\right)} \right)\tanh\!\left(c_{f}\hat{x}_{2}\right)\\ & f_{20}=-\frac{4D_{m}^{2}\beta_{c0}}{JV_{10}}\hat{x}_{2}-\frac{ 4\beta_{c0}J\Gamma_{20}}{V_{10}D_{m}}x_{3}\\ & g_{20}=\frac{4\beta_{c0}k_{10}}{V_{10}}\sqrt{P_{s}-\text{sign }(u)\frac{J}{D_{m}}x_{3}}\end{split} \] The two ESOs are designed as \[ \begin{split}&\begin{cases}\hat{x}_{1}=\hat{x}_{2}+3\omega_{c1} (x_{1}-\hat{x}_{1})\\ \hat{x}_{2}=x_{3}+f_{10}(\hat{x}_{2})+\hat{d}_{1}+3\omega_{1}^{2}(x_{1}-\hat{ x}_{1})\\ &\hat{d}_{1}=\omega_{1}^{3}(x_{1}-\hat{x}_{1})\\ &\hat{x}_{3}=f_{20}(\hat{x}_{2},x_{3})+g_{20}(x_{3},u)u+2\omega_{2}(x _{3}-\hat{x}_{3})\\ &\hat{d}_{2}=\omega_{2}^{2}(x_{3}-\hat{x}_{3})\end{cases}\\ \end{split} \] (60)  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Parameter** & **Unit** & **Value** & **Parameter** & **Unit** & **Value** \\ \hline \(J\) & \(\text{kg}\cdot\text{m}^{2}\) & 0.2 & \(C_{0}\) & \(\text{m}^{3}\cdot\text{s}^{-1}\cdot\text{Pa}^{-2}\) & 0 \\ \(B\) & \(\text{N}\cdot\text{m}\cdot\text{s}\cdot\text{rad}^{-1}\) & 90 & \(C_{1}\) & \(\text{m}^{3}\cdot\text{s}^{-1}\cdot\text{Pa}^{-1}\) & \(1\times 10^{-12}\) \\ \(D_{m}\) & \(\text{m}^{3  
where \(f_{10}(x_{2})\), \(f_{20}(x_{2},x_{3})\), and \(g_{20}(x_{3},u)\) are nominal system dynamic functions using known nominal system parameters, \(\omega_{1}\) and \(\omega_{2}\) are the observer gains, \(\hat{d}_{1}\) and \(\hat{d}_{2}\) are the estimates of mismatched and matched disturbances, respectively, and \(\hat{x}_{i}\) is the estimated value of the system state \(x_{i}\). The control parameters and observer gains are chosen as \(k_{1}=450\), \(k_{2}=350\), and \(k_{3}=350\); \(\omega_{1}=1000\) and \(\omega_{2}=1000\), respectively, to make the tracking performance as good as possible without chattering issues.