## 7 Simulation Example  
Consider the servo-hydraulic system of [28], as illustrated in Figure 2.  
Figure 2: Schematic diagram of servo-hydraulic system.  
The system has typical strict-feedback non-linear characteristics where \(x_{q}\) is the output displacement, \(F_{a}\) is the output driving force of the hydraulic drive, \(P_{i}\) is the pressure, \(m\) is the mass of the load, \(k_{s}\) is the spring coefficient, and \(c\) is the damping coefficient.  
The system model is as follows:  
\[\left\{\begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=-f_{11}(x)x_{1}-f_{12}(x)x_{2}+x_{3}\\ \dot{x}_{3}=-f_{21}(x)x_{2}-f_{22}(x)x_{3}+f_{23}(x)u\\ y=x_{1}\end{array}\right. \tag{53}\]  
where \(f_{11}(x)=\frac{k_{s}}{m}\), \(f_{12}(x)=\frac{c}{m}\), \(f_{21}(x)=\frac{4\beta_{e}\omega^{2}}{V_{t}m}\), \(f_{22}(x)=\frac{4\beta_{e}}{V_{t}m}C_{t}\), and \(f_{23}(x)=\frac{4\beta_{e}\omega}{V_{t}m}\chi\).  
\(V_{t}\) is the total volume of the hydraulic cylinder, \(\beta_{e}\) is the elastic modulus of the hydraulic fluid, \(\omega\) is the effective acting area of the piston in the hydraulic cylinder, and \(\chi\) is the effective conversion ratio of the servo valve input and output.  
In order to verify the validity, we select data close to reality:  
\(m=38\) kg; \(k_{s}=1.425\times 10^{4}\) N/m; \(c=1.425\times 10^{3}\) N\(\cdot\)s/m; \(V_{t}=5.6\times 10^{-5}\) m\({}^{3}\); \(\beta_{e}=600\) MPa; \(\omega=2.8\times 10^{-4}\) m\({}^{2}\); \(C_{t}=4\times 10^{-13}\) m\({}^{3}\cdot\)Pa/s; \(\chi=1\times 10^{-2}\) m\({}^{3}\cdot\)V/s.  
The unit step response curve of the MTN is illustrated in Figure 3.  
In Figure 3, the BPNN controller and RBFNN controller are given. As traditional control methods, the neural network controllers work well and have the ability to resist disturbance. From these experimental results, it is shown that the method proposed in this paper is faster than another.  
To accurately assess the performance of the three control methods, we employed three error metrics: the (1) Root Mean Square Error (RMSE), which represents the square root of the ratio of the squared differences between the actual values and the predicted values and is sensitive to outliers in the data; (2) the Mean Absolute Error (MAE), which measures the average distance between the model's predicted values and the actual values and is less sensitive to outliers; and (3) the Mean Absolute Percentage Error (MAPE), which is a relative measure that quantifies the accuracy of predictions using relative error. The values of these metrics obtained using the three control methods are shown in Table 1.  
Figure 3: Output comparison.  
As shown in Table 1, the proposed method outperforms the NN and the RBF in terms of most metrics.  
In order to verify the tracking performance of the system, \(y_{d}=1+0.1\sin(t)\) was chosen as the desired signal, and the system outputs are illustrated in Figure 4.  
Similarly, the results of the three indicators are shown in Table 2.  
The latter figure presents the system output and the ideal tracking signal, highlighting that the system has a good tracking performance, i.e., the effectiveness of the proposed method is highlighted from the simulation results.