## 5 Experiments
### Position/Pressure Tracking Experiments  
Firstly, compared to the ADRC algorithm and the PID algorithm, the TDDSC algorithm was tested to evaluate the tracking accuracy of the displacement controller. The back pressure was set to be 4 MPa, whereas the position reference curve and the load curve during the test are shown in Figure 8. A large number of parameter sets were verified in order to make sure that the TDDSC controller worked with the highest performance. The TDDSC controller parameters have the values shown as following: \(k_{1}=50\), \(k_{2}=40\), \(k_{3}=400\), \(R=10\), \(a_{1}=0.9\), \(a_{2}=0.8\), \(f_{1}=0.8\), \(f_{2}=0.5\), and \(\eta_{01}=100\). The PID controller parameters were tuned using the Ziegler-Nichols tuning rule [34]. The parameters of PID are \([40,25,0]\). The control law of the ADRC is given by Equations (40) and (41):  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}-\varphi_{1}e\\ \dot{x}_{2}=-\varphi_{2}\cdot fal(e,\alpha,\delta)\\ \dot{x}_{3}=x_{4}-\varphi_{3}\cdot fal(e,\beta,\delta)+b\cdot u\\ \dot{x}_{4}=-\varphi_{4}\cdot fal(e,\gamma,\delta)\end{array} \right., \tag{40}\]  
\[u=\lambda_{1}\cdot fal(e_{1},\alpha,\delta)+\lambda_{2}\cdot fal(e_{2},\beta, \delta)+\lambda_{3}\cdot fal(e_{3},\gamma,\delta), \tag{41}\]  
where \(e_{1}\), \(e_{2}\), and \(e_{3}\) are the state variable errors and \(fal\) function is given by Equation (42):  
\[fal=\left\{ \begin{array}{l}|e|^{\alpha}\cdot sign(e),|e|\geq\delta\\ \frac{e}{\delta^{\alpha-\alpha}},|e|<\delta\end{array} \right.. \tag{42}\]  
The initial values of parameters for the ADRC were determined by the method mentioned in the literature [35], and minor adjustments were required to make sure that the ADRC worked with the highest performance. The parameters of the ADRC are as follows:\(\alpha=0.75\), \(\beta=1.5\), \(\gamma=1.5\), \(\delta=0.1\), \(\lambda_{1}=6\), \(\lambda_{2}=1.5\), \(\lambda_{3}=0.05\), \(\varphi_{1}=100\), \(\varphi_{2}=300\), and \(\varphi_{3}=1000\). The tracking errors of the TDDSC, ADRC, and PID algorithms are shown in Figure 9. Since the supply pressure causes disturbance to the system, the ADRC algorithm oscillates, when the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Value** & **Units** \\ \hline Pump displacement (1 and 10) & 40 & mL/r \\ Motor speed & 1500 & rpm \\ Servo valve rated flow (3.1) & 63 & L/min \\ Servo valve rated flow (3.2) & 38 & L/min \\ Piston diameter of cylinders (5 and 8) & 63 & mm \\ Rod diameter of cylinders (5 and 8) & 36 & mm \\ Weight of the moving part & 100 & Kg \\ Cylinder strokes (5 and 8) & 300 & mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of the test rig. Figure 8: Displacement and load force as a function of time. cylinder retracts. Although the PID algorithm does not oscillate, the tracking error is still large. The root-mean-square errors (RMSEs) of the TDDSC, ADRC, and PID algorithms have values of 0.47, 1.25 and 2.24 mm, respectively. Since the uncertainty in the model is observed by the NDO, the TDDCS algorithm has higher tracking accuracy, which indicates that the designed position controller meets the requirements. Figures 10 and 11 show the estimated values for \(\hat{\theta}_{i2}\) and \(\hat{\theta}_{i3}\), respectively, which indicated that the designed NDO works properly. In another experiment, the tracking accuracies of the TDDSC and DSC algorithms were evaluated. Their tracking errors are shown in Figure 12. Since a TD is applied to the TDDSC controller, the performance of the TD is better than that of the first-order filter in the DSC. Therefore, the tracking accuracy of the TDDCS control algorithm is higher, and it is not easy to generate oscillations. Figure 11: Estimate of \(\theta_{i3}\) as a function of time. Figure 9: Tracking errors of the TDDSC, active disturbance rejection controller (ADRC), and proportional–integral–derivative (PID) algorithms as a function of time.