## 4 Case Studies  
In the case studies described in this section, the presented controller was applied to the electrohydraulic system of a hydraulic press, the maximum load of which is 700 kN. The dynamics of the system are defined by (9), and the system parameters of the hydraulic press are given in Table 1.  
To show the efficiency of the suggested method, two case studies were carried out in a simulation, and the following three controllers were compared in the two cases:  
1. The proposed smooth reference governor-based controller (SRG);
2. The nominal reference governor-based controller (RG);
3. A third controller conducting pressure tracking without a reference governor. The impact of the reference governor was simulated by a first-order low-pass filter (LPF) with a 0.17 s time constant. The selection of the control parameters of the proposed controller is based on the following rules: (1) control gains \(k_{1}\), \(k_{2}\) and \(c_{1}\) are chosen based on the saturation limits and bandwidth of the actuator; (2) filter parameters \(T_{du}\) and \(\xi_{du}\) depend on the saturation value of the valve displacement; and (3) decreasing the sampling time \(\mathrm{T_{s}}\) and increasing  
Figure 4: Block diagram of the controller structure. Figure 3: The schematic diagram of the smooth reference governor. the prediction time \(T_{p}\) can improve control performance, but more computing resources are required. According to the above rules and through trial and error, the control parameters of the proposed SRG controller were chosen for the simulation, as shown in Table 2.  
Case 1: Mild Pressure Trajectory Test. The three controllers were first tested with the trajectory shown in Figure 5. The constraints for the pump and valve were satisfied for all three controllers, and these three controllers showed almost the same performance, which can be seen in Figure 6. This is due to the relatively mild pressure trajectory, which can be tracked without aggressive actuator control. It also shows that the proposed smooth reference governor does not slow down the tracking results when the actuator constraints are not violated. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \(m\) & Load mass & 1200 kg \\ \(A_{1}\) & Driving cylinder piston side chamber area & 3.8 \(\times\) 10\({}^{-2}\) m\({}^{2}\) \\ \(P_{s}\) & Pump supply pressure & 2.0 \(\times\) 10\({}^{7}\) Pa \\ \(C_{t}\) & Driving cylinder leakage coefficient & 0 m\({}^{3}\)/(s-Pa) \\ \(\beta_{e}\) & Oil bulk modulus & 1 \(\times\) 10\({}^{9}\) Pa \\ \(k_{q}\)-\(k_{x}\) & Servo valve flow gain coefficient & 1.782 \(\times\) 10\({}^{-8}\) m\({}^{3}\)/(s-V-\(\sqrt{\text{Pa}}\)) \\ \(k_{p}\) & Pump flow coefficient & 6.56 \(\times\) 10\({}^{-3}\) m\({}^{3}\)/(s-rad) \\ \(\omega_{x}\) & Pump dynamics natural frequency & 25 rad/s \\ \(\xi_{a}\) & Pump dynamics damping ratio & 1 \\ \(\omega_{v}\) & Valve dynamics natural frequency & 200 rad/s \\ \(\xi_{v}\) & Valve dynamics damping ratio & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the electrohydraulic system. Figure 5: Tracking performance for a step trajectory from 10 bar to 30 bar. (**a**) The pressure trajectory. (**b**) The tracking error of the three controllers. Case 2: Violent Pressure Trajectory Test  
Figure 7 shows the performance when a step trajectory from 10 bar to 70 bar is used. The variable displacement pump's angle rate constraint is strongly violated, leading to oscillating tracking performance if no load governor is used. Compared with the smooth reference governor, the nominal reference governor applies the pressure trajectory too aggressively, which leads to intermediate steps when \(\alpha_{k}\) cannot be optimized.