# Research on Fractional-Order Control of Anchor Drilling Machine Optimized by Intelligent Algorithms
## 4 Mathematical Modeling and Simulation Test  
### Modeling of Middle Valve-Controlled Hydraulic Cylinder  
This study focuses on the electro-hydraulic cylinder at the center of the robotic arm driven by the drill rig, aiming to improve its control accuracy for precise alignment of the  
Figure 4: Flowchart of the enhanced PSO algorithm for FOPID controller tuning. Wherein, the yellow modules represent the system modeling and error calculation phases; the green diamond indicates the determination of the early stopping condition; and the blue modules describe the elite reverse learning mechanism used for solution improvement.  
drill rod with the power head. The transfer function model for the position control of the electro-hydraulic proportional valve-controlled hydraulic cylinder, shown in Figure 5, is used [22]. A typical small hydraulic cylinder and proportional valve (Bosch 4 WRPEH6, manufactured by Bosch Rexroth (Changzhou) Co., Ltd., located at No. 17 Longmen Road, Wujin National High-Tech Industrial Development Zone, Changzhou, China) are selected, considering a 50 kg equivalent load. The system parameters are obtained from specifications, as detailed in Table 1.  
Based on the identification using the data from Figure 5 and Table 1, the transfer function of the valve-controlled hydraulic cylinder is obtained as follows:  
\[\begin{array}{ll}G(s)&=K_{a}\frac{K_{b}}{\frac{s^{2}}{\omega_{2\pi}^{2}}+ \frac{2\pi s}{\omega_{2\pi}}+1}\frac{A_{a}}{s\left(\frac{s^{2}}{\omega_{b}^{2}} +\frac{2\pi s}{\omega_{b}}+1\right)}\\ &=\frac{1}{s\left(59\times 10^{-10}s^{348}+1.81\times 10^{-6}s^{30001}+9.6\times 1 0^{-4}s^{1.9922}+1.6\times 10^{-3}s^{1.001}+0.006\right)}\end{array} \tag{4}\]  
A simulation model of the hydraulic system with an POPID controller was constructed in MATLAB-Simulink (Version 2023a). For this fractional-order controlled object, optimal PID, PI\({}^{\lambda}\)D\({}^{\mu}\), and PID\({}^{\mu}\) controllers were designed. The step response curves of the systems with these controllers are shown in Figure 6.  
The results indicate that the optimal integer-order PID controller exhibits significant oscillation, noticeable overshoot, and relatively slow convergence in the initial step response. In contrast, both fractional-order controllers demonstrate faster response speeds and smaller oscillation amplitudes, indicating that they outperform the integer-order PID controller in terms of dynamic characteristics and steady-state accuracy.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Parameter Meaning** & **Parameter Value** & **Parameter Unit** \\ \hline \(A_{e}\) & Effective area of hydraulic cylinder piston & \(3.142\times 10^{-4}\) & m\({}^{2}\) \\ \(V_{2}\) & Hydraulic cylinder rod chamber volume & \(1.131\times 10^{-5}\) & m\({}^{3}\) \\ \(\beta_{e}\) & Integrated modulus of elasticity of oil & \(7\times 10^{8}\) & Pa \\ \(\rho\) & Fluid density & 850 & kg/m\({}^{-3}\) \\ \(B\) & Piston and load viscous damping factor & 628 & N \(\cdot\) s/m \\ \(C_{tip}\) & Hydraulic internal leakage coefficient & \(5.0\times 10^{-12}\) & m\({}^{5}\)/(N \(\cdot\) s) \\ \(K_{a}\) & Amplifier gain & 0.27 & A/V \\ \(K_{cr}\) & Proportional directional valve flow–pressure coefficients & \(4.74\times 10^{-12}\) & m\({}^{5}\)/(N \(\cdot\) s) \\ \(K_{d}\) & Proportional directional valve flow Gain & \(4.8\times 10^{-2}\) & m\({}^{3}\)/(s \(\cdot\) A) \\ \(\omega_{sq}\) & Proportional directional valves inherent frequency & 75 & rad/s \\ \(\xi_{sq}\) & Proportional directional valve damping ratio & 0.5 & - \\ \(\omega_{h}\) & Intrinsic frequency of hydraulic cylinders & 42.8 & rad/s \\ \(\xi_{h}\) & Hydraulic cylinder damping ratio & 2.3 & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mathematical modeling parameters of valve-regulated hydraulic cylinders.  
Figure 5: Position control model of electro-hydraulic proportional valve-controlled hydraulic cylinder.