# Research on Fractional-Order Control of Anchor Drilling Machine Optimized by Intelligent Algorithms
## 4 Mathematical Modeling and Simulation Test
### Simulation of Integer-Order and Fractional-Order Controllers Based on Intelligent Algorithms  
The electro-hydraulic servo model established above was simulated in MATLAB-Simulink using intelligent optimization algorithms. Based on preliminary tests, a population of 30 search agents and an iteration limit of 60 were chosen. The selection of 30 agents offered an effective balance between sufficient exploration and computational efficiency. The iteration limit of 60 was set because preliminary runs indicated short computation times without significant improvement in dynamic parameters beyond this number, aligning well with the requirements of embedded real-time control in mining environments. Optimization parameter bounds for the PID controller were set from [0] to [20], and for the fractional-order PID (FOPID) from [0] to [2,20]. These ranges were informed by engineering knowledge of typical hydraulic servo controllers and preliminary algorithm evaluations, ensuring that the search space was both feasible and likely to contain optimal solutions. The Integral of Time-Weighted Absolute Error (ITAE) was chosen as the fitness function because preliminary comparative experiments (Figure 7) showed its superior convergence behavior and ability to achieve lower final fitness values, indicating desirable transient responses (low overshoot and rapid settling) suitable for the hydraulic positioning system of the anchor drilling rig. To validate the effectiveness of the proposed enhanced ILPSO algorithm and provide a clear rationale for its selection over other optimization approaches, a comprehensive comparative study was conducted. To ensure consistency in initial conditions, various intelligent optimization algorithms, including the standard PSO, GA, WOA, GWO, and  
Figure 7: Fitness curves of algorithms under different performance metrics. (**a**) Fitness curves of the WOA under different performance metrics; (**b**) fitness curves of the GWO under different performance metrics; (**c**) fitness curves of the PSO under different performance metrics; (**d**) fitness curves of the ILPSO under different performance metrics. This section presents a comparison of the system's output response under different intelligent algorithms combined with both PID and FOPID control strategies. As shown in Figure 8 and Table 2, the step response of various intelligent algorithms with the PID control strategy demonstrates that the system generally exhibits overshoot values above 22% and settling times exceeding 0.07 s. In contrast, the FOPID control strategy, as shown in Figure 9 and Table 2, significantly improves these metrics by reducing overshoot to approximately 1-2% and shortening the settling time to below 0.02 s, which greatly enhances the system's dynamic response and stability. The comparison between PID and FOPID control strategies indicates that FOPID consistently outperforms PID in both dynamic and steady-state performance. The reduction in overshoot and settling time is particularly notable, demonstrating the effectiveness of the fractional-order controller in improving system stability and response. As demonstrated by the comprehensive comparative results presented in Figures 8 and 9, and quantitatively summarized by the performance metrics in Table 2, the proposed ILPSO algorithm exhibits superior performance in optimizing controller parameters compared to other intelligent optimization methods investigated for this application, including the genetic algorithm (GA) and other tested swarm optimization algorithms. Specifically, when combined with the FOPID controller, the ILPSO-FOPID pairing achieved the most favorable dynamic response characteristics among all tested algorithm-controller pairings, exhibiting the lowest overshoot (1.186%), the shortest settling time (0.0098 s), and a very low standard deviation (0.013).