# Research on Fractional-Order Control of Anchor Drilling Machine Optimized by Intelligent Algorithms
## 4 Mathematical Modeling and Simulation Test
### The Stability Analysis of the Model Under Fractional-Order Controllers
The filter order N is increased to more precisely match the high-frequency response and enhance phase tracking capability. The frequency-domain comparison of the approximate model and controller with the original fractional-order object is shown in Figure 12. The high-order integer-order approximate model effectively replicates the magnitude-frequency characteristics of the original fractional-order model. The phase difference of 360\({}^{\circ}\) is considered equivalent due to its periodic nature. The frequency response of the approximated integer-order controller also matches that of the original controller well, with small-magnitude errors in the main frequency band and phase deviations kept within \(\pm 5^{\circ}\). Although there is slight phase lag in the high-frequency range, the overall fitting performance is satisfactory. We now analyze the stability of the approximated closed-loop system in the frequency domain. The Bode plot is shown in Figure 13.  
As shown in Figure 13, the gain is near 0 dB in the low/middle frequencies but shows a relatively low amplitude overall, suggesting signal attenuation. There is a slight gain increase at high frequencies, potentially amplifying noise. The phase remains near 0\({}^{\circ}\) at low frequencies and shows insufficient phase compensation across the relevant frequency range. The frequency-domain analysis highlights potential stability concerns despite the earlier numerical pole analysis suggesting stability. The low phase margin indicated by the phase response staying near 0\({}^{\circ}\) suggests poor robustness to disturbances and potential instability. The high-frequency gain increase indicates susceptibility to high-frequency noise amplification. These findings suggest that while the system might be nominally stable according to pole locations, its practical stability margins are inadequate, necessitating further controller refinement or compensation to improve robustness and performance. Figure 12: Comparison diagram in the frequency domain between the fitted model and the controller (**a**) Comparison of Bode plots between the original fractional-order model and the fitted high-order integer-order model; (**b**) comparison of Bode plots between the original fractional-order controller and the fitted high-order integer-order controller. To enhance the anchor drilling machine's robustness against disturbances and model uncertainties under complex operating conditions, particularly those identified during frequency analysis, the Ho control methodology is employed [31]. The objective is to synthesize a feedback controller u(t) = K\({}_{\text{x}}\)(t) that minimizes the Ho norm of the disturbance-to-output transfer function (T\({}_{\text{yw}}\)) below a specified threshold \(\gamma\) ( \(\mid\) 1 T\({}_{\text{yw}}\mid\)\(\mid\)\(\infty\)\(<\)\(\gamma\)). \[E_{\alpha}(Z)=\sum_{K=0}^{\infty}\frac{Z^{K}}{\Gamma(\alpha K+1)},\alpha>0 \tag{7}\]0  
For this purpose, the existing FOPID controller combined with the plant is treated as a generalized plant for Ho synthesis. Sensitivity (W\({}_{1}\)) and control (W\({}_{2}\)) weighting functions are introduced to shape the system's response, attenuating external disturbances while limiting control effort for energy efficiency. Specifically, W\({}_{1}\) primarily influences the controller's ability to reject low-frequency disturbances and track commands, while W\({}_{2}\) impacts the magnitude and bandwidth of the control input, affecting energy consumption and sensitivity to high-frequency noise. An optimal Ho controller is then designed within this weighted framework. This design process often involves iterative tuning of the weighting function parameters; starting with initial parameters, the corresponding Ho controller is synthesized, and the closed-loop performance is evaluated, particularly checking if the Ho norm meets the requirement \(\gamma\leq\gamma_{target}\). The weighting function parameters are then adjusted based on this evaluation, and the process is repeated until satisfactory performance is achieved.