# 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  
Stability analysis for fractional-order (FO) systems differs from traditional linear time-invariant (LTI) integer-order systems. While LTI system stability requires all characteristic equation poles to be in the left half of the complex plane, FO systems can exhibit stable behavior even with poles outside this region [27; 28; 29]. The stability is often assessed relative to a wedge-shaped region in the complex plane defined by the fractional order, \(\alpha\). A key concept is Mittag-Leffler stability. A fractional-order dynamical system \(\mathrm{x}^{\prime}(\mathrm{t})=\mathrm{Ax}(\mathrm{t})\) (where the derivative is fractional) is Mittag-Leffler-stable if its state trajectory \(\mathrm{x}(\mathrm{t})\) decays towards zero bounded by the Mittag-Leffler function:  
\[\|x(t)\|\leq\mathrm{CE}_{\alpha}(-\gamma t^{\alpha}),t\geq t_{0} \tag{6}\]  
where \(\mathrm{E}\alpha(\mathrm{z})\) is the Mittag-Leffler function [See Equation (6)], \(\gamma>0\), and \(\mathrm{C}>0\) depends on initial conditions. \[E_{\alpha}(Z)=\sum_{K=0}^{\infty}\frac{Z^{K}}{\Gamma(\alpha K+1)},\alpha>0 \tag{7}\]Matignon's Theorem provides a practical criterion for assessing the asymptotic stability (in the Mittag-Leffler sense) of an FO system represented by the transfer function G(s) = N(s)/D(s\(\alpha\)). The system is stable if and only if all roots (poles) \(\lambda_{\text{j}}\) of the characteristic polynomial D(\(\lambda\)) satisfy the following:  
\[\left|\arg\!\left(\Lambda_{(j)}\right)\right|>\alpha\frac{\pi}{2},\forall j \tag{8}\]  
where \(0<\alpha<1\) is the fractional order. This means all poles must lie outside the closed angular sector defined by \(\pm\alpha\pi/2\), as illustrated conceptually in Figure 10.  
We analyze the closed-loop system incorporating the fractional-order controller G(s) obtained via the ILPSO algorithm. The characteristic roots of the polynomial equation for the variable \(\lambda\) can be directly calculated, as shown in Figure 11. These poles also reside within the stable sector defined by [PERSON]'s criterion, with the closest pole having a slope of approximately 0.01816. Based on this numerical analysis, the closed-loop system appears to be Mittag-Leffler-stable. However, this direct numerical root-finding approach for FO systems has significant limitations. Standard numerical solvers are designed for integer-order polynomials. Applying them to FO characteristic equations, often through polynomial approximation, can introduce substantial numerical errors. In this case, the reported root-finding error norm reached values as high as \(\sim\)10\({}^{40}\). Such large errors cast doubt on the accuracy of the com  
Figure 11: Region of stability for fractional-order systems. Figure 10: Stability regions in the complex plane for different fractional orders \(\alpha\). (**a**) When \(0<\alpha<1\), the stable region is narrower and closer to the negative real axis. (**b**) When \(\alpha=1\), the boundary aligns with the imaginary axis, corresponding to classical integer-order systems. (**c**) When \(1<\alpha<2\), the stable region expands, covering a wider angle around the negative real axis. The angle of the stable region is defined by \(\alpha\pi\)2.  
puted pole locations and, consequently, the reliability of the stability assessment based solely on this method. Therefore, while indicative, this numerical result lacks rigorous theoretical robustness and should be interpreted with caution. Due to the error accumulation problem when directly using integer-order methods to analyze fractional-order systems, this study adopts an improved frequency-domain fitting method based on frequency response. The optimal integer-order transfer function model is obtained by minimizing the error function. To further improve the fitting accuracy, a dynamic frequency band adjustment strategy is employed, where the start and stop frequencies of the band are adjusted according to the system's dominant frequency [30].