# Research on Fractional-Order Control of Anchor Drilling Machine Optimized by Intelligent Algorithms
## 3 A Fractional-Order FOPID Controller Based on Intelligent Optimization Algorithms
### Particle Swarm Optimization Algorithm Based on Improved Levy Flight and Chaotic Mapping  
Tuning the parameters of controllers, especially those with multiple parameters like fractional-order PID (FOPID) controllers, is a complex and challenging task. Manual tuning methods are often time-consuming and may not yield optimal parameter sets that guarantee high performance, particularly when the controlled system operates under varying conditions or in demanding environments. Intelligent optimization algorithms provide powerful tools for automatically searching the parameter space to find optimal or near-optimal controller gains that can significantly enhance system performance, robustness, and dynamic response. To effectively address these challenges and enhance the performance of the controller tuning process for the anchor drilling rig's hydraulic system, particularly to ensure the high precision and strong robustness required for unmanned mining operations, this study proposes an enhanced Particle Swarm Optimization (PSO) algorithm incorporating multiple chaotic mappings and an improved Levy flight mechanism [20]. This specific algorithm is designed to overcome the limitations of standard optimization methods by incorporating advanced search mechanisms tailored for complex parameter spaces.  
Building upon the standard PSO algorithm, inspired by social behavior patterns such as bird flocking and fish schooling, which updates particle positions based on individual and global bests, ILPSO incorporates advanced mechanisms to improve search efficiency and convergence quality. For instance, the initial particle population is generated using chaotic mappings--Logistic, Tent and Chebyshev--to increase diversity and ensure a uniform distribution within the search space, preventing premature convergence in the early stages. The iterative equations for the chaotic mappings are as follows:  
\[\begin{array}{ll}\text{Logistic}:&x_{n+1}=\mu x_{n}(1-x_{n})\\ \text{Tent}:&x_{n+1}=1-2|x_{n}-0.5\end{array} \tag{1}\]  
Figure 3: Intelligent optimization algorithm control system.  
\[\text{Chebyshev}:\quad x_{n+1}=\cos(k\text{-}\arccos(x_{n})) \tag{2}\]  
During the particle position update process, an adaptive chaotic perturbation mechanism is introduced, coupled with the Levy flight strategy [21]. The Levy flight mechanism is defined as follows:  
\[\text{Levy flight}:L(\lambda)\sim t^{-\lambda}\text{sin}\bigg{(}\frac{\pi\lambda }{2}\bigg{)} \tag{3}\]  
This dual mechanism enhances the global search capability and optimization precision of the PSO algorithm. To avoid local optima in complex search spaces, an Elite Reverse Learning (ERL) strategy is introduced. This step modifies the elite solutions to explore new regions of the search space by applying reverse learning principles. If no improvement is observed after several iterations, the Elite Reverse Learning mechanism adjusts the elite particle's position, effectively guiding the search process towards better solutions.  
The enhanced PSO algorithm combines chaotic perturbations and the Levy flight strategy with the Elite Reverse Learning mechanism, ensuring efficient optimization of PID and FOPID controller parameters. The following figure, Figure 4, illustrates the flow of the enhanced PSO algorithm in collaboration with the FOPID controller, highlighting the key steps, including parameter initialization, chaotic mapping, Levy flight, and Elite Reverse Learning.