# Hybrid Optimization Method for Correcting Synchronization Errors in Tapping Center Machines
## 3 Related Research
### Fractional-Order Particle Swarm Optimization (FPSO)  
The FPSO was first proposed by [PERSON] et al. [27] and was derived from PSO by introducing the Grunwald-Letnikov fractional-order derivative to enhance its diversity and algorithmic effectiveness. The PSO algorithm [37; 38; 39; 40; 41] was inspired by the group behavior of foraging birds and seeks the best solution in the solution space by maximizing or minimizing fitness value, which is analogous to the process in which foraging birds maximize the amount of food they consume. Evolution of position and evolution of speed occur as the optimal parameters for position and speed are updated after each iteration. The calculation formula for PSO is as follows:  
\[V_{i}(k+1)=\omega\cdot V_{i}(k)+c_{1}\cdot r_{1}\cdot(P_{i}^{J}(k)-p_{i}(k))+c_{ 2}\cdot r_{2}\cdot(P^{g}(k)-p_{i}(k)) \tag{4}\]  
\[p_{i}(k+1)=p_{i}(k)+V_{i}(k+1) \tag{5}\]  
where \(i=1\), \(2\),..., \(m\) (\(m\) is the number of particles); \(k\) is the number of iterations; \(w\) is the inertial weight; \(V\) is the velocity; \(c_{1}\) and \(c_{2}\) are individual and group parameters, respectively; \(p_{i}(k)\) is the position vector; \(P_{i}^{J}(k)\) is the best local solution in the current iteration; \(P^{g}(k)\) is the best global solution in the current iteration, and \(r_{1}\) and \(r_{2}\) are random numbers from 0 to 1.  
Recently, many studies enhanced the performance and effectiveness of PSO, including adjusting the strategies [42; 43], particle grouping [44; 45; 46], inertia weights [47; 48; 49], and other optimization techniques [50; 51; 52; 53]. However, PSO has its main drawbacks, such as lack of robustness and tendency to fall into the local optimum. Additionally, the fractional-order derivative describes the real world more than the integer-order one. Therefore, [PERSON] et al. [27] involved the fractional-order derivative in PSO to improve the performance and effectiveness, called FPSO. After that, [PERSON] et al. [28] took a nonlinear time-varying inertia weight in FODPSO (Fractional-order Darwinian PSO) to improve. In this paper, the inertial weight was applied to FPSO, and the calculation formula is as follows:  
\[V_{i}(k+1)=\omega\cdot V_{i}(k)+c_{1}\cdot r_{1}\cdot(P_{i}^{J}(k)-\lambda \cdot p_{i}(k)-\lambda/2\cdot(1-\lambda)\cdot p_{i}(k-1)-\lambda/6\cdot(1- \lambda)\cdot(2-\lambda)\cdot p_{i}(k-2)-\lambda/24\cdot(1-\lambda)\cdot(2- \lambda)\cdot(3-\lambda)\cdot p_{i}(k-3))+c_{2}\cdot r_{2}\cdot(P^{g}(k)-p_{i}( k))\analyzed to maximize product quality and minimize production costs. A large number of design variables are studied in a small number of experiments. The best combination of design variables can be expressed by the orthogonal table and the signal-noise ratio (SNR). The basic concept of the Taguchi method is to maximize the performance index by using the orthogonal table to perform experiments. Using the orthogonal table reduces experiment time and increases convergence speed. In the Taguchi method, orthogonal arrays and SNR are tools used to optimize the design of engineering parameters in the experimental plan, to reduce variation in important quality characteristics, and to achieve the goal of total cost reduction. Table 2 shows an example of an \(L_{9}(3^{4})\) orthogonal array that accommodates a maximum of 4 factors with 3 levels per factor.  
The SNR was calculated for quality characteristics, and the goal of this study was the smaller, the better characteristics. The quality function characteristic \(Y\) is a non-negative value with a smaller-the-better characteristic; \(Y\) can be equal to zero. The SNR of the smaller-the better characteristic is defined as  
\[SNR=-10 log_{10}\frac{1}{n}\sum_{i=1}^{n}\left(Y_{i}^{2}\right) \tag{8}\]