## 1 Introduction  
Optimization methods involve automatically finding the best solution in a problem's solution space set. When converting a real problem into a mathematical model, simulating the actual physical characteristics of the problem requires a detailed description of the conversion process. Moreover, the mathematical model of the problem becomes more complex. Currently, three methods exist for solving optimization problems: numerical method, enumerative, and random search. Numerical methods use the derivative as a technique to find the best value in the space. For example, the traditional neural network series of algorithms is based on this method's gradient descent to find the best parameters [1]. However, the numerical method has two shortcomings. First, it searches for the best solution from a local point of view, so there is no guarantee that the solution found is globally optimal. Second, numerical methods are not applicable for search spaces that are not smooth or continuous [2,3]. However, because usually many regional optimal solutions exist in a search space that is not smooth or continuous, it is easy to converge early in the search process and find the optimal local solution. The enumeration method, such as grid search, uses the objective function to test all solutions in the search space at that level when the segmentation level is selected. This method has a better chance of obtaining the best solution, but it requires considerable computation time. Therefore, when the search space is ample, the enumeration method is inefficient. The random search method is currently a commonly used optimization method, which finds the best solution space by imitating natural or biological behavior, and particle swarm optimization (PSO) [4] is one of the optimization methods that imitate biological behavior. PSO is an intelligent swarm algorithm that observes the behavior of swarm creatures. This method is used to find the best position in the current space. After PSO was published, many scholars proposed different methods to improve the algorithm. These methods have been applied in many fields [5, 6, 7]. For example, [PERSON] and [PERSON] [8] proposed a constant inertia weight to improve the moving direction of particles. Different inertial weights make it possible to find a balance between local and global searches. Thus, the algorithm considers the best solution in the whole domain. [PERSON] and [PERSON] [9] again proposed linearly decreasing inertia weights. In the same year, [PERSON] [10] proposed a linear decrease with dynamic characteristics applied to individual learning parameters \(c_{1}\) and group learning parameters \(c_{2}\), which effectively improved global search. [PERSON] [11,12] put forward the concept of the shrinkage coefficient, and its idea is to change the moving direction of particles to increase local search. [PERSON] and [PERSON] [13] proposed the maximum speed method to improve search. [PERSON] et al. [14] improved the method proposed by [PERSON] by changing the swarm learning parameter from linearly decreasing to linearly increasing. [PERSON] and [PERSON] [15] proposed to change inertia weights in a nonlinearly decreasing way. [PERSON] et al. [16] extended the concept of nonlinear change to individual and group learning parameters. They changed the individual learning parameters to nonlinear decreasing individual learning parameters and the group learning parameters to nonlinear increasing group learning parameters. With many scholars proposing methods to improve the original algorithm, the particle swarm algorithm search has been dramatically improved. Since 2002, [PERSON] and [PERSON] [12] have used the dynamic system notation in control theory to explore the internal operation of the particle swarm algorithm. Many scholars have also proposed the stability of particle swarm algorithms under different conditions based on a dynamic system representation [17, 18, 19, 20]. In particular, in 2014, [PERSON] [21] proposed the PSO algorithm with controllability.