Fractional-order PID-based search algorithms: A math-inspired meta-heuristic technique with historical information consideration

Abstract

The PID-based Search Algorithm (PSA) is a novel math-inspired metaheuristic algorithm. However, the traditional PSA, based on PID principles, only considers the current population information. To investigate the influence of Population Historical Information (PHI) on the convergence performance of PSA and design a more effective population evolution mechanism, we drew inspiration from the fractional-order PID and introduced the fractional-order Nabla operator, which is well-suited for modeling discrete systems characterized by memory and heredity, to improve PSA. We proposed three fractional-order variants of PSA, named FoPSA-I, FoPSA-II, and FoPSA-III, based on three types of historical information in the population update process: error, input, and position. Through fractional-order sensitivity analysis on CEC benchmark test functions and numerical experiments in relevant engineering applications, we found that among the three FoPSA variants, FoPSA-III, which considers historical position information, showed significant differences in convergence performance compared to PSA, whereas FoPSA-I and FoPSA-II showed minimal differences from PSA. Additionally, the p-values obtained from the Wilcoxon test further validated the differences among the three FoPSAs and PSA, with p-values for FoPSA-I, FoPSA-II, and FoPSA-III being 0.1446, 0.0475, and 0.0019, respectively. Finally, through mathematical analysis, we qualitatively explored the reasons for the differing convergence performance of the three FoPSA variants. The results indicate that considering historical position information in the PSA population update process can enhance population diversity and the algorithm’s convergence performance. This provides new insights into the design of population update mechanisms in metaheuristic algorithms.