Parameter estimation for Fractional-Order Systems based on a Mixed-difference Wild Horse Optimizer

Although the Wild Horse Optimizer (WHO) is recognized for its strong convergence performance, it suffers from inherent instability and premature convergence when it is applied to parameter estimation for fractional-order systems. To address these drawbacks, this study proposes a novel Mixed-Difference Wild Horse Optimizer (MWHO). Three strategic improvements are integrated into MWHO: (1) a boundary-aware reinitialization mechanism to enhance population diversity; (2) an elite mating strategy to preserve the competitive advantages of superior individuals; and (3) a Differential Evolution-based mutation strategy to promote escaping from local optima. The proposed algorithm is systematically evaluated on ten benchmark functions and applied to parameter estimation of fractional-order chaotic systems, with comparative experiments conducted against four state-of-the-art algorithms (the original WHO, Grey Wolf Optimizer, Particle Swarm Optimization, and Genetic Algorithm) in both scenarios. Experimental results demonstrate that MWHO significantly outperforms peer algorithms. In benchmark tests, MWHO improves convergence accuracy by more than 12 orders of magnitude compared to the original WHO. For the fractional-order Chen system, MWHO achieves an average fitness function value of approximately 10 −30 , vastly outperforming the original WHO (10 −18 ) and other compared algorithms (10 −6 ). Furthermore, it exhibits exceptional convergence efficiency, with its fitness value stabilizing below 10 −20 within 100 iterations. These findings confirm that MWHO achieves an effective balance between global exploration and local exploitation, making it a highly competitive approach for high-dimensional, nonlinear parameter optimization in fractional-order systems.