Abstract: Fractional calculus is considered a useful tool for gaining deeper insights into systems with memory effects or history. Fractional-order modeling of nonlinear systems may increase the stiffness and complexity of the system, but also provides better insights. This study introduces a swarm intelligence-based parameter estimation of the fractional Hammerstein autoregressive exogenous noise (fractional-HARX) model. The Grünwald–Letnikov finite difference formula is used to develop the fractional-HARX model from the standard HARX model. This study presents the design of a swarm intelligence-based electric eel foraging optimization algorithm (EEFOA) for parameter estimation of the fractional-HARX model under multiple noise scenarios for second- and third-order polynomial type nonlinearity. The key-term separation principle is also incorporated in the system model to reduce the occurrence of redundant parameters due to cross-product terms in the information vector. The designed methodology is examined, and the superiority of EEFOA is endorsed in terms of convergence, robustness, stiff parameter estimation, and deviation from the mean point in comparison with state-of-the-art optimization heuristics such as the whale optimization algorithm, the African vulture optimization algorithm, Harris hawk’s optimizer, and the reptile search algorithm. The statistical significance of the EEFOA for the estimation of fractional-HARX models is also established using statistical indices of best, mean, and worst fitness values along with standard deviation for multiple noise scenarios.

[1]Abdollahzadeh B, Gharehchopogh FS, Mirjalili S, 2021. African vultures optimization algorithm: a new nature-inspired metaheuristic algorithm for global optimization problems. Comput Ind Eng, 158:107408.

[2]Abdollahzadeh B, Khodadadi N, Barshandeh S, et al., 2024. Puma optimizer (PO): a novel metaheuristic optimization algorithm and its application in machine learning. Cluster Comput, 27(4):5235-5283.

[3]Abualigah L, Abd Elaziz M, Sumari P, et al., 2022. Reptile search algorithm (RSA): a nature-inspired meta-heuristic optimizer. Expert Syst Appl, 191:116158.

[4]Ala A, Goli A, Mirjalili S, et al., 2024. A fuzzy multi-objective optimization model for sustainable healthcare supply chain network design. Appl Soft Comput, 150:111012.

[5]Ali R, Asjad MI, Akgül A, 2021. An analysis of a mathematical fractional model of hybrid viscous nanofluids and its application in heat and mass transfer. J Comput Appl Math, 383:113096.

[6]Altaf F, Chang CL, Chaudhary NI, et al., 2025. Astrophysical expedition: transit search heuristics for fractional Hammerstein control autoregressive models. Mod Phys Lett B, 39(7):2450417.

[7]Dong RY, Sun LX, Ma L, et al., 2023. Boosting kernel search optimizer with slime mould foraging behavior for combined economic emission dispatch problems. J Bionic Eng, 20(6):2863-2895.

[8]El-Hasnony IM, Barakat SI, Mostafa RR, 2020. Optimized ANFIS model using hybrid metaheuristic algorithms for Parkinson’s disease prediction in IoT environment. IEEE Access, 8:119252-119270.

[9]Fang LL, Liang XY, 2023. A novel method based on nonlinear binary grasshopper whale optimization algorithm for feature selection. J Bionic Eng, 20(1):237-252.

[10]Frikh ML, Boutasseta N, 2024. Pitch angle control of wind turbines using model-free auto-tuned fractional order proportional derivative ATFOPD controller. Comput Electr Eng, 116:109199.

[11]Gamini S, Kumar SS, 2023. Homomorphic filtering for the image enhancement based on fractional-order derivative and genetic algorithm. Comput Electr Eng, 106:108566.

[12]González-Patiño D, Villuendas-Rey Y, Argüelles-Cruz AJ, et al., 2019. A novel bio-inspired method for early diagnosis of breast cancer through mammographic image analysis. Appl Sci, 9(21):4492.

[13]Han Y, Chen WB, Heidari AA, et al., 2023. Multi-verse optimizer with Rosenbrock and diffusion mechanisms for multilevel threshold image segmentation from COVID-19 chest X-ray images. J Bionic Eng, 20(3):1198-1262.

[14]Heidari AA, Mirjalili S, Faris H, et al., 2019. Harris hawks optimization: algorithm and applications. Fut Gener Comput Syst, 97:849-872.

[15]Hu HY, Xie ZK, Wang DQ, 2024. Temporal pattern attention based Hammerstein model for estimating battery SOC. J Energy Storage, 100:113666.

[16]Ionescu C, Lopes A, Copot D, et al., 2017. The role of fractional calculus in modeling biological phenomena: a review. Commun Nonl Sci Numer Simul, 51:141-159.

[17]Jakšić Z, Devi S, Jakšić O, et al., 2023. A comprehensive review of bio-inspired optimization algorithms including applications in microelectronics and nanophotonics. Biomimetics, 8(3):278.

[18]Karaca Y, 2023. Fractional calculus operators–Bloch–Torrey partial differential equation–artificial neural networks–computational complexity modeling of the micro–macrostructural brain tissues with diffusion MRI signal processing and neuronal multi-components. Fractals, 31(10):2340204.

[19]Kausar A, Chang CY, Raja MAZ, et al., 2025. A novel design of layered recurrent neural networks for fractional order Caputo–Fabrizio stiff electric circuit models. Mod Phys Lett B, 39(2):2450393.

[20]Khan TA, Chaudhary NI, Khan ZA, et al., 2024a. Design of Runge–Kutta optimization for fractional input nonlinear autoregressive exogenous system identification with key-term separation. Chaos Sol Fract, 182:114723.

[21]Khan TA, Chaudhary NI, Hsu CC, et al., 2024b. A gazelle optimization expedition for key term separated fractional nonlinear systems with application to electrically stimulated muscle modeling. Chaos Sol Fract, 185:115111.

[22]Li F, Liang MJ, Luo YS, 2023a. Correlation analysis-based parameter learning of Hammerstein nonlinear systems with output noise. Eur J Contr, 72:100819.

[23]Li F, Zheng T, Cao QF, 2023b. Modeling and identification for practical nonlinear process using neural fuzzy network–based Hammerstein system. Trans Inst Meas Contr, 45(11):2091-2102.

[24]Li F, Zhu XJ, Cao QF, 2023c. Parameter learning for the nonlinear system described by a class of Hammerstein models. Circ Syst Signal Process, 42(5):2635-2653.

[25]Li F, Zhou SB, Liu RR, 2024. Parameter estimation for the Hammerstein–Wiener nonlinear system and application in lithium-ion batteries. J Energy Storage, 102:114265.

[26]Li S, Huang CD, Song XS, 2023. Novel method to detect Hopf bifurcation in a delayed fractional-order network model with bidirectional ring structure. Int J Biomath, 16(6):2250117.

[27]Li ZQ, Wang WW, Zhang CL, et al., 2023. Fault-tolerant control based on fractional sliding mode: crawler plant protection robot. Comput Electr Eng, 105:108527.

[28]Li ZX, Yang Y, Li LW, et al., 2023. A weighted Pearson correlation coefficient based multi-fault comprehensive diagnosis for battery circuits. J Energy Storage, 60:106584.

[29]Li ZX, Li LW, Chen J, et al., 2024. A multi-head attention mechanism aided hybrid network for identifying batteries’ state of charge. Energy, 286:129504.

[30]Liu X, Wang C, Dai W, 2024. Probability-based identification of Hammerstein systems with asymmetric noise characteristics. IEEE Trans Instrum Meas, 73:6500611.

[31]Machado JAT, Lopes AM, 2015. Analysis of natural and artificial phenomena using signal processing and fractional calculus. Fract Calc Appl Anal, 18(2):459-478.

[32]Malik MF, Chang CL, Aslam MS, et al., 2022. Fuzzy-evolution computing paradigm for fractional Hammerstein control autoregressive systems. Int J Fuzzy Syst, 24(5):2447-2475.

[33]Malik MF, Chang CL, Chaudhary NI, et al., 2023. Swarming intelligence heuristics for fractional nonlinear autoregressive exogenous noise systems. Chaos Sol Fract, 167:113085.

[34]Malik NA, Chang CL, Chaudhary NI, et al., 2024. Astrophysics-based transit search optimization heuristics for parameter estimation of multi-frequency sinusoidal signals. Mod Phys Lett B, 38(34):2450342.

[35]Mathi MT, Baburaj E, 2022. Comparative analysis of bio-inspired optimization algorithms in neural network-based data mining classification. Int J Swarm Intell Res, 13(1):25.

[36]Megherbi O, Hamiche H, Bettayeb M, 2024. Implementation of a wireless text data transmission based on the impulsive control of fractional-order chaotic systems. Comput Electr Eng, 116:109224.

[37]Mehmood K, Chaudhary NI, Khan ZA, et al., 2024. Atomic physics-inspired atom search optimization heuristics integrated with chaotic maps for identification of electro-hydraulic actuator systems. Mod Phys Lett B, 38(30):2450308.

[38]Mehmood N, Abbas A, Akgül A, et al., 2023. Existence and stability results for coupled system of fractional differential equations involving AB-Caputo derivative. Fractals, 31(2):2340023.

[39]Mirjalili S, Lewis A, 2016. The whale optimization algorithm. Adv Eng Softw, 95:51-67.

[40]Mukhtar R, Chang CY, Raja MAZ, et al., 2024. Novel nonlinear fractional order Parkinson’s disease model for brain electrical activity rhythms: intelligent adaptive Bayesian networks. Chaos Sol Fract, 180:114557.

[41]Padhi JR, Deeb MA, Tripathy S, et al., 2023. Fractional calculus based PI-FOPID controller for frequency deviation control in integrated power system. Proc Electric Power and Renewable Energy Conf, p.213-224.

[42]Partohaghighi M, Yusuf A, Alshomrani AS, et al., 2024. Fractional hyper-chaotic system with complex dynamics and high sensitivity: applications in engineering. Int J Mod Phys B, 38(1):2450012.

[43]Rahmanshahi M, Jafari-Asl J, Fathi-Moghadam M, et al., 2024. Metaheuristic learning algorithms for accurate prediction of hydraulic performance of porous embankment weirs. Appl Soft Comput, 151:111150.

[44]Šapina M, Garcin M, Kramarić K, et al., 2020. The Hurst exponent of heart rate variability in neonatal stress, based on a mean-reverting fractional Lévy stable motion. Fluctuat Noise Lett, 19(3):2050026.

[45]Sowa M, Majka Ł, Wajda K, 2023. Excitation system voltage regulator modeling with the use of fractional calculus. AEU-Int J Electron Commun, 159:154471.

[46]Sweis H, Arqub OA, Shawagfeh N, 2023. Fractional delay integrodifferential equations of nonsingular kernels: existence, uniqueness, and numerical solutions using Galerkin algorithm based on shifted Legendre polynomials. Int J Mod Phys C, 34(4):2350052.

[47]Tubishat M, Al-Obeidat F, Sadiq AS, et al., 2023. An improved dandelion optimizer algorithm for spam detection: next-generation email filtering system. Computers, 12(10):196.

[48]Vyawahare VA, Nataraj PSV, 2013. Fractional-order modeling of neutron transport in a nuclear reactor. Appl Math Modell, 37(23):9747-9767.

[49]Wang DQ, 2024. Key-term separation based hierarchical gradient approach for NN based Hammerstein battery model. Appl Math Lett, 157:109207.

[50]Wang PF, Xu ZK, Chen DY, 2023a. An integrated framework for reliability prediction and condition-based maintenance policy for a hydropower generation unit using GPHM and SMDP. Reliab Eng Syst Saf, 238:109419.

[51]Wang PF, Guo YX, Xu ZK, et al., 2023b. A novel approach of full state tendency measurement for complex systems based on information causality and PageRank: a case study of a hydropower generation system. Mech Syst Signal Process, 187:109956.

[52]Wang SW, Xiao XP, Ding Q, 2024. A novel fractional system grey prediction model with dynamic delay effect for evaluating the state of health of lithium battery. Energy, 290:130057.

[53]Ye SQ, Zhou KQ, Zain AM, et al., 2023. A modified harmony search algorithm and its applications in weighted fuzzy production rule extraction. Front Inform Technol Electron Eng, 24(11):1574-1590.

[54]Zhang MG, Li F, Yu Y, et al., 2024. Estimation of Hammerstein nonlinear systems with noises using filtering and recursive approaches for industrial control. Front Inform Technol Electron Eng, 25(2):260-271.

[55]Zhang XF, Boutat D, Liu DY, 2023. Applications of fractional operator in image processing and stability of control systems. Fract Fract, 7(5):359.

[56]Zhao WG, Wang LY, Zhang ZX, et al., 2024. Electric eel foraging optimization: a new bio-inspired optimizer for engineering applications. Expert Syst Appl, 238:122200.

[57]Zhao ZW, Yuan YC, He MJ, et al., 2022. Stability and efficiency performance of pumped hydro energy storage system for higher flexibility. Renew Energy, 199:1482-1494.