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Optimal Design of IIR-Type Fractional Order Digital Integrator Using Mayfly Optimization Algorithm

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Abstract

This paper represents the execution of a reliable, steady and wideband fractional order digital integrator (FODI) in respect of infinite impulse response (IIR) filter using a recent nature inspired meta-heuristic optimization technique known as mayfly optimization algorithm (MOA). MOA significantly functions stronger against some of the popular algorithms like real-coded genetic algorithm (RGA), particle swarm optimization (PSO), differential evolution (DE), improved particle swarm optimization (IPSO), whale optimization algorithm (WOA) and chaotic oppositional whale optimization algorithm (COWOA) with reference to different magnitude error performances, explanation character accuracy, convergence profile and computational time required to find the optimal solution. In this paper, the authors thoroughly analyze the convergence behavior and magnitude error metrices of the different order FODI’s using WOA, COWOA and MOA. MATLAB simulation results also powerfully uphold the magnitude response efficiency and stability of the proposed different order FODI’s over some recently published research papers.

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References

  1. R.S. Barbosa, J.T. Machado, M.F. Silva, Time domain design of fractional differintegrators using least-squares. Signal Process. 86(10), 2567–2581 (2006)

    Article  MATH  Google Scholar 

  2. M. Benmalek, A. Charef, Digital fractional order operators for r-wave detection in electrocardiogram signal. IET Signal Process. 3(5), 381391 (2009)

    Article  Google Scholar 

  3. Y.Q. Chen, K.L. Moore, Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 49(3), 363–367 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Y. Chen, Vinagre BM (2003) A new iir-type digital fractional order differentiator. Signal Process. 83(11), 2359–2365 (2003)

    Article  MATH  Google Scholar 

  5. Y. Chen, B.M. Vinagre, I. Podlubny, Continued fraction expansion approaches to discretizing fractional order derivatives–an expository review. Nonlinear Dyn. 38(1), 155–170 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. L. Debnath, Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 2003(54), 3413–3442 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. N. Engheta, On fractional calculus and fractional multipoles in electromagnetism. IEEE Trans. Antennas Prop. 44(4), 554–566 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Y. Ferdi, Computation of fractional order derivative and integral via power series expansion and signal modelling. Nonlinear Dyn. 46(1), 1–15 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. R.K.H. Galvao, S. Hadjiloucas, K.H. Kienitz, H.M. Paiva, R.J.M. Afonso, Fractional order modeling of large three-dimensional rc networks. IEEE Trans. Circ. Syst. I Regul. Pap. 60(3), 624–637 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. D.E. Golberg, Genetic algorithms in search, optimization, and machine learning. Addion Wesley 1989(102), 36 (1989)

    Google Scholar 

  11. M. Gupta, R. Yadav, Design of improved fractional order integrators using indirect discretization method. Int. J. Comput. Appl. 59(14), 19–24 (2012)

    Google Scholar 

  12. P. Igor, Fractional-order systems and \(\Pi \lambda d\mu \) controllers. IEEE Trans. Autom. Control 44(1), 208–214 (1999)

    Article  MathSciNet  Google Scholar 

  13. J. Kennedy, R. Eberhart, R, in Particle swarm optimization. Proceedings Oficnn’95-Int. Conf. on Neur. Net., vol. 4 (1995), pp. 1942–1948. IEEE

  14. J.T. Machado, Analysis and design of fractional-order digital control systems. Syst. Anal. Model. Simul. 27(2–3), 107–122 (1997)

    MATH  Google Scholar 

  15. S. Mahata, S.K. Saha, R. Kar, D. Mandal, Optimal and accurate design of fractional-order digital differentiator-an evolutionary approach. IET Signal Process. 11(2), 181–196 (2017)

    Article  Google Scholar 

  16. S. Mahata, S.K. Saha, R. Kar, D. Mandal, Improved iir-type fractional order digital integrators using cat swarm optimization. Turkish J. Electr. Eng. Comput. Sci. 26(2), 856–866 (2018)

    Article  Google Scholar 

  17. S. Mahata, S.K. Saha, R. Kar, D. Mandal, Optimal design of iir digital foi using ipso. Int. J. Electron. Lett. 6(2), 181–191 (2018)

    Article  Google Scholar 

  18. G. Maione, Continued fractions approximation of the impulse response of fractional-order dynamic systems. IET Control Theory Appl. 2(7), 564–572 (2008)

    Article  MathSciNet  Google Scholar 

  19. Maione, G (2012). Thiele’s continued fractions in digital implementation of noninteger differintegrators. Signal Image Video Process. 6(3), 401-410

  20. B. Mathieu, P. Melchior, A. Oustaloup, C. Ceyral, Fractional differentiation for edge detection. Signal Process. 83(11), 2421–2432 (2003)

    Article  MATH  Google Scholar 

  21. S. Mirjalili, A. Lewis, The whale optimization algorithm. Adv. Eng. Softw. 95, 51–67 (2016)

    Article  Google Scholar 

  22. J. Nasiri, F.M. Khiyabani, A whale optimization algorithm (woa) approach for clustering. Cogent Math. Stat. 5(1), 1483565 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Oustaloup, F. Levron, B. Mathieu, F.M. Nanot, Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 47(1), 25–39 (2000)

    Article  Google Scholar 

  24. D. Panigrahy, P. Sahu, F. Albu, Detection of ventricular fibrillation rhythm by using boosted support vector machine with an optimal variable combination. Comput. Electr. Eng. 91, 107035 (2021)

    Article  Google Scholar 

  25. S. Rahnamayan, H.R. Tizhoosh, M.M. Salama, Opposition-based differential evolution. IEEE Trans. Evol. Comput. 12(1), 64–79 (2008)

    Article  Google Scholar 

  26. S. Rani, A. Singh, On the design and optimisation of new fractal antenna using pso. Int. J. Electron. 100(10), 1383–1397 (2013)

    Article  Google Scholar 

  27. M. Romero, A. De Madrid, C. Manoso, B. Vinagre, Iir approximations to the fractional differentiator/integrator using Cheby-Shev polynomials theory. ISA Trans. 52(4), 461–468 (2013)

    Article  Google Scholar 

  28. P.K. Roy, C. Paul, S. Sultana, Oppositional teaching learning based optimization approach for combined heat and power dispatch. Int. J. Electr. Power Energy Syst. 57, 392–403 (2014)

    Article  Google Scholar 

  29. R. Storn, K. Price, Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11(4), 341–359 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Sundaram, Multiobjective multi-verse optimization algorithm to solve combined economic, heat and power emission dispatch problems. Appl. Soft Comput. 91, 106195 (2020)

    Article  Google Scholar 

  31. C.-C. Tseng, Design of fir and iir fractional order Simpson digital integrators. Signal Process. 87(5), 1045–1057 (2007)

    Article  MATH  Google Scholar 

  32. B.M. Vinagre, I. Petras, P. Merchan, L. Dorcak, in Two digital realizations of fractional controllers: application to temperature control of a solid. 2001 European Contr. Conf. (ECC) (2001), pp. 1764–1767. IEEE

  33. B. Vinagre, I. Podlubny, A. Hernandez, V. Feliu, V, in On realization of fractional-order controllers. Proc. of the Conf. Inter. Francophone d’automatique (200a), pp. 5–8

  34. B. Vinagre, I. Podlubny, A. Hernandez, V. Feliu, Some approximations of fractional order operators used in control theory and applications. Fract. Calc. Appl. Anal. 3(3), 231–248 (2000)

    MathSciNet  MATH  Google Scholar 

  35. F. Wang, X. Ma, Fractional order buck-boost converter in ccm: modelling, analysis and simulations. Int. J. Electron. 101(12), 16711682 (2014)

    Article  Google Scholar 

  36. B.J. West, M. Bologna, P. Grigolini, Physics of Fractal Operators (Springer, Berlin, 2003)

    Book  Google Scholar 

  37. R. Yadav, M. Gupta, Approximations of higher-order fractional differentiators and integrators using indirect discretization. Turkish J. Electr. Eng. Comput. Sci. 23(3), 666–680 (2015)

    Article  Google Scholar 

  38. R. Yadav, M. Gupta, New improved fractional order integrators using pso optimisation. Int. J. Electron. 102(3), 490–499 (2015)

    Article  Google Scholar 

  39. X.S. Yang, C. He, Firefly algorithm: recent advances and applications. arXiv preprintarXiv:1308.3898 (2013)

  40. K. Zervoudakis, S. Tsafarakis, A mayfly optimization algorithm. Comput. Ind. Eng. 145, 106559 (2020)

    Article  Google Scholar 

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Correspondence to Souvik Dey.

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Dey, S., Roy, P.K. & Chakraborty, S. Optimal Design of IIR-Type Fractional Order Digital Integrator Using Mayfly Optimization Algorithm. Circuits Syst Signal Process 42, 913–942 (2023). https://doi.org/10.1007/s00034-022-02141-0

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