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RETRACTED ARTICLE: On the Algorithmic Stability of Optimal Control with Derivative Operators

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This article was retracted on 16 December 2023

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Abstract

The aim of this paper is to develop a productive numerical technique to deal with a class of time partial ideal AI control issues. The classical fuzzy inference methods cannot work to their full potential in such circumstances, because the given knowledge does not cover the entire problem domain. In addition, the requirements of fuzzy systems may change over time. The use of a static rule base may affect the effectiveness of fuzzy rule interpolation due to the absence of the most concurrent (dynamic) rules. The experimental result indicates that evolved bat algorithm with our proposed fitness function presents a 93.77% success rate in average for finding the feasible solutions. The contribution of this study is that near outcomes likewise confirm that the partial administrator for a Mittag–Leffler circuit in the Caputo sense improves the execution of the AI controlled framework as far as the transient reaction, in contrast with the other fragmentary and whole number subordinate administrators.

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References

  1. T. Abdeljawad, D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag–Leffler nonsingular kernel. J. Nonlinear Sci. Appl. 10(3), 1098–1107 (2017)

    Article  MathSciNet  Google Scholar 

  2. O.P. Agrawal, General formulation for the numerical solution of optimal control problems. Int. J. Control 50, 627–638 (1989)

    Article  MathSciNet  Google Scholar 

  3. A. Akbarian, M. Keyanpour, A new approach to the numerical solution of fractional order optimal control problems. Appl.Appl. Math. 8(2), 523–534 (2013)

    MathSciNet  Google Scholar 

  4. A. Alizadeh, S. Effati, An iterative approach for solving fractional optimal control problems. J. Vib. Control 24(1), 18–36 (2018)

    Article  MathSciNet  Google Scholar 

  5. R. Almeida, D.F.M. Torres, A discrete method to solve fractional optimal control problems. Nonlinear Dyn. 80(4), 1811–1816 (2015)

    Article  MathSciNet  Google Scholar 

  6. A. Atangana, J.F. Gómez-Aguilar, A new derivative with normal distribution kernel: theory, methods and applications. Phys. A 476, 1–14 (2017)

    Article  MathSciNet  Google Scholar 

  7. A. Atangana, J.F. Gómez-Aguilar, Numerical approximation of Riemann-Liouville definition of fractional derivative: from Riemann-Liouville to Atangana-Baleanu. Numer. Methods Partial Differ. Equ. (2017). https://doi.org/10.1002/num.22195

    Article  Google Scholar 

  8. A. Atangana, J.F. Gómez-Aguilar, Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag–Leffler laws. Chaos, Solitons Fractals (2017). https://doi.org/10.1016/j.chaos.2017.03.022

    Article  MathSciNet  Google Scholar 

  9. A. Atangana, J.F. Gómez-Aguilar, Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur. Phys. J. Plus 133, 1–23 (2018)

    Article  Google Scholar 

  10. L.F. Ávalos-Ruiz, C.J. Zúñiga-Aguilar, J.F. Gómez-Aguilar, R.F. Escobar-Jiménez, H.M. Romero-Ugalde, FPGA implementation and control of chaotic systems involving the variable-order fractional operator with Mittag–Leffler law. Chaos, Solitons Fractals 115, 177–189 (2018)

    Article  MathSciNet  Google Scholar 

  11. D. Baleanu, A. Jajarmi, M. Hajipour, On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel. Nonlinear Dyn. 94(1), 397–414 (2018)

    Article  Google Scholar 

  12. R.K. Biswas, S. Sen, Fractional optimal control problems with specified final time. J. Comput. Nonlinear Dyn. 6(2), 021009–6 (2010)

    Google Scholar 

  13. C.W. Chen, Interconnected TS fuzzy technique for nonlinear time-delay structural systems. Nonlinear Dyn. 76(1), 13–22 (2014)

    Article  MathSciNet  Google Scholar 

  14. C.W. Chen, A criterion of robustness intelligent nonlinear control for multiple time-delay systems based on fuzzy Lyapunov methods. Nonlinear Dyn. 76(1), 23–31 (2014)

    Article  MathSciNet  Google Scholar 

  15. T. Chen, S. Rao, R.T. Sabitovich, An intelligent algorithm optimum for building design of fuzzy structures. Iran. J. Sci. Technol. Trans. Civ. Eng. 44, 523–531 (2020)

    Article  Google Scholar 

  16. T. Chen, LMI based criterion for reinforced concrete frame structures. Adv. Concrete Constr. 9(4), 407–412 (2020)

    Google Scholar 

  17. T. Chen, A. Babanin, A. Muhammad, B. Chapron, C. Y. J. Chen, Evolved fuzzy NN control for discrete-time nonlinear systems. J. Circuits Syst. Comput. 29(1), 2050015 (2020)

    Article  Google Scholar 

  18. S.H. Cheng, S.M. Chen, C.L. Chen, Weighted fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on piecewise fuzzy entropies of fuzzy sets. Inf. Sci. 329, 503–523 (2016)

    Article  Google Scholar 

  19. A. Coronel-Escamilla, J.F. Gómez-Aguilar, M.G. López-Lópeza, V.M. Alvarado-Martínez, G.V. Guerrero-Ramíreza, Triple pendulum model involving fractional derivatives with different kernels. Chaos, Solitons Fractals 91, 248–261 (2016)

    Article  MathSciNet  Google Scholar 

  20. A. Dabiri, B.P. Moghaddam, J.A.T. Machado, Optimal variable-order fractional PID controllers for dynamical systems. J. Comput. Appl. Math. 339, 40–48 (2018)

    Article  MathSciNet  Google Scholar 

  21. S.A. David, C. Fischer, J.A.T. Machado, Fractional electronic circuit simulation of a nonlinear macroeconomic model. AEU Int. J. Electron. Commun. 84, 210–220 (2018)

    Article  Google Scholar 

  22. H. Fatoorehchi, M. Alidadi, R. Rach, Theoretical and experimental investigation of thermal dynamics of steinhart-hart negative temperature coefficient thermistors. Trans. ASME J. Heat Transf. 141(7), 072003 (2019)

    Article  Google Scholar 

  23. J.F. Gómez-Aguilar, A. Atangana, New insight in fractional differentiation: power, exponential decay and Mittag–Leffler laws and applications. Eur. Phys. J. Plus 132(1), 1–23 (2017)

    Article  Google Scholar 

  24. W. Hackbush, A numerical method for solving parabolic equations with opposite orientations. Computing 20(3), 229–240 (1978)

    Article  MathSciNet  Google Scholar 

  25. A. Jajarmi, M. Hajipour, D. Baleanu, New aspects of the adaptive synchronization and hyper chaos suppression of a financial model. Chaos, Solitons Fractals 99, 285–296 (2017)

    Article  MathSciNet  Google Scholar 

  26. A. Jajarmi, M. Hajipour, E. Mohammadzadeh, D. Baleanu, A new approach for the nonlinear fractional optimal control problems with external persistent disturbances. J. Franklin Inst. 335(9), 3938–3967 (2018)

    Article  MathSciNet  Google Scholar 

  27. H.K. Lam, Stability analysis of T-S fuzzy control systems using parameter-dependent Lyapunov function. IET Control Theory Appl. 3, 750–762 (2009)

    Article  MathSciNet  Google Scholar 

  28. C. Li, F. Zeng, The finite difference methods for fractional ordinary differential equations. Numer. Funct. Anal. Optim. 34(2), 149–179 (2013)

    Article  MathSciNet  Google Scholar 

  29. S. Mashayekhi, M. Razzaghi, An approximate method for solving fractional optimal control problems by hybrid functions. J. Vib. Control 24(9), 1621–1631 (2018)

    Article  MathSciNet  Google Scholar 

  30. H.S. Nik, P. Rebelo, M.S. Zahedi, Solution of infinite horizon nonlinear optimal control problems by piecewise adomian decomposition method. Math. Model. Anal. 18(4), 543–560 (2013)

    Article  MathSciNet  Google Scholar 

  31. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications (Academic Press, New York, 1999)

    Google Scholar 

  32. P.K. Sahu, S.S. Ray, Comparison on wavelets techniques for solving fractional optimal control problems. J. Vib. Control 24(6), 1185–1201 (2018)

    Article  MathSciNet  Google Scholar 

  33. N. Singha, C. Nahak, An efficient approximation technique for solving a class of fractional optimal control problems. J. Optim. Theory Appl. 174(3), 785–802 (2017)

    Article  MathSciNet  Google Scholar 

  34. P. Su, C. Shang, T. Chen, Q. Shen, Exploiting data reliability and fuzzy clustering for journal ranking. IEEE Trans. Fuzzy Syst. 25(5), 1306–1319 (2017)

    Article  Google Scholar 

  35. N.H. Sweilam, S.M. Al-Mekhlafi, On the optimal control for fractional multi-strain TB model. Optim. Control Appl. Methods 37(6), 1355–1374 (2016)

    Article  MathSciNet  Google Scholar 

  36. P.W. Tsai, T. Hayat, B. Ahmad, Structural system simulation and control via NN based fuzzy model. Struct. Eng. Mech. 56(3), 385–407 (2015). https://doi.org/10.12989/sem.2015.56.3.385

    Article  Google Scholar 

  37. H.O. Wang, K. Tanaka, M. Griffin, An approach to fuzzy control of nonlinear systems: stability and design issues. IEEE Trans. Fuzzy Syst. 4, 14–23 (1996)

    Article  Google Scholar 

  38. C.J. Zuñiga-Aguilar, J.F. Gómez-Aguilar, R.F. Escobar-Jiménez, H.M. Romero-Ugalde, Robust control for fractional variable-order chaotic systems with non-singular kernel. Eur. Phys. J. Plus 133(1), 1–13 (2018)

    Article  Google Scholar 

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Acknowledgements

This research was completed without funding.

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Authors and Affiliations

  1. AI LAB, Faculty of Information Technology, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Tim Chen

  2. Covenant University, 10 Idiroko Road, Canaan Land, Ota, Ogun State, Nigeria

    J. C.-Y. Cheng

  3. NAAM Research Group, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    J. C.-Y. Cheng

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  1. Tim Chen
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Correspondence to J. C.-Y. Cheng.

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This article has been retracted. Please see the retraction notice for more detail: https://doi.org/10.1007/s00034-023-02569-y

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Chen, T., Cheng, J.CY. RETRACTED ARTICLE: On the Algorithmic Stability of Optimal Control with Derivative Operators. Circuits Syst Signal Process 39, 5863–5881 (2020). https://doi.org/10.1007/s00034-020-01447-1

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