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Reset control approximates complex order transfer functions

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Abstract

A controller with the frequency response of a complex order derivative may have a gain that decreases with frequency, while the phase increases. This behaviour may be desirable to ensure simultaneous rejection of high-frequency noise and robustness to variations of the open-loop gain. Implementations of such complex order controllers found in the literature are unsatisfactory for several reasons: the desired behaviour of the gain may be difficult or impossible to obtain, or non-minimum phase zeros may appear, or even unstable open-loop poles. We propose an alternative nonlinear approximation, combining a CRONE approximation of a fractional derivative with reset control, which does not suffer from these problems. An experimental proof of concept confirms the good results of this approximation and shows that nonlinear effects do not preclude the desired performance.

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Notes

  1. CRONE is the French acronym for non-integer order robust control.

References

  1. Chen, L., Saikumar, N., HosseinNia, S.H.: Development of robust fractional-order reset control. IEEE Trans. Control Syst. Technol. (2019). https://doi.org/10.1109/TCST.2019.2913534

    Google Scholar 

  2. Guo, Y., Wang, Y., Xie, L.: Frequency-domain properties of reset systems with application in hard-disk-drive systems. IEEE Trans. Control Syst. Technol. 17(6), 1446–1453 (2009)

    Article  Google Scholar 

  3. https://www.mathworks.com/matlabcentral/fileexchange/70366-complex-order-reset-controllers

  4. Moghadam, M.G., Padula, F., Ntogramatzidis, L.: Tuning and performance assessment of complex fractional-order PI controllers. In: 3rd IFAC Conference on Advances in Proportional-Integral-Derivative Control, vol. 51, pp. 757–762 (2018)

  5. Moreau, X., Altet, O., Oustaloup, A.: The crone suspension: management of the dilemma comfort-road holding. Nonlinear Dyn. 38, 461–484 (2004)

    Article  MATH  Google Scholar 

  6. Ogata, K.: Modern Control Engineering. Prentice Hall, Upper Saddle River (2010)

    MATH  Google Scholar 

  7. Oustaloup, A.: La commande CRONE—commande robuste d’ordre non-entier. Hermès (1991)

  8. Oustaloup, A., Bansard, M.: First generation CRONE control. In: International Conference on Systems, Man and Cybernetics. IEEE, Le Touquet (1993)

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

    Article  Google Scholar 

  10. Oustaloup, A., Mathieu, B., Lanusse, P.: Second generation CRONE control. In: International Conference on Systems, Man and Cybernetics. IEEE, Le Touquet (1993)

  11. Palanikumar, A., Saikumar, N., HosseinNia, S.H.: No more differentiator in PID: development of nonlinear lead for precision mechatronics. In: 17th European Control Conference, pp. 991–996 (2018)

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

    MATH  Google Scholar 

  13. Saikumar, N., HosseinNia, S.: Generalized fractional order reset element (GFRORE). In: 9th European Nonlinear Dynamics Conference (ENOC) (2017)

  14. Saikumar, N., Sinha, R.K., HosseinNia, S.H.: ‘Constant in gain lead in phase’ element: application in precision motion control. IEEE ASME Trans. Mechatron. (2019). https://doi.org/10.1109/TMECH.2019.2909082

    Google Scholar 

  15. Saikumar, N., Sinha, R.K., HosseinNia, S.H.: Resetting disturbance observers with application in compensation of bounded nonlinearities like hysteresis in piezo-actuators. Control Eng. Pract. 82, 36–49 (2019)

    Article  Google Scholar 

  16. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach, Yverdon (1993)

    MATH  Google Scholar 

  17. Shah, P., Agashe, S.: Review of fractional PID control. Mechatronics 38, 29–41 (2016)

    Article  Google Scholar 

  18. Valério, D., Sá da Costa, J.: An Introduction to Fractional Control. IET, Stevenage (2013)

    MATH  Google Scholar 

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Acknowledgements

This work was supported by NWO, through OTP TTW project #16335, by FCT, through IDMEC, under LAETA, project UID/EMS/50022/2019, and grant SFRH/BSAB/142920/2018 attributed to the first author.

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Authors and Affiliations

  1. IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

    Duarte Valério

  2. Department of Precision and Microsystems Engineering, Delft University of Technology, Delft, The Netherlands

    Niranjan Saikumar, Ali Ahmadi Dastjerdi, Nima Karbasizadeh & S. Hassan HosseinNia

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  1. Duarte Valério
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  2. Niranjan Saikumar
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  3. Ali Ahmadi Dastjerdi
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  4. Nima Karbasizadeh
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  5. S. Hassan HosseinNia
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Correspondence to Duarte Valério.

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Valério, D., Saikumar, N., Dastjerdi, A.A. et al. Reset control approximates complex order transfer functions. Nonlinear Dyn 97, 2323–2337 (2019). https://doi.org/10.1007/s11071-019-05130-2

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