Advertisement

Realization of Fractional-Order Proportional Derivative Controller for a Class of Fractional-Order System in Delta Domain

  • Jaydeep SwarnakarEmail author
  • Prasanta Sarkar
  • Lairenlakpam Joyprakash Singh
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 989)

Abstract

The study of the fractional-order controller (FOC) is an emerging area of research in system and control at the present scenario. In this paper, a fractional-order proportional derivative \( \left( {PD^{\eta } } \right) \) controller is implemented for a class of fractional-order plant from certain design specifications of the frequency domain. The entire work has been accomplished in two stages. The first stage deals with the design of continuous-time FOC to satisfy the desired design criterions. In the second stage, a continued fraction expansion (CFE) based direct discrete-time approximation method has been employed to realize the continuous-time FOC in delta domain. The realization of the discrete-time FOC employs the transformed delta operator as a generating function. The efficacy of the controller design method is verified by engaging the discretized FOC with the fractional-order plant and subsequently verifying the frequency response of the discretized open-loop system with respect to the desired frequency response of the original open-loop fractional-order system. The robustness of the overall controlled system is also investigated by altering the plant gain. Simulation results are presented to justify the efficacy of the proposed FOC realization method.

Keywords

Fractional-order controller Continued fraction expansion (CFE) Delta operator Delta domain 

References

  1. 1.
    Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-Order Systems and Controls. Springer, London (2010)CrossRefGoogle Scholar
  2. 2.
    Oustaloup, A.: CRONE Control: Robust Control of Non-integer order. Hermes, Paris (1991)Google Scholar
  3. 3.
    Podlubny, I.: Fractional-order systems and PIλDμ controllers. IEEE Trans. Autom. Control 44(1), 208–214 (1999)Google Scholar
  4. 4.
    Folea, S., De Keyser, R., Birs, I.R., Muresan, C.I., Ionescu, C.: Discrete-time implementation and experimental validation of a fractional order PD controller for vibration suppression in airplane wings. Acta Polytechnica Hungarica 14(1), 191–206 (2017)Google Scholar
  5. 5.
    Sun, J., Wang, C., Xin, R.: Design of fractional order proportional differentiation controller for second order position servo system. In: Proceedings of the IEEE Chinese Control and Decision Conference (CCDC), pp. 5939–5944 (2018)Google Scholar
  6. 6.
    Luo, Y., Li, H., Chen, Y.: Fractional order proportional and derivative controller synthesis for a class of fractional order systems: tuning rule and hardware-in-the-loop experiment. In: Proceedings of the 48th IEEE Conference on Decision and Control Held Jointly with 28th Chinese Control Conference, pp. 5460–5465 (2009)Google Scholar
  7. 7.
    Luo, Y., Chen, Y.Q., Wang, C.Y., Pi, Y.G.: Tuning fractional order proportional integral controllers for fractional order systems. J. Process Control 20(7), 823–831 (2010)CrossRefGoogle Scholar
  8. 8.
    Wang, C., Fu, W., Shi, Y.: Tuning fractional order proportional integral differentiation controller for fractional order system. In: Proceedings of the 32nd IEEE Chinese Control Conference (CCC), pp. 552–555 (2013)Google Scholar
  9. 9.
    Zamojski, M.: Implementation of fractional order PID controller based on recursive oustaloup’e filter. In: Proceedings of the IEEE International Interdisciplinary Ph.D. Workshop, pp. 414–417 (2018)Google Scholar
  10. 10.
    Chen, Y.Q., Vinagre, B.M., Podlubny, I.: Continued fraction expansion approach to discretizing fractional order derivatives-an expository review. Nonlinear Dyn. 38(1–4), 155–170 (2004)CrossRefGoogle Scholar
  11. 11.
    Song, B., Xu, L., Lu, X.: A comparative study on Tustin rule based discretization methods for fractional-order differentiator. In: Proceedings of the 4th IEEE International Conference on Information science and Technology, Shenzhen, China, pp. 515–518 (2014)Google Scholar
  12. 12.
    Middleton, R.H., Goodwin, G.C.: Digital Control and Estimation: A Unified Approach. Prentice Hall, Englewood Cliffs, NJ (1990)zbMATHGoogle Scholar
  13. 13.
    Swarnakar, J., Sarkar, P., Singh, L.J.: Realization of fractional-order operator in complex domains—a comparative study. In: Bera, R., Sarkar, S., Chakraborty, S. (eds.) Lecture Notes in Electrical Engineering, vol. 462, pp. 711–718. Springer, Singapore (2018)CrossRefGoogle Scholar
  14. 14.
    Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publishers Inc., Redding (2006)Google Scholar
  15. 15.
    Swarnakar, J., Sarkar, P., Singh, L.J.: A unified direct approach for discretizing fractional-order differentiator in delta-domain. Int. J. Modeling, Simul. Sci. Comput. 9(6), 185055-1–185055-20 (2018)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  • Jaydeep Swarnakar
    • 1
    Email author
  • Prasanta Sarkar
    • 2
  • Lairenlakpam Joyprakash Singh
    • 1
  1. 1.Department of Electronics & Communication EngineeringSchool of Technology, North-Eastern Hill UniversityShillongIndia
  2. 2.Department of Electrical EngineeringNational Institute of Technical Teachers’ Training and ResearchKolkataIndia

Personalised recommendations