Stabilizing and Trajectory Tracking of Inverted Pendulum Based on Fractional Order PID Control

  • Akshaya Kumar Patra
  • Alok Kumar Mishra
  • Anuja Nanda
  • Dillip Kumar Subudhi
  • Ramachandra Agrawal
  • Abhishek Patra
Conference paper
Part of the Lecture Notes in Networks and Systems book series (LNNS, volume 109)


This manuscript presents a Simulink model of inverted pendulum (IP) and design of a fractional order proportional-integral-derivative controller (FOPIDC) to control of cart position (CP) and angular position (AP) of the pendulum under uncertainties and disturbances. In this control strategy, the conventional PID controller (CPIDC) is re-formulated with fractional orders of the integrator and differentiator to improve the control performance. The FOPIDC is a novel approach whose gains dynamically vary with respect to the error signal. The validation of the improved control performance of FOPIDC is established by comparative result investigation with other published control algorithms. The comparative results clearly reveal the better response of the proposed approach to control the system dynamics within the stable range with respect to accuracy, robustness, and ability to handle uncertainties.


Inverted pendulum Angular displacement Angular velocity FOPIDC 


  1. 1.
    Iqbal J, Ajwad SA (2016) Automating industrial tasks through mechatronic systems – a review of robotics in industrial perspective. Tehnic ki vjesnik – Technical Gazette 23:917–924Google Scholar
  2. 2.
    Ajwad SA, Iqbal J (2017) Role and review of educational robotic platforms in preparing engineers for industry. Maejo Int J Sci Technol 11:17–34Google Scholar
  3. 3.
    Bettayeb M, Al-Saggaf U (2014) Stabilization of an inverted pendulum-cart system by fractional PI-state feedback. ISA Trans 53:508–516CrossRefGoogle Scholar
  4. 4.
    Iqbal J, Khan SG, Ukovic SC (2017) Nonlinear control systems -a brief overview of historical and recent advances. Nonlinear Eng 6:301–312CrossRefGoogle Scholar
  5. 5.
    Ghosh A, Subudhi B (2012) Brief paper – robust proportional-integral-derivative compensation of an inverted Cart-Pendulum System: an experimental study. IET Control Theory Appl 6(8):1145–1152MathSciNetCrossRefGoogle Scholar
  6. 6.
    Wang C, Liu C, Fu W (2016) Design and simulation of IP system based on the fractional PID controller. In: IEEE 11th Conference on Industrial Electronics and Applications (ICIEA), pp 1760–1764Google Scholar
  7. 7.
    Magana ME, Holzapfel F (1998) Fuzzy-logic control of an inverted pendulum with vision feedback. IEEE Trans Educ 41(2):165–170CrossRefGoogle Scholar
  8. 8.
    Ozana S, Hajovsky R (2012) Design and implementation of LQR controller for inverted pendulum by use of REX control system. In: IEEE International Conference on Circuits and Systems. vol 1, pp 343–347Google Scholar
  9. 9.
    Kumar EV, Jerome J (2013) Robust LQR controller design for stabilizing and trajectory tracking of inverted pendulum. Procedia Eng 64:169–178CrossRefGoogle Scholar
  10. 10.
    Prasad LB, Gupta HO (2012) Modelling and simulation for optimal control of nonlinear inverted pendulum dynamical system using PID controller and LQR. In: Sixth Asia Modelling Symposium (AMS), pp 138–143Google Scholar
  11. 11.
    Pasemann F (1998) Evolving neurocontrollers for balancing an inverted pendulum. Netw Comput Neural Syst 9:1–4CrossRefGoogle Scholar
  12. 12.
    Deng L, Gao S (2011) The design for the controller of the linear IP based on backstepping. In: International conference on electronic and mechanical engineering and information technology (EMEIT). vol 6, pp 2892–2895Google Scholar
  13. 13.
    Jörgl M, Gattringer H (2013) Passivity based control of a cart with inverted pendulum. Appl Mech Mater 332:339–344CrossRefGoogle Scholar
  14. 14.
    Žilić T, Pavković D (2009) Modeling and control of a pneumatically actuated inverted pendulum. ISA Trans 48:327–335CrossRefGoogle Scholar
  15. 15.
    Lambrecht P, Vander G (1988) H-infinity control of an experimental inverted pendulum with dry friction. IEEE Control Syst Mag 13(4):44–50Google Scholar
  16. 16.
    Wai RJ, Chang LJ (2006) Adaptive stabilizing and tracking control for a nonlinear inverted-pendulum system via sliding-mode technique. IEEE Trans Ind Electron 53:674–692CrossRefGoogle Scholar
  17. 17.
    Tao CW, Taur J, Chang J (2010) Adaptive fuzzy switched swing-up and sliding control for the double-pendulum-and-cart system. IEEE Trans Syst Man Cybern B Cybern 40(1):241–252CrossRefGoogle Scholar
  18. 18.
    Chen CS, Chen WL (1998) Robust adaptive sliding-mode control using fuzzy modelling for an inverted-pendulum system. IEEE Trans Ind Electron 45(2):297–306CrossRefGoogle Scholar
  19. 19.
    Patra AK, Rout PK (2019) Backstepping linear quadratic gaussian controller design for balancing an inverted pendulum. IETE J Res 1:1–15CrossRefGoogle Scholar
  20. 20.
    Khalil HK (2002) Nonlinear systems, 3rd edn. Prentice Hall, Upper Saddle River, New JerseyzbMATHGoogle Scholar
  21. 21.
    Patra AK, Rout PK (2017) Adaptive sliding mode Gaussian controller for artificial pancreas in TIDM patient. J Process Control 58:23–27CrossRefGoogle Scholar
  22. 22.
    Patra AK, Rout PK (2018) Backstepping sliding mode Gaussian insulin injection control for blood glucose regulation in TIDM patient. J Dyn Sys Meas Control 140(9):091006CrossRefGoogle Scholar
  23. 23.
    Irfan S, Razzaq MT, Iqbal J (2018) Advanced sliding mode control techniques for IP: modelling and simulation. Eng Sci Tech Int J. Scholar
  24. 24.
    Ronquillo-Lomeli G, Ríos-Moreno GJ (2016) Nonlinear identification of inverted pendulum system using Volterra polynomials. Mech Based Des Struct Mach 44(1):5–15CrossRefGoogle Scholar
  25. 25.
    Kajita S, Fujiwara K (2003) Biped walking pattern generation by a simple three-dimensional inverted pendulum model. Adv Robot 17(2):131–147CrossRefGoogle Scholar
  26. 26.
    Bingul Z, Karahan O (2018) Comparison of PID and FOPID controllers tuned by PSO and ABC algorithms for unstable and integrating systems with time delay. Optimal Control Appl Meth 39(4):1431–1450MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  • Akshaya Kumar Patra
    • 1
  • Alok Kumar Mishra
    • 1
  • Anuja Nanda
    • 1
  • Dillip Kumar Subudhi
    • 2
  • Ramachandra Agrawal
    • 2
  • Abhishek Patra
    • 2
  1. 1.Department of EEEITER, S‘O’A University, Deemed to be UniversityBhubaneswarIndia
  2. 2.Department of CSITITER, S‘O’A University, Deemed to be UniversityBhubaneswarIndia

Personalised recommendations