The Application of an Adaptive Controller Combined with the LQR Controller for the Inverted Pendulum

eng Article in English DOI: 10.14313/PAR_234/47

send Jakub Żegleń-Włodarczyk AGH Akademia Górniczo-Hutnicza, Wydział Elektrotechniki, Automatyki, Informatyki i Inżynierii Biomedycznej

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Abstract

The inverted pendulum is an unstable system with nonlinear dynamics. The task of controlling the inverted pendulum is complex. Therefore, the inverted pendulum over the years has become one of the most important systems on which every controller is tested. Here the objective is to control the system in such a way that the inverted pendulum stabilizes in the upright position. This analysis proposes a non-linear Lyapunov-based controller. The controller at hand, only provides the pendulum’s stabilization, therefore an additional module is needed – in this case the LQR controller. Both modules are combined with each other by using a two-loop parallel design. The newly designed controller has been experimentally tested and compared to the single LQR controller.

Keywords

adaptive controller, inverted pendulum, LQR control

Zastosowanie regulatora adaptacyjnego w połączeniu z regulatorem LQR dla wahadła odwróconego

Streszczenie

Odwrócone wahadło jest niestabilnym systemem o nieliniowej dynamice. Zadanie sterowania wahadłem odwróconym jest trudne, dlatego też układ ten przez lata stał się jednym z najważniejszych systemów, na których testowane są wszelkiego rodzaju regulatory. Celem sterowania systemem jest ustabilizowanie wahadła odwróconego w pozycji pionowo skierowanej ku górze. W artykule zaproponowano nowy algorytm adaptacyjny dla wahadła, będący kombinacją regulatora LQR oraz regulatora nieliniowego bazującego na twierdzeniu Lapunova. Oba moduły są połączone za pomocą dwupętlowej konstrukcji równoległej. Nowo zaprojektowany regulator został przetestowany eksperymentalnie i porównany z niezależnym modułem LQR.

Słowa kluczowe

LQR control, regulator adaptacyjny, wahadło odwrócone

Bibliography

  1. Aguilar-Ibánez C., Gutiérrez Frias O., Suárez Castanón M., Lyapunov-Based Controller for the Inverted Pendulum Cart System, “Nonlinear Dynamics”, Vol. 40, No. 4, 2005, 367–374, DOI: 10.1007/s11071-005-7290-y.
  2. Aguilar-Ibánez C., Suárez-Castanón M., Stabilization of the Inverted Pendulum via a Constructive Lyapunov Function, “Acta Applicandae Mathematicae”, Vol. 111, No. 1, 2010, 15–26, DOI: 10.1007/s10440-009-9527-0.
  3. Benaskeur A., Desbiens A., Application of adaptive backstepping to the stabilization of the inverted pendulum, Conference Proceedings of IEEE Canadian Conference on Electrical and Computer Engineering, 1998, DOI: 10.1109/CCECE.1998.682564.
  4. Bertrand D.J.S.R., Collins D.J., Neural network controllers for the X29 aircraft, Proceedings of IJCNN International Joint Conference on Neural Networks, Baltimore, MD, USA, 1992, DOI: 10.1109/IJCNN.1992.287191.
  5. Charney D.M., Josin G.M., Neural network servo control of a robot manipulator joint in real-time, Proceedings of IEEE International Joint Conference on Neural Networks, Singapore, 1991, DOI: 10.1109/IJCNN.1991.170673.
  6. Elsayed B.A., Hassan M.A., Mekhilef S., Fuzzy swinging-up with sliding mode control for third order cart-inverted pendulum system, “International Journal of Control, Automation and Systems”, Vol. 13, No. 1, 2015, 238–248, DOI: 10.1007/s12555-014-0033-4.
  7. Gupta H.O., Prasad L.B., Tyagi B., Optimal Control of Nonlinear Inverted Pendulum System Using PID Controller and LQR: Performance Analysis Without and With Disturbance input, “International Journal of Automation and Computing”, Vol. 11, Issue 6, 2014, 661–670, DOI: 10.1007/s11633-014-0818-1.
  8. Hassan H.A., El-Metwally M., Bendary A.F., Dead-zone robust adaptive controller for FACTS using Quadratic and Non-Quadratic Lyapunov Functions, International Conference on Power Electronics and Drive Systems (PEDS), Taipei, 2009, DOI: 10.1109/PEDS.2009.5385747.
  9. Inline measurement terminal for position encoder – IB IL INC-IN-PAC – 2861755, https://www.phoenixcontact.com/online/portal/us?uri=pxc-oc-itemdetail:pid=2861755&library=usen&tab=1
  10. Inline, function terminal for pulse width modulation and frequency modulation – IB IL PWM/2-PAC – 2861632, https://www.phoenixcontact.com/online/portal/pl?uri=pxc-oc-itemdetail:pid=2861632\&library=plpl\&tab=1
  11. Jia-Jun W., Position and speed tracking control of inverted pendulum based on double PID controllers, 34th Chinese Control Conference (CCC), Hangzhou, 2015, DOI: 10.1109/ChiCC.2015.7260286.
  12. Jia-Jun W., Stabilization and tracking control of X-Z inverted pendulum based on PID controllers, 34th Chinese Control Conference (CCC), Hangzhou, 2015, DOI: 10.1109/ChiCC.2015.7260287.
  13. Lal Bahadur P., Barjeev T., Hari Om G., Optimal Control of Nonlinear Inverted Pendulum System Using PID Controller and LQR: Performance Analysis Without and With Disturbance Input, “International Journal of Automation and Computing”, Vol. 11, No. 6, 2014, 661–670, DOI: 10.1007/s11633-014-0818-1.
  14. Luhao W., Zhanshi S., LQR-Fuzzy Control for Double Inverted Pendulum, International Conference on Digital Manufacturing & Automation, Changsha, 2010, DOI: 10.1109/ICDMA.2010.170.
  15. Maruki Y., Kawano K., Suemitsu H., Matsuo T., Adaptive backstepping control of wheeled inverted pendulum with velocity estimator, “International Journal of Control, Automation and Systems”, Vol. 12, No. 5, 2014, 1040–1048, DOI: 10.1007/s12555-013-0402-4.
  16. Moysis L., Balancing a double inverted pendulum using optimal control an Laguerre functions, Technical Reviews, 2016, http://ikee.lib.auth.gr/record/282764.
  17. Paliwal S., Pathak V.K., Analysis & Control of Inverted Pendulum System Using PID Controller, “Journal of Engineering Research and Application”, Vol. 7, No. 5, DOI: 10.9790/9622-0705040104.
  18. PLC controller – ILC 350 PN – 2876928, https://www.phoenixcontact.com/online/portal/pl?uri=pxc-oc-itemdetail:pid=2876928\&library=plpl\&tab=1
  19. Rusu C., Birou I., Szoke E., Model based design controller for the stepper motor, IEEE International Conference on Automation, Quality and Testing, Robotics, Cluj-Napoca, 2008, DOI: 10.1109/AQTR.2008.4588816.
  20. Safaei A., Mahyuddin M.N., Lyapunov-based nonlinear controller for quadrotor position and attitude tracking with GA optimization, IEEE Industrial Electronics and Applications Conference (IEACon), Kota Kinabalu, 2016, DOI: 10.1109/IEACON.2016.8067402.
  21. User manual – UM EN ILC 330/350 – 2699370, https://www.phoenixcontact.com/online/portal/us?uri=pxc-oc-itemdetail:pid=2699370&library=usen&tab=1.
  22. Using the Lagrangian to obtain Equations of Motion, http://et.engr.iupui.edu//~skoskie/ECE680/ECE680_l3notes.pdf.
  23. Wang H., Dong H., He L., Shi Y., Zhang Y., Design and Simulation of LQR Controller with the Linear Inverted Pendulum, International Conference on Electrical and Control Engineering, Wuhan, 2010, DOI: 10.1109/iCECE.2010.178.
  24. Wu J., Wang Z., Research on Fuzzy Control of Inverted Pendulum, First International Conference on Instrumentation, Measurement, Computer, Communication and Control, Beijing, 2011, DOI: 10.1109/IMCCC.2011.219.
  25. Yu L. H., Jian F., An Inverted Pendulum Fuzzy Controller Design and Simulation, International Symposium on Computer, Consumer and Control, Taichung, 2014, DOI: 10.1109/IS3C.2014.151.
  26. Yu-Sheng L., Hua-Hsu C., Shu-Fen L., An improved backstepping design for the control of an underactuated inverted pendulum, “Journal of Mechanical Science and Technology”, Vol. 27, No. 3, 2013, 865-873, DOI: 10.1007/s12206-013-0203-y.