Journal of Applied Science and Engineering

Published by Tamkang University Press


Impact Factor



Thi-Van-Anh NguyenThis email address is being protected from spambots. You need JavaScript enabled to view it., Ngoc-Hiep Tran

School of Electrical and Electronic Engineering, Hanoi University of Science and Technology


Received: June 8, 2023
Accepted: July 31, 2023
Publication Date: October 3, 2023

 Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.

Download Citation: ||  

This paper addresses the challenging control problem of stabilizing an inverted pendulum on a cart. The inherent nonlinearity, instability, and underactuation of the system pose significant difficulties in achieving simultaneous pendulum stabilization and cart movement. To overcome these challenges, we propose an integrated approach that combines Linear Quadratic Regulator (LQR) and fuzzy logic control methods. This integrated control strategy effectively stabilizes the pendulum and controls the cart’s position. Notably, the integrated control outperforms the LQR control in terms of convergence speed. Furthermore, we explore the use of observers for state estimation, specifically the high-order integral-chain differentiator and the extended state observer, to accurately estimate pendulum angular velocity. Simulation results, along with detailed discussions, are presented to validate the accuracy and effectiveness of the proposed control methods and observers.

Keywords: Inverted pendulum; Fuzzy logic control; Linear quadratic regulator; Extended state observer; High-order integral-chain differentiator

  1. [1] S. Zeghlache, M. Z. Ghellab, A. Djerioui, B. Bouderah, and M. F. Benkhoris, (2023) “Adaptive fuzzy fast terminal sliding mode control for inverted pendulum-cart system with actuator faults" Mathematics and Computers in Simulation 210: 207–234. DOI:
  2. [2] V.-A. Nguyen, D.-B. Pham, D.-T. Pham, N.-T. Bui, and Q.-T. Dao. “A Hybrid Energy Sliding Mode Controller for the Rotary Inverted Pendulum”. In: Advances in Engineering Research and Application: Proceedings of the International Conference on Engineering Research and Applications, ICERA 2022. Springer. 2022, 34–41. DOI:  
  3. [3] T. Zielinska, G. R. Rivera Coba, and W. Ge, (2021) “Variable Inverted Pendulum Applied to Humanoid Motion Design" Robotica 39(8): 1368–1389. DOI: 10.1017/S0263574720001228.
  4. [4] S.-J. Huang, S.-S. Chen, and S.-C. Lin, (2019) “Design and Motion Control of a Two-Wheel Inverted Pendulum Robot" International Journal of Mechanical and Mechatronics Engineering 13(3): 194–201. DOI:
  5. [5] L. B. Prasad, B. Tyagi, and H. O. Gupta, (2014) “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 11: 661–670. DOI:
  6. [6] L. Messikh, E.-H. Guechi, and S. Blazic, (2021) “Stabilization of the cart–inverted-pendulum system using state-feedback pole-independent MPC controllers" Sensors 22(1): 243. DOI:
  7. [7] M. S. Mahmoud, R. A. Saleh, and A. Ma’arif, (2022) “Stabilizing of inverted pendulum system using Robust sliding mode control" International Journal of Robotics and Control Systems 2(2): 230–239. DOI:
  8. [8] A. Jose, C. Augustine, S. M. Malola, K. Chacko, et al., (2015) “Performance study of PID controller and LQR technique for inverted pendulum" World Journal of Engineering and Technology 3(02): 76. DOI: 10.4236/wjet.2015.32008.
  9. [9] , (2017) “Fuzzy-logic control of an inverted pendulum on a cart" Computers Electrical Engineering 61: 31–47. DOI:
  10. [10] D.-B. Pham, D.-T. Pham, Q.-T. Dao, and V.-A. Nguyen. “Takagi-Sugeno fuzzy control for stabilizing nonlinear inverted pendulum”. In: Intelligent Systems and Networks: Selected Articles from ICISN 2022, Vietnam. Springer, 2022, 333–341. DOI:
  11. [11] B. Sharma and B. Tyagi. “Lqr-based ts-fuzzy logic controller design for inverted pendulum-coupled cart system”. In: Systems Thinking Approach for Social Problems: Proceedings of 37th National Systems Conference, December 2013. Springer. 2015, 207–219. DOI:
  12. [12] N. S. Bhangal, (2013) “Design and performance of LQR and LQR based fuzzy controller for double inverted pendulum system" Journal of Image and Graphics 1(3): 143–146. DOI: 10.12720/joig.1.3.143-146.
  13. [13] S. Zhang, R. Liu, X. Qian, et al., (2020) “Control of a Flexible Manipulator System with Finite-Time Disturbance Observer" Journal of Applied Science and Engineering 23(2): 271–278. DOI:
  14. [14] N. J. Mathew, K. K. Rao, and N. Sivakumaran, (2013) “Swing up and stabilization control of a rotary inverted pendulum" IFAC Proceedings Volumes 46(32): 654– 659. DOI:
  15. [15] H. O. Wang and K. Tanaka. Fuzzy control systems design and analysis: a linear matrix inequality approach. John Wiley & Sons, 2004.
  16. [16] J. Liu, X. Wang, J. Liu, and X. Wang. Advanced sliding mode control. Springer, 2011.
  17. [17] X. Wang, Z. Chen, and G. Yang, (2007) “Finite-timeconvergent differentiator based on singular perturbation technique" IEEE Transactions on Automatic Control 52(9): 1731–1737. DOI: 10.1109/TAC.2007.904290.
  18. [18] B. Wenyan, C. Sen, C. HUANG, L. Kunfeng, and H. ZHONG. “A new design of extended state observer for a class of uncertain nonlinear systems with sampled-data measurement”. In: 2020 Chinese Automation Congress (CAC). IEEE. 2020, 7538–7543. DOI: 10.1109/CAC51589.2020.9326733



69th percentile
Powered by  Scopus

SCImago Journal & Country Rank

Enter your name and email below to receive latest published articles in Journal of Applied Science and Engineering.