Arabian Journal for Science and Engineering

, Volume 44, Issue 8, pp 6783–6793 | Cite as

Nonlinear SPKF-Based Time-Varying LQG for Inverted Pendulum System

  • R. ShalabyEmail author
  • M. El-Hossainy
  • B. Abo-Zalam
Research Article - Electrical Engineering


This paper deals with the holing issue of nonlinear inverted pendulum (IP) system. Close to the equilibrium point, IP is considered as a linear system without disruption and linear control is sufficient. However, when the IP swings over a wide range, its nonlinear dynamics becomes significant and the stabilization of IP becomes challenging task due to the inconsistency between its nonlinear dynamics and the controllers designed based on linearized models. Hence, the need for sophisticated control becomes highly demanding. This paper proposes an optimal time-varying linear quadratic Gaussian controller (TV-LQG) that is able to overcome this inconsistency problem. The proposed TV-LQG utilizes a sigma-point Kalman filter (SPKF) and a linear quadratic regulator with a prescribed degree of stability. SPKF is a highly accurate nonlinear state estimator since it does not use any linearization for calculating the state prediction covariance and Kalman gains. This leads, instantaneously, to a more exact nonlinear state estimation. The estimated states are fed to the Jacobian method, which updates, at once, the system dynamics accordingly. Thus, the parameters of the proposed TV-LQG are optimized based on the updated system dynamics. The proposed controller is intensively tested using simulation experiments and compared to the traditional LQG, LQR self-adjusting control and LQR-fuzzy control. The results proved the robustness and competitiveness of the proposed scheme to enhance the performance of the nonlinear IP system.


TV-LQG Nonlinear IP Optimal control SPKF Nonlinear state estimation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Shah, I.; Rehman, F.: Smooth higher-order sliding mode control of a class of underactuated mechanical systems. Arab. J. Sci. Eng. 42(12), 5147–5164 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Glück, T.; Eder, A.; Kugi, A.: Swing-up control of a triple pendulum on a cart with experimental validation. Automatica 49(3), 801–808 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Huang, J.; Ding, F.; Fukuda, T.; Matsuno, T.: Modeling and velocity control for a novel narrow vehicle based on mobile wheeled inverted pendulum. IEEE Trans. Control Syst. Technol. 21(5), 1607–1617 (2013)CrossRefGoogle Scholar
  4. 4.
    Muskinja, N.; Tovornik, B.: Swinging up and stabilization of a real inverted pendulum. IEEE Trans. Ind. Electron. 53(2), 631–639 (2006)CrossRefGoogle Scholar
  5. 5.
    Singh, A.P.; Agarwal, H.; Srivastava, P.: Fractional order controller design for inverted pendulum on a cart system (POAC). WSEAS Trans. Syst. Control 10, 172–178 (2015)Google Scholar
  6. 6.
    Mircea, D.; Adrian, G.; Tudor-Mircea, D.: Fractional order controllers versus integer order controllers. Proc. Eng. 181, 538–545 (2017)CrossRefGoogle Scholar
  7. 7.
    Wanli, Z.; Guoxin, L.; Lirong, W.: Research on the control method of inverted pendulum based on Kalman filter. In: Dependable, Autonomic and Secure Computing (DASC), IEEE 12th International Conference, pp. 520–523 (2014)Google Scholar
  8. 8.
    Kumar, E.V.; Jerome, J.: Robust LQR controller design for stabilizing and trajectory tracking of inverted pendulum. Proc. Eng. 64, 169–178 (2013)CrossRefGoogle Scholar
  9. 9.
    Yadav, S.K.; Sharma, S.; Singh, N.: Optimal control of double inverted pendulum using LQR controller. Int. J. Adv. Res. Comput. Sci. Softw. Eng. 2(2), 189–192 (2012)Google Scholar
  10. 10.
    Shehu, M.; Ahmad, M.R.; Shehu, A.; Alhassan, A.: LQR, double-PID and pole placement stabilization and tracking control of single link inverted pendulum. In: Control System, Computing and Engineering (ICCSCE), IEEE International Conference, pp 218–223 (2015)Google Scholar
  11. 11.
    Li, W.; Ding, H.; Cheng, K.: An investigation on the design and performance assessment of double-PID and LQR controllers for the inverted pendulum. In: UKACC International Conference on Control, pp. 190–196 (2012)Google Scholar
  12. 12.
    Prasad, L.B.; Tyagi, B.; Gupta, H.O.: Optimal control of nonlinear inverted pendulum system using PID controller and LQR: performance analysis without and with disturbance input. Int. J. Autom. Comput. 11(6), 661–670 (2014)CrossRefGoogle Scholar
  13. 13.
    Ahmad, N.B.: Linear Quadratic Gaussian (LQG) for Inverted Pendulum. M.S. thesis, Elect. Eng., Univ. Tun Hussein Onn, Johor, Malaysia (2013)Google Scholar
  14. 14.
    Da Fonseca Neto, J.V.; Abreu, I.S.; Da Silva, F.N.: Neural-genetic synthesis for state-space controllers based on linear quadratic regulator design for eigen structure assignment. IEEE Trans. Syst. Man Cybern. B (Cybern.) 40(2), 266–285 (2010)CrossRefGoogle Scholar
  15. 15.
    Mobayen, S.; Rabiei, A.; Moradi, M.; Mohammady, B.: Linear quadratic optimal control system design using particle swarm optimization algorithm. Int. J. Phys. Sci. 6(30), 6958–6966 (2011)CrossRefGoogle Scholar
  16. 16.
    Karthick, S.; Jerome, J.; Kumar, E. V.; Raaja, G.: APSO based weighting matrices selection of LQR applied to tracking control of SIMO system. In: Proceedings of 3rd International Conference on Advanced Computing, Networking and Informatics. Springer, New Delhi, pp 11–20 (2016)Google Scholar
  17. 17.
    Ghoreishi, S.A.; Nekoui, M.A.; Basiri, S.O.: Optimal design of LQR weighting matrices based on intelligent optimization methods. Int. J. Intell. Inf. Process. 2(1), 63–74 (2011)Google Scholar
  18. 18.
    Trimpe, S.; Millane, A.; Doessegger, S.; D’Andrea, R.: A self-tuning LQR approach demonstrated on an inverted pendulum. IFAC Proc. Vol. 47(3), 11281–11287 (2014)CrossRefGoogle Scholar
  19. 19.
    Chakraborty, K.; Mukherjee, R.; Mukherjee, S.: Tuning of PID controller of inverted pendulum using genetic algorithm. Int. J. Soft Comput. Eng. 3(1), 21–24 (2013)Google Scholar
  20. 20.
    Goel, A.; Kumar, R.; Narayan, S.: Design of MRAC augmented with PID controller using genetic algorithm. In: Power Electronics, Intelligent Control and Energy Systems (ICPEICES), IEEE International Conference, pp. 1–5 (2016)Google Scholar
  21. 21.
    Tabari, M.Y.; Kamyad, D.A.V.: Design optimal fractional PID controller for inverted pendulum with genetic algorithm. Int. J. Sci. Eng. Res. 4(2), 1–4 (2013)Google Scholar
  22. 22.
    Jacknoon, A.; Abido, M.A.: Ant colony based LQR and PID tuned parameters for controlling Inverted Pendulum. In: Communication, Control, Computing and Electronics Engineering (ICCCCEE), IEEE International Conference, pp. 1–8 (2017)Google Scholar
  23. 23.
    Li, B.; Sinha, U.; Sankaranarayanan, G.: Modelling and control of nonlinear tissue compression and heating using an LQG controller for automation in robotic surgery. Trans. Inst. Meas. Control 38(12), 1491–1499 (2016)CrossRefGoogle Scholar
  24. 24.
    Oróstica, R.; Duarte-Mermoud, M. A.; Jáuregui, C.: Stabilization of inverted pendulum using LQR, PID and fractional order PID controllers: a simulated study. In: Automatica (ICA-ACCA), IEEE International Conference, pp. 1–7 (2016)Google Scholar
  25. 25.
    Lim, W.H.; Isa, N.A.M.: Teaching and peer-learning particle swarm optimization. Appl. Soft Comput. 18, 39–58 (2014)CrossRefGoogle Scholar
  26. 26.
    Owczarkowski, A.; Horla, D.: Robust LQR and LQI control with actuator failure of a 2DOF unmanned bicycle robot stabilized by an inertial wheel. Int. J. Appl. Math. Comput. Sci. 26(2), 325–334 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kandepu, R.; Foss, B.; Imsland, L.: Applying the unscented Kalman filter for nonlinear state estimation. J. Process Control 18(7–8), 753–768 (2008)CrossRefGoogle Scholar
  28. 28.
    Alkaya, A.: Unscented Kalman filter performance for closed-loop nonlinear state estimation: a simulation case study. Electr. Eng. 96(4), 299–308 (2014)CrossRefGoogle Scholar
  29. 29.
    Zhang, J.L.; Zhang, W.: LQR self-adjusting based control for the planar double inverted pendulum. Phys. Proc. 24, 1669–1676 (2012)CrossRefGoogle Scholar
  30. 30.
    Luhao, W.; Zhanshi, S.: LQR-Fuzzy control for double inverted pendulum. In: International Conference on, Digital Manufacturing and Automation (ICDMA) vol. 1, pp. 900–903 (2010)Google Scholar
  31. 31.
    Mandal, O.; Mondal, K.: Design of state feedback controller for inverted pendulum and fine tune by GA. Int. J. Electron. Electr. Comput. Syst. 5(11), 46–54 (2016)Google Scholar
  32. 32.
    Dallagi, H.; Nejim, S.: Optimal control of double star synchronous machine for the ship electric propulsion system. In: 3rd International Conference on, Automation, Control, Engineering and Computer Science, vol. 4, pp. 119–126 (2016)Google Scholar
  33. 33.
    Varghese, E.S.; Vincent, A.K.; Bagyaveereswaran, V.: Optimal control of inverted pendulum system using PID controller, LQR and MPC. IOP Conf. Ser. Mater. Sci. Eng. 263(5), 052007 (2017)CrossRefGoogle Scholar
  34. 34.
    Gajic, Z.: Linear Dynamic Systems and Signals. Prentice Hall/Pearson Education, Upper Saddle River (2003)Google Scholar
  35. 35.
    Julier, S. J.; Uhlmann, J. K.: A general method for approximating nonlinear transformations of probability distributions. Technical Report, Robotics Research Group, Department of Engineering Science, University of Oxford, pp. 1-27 (1996)Google Scholar
  36. 36.
    Julier, S.J.; Uhlmann, J.K.: New extension of the Kalman filter to nonlinear systems. In: Signal Processing, Sensor Fusion, and Target Recognition VI. International Society for Optics and Photonics, vol. 3068, pp. 182–194 (1997)Google Scholar
  37. 37.
    Lewis, F.L.; Vrabie, D.L.; Syrmos, V.L.: Optimal Control. Wiley, New York (2012)CrossRefzbMATHGoogle Scholar
  38. 38.
    Naidu, D.S.: Optimal Control Systems. CRC Press, Boca Raton (2002)Google Scholar
  39. 39.
    Mustafa, A.; Munawar, K.; Malik, F.M.; Malik, M.B.; Salman, M.; Amin, S.: Reduced order observer design with DMPC and LQR for system with backlash nonlinearity. Arab. J. Sci. Eng. 39(8), 6521–6530 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  1. 1.Department of Industrial Electronics and Control Engineering, Faculty of Electronic EngineeringMenoufia UniversityMenoufEgypt

Personalised recommendations