Skip to main content
SpringerLink
Log in

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

  • Research Article - Electrical Engineering
  • Published:
Arabian Journal for Science and Engineering Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  4. Muskinja, N.; Tovornik, B.: Swinging up and stabilization of a real inverted pendulum. IEEE Trans. Ind. Electron. 53(2), 631–639 (2006)

    Article  Google Scholar 

  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. Mircea, D.; Adrian, G.; Tudor-Mircea, D.: Fractional order controllers versus integer order controllers. Proc. Eng. 181, 538–545 (2017)

    Article  Google Scholar 

  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)

  8. Kumar, E.V.; Jerome, J.: Robust LQR controller design for stabilizing and trajectory tracking of inverted pendulum. Proc. Eng. 64, 169–178 (2013)

    Article  Google Scholar 

  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. 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)

  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)

  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)

    Article  Google Scholar 

  13. Ahmad, N.B.: Linear Quadratic Gaussian (LQG) for Inverted Pendulum. M.S. thesis, Elect. Eng., Univ. Tun Hussein Onn, Johor, Malaysia (2013)

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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. 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)

    Article  Google Scholar 

  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. 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)

  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. 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)

  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)

    Article  Google Scholar 

  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)

  25. Lim, W.H.; Isa, N.A.M.: Teaching and peer-learning particle swarm optimization. Appl. Soft Comput. 18, 39–58 (2014)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  28. Alkaya, A.: Unscented Kalman filter performance for closed-loop nonlinear state estimation: a simulation case study. Electr. Eng. 96(4), 299–308 (2014)

    Article  Google Scholar 

  29. Zhang, J.L.; Zhang, W.: LQR self-adjusting based control for the planar double inverted pendulum. Phys. Proc. 24, 1669–1676 (2012)

    Article  Google Scholar 

  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)

  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. 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)

  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)

    Article  Google Scholar 

  34. Gajic, Z.: Linear Dynamic Systems and Signals. Prentice Hall/Pearson Education, Upper Saddle River (2003)

    Google Scholar 

  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)

  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)

  37. Lewis, F.L.; Vrabie, D.L.; Syrmos, V.L.: Optimal Control. Wiley, New York (2012)

    Book  MATH  Google Scholar 

  38. Naidu, D.S.: Optimal Control Systems. CRC Press, Boca Raton (2002)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Industrial Electronics and Control Engineering, Faculty of Electronic Engineering, Menoufia University, Menouf, 32952, Egypt

    R. Shalaby, M. El-Hossainy & B. Abo-Zalam

Authors
  1. R. Shalaby
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. El-Hossainy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Abo-Zalam
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Shalaby.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shalaby, R., El-Hossainy, M. & Abo-Zalam, B. Nonlinear SPKF-Based Time-Varying LQG for Inverted Pendulum System. Arab J Sci Eng 44, 6783–6793 (2019). https://doi.org/10.1007/s13369-018-3631-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13369-018-3631-2

Keywords

Search

Navigation