Skip to main content
SpringerLink
Log in

Adaptive State-space Control of Under-actuated Systems Using Error-magnitude Dependent Self-tuning of Cost Weighting-factors

  • Regular Papers
  • Control Theory and Applications
  • Published:
International Journal of Control, Automation and Systems Aims and scope Submit manuscript

Abstract

This article methodically constructs a novel adaptive self-tuning state-space controller that enhances the robustness of under-actuated systems against bounded exogenous disturbances. The generic Linear-Quadratic-Regulator (LQR) is employed as the baseline controller. The main contribution of this article is the formulation of a hierarchical online gain-adjustment mechanism that adaptively modulates the weighting-factors of LQR’s quadratic-performance-index by using pre-calibrated continuous hyperbolic scaling functions. The hyperbolic scaling functions are driven by the magnitude of system’s state-error variables. This augmentation dynamically updates the solution of the Matrix-Riccati-Equation which modifies the state-feedback gains after every sampling interval. The efficacy of the proposed adaptive controller is validated by conducting hardware-in-the-loop experiments on QNET Rotary Pendulum setup. The experimental outcomes show that the proposed adaptive control scheme yields stronger damping against oscillations and faster error-convergence rate, while maintaining the controller’s asymptotic-stability, under the influence of parametric uncertainties.

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. M. Zhang, Y. Zhang, and X. Cheng, “An enhanced coupling PD with sliding mode control method for underactuated double-pendulum overhead crane systems,” International Journal of Control, Automation and Systems, vol. 17, pp. 1579–1588, May 2019.

    Article  Google Scholar 

  2. H. Gritli and S. Belghit, “Robust feedback control of the underactuated inertia wheel inverted pendulum under parametric uncertainties and subject to external disturbances: LMI formulation,” Journal of the Franklin Institute, vol. 355, no. 18, pp. 9150–9191, December 2018.

    Article  MathSciNet  Google Scholar 

  3. H. Zhang, J. Wang, and G. Lu, “Self-organizing fuzzy optimal control for under-actuated systems,” Journal of Systems and Control Engineering, vol. 228, no. 8, pp. 578–590, May 2014.

    Google Scholar 

  4. O. S. Bhatti, K. Mehmood-ul-Hasan, and M. A. Imtiaz, “Attitude control and stabilization of a two-wheeled self balancing robot,” Control Engineering and Applied Informatics, vol. 17, no. 3 pp. 98–104, October 2015.

    Google Scholar 

  5. H. Freire, P. B. M. Oliveira, and E. J. S. Pires, “From single to many-objective PID controller design using particle swarm optimization,” International Journal of Control, Automation and Systems, vol. 15, pp. 918–932, March 2017.

    Article  Google Scholar 

  6. O. S. Bhatti, O. B. Tariq, A. Manzar, and O. A. Khan, “Adaptive intelligent cascade control of a ball-riding robot for optimal balancing and station-keeping,” Advanced Robotics, vol. 32, no. 2, pp. 63–76, 2018.

    Article  Google Scholar 

  7. M. Taghizadeh and M. J. Yarmohammadi, “Development of a self-tuning PID controller on hydraulically actuated Stewart platform stabilizer with base excitation,” International Journal of Control, Automation and Systems, vol. 16, no. 11, pp. 2990–2999, November 2018.

    Article  Google Scholar 

  8. J. J. Wang, “Stabilization and tracking control of X-Z inverted pendulum with sliding-mode control,” ISA Trans., vol. 51, no. 6, pp. 763–770, November 2012.

    Article  Google Scholar 

  9. X. Liu, Y. Wu, Y. Zhang, and S. Xiao, “A control method to make LQR robust: A planes cluster approaching mode,” International Journal of Control, Automation and Systems, vol. 12, pp. 302–308, April 2014.

    Article  Google Scholar 

  10. M. K. Ghartemani, S. A. Khajehoddin, P. Jain, and A. Bakhshai, “Linear quadratic output tracking and disturbance rejection,” International Journal of Control, vol. 84, no. 8, pp. 1442–1449, August 2011.

    Article  MathSciNet  Google Scholar 

  11. H. Chen and N. Sun, “Nonlinear control of underactuated systems subject to both actuated and unactuated state constraints with experimental verification,” IEEE Trans. on Industrial Electronics, vol. 67, no. 9, pp. 7702–7714, September 2020.

    Article  MathSciNet  Google Scholar 

  12. J. Shi, F. Lv, and B. Liu, “Self-tuning speed control of ultrasonic motor combined with efficiency optimization,” International Journal of Control, Automation and Systems, vol. 12, pp. 93–101, February 2014.

    Article  Google Scholar 

  13. J. Yang, H. Su, Z. Li, D. Ao, and R. Song, “Adaptive control with a fuzzy tuner for cable-based rehabilitation robot,” International Journal of Control, Automation and Systems, vol. 14, pp. 865–875, June 2016.

    Article  Google Scholar 

  14. M. Szuster and Z. Hendzel, Intelligent Optimal Adaptive Control for Mechatronic Systems, Springer, Cham, December 2017.

    MATH  Google Scholar 

  15. B. Gruenwald and T. Yucelen, “On transient performance improvement of adaptive control architectures,” International Journal of Control, vol. 88, no. 11, pp. 2305–2315, May 2015.

    Article  MathSciNet  Google Scholar 

  16. M. Farahani and S. Ganjefar, “Intelligent control of static synchronous series compensator via an adaptive self-tuning PID controller for suppression of torsional oscillations,” International Journal of Control, Automation and Systems, vol. 10, pp. 744–752, August 2012.

    Article  Google Scholar 

  17. W. Sun, J. W. Lin, S. F. Su, N. Wang, and M. J. Er, “Reduced adaptive fuzzy decoupling control for lower limb exoskeleton,” IEEE Trans. on Cybernetics, pp. 1–11, February 2020. DOI: https://doi.org/10.1109/TCYB.2020.2972582

  18. W. Sun, S. F. Su, J. Xia, and Y. Wu, “Adaptive tracking control of wheeled inverted pendulums with periodic disturbances,” IEEE Trans. on Cybernetics, vol. 50, no. 5, pp. 1867–1876, May 2020.

    Article  Google Scholar 

  19. H. Jafarnejadsani and J. Pieper, “Gain-scheduled ℓ1-optimal control of variable-speed-variable-pitch wind turbines,” IEEE Trans. on Control Systems Technology, vol. 23, no. 1, pp. 372–379, January 2015.

    Article  Google Scholar 

  20. Y. Bolea, V. Puig, and J. Blesa, “Gain-scheduled Smith predictor PID-based LPV controller for open-flow canal control,” IEEE Trans. on Control Systems Technology, vol. 22, no. 2, pp. 468–477, March 2014.

    Article  Google Scholar 

  21. X. Wang, X. Chen, and L. Wen, “The LQR baseline with adaptive augmentation rejection of unmatched input disturbance,” International Journal of Control, Automation and Systems, vol. 15, pp. 1302–1313, May 2017.

    Article  Google Scholar 

  22. O. Saleem, M. Rizwan, K. Mahmood-ul-Hasan, and M. Ahmad, “Performance enhancement of multivariable model-reference optimal adaptive motor speed controller using error-dependent hyperbolic gain functions,” Automatika, vol. 61, no. 1, pp. 117–131, January 2020.

    Article  Google Scholar 

  23. M. Zhou and Y. Fu, “Stability and stabilization for discrete-time Markovian jump stochastic systems with piecewise homogeneous transition probabilities,” International Journal of Control, Automation and Systems, vol. 17, pp. 2165–2173, May 2019.

    Article  Google Scholar 

  24. W. Qi and X. Gao, “L1 control for positive Markovian jump systems with partly known transition rates,” International Journal of Control, Automation and Systems, vol. 15, pp. 274–280, January 2017.

    Article  Google Scholar 

  25. M. Zhuang, Y. Fang, H. Zheng, and L. Liu, “Active disturbance rejection control with self-adjusting parameters for vibration displacement system of continuous casting mold,” IEEE ACCESS, vol. 7, pp. 52498–52507, April 2019.

    Article  Google Scholar 

  26. O. Saleem and U. Omer, “EKF-based self-regulation of an adaptive nonlinear PI speed controller for a DC motor,” Turkish Journal of Electrical Engineering and Computer Sciences, vol. 25, no. 5, pp. 4131–4141, May 2017.

    Article  Google Scholar 

  27. O. Saleem and K. Mahmood-ul-Hasan, “Adaptive collaborative speed control of PMDC motor using hyperbolic secant functions and particle swarm optimization,” Turkish Journal of Electrical Engineering and Computer Sciences, vol. 26, no. 3, pp. 1612–1622, May 2018.

    Google Scholar 

  28. W. W. Shang, S. Cong, Z. X. Li, and S. L. Jiang, “Augmented nonlinear PD controller for a redundantly actuated parallel manipulator,” Advanced Robotics, vol. 12, no. 12–13, pp. 1725–1742, December 2009.

    Article  Google Scholar 

  29. O. Saleem and K. Mahmood-ul-Hasan, “Robust stabilisation of rotary inverted pendulum using intelligently optimised nonlinear self-adaptive dual fractional order PD controllers,” International Journal of Systems Science, vol. 50, no. 7, pp. 1399–1414, June 2019.

    Article  MathSciNet  Google Scholar 

  30. O. Boubaker and R. Iriarte, The Inverted Pendulum in Control Theory and Robotics: From Theory to New Innovations, Institution of Engineering and Technology, October 2017.

  31. S. Kurode, A. Chalanga, and B. Bandyopadhyay, “Swing-up and stabilization of rotary inverted pendulum using sliding modes,” IFAC Proceedings Volumes, vol. 44, no. 1, pp. 10685–10690, January 2011.

    Article  Google Scholar 

  32. F. L. Lewis, D. Vrabie, and V. L. Syrmos, Optimal Control, John Wiley and Sons, pp. 110–167, January 2012.

  33. I. Filip, C. Vasar, I. Szeidert, and O. Prostean, “Self-tuning strategy for a minimum variance control system of a highly disturbed process,” European Journal of Control, vol. 46, pp. 49–62, March 2019.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, National University of Computer and Emerging Sciences, 852 B-Block, Faisal Town, Lahore, Pakistan

    Omer Saleem

  2. Department of Electrical Engineering, University of Engineering and Technology, G.T. Road, Lahore, Pakistan

    Khalid Mahmood-ul-Hasan

Authors
  1. Omer Saleem
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Khalid Mahmood-ul-Hasan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Omer Saleem.

Additional information

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Recommended by Associate Editor Ning Sun under the direction of Editor Euntai Kim.

Omer Saleem received his Bachelor’s and Master’s degree in electrical engineering with specialization in control systems from University of Engineering and Technology (UET), Lahore, Pakistan. He is also serving as an Assistant Professor at the Department of Electrical Engineering, National University of Computer and Emerging Sciences (NUCES), Lahore, Pakistan. He has published several research papers in SCIE/SCI-indexed journals. His research interests include the design and formulation of adaptive and self-tuning control mechanisms for under-actuated electro-mechanical systems and power electronic converters.

Khalid Mahmood-ul-Hasan received his Ph.D. degree in electrical engineering with specialization in control systems from University of Bradford, UK. He is currently serving as a Professor as well as the Chairman at the Department of Electrical Engineering, University of Engineering and Technology, Lahore, Pakistan. His research interests include linear systems theory, digital control systems, and control of electrical machine drives.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saleem, O., Mahmood-ul-Hasan, K. Adaptive State-space Control of Under-actuated Systems Using Error-magnitude Dependent Self-tuning of Cost Weighting-factors. Int. J. Control Autom. Syst. 19, 931–941 (2021). https://doi.org/10.1007/s12555-020-0209-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12555-020-0209-z

Keywords

Search

Navigation