Skip to main content

PID Principles to Obtain Adaptive Variable Gains for a Bi-order Sliding Mode Control

Abstract

A new model to obtain adaptive variable gains for a bi-order Sliding Mode Control is proposed in this work. The variable gains for the controller are designed to dynamically adapt their values, using principles of the well-known Proportional-Integral-Derivative control technique, where the magnitude of the tracking error is the signal feedback. According the way to tune parameters, it can become a first-order or a second-order controller. This design takes into account the actuators constraints (operational limits of the plant to control). As a result of the adaptive properties of the proposed scheme, the new controller significantly reduces the energy consumption in control processes, and it rejects the so called chattering-effect, simultaneously maintaining the main robust properties of the Sliding Mode strategy. In order to show the feasibility and effectiveness of the proposed design, simulation results are presented, where the performance of the proposed controller is compared with the conventional Sliding Modes of order one and two, and also with the classical PID controller. A strong stability analysis in the sense of Lyapunov is presented, showing global exponential stability for the equilibrium point of the closed-loop control system when the proposed control design is used.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    B. Bandyopadhyay, S. Janardhanan, and S. Spurgeon, Advances in Sliding Mode Control: Concept, Theory and Implementation, Springer-Verlag, Berlin Heidelberg, 2013.

    Book  Google Scholar 

  2. [2]

    Y. Fang and T. Chow, “Chattering free sliding mode control based on recurrent neural network,” Proc. of IEEE International Conference on Systems, Man, and Cybernetics, San Diego, October 1998.

  3. [3]

    K. Jezernik, “Robust chattering-free sliding mode control of servo drives,” International Journal of Electronics, vol. 80, no. 2, pp. 169–179, February 1996.

    Article  Google Scholar 

  4. [4]

    L. Fridman, “An averaging approach to chattering,” IEEE Transactions on Automatic Control, vol. 46, no. 8, pp. 1260–1265, August 2001.

    MathSciNet  Article  Google Scholar 

  5. [5]

    A. Levant, “Chattering analysis,” IEEE Transactions on Automatic Control, vol. 55, no. 6, pp. 1380–1389, February 2010.

    MathSciNet  Article  Google Scholar 

  6. [6]

    L. Fridman, J. Moreno, and R. Iriarte, Sliding Modes after the First Decade of the 21st Century: State of the Art, Springer-Verlag, Berlin Heidelbert, 2011.

    Google Scholar 

  7. [7]

    G. Bartolini, “Chattering phenomena in discontinuous control systems,” Int. J. Systems Sci., vol. 20, no. 12, pp. 2471–2481, December 1989.

    MathSciNet  Article  Google Scholar 

  8. [8]

    I. Boiko and L. Fridman, “Analysis of chattering in continuous sliding-mode controllers,” IEEE Transactions on Automatic Control, vol. 50, no. 9, pp. 1442–1446, September 2005.

    MathSciNet  Article  Google Scholar 

  9. [9]

    A. Levant, “Sliding order and sliding accuracy in sliding mode control,” International Journal of Control, vol. 58, no. 6, pp. 1247–1263, December 1993.

    MathSciNet  Article  Google Scholar 

  10. [10]

    L. Dorel and A. Levant, “On chattering-free,” Proceedings of the 47th IEEE Conference on Decision and Control, Cancún, México, pp. 2196–2201, December 2008.

    Google Scholar 

  11. [11]

    Y.-J. Huang, T.-C. Kuo, and S.-H. Chang, “Adaptive sliding-mode control for nonlinear systems with uncertain parameters,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 38, no. 2, pp. 534–539, April 2008.

    Article  Google Scholar 

  12. [12]

    F. Plestan, Y. Shtessel, V. Brégeault, and A. Poznyak, “New methodologies for adaptive sliding mode control,” International Journal of Control, vol. 83, no. 9, pp. 1907–1919, June 2010.

    MathSciNet  Article  Google Scholar 

  13. [13]

    T. González, J. Moreno, and L. Fridman, “Variable and adaptive gain super-twisting sliding mode control,” IEEE Transactions on Automatic Control, vol. 57, no. 8, pp. 2100–2105, August 2012.

    MathSciNet  Article  Google Scholar 

  14. [14]

    E. Cruz-Zavala, J. Moreno, and L. Fridman, “Adaptive gains super-twisting algorithm for systems with growing perturbations,” Proc. of the 18th IFAC World Congress, Milano, pp. 3039–3044, January 2011.

  15. [15]

    M. Taleb, A. Levant, and F. Plestan, “Twisting algorithm adaptation for control of electropneumatic actuators,” Proc. of IEEE International Workshop on Variable Structure Systems (VSS), Mumbai, pp. 178–183, January 2012.

  16. [16]

    Y. Shtessel, M. Taleb, and F. Plestan, “A novel adaptive-gain supertwisting sliding mode controller: Methodology and application,” Automatica, vol. 48, no. 5, pp. 759–769, May 2012.

    MathSciNet  Article  Google Scholar 

  17. [17]

    L. Yu, M. Zhang, and Z. Fei, “Nonlinear adaptive sliding mode switching control with average dwell-time,” International Journal of System Science, vol. 44, no. 3, pp. 471–478, March 2013.

    Article  Google Scholar 

  18. [18]

    J. Moreno, Y. Negrete, V. Torres-González, and L. Fridman, “Adaptive continuous twisting algorithm,” International Journal of Control, vol. 89, no. 9, pp. 1798–1806, September 2016.

    MathSciNet  Article  Google Scholar 

  19. [19]

    S. Mobayen, “An adaptive chattering-free PID sliding mode control based on dynamic sliding manifolds for a class of uncertain nonlinear systems,” Nonlinear Dynamics, vol. 82, no. 1–2, pp. 53–60, October 2015

    MathSciNet  Article  Google Scholar 

  20. [20]

    M. A. Golkani, L. Fridman, S. Koch, M. Reichhartinger, and M. Horn, “Observer-output saturated output feedback control using twisting algorithm,” Proc. of 14th International Workshop on Variable Structure Systems (VSS), Nanjing, China, June 2016.

    Google Scholar 

  21. [21]

    S. Roy, S. B. Roy, and I. N. Kar, “A new design methodology of adaptive sliding mode control for a class of nonlinear systems with state dependent uncertainty bound,” Proc. of 15th International Workshop on Variable Structure Systems, Austria, July 2018.

  22. [22]

    M. Rahmani, H. Komijani, A. Ghanbari, and M. M. Ettefagh, “Optimal novel super-twisting PID sliding mode control of a MEMS gyroscope based on multi-objective bat algorithm,” Microsystem Technologies, vol. 24, no. 6, pp. 2835–2846, June 2018.

    Article  Google Scholar 

  23. [23]

    V. T. Yen, W. Y. Nan, and P. V. Cuong, “Robust adaptive sliding mode neural networks control for industrial robot manipulators,” International Journal of Control, Automation and Systems, vol. 17, no. 3, pp. 783–792, March 2019.

    Article  Google Scholar 

  24. [24]

    S. Alvarez-Rodríguez, G. Flores, and N. Alcalá Ochoa, “Variable gains sliding mode control,” International Journal of Control, Automation and Systems, vol. 17, No. 3, pp. 555–564, March 2019.

    Article  Google Scholar 

  25. [25]

    A. V. Starbino and S. Sathiyavathi, “Real-time implementation of SMC-PID for magnetic levitation system,” Sādhanā, vol. 44, no. 115, pp. 1–13, May 2019.

    MathSciNet  Google Scholar 

  26. [26]

    H. Hettrick and J. Todd, “In-flight adaptive PID sliding mode position and attitude controller,” Proc. of IEEE Aerospace Conference, pp. 1–9, Big Sky, MT, USA, March 2019.

    Google Scholar 

  27. [27]

    M. Van, X. P. Do, and M. Mavrovouniotis, “Self-tuning fuzzy PID-nonsingular fast terminal sliding mode control for robust fault tolerant control of robot manipulators,” ISA Transactions, 2019. DOI: https://doi.org/10.1016/j.isatra.2019.06.017

  28. [28]

    W. Qi, G. Zong, and H. R. Karimi, “Sliding mode control for nonlinear stochastic semi-Markov switching systems with application to space robot manipulator model,” IEEE Transactions on Industrial Electronics, vol. 67, no. 5, pp. 3955–3966, 2020.

    Article  Google Scholar 

  29. [29]

    B. Jiang, H. R. Karimi, Y. Kao, and C. Gao, “Takagi-Sugeno model based event-triggered fuzzy sliding mode control of networked control systems with semi-Markovian switchings,” IEEE Transactions on Fuzzy Systems, 2019. DOI: https://doi.org/10.1109/TFUZZ.2019.2914005

  30. [30]

    Z. Liu, H. R. Karimi, and J. Yu, “Passivity-based robust sliding mode synthesis for uncertain delayed stochastic systems via state observer,” Automatica, vol. 111, 2020.

  31. [31]

    V. Utkin and A. Pozniak, “Adaptive sliding mode control with application to super-twisting algorithm: Equivalent control method,” Automatica, vol. 49, no. 1, pp. 39–47, January 2013.

    MathSciNet  Article  Google Scholar 

  32. [32]

    V. Utkin, J. Guldner, and J. Shi, Sliding Mode Control in Electromechanical Systems, CRC Press, New York, 1999.

    Google Scholar 

  33. [33]

    Y. Dvir and A. Levant, “Accelerated twisting algorithm,” IEEE Transactions on Automatic Control, vol. 60, no. 10, pp. 2803–2807, October 2015.

    MathSciNet  Article  Google Scholar 

  34. [34]

    Y. Orlov, “Finite time stability and robust control synthesis of uncertain switched systems,” SIAM J. Control Optim., vol. 43, no. 4, pp. 1253–1271, July 2006.

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sergio Alvarez-Rodríguez.

Additional information

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

Recommended by Associate Editor Guangdeng Zong under the direction of Editor Hamid Reza Karimi. This work was partially supported by the FORDECYT-CONACYT under grant 292399 of the National Council of Science and Technology in Mexico (CONACYT).

Sergio Alvarez-Rodríguez received his post-graduated degree in mechatronics engineering from Centro Nacional de Investigación y Desarrollo Tecnológico, México, in 2007. He received a Ph.D. degree in Science and Technology from Universidad de Guadalajara, México, in 2014, and from February 2016 to July 2017 he made a post-doctoral stay at Optical Research Center, AC. He is currenlty a full time researcher professor at TecMM, Lagos de Moreno, Jalisco, México. His areas of interest are on robotics, control theory, and sensors for instrumentation.

Gerardo Flores received his B.S. degree in Electronic Engineering with honors from the Instituto Tecnológico de Saltillo, México in 2000; an M.S. degree in Automatic Control from CINVESTAV-IPN, Mexico City, in 2010; and a Ph.D. degree in Systems and Information Technology from the Heudiasyc Laboratory of the Université de Technologie de Compiègne — Sorbonne Universités, France in October 2014. Since August 2016, he has been a full time researcher and the head of the Perception and Robotics LAB with the Center for Research in Optics, León Guanajuato, Mexico. His research interests are focused on the theoretical and practical problems arising from the development of autonomous robotic systems and vision systems. Dr. Flores has published more than 40 papers in the areas of control systems, computer vision and robotics.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alvarez-Rodríguez, S., Flores, G. PID Principles to Obtain Adaptive Variable Gains for a Bi-order Sliding Mode Control. Int. J. Control Autom. Syst. 18, 2456–2467 (2020). https://doi.org/10.1007/s12555-019-0343-7

Download citation

Keywords

  • Chattering-effect
  • Lyapunov stability approach
  • PID control
  • sliding mode control