Approximate feedback linearization based optimal robust control for an inverted pendulum system with time-varying uncertainties

Abstract

In the present study, to attain an approximate feedback linearization based optimal robust control of an under-actuated cart-type inverted pendulum system with two-degree-of-freedom (2DOF) having time-varying uncertainties is desirable. To reach such a goal, at first, the governing dynamical equations of the cart-type inverted pendulum are presented. Then, using an approximate feedback linearization method, the dynamics of the nonlinear system are changed to those of the linear one. In order to simultaneously stabilize the both degrees of freedom, a robust sliding mode controller is designed for this under-actuated system. Finally, the control law is improved with aid of a novel adaptive approach based on an approximating function found via a multiple-crossover genetic algorithm so that the system always will be optimally robust against time-varying parametric uncertainties. The results are displayed to show how the proposed controller improves the settling time and overshoot of the system outputs. In addition, such an improvement in the values of the selected objective functions is clearly evident.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

References

  1. 1.

    Chen D, Zhao M, Sun D, Zheng L, Chen J (2019) Robust H∞ control of cooperative driving system with external disturbances and communication delays in the vicinity of traffic signals, Physica A: Statistical Mechanics and its Applications. In press, corrected proof, Available online 2019; Article 123385

  2. 2.

    Yin H, Chen Y. H, Yu D (2019) Fuzzy dynamical system approach for a dual-parameter hybrid-order robust control design. Fuzzy Sets Syst. In press, corrected proof, Available online 2019

  3. 3.

    Qi D, Yang L, Zhang Y, Cai W (2019) Indirect robust suboptimal control of two-satellite electromagnetic formation reconfiguration with geomagnetic effect. Adv Space Res 64(11):2331–2344

    Article  Google Scholar 

  4. 4.

    Zhang X, Han X, Guan W, Zhang G (2019) Improvement of integrator backstepping control for ships with concise robust control and nonlinear decoration. Ocean Eng 189:106349

    Article  Google Scholar 

  5. 5.

    Rahmani B (2019) Sliding mode based-variable selective control for robust tracking of uncertain Internet-based systems. ISA Trans, In press, corrected proof, Available online 2019, vol 28

  6. 6.

    Marti K, Stein I (2012) Optimal structural control under stochastic uncertainty: robust optimal open-loop feedback control. IFAC Proceed Vol 45(2):270–275

    Article  Google Scholar 

  7. 7.

    Lee K, Jeon Y (2020) Measuring Chinese consumers’ perceived uncertainty. Int Rev Econ Finance 66:51–70

    Article  Google Scholar 

  8. 8.

    Wang W, Nguong SK, Zhong S, Liu F (2014) Novel delay-dependent stability criterion for time-varying delay systems with parameter uncertainties and nonlinear perturbations. Inf Sci 281:321–333

    MathSciNet  Article  Google Scholar 

  9. 9.

    Zeinali M, Notash L (2010) Adaptive sliding mode control with uncertainty estimator for robot manipulators. Mech Mach Theory 45(1):80–90

    MathSciNet  Article  Google Scholar 

  10. 10.

    Chaoui H, Sicard P (2010) Adaptive fuzzy logic motion and posture control of inverted. In: Proceedings of the 6th annual IEEE conference on automation science and engineering, Toronto, Ontario, Canada 2010, pp 638–643

  11. 11.

    Shojaei K, Mohammad Shahri A, Tarakameh A (2011) Adaptive feedback linearizing control of nonholonomic wheeled mobile robots in presence of parametric and nonparametric uncertainties. Robot Comput Integr Manuf 27(1):194–204

    Article  Google Scholar 

  12. 12.

    Ricardez-Sandoval LA (2012) Optimal design and control of dynamic systems under uncertainty: a probabilistic approach. Comput Chem Eng 43:91–107

    Article  Google Scholar 

  13. 13.

    Figueroa JL, Biagiola SI (2013) Modelling and uncertainties characterization for robust control. J Process Control 23(3):415–428

    Article  Google Scholar 

  14. 14.

    Li Y, Chen J, Feng L (2013) Dealing with uncertainty: a survey of theories and practices. IEEE Trans Knowl Data Eng 25(11):2463–2482

    Article  Google Scholar 

  15. 15.

    Yoon H, Eun Y, Park C (2014) Adaptive tracking control of spacecraft relative motion with mass and thruster uncertainties. Aerosp Sci Technol 34:75–83

    Article  Google Scholar 

  16. 16.

    Lu L, Yao B (2014) Online constrained optimization based adaptive robust control of a class of MIMO nonlinear systems with matched uncertainties and input/state constraints. Automatica 50:864–873

    MathSciNet  Article  Google Scholar 

  17. 17.

    Petersen IR, Tempo R (2014) Robust control of uncertain systems: classical results and recent developments. Automatica 50(5):1315–1335

    MathSciNet  Article  Google Scholar 

  18. 18.

    Slotine JE, Li W (1991) Applied nonlinear control. Prentice hall, Englewood Cliffs

    Google Scholar 

  19. 19.

    Khalil HK (1996) Nonlinear systems. Prentice Hall, Upper Saddle River

    Google Scholar 

  20. 20.

    Sastry S (1999) Nonlinear systems: analysis, stability and control. Springer, New York

    Google Scholar 

  21. 21.

    Andalib Sahnehsaraei M, Mahmoodabadi MJ, Bagheri A (2014) Pareto optimum control of a 2-DOF inverted pendulum using approximate feedback linearization and sliding mode control. Trans Inst Measur Control 36(4):496–505

    Article  Google Scholar 

  22. 22.

    Mahmoodabadi MJ, Soleymani T, Andalib Sahnehsaraei M (2018) A hybrid optimal controller based on the robust decoupled sliding mode and adaptive feedback linearization. Inf Technol Control 47(2):295–309

    Google Scholar 

  23. 23.

    Mahmoodabadi MJ, Soleymani T (2020) Optimum fuzzy combination of robust decoupled sliding mode and adaptive feedback linearization controllers for uncertain under-actuated nonlinear systems. Chin J Phys 64:241–250

    MathSciNet  Article  Google Scholar 

  24. 24.

    Sadafi MH, Hosseini R, Safikhani H, Bagheri A, Mahmoodabadi MJ (2011) Multi-objective optimization of solar thermal energy storage using hybrid of particle swarm optimization, multiple crossover and mutation operator. Int J Eng Trans B 24(3):367–376

    Article  Google Scholar 

  25. 25.

    Andalib Sahnehsaraei M, Mahmoodabadi MJ, Taherkhorsandi M, Castillo-Villar KK, Mortazavi Yazdi SM (2015) A hybrid global optimization algorithm: particle swarm optimization in association with a genetic algorithm, complex system modelling and control through intelligent soft computations, studies in fuzziness and soft computing, vol 319. Springer, Berlin, pp 45–86

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Mohammad Javad Mahmoodabadi.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sahnehsaraei, M.A., Mahmoodabadi, M.J. Approximate feedback linearization based optimal robust control for an inverted pendulum system with time-varying uncertainties. Int. J. Dynam. Control 9, 160–172 (2021). https://doi.org/10.1007/s40435-020-00651-w

Download citation

Keywords

  • Adaptive optimal control
  • Feedback linearization
  • Genetic algorithm
  • Inverted pendulum
  • Time-varying uncertainty