International Journal of Dynamics and Control

, Volume 7, Issue 4, pp 1443–1452 | Cite as

A robust controller design for uncertain nonlinear non-affine systems

  • Hamzeh Ansari
  • Alireza AlfiEmail author


In this paper, a robust controller for a general class of uncertain nonlinear non-affine systems is designed. An affine virtual description with lumped uncertainty is provided for such systems using the pulse width modulation (PWM) and the Filippov’s average model. The uncertainty is estimated by means of an estimator, and a robust controller is then designed to cope with this uncertainty. Stability analysis of the whole system is derived based on the Lyapunov theorem. Finally, theoretical results are tested by simulations.


Robust controller Uncertainty Estimator Non-linear non-affine systems Filippov’s average model Pulse width modulation 


  1. 1.
    Asada H, Slotine JJ, Slotine JJ (1986) Robot analysis and control. Wiley, New YorkGoogle Scholar
  2. 2.
    Spong MW, Hutchinson S, Vidyasagar M et al (2006) Robot modeling and control, vol 3. Wiley, New YorkGoogle Scholar
  3. 3.
    Ansari H, Mohammad MP, Ghanbari A (2014) Adaptive fuzzy computed torque controller for under actuated bipedal robot. In: ASME 2014 international mechanical engineering congress and exposition, pp. V04BT04A033–V04BT04A033. American Society of Mechanical Engineers.
  4. 4.
    Khalil HK (2015) Nonlinear control. Pearson, New YorkzbMATHGoogle Scholar
  5. 5.
    Slotine JJE, Li W et al (1991) Applied nonlinear control, vol 199. Prentice Hall, Englewood CliffszbMATHGoogle Scholar
  6. 6.
    Bryson AE (2018) Applied optimal control: optimization, estimation and control. Routledge, New YorkCrossRefGoogle Scholar
  7. 7.
    Wang Y, Gao F, Doyle FJ III (2009) Survey on iterative learning control, repetitive control, and run-to-run control. J Process Control 19(10):1589–1600CrossRefGoogle Scholar
  8. 8.
    Vaidyanathan S, Volos C et al (2016) Advances and applications in nonlinear control systems, vol 635. Springer, New YorkCrossRefGoogle Scholar
  9. 9.
    Bu X, He G, Wang K (2018) Tracking control of air-breathing hypersonic vehicles with non-affine dynamics via improved neural back-stepping design. ISA Trans 75:88–100CrossRefGoogle Scholar
  10. 10.
    Mobayen S, Tchier F (2016) An lmi approach to adaptive robust tracker design for uncertain nonlinear systems with time-delays and input nonlinearities. Nonlinear Dyn 85(3):1965–1978MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wang Y, Chen M, Wu Q, Zhang J (2018) Fuzzy adaptive non-affine attitude tracking control for a generic hypersonic flight vehicle. Aerosp Sci Technol 80:56–66CrossRefGoogle Scholar
  12. 12.
    Bu X (2018) Guaranteeing prescribed performance for air-breathing hypersonic vehicles via an adaptive non-affine tracking controller. Acta Astronaut. CrossRefzbMATHGoogle Scholar
  13. 13.
    Bu X (2018) Air-breathing hypersonic vehicles funnel control using neural approximation of non-affine dynamics. IEEE/ASME Trans Mechatron 23(5):2099–2108CrossRefGoogle Scholar
  14. 14.
    Haseltalab A, Negenborn RR (2017) Adaptive control for a class of partially unknown non-affine systems: applied to autonomous surface vessels. IFAC-PapersOnLine 50(1):4252–4257CrossRefGoogle Scholar
  15. 15.
    Geranmehr B, Nekoo SR (2015) Nonlinear suboptimal control of fully coupled non-affine six-dof autonomous underwater vehicle using the state-dependent riccati equation. Ocean Eng 96:248–257CrossRefGoogle Scholar
  16. 16.
    Zong-Cheng L, Xin-Min D, Jian-Ping X (2014) Adaptive neural control for a class of uncertain chaotic system with non-affine input. In: Control conference (CCC), 2014 33rd Chinese, pp 2069–2074. IEEE.
  17. 17.
    Tombul GS, Banks SP, Akturk N (2009) Sliding mode control for a class of non-affine nonlinear systems. Nonlinear Anal Theory Methods Appl 71(12):e1589–e1597MathSciNetCrossRefGoogle Scholar
  18. 18.
    Bogkovic J, Chen L, Mehra RK (2001) Adaptive tracking control of a class of non-affine plants using dynamic feedback. In: American control conference, 2001. Proceedings of the 2001, vol 3, pp 2450–2455. IEEEGoogle Scholar
  19. 19.
    Labiod S, Boucherit MS, Guerra TM (2005) Adaptive fuzzy control of a class of mimo nonlinear systems. Fuzzy Sets Syst 151(1):59–77MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mobayen S, Tchier F (2015) A new LMI-based robust finite-time sliding mode control strategy for a class of uncertain nonlinear systems. Kybernetika 51(6):1035–1048MathSciNetzbMATHGoogle Scholar
  21. 21.
    Mobayen S, Tchier F (2017) A novel robust adaptive second-order sliding mode tracking control technique for uncertain dynamical systems with matched and unmatched disturbances. Int J Control Autom Syst 15(3):1097–1106CrossRefGoogle Scholar
  22. 22.
    Esmaeili N, Alfi A, Khosravi H (2019) Balancing and trajectory tracking of two-wheeled mobile robot using backstepping sliding mode control: design and experiments. J Franklin Inst 87Google Scholar
  23. 23.
    Deng Y, Wang J, Li H, Liu J, Tian D (2018) Adaptive sliding mode current control with sliding mode disturbance observer for pmsm drives. ISA Trans. CrossRefGoogle Scholar
  24. 24.
    Guo J (2019) Application of full order sliding mode control based on different areas power system with load frequency control. ISA Trans. CrossRefGoogle Scholar
  25. 25.
    Adhikary N, Mahanta C (2018) Sliding mode control of position commanded robot manipulators. Control Eng Pract 81:183–198CrossRefGoogle Scholar
  26. 26.
    Edwards C, Shtessel Y (2019) Enhanced continuous higher order sliding mode control with adaptation. J Franklin Inst. MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Gu J, Wang Z, Kuen J, Ma L, Shahroudy A, Shuai B, Liu T, Wang X, Wang G, Cai J et al (2018) Recent advances in convolutional neural networks. Pattern Recognit 77:354–377CrossRefGoogle Scholar
  28. 28.
    Tabatabaei SM, Arefi MM (2016) Adaptive neural control for a class of uncertain non-affine nonlinear switched systems. Nonlinear Dyn 83(3):1773–1781MathSciNetCrossRefGoogle Scholar
  29. 29.
    Ramezani Z, Arefi MM, Zargarzadeh H, Jahed-Motlagh MR (2016) Neuro-adaptive backstepping control of siso non-affine systems with unknown gain sign. ISA Trans 65:199–209CrossRefGoogle Scholar
  30. 30.
    Ramezani Z, Arefi MM, Zargarzadeh H, Jahed-Motlagh MR (2016) Neuro observer-based control of pure feedback mimo systems with unknown control direction. IET Control Theory Appl 11(2):213–224MathSciNetCrossRefGoogle Scholar
  31. 31.
    Wang C, Hill DJ, Ge SS, Chen G (2006) An iss-modular approach for adaptive neural control of pure-feedback systems. Automatica 42(5):723–731MathSciNetCrossRefGoogle Scholar
  32. 32.
    Zhang T, Ge S, Hang C (1998) Direct adaptive control of non-affine nonlinear systems using multilayer neural networks. In: American control conference, 1998. Proceedings of the 1998, vol 1, pp 515–519. IEEEGoogle Scholar
  33. 33.
    Boulkroune A, M’Saad M, Farza M (2012) Fuzzy approximation-based indirect adaptive controller for multi-input multi-output non-affine systems with unknown control direction. IET Control Theory Appl 6(17):2619–2629MathSciNetCrossRefGoogle Scholar
  34. 34.
    Boukezzoula R, Galichet S, Foulloy L (2003) Fuzzy adaptive linearizing control for non-affine systems. In: Fuzzy Systems, 2003. FUZZ’03. The 12th IEEE international conference on, vol 1, pp 543–548. IEEEGoogle Scholar
  35. 35.
    Zhang Q, Wang C, Xu D (2018) Finite-time stabilization for a class of non-affine nonlinear systems with input saturation and time-varying output constraints. IEEE Access 6:23529–23539CrossRefGoogle Scholar
  36. 36.
    Zhang W, Ge SS (2006) A global implicit function theorem without initial point and its applications to control of non-affine systems of high dimensions. J Math Anal Appl 313(1):251–261MathSciNetCrossRefGoogle Scholar
  37. 37.
    Arefi MM, Zarei J, Karimi HR (2014) Adaptive output feedback neural network control of uncertain non-affine systems with unknown control direction. J Franklin Inst 351(8):4302–4316MathSciNetCrossRefGoogle Scholar
  38. 38.
    Nekoo SR, Geranmehr B (2014) Nonlinear observer-based optimal control using the state-dependent Riccati equation for a class of non-affine control systems. J Control Eng Appl Inf 16(2):5–13Google Scholar
  39. 39.
    Rong-Hu C, Zhong-Sheng H (2007) Dual-stage optimal iterative learning control for nonlinear non-affine discrete-time systems. Acta Autom Sin 33(10):1061–1065MathSciNetGoogle Scholar
  40. 40.
    Arefi MM, Ramezani Z, Jahed-Motlagh MR (2014) Observer-based adaptive robust control of nonlinear nonaffine systems with unknown gain sign. Nonlinear Dyn 78(3):2185–2194MathSciNetCrossRefGoogle Scholar
  41. 41.
    Shahri ESA, Alfi A, Machado JT (2015) An extension of estimation of domain of attraction for fractional order linear system subject to saturation control. Appl Math Lett 47:26–34MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Electrical and Robotic EngineeringShahrood University of TechnologyShahroodIran

Personalised recommendations