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Adaptive RBFNN finite-time control of normal forms for underactuated mechanical systems

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A Correction to this article was published on 28 December 2017

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Abstract

This paper presents a constructive design of a continuous finite-time controller for a class of mechanical systems known as underactuated systems that satisfy the symmetry properties. An adaptive radial basis function neural network (RBFNN) finite-time control scheme is proposed to stabilize the underactuated system at a given equilibrium, regardless of the various uncertainties and disturbances that the system contains. First, a coordinate transformation is introduced to decouple the control input so that an n-th order underactuated system can be represented into a special cascade form. Next, an adaptive robust finite-time controller is derived from adding a power integrator technique and the RBFNN to approximate the nonlinear unknown dynamics in the new space, whose bounds are supposedly unknown. The stability and finite-time convergence of the closed-loop system are established by using Lyapunov theory. To show the effectiveness of the proposed method, simulations are carried out on the rotary inverted pendulum, a typical example of an underactuated mechanical system.

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  • 28 December 2017

    The list of authors in the original publication was incomplete. The complete list of authors is shown here, including the third author, Faiçal Mnif.

References

  1. Adhikary, N., Mahanta, C.: Integral backstepping sliding mode control for underactuated systems: swing-up and stabilization of the cart-pendulum system. ISA Trans. 58, 870880 (2013)

    Google Scholar 

  2. Amato, F., Ariola, M., Dorato, P.: Finite-time control of linear systems subject to parametric uncertainties and disturbances. Automatica 37(9), 1459–1463 (2001)

    Article  MATH  Google Scholar 

  3. Bhat, S.P., Bernstein, D.S.: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751766 (2000)

    Article  MathSciNet  Google Scholar 

  4. Bortoff, S.A., Spong, M.W.: Pseudolinearization of the acrobot using spline functions. In: Proc. 31st. IEEE Conf. Decision Contr., pp. 593–598, Tucson (1992)

  5. Carlos, C.A., Julio, I.M.-M., Jorge, D.: Stabilization of the cart pole system: by sliding mode control. Nonlinear Dyn. 78(4), 2769–2777 (2014)

    Article  MathSciNet  Google Scholar 

  6. Chen, Y.-F., Huang, A.-C.: Adaptive control of rotary inverted pendulum system with time-varying uncertainties. Nonlinear Dyn. 76, 95–102 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheng, D., Lin, W.: On normal form of nonlinear systems. IEEE Trans. Automat. Control 48(11), 1242–1248 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Choukchou-Braham, A., Cherki, B., Djemai, M., Busawon, K.: Analysis and Control of Underactuated Mechanical Systems. Springer, London (2014)

    Book  MATH  Google Scholar 

  9. Fantoni, I., Lozano, R.: Non-linear Control for Underactuated Mechanical Systems. Springer, New York (2001)

    MATH  Google Scholar 

  10. Haimo, V.T.: Finite-time controllers. SIAM J. Control Optim. 24, 760–770 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hong, Y.: Finite-time stabilization and stabilizability of a class of controllable systems. Syst. Control Lett. 46, 231–236 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hong, Y., Wang, J., Xi, Z.: Stabilization of uncertain chained form systems within finite settling time. IEEE Trans. Autom. Control 50(9), 1379–1384 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hong, Y.G., Huang, J., Xu, Y.: On an output feedback finite-time stabilization problem. IEEE Trans. Autom. Control 46, 305–309 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hong, Y.G., Jiang, Z.P.: Finite-time stabilization of nonlinear systems with parametric and dynamic uncertainties. IEEE Trans. Autom. Control 51, 1950–1956 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hong, Y.G., Wang, K., Cheng, D.: Adaptive finite-time control of nonlinear systems with parametric uncertainty. IEEE Trans. Autom. Control 51, 858–862 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang, A.-C., Chen, Y.-F., Kai, C.-Y.: Adaptive Control of Underactuated Mechanical Systems. World Scientific, New York (2014)

    Google Scholar 

  17. Huang, X., Lin, W., Yang, B.: Global finite-time stabilization of a class of uncertain nonlinear systems. Automatica 41, 881–888 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Huang, X., Lin, W., Yang, B.: Global finite-time stabilization of a class of uncertain nonlinear systems. Automatica 41(5), 881888 (2005)

    Article  MathSciNet  Google Scholar 

  19. Jadlovska, S., Sarnovsky, J.: A complex overview of modeling and control of the rotary single inverted pendulum system. Power Eng. Electr. Eng. 11(2), 73–85 (2013)

    Google Scholar 

  20. Jadlovska, S., Sarnovsky, J.: Modeling and optimal control of nonlinear underactuated mechanical systems a survey. In: 13th Scientific Conference of Young Researchers FEI TU of Kosice, pp. 32–35, Herlany (2013)

  21. Khalil, H.: Nonlinear Systems, 2nd edn. Prentice Hall, Upper Saddle River (1996)

    Google Scholar 

  22. Krstic, M., Kanellakopoulos, I., Kokotovic, P.V.: Nonlinear and Adaptive Control Design. Wiley, New York (1995)

    MATH  Google Scholar 

  23. Lee, J., Mukherjee, R., Khalil, H.K.: Output feedback stabilization of inverted pendulum on a cart in the presence of uncertainties. Automatica 54, 146–157 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lee, S.H., Park, J.B., Choi, Y.H.: Finite time control of nonlinear underactuated systems using terminal sliding surface. In: IEEE International Symposium on Industrial Electronics, Seoul Olympic Parktel, Seoul (2009)

  25. Li, J., Qian, C., Ding, S.: Global finite-time stabilisation by output feedback for a class of uncertain nonlinear systems. Int. J. Control 83, 2241–2252 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, T., Li, G., Zhao, Q.: Adaptive fault-tolerant stochastic shape control with application to particle distribution control. IEEE Trans. Syst. Man Cybern. Syst. 12, 1592–1604 (2015)

    Article  Google Scholar 

  27. Lin, W., Qian, C.: Adaptive regulation of high-order lower-triangular systems: adding a power integrator technique. Syst. Control Lett. 39, 353–364 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lin, W., Qian, C.: Adding one power integrator: a tool for global stabilization of high order lower-triangular systems. Syst. Control Lett. 39, 339–351 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  29. De Luca, A., Mattone, R., Oriolo, G.: Stabilization of underactuated planar 2R manipulator. Int. J. Robust Nonlinear Control 10(4), 181–198 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ming, Y., Xing, W., Li, Zhijun: Adaptive sliding-mode control for two-wheeled inverted pendulum vehicle based on zero-dynamics theory. Nonlinear Dyn. 76(1), 459–471 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Moulay, E., Perruquetti, W.: Finite time stability conditions for non autonomous continuous systems. Int. J. Control 81, 797–803 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ibanez, C.A., Frias, O.G.: Controlling the inverted pendulum by means of a nested saturation function. Nonlinear Dyn. 53, 273–280 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Olfati-Saber, R.: Fixed point controllers and stabilization of the cart-pole system and the rotating pendulum. In: Proc. 38th. IEEE Conf. Decision Contr. (CDC’99), pp. 1174–1181, Phoenix (1999)

  34. Olfati-Saber, R.: Nonlinear control of underactuated mechanical systems with application to robotics and aerospace vehicles. Ph.D. Thesis, Massachusetts Institute of Technology (2001)

  35. Park, M., Chwa, D.: Swing-up and stabilization control of inverted pendulum systems via coupled sliding-mode control method. IEEE Trans. Ind. Electron. 56, 3541–3555 (2009)

    Article  Google Scholar 

  36. Qian, C., Lin, W.: A continuous feedback approach to global strong stabilization of nonlinear systems. IEEE Trans. Autom. Control 46(7), 1061–1079 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. Qian, C., Lin, W.: A continuous feedback approach to global strong stabilization of nonlinear systems. IEEE Trans. Autom. Control 46(7), 1061–1079 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  38. Qian, C.J., Li, J.: Global finite-time stabilization by output feedback for planar systems without observable linearization. IEEE Trans. Autom. Control 50, 885–890 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  39. Qian, D., Yi, J., Zhao, D.: Hierarchical sliding mode control for a class of simo under-actuated systems. Control Cybern. 37(1), 1–18 (2008)

    MathSciNet  MATH  Google Scholar 

  40. Ravichandran, M.T., Mahindrakar, A.D.: Robust stabilization of a class of underactuated mechanical systems using time scaling and lyapunov redesign. IEEE Trans. Ind. Electron. 58, 4299–4313 (2011)

    Article  Google Scholar 

  41. Reyhanoglu, M., Van der Schaft, A., McClamroch, N.H., Kolmanovsky, I.: Dynamics and control of a class of underactuated mechanical systems. IEEE Trans. Autom. Control 44, 1663–1671 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  42. She, J., Zhang, A., Lai, X., Wu, M.: Global stabilization of 2-DOF underactuated mechanical systems—an equivalent-input-disturbance approach. Nonlinear Dyn. 69(1), 495–509 (2012)

    Article  Google Scholar 

  43. Shen, M., Park, J.H., Ye, D.: A separated approach to control of markov jump nonlinear systems with general transition probabilities. IEEE Trans. Syst. Man Cybern. 46, 2010–2018 (2016)

    Google Scholar 

  44. Spong, M.: The swing up control problem for the acrobot. IEEE Control Syst. Mag. 15(1), 49–55 (1995)

    Article  Google Scholar 

  45. Spong, M.: Energy based control of a class of underactuated mechanical system. In: 13th IFAC World Congress, pp. 431–436, San Francisco (1996)

  46. Spong, M.: Underactuated mechanical systems. Control problems in robotics and automation. In: Siciliano, B., Valavanis, K.P. (eds.) LNCIS, vol. 230, pp. 135–150. Springer, Berlin (1998)

    Google Scholar 

  47. Wang, Y., Song, Y., Krstic, M., Wen, C.: Adaptive finite time coordinated consensus for high-order multi-agent systems: adjustable fraction power feedback approach. Inf. Sci. 372, 392–406 (2016)

  48. Xu, J., Guo, Z., Lee, T.H.: Design and implementation of integral sliding mode control on an underactuated two-wheeled mobile robot. ISA Trans. 58, 870–880 (2013)

    Google Scholar 

  49. Yang, K.S., Cheng, C.C., Yang, J.H.: Robust finite time controller design for second order underactuated mechanical systems. Appl. Mech. Mater. 284–287, 2310–2314 (2013)

    Article  Google Scholar 

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Authors and Affiliations

  1. CEM-Lab, Department of Electrical Engineering, National Institute of Applied Sciences and Technology, Tunis, Tunisia

    Jawhar Ghommam

  2. LIRMM, Université Montpellier 2-CNRS, 161 rue Ada, 34392, Montpellier, France

    Ahmed Chemori

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  1. Jawhar Ghommam
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  2. Ahmed Chemori
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Correspondence to Jawhar Ghommam.

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A correction to this article is available online at https://doi.org/10.1007/s11071-017-4024-x.

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Ghommam, J., Chemori, A. Adaptive RBFNN finite-time control of normal forms for underactuated mechanical systems. Nonlinear Dyn 90, 301–315 (2017). https://doi.org/10.1007/s11071-017-3662-3

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  • DOI: https://doi.org/10.1007/s11071-017-3662-3

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