International Journal of Control, Automation, and Systems

  • 제5권5호
  • /
  • Pages.547-558
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  • 2007
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  • 1598-6446(pISSN)
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  • 2005-4092(eISSN)
제어로봇시스템학회 (Institute of Control, Robotics and Systems)
권호 표지

Exponential Stabilization of a Class of Underactuated Mechanical Systems using Dynamic Surface Control

  • Qaiser, Nadeem (Dept. of Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences Islamabad) ;
  • Iqbal, Naeem (Dept. of Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences Islamabad) ;
  • Hussain, Amir (Dept. of Computer Science and Mathematics, University of Sterling) ;
  • Qaiser, Naeem (Dept. of Computer Science and Information Technology, Pakistan Institute of Engineering and Applied Sciences Islamabad)
  • 발행 : 2007.10.31
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초록

This paper proposes a simpler solution to the stabilization problem of a special class of nonlinear underactuated mechanical systems which includes widely studied benchmark systems like Inertia Wheel Pendulum, TORA and Acrobot. Complex internal dynamics and lack of exact feedback linearizibility of these systems makes design of control law a challenging task. Stabilization of these systems has been achieved using Energy Shaping and damping injection and Backstepping technique. Former results in hybrid or switching architectures that make stability analysis complicated whereas use of backstepping some times requires closed form explicit solutions of highly nonlinear equations resulting from partial feedback linearization. It also exhibits the phenomenon of explosions of terms resulting in a highly complicated control law. Exploiting recently introduced Dynamic Surface Control technique and using control Lyapunov function method, a novel nonlinear controller design is presented as a solution to these problems. The stability of the closed loop system is analyzed by exploiting its two-time scale nature and applying concepts from Singular Perturbation Theory. The design procedure is shown to be simpler and more intuitive than existing designs. Design has been applied to important benchmark systems belonging to the class demonstrating controller design simplicity. Advantages over conventional Energy Shaping and Backstepping controllers are analyzed theoretically and performance is verified using numerical simulations.

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참고문헌

  1. M. W. Spong, 'Underactuated mechanical systems,' In B. Siciliano and K. P. Valavanis, editor, Control Problems in Robotics and Automation, SpringerVerlag, London, UK, 1997
  2. Alberto Isidori, Nonlinear Control Systems, 2nd edition, Springer-Verlag, Berlin, 1989
  3. M. W. Spong, P. Corke, and R. Lozano, 'Nonlinear control of the inertia wheel pendulum,' Automatica, vol. 37, pp. 1845-1851, 2001 https://doi.org/10.1016/S0005-1098(01)00145-5
  4. I. Fantoni, R. Lozano, and M. W. Spong, 'Energy based control of the Pendubot,' IEEE Trans. on Automatic Control, vol. 45, no. 4, pp. 725-729, 2000 https://doi.org/10.1109/9.847110
  5. E. D. Sontag and H. J. Sussman, 'Further comments on the stability of the angular velocity of a rigid body,' Systems and Control Letters, vol. 12, pp. 213-217, 1988
  6. J. K. Hedrick and Y. Y. Yip, 'Multiple sliding surface control: Theory and application,' Trans. ASME, Journal of Dynamic Systems, Measurement, and Control, vol. 122, no. 4, pp. 586-593, 2000 https://doi.org/10.1115/1.1321268
  7. D. Swaroop, J. K. Hedrick, P. P. Yip, and J. C. Gerdes, 'Dynamic surface control for a class of nonlinear systems,' IEEE Trans. on Automatic Control, vol. 45, no. 11, pp. 1893-1899, Oct. 2000 https://doi.org/10.1109/TAC.2000.880994
  8. X. Lai, J.-H. She, Y. Ohyama, and Z. Cai, 'Fuzzy control strategy for acrobots combining model-free and model-based control,' IEE Proceedings-Control Theory & Applications, vol. 146, no. 6, pp. 505-510, November 1999 https://doi.org/10.1049/ip-cta:19990749
  9. H. Goldstein, Classical Mechanics, 2nd edition, Addison Wesley, USA, 1981
  10. C. J. Wan, D. D. Bernstein, and T. V. Coppola, 'Global stabilization of the oscillating eccentric rotor,' Proc. of the 33rd Decision & Control Conference, Florida, USA, pp. 4024-4029, Dec. 1994
  11. R. Olfati-Saber, 'Normal forms for underactuated mechanical systems with symmetry,' IEEE Trans. on Automatic Control, vol. 47, no. 2, pp. 305-308, Feb. 2002 https://doi.org/10.1109/9.983365
  12. M. W. Spong, 'Energy based control of a class of under actuated mechanical systems,' Proc. of IFAC World Congress, vol. F, pp. 431-435 July 1996
  13. R. Olfati-Saber, 'Control of underactuated mechanical systems with two degrees of freedom and symmetry,' Proc. Amer. Control Conf., Chicago, IL, vol. 6, pp. 4092-4096, June 2000
  14. H. K. Khalil, Nonlinear Systems, 3rd edition, p. 165, pp. 423-460, Prentice-Hall, USA, 2000
  15. M. Jankovic, D. Fontaine, and P. V. Kokotovic, 'TORA example: Cascade and passivity based control designs,' IEEE Trans. on Control System Technology, vol. 34, no. 7, pp. 907-915, 1996
  16. R. Olfati-Saber, Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles, Ph.D. Thesis, Department of Electrical Engineering and Computer Science, MIT, 2001
  17. R. M. Murray and J. Hauser, 'A case study in approximate linearization: The Acrobot example,' Proc. of American Control Conference, 1990
  18. G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer Methods for Mathematical Computations, Prentice-Hall, 1976
  19. N. Qaiser, A. Hussain, N. Iqbal, and N. Qaiser, 'Dynamic surface control for stabilization of the oscillating eccentric rotor,' Proc. IMechE Part I: Journal of Systems and Control Engineering, vol. 221, no. 3, pp. 311-319, 2007 https://doi.org/10.1243/09596518JSCE222
  20. M. Jankovic, D. Fontaine, and P. V. Kokotovic, 'TORA example: Cascade and passivity based control designs,' IEEE Trans. on Control System Technology, vol. 34, no. 7, pp. 907-915, 1996
  21. M. W. Spong, 'The swing up control problem for the Acrobot,' IEEE Control Systems Magazine, vol. 15, no. 1, pp. 49-55, Feb. 1995