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Computación y Sistemas

On-line version ISSN 2007-9737Print version ISSN 1405-5546

Comp. y Sist. vol.7 n.4 Ciudad de México Apr./Jun. 2004





A Stabilizable Control Laws For a Rotational Pendulum: A Trajectory Planning Approach


Leyes de control Estabilizadoras para un péndulo rotacional: Una Planificación de Trayectorias


Carlos F. Aguilar Ibañez1 and Oscar Chavoya A.2


1 Centro de investigación en Computación del IPN. Laboratorio de Metrología y Control. Av. Dios Bátiz s/n; México D.F., C.P. 07738; México. E–mail:

2 Camelback High School, Phoenix, AZ85016, U.S.A.


Article received on April 27, 2000
Accepted on May 17, 2004



We propose two simple controls for the regulation of an under actuated rotational pendulum. Both controllers are based on the Lyapunov approach; the first is a simple passive control which makes the closed–loop solution converges asymptotically to an equilibrium manifold. The second approach is a combination of the Lyapunov and the off–line trajectory planning approaches to move the pendulum from an equilibrium point to another equilibrium point, both point belonging to an equilibrium manifold. The last task is accomplished in an approximated fashion. The results are illustrated by means of digital computer simulations.

Keywords: Lyapunov–based control, Trajectory Planning and Under Actuated Systems.



Se proponen dos controles simples para la regulación de un péndulo rotacional sub–actuado. Ambos controles están basados en el enfoque de Lyapunov, el primero es un control pasivo simple que hace que la solución de lazo cerrado converja asintóticamente a una variedad (manifold) de equilibrio. El segundo enfoque es una combinación de los enfoques de Lyapunov y el de planeación de trayectoria fuera de línea para mover el péndulo de un punto de equilibrio a otro punto de equilibrio, ambos pertenecientes a una variedad (manifold) de equilibrio. La última tarea se logra de forma aproximada. Los resultados se ilustran mediante simulaciones hechas en una computadora digital.

Palabras clave: Control basado en el enfoque de Lyapunov, Planeación de Trayectoria y Sistemas Subactuados.





This work was supported by CIC–IPN, and by the Coordinación General de Posgrado e Investigación (CGPI–IPN) under research Grant 20040877. Also this paper is dedicated in memory of Professor Leopoldo Arostegui (CBTIS 78).



1. Bloch A. M., Chang D., Leonard N., and Marsden J. E., "Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping." Trans IEEE on Auto. Control, Vol. 46, 1556–1571. 2002.         [ Links ]

2. Bloch A. M., Leonard N. E., and Marsden J. E., "Controlled lagrangians and the Stabilization of Mechanical Systems I: The First matching theorem." IEEE Trans. on Sytems and Control, Vol. 45, pp. 2253–2270, 2000.         [ Links ]

3. Brockett R. W., "Asymptotic stabilty and feedback stabilization." In R.S. Millman R.W. Brockett and H.J. Sussman, editors, Differential Geometric Control Theory, 181–191, Birkhauser, (1983).        [ Links ]

4. Bupp R.T., Bernstein D.S. and Coppola V.T., "A benchmark problem for nonlinear control design: problem statement, experimental test bed and pasive nonlinear compensation."1995 Proccedings American Control Conference. Seattle, W.A., pp. 4363–4367, 1995.         [ Links ]

5. Christopher E., and Spurgeon S., Sliding Mode Control: Theory and Applications, Taylor and Francis Ltd., UK, 1998.         [ Links ]

6. de Jager B. and Nijmeijer, H. (Eds.), Special Issue on Control of Underactuated Nonlinear Systems, Int. J. Robust Nonlinear Control, Vol. 10, No. 4, 2000.         [ Links ]

7. Escobar G., Ortega R, Sira H., "Output feedback global stabilization of nonlinear benchmark system using satured passivity based controller." IEEE Transactions Control, Systems and Technology. Vol. 7, pp. 83–95. 1999.        [ Links ]

8. Fantoni I., Lozano R., "Non–Linear control for underactuated mechanical systems". Springer, 2002.        [ Links ]

9. Furuta K., Yamakita M. and Kobayashi S., "Swing up control of inverted pendulum using pseudo–state feedback." Journal of System and Control Engineering, Vol. 206, pp. 263–269, 1992.        [ Links ]

10. Hauser J., Sastry S., Kokotovic P., "Nonlinear Control Via Approximate Input–Output Linearization: The Ball and Beam Example", IEEE Transactions on Automatic Control, Vol. 37, pp. 392–398, 1992.         [ Links ]

11. Isidori A., "Nonlinear Control Systems". Springer, 1995.         [ Links ]

12. Jagger D. B. and Nijmeijer H. (edd) "Special issue on Control underactuacted system." International Journal of Robust and Nonlinear. Vol. 10, pp. 283–300, 2000.        [ Links ]

13. Jakubczyk B, Respondek W., "On the linearization of control systems", Bull. Acad. Polon. Sci. Math., Vol. 28, 1980. pp. 517–522.         [ Links ]

14. Khalil H.K., Non Linear Systems, Prentice Hall second edition, 1996.        [ Links ]

15. Krstic M., Kanellakopoulos I., and Kokotovic P., "Nonlinear and Adaptive Control Design". John Wiley & Sons, 1995.        [ Links ]

16. Lozano R., Fantoni I., and Block D. J., "Stabilization of the inverted pendulum around its homoclinics orbit." Systems and Control Letters, Vol. 40, pp. 197–204, 2000.         [ Links ]

17. Mazenc F., and Praly L., "Adding integrations, saturated controls, and stabilization for feedforward Systems." IEEE Transactions on Automatic Control. Vol. 40, pp. 1559–1578., 1996.         [ Links ]

18. Olfati S. R., "Fixed Point Controllers and Stabilization of the Cart–Pole and Rotating Pendulum" .Proc. of the 38th Conf. on Decision and Control, pp. 1174–1181, Phoenix, Az. Dec. 1999.         [ Links ]

19. Reyhanoglu M., van der Schaft A., McClamroch N. H., and Kolmanovsky I., "Dynamics and control of a class of underactuated mechanical systems." IEEE Transactions on Automatic Control. Vol. 44, 1663–1671, 1999.         [ Links ]

20. Sepulchre R., Jankovic M., and Kokotovic P., Constructive Nonlinear Control, Springer–Verlag, 1997.        [ Links ]

21. Shiriaev A. S., Pogromsky A., Ludvigsen H., and Egeland O. "On global properties of passivity–based control of an inverted pendulum" International Journal of Robust and Nonlinear. Vol. 10, pp. 283–300, 2000.        [ Links ]

22. Sira R. H., "On the Control of the Ball and Beam System: A trajectory Planning Approach", Proceedings of the 39IEEE Conference on Decision and Control, Australia, December, pp. 4042–4047, 2000.         [ Links ]

23. Sira R. H., Llanes O., "Sliding mode control of nonlinear mechanical vibrations."Journal of Dinamics, Systems, Measurements and Control. Vol 122, pp. 674–678, 2000.         [ Links ]

24. Spong M. W., "Control Problems in Robotics and Automation" Underactuated mechanical systems. In B. Siciliano and K. P. Valavanis, editors. UK: Springer–Verlag, 1997.         [ Links ]

25. Teel A. R., "A nonlinear small gain theorem for the analysis of control systems with saturation." IEEE Transactions on Automatic Contro. Vol. 40, pp. 1256–1270, 1996.         [ Links ]

26. Utkin V. I., Sliding Modes in Control and Optimization, Springer– Verlag, Berlin, 1992.        [ Links ]

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