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Authors
Ilya V. Burkov

Digital Object Identifier (DOI)
10.3182/20070829-3-RU-4912.00012

Page Numbers:
77-82

Index Terms
Lyapunov methods,nonlinear systems,saturation,cranes,robots,damping

Abstract
In some cases the desired uniform motion may be described by a pair of first integrals of the system with zero control input. These two integrals and the integration of nonlinear function of saturation are used to construct Lyapunov function The control is designed from the condition of decreasing Lyapunov function on the trajectories of the closed loop system. This control may be chosen a priori bounded and linear in a small viciniti of the desired motion. This method is applied to stabilize rotating body beam, for damping the oscillations of blades of an elastic propeller and for stabilization of the uniform transition of the pendulum on a cart.

References

[1] D'Andrea-Novel B., Coron J.M. (2000) Exponential
    stabilization of an overhead crane with flexible
    cable via a backstepping approach. Automatica
    . 36: 587-593.
[2] Burkov I.V. (2005) Asymptotic stabilization of desired
     rotation in multidimensional Hamiltonian
    systems by Chetaev method. 4th. Internat.
    Workshop on Multidimensional Systems, Wuppertal.
    
[3] Coron J.-M., d'Andrea-Novel B. (1998) Stabilization
    of rotating body beam without damping. IEEE
    Tr. AC 43: 608-618.
[4] Dimentberg F.M. (1959) Oscillations of rotating shafts.
    Moscow, USSR Acad. Sci. Engl. transl. exists.
    
[5] Dunskaya N.V., Pyatnitskii E.S. (1988). Stabilization
    of mechanical and electromechanical systems.
    Avtomatika i telemekhanika. No. 12, pp. 40-51.
    Engl. transl. in: Automation and Remote Control
    .
[6] Fantoni I., Lozano R., Spong M.W. (2000) Energy
    based control of the pendubot. IEEE Tr. Aut.
    Control. Vol. 45. pp. 725-729.
[7] Kolesnichenko O., Shiriaev A.S., Robertson A. (2002)
    Extension of Pozharitsky theorem for partial
    stabilization of a system with several first integrals.
     IEEE Conf. CDC. pp. 3512-3517.
[8] Mazenc F., Bowong S. (2003) Tracking trajectories of
    the cart-pendulum system. Automatica. 39:
    677-684.
[9] Rouche N., Habets P., Laloy M. (1977) Stability Theory
     by Liapunov's Direct Method. Berlin, Springer.
    Russ. transl.: Moscow, Nauka, 1980.
[10] Tondl A. (1965) Some problems of rotor dynamics.
     Prague, Cz. Acad. Sci.