THE EXTENDED Λ-METHOD FOR CONTROLLED LAGRANGIAN SYSTEMS

Abstract This paper extends the Λ-method, which was developed in (Auckly et al., 2000; Auckly and Kapitanski, 2003) to solve effectively PDE's involved in the method of controlled Lagrangian systems, by taking into account a gyroscopic term. The gyroscopic force provides more free parameters when one designs stabilizing controllers in the method of controlled Lagrangian systems. We illustrate the extended Λ-method in the example of the Furuta pendulum.

[1]  J. Hamberg.  Controlled Lagrangians, Symmetries and Conditions for Strong Matching , 2000 .

[2]  Naomi Ehrich Leonard,et al.  Stabilization of mechanical systems using controlled Lagrangians , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[3]  Romeo Ortega,et al.  Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment , 2002, IEEE Trans. Autom. Control..

[4]  Naomi Ehrich Leonard,et al.  Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem , 2000, IEEE Trans. Autom. Control..

[5]  Naomi Ehrich Leonard,et al.  The equivalence of controlled lagrangian and controlled hamiltonian systems , 2002 .

[6]  Warren White,et al.  Control of nonlinear underactuated systems , 1999 .

[7]  David Auckly,et al.  On the Lambda-Equations for Matching Control Laws , 2002, SIAM J. Control. Optim..

[8]  Naomi Ehrich Leonard,et al.  Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping , 2001, IEEE Trans. Autom. Control..

[9]  D. Auckly,et al.  On the Λ-equations for Matching Control Laws * , 2001 .

[10]  Naomi Ehrich Leonard,et al.  Stabilization of the pendulum on a rotor arm by the method of controlled Lagrangians , 1999, Proceedings 1999 IEEE International Conference on Robotics and Automation (Cat. No.99CH36288C).

[11]  R. Ortega,et al.  The matching conditions of controlled Lagrangians and IDA-passivity based control , 2002 .

[12]  Dong Eui Chang,et al.  Controlled Lagrangian and Hamiltonian systems , 2002 .

[13]  Jerrold E. Marsden,et al.  Reduction of Controlled Lagrangian and Hamiltonian Systems with Symmetry , 2004, SIAM J. Control. Optim..

[14]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .