Physical dissipation and the method of controlled Lagrangians

We describe the effect of physical dissipation on stability of equilibria which have been stabilized, in the absence of damping, using the method of controlled Lagrangians. This method applies to a class of underactuated mechanical systems including "balance" systems such as the pendulum on a cart. Since the method involves modifying a system's kinetic energy metric through feedback, the effect of dissipation is obscured. In particular, it is not generally true that damping makes a feedback-stabilized equilibrium asymptotically stable. Damping in the unactuated directions does tend to enhance stability, however damping in the controlled directions must be "reversed" through feedback. In this paper, we suggest a choice of feedback dissipation to locally exponentially stabilize a class of controlled Lagrangian systems.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  A. J. Schaft,et al.  Stabilization of Hamiltonian systems , 1986 .

[3]  Naomi Ehrich Leonard,et al.  Global asymptotic stabilization of an underwater vehicle using internal rotors , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[4]  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).

[5]  R. Ortega,et al.  Energy-shaping of port-controlled Hamiltonian systems by interconnection , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

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

[7]  J. Hamberg.  General matching conditions in the theory of controlled Lagrangians , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

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

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