Energy Shaping and Dissipation: The Effect of Damping on Controlled Lagrangian Systems

Prof. Craig Woolsey, Department of Aerospace and Ocean Engineering, Virginia Tech

The method of controlled Lagrangians is a technique for stabilizing a class of underactuated mechanical systems with symmetry. This method provides a feedback control law which preserves the Lagrangian structure but which shapes the energy of the closed-loop system. Using the control-modified energy and other conserved quantities, one may construct a control Lyapunov function for the desired equilibrium. In the conservative setting, this candidate Lyapunov function indicates conditions on control gains for stability; it also suggests a choice of feedback dissipation for asymptotic stability.

Since the method of controlled Lagrangians involves modifying a system's kinetic energy metric through feedback, the effect of unmodeled physical dissipation is not intuitive. For ``balance systems,'' such as the pendulum on a cart, it is not generally true that damping makes the feedback-stabilized equilibrium asymptotically stable. Generic damping in the unactuated directions makes the Lyapunov rate indefinite, destroying the nonlinear stability result. Surprisingly, local analysis shows that damping in the unactuated directions actually enhances stability. However, damping in the controlled directions tends to destabilize the equilibrium and must therefore be ``reversed'' through feedback. These results have been verified through simulation and experiment.