Some remarks on interconnection and damping assignment passivity-based control of mechanical systems,” in Taming Heterogeneity and Complexity of Embedded Control

Total Energy Shaping is a controller design methodology that achieves (asymptotic) stabilization of mechanical systems endowing the closed–loop system with a Lagrangian or Hamiltonian structure with a desired energy function—that qualifies as Lyapunov function for the desired equilibrium. The success of the method relies on the possibility of solving two PDEs which identify the kinetic and potential energy functions that can be assigned to the closed–loop. Particularly troublesome is the PDE associated to the kinetic energy which is nonlinear and non–homogeneous and the solution, ∗The work of Giuseppe Viola was partially sponsored by the Control Training Site programme of the European Com-mission. †Corresponding author. 1 that defines the desired inertia matrix, must be positive definite. In this paper we prove that we can eliminate or simplify the forcing term in this PDE modifying the target dynamics and introducing a change of coordinates in the original system. Furthermore, it is shown that, in the particular case of transformation to the Lagrangian coordinates, the possibility of simplifying the PDEs is determined by the interaction between the Coriolis and centrifugal forces and the actuation structure. The examples of pendulum on a cart and Furuta’s pendulum are used to illustrate the results