On the matching equations of energy shaping controllers for mechanical systems

Abstract : Total energy shaping is a controller design methodology that achieves (asymptotic) stabilisation of mechanical systems endowing the closed-loop system with a Lagrangian or Hamiltonian structure with a desired energy function. The success of the method relies on the possibility of solving two partial differential equations (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 (KE) which is quasi-linear and non-homogeneous, and the solution that defines the desired inertia matrix must be positive definite. This task is simplified by the inclusion of gyroscopic forces in the target dynamics, which translates into the presence of a free skew-symmetric matrix in the KE matching equation that reduces the number of PDEs to be solved. Recently, it has been claimed that considering a more general form for the target dynamic forces that relax the skew-symmetry condition further reduces the number of KE PDEs. The purpose of this paper is to prove that this claim is wrong.