Energy-balancing passivity-based control is equivalent to dissipation and output invariance

Passivity-based controllers (PBCs) achieve stabilization of nonlinear systems, rendering the closed-loop passive with a desired energy (storage) function. A natural question is, under which conditions is it possible to make this function equal to the difference between the plant and controller energies-when the controller is said to be energy-balancing. In this paper we prove that a necessary and sufficient condition for energy-balancing is that the open and the closed-loop systems have the same dissipation functions and passive outputs. A second contribution of our work is the identification of a new passive output for Port-Hamiltonian systems, which is invariant to the action of PBCs that modify only the energy function-so-called basic interconnection and damping assignment PBCs-proving that they are energy-balancing. To establish these results a new algebraic framework for analysis and design of PBCs, centered around the principles of output and dissipation invariance, is developed. Using this framework several PBC schemes reported in the literature are compared. Also, we present a systematic procedure to generate new passive outputs, this result is of interest on its own, since it allows to extend the applicability of PBC to systems that are non-minimum phase and/or have relative degree larger than one.