Passivity-based control for hybrid systems with applications to mechanical systems exhibiting impacts

Motivated by applications of systems interacting with their environments, we study the design of passivity-based controllers for a class of hybrid systems in which the energy dissipation may only happen along either the continuous or the discrete dynamics. A general definition of passivity, encompassing the said special cases, is introduced and, along with detectability and solution conditions, linked to stability and asymptotic stability of compact sets. The proposed results allow us to take advantage of the passivity property of the system at flows or at jumps and are employed to design passivity-based controllers for the class of hybrid systems of interest. Two applications, one pertaining to a point mass physically interacting with a wall and another about controlling a ball bouncing on an actuated surface, illustrate the definitions and results throughout the paper.