Reference mirroring for control with impacts

We illustrate a peculiar concept of mirrored reference to address reference tracking control systems for plants subject to impacts. The idea is based on the constructions of artificial references corresponding to suitable reflections of the actual reference performed with respect to the impacting surfaces. We illustrate the application of this idea to reference tracking and state observation for a number of examples of control systems subject to impacts: a one-dimensional hammer hitting a surface, a two-dimensional Newton’s cradle, a two-dimensional mass moving in a planar equilateral triangle and a mass moving in an n-dimensional polyhedral region. For each one of these examples, we provide controllers inducing global exponential stability of the error dynamics. Despite the apparently different nature among the examples, we show that the controller/observer design follows the same paradigm based on mirrored references and we highlight the similarities of the different proofs by showing that they are all based on a specific result about Lyapunov conditions for global exponential stability of hybrid dynamical systems.