Modelling, Control and Stability Analysis of Flexible Rotating Beam's Impacts During Contact Scenario

This paper considers the problem of a rotating flexible beam in collision with an external object. The flexible beam's colliding equations exhibit instant changes during impact times, therefore the model is cast in the class of switched infinite dimensional operator systems. The aim is to study the stability of the closed loop system with a PD control law, making use of the semigroup formalism together with the Lyapunov stability theory. To this end, we present a new stability result making use of multiple Lyapunov functions obtained as an adaptation of a theorem from finite dimensional hybrid systems theory. We show the port-Hamiltonian modelling procedure for a controlled rotating flexible beam in impact scenario, using distributed parameter equations to describe the beam's dynamic. Then, we compute the equilibrium position of the closed loop system and using the shifted variables with respect to the equilibrium position, we cast the system in the class of switched infinite dimensional operator systems. Finally we select the Lyapunov functions for the contact and non-contact phases and we show, through numerical simulations, that they respect the assumptions of the proposed stability theorem.