Barriers and potentially safe sets in hybrid systems: Pendulum with non-rigid cable

Abstract

This paper deals with an application of the notion of barrier in mixed constrained nonlinear systems to an example of a pendulum mounted on a cart with non-rigid cable, whose dynamics may switch to free-fall when the tension of the cable vanishes. We present a direct construction of the boundary of the potentially safe set in which there always exists a control such that the cable never goes slack. A discussion on the dependence of this set with respect to the pendulum and cart masses is then sketched.

Bracketing backward reach sets of a dynamical system

“…Lyapunov-based methods (Ratschan and She, 2010;Delanoue, Jaulin, and Cottenceau, 2006;Gonzaga, Jungers, and Daafouz, 2012;Barreiro, Aracil, and Pagano, 2002), levelset methods (Mitchell, Bayen, and Tomlin, 2001;Biemond and Michiels, 2014), or barrier functions Bouissou et al 2014;Esterhuizen and Lévine 2016 can also be considered as Eulerian since they only check the constraints on the state space and do not need to perform any integration through time. Now, these methods require a parametric expression for candidate Lyapunov-like functions She and Xue 2013.…”