Convenient Model for Systems with Hystereses-Control

We establish a model of a system with hystereses, which allows for standard stability analysis of fixed points and closed orbits. To this end, we represent a system with hystereses as a piecewise-affine switched system that consists of a family of dynamical systems defined on disjoint polyhedral sets. The discrete transitions are realized by reset maps defined on the facets of these polyhedral sets. We have shown that the state space of a resulting switched system is a smooth manifold, the Cartesian product of a torus with an Euclidean space. Additionally, we construct the charts explicitly. Thereby, the analysis of a system with hystereses can be seen as the analysis of a dynamical system on a manifold. This dynamical system in a chart corresponds to a differential equation with discontinuous right hand side, which solution is shown to exist and to be unique.