Necessary and sufficient stability conditions for equilibria of linear SISO feedbacks with a play operator

We consider the feedback interconnection of a strictly proper single input single output plant with a play (equivalently backlash) operator. Under the standard assumption that the linear feedback is exponentially stable we characterize the set of equilibria of the arising nonlinear closed-loop system and show that it is a bounded set containing the origin. Then we provide necessary and sufficient conditions for global exponential stability of this set, that correspond to exponential stability of the open-loop dynamics. We prove our main result by proposing a novel model for the play operator, corresponding to a constrained differential inclusion. With this representation, we also show that the nonlinear closed loop under consideration can be projected to a subspace where it evolves like a switching linear system. We illustrate our results by some numerical simulations illustrating a few possible scenarios.