Input–output stability of systems with backlash

The objective of this work is to study the stability of systems with backlash, from an input–output point of view. In the first part, an L∞L∞ analysis is addressed. This analysis provides not only conditions for boundedness of the loop signals, but also results on existence of solutions, based on Schauder fixed point theorem. In the second part, the system is studied using an L2L2 or conic sector approach. The backlash graph is confined into certain conic sector, which is shown to be optimum or maximal. The conic inequality induces frequencial conditions on the linear part, which are further relaxed introducing dynamic multipliers. In the third part, both L∞L∞ and L2L2 techniques are combined reaching a final criterion which results in a Popov-like stability condition.