# Modelling a system with multiple hysteresis behaviours and mutual interference

I'm trying to model a non-linear, static SISO system with

• input variable $u$
• disturbance variable $z$ (can be controlled for identification purposes but not during actual operation)
• output variable $x$
• system behaviour $N\{x,u,z\}$ is hysteresis afflicted for both $u$ and $z$
• $u$ and $z$ influence the hysteresis characteristics of each other

I'm searching for any literature that deals with this kind of problem. It should contain approaches on how to model such a system and if possible a way to invert it, because ultimately I'm trying to linearize the overall behaviour $N^{-1}\{N\} = 1$.

2. If the hysteresis of the disturbance value $z$ is negligible, one might try the modified Prandtl-Ishlinksi approach, in which the influence of $z$ on the $x$-$u$ hysteresis is modelled by an unambiguous characteristic. Furthermore this approach is highly suited for compensation purposes, since the model can be analytically inverted.